Y combinator

Y combinator
You are encouraged to solve this task according to the task description, using any language you may know.

In strict functional programming and the lambda calculus, functions (lambda expressions) don't have state and are only allowed to refer to arguments of enclosing functions.

This rules out the usual definition of a recursive function wherein a function is associated with the state of a variable and this variable's state is used in the body of the function.

The   Y combinator   is itself a stateless function that, when applied to another stateless function, returns a recursive version of the function.

The Y combinator is the simplest of the class of such functions, called fixed-point combinators.

Task

Define the stateless   Y combinator   and use it to compute factorials and Fibonacci numbers from other stateless functions or lambda expressions.

Cf

• Jim Weirich: Adventures in Functional Programming

AArch64 Assembly

Works with: as version Raspberry Pi 3B version Buster 64 bits

/* ARM assembly AARCH64 Raspberry PI 3B */
/*  program Ycombi64.s   */
 
/*******************************************/
/* Constantes file                         */
/*******************************************/
/* for this file see task include a file in language AArch64 assembly*/
.include "../includeConstantesARM64.inc"

/*******************************************/
/* Structures                               */
/********************************************/
/* structure function*/
    .struct  0
func_fn:                    // next element
    .struct  func_fn + 8 
func_f_:                    // next element
    .struct  func_f_ + 8 
func_num:
    .struct  func_num + 8 
func_fin:
 
/* Initialized data */
.data
szMessStartPgm:            .asciz "Program start \n"
szMessEndPgm:              .asciz "Program normal end.\n"
szMessError:               .asciz "\033[31mError Allocation !!!\n"
 
szFactorielle:             .asciz "Function factorielle : \n"
szFibonacci:               .asciz "Function Fibonacci : \n"
szCarriageReturn:          .asciz "\n"
 
/* datas message display */
szMessResult:            .ascii "Result value : @ \n"
 
/* UnInitialized data */
.bss 
sZoneConv:                .skip 100
/*  code section */
.text
.global main 
main:                                           // program start
    ldr x0,qAdrszMessStartPgm                   // display start message
    bl affichageMess
    adr x0,facFunc                              // function factorielle address
    bl YFunc                                    // create Ycombinator
    mov x19,x0                                   // save Ycombinator
    ldr x0,qAdrszFactorielle                    // display message
    bl affichageMess
    mov x20,#1                                   // loop counter
1:  // start loop
    mov x0,x20
    bl numFunc                                  // create number structure
    cmp x0,#-1                                  // allocation error ?
    beq 99f
    mov x1,x0                                   // structure number address
    mov x0,x19                                  // Ycombinator address
    bl callFunc                                 // call 
    ldr x0,[x0,#func_num]                       // load result
    ldr x1,qAdrsZoneConv                        // and convert ascii string
    bl conversion10S                            // decimal conversion
    ldr x0,qAdrszMessResult
    ldr x1,qAdrsZoneConv
    bl strInsertAtCharInc                       // insert result at @ character
    bl affichageMess                            // display message final

    add x20,x20,#1                              // increment loop counter
    cmp x20,#10                                 // end ?
    ble 1b                                      // no -> loop
/*********Fibonacci  *************/
    adr x0,fibFunc                              // function fibonacci address
    bl YFunc                                    // create Ycombinator
    mov x19,x0                                  // save Ycombinator
    ldr x0,qAdrszFibonacci                      // display message
    bl affichageMess
    mov x20,#1                                  // loop counter
2:  // start loop
    mov x0,x20
    bl numFunc                                  // create number structure
    cmp x0,#-1                                  // allocation error ?
    beq 99f
    mov x1,x0                                   // structure number address
    mov x0,x19                                   // Ycombinator address
    bl callFunc                                 // call 
    ldr x0,[x0,#func_num]                       // load result
    ldr x1,qAdrsZoneConv                        // and convert ascii string
    bl conversion10S
    ldr x0,qAdrszMessResult
    ldr x1,qAdrsZoneConv
    bl strInsertAtCharInc                       // insert result at @ character
    bl affichageMess
    add x20,x20,#1                                   // increment loop counter
    cmp x20,#10                                  // end ?
    ble 2b                                      // no -> loop
    ldr x0,qAdrszMessEndPgm                     // display end message
    bl affichageMess
    b 100f
99:                                             // display error message 
    ldr x0,qAdrszMessError
    bl affichageMess
100:                                            // standard end of the program
    mov x0,0                                    // return code
    mov x8,EXIT                                 // request to exit program
    svc 0                                       // perform system call
qAdrszMessStartPgm:        .quad szMessStartPgm
qAdrszMessEndPgm:          .quad szMessEndPgm
qAdrszFactorielle:         .quad szFactorielle
qAdrszFibonacci:           .quad szFibonacci
qAdrszMessError:           .quad szMessError
qAdrszCarriageReturn:      .quad szCarriageReturn
qAdrszMessResult:          .quad szMessResult
qAdrsZoneConv:             .quad sZoneConv
/******************************************************************/
/*     factorielle function                         */ 
/******************************************************************/
/* x0 contains the Y combinator address  */
/* x1 contains the number structure  */
facFunc:
    stp x1,lr,[sp,-16]!            // save  registers
    stp x2,x3,[sp,-16]!            // save  registers
    mov x2,x0                   // save Y combinator address
    ldr x0,[x1,#func_num]       // load number
    cmp x0,#1                   // > 1 ?
    bgt 1f                      // yes
    mov x0,#1                   // create structure number value 1
    bl numFunc
    b 100f
1:
    mov x3,x0                   // save number
    sub x0,x0,#1                   // decrement number
    bl numFunc                  // and create new structure number
    cmp x0,#-1                  // allocation error ?
    beq 100f
    mov x1,x0                   // new structure number -> param 1
    ldr x0,[x2,#func_f_]        // load function address to execute
    bl callFunc                 // call
    ldr x1,[x0,#func_num]       // load new result
    mul x0,x1,x3                // and multiply by precedent
    bl numFunc                  // and create new structure number
                                // and return her address in x0
100:
    ldp x2,x3,[sp],16              // restaur  2 registers
    ldp x1,lr,[sp],16              // restaur  2 registers
    ret                            // return to address lr x30
/******************************************************************/
/*     fibonacci function                         */ 
/******************************************************************/
/* x0 contains the Y combinator address  */
/* x1 contains the number structure  */
fibFunc:
    stp x1,lr,[sp,-16]!            // save  registers
    stp x2,x3,[sp,-16]!            // save  registers
    stp x4,x5,[sp,-16]!            // save  registers
    mov x2,x0                   // save Y combinator address
    ldr x0,[x1,#func_num]       // load number
    cmp x0,#1                   // > 1 ?
    bgt 1f                      // yes
    mov x0,#1                   // create structure number value 1
    bl numFunc
    b 100f
1:
    mov x3,x0                   // save number
    sub x0,x0,#1                // decrement number
    bl numFunc                  // and create new structure number
    cmp x0,#-1                  // allocation error ?
    beq 100f
    mov x1,x0                   // new structure number -> param 1
    ldr x0,[x2,#func_f_]        // load function address to execute
    bl callFunc                 // call
    ldr x4,[x0,#func_num]       // load new result
    sub x0,x3,#2                // new number - 2
    bl numFunc                  // and create new structure number
    cmp x0,#-1                  // allocation error ?
    beq 100f
    mov x1,x0                   // new structure number -> param 1
    ldr x0,[x2,#func_f_]        // load function address to execute
    bl callFunc                 // call
    ldr x1,[x0,#func_num]       // load new result
    add x0,x1,x4                // add two results
    bl numFunc                  // and create new structure number
                                // and return her address in x0
100:
    ldp x4,x5,[sp],16              // restaur  2 registers
    ldp x2,x3,[sp],16              // restaur  2 registers
    ldp x1,lr,[sp],16              // restaur  2 registers
    ret                            // return to address lr x30
/******************************************************************/
/*     call function                         */ 
/******************************************************************/
/* x0 contains the address of the function  */
/* x1 contains the address of the function 1 */
callFunc:
    stp x2,lr,[sp,-16]!            // save  registers
    ldr x2,[x0,#func_fn]           // load function address to execute
    blr x2                         // and call it
    ldp x2,lr,[sp],16              // restaur  2 registers
    ret                            // return to address lr x30
/******************************************************************/
/*     create Y combinator function                         */ 
/******************************************************************/
/* x0 contains the address of the function  */
YFunc:
    stp x1,lr,[sp,-16]!            // save  registers
    mov x1,#0
    bl newFunc
    cmp x0,#-1                     // allocation error ?
    beq 100f
    str x0,[x0,#func_f_]           // store function and return in x0
100:
    ldp x1,lr,[sp],16              // restaur  2 registers
    ret                            // return to address lr x30
/******************************************************************/
/*     create structure number function                         */ 
/******************************************************************/
/* x0 contains the number  */
numFunc:
    stp x1,lr,[sp,-16]!            // save  registers
    stp x2,x3,[sp,-16]!            // save  registers
    mov x2,x0                      // save number
    mov x0,#0                      // function null
    mov x1,#0                      // function null
    bl newFunc
    cmp x0,#-1                     // allocation error ?
    beq 100f
    str x2,[x0,#func_num]          // store number in new structure
100:
    ldp x2,x3,[sp],16              // restaur  2 registers
    ldp x1,lr,[sp],16              // restaur  2 registers
    ret                            // return to address lr x30
/******************************************************************/
/*     new function                                               */ 
/******************************************************************/
/* x0 contains the function address   */
/* x1 contains the function address 1   */
newFunc:
    stp x1,lr,[sp,-16]!            // save  registers
    stp x3,x4,[sp,-16]!            // save  registers
    stp x5,x8,[sp,-16]!            // save  registers
    mov x4,x0                      // save address
    mov x5,x1                      // save adresse 1
                                   // allocation place on the heap
    mov x0,#0                      // allocation place heap
    mov x8,BRK                     // call system 'brk'
    svc #0
    mov x6,x0                      // save address heap for output string
    add x0,x0,#func_fin            // reservation place one element
    mov x8,BRK                     // call system 'brk'
    svc #0
    cmp x0,#-1                     // allocation error
    beq 100f
    mov x0,x6
    str x4,[x0,#func_fn]           // store address
    str x5,[x0,#func_f_]
    str xzr,[x0,#func_num]         // store zero to number
100:
    ldp x5,x8,[sp],16              // restaur  2 registers
    ldp x3,x4,[sp],16              // restaur  2 registers
    ldp x1,lr,[sp],16              // restaur  2 registers
    ret                            // return to address lr x30
/********************************************************/
/*        File Include fonctions                        */
/********************************************************/
/* for this file see task include a file in language AArch64 assembly */
.include "../includeARM64.inc"

ALGOL 68

Translation of: Python

Note: This specimen retains the original Python coding style.

Works with: ALGOL 68S version from Amsterdam Compiler Kit ( Guido van Rossum's teething ring) with runtime scope checking turned off.

BEGIN
  MODE F = PROC(INT)INT;
  MODE Y = PROC(Y)F;

# compare python Y = lambda f: (lambda x: x(x)) (lambda y: f( lambda *args: y(y)(*args)))#
  PROC y =      (PROC(F)F f)F: (  (Y x)F: x(x)) (  (Y z)F: f((INT arg )INT: z(z)( arg )));

  PROC fib = (F f)F: (INT n)INT: CASE n IN n,n OUT f(n-1) + f(n-2) ESAC;

  FOR i TO 10 DO print(y(fib)(i)) OD
END

The version below works with ALGOL 68 Genie 3.5.0 tested with Linux kernel release 6.7.4-200.fc39.x86_64

N.B. 4 warnings are issued of the form
a68g: warning: 1: declaration hides a declaration of "..." with larger reach, in closed-clause starting at "(" in line dd.
These could easily be fixed by changing names, but I believe that doing so would make the code harder to follow.

BEGIN

# This version needs partial parameterisation in order to work #
# The commented code is JavaScript aka ECMAScript ES6 #


MODE F = PROC( INT ) INT ;
MODE X = PROC( X ) F ;


#
Y_combinator =
  func_gen => ( x => x( x ) )( x => func_gen( arg => x( x )( arg ) ) )
#

PROC y combinator = ( PROC( F ) F func gen ) F:
( ( X x ) F:  x( x ) )
  (
    (
      ( PROC( F ) F func gen , X x ) F:
        func gen( ( ( X x , INT arg ) INT: x( x )( arg ) )( x , ) )
    )( func gen , )
  )
;


#
fac_gen = fac => (n => ( ( n === 0 ) ? 1 : n * fac( n - 1 ) ) )
#

PROC fac gen = ( F fac ) F:
( ( F fac , INT n ) INT: IF n = 0 THEN 1 ELSE n * fac( n - 1 ) FI )( fac , )
;


#
factorial = Y_combinator( fac_gen )
#

F factorial = y combinator( fac gen ) ;


#
fib_gen =
  fib =>
    ( n => ( ( n === 0 ) ? 0 : ( n === 1 ) ? 1 : fib( n - 2 ) + fib( n - 1 ) ) )
#

PROC fib gen = ( F fib ) F:
(
  ( F fib , INT n ) INT:
  CASE n + 1 IN 0 , 1 OUT fib( n - 2 ) + fib( n - 1 ) ESAC
)( fib , )
;


#
fibonacci = Y_combinator( fib_gen )
#

F fibonacci = y combinator( fib gen ) ;


#
for ( i = 1 ; i <= 12 ; i++) { process.stdout.write( " " + factorial( i ) ) }
#

INT nofacs = 12 ;
printf( ( $ l , "Here are the first " , g( 0 ) , " factorials." , l $ , nofacs ) ) ;
FOR i TO nofacs
DO
  printf( ( $ "  " , g( 0 ) $ , factorial( i ) ) )
OD ;
print( newline ) ;


#
for ( i = 1 ; i <= 12 ; i++) { process.stdout.write( " " + fibonacci( i ) ) }
#

INT nofibs = 12 ;
printf( (
  $ l , "Here are the first " , g( 0 ) , " fibonacci numbers." , l $
, nofibs
      ) )
;
FOR i TO nofibs
DO
  printf( ( $ "  " , g( 0 ) $ , fibonacci( i ) ) )
OD ;
print( newline )

END

AppleScript

AppleScript is not particularly "functional" friendly. It can, however, support the Y combinator.

AppleScript does not have anonymous functions, but it does have anonymous objects. The code below implements the latter with the former (using a handler (i.e. function) named 'lambda' in each anonymous object).

Unfortunately, an anonymous object can only be created in its own statement ('script'...'end script' can not be in an expression). Thus, we have to apply Y to the automatic 'result' variable that holds the value of the previous statement.

The identifier used for Y uses "pipe quoting" to make it obviously distinct from the y used inside the definition.

-- Y COMBINATOR ---------------------------------------------------------------

on |Y|(f)
    script
        on |λ|(y)
            script
                on |λ|(x)
                    y's |λ|(y)'s |λ|(x)
                end |λ|
            end script
            
            f's |λ|(result)
        end |λ|
    end script
    
    result's |λ|(result)
end |Y|


-- TEST -----------------------------------------------------------------------
on run
    
    -- Factorial
    script fact
        on |λ|(f)
            script
                on |λ|(n)
                    if n = 0 then return 1
                    n * (f's |λ|(n - 1))
                end |λ|
            end script
        end |λ|
    end script
    
    
    -- Fibonacci
    script fib
        on |λ|(f)
            script
                on |λ|(n)
                    if n = 0 then return 0
                    if n = 1 then return 1
                    (f's |λ|(n - 2)) + (f's |λ|(n - 1))
                end |λ|
            end script
        end |λ|
    end script
    
    {facts:map(|Y|(fact), enumFromTo(0, 11)), fibs:map(|Y|(fib), enumFromTo(0, 20))}
    
    --> {facts:{1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800}, 
    
    --> fibs:{0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 
    --           1597, 2584, 4181, 6765}} 
    
end run


-- GENERIC FUNCTIONS FOR TEST -------------------------------------------------

-- map :: (a -> b) -> [a] -> [b]
on map(f, xs)
    tell mReturn(f)
        set lng to length of xs
        set lst to {}
        repeat with i from 1 to lng
            set end of lst to |λ|(item i of xs, i, xs)
        end repeat
        return lst
    end tell
end map

-- enumFromTo :: Int -> Int -> [Int]
on enumFromTo(m, n)
    if n < m then
        set d to -1
    else
        set d to 1
    end if
    set lst to {}
    repeat with i from m to n by d
        set end of lst to i
    end repeat
    return lst
end enumFromTo

-- Lift 2nd class handler function into 1st class script wrapper 
-- mReturn :: Handler -> Script
on mReturn(f)
    if class of f is script then
        f
    else
        script
            property |λ| : f
        end script
    end if
end mReturn

Output:

{facts:{1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800}, 
fibs:{0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765}}

ARM Assembly

Works with: as version Raspberry Pi

/* ARM assembly Raspberry PI  */
/*  program Ycombi.s   */

/* REMARK 1 : this program use routines in a include file 
   see task Include a file language arm assembly 
   for the routine affichageMess conversion10 
   see at end of this program the instruction include */

/* Constantes    */
.equ STDOUT, 1                           @ Linux output console
.equ EXIT,   1                           @ Linux syscall
.equ WRITE,  4                           @ Linux syscall


/*******************************************/
/* Structures                               */
/********************************************/
/* structure function*/
    .struct  0
func_fn:                    @ next element
    .struct  func_fn + 4 
func_f_:                    @ next element
    .struct  func_f_ + 4 
func_num:
    .struct  func_num + 4 
func_fin:

/* Initialized data */
.data
szMessStartPgm:            .asciz "Program start \n"
szMessEndPgm:              .asciz "Program normal end.\n"
szMessError:               .asciz "\033[31mError Allocation !!!\n"

szFactorielle:             .asciz "Function factorielle : \n"
szFibonacci:               .asciz "Function Fibonacci : \n"
szCarriageReturn:          .asciz "\n"

/* datas message display */
szMessResult:            .ascii "Result value :"
sValue:                  .space 12,' '
                         .asciz "\n"

/* UnInitialized data */
.bss 

/*  code section */
.text
.global main 
main:                                           @ program start
    ldr r0,iAdrszMessStartPgm                   @ display start message
    bl affichageMess
    adr r0,facFunc                              @ function factorielle address
    bl YFunc                                    @ create Ycombinator
    mov r5,r0                                   @ save Ycombinator
    ldr r0,iAdrszFactorielle                    @ display message
    bl affichageMess
    mov r4,#1                                   @ loop counter
1:  @ start loop
    mov r0,r4
    bl numFunc                                  @ create number structure
    cmp r0,#-1                                  @ allocation error ?
    beq 99f
    mov r1,r0                                   @ structure number address
    mov r0,r5                                   @ Ycombinator address
    bl callFunc                                 @ call 
    ldr r0,[r0,#func_num]                       @ load result
    ldr r1,iAdrsValue                           @ and convert ascii string
    bl conversion10
    ldr r0,iAdrszMessResult                     @ display result message
    bl affichageMess
    add r4,#1                                   @ increment loop counter
    cmp r4,#10                                  @ end ?
    ble 1b                                      @ no -> loop
/*********Fibonacci  *************/
    adr r0,fibFunc                              @ function factorielle address
    bl YFunc                                    @ create Ycombinator
    mov r5,r0                                   @ save Ycombinator
    ldr r0,iAdrszFibonacci                      @ display message
    bl affichageMess
    mov r4,#1                                   @ loop counter
2:  @ start loop
    mov r0,r4
    bl numFunc                                  @ create number structure
    cmp r0,#-1                                  @ allocation error ?
    beq 99f
    mov r1,r0                                   @ structure number address
    mov r0,r5                                   @ Ycombinator address
    bl callFunc                                 @ call 
    ldr r0,[r0,#func_num]                       @ load result
    ldr r1,iAdrsValue                           @ and convert ascii string
    bl conversion10
    ldr r0,iAdrszMessResult                     @ display result message
    bl affichageMess
    add r4,#1                                   @ increment loop counter
    cmp r4,#10                                  @ end ?
    ble 2b                                      @ no -> loop
    ldr r0,iAdrszMessEndPgm                     @ display end message
    bl affichageMess
    b 100f
99:                                             @ display error message 
    ldr r0,iAdrszMessError
    bl affichageMess
100:                                            @ standard end of the program
    mov r0, #0                                  @ return code
    mov r7, #EXIT                               @ request to exit program
    svc 0                                       @ perform system call
iAdrszMessStartPgm:        .int szMessStartPgm
iAdrszMessEndPgm:          .int szMessEndPgm
iAdrszFactorielle:         .int szFactorielle
iAdrszFibonacci:           .int szFibonacci
iAdrszMessError:           .int szMessError
iAdrszCarriageReturn:      .int szCarriageReturn
iAdrszMessResult:          .int szMessResult
iAdrsValue:                .int sValue
/******************************************************************/
/*     factorielle function                         */ 
/******************************************************************/
/* r0 contains the Y combinator address  */
/* r1 contains the number structure  */
facFunc:
    push {r1-r3,lr}             @ save  registers 
    mov r2,r0                   @ save Y combinator address
    ldr r0,[r1,#func_num]       @ load number
    cmp r0,#1                   @ > 1 ?
    bgt 1f                      @ yes
    mov r0,#1                   @ create structure number value 1
    bl numFunc
    b 100f
1:
    mov r3,r0                   @ save number
    sub r0,#1                   @ decrement number
    bl numFunc                  @ and create new structure number
    cmp r0,#-1                  @ allocation error ?
    beq 100f
    mov r1,r0                   @ new structure number -> param 1
    ldr r0,[r2,#func_f_]        @ load function address to execute
    bl callFunc                 @ call
    ldr r1,[r0,#func_num]       @ load new result
    mul r0,r1,r3                @ and multiply by precedent
    bl numFunc                  @ and create new structure number
                                @ and return her address in r0
100:
    pop {r1-r3,lr}              @ restaur registers
    bx lr                       @ return
/******************************************************************/
/*     fibonacci function                         */ 
/******************************************************************/
/* r0 contains the Y combinator address  */
/* r1 contains the number structure  */
fibFunc:
    push {r1-r4,lr}             @ save  registers 
    mov r2,r0                   @ save Y combinator address
    ldr r0,[r1,#func_num]       @ load number
    cmp r0,#1                   @ > 1 ?
    bgt 1f                      @ yes
    mov r0,#1                   @ create structure number value 1
    bl numFunc
    b 100f
1:
    mov r3,r0                   @ save number
    sub r0,#1                   @ decrement number
    bl numFunc                  @ and create new structure number
    cmp r0,#-1                  @ allocation error ?
    beq 100f
    mov r1,r0                   @ new structure number -> param 1
    ldr r0,[r2,#func_f_]        @ load function address to execute
    bl callFunc                 @ call
    ldr r4,[r0,#func_num]       @ load new result
    sub r0,r3,#2                @ new number - 2
    bl numFunc                  @ and create new structure number
    cmp r0,#-1                  @ allocation error ?
    beq 100f
    mov r1,r0                   @ new structure number -> param 1
    ldr r0,[r2,#func_f_]        @ load function address to execute
    bl callFunc                 @ call
    ldr r1,[r0,#func_num]       @ load new result
    add r0,r1,r4                @ add two results
    bl numFunc                  @ and create new structure number
                                @ and return her address in r0
100:
    pop {r1-r4,lr}              @ restaur registers
    bx lr                       @ return
/******************************************************************/
/*     call function                         */ 
/******************************************************************/
/* r0 contains the address of the function  */
/* r1 contains the address of the function 1 */
callFunc:
    push {r2,lr}                                @ save  registers 
    ldr r2,[r0,#func_fn]                        @ load function address to execute
    blx r2                                      @ and call it
    pop {r2,lr}                                 @ restaur registers
    bx lr                                       @ return
/******************************************************************/
/*     create Y combinator function                         */ 
/******************************************************************/
/* r0 contains the address of the function  */
YFunc:
    push {r1,lr}                                @ save  registers 
    mov r1,#0
    bl newFunc
    cmp r0,#-1                                  @ allocation error ?
    strne r0,[r0,#func_f_]                      @ store function and return in r0
    pop {r1,lr}                                 @ restaur registers
    bx lr                                       @ return
/******************************************************************/
/*     create structure number function                         */ 
/******************************************************************/
/* r0 contains the number  */
numFunc:
    push {r1,r2,lr}                             @ save  registers 
    mov r2,r0                                   @ save number
    mov r0,#0                                   @ function null
    mov r1,#0                                   @ function null
    bl newFunc
    cmp r0,#-1                                  @ allocation error ?
    strne r2,[r0,#func_num]                     @ store number in new structure
    pop {r1,r2,lr}                              @ restaur registers
    bx lr                                       @ return
/******************************************************************/
/*     new function                                               */ 
/******************************************************************/
/* r0 contains the function address   */
/* r1 contains the function address 1   */
newFunc:
    push {r2-r7,lr}                             @ save  registers 
    mov r4,r0                                   @ save address
    mov r5,r1                                   @ save adresse 1
    @ allocation place on the heap
    mov r0,#0                                   @ allocation place heap
    mov r7,#0x2D                                @ call system 'brk'
    svc #0
    mov r3,r0                                   @ save address heap for output string
    add r0,#func_fin                            @ reservation place one element
    mov r7,#0x2D                                @ call system 'brk'
    svc #0
    cmp r0,#-1                                  @ allocation error
    beq 100f
    mov r0,r3
    str r4,[r0,#func_fn]                        @ store address
    str r5,[r0,#func_f_]
    mov r2,#0
    str r2,[r0,#func_num]                       @ store zero to number
100:
    pop {r2-r7,lr}                              @ restaur registers
    bx lr                                       @ return
/***************************************************/
/*      ROUTINES INCLUDE                 */
/***************************************************/
.include "../affichage.inc"

Output:

Program start
Function factorielle :
Result value :1
Result value :2
Result value :6
Result value :24
Result value :120
Result value :720
Result value :5040
Result value :40320
Result value :362880
Result value :3628800
Function Fibonacci :
Result value :1
Result value :2
Result value :3
Result value :5
Result value :8
Result value :13
Result value :21
Result value :34
Result value :55
Result value :89
Program normal end.

ATS

(* ****** ****** *)
//
#include "share/atspre_staload.hats"
//
(* ****** ****** *)
//
fun
myfix
{a:type}
(
 f: lazy(a) -<cloref1> a
) : lazy(a) = $delay(f(myfix(f)))
//
val
fact =
myfix{int-<cloref1>int}
(
lam(ff) => lam(x) => if x > 0 then x * !ff(x-1) else 1
)
(* ****** ****** *)
//
implement main0 () = println! ("fact(10) = ", !fact(10))
//
(* ****** ****** *)

BASIC

FreeBASIC

FreeBASIC does not support nested functions, lambda expressions or functions inside nested types

Function Y(f As String) As String
    Y = f
End Function

Function fib(n As Long) As Long
    Dim As Long n1 = 0, n2 = 1, k, sum
    For k = 1 To Abs(n)
        sum = n1 + n2
        n1 = n2
        n2 = sum
    Next k
    Return Iif(n < 0, (n1 * ((-1) ^ ((-n)+1))), n1)
End Function

Function fac(n As Long) As Long
    Dim As Long r = 1, i
    For i = 2 To n
        r *= i
    Next i
    Return r
End Function

Function execute(s As String, n As Integer) As Long
    Return Iif (s = "fac", fac(n), fib(n))
End Function

Sub test(nombre As String)
    Dim f As String: f = Y(nombre)
    Print !"\n"; f; ":";
    For i As Integer = 1 To 10
        Print execute(f, i);
    Next i
End Sub

test("fac")
test("fib")
Sleep

Output:

fac: 1 2 6 24 120 720 5040 40320 362880 3628800
fib: 1 1 2 3 5 8 13 21 34 55

VBA

Translation of: Phix

The IIf as translation of Iff can not be used as IIf executes both true and false parts and will cause a stack overflow.

Private Function call_fn(f As String, n As Long) As Long
    call_fn = Application.Run(f, f, n)
End Function
 
Private Function Y(f As String) As String
    Y = f
End Function
 
Private Function fac(self As String, n As Long) As Long
    If n > 1 Then
        fac = n * call_fn(self, n - 1)
    Else
        fac = 1
    End If
End Function
 
Private Function fib(self As String, n As Long) As Long
    If n > 1 Then
        fib = call_fn(self, n - 1) + call_fn(self, n - 2)
    Else
        fib = n
    End If
End Function
 
Private Sub test(name As String)
    Dim f As String: f = Y(name)
    Dim i As Long
    Debug.Print name
    For i = 1 To 10
        Debug.Print call_fn(f, i);
    Next i
    Debug.Print
End Sub

Public Sub main()
    test "fac"
    test "fib"
End Sub

Output:

fac
 1  2  6  24  120  720  5040  40320  362880  3628800 
fib
 1  1  2  3  5  8  13  21  34  55 

uBasic/4tH

Translation of: Yabasic

Proc _Test("fac")
Proc _Test("fib")
End

_fac
  Param (2)
  If b@ > 1 Then Return (b@ * FUNC (a@ (a@, b@-1)))
Return (1)

_fib
  Param (2)
  If b@ > 1 Then Return (FUNC (a@ (a@, b@-1)) + FUNC (a@ (a@, b@-2)))
Return (b@)  
 
_Test
  Param (1)
  Local (1)
  
  Print Show (a@), ": "; : a@ = Name (a@)
  For b@ = 1 to 10 : Print FUNC (a@ (a@, b@)), : Next : Print
Return

Output:

fac     : 1     2       6       24      120     720     5040    40320   362880  3628800 
fib     : 1     1       2       3       5       8       13      21      34      55      

0 OK, 0:39 

Yabasic

sub fac(self$, n)
    if n > 1 then
        return n * execute(self$, self$, n - 1)
    else
        return 1
    end if
end sub
 
sub fib(self$, n)
    if n > 1 then
        return execute(self$, self$, n - 1) + execute(self$, self$, n - 2)
    else
        return n
    end if
end sub
 
sub test(name$)
    local i
    
    print name$, ": ";
    for i = 1 to 10
        print execute(name$, name$, i);
    next
    print
end sub

test("fac")
test("fib")

Binary Lambda Calculus

This BLC program outputs 6!, as computed with the Y combinator, in unary (generated from https://github.com/tromp/AIT/blob/master/rosetta/facY.lam) :

11 c2 a3 40 b0 bf 65 ee 05 7c 0c ef 18 89 70 39 d0 39 ce 81 6e c0 3c e8 31

BlitzMax

BlitzMax doesn't support anonymous functions or classes, so everything needs to be explicitly named.

SuperStrict

'Boxed type so we can just use object arrays for argument lists
Type Integer
	Field val:Int
	Function Make:Integer(_val:Int)
		Local i:Integer = New Integer
		i.val = _val
		Return i
	End Function
End Type


'Higher-order function type - just a procedure attached to a scope
Type Func Abstract
	Method apply:Object(args:Object[]) Abstract
End Type

'Function definitions - extend with fields as locals and implement apply as body
Type Scope Extends Func Abstract
	Field env:Scope

	'Constructor - bind an environment to a procedure
	Function lambda:Scope(env:Scope) Abstract

	Method _init:Scope(_env:Scope)	'Helper to keep constructors small
		env = _env ; Return Self
	End Method
End Type


'Based on the following definition:
'(define (Y f)
'    (let ((_r (lambda (r) (f (lambda a (apply (r r) a))))))
'      (_r _r)))

'Y (outer)
Type Y Extends Scope
	Field f:Func	'Parameter - gets closed over

	Function lambda:Scope(env:Scope)	'Necessary due to highly limited constructor syntax
		Return (New Y)._init(env)
	End Function

	Method apply:Func(args:Object[])
		f = Func(args[0])
		Local _r:Func = YInner1.lambda(Self)
		Return Func(_r.apply([_r]))
	End Method
End Type

'First lambda within Y
Type YInner1 Extends Scope
	Field r:Func	'Parameter - gets closed over

	Function lambda:Scope(env:Scope)
		Return (New YInner1)._init(env)
	End Function

	Method apply:Func(args:Object[])
		r = Func(args[0])
		Return Func(Y(env).f.apply([YInner2.lambda(Self)]))
	End Method
End Type

'Second lambda within Y
Type YInner2 Extends Scope
	Field a:Object[]	'Parameter - not really needed, but good for clarity

	Function lambda:Scope(env:Scope)
		Return (New YInner2)._init(env)
	End Function

	Method apply:Object(args:Object[])
		a = args
		Local r:Func = YInner1(env).r
		Return Func(r.apply([r])).apply(a)
	End Method
End Type


'Based on the following definition:
'(define fac (Y (lambda (f)
'                 (lambda (x)
'                   (if (<= x 0) 1 (* x (f (- x 1)))))))

Type FacL1 Extends Scope
	Field f:Func	'Parameter - gets closed over

	Function lambda:Scope(env:Scope)
		Return (New FacL1)._init(env)
	End Function

	Method apply:Object(args:Object[])
		f = Func(args[0])
		Return FacL2.lambda(Self)
	End Method
End Type

Type FacL2 Extends Scope
	Function lambda:Scope(env:Scope)
		Return (New FacL2)._init(env)
	End Function

	Method apply:Object(args:Object[])
		Local x:Int = Integer(args[0]).val
		If x <= 0 Then Return Integer.Make(1) ; Else Return Integer.Make(x * Integer(FacL1(env).f.apply([Integer.Make(x - 1)])).val)
	End Method
End Type


'Based on the following definition:
'(define fib (Y (lambda (f)
'                 (lambda (x)
'                   (if (< x 2) x (+ (f (- x 1)) (f (- x 2)))))))

Type FibL1 Extends Scope
	Field f:Func	'Parameter - gets closed over

	Function lambda:Scope(env:Scope)
		Return (New FibL1)._init(env)
	End Function

	Method apply:Object(args:Object[])
		f = Func(args[0])
		Return FibL2.lambda(Self)
	End Method
End Type

Type FibL2 Extends Scope
	Function lambda:Scope(env:Scope)
		Return (New FibL2)._init(env)
	End Function

	Method apply:Object(args:Object[])
		Local x:Int = Integer(args[0]).val
		If x < 2
			Return Integer.Make(x)
		Else
			Local f:Func = FibL1(env).f
			Local x1:Int = Integer(f.apply([Integer.Make(x - 1)])).val
			Local x2:Int = Integer(f.apply([Integer.Make(x - 2)])).val
			Return Integer.Make(x1 + x2)
		EndIf
	End Method
End Type


'Now test
Local _Y:Func = Y.lambda(Null)

Local fac:Func = Func(_Y.apply([FacL1.lambda(Null)]))
Print Integer(fac.apply([Integer.Make(10)])).val

Local fib:Func = Func(_Y.apply([FibL1.lambda(Null)]))
Print Integer(fib.apply([Integer.Make(10)])).val

Bracmat

The lambda abstraction

 (λx.x)y

translates to

 /('(x.$x))$y

in Bracmat code. Likewise, the fixed point combinator

Y := λg.(λx.g (x x)) (λx.g (x x))

the factorial

G := λr. λn.(1, if n = 0; else n × (r (n−1)))

the Fibonacci function

H := λr. λn.(1, if n = 1 or n = 2; else (r (n−1)) + (r (n−2)))

and the calls

(Y G) i

and

(Y H) i

where i varies between 1 and 10, are translated into Bracmat as shown below

(   ( Y
    = /(
       ' ( g
         .   /('(x.$g'($x'$x)))
           $ /('(x.$g'($x'$x)))
         )
       )
    )
  & ( G
    = /(
       ' ( r
         . /(
            ' ( n
              .   $n:~>0&1
                | $n*($r)$($n+-1)
              )
            )
         )
       )
    )
  & ( H
    = /(
       ' ( r
         . /(
            ' ( n
              .   $n:(1|2)&1
                | ($r)$($n+-1)+($r)$($n+-2)
              )
            )
         )
       )
    )
  & 0:?i
  &   whl
    ' ( 1+!i:~>10:?i
      & out$(str$(!i "!=" (!Y$!G)$!i))
      )
  & 0:?i
  &   whl
    ' ( 1+!i:~>10:?i
      & out$(str$("fib(" !i ")=" (!Y$!H)$!i))
      )
  &
)

Output:

1!=1
2!=2
3!=6
4!=24
5!=120
6!=720
7!=5040
8!=40320
9!=362880
10!=3628800
fib(1)=1
fib(2)=1
fib(3)=2
fib(4)=3
fib(5)=5
fib(6)=8
fib(7)=13
fib(8)=21
fib(9)=34
fib(10)=55

Bruijn

As defined in std/Combinator:

:import std/Number .

# sage bird combinator
y [[1 (0 0)] [1 (0 0)]]

# factorial using y
factorial y [[=?0 (+1) (0 ⋅ (1 --0))]]

:test ((factorial (+6)) =? (+720)) ([[1]])

# (very slow) fibonacci using y
fibonacci y [[0 <? (+1) (+0) (0 <? (+2) (+1) rec)]]
	rec (1 --0) + (1 --(--0))

:test ((fibonacci (+6)) =? (+8)) ([[1]])

C

C doesn't have first class functions, so we demote everything to second class to match.

#include <stdio.h>
#include <stdlib.h>

/* func: our one and only data type; it holds either a pointer to
   a function call, or an integer.  Also carry a func pointer to
   a potential parameter, to simulate closure                   */
typedef struct func_t *func;
typedef struct func_t {
        func (*fn) (func, func);
        func _;
        int num;
} func_t;

func new(func(*f)(func, func), func _) {
        func x = malloc(sizeof(func_t));
        x->fn = f;
        x->_ = _;       /* closure, sort of */
        x->num = 0;
        return x;
}

func call(func f, func n) {
        return f->fn(f, n);
}

func Y(func(*f)(func, func)) {
        func g = new(f, 0);
        g->_ = g;
        return g;
}

func num(int n) {
        func x = new(0, 0);
        x->num = n;
        return x;
}


func fac(func self, func n) {
        int nn = n->num;
        return nn > 1   ? num(nn * call(self->_, num(nn - 1))->num)
                        : num(1);
}

func fib(func self, func n) {
        int nn = n->num;
        return nn > 1
                ? num(  call(self->_, num(nn - 1))->num +
                        call(self->_, num(nn - 2))->num )
                : num(1);
}

void show(func n) { printf(" %d", n->num); }

int main() {
        int i;
        func f = Y(fac);
        printf("fac: ");
        for (i = 1; i < 10; i++)
                show( call(f, num(i)) );
        printf("\n");

        f = Y(fib);
        printf("fib: ");
        for (i = 1; i < 10; i++)
                show( call(f, num(i)) );
        printf("\n");

        return 0;
}

Output:

fac:  1 2 6 24 120 720 5040 40320 362880
fib:  1 2 3 5 8 13 21 34 55

C#

Like many other statically typed languages, this involves a recursive type, and like other strict languages, it is the Z-combinator instead.

The combinator here is expressed entirely as a lambda expression and is a static property of the generic YCombinator class. Both it and the RecursiveFunc type thus "inherit" the type parameters of the containing class—there effectively exists a separate specialized copy of both for each generic instantiation of YCombinator.

Note: in the code, Func<T, TResult> is a delegate type (the CLR equivalent of a function pointer) that has a parameter of type T and return type of TResult. See Higher-order functions#C# or the documentation for more information.

using System;

static class YCombinator<T, TResult>
{
    // RecursiveFunc is not needed to call Fix() and so can be private.
    private delegate Func<T, TResult> RecursiveFunc(RecursiveFunc r);

    public static Func<Func<Func<T, TResult>, Func<T, TResult>>, Func<T, TResult>> Fix { get; } =
        f => ((RecursiveFunc)(g => f(x => g(g)(x))))(g => f(x => g(g)(x)));
}

static class Program
{
    static void Main()
    {
        var fac = YCombinator<int, int>.Fix(f => x => x < 2 ? 1 : x * f(x - 1));
        var fib = YCombinator<int, int>.Fix(f => x => x < 2 ? x : f(x - 1) + f(x - 2));

        Console.WriteLine(fac(10));
        Console.WriteLine(fib(10));
    }
}

Output:

3628800
55

Alternatively, with a non-generic holder class (note that Fix is now a method, as properties cannot be generic):

static class YCombinator
{
    private delegate Func<T, TResult> RecursiveFunc<T, TResult>(RecursiveFunc<T, TResult> r);

    public static Func<T, TResult> Fix<T, TResult>(Func<Func<T, TResult>, Func<T, TResult>> f)
        => ((RecursiveFunc<T, TResult>)(g => f(x => g(g)(x))))(g => f(x => g(g)(x)));
}

Using the late-binding offered by dynamic to eliminate the recursive type:

static class YCombinator<T, TResult>
{
    public static Func<Func<Func<T, TResult>, Func<T, TResult>>, Func<T, TResult>> Fix { get; } =
        f => ((Func<dynamic, Func<T, TResult>>)(g => f(x => g(g)(x))))((Func<dynamic, Func<T, TResult>>)(g => f(x => g(g)(x))));
}

The usual version using recursion, disallowed by the task (implemented as a generic method):

static class YCombinator
{
    static Func<T, TResult> Fix<T, TResult>(Func<Func<T, TResult>, Func<T, TResult>> f) => x => f(Fix(f))(x);
}

Translations

To compare differences in language and runtime instead of in approaches to the task, the following are translations of solutions from other languages. Two versions of each translation are provided, one seeking to resemble the original as closely as possible, and another that is identical in program control flow but syntactically closer to idiomatic C#.

C++

std::function<TResult(T)> in C++ corresponds to Func<T, TResult> in C#.

Verbatim

using Func = System.Func<int, int>;
using FuncFunc = System.Func<System.Func<int, int>, System.Func<int, int>>;

static class Program {
    struct RecursiveFunc<F> {
        public System.Func<RecursiveFunc<F>, F> o;
    }

    static System.Func<A, B> Y<A, B>(System.Func<System.Func<A, B>, System.Func<A, B>> f) {
        var r = new RecursiveFunc<System.Func<A, B>>() {
            o = new System.Func<RecursiveFunc<System.Func<A, B>>, System.Func<A, B>>((RecursiveFunc<System.Func<A, B>> w) => {
                return f(new System.Func<A, B>((A x) => {
                    return w.o(w)(x);
                }));
            })
        };
        return r.o(r);
    }

    static FuncFunc almost_fac = (Func f) => {
        return new Func((int n) => {
            if (n <= 1) return 1;
            return n * f(n - 1);
        });
    };

    static FuncFunc almost_fib = (Func f) => {
        return new Func((int n) => {
            if (n <= 2) return 1;
            return f(n - 1) + f(n - 2);
        });
    };

    static int Main() {
        var fib = Y(almost_fib);
        var fac = Y(almost_fac);
        System.Console.WriteLine("fib(10) = " + fib(10));
        System.Console.WriteLine("fac(10) = " + fac(10));
        return 0;
    }
}

Semi-idiomatic

using System;

using FuncFunc = System.Func<System.Func<int, int>, System.Func<int, int>>;

static class Program {
    struct RecursiveFunc<F> {
        public Func<RecursiveFunc<F>, F> o;
    }

    static Func<A, B> Y<A, B>(Func<Func<A, B>, Func<A, B>> f) {
        var r = new RecursiveFunc<Func<A, B>> {
            o = w => f(x => w.o(w)(x))
        };
        return r.o(r);
    }

    static FuncFunc almost_fac = f => n => n <= 1 ? 1 : n * f(n - 1);

    static FuncFunc almost_fib = f => n => n <= 2 ? 1 : f(n - 1) + f(n - 2);

    static void Main() {
        var fib = Y(almost_fib);
        var fac = Y(almost_fac);
        Console.WriteLine("fib(10) = " + fib(10));
        Console.WriteLine("fac(10) = " + fac(10));
    }
}

Ceylon

TResult(T) in Ceylon corresponds to Func<T, TResult> in C#.

Since C# does not have local classes, RecursiveFunc and y1 are declared in a class of their own. Moving the type parameters to the class also prevents type parameter inference.

Verbatim

using System;
using System.Diagnostics;

class Program {
    public delegate TResult ParamsFunc<T, TResult>(params T[] args);

    static class Y<Result, Args> {
        class RecursiveFunction {
            public Func<RecursiveFunction, ParamsFunc<Args, Result>> o;
            public RecursiveFunction(Func<RecursiveFunction, ParamsFunc<Args, Result>> o) => this.o = o;
        }

        public static ParamsFunc<Args, Result> y1(
                Func<ParamsFunc<Args, Result>, ParamsFunc<Args, Result>> f) {

            var r = new RecursiveFunction((RecursiveFunction w)
                => f((Args[] args) => w.o(w)(args)));

            return r.o(r);
        }
    }

    static ParamsFunc<Args, Result> y2<Args, Result>(
            Func<ParamsFunc<Args, Result>, ParamsFunc<Args, Result>> f) {

        Func<dynamic, ParamsFunc<Args, Result>> r = w => {
            Debug.Assert(w is Func<dynamic, ParamsFunc<Args, Result>>);
            return f((Args[] args) => w(w)(args));
        };

        return r(r);
    }

    static ParamsFunc<Args, Result> y3<Args, Result>(
            Func<ParamsFunc<Args, Result>, ParamsFunc<Args, Result>> f)
        => (Args[] args) => f(y3(f))(args);

    static void Main() {
        var factorialY1 = Y<int, int>.y1((ParamsFunc<int, int> fact) => (int[] x)
            => (x[0] > 1) ? x[0] * fact(x[0] - 1) : 1);

        var fibY1 = Y<int, int>.y1((ParamsFunc<int, int> fib) => (int[] x)
            => (x[0] > 2) ? fib(x[0] - 1) + fib(x[0] - 2) : 2);

        Console.WriteLine(factorialY1(10)); // 362880
        Console.WriteLine(fibY1(10));       // 110
    }
}

Semi-idiomatic

using System;
using System.Diagnostics;

static class Program {
    delegate TResult ParamsFunc<T, TResult>(params T[] args);

    static class Y<Result, Args> {
        class RecursiveFunction {
            public Func<RecursiveFunction, ParamsFunc<Args, Result>> o;
            public RecursiveFunction(Func<RecursiveFunction, ParamsFunc<Args, Result>> o) => this.o = o;
        }

        public static ParamsFunc<Args, Result> y1(
                Func<ParamsFunc<Args, Result>, ParamsFunc<Args, Result>> f) {

            var r = new RecursiveFunction(w => f(args => w.o(w)(args)));

            return r.o(r);
        }
    }

    static ParamsFunc<Args, Result> y2<Args, Result>(
            Func<ParamsFunc<Args, Result>, ParamsFunc<Args, Result>> f) {

        Func<dynamic, ParamsFunc<Args, Result>> r = w => {
            Debug.Assert(w is Func<dynamic, ParamsFunc<Args, Result>>);
            return f(args => w(w)(args));
        };

        return r(r);
    }

    static ParamsFunc<Args, Result> y3<Args, Result>(
            Func<ParamsFunc<Args, Result>, ParamsFunc<Args, Result>> f)
        => args => f(y3(f))(args);

    static void Main() {
        var factorialY1 = Y<int, int>.y1(fact => x => (x[0] > 1) ? x[0] * fact(x[0] - 1) : 1);
        var fibY1 = Y<int, int>.y1(fib => x => (x[0] > 2) ? fib(x[0] - 1) + fib(x[0] - 2) : 2);

        Console.WriteLine(factorialY1(10));
        Console.WriteLine(fibY1(10));
    }
}

Go

func(T) TResult in Go corresponds to Func<T, TResult> in C#.

Verbatim

using System;

// Func and FuncFunc can be defined using using aliases and the System.Func<T, TReult> type, but RecursiveFunc must be a delegate type of its own.
using Func = System.Func<int, int>;
using FuncFunc = System.Func<System.Func<int, int>, System.Func<int, int>>;

delegate Func RecursiveFunc(RecursiveFunc f);

static class Program {
    static void Main() {
        var fac = Y(almost_fac);
        var fib = Y(almost_fib);
        Console.WriteLine("fac(10) = " + fac(10));
        Console.WriteLine("fib(10) = " + fib(10));
    }

    static Func Y(FuncFunc f) {
        RecursiveFunc g = delegate (RecursiveFunc r) {
            return f(delegate (int x) {
                return r(r)(x);
            });
        };
        return g(g);
    }

    static Func almost_fac(Func f) {
        return delegate (int x) {
            if (x <= 1) {
                return 1;
            }
            return x * f(x-1);
        };
    }

    static Func almost_fib(Func f) {
        return delegate (int x) {
            if (x <= 2) {
                return 1;
            }
            return f(x-1)+f(x-2);
        };
    }
}

Recursive:

    static Func Y(FuncFunc f) {
        return delegate (int x) {
            return f(Y(f))(x);
        };
    }

Semi-idiomatic

using System;

delegate int Func(int i);
delegate Func FuncFunc(Func f);
delegate Func RecursiveFunc(RecursiveFunc f);

static class Program {
    static void Main() {
        var fac = Y(almost_fac);
        var fib = Y(almost_fib);
        Console.WriteLine("fac(10) = " + fac(10));
        Console.WriteLine("fib(10) = " + fib(10));
    }

    static Func Y(FuncFunc f) {
        RecursiveFunc g = r => f(x => r(r)(x));
        return g(g);
    }

    static Func almost_fac(Func f) => x => x <= 1 ? 1 : x * f(x - 1);

    static Func almost_fib(Func f) => x => x <= 2 ? 1 : f(x - 1) + f(x - 2);
}

Recursive:

    static Func Y(FuncFunc f) => x => f(Y(f))(x);

Java

Verbatim

Since Java uses interfaces and C# uses delegates, which are the only type that the C# compiler will coerce lambda expressions to, this code declares a Functions class for providing a means of converting CLR delegates to objects that implement the Function and RecursiveFunction interfaces.

using System;

static class Program {
    interface Function<T, R> {
        R apply(T t);
    }

    interface RecursiveFunction<F> : Function<RecursiveFunction<F>, F> {
    }

    static class Functions {
        class Function<T, R> : Program.Function<T, R> {
            readonly Func<T, R> _inner;

            public Function(Func<T, R> inner) => this._inner = inner;

            public R apply(T t) => this._inner(t);
        }

        class RecursiveFunction<F> : Function<Program.RecursiveFunction<F>, F>, Program.RecursiveFunction<F> {
            public RecursiveFunction(Func<Program.RecursiveFunction<F>, F> inner) : base(inner) {
            }
        }

        public static Program.Function<T, R> Create<T, R>(Func<T, R> inner) => new Function<T, R>(inner);
        public static Program.RecursiveFunction<F> Create<F>(Func<Program.RecursiveFunction<F>, F> inner) => new RecursiveFunction<F>(inner);
    }

    static Function<A, B> Y<A, B>(Function<Function<A, B>, Function<A, B>> f) {
        var r = Functions.Create<Function<A, B>>(w => f.apply(Functions.Create<A, B>(x => w.apply(w).apply(x))));
        return r.apply(r);
    }

    static void Main(params String[] arguments) {
        Function<int, int> fib = Y(Functions.Create<Function<int, int>, Function<int, int>>(f => Functions.Create<int, int>(n =>
            (n <= 2)
              ? 1
              : (f.apply(n - 1) + f.apply(n - 2))))
        );
        Function<int, int> fac = Y(Functions.Create<Function<int, int>, Function<int, int>>(f => Functions.Create<int, int>(n =>
            (n <= 1)
              ? 1
              : (n * f.apply(n - 1))))
        );

        Console.WriteLine("fib(10) = " + fib.apply(10));
        Console.WriteLine("fac(10) = " + fac.apply(10));
    }
}

"Idiomatic"

For demonstrative purposes, to completely avoid using CLR delegates, lambda expressions can be replaced with explicit types that implement the functional interfaces. Closures are thus implemented by replacing all usages of the original local variable with a field of the type that represents the lambda expression; this process, called "hoisting" is actually how variable capturing is implemented by the C# compiler (for more information, see this Microsoft blog post.

using System;

static class YCombinator {
    interface Function<T, R> {
        R apply(T t);
    }

    interface RecursiveFunction<F> : Function<RecursiveFunction<F>, F> {
    }

    static class Y<A, B> {
        class __1 : RecursiveFunction<Function<A, B>> {
            class __2 : Function<A, B> {
                readonly RecursiveFunction<Function<A, B>> w;

                public __2(RecursiveFunction<Function<A, B>> w) {
                    this.w = w;
                }

                public B apply(A x) {
                    return w.apply(w).apply(x);
                }
            }

            Function<Function<A, B>, Function<A, B>> f;

            public __1(Function<Function<A, B>, Function<A, B>> f) {
                this.f = f;
            }

            public Function<A, B> apply(RecursiveFunction<Function<A, B>> w) {
                return f.apply(new __2(w));
            }
        }

        public static Function<A, B> _(Function<Function<A, B>, Function<A, B>> f) {
            var r = new __1(f);
            return r.apply(r);
        }
    }

    class __1 : Function<Function<int, int>, Function<int, int>> {
        class __2 : Function<int, int> {
            readonly Function<int, int> f;

            public __2(Function<int, int> f) {
                this.f = f;
            }

            public int apply(int n) {
                return
                    (n <= 2)
                  ? 1
                  : (f.apply(n - 1) + f.apply(n - 2));
            }
        }

        public Function<int, int> apply(Function<int, int> f) {
            return new __2(f);
        }
    }

    class __2 : Function<Function<int, int>, Function<int, int>> {
        class __3 : Function<int, int> {
            readonly Function<int, int> f;

            public __3(Function<int, int> f) {
                this.f = f;
            }

            public int apply(int n) {
                return
                    (n <= 1)
                  ? 1
                  : (n * f.apply(n - 1));
            }
        }

        public Function<int, int> apply(Function<int, int> f) {
            return new __3(f);
        }
    }

    static void Main(params String[] arguments) {
        Function<int, int> fib = Y<int, int>._(new __1());
        Function<int, int> fac = Y<int, int>._(new __2());

        Console.WriteLine("fib(10) = " + fib.apply(10));
        Console.WriteLine("fac(10) = " + fac.apply(10));
    }
}

C# 1.0

To conclude this chain of decreasing reliance on language features, here is above code translated to C# 1.0. The largest change is the replacement of the generic interfaces with the results of manually substituting their type parameters.

using System;

class Program {
    interface Func {
        int apply(int i);
    }

    interface FuncFunc {
        Func apply(Func f);
    }

    interface RecursiveFunc {
        Func apply(RecursiveFunc f);
    }

    class Y {
        class __1 : RecursiveFunc {
            class __2 : Func {
                readonly RecursiveFunc w;

                public __2(RecursiveFunc w) {
                    this.w = w;
                }

                public int apply(int x) {
                    return w.apply(w).apply(x);
                }
            }

            readonly FuncFunc f;

            public __1(FuncFunc f) {
                this.f = f;
            }

            public Func apply(RecursiveFunc w) {
                return f.apply(new __2(w));
            }
        }

        public static Func _(FuncFunc f) {
            __1 r = new __1(f);
            return r.apply(r);
        }
    }

    class __fib : FuncFunc {
        class __1 : Func {
            readonly Func f;

            public __1(Func f) {
                this.f = f;
            }

            public int apply(int n) {
                return
                    (n <= 2)
                  ? 1
                  : (f.apply(n - 1) + f.apply(n - 2));
            }

        }

        public Func apply(Func f) {
            return new __1(f);
        }
    }

    class __fac : FuncFunc {
        class __1 : Func {
            readonly Func f;

            public __1(Func f) {
                this.f = f;
            }

            public int apply(int n) {
                return
                    (n <= 1)
                  ? 1
                  : (n * f.apply(n - 1));
            }
        }

        public Func apply(Func f) {
            return new __1(f);
        }
    }

    static void Main(params String[] arguments) {
        Func fib = Y._(new __fib());
        Func fac = Y._(new __fac());

        Console.WriteLine("fib(10) = " + fib.apply(10));
        Console.WriteLine("fac(10) = " + fac.apply(10));
    }
}

Modified/varargs (the last implementation in the Java section)

Since C# delegates cannot declare members, extension methods are used to simulate doing so.

using System;
using System.Collections.Generic;
using System.Linq;
using System.Numerics;

static class Func {
    public static Func<T, TResult2> andThen<T, TResult, TResult2>(
            this Func<T, TResult> @this,
            Func<TResult, TResult2> after)
        => _ => after(@this(_));
}

delegate OUTPUT SelfApplicable<OUTPUT>(SelfApplicable<OUTPUT> s);
static class SelfApplicable {
    public static OUTPUT selfApply<OUTPUT>(this SelfApplicable<OUTPUT> @this) => @this(@this);
}

delegate FUNCTION FixedPoint<FUNCTION>(Func<FUNCTION, FUNCTION> f);

delegate OUTPUT VarargsFunction<INPUTS, OUTPUT>(params INPUTS[] inputs);
static class VarargsFunction {
    public static VarargsFunction<INPUTS, OUTPUT> from<INPUTS, OUTPUT>(
            Func<INPUTS[], OUTPUT> function)
        => function.Invoke;

    public static VarargsFunction<INPUTS, OUTPUT> upgrade<INPUTS, OUTPUT>(
            Func<INPUTS, OUTPUT> function) {
        return inputs => function(inputs[0]);
    }

    public static VarargsFunction<INPUTS, OUTPUT> upgrade<INPUTS, OUTPUT>(
            Func<INPUTS, INPUTS, OUTPUT> function) {
        return inputs => function(inputs[0], inputs[1]);
    }

    public static VarargsFunction<INPUTS, POST_OUTPUT> andThen<INPUTS, OUTPUT, POST_OUTPUT>(
            this VarargsFunction<INPUTS, OUTPUT> @this,
            VarargsFunction<OUTPUT, POST_OUTPUT> after) {
        return inputs => after(@this(inputs));
    }

    public static Func<INPUTS, OUTPUT> toFunction<INPUTS, OUTPUT>(
            this VarargsFunction<INPUTS, OUTPUT> @this) {
        return input => @this(input);
    }

    public static Func<INPUTS, INPUTS, OUTPUT> toBiFunction<INPUTS, OUTPUT>(
            this VarargsFunction<INPUTS, OUTPUT> @this) {
        return (input, input2) => @this(input, input2);
    }

    public static VarargsFunction<PRE_INPUTS, OUTPUT> transformArguments<PRE_INPUTS, INPUTS, OUTPUT>(
            this VarargsFunction<INPUTS, OUTPUT> @this,
            Func<PRE_INPUTS, INPUTS> transformer) {
        return inputs => @this(inputs.AsParallel().AsOrdered().Select(transformer).ToArray());
    }
}

delegate FixedPoint<FUNCTION> Y<FUNCTION>(SelfApplicable<FixedPoint<FUNCTION>> y);

static class Program {
    static TResult Cast<TResult>(this Delegate @this) where TResult : Delegate {
        return (TResult)Delegate.CreateDelegate(typeof(TResult), @this.Target, @this.Method);
    }

    static void Main(params String[] arguments) {
        BigInteger TWO = BigInteger.One + BigInteger.One;

        Func<IFormattable, long> toLong = x => long.Parse(x.ToString());
        Func<IFormattable, BigInteger> toBigInteger = x => new BigInteger(toLong(x));

        /* Based on https://gist.github.com/aruld/3965968/#comment-604392 */
        Y<VarargsFunction<IFormattable, IFormattable>> combinator = y => f => x => f(y.selfApply()(f))(x);
        FixedPoint<VarargsFunction<IFormattable, IFormattable>> fixedPoint =
            combinator.Cast<SelfApplicable<FixedPoint<VarargsFunction<IFormattable, IFormattable>>>>().selfApply();

        VarargsFunction<IFormattable, IFormattable> fibonacci = fixedPoint(
            f => VarargsFunction.upgrade(
                toBigInteger.andThen(
                    n => (IFormattable)(
                        (n.CompareTo(TWO) <= 0)
                        ? 1
                        : BigInteger.Parse(f(n - BigInteger.One).ToString())
                            + BigInteger.Parse(f(n - TWO).ToString()))
                )
            )
        );

        VarargsFunction<IFormattable, IFormattable> factorial = fixedPoint(
            f => VarargsFunction.upgrade(
                toBigInteger.andThen(
                    n => (IFormattable)((n.CompareTo(BigInteger.One) <= 0)
                        ? 1
                        : n * BigInteger.Parse(f(n - BigInteger.One).ToString()))
                )
            )
        );

        VarargsFunction<IFormattable, IFormattable> ackermann = fixedPoint(
            f => VarargsFunction.upgrade(
                (BigInteger m, BigInteger n) => m.Equals(BigInteger.Zero)
                    ? n + BigInteger.One
                    : f(
                        m - BigInteger.One,
                        n.Equals(BigInteger.Zero)
                            ? BigInteger.One
                            : f(m, n - BigInteger.One)
                    )
            ).transformArguments(toBigInteger)
        );

        var functions = new Dictionary<String, VarargsFunction<IFormattable, IFormattable>>();
        functions.Add("fibonacci", fibonacci);
        functions.Add("factorial", factorial);
        functions.Add("ackermann", ackermann);

        var parameters = new Dictionary<VarargsFunction<IFormattable, IFormattable>, IFormattable[]>();
        parameters.Add(functions["fibonacci"], new IFormattable[] { 20 });
        parameters.Add(functions["factorial"], new IFormattable[] { 10 });
        parameters.Add(functions["ackermann"], new IFormattable[] { 3, 2 });

        functions.AsParallel().Select(
            entry => entry.Key
                + "[" + String.Join(", ", parameters[entry.Value].Select(x => x.ToString())) + "]"
                + " = "
                + entry.Value(parameters[entry.Value])
        ).ForAll(Console.WriteLine);
    }
}

Swift

T -> TResult in Swift corresponds to Func<T, TResult> in C#.

Verbatim

The more idiomatic version doesn't look much different.

using System;

static class Program {
    struct RecursiveFunc<F> {
        public Func<RecursiveFunc<F>, F> o;
    }

    static Func<A, B> Y<A, B>(Func<Func<A, B>, Func<A, B>> f) {
        var r = new RecursiveFunc<Func<A, B>> { o = w => f(_0 => w.o(w)(_0)) };
        return r.o(r);
    }

    static void Main() {
        // C# can't infer the type arguments to Y either; either it or f must be explicitly typed.
        var fac = Y((Func<int, int> f) => _0 => _0 <= 1 ? 1 : _0 * f(_0 - 1));
        var fib = Y((Func<int, int> f) => _0 => _0 <= 2 ? 1 : f(_0 - 1) + f(_0 - 2));

        Console.WriteLine($"fac(5) = {fac(5)}");
        Console.WriteLine($"fib(9) = {fib(9)}");
    }
}

Without recursive type:

    public static Func<A, B> Y<A, B>(Func<Func<A, B>, Func<A, B>> f) {
        Func<dynamic, Func<A, B>> r = z => { var w = (Func<dynamic, Func<A, B>>)z; return f(_0 => w(w)(_0)); };
        return r(r);
    }

Recursive:

    public static Func<In, Out> Y<In, Out>(Func<Func<In, Out>, Func<In, Out>> f) {
        return x => f(Y(f))(x);
    }

C++

Works with: C++11

Known to work with GCC 4.7.2. Compile with

g++ --std=c++11 ycomb.cc

#include <iostream>
#include <functional>

template <typename F>
struct RecursiveFunc {
	std::function<F(RecursiveFunc)> o;
};

template <typename A, typename B>
std::function<B(A)> Y (std::function<std::function<B(A)>(std::function<B(A)>)> f) {
	RecursiveFunc<std::function<B(A)>> r = {
		std::function<std::function<B(A)>(RecursiveFunc<std::function<B(A)>>)>([f](RecursiveFunc<std::function<B(A)>> w) {
			return f(std::function<B(A)>([w](A x) {
				return w.o(w)(x);
			}));
		})
	};
	return r.o(r);
}

typedef std::function<int(int)> Func;
typedef std::function<Func(Func)> FuncFunc;
FuncFunc almost_fac = [](Func f) {
	return Func([f](int n) {
		if (n <= 1) return 1;
		return n * f(n - 1);
	});
};

FuncFunc almost_fib = [](Func f) {
	return Func([f](int n) {
	 	if (n <= 2) return 1;
		return  f(n - 1) + f(n - 2);
	});
};

int main() {
	auto fib = Y(almost_fib);
	auto fac = Y(almost_fac);
	std::cout << "fib(10) = " << fib(10) << std::endl;
	std::cout << "fac(10) = " << fac(10) << std::endl;
	return 0;
}

Output:

fib(10) = 55
fac(10) = 3628800

Works with: C++14

A shorter version, taking advantage of generic lambdas. Known to work with GCC 5.2.0, but likely some earlier versions as well. Compile with

g++ --std=c++14 ycomb.cc

#include <iostream>
#include <functional>
int main () {
  auto y = ([] (auto f) { return
              ([] (auto x) { return x (x); }
                 ([=] (auto y) -> std:: function <int (int)> { return
                    f ([=] (auto a) { return
                          (y (y)) (a) ;});}));});

  auto almost_fib = [] (auto f) { return
                       [=] (auto n) { return
                         n < 2? 1: f (n - 1) + f (n - 2) ;};};
  auto almost_fac = [] (auto f) { return 
                       [=] (auto n) { return 
                         n <= 1? n: n * f (n - 1); };};

  auto fib = y (almost_fib);
  auto fac = y (almost_fac);
  std:: cout << fib (10) << '\n' 
             << fac (10) << '\n';
}

Output:

fib(10) = 55
fac(10) = 3628800

The usual version using recursion, disallowed by the task:

template <typename A, typename B>
std::function<B(A)> Y (std::function<std::function<B(A)>(std::function<B(A)>)> f) {
	return [f](A x) {
		return f(Y(f))(x);
	};
}

Another version which is disallowed because a function passes itself, which is also a kind of recursion:

template <typename A, typename B>
struct YFunctor {
  const std::function<std::function<B(A)>(std::function<B(A)>)> f;
  YFunctor(std::function<std::function<B(A)>(std::function<B(A)>)> _f) : f(_f) {}
  B operator()(A x) const {
    return f(*this)(x);
  }
};

template <typename A, typename B>
std::function<B(A)> Y (std::function<std::function<B(A)>(std::function<B(A)>)> f) {
  return YFunctor<A,B>(f);
}

Ceylon

Using a class for the recursive type:

Result(*Args) y1<Result,Args>(
        Result(*Args)(Result(*Args)) f)
        given Args satisfies Anything[] {

    class RecursiveFunction(o) {
        shared Result(*Args)(RecursiveFunction) o;
    }

    value r = RecursiveFunction((RecursiveFunction w)
        =>  f(flatten((Args args) => w.o(w)(*args))));

    return r.o(r);
}

value factorialY1 = y1((Integer(Integer) fact)(Integer x)
    =>  if (x > 1) then x * fact(x - 1) else 1);

value fibY1 = y1((Integer(Integer) fib)(Integer x)
    =>  if (x > 2) then fib(x - 1) + fib(x - 2) else 2);

print(factorialY1(10)); // 3628800
print(fibY1(10));       // 110

Using Anything to erase the function type:

Result(*Args) y2<Result,Args>(
        Result(*Args)(Result(*Args)) f)
        given Args satisfies Anything[] {

    function r(Anything w) {
        assert (is Result(*Args)(Anything) w);
        return f(flatten((Args args) => w(w)(*args)));
    }

    return r(r);
}

Using recursion, this does not satisfy the task requirements, but is included here for illustrative purposes:

Result(*Args) y3<Result, Args>(
        Result(*Args)(Result(*Args)) f)
        given Args satisfies Anything[]
    =>  flatten((Args args) => f(y3(f))(*args));

Chapel

Strict (non-lazy = non-deferred execution) languages will race with the usually defined Y combinator (call-by-name) so most implementations are the Z combinator which lack one Beta Reduction from the true Y combinator (they are call-by-value). Although one can inject laziness so as to make the true Y combinator work with strict languages, the following code implements the usual Z call-by-value combinator using records to represent closures as Chapel does not have First Class Functions that can capture bindings from outside their scope other than from global scope:

Works with: Chapel version 1.24.1

proc fixz(f) {
  record InnerFunc {
    const xi;
    proc this(a) { return xi(xi)(a); }
  }
  record XFunc {
    const fi;
    proc this(x) { return fi(new InnerFunc(x)); }
  }
  const g = new XFunc(f);
  return g(g);
}

record Facz {
  record FacFunc {
    const fi;
    proc this(n: int): int {
      return if n <= 1 then 1 else n * fi(n - 1); }
  }
  proc this(f) { return new FacFunc(f); }
}

record Fibz {
  record FibFunc {
    const fi;
    proc this(n: int): int {
      return if n <= 1 then n else fi(n - 2) + fi(n - 1); }
  }
  proc this(f) { return new FibFunc(f); }
}

const facz = fixz(new Facz());
const fibz = fixz(new Fibz());

writeln(facz(10));
writeln(fibz(10));

Output:

3628800
55

One can write a true call-by-name Y combinator by injecting one level of laziness or deferred execution at the defining function level as per the following code:

Works with: Chapel version 1.24.1

// this is the longer version...
/*
proc fixy(f) {
  record InnerFunc {
    const xi;
    proc this() { return xi(xi); }
  }
  record XFunc {
    const fi;
    proc this(x) { return fi(new InnerFunc(x)); }
  }
  const g = new XFunc(f);
  return g(g);
}
// */

// short version using direct recursion as Chapel has...
// note that this version of fix uses function recursion in its own definition;
// thus its use just means that the recursion has been "pulled" into the "fix" function,
// instead of the function that uses it...
proc fixy(f) {
  record InnerFunc { const fi; proc this() { return fixy(fi); } }
  return f(new InnerFunc(f));
}

record Facy {
  record FacFunc {
    const fi;
    proc this(n: int): int {
        return if n <= 1 then 1 else n * fi()(n - 1); }
  }
  proc this(f) { return new FacFunc(f); }
}

record Fiby {
  record FibFunc {
    const fi;
    proc this(n: int): int {
      return if n <= 1 then n else fi()(n - 2) + fi()(n - 1); }
  }
  proc this(f) { return new FibFunc(f); }
}

const facy = fixy(new Facy());
const fibz = fixy(new Fiby());

writeln(facy(10));
writeln(fibz(10));

The output is the same as the above.

Clojure

(defn Y [f]
  ((fn [x] (x x))
   (fn [x]
     (f (fn [& args]
          (apply (x x) args))))))

(def fac
     (fn [f]
       (fn [n]
         (if (zero? n) 1 (* n (f (dec n)))))))

(def fib
     (fn [f]
       (fn [n]
         (condp = n
           0 0
           1 1
           (+ (f (dec n))
              (f (dec (dec n))))))))

Output:

user> ((Y fac) 10)
3628800
user> ((Y fib) 10)
55

Y can be written slightly more concisely via syntax sugar:

(defn Y [f]
  (#(% %) #(f (fn [& args] (apply (% %) args)))))

CoffeeScript

Y = (f) -> g = f( (t...) -> g(t...) )

or

Y = (f) -> ((h)->h(h))((h)->f((t...)->h(h)(t...)))

fac = Y( (f) -> (n) -> if n > 1 then n * f(n-1) else 1 )
fib = Y( (f) -> (n) -> if n > 1 then f(n-1) + f(n-2) else n )

Common Lisp

(defun Y (f)
  ((lambda (g) (funcall g g))
   (lambda (g)
     (funcall f (lambda (&rest a)
		  (apply (funcall g g) a))))))

(defun fac (n)
  (funcall
   (Y (lambda (f)
       (lambda (n)
         (if (zerop n)
	   1
	   (* n (funcall f (1- n)))))))
   n))

(defun fib (n)
  (funcall
   (Y (lambda (f)
       (lambda (n a b)
         (if (< n 1)
           a
           (funcall f (1- n) b (+ a b))))))
   n 0 1))

? (mapcar #'fac '(1 2 3 4 5 6 7 8 9 10))
(1 2 6 24 120 720 5040 40320 362880 3628800))

? (mapcar #'fib '(1 2 3 4 5 6 7 8 9 10))
(1 1 2 3 5 8 13 21 34 55)

Crystal

Although Crystal is very much an OOP language, it does have "Proc"'s that can be used as lambda functions and even as closures where they capture state from the environment external to the body, and these can be used to implement the Y-Combinator. Note that many of the other static strict languages don't implement the true Y-Combinator but rather the Z-Combinator, which lacks one Beta reduction from the Y-Combinator and is more limiting in use. For strict languages such as Crystal, all that is needed to implement the true Y-Combinator is to inject some laziness by deferring execution using a "Thunk" - a function without any arguments returning a deferred value, which requires that functions can also be closures.

The following Crystal code implements the name-recursion Y-Combinator without assuming that functions are recursive (which in Crystal they actually are):

require "big"

struct RecursiveFunc(T) # a generic recursive function wrapper...
  getter recfnc : RecursiveFunc(T) -> T
  def initialize(@recfnc) end
end

struct YCombo(T) # a struct or class needs to be used so as to be generic...
  def initialize (@fnc : Proc(T) -> T) end
  def fixy
    g = -> (x : RecursiveFunc(T)) {
              @fnc.call(-> { x.recfnc.call(x) }) }
    g.call(RecursiveFunc(T).new(g))
  end
end

def fac(x) # horrendouly inefficient not using tail calls...
  facp = -> (fn : Proc(BigInt -> BigInt)) {
               -> (n : BigInt) { n < 2 ? n : n * fn.call.call(n - 1) } }
  YCombo.new(facp).fixy.call(BigInt.new(x))
end

def fib(x) # horrendouly inefficient not using tail calls...
  facp = -> (fn : Proc(BigInt -> BigInt)) {
               -> (n : BigInt) {
                     n < 3 ? n - 1 : fn.call.call(n - 2) + fn.call.call(n - 1) } }
  YCombo.new(facp).fixy.call(BigInt.new(x))
end

puts fac(10)
puts fib(11) # starts from 0 not 1!

The "horrendously inefficient" massively repetitious implementations can be made much more efficient by changing the implementation for the two functions as follows:

def fac(x) # the more efficient tail recursive version...
  facp = -> (fn : Proc(BigInt -> (Int32 -> BigInt))) {
               -> (n : BigInt) { -> (i : Int32) {
                     i < 2 ? n : fn.call.call(i * n).call(i - 1) } } }
  YCombo.new(facp).fixy.call(BigInt.new(1)).call(x)
end

def fib(x) # the more efficient tail recursive version...
  fibp = -> (fn : Proc(BigInt -> (BigInt -> (Int32 -> BigInt)))) {
    -> (f : BigInt) { -> (s : BigInt) { -> (i : Int32) {
          i < 2 ? f : fn.call.call(s).call(f + s).call(i - 1) } } } }
  YCombo.new(fibp).fixy.call(BigInt.new).call(BigInt.new(1)).call(x)
end

Finally, since Crystal function's/"def"'s can call themselves recursively, the implementation of the Y-Combinator can be changed to use this while still being "call by name" (not value/variable recursion) as follows; this uses the identical lambda "Proc"'s internally with just the application to the Y-Combinator changed:

def ycombo(f)
  f.call(-> { ycombo(f) })
end

def fac(x) # the more efficient tail recursive version...
  facp = -> (fn : Proc(BigInt -> (Int32 -> BigInt))) {
               -> (n : BigInt) { -> (i : Int32) {
                     i < 2 ? n : fn.call.call(i * n).call(i - 1) } } }
  ycombo(facp).call(BigInt.new(1)).call(x)
end

def fib(x) # the more efficient tail recursive version...
  fibp = -> (fn : Proc(BigInt -> (BigInt -> (Int32 -> BigInt)))) {
    -> (f : BigInt) { -> (s : BigInt) { -> (i : Int32) {
          i < 2 ? f : fn.call.call(s).call(f + s).call(i - 1) } } } }
  ycombo(fibp).call(BigInt.new).call(BigInt.new(1)).call(x)
end

All versions produce the same output:

Output:

3628800
55

D

A stateless generic Y combinator:

import std.stdio, std.traits, std.algorithm, std.range;

auto Y(S, T...)(S delegate(T) delegate(S delegate(T)) f) {
    static struct F {
        S delegate(T) delegate(F) f;
        alias f this;
    }
    return (x => x(x))(F(x => f((T v) => x(x)(v))));
}

void main() { // Demo code:
    auto factorial = Y((int delegate(int) self) =>
        (int n) => 0 == n ? 1 : n * self(n - 1)
    );

    auto ackermann = Y((ulong delegate(ulong, ulong) self) =>
        (ulong m, ulong n) {
            if (m == 0) return n + 1;
            if (n == 0) return self(m - 1, 1);
            return self(m - 1, self(m, n - 1));
    });

    writeln("factorial: ", 10.iota.map!factorial);
    writeln("ackermann(3, 5): ", ackermann(3, 5));
}

Output:

factorial: [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
ackermann(3, 5): 253

Delphi

Works with: Delphi XE and higher

May work with Delphi 2009 and 2010 too.

Translation of: C++

(The translation is not literal; e.g. the function argument type is left unspecified to increase generality.)

program Y;

{$APPTYPE CONSOLE}

uses
  SysUtils;

type
  YCombinator = class sealed
    class function Fix<T> (F: TFunc<TFunc<T, T>, TFunc<T, T>>): TFunc<T, T>; static;
  end;

  TRecursiveFuncWrapper<T> = record // workaround required because of QC #101272 (http://qc.embarcadero.com/wc/qcmain.aspx?d=101272)
    type
      TRecursiveFunc = reference to function (R: TRecursiveFuncWrapper<T>): TFunc<T, T>;
    var
      O: TRecursiveFunc;
  end;

class function YCombinator.Fix<T> (F: TFunc<TFunc<T, T>, TFunc<T, T>>): TFunc<T, T>;
var
  R: TRecursiveFuncWrapper<T>;
begin
  R.O := function (W: TRecursiveFuncWrapper<T>): TFunc<T, T>
    begin
      Result := F (function (I: T): T
        begin
          Result := W.O (W) (I);
        end);
    end;
  Result := R.O (R);
end;


type
  IntFunc = TFunc<Integer, Integer>;

function AlmostFac (F: IntFunc): IntFunc;
begin
  Result := function (N: Integer): Integer
    begin
      if N <= 1 then
        Result := 1
      else
        Result := N * F (N - 1);
    end;
end;

function AlmostFib (F: TFunc<Integer, Integer>): TFunc<Integer, Integer>;
begin
  Result := function (N: Integer): Integer
    begin
      if N <= 2 then
        Result := 1
      else
        Result := F (N - 1) + F (N - 2);
    end;
end;

var
  Fib, Fac: IntFunc;
begin
  Fib := YCombinator.Fix<Integer> (AlmostFib);
  Fac := YCombinator.Fix<Integer> (AlmostFac);
  Writeln ('Fib(10) = ', Fib (10));
  Writeln ('Fac(10) = ', Fac (10));
end.

Dhall

Dhall is not a turing complete language, so there's no way to implement the real Y combinator. That being said, you can replicate the effects of the Y combinator to any arbitrary but finite recursion depth using the builtin function Natural/Fold, which acts as a bounded fixed-point combinator that takes a natural argument to describe how far to recurse.

Here's an example using Natural/Fold to define recursive definitions of fibonacci and factorial:

let const
    : ∀(b : Type) → ∀(a : Type) → a → b → a
    = λ(r : Type) → λ(a : Type) → λ(x : a) → λ(y : r) → x

let fac
    : ∀(n : Natural) → Natural
    = λ(n : Natural) →
        let factorial =
              λ(f : Natural → Natural → Natural) →
              λ(n : Natural) →
              λ(i : Natural) →
                if Natural/isZero i then n else f (i * n) (Natural/subtract 1 i)

        in  Natural/fold
              n
              (Natural → Natural → Natural)
              factorial
              (const Natural Natural)
              1
              n

let fib
    : ∀(n : Natural) → Natural
    = λ(n : Natural) →
        let fibFunc = Natural → Natural → Natural → Natural

        let fibonacci =
              λ(f : fibFunc) →
              λ(a : Natural) →
              λ(b : Natural) →
              λ(i : Natural) →
                if    Natural/isZero i
                then  a
                else  f b (a + b) (Natural/subtract 1 i)

        in  Natural/fold
              n
              fibFunc
              fibonacci
              (λ(a : Natural) → λ(_ : Natural) → λ(_ : Natural) → a)
              0
              1
              n

in [fac 50, fib 50]

The above dhall file gets rendered down to:

[ 30414093201713378043612608166064768844377641568960512000000000000
, 12586269025
]

Déjà Vu

Translation of: Python

Y f:
	labda y:
		labda:
			call y @y
		f
	labda x:
		x @x
	call

labda f:
	labda n:
		if < 1 n:
			* n f -- n
		else:
			1
set :fac Y

labda f:
	labda n:
		if < 1 n:
			+ f - n 2 f -- n
		else:
			1
set :fib Y

!. fac 6
!. fib 6

Output:

720
13

E

Translation of: Python

def y := fn f { fn x { x(x) }(fn y { f(fn a { y(y)(a) }) }) }
def fac := fn f { fn n { if (n<2) {1} else { n*f(n-1) } }}
def fib := fn f { fn n { if (n == 0) {0} else if (n == 1) {1} else { f(n-1) + f(n-2) } }}

? pragma.enable("accumulator")
? accum [] for i in 0..!10 { _.with(y(fac)(i)) }
[1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]

? accum [] for i in 0..!10 { _.with(y(fib)(i)) }
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]

EchoLisp

;; Ref : http://www.ece.uc.edu/~franco/C511/html/Scheme/ycomb.html

 (define Y
    (lambda (X)
      ((lambda (procedure)
         (X (lambda (arg) ((procedure procedure) arg))))
       (lambda (procedure)
         (X (lambda (arg) ((procedure procedure) arg)))))))

; Fib
(define Fib* (lambda (func-arg) 
    (lambda (n) (if (< n 2) n (+ (func-arg (- n 1)) (func-arg (- n 2)))))))
(define fib (Y Fib*))
(fib 6)
    → 8

; Fact
(define F*
   (lambda (func-arg) (lambda (n) (if (zero? n) 1 (* n (func-arg (- n 1)))))))
(define fact (Y F*))

(fact 10)
    → 3628800

Eero

Translated from Objective-C example on this page.

#import <Foundation/Foundation.h>

typedef int (^Func)(int)
typedef Func (^FuncFunc)(Func)
typedef Func (^RecursiveFunc)(id) // hide recursive typing behind dynamic typing

Func fix(FuncFunc f)
  Func r(RecursiveFunc g)
    int s(int x)
      return g(g)(x)
    return f(s)
  return r(r)

int main(int argc, const char *argv[])
  autoreleasepool

    Func almost_fac(Func f)
      return (int n | return n <= 1 ? 1 : n * f(n - 1))

    Func almost_fib(Func f)
      return (int n | return n <= 2 ? 1 : f(n - 1) + f(n - 2))

    fib := fix(almost_fib)
    fac := fix(almost_fac)

    Log('fib(10) = %d', fib(10))
    Log('fac(10) = %d', fac(10))

  return 0

Ela

fix = \f -> (\x -> & f (x x)) (\x -> & f (x x))

fac _ 0 = 1
fac f n = n * f (n - 1)

fib _ 0 = 0
fib _ 1 = 1
fib f n = f (n - 1) + f (n - 2)

(fix fac 12, fix fib 12)

Output:

(479001600,144)

Elena

Translation of: Smalltalk

ELENA 6.x :

import extensions;
 
singleton YCombinator
{
    fix(func)
        = (f){(x){ x(x) }((g){ f((x){ (g(g))(x) })})}(func);
}
 
public program()
{
    var fib := YCombinator.fix::(f => (i => (i <= 1) ? i : (f(i-1) + f(i-2)) ));
    var fact := YCombinator.fix::(f => (i => (i == 0) ? 1 : (f(i-1) * i) ));
 
    console.printLine("fib(10)=",fib(10));
    console.printLine("fact(10)=",fact(10));
}

Output:

fib(10)=55
fact(10)=3628800

Elixir

Translation of: Python

iex(1)> yc = fn f -> (fn x -> x.(x) end).(fn y -> f.(fn arg -> y.(y).(arg) end) end) end
#Function<6.90072148/1 in :erl_eval.expr/5>
iex(2)> fac = fn f -> fn n -> if n < 2 do 1 else n * f.(n-1) end end end
#Function<6.90072148/1 in :erl_eval.expr/5>
iex(3)> for i <- 0..9, do: yc.(fac).(i)
[1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
iex(4)> fib = fn f -> fn n -> if n == 0 do 0 else (if n == 1 do 1 else f.(n-1) + f.(n-2) end) end end end
#Function<6.90072148/1 in :erl_eval.expr/5>
iex(5)> for i <- 0..9, do: yc.(fib).(i)
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]

Elm

This is similar to the Haskell solution below, but the first `fixz` is a strict fixed-point combinator lacking one beta reduction as compared to the Y-combinator; the second `fixy` injects laziness using a "thunk" (a unit argument function whose return value is deferred until the function is called/applied).

Note: the Fibonacci sequence is defined to start with zero or one, with the first exactly the same but with a zero prepended; these Fibonacci calculations use the second definition.

module Main exposing ( main )

import Html exposing ( Html, text )
 
-- As with most of the strict (non-deferred or non-lazy) languages,
-- this is the Z-combinator with the additional value parameter...

-- wrap type conversion to avoid recursive type definition...
type Mu a b = Roll (Mu a b -> a -> b)
 
unroll : Mu a b -> (Mu a b -> a -> b) -- unwrap it...
unroll (Roll x) = x
 
-- note lack of beta reduction using values...
fixz : ((a -> b) -> (a -> b)) -> (a -> b)
fixz f = let g r = f (\ v -> unroll r r v) in g (Roll g)
 
facz : Int -> Int
-- facz = fixz <| \ f n -> if n < 2 then 1 else n * f (n - 1) -- inefficient recursion
facz = fixz (\ f n i -> if i < 2 then n else f (i * n) (i - 1)) 1 -- efficient tailcall
 
fibz : Int -> Int
-- fibz = fixz <| \ f n -> if n < 2 then n else f (n - 1) + f (n - 2) -- inefficient recursion
fibz = fixz (\ fn f s i -> if i < 2 then f else fn s (f + s) (i - 1)) 1 1 -- efficient tailcall
 
-- by injecting laziness, we can get the true Y-combinator...
-- as this includes laziness, there is no need for the type wrapper!
fixy : ((() -> a) -> a) -> a
fixy f = f <| \ () -> fixy f -- direct function recursion
-- the above is not value recursion but function recursion!
-- fixv f = let x = f x in x -- not allowed by task or by Elm!
-- we can make Elm allow it by injecting laziness...
-- fixv f = let x = f () x in x -- but now value recursion not function recursion
 
facy : Int -> Int
-- facy = fixy <| \ f n -> if n < 2 then 1 else n * f () (n - 1) -- inefficient recursion
facy = fixy (\ f n i -> if i < 2 then n else f () (i * n) (i - 1)) 1 -- efficient tailcall
 
fiby : Int -> Int
-- fiby = fixy <| \ f n -> if n < 2 then n else f () (n - 1) + f (n - 2) -- inefficient recursion
fiby = fixy (\ fn f s i -> if i < 2 then f else fn () s (f + s) (i - 1)) 1 1 -- efficient tailcall
 
-- something that can be done with a true Y-Combinator that
-- can't be done with the Z combinator...
-- given an infinite Co-Inductive Stream (CIS) defined as...
type CIS a = CIS a (() -> CIS a) -- infinite lazy stream!
 
mapCIS : (a -> b) -> CIS a -> CIS b -- uses function to map
mapCIS cf cis =
  let mp (CIS head restf) = CIS (cf head) <| \ () -> mp (restf()) in mp cis
 
-- now we can define a Fibonacci stream as follows...
fibs : () -> CIS Int
fibs() = -- two recursive fix's, second already lazy...
  let fibsgen = fixy (\ fn (CIS (f, s) restf) ->
        CIS (s, f + s) (\ () -> fn () (restf())))
  in fixy (\ cisthnk -> fibsgen (CIS (0, 1) cisthnk))
       |> mapCIS (\ (v, _) -> v)
 
nCISs2String : Int -> CIS a -> String -- convert n CIS's to String
nCISs2String n cis =
  let loop i (CIS head restf) rslt =
        if i <= 0 then rslt ++ " )" else
        loop (i - 1) (restf()) (rslt ++ " " ++ Debug.toString head)
  in loop n cis "("
 
-- unfortunately, if we need CIS memoization so as
-- to make a true lazy list, Elm doesn't support it!!!
 
main : Html Never
main =
  String.fromInt (facz 10) ++ " " ++ String.fromInt (fibz 10)
    ++ " " ++ String.fromInt (facy 10) ++ " " ++ String.fromInt (fiby 10)
    ++ " " ++ nCISs2String 20 (fibs())
      |> text

Output:

3628800 55 3628800 55 ( 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 )

Erlang

Y = fun(M) -> (fun(X) -> X(X) end)(fun (F) -> M(fun(A) -> (F(F))(A) end) end) end.

Fac = fun (F) ->
          fun (0) -> 1;
              (N) -> N * F(N-1)
          end
      end.
Fib = fun(F) ->
          fun(0) -> 0;
             (1) -> 1;
             (N) -> F(N-1) + F(N-2)
          end
      end.
(Y(Fac))(5). %% 120
(Y(Fib))(8). %% 21

F#

March 2024

In spite of everything that follows I am going to go with this.

// Y combinator. Nigel Galloway: March 5th., 2024
type Y<'T> = { eval: Y<'T> -> ('T -> 'T) }
let Y n g=let l = { eval = fun l -> fun x -> (n (l.eval l)) x } in  (l.eval l) g
let fibonacci=function 0->1 |x->let fibonacci f= function 0->0 |1->1 |x->f(x - 1) + f(x - 2) in Y fibonacci x
let factorial n=let factorial f=function 0->1 |x->x*f(x-1) in Y factorial n
printfn "fibonacci 10=%d\nfactorial 5=%d" (fibonacci 10) (factorial 5)

Output:

fibonacci 10=55
factorial 5=120

Pre March 2024

type 'a mu = Roll of ('a mu -> 'a)  // ' fixes ease syntax colouring confusion with
 
let unroll (Roll x) = x
// val unroll : 'a mu -> ('a mu -> 'a)
 
// As with most of the strict (non-deferred or non-lazy) languages,
// this is the Z-combinator with the additional 'a' parameter...
let fix f = let g = fun x a -> f (unroll x x) a in g (Roll g)
// val fix : (('a -> 'b) -> 'a -> 'b) -> 'a -> 'b = <fun>
 
// Although true to the factorial definition, the
// recursive call is not in tail call position, so can't be optimized
// and will overflow the call stack for the recursive calls for large ranges...
//let fac = fix (fun f n -> if n < 2 then 1I else bigint n * f (n - 1))
// val fac : (int -> BigInteger) = <fun>
 
// much better progressive calculation in tail call position...
let fac = fix (fun f n i -> if i < 2 then n else f (bigint i * n) (i - 1)) <| 1I
// val fac : (int -> BigInteger) = <fun>
 
// Although true to the definition of Fibonacci numbers,
// this can't be tail call optimized and recursively repeats calculations
// for a horrendously inefficient exponential performance fib function...
// let fib = fix (fun fnc n -> if n < 2 then n else fnc (n - 1) + fnc (n - 2))
// val fib : (int -> BigInteger) = <fun>
 
// much better progressive calculation in tail call position...
let fib = fix (fun fnc f s i -> if i < 2 then f else fnc s (f + s) (i - 1)) 1I 1I
// val fib : (int -> BigInteger) = <fun>
 
[<EntryPoint>]
let main argv =
  fac 10 |> printfn "%A" // prints 3628800
  fib 10 |> printfn "%A" // prints 55
  0 // return an integer exit code

Output:

3628800
55

Note that the first `fac` definition isn't really very good as the recursion is not in tail call position and thus will build stack, although for these functions one will likely never use it to stack overflow as the result would be exceedingly large; it is better defined as per the second definition as a steadily increasing function controlled by an `int` indexing argument and thus be in tail call position as is done for the `fib` function.

Also note that the above isn't the true fix point Y-combinator which would race without the beta conversion to the Z-combinator with the included `a` argument; the Z-combinator can't be used in all cases that require a true Y-combinator such as in the formation of deferred execution sequences in the last example, as follows:

// same as previous...
type 'a mu = Roll of ('a mu -> 'a)  // ' fixes ease syntax colouring confusion with
 
// same as previous...
let unroll (Roll x) = x
// val unroll : 'a mu -> ('a mu -> 'a)
 
// break race condition with some deferred execution - laziness...
let fix f = let g = fun x -> f <| fun() -> (unroll x x) in g (Roll g)
// val fix : ((unit -> 'a) -> 'a -> 'a) = <fun>

// same efficient version of factorial functionb with added deferred execution...
let fac = fix (fun f n i -> if i < 2 then n else f () (bigint i * n) (i - 1)) <| 1I
// val fac : (int -> BigInteger) = <fun>

// same efficient version of Fibonacci function with added deferred execution...
let fib = fix (fun fnc f s i -> if i < 2 then f else fnc () s (f + s) (i - 1)) 1I 1I
// val fib : (int -> BigInteger) = <fun>

// given the following definition for an infinite Co-Inductive Stream (CIS)...
type CIS<'a> = CIS of 'a * (unit -> CIS<'a>) // ' fix formatting

// Using a double Y-Combinator recursion...
// defines a continuous stream of Fibonacci numbers; there are other simpler ways,
// this way implements recursion by using the Y-combinator, although it is
// much slower than other ways due to the many additional function calls,
// it demonstrates something that can't be done with the Z-combinator...
let fibs() =
  let fbsgen = fix (fun fnc (CIS((f, s), rest)) ->
                 CIS((s, f + s), fun() -> fnc () <| rest()))
  Seq.unfold (fun (CIS((v, _), rest)) -> Some(v, rest()))
               <| fix (fun cis -> fbsgen (CIS((1I, 0I), cis))) // cis is a lazy thunk!

[<EntryPoint>]
let main argv =
  fac 10 |> printfn "%A" // prints 3628800
  fib 10 |> printfn "%A" // prints 55
  fibs() |> Seq.take 20 |> Seq.iter (printf "%A ")
  printfn ""
  0 // return an integer exit code

Output:

3628800
55
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 

The above would be useful if F# did not have recursive functions (functions that can call themselves in their own definition), but as for most modern languages, F# does have function recursion by the use of the `rec` keyword before the function name, thus the above `fac` and `fib` functions can be written much more simply (and to run faster using tail recursion) with a recursion definition for the `fix` Y-combinator as follows, with a simple injected deferred execution to prevent race:

let rec fix f = f <| fun() -> fix f
// val fix : f:((unit -> 'a) -> 'a) -> 'a

// the application of this true Y-combinator is the same as for the above non function recursive version.

Using the Y-combinator (or Z-combinator) as expressed here is pointless as in unnecessary and makes the code slower due to the extra function calls through the call stack, with the first non-function recursive implementation even slower than the second function recursion one; a non Y-combinator version can use function recursion with tail call optimization to simplify looping for about 100 times the speed in the actual loop overhead; thus, this is primarily an intellectual exercise.

Also note that these Y-combinators/Z-combinator are the non sharing kind; for certain types of algorithms that require that the input and output recursive values be the same (such as the same sequence or lazy list but made reference at difference stages), these will work but may be many times slower as in over 10 times slower than using binding recursion if the language allows it; F# allows binding recursion with a warning.

Factor

In rosettacode/Y.factor

USING: fry kernel math ;
IN: rosettacode.Y
: Y ( quot -- quot )
    '[ [ dup call call ] curry @ ] dup call ; inline

: almost-fac ( quot -- quot )
    '[ dup zero? [ drop 1 ] [ dup 1 - @ * ] if ] ;

: almost-fib ( quot -- quot )
    '[ dup 2 >= [ 1 2 [ - @ ] bi-curry@ bi + ] when ] ;

In rosettacode/Y-tests.factor

USING: kernel tools.test rosettacode.Y ;
IN: rosettacode.Y.tests

[ 120 ] [ 5 [ almost-fac ] Y call ] unit-test
[ 8 ]   [ 6 [ almost-fib ] Y call ] unit-test

running the tests :

 ( scratchpad - auto ) "rosettacode.Y" test
Loading resource:work/rosettacode/Y/Y-tests.factor
Unit Test: { [ 120 ] [ 5 [ almost-fac ] Y call ] }
Unit Test: { [ 8 ]   [ 6 [ almost-fib ] Y call ] }

Falcon

Y = { f => {x=> {n => f(x(x))(n)}} ({x=> {n => f(x(x))(n)}}) }
facStep = { f => {x => x < 1 ? 1 : x*f(x-1) }}
fibStep = { f => {x => x == 0 ? 0 : (x == 1 ? 1 : f(x-1) + f(x-2))}}

YFac = Y(facStep)
YFib = Y(fibStep)

> "Factorial 10: ", YFac(10)
> "Fibonacci 10: ", YFib(10)

Forth

\ Begin of approach. Depends on 'latestxt' word of GForth implementation.

: self-parameter  ( xt -- xt' )
  >r :noname  latestxt postpone literal r> compile, postpone ;
;
: Y  ( xt -- xt' )
  dup self-parameter 2>r
  :noname 2r> postpone literal compile, postpone ;
;

Usage:

\ Fibonnacci test
10 :noname ( u xt -- u' ) over 2 < if drop exit then >r 1- dup r@ execute swap 1- r> execute + ; Y execute . 55  ok
\ Factorial test
10 :noname  ( u xt -- u' )  over 2 < if 2drop 1 exit then  over 1- swap execute * ; Y execute . 3628800  ok

\ End of approach.

\ Address of an xt.
variable 'xt
\ Make room for an xt.
: xt, ( -- ) here 'xt !  1 cells allot ;
\ Store xt.
: !xt ( xt -- ) 'xt @ ! ;
\ Compile fetching the xt.
: @xt, ( -- ) 'xt @ postpone literal postpone @ ;
\ Compile the Y combinator.
: y, ( xt1 -- xt2 ) >r :noname @xt, r> compile, postpone ; ;
\ Make a new instance of the Y combinator.
: y ( xt1 -- xt2 ) xt, y, dup !xt ;

Samples:

\ Factorial
10 :noname ( u1 xt -- u2 ) over ?dup if 1- swap execute * else 2drop 1 then ;
y execute . 3628800 ok

\ Fibonacci
10 :noname ( u1 xt -- u2 ) over 2 < if drop else >r 1- dup r@ execute swap 1- r> execute + then ;
y execute . 55 ok

GAP

Y := function(f)
    local u;
    u := x -> x(x);
    return u(y -> f(a -> y(y)(a)));
end;

fib := function(f)
    local u;
    u := function(n)
        if n < 2 then
            return n;
        else
            return f(n-1) + f(n-2);
        fi;
    end;
    return u;
end;

Y(fib)(10);
# 55

fac := function(f)
    local u;
    u := function(n)
        if n < 2 then
            return 1;
        else
            return n*f(n-1);
        fi;
    end;
    return u;
end;

Y(fac)(8);
# 40320

Genyris

Translation of: Scheme

def fac (f)
    function (n)
      if (equal? n 0) 1
        * n (f (- n 1))
def fib (f)
  function (n)
    cond
      (equal? n 0) 0
      (equal? n 1) 1
      else (+ (f (- n 1)) (f (- n 2)))

def Y (f)
  (function (x) (x x))
      function (y)
          f
             function (&rest args) (apply (y y) args)

assertEqual ((Y fac) 5) 120
assertEqual ((Y fib) 8) 21

Go

package main

import "fmt"

type Func func(int) int
type FuncFunc func(Func) Func
type RecursiveFunc func (RecursiveFunc) Func

func main() {
	fac := Y(almost_fac)
	fib := Y(almost_fib)
	fmt.Println("fac(10) = ", fac(10))
	fmt.Println("fib(10) = ", fib(10))
}

func Y(f FuncFunc) Func {
	g := func(r RecursiveFunc) Func {
		return f(func(x int) int {
			return r(r)(x)
		})
	}
	return g(g)
}

func almost_fac(f Func) Func {
	return func(x int) int {
		if x <= 1 {
			return 1
		}
		return x * f(x-1)
	}
}

func almost_fib(f Func) Func {
	return func(x int) int {
		if x <= 2 {
			return 1
		}
		return f(x-1)+f(x-2)
	}
}

Output:

fac(10) =  3628800
fib(10) =  55

The usual version using recursion, disallowed by the task:

func Y(f FuncFunc) Func {
	return func(x int) int {
		return f(Y(f))(x)
	}
}

Groovy

Here is the simplest (unary) form of applicative order Y:

def Y = { le -> ({ f -> f(f) })({ f -> le { x -> f(f)(x) } }) }

def factorial = Y { fac ->
    { n -> n <= 2 ? n : n * fac(n - 1) }
}

assert 2432902008176640000 == factorial(20G)

def fib = Y { fibStar ->
    { n -> n <= 1 ? n : fibStar(n - 1) + fibStar(n - 2) }
}

assert fib(10) == 55

This version was translated from the one in The Little Schemer by Friedman and Felleisen. The use of the variable name le is due to the fact that the authors derive Y from an ordinary recursive length function.

A variadic version of Y in Groovy looks like this:

def Y = { le -> ({ f -> f(f) })({ f -> le { Object[] args -> f(f)(*args) } }) }

def mul = Y { mulStar -> { a, b -> a ? b + mulStar(a - 1, b) : 0 } }

1.upto(10) {
    assert mul(it, 10) == it * 10
}

Haskell

The obvious definition of the Y combinator (\f-> (\x -> f (x x)) (\x-> f (x x))) cannot be used in Haskell because it contains an infinite recursive type (a = a -> b). Defining a data type (Mu) allows this recursion to be broken.

newtype Mu a = Roll
  { unroll :: Mu a -> a }
 
fix :: (a -> a) -> a
fix = g <*> (Roll . g)
  where
    g = (. (>>= id) unroll)
 
- this version is not in tail call position...
-- fac :: Integer -> Integer
-- fac =
--   fix $ \f n -> if n <= 0 then 1 else n * f (n - 1)

-- this version builds a progression from tail call position and is more efficient...
fac :: Integer -> Integer
fac =
  (fix $ \f n i -> if i <= 0 then n else f (i * n) (i - 1)) 1
 
-- make fibs a function, else memory leak as
-- head of the list can never be released as per:
--   https://wiki.haskell.org/Memory_leak, type 1.1
-- overly complex version...
{--
fibs :: () -> [Integer]
fibs() =
  fix $
    (0 :) . (1 :) .
      (fix
        (\f (x:xs) (y:ys) ->
          case x + y of n -> n `seq` n : f xs ys) <*> tail)
--}

-- easier to read, simpler (faster) version...
fibs :: () -> [Integer]
fibs() = 0 : 1 : fix fibs_ 0 1
  where
    fibs_ fnc f s =
      case f + s of n -> n `seq` n : fnc s n
 
main :: IO ()
main =
  mapM_
    print
    [ map fac [1 .. 20]
    , take 20 $ fibs()
    ]

The usual version uses recursion on a binding, disallowed by the task, to define the fix itself; but the definitions produced by this fix does not use recursion on value bindings although it does use recursion when defining a function (not possible in all languages), so it can be viewed as a true Y-combinator too:

-- note that this version of fix uses function recursion in its own definition;
-- thus its use just means that the recursion has been "pulled" into the "fix" function,
-- instead of the function that uses it...
fix :: (a -> a) -> a
fix f = f (fix f) -- _not_ the {fix f = x where x = f x}

fac :: Integer -> Integer
fac =
  (fix $
    \f n i ->
      if i <= 0 then n
      else f (i * n) (i - 1)) 1
 
fib :: Integer -> Integer
fib =
  (fix $
    \fnc f s i ->
      if i <= 1 then f
      else case f + s of n -> n `seq` fnc s n (i - 1)) 0 1

{--
-- compute a lazy infinite list. This is
-- a Y-combinator version of: fibs() = 0:1:zipWith (+) fibs (tail fibs)
-- which is the same as the above version but easier to read...
fibs :: () -> [Integer]
fibs() = fix fibs_
  where
    zipP f (x:xs) (y:ys) =
      case x + y of n -> n `seq` n : f xs ys
    fibs_ a = 0 : 1 : fix zipP a (tail a)
--}

-- easier to read, simpler (faster) version...
fibs :: () -> [Integer]
fibs() = 0 : 1 : fix fibs_ 0 1
  where
    fibs_ fnc f s =
      case f + s of n -> n `seq` n : fnc s n

-- This code shows how the functions can be used:
main :: IO ()
main =
  mapM_
    print
    [ map fac [1 .. 20] 
    , map fib [1 .. 20] 
    , take 20 fibs()
    ]

Now just because something is possible using the Y-combinator doesn't mean that it is practical: the above implementations can't compute much past the 1000th number in the Fibonacci list sequence and is quite slow at doing so; using direct function recursive routines compute about 100 times faster and don't hang for large ranges, nor give problems compiling as the first version does (GHC version 8.4.3 at -O1 optimization level).

If one has recursive functions as Haskell does and as used by the second `fix`, there is no need to use `fix`/the Y-combinator at all since one may as well just write the recursion directly. The Y-combinator may be interesting mathematically, but it isn't very practical when one has any other choice.

J

See also: j:Essays/Combinators

Non-tacit version

Unfortunately, in principle, J functions cannot take functions of the same type as arguments. In other words, verbs (functions) cannot take verbs, and adverbs or conjunctions (higher-order functions) cannot take adverbs or conjunctions. This implementation uses the body, a literal (string), of an explicit adverb (a higher-order function with a left argument) as the argument for Y, to represent the adverb for which the product of Y is a fixed-point verb; Y itself is also an adverb.

Y=. '('':''<@;(1;~":0)<@;<@((":0)&;))'(2 : 0 '')
  (1 : (m,'u'))(1 : (m,'''u u`:6('',(5!:5<''u''),'')`:6 y'''))(1 :'u u`:6')
)

This Y combinator follows the standard method: it produces a fixed-point which reproduces and transforms itself anonymously according to the adverb represented by Y's argument. All names (variables) refer to arguments of the enclosing adverbs and there are no assignments.

The factorial and Fibonacci examples follow:

   'if. * y do. y * u <: y else. 1 end.' Y 10 NB. Factorial
3628800   
          '(u@:<:@:<: + u@:<:)^:(1 < ])' Y 10 NB. Fibonacci
55

The names u, x, and y are J's standard names for arguments; the name y represents the argument of u and the name u represents the verb argument of the adverb for which Y produces a fixed-point. Any verb can also be expressed tacitly, without any reference to its argument(s), as in the Fibonacci example.

A structured derivation of a Y with states follows (the stateless version can be produced by replacing all the names by its referents):

   arb=. ':'<@;(1;~":0)<@;<@((":0)&;)                     NB. AR of an explicit adverb from its body 
   
   ara=. 1 :'arb u'                                       NB. The verb arb as an adverb
   srt=. 1 :'arb ''u u`:6('' , (5!:5<''u'') , '')`:6 y''' NB. AR of the self-replication and transformation adverb
   gab=. 1 :'u u`:6'                                      NB. The AR of the adverb and the adverb itself as a train
   
   Y=. ara srt gab                                        NB. Train of adverbs

The adverb Y, apart from using a representation as Y's argument, satisfies the task's requirements. However, it only works for monadic verbs (functions with a right argument). J's verbs can also be dyadic (functions with a left and right arguments) and ambivalent (almost all J's primitive verbs are ambivalent; for example - can be used as in - 1 and 2 - 1). The following adverb (XY) implements anonymous recursion of monadic, dyadic, and ambivalent verbs (the name x represents the left argument of u),

XY=. (1 :'('':''<@;(1;~":0)<@;<@((":0)&;))u')(1 :'('':''<@;(1;~":0)<@;<@((":0)&;))((''u u`:6('',(5!:5<''u''),'')`:6 y''),(10{a.),'':'',(10{a.),''x(u u`:6('',(5!:5<''u''),'')`:6)y'')')(1 :'u u`:6')

The following are examples of anonymous dyadic and ambivalent recursions,

   1 2 3 '([:`(>:@:])`(<:@:[ u 1:)`(<:@[ u [ u <:@:])@.(#.@,&*))'XY"0/  1 2 3 4 5 NB. Ackermann function...
 3  4  5   6   7
 5  7  9  11  13
13 29 61 125 253
                               '1:`(<: u <:)@.* : (+ + 2 * u@:])'XY"0/~ i.7       NB. Ambivalent recursion...
2  5 14 35 80 173 362
3  6 15 36 81 174 363
4  7 16 37 82 175 364
5  8 17 38 83 176 365
6  9 18 39 84 177 366
7 10 19 40 85 178 367
8 11 20 41 86 179 368
   NB. OEIS A097813 - main diagonal   
   NB. OEIS A050488 = A097813 - 1 - adyacent upper off-diagonal

J supports directly anonymous tacit recursion via the verb $: and for tacit recursions, XY is equivalent to the adverb,

YX=. (1 :'('':''<@;(1;~":0)<@;<@((":0)&;))u')($:`)(`:6)

Tacit version

The Y combinator can be implemented indirectly using, for example, the linear representations of verbs (Y becomes a wrapper which takes an ad hoc verb as an argument and serializes it; the underlying self-referring system interprets the serialized representation of a verb as the corresponding verb):

Y=. ((((&>)/)((((^:_1)b.)(`(<'0';_1)))(`:6)))(&([ 128!:2 ,&<)))

The factorial and Fibonacci examples:

   u=. [ NB. Function (left)
   n=. ] NB. Argument (right)
   sr=. [ apply f. ,&< NB. Self referring
    
   fac=. (1:`(n * u sr n - 1:)) @. (0 < n)
   fac f. Y 10
3628800
       
   Fib=. ((u sr n - 2:) + u sr n - 1:) ^: (1 < n)
   Fib f. Y 10
55

The stateless functions are shown next (the f. adverb replaces all embedded names by its referents):

   fac f. Y NB. Factorial...
'1:`(] * [ ([ 128!:2 ,&<) ] - 1:)@.(0 < ])&>/'&([ 128!:2 ,&<)

   fac f.   NB. Factorial step...
1:`(] * [ ([ 128!:2 ,&<) ] - 1:)@.(0 < ])

   
   Fib f. Y NB. Fibonacci...
'(([ ([ 128!:2 ,&<) ] - 2:) + [ ([ 128!:2 ,&<) ] - 1:)^:(1 < ])&>/'&([ 128!:2 ,&<)

   Fib f.   NB. Fibonacci step...
(([ ([ 128!:2 ,&<) ] - 2:) + [ ([ 128!:2 ,&<) ] - 1:)^:(1 < ])

A structured derivation of Y follows:

   sr=. [ apply f.,&<                 NB. Self referring
   lv=. (((^:_1)b.)(`(<'0';_1)))(`:6) NB. Linear representation of a verb argument
   Y=. (&>)/lv(&sr)                   NB. Y with embedded states
   Y=. 'Y'f.                          NB. Fixing it...
   Y                                  NB. ... To make it stateless (i.e., a combinator)
((((&>)/)((((^:_1)b.)(`_1))(`:6)))(&([ 128!:2 ,&<)))

Explicit alternate implementation

Another approach:

Y=:1 :0
  f=. u Defer
  (5!:1<'f') f y
)

Defer=: 1 :0
:
  g=. x&(x`:6)
  (5!:1<'g') u y
)

almost_factorial=: 4 :0
  if. 0 >: y do. 1
  else. y * x`:6 y-1 end.
)

almost_fibonacci=: 4 :0
  if. 2 > y do. y
  else. (x`:6 y-1) + x`:6 y-2 end.
)

Example use:

   almost_factorial Y 9
362880
   almost_fibonacci Y 9
34
   almost_fibonacci Y"0 i. 10
0 1 1 2 3 5 8 13 21 34

Or, if you would prefer to not have a dependency on the definition of Defer, an equivalent expression would be:

Y=:2 :0(0 :0)
NB. this block will be n in the second part
:
  g=. x&(x`:6)
  (5!:1<'g') u y
)
  f=. u (1 :n)
  (5!:1<'f') f y
)

That said, if you think of association with a name as state (because in different contexts the association may not exist, or may be different) you might also want to remove that association in the context of the Y combinator.

For example:

   almost_factorial f. Y 10
3628800

Java

Works with: Java version 8+

import java.util.function.Function;

public interface YCombinator {
  interface RecursiveFunction<F> extends Function<RecursiveFunction<F>, F> { }
  public static <A,B> Function<A,B> Y(Function<Function<A,B>, Function<A,B>> f) {
    RecursiveFunction<Function<A,B>> r = w -> f.apply(x -> w.apply(w).apply(x));
    return r.apply(r);
  }

  public static void main(String... arguments) {
    Function<Integer,Integer> fib = Y(f -> n ->
      (n <= 2)
        ? 1
        : (f.apply(n - 1) + f.apply(n - 2))
    );
    Function<Integer,Integer> fac = Y(f -> n ->
      (n <= 1)
        ? 1
        : (n * f.apply(n - 1))
    );

    System.out.println("fib(10) = " + fib.apply(10));
    System.out.println("fac(10) = " + fac.apply(10));
  }
}

Output:

fib(10) = 55
fac(10) = 3628800

The usual version using recursion, disallowed by the task:

    public static <A,B> Function<A,B> Y(Function<Function<A,B>, Function<A,B>> f) {
        return x -> f.apply(Y(f)).apply(x);
    }

Another version which is disallowed because a function passes itself, which is also a kind of recursion:

    public static <A,B> Function<A,B> Y(Function<Function<A,B>, Function<A,B>> f) {
        return new Function<A,B>() {
	    public B apply(A x) {
		return f.apply(this).apply(x);
	    }
	};
    }

Works with: Java version pre-8

We define a generic function interface like Java 8's Function.

interface Function<A, B> {
    public B call(A x);
}

public class YCombinator {
    interface RecursiveFunc<F> extends Function<RecursiveFunc<F>, F> { }

    public static <A,B> Function<A,B> fix(final Function<Function<A,B>, Function<A,B>> f) {
        RecursiveFunc<Function<A,B>> r =
            new RecursiveFunc<Function<A,B>>() {
            public Function<A,B> call(final RecursiveFunc<Function<A,B>> w) {
                return f.call(new Function<A,B>() {
                        public B call(A x) {
                            return w.call(w).call(x);
                        }
                    });
            }
        };
        return r.call(r);
    }

    public static void main(String[] args) {
        Function<Function<Integer,Integer>, Function<Integer,Integer>> almost_fib =
            new Function<Function<Integer,Integer>, Function<Integer,Integer>>() {
            public Function<Integer,Integer> call(final Function<Integer,Integer> f) {
                return new Function<Integer,Integer>() {
                    public Integer call(Integer n) {
                        if (n <= 2) return 1;
                        return f.call(n - 1) + f.call(n - 2);
                    }
                };
            }
        };

        Function<Function<Integer,Integer>, Function<Integer,Integer>> almost_fac =
            new Function<Function<Integer,Integer>, Function<Integer,Integer>>() {
            public Function<Integer,Integer> call(final Function<Integer,Integer> f) {
                return new Function<Integer,Integer>() {
                    public Integer call(Integer n) {
                        if (n <= 1) return 1;
                        return n * f.call(n - 1);
                    }
                };
            }
        };

        Function<Integer,Integer> fib = fix(almost_fib);
        Function<Integer,Integer> fac = fix(almost_fac);

        System.out.println("fib(10) = " + fib.call(10));
        System.out.println("fac(10) = " + fac.call(10));
    }
}

The following code modifies the Function interface such that multiple parameters (via varargs) are supported, simplifies the y function considerably, and the Ackermann function has been included in this implementation (mostly because both D and PicoLisp include it in their own implementations).

import java.util.function.Function;

@FunctionalInterface
public interface SelfApplicable<OUTPUT> extends Function<SelfApplicable<OUTPUT>, OUTPUT> {
  public default OUTPUT selfApply() {
    return apply(this);
  }
}

import java.util.function.Function;
import java.util.function.UnaryOperator;

@FunctionalInterface
public interface FixedPoint<FUNCTION> extends Function<UnaryOperator<FUNCTION>, FUNCTION> {}

import java.util.Arrays;
import java.util.Optional;
import java.util.function.Function;
import java.util.function.BiFunction;

@FunctionalInterface
public interface VarargsFunction<INPUTS, OUTPUT> extends Function<INPUTS[], OUTPUT> {
  @SuppressWarnings("unchecked")
  public OUTPUT apply(INPUTS... inputs);

  public static <INPUTS, OUTPUT> VarargsFunction<INPUTS, OUTPUT> from(Function<INPUTS[], OUTPUT> function) {
    return function::apply;
  }

  public static <INPUTS, OUTPUT> VarargsFunction<INPUTS, OUTPUT> upgrade(Function<INPUTS, OUTPUT> function) {
    return inputs -> function.apply(inputs[0]);
  }

  public static <INPUTS, OUTPUT> VarargsFunction<INPUTS, OUTPUT> upgrade(BiFunction<INPUTS, INPUTS, OUTPUT> function) {
    return inputs -> function.apply(inputs[0], inputs[1]);
  }

  @SuppressWarnings("unchecked")
  public default <POST_OUTPUT> VarargsFunction<INPUTS, POST_OUTPUT> andThen(
      VarargsFunction<OUTPUT, POST_OUTPUT> after) {
    return inputs -> after.apply(apply(inputs));
  }

  @SuppressWarnings("unchecked")
  public default Function<INPUTS, OUTPUT> toFunction() {
    return input -> apply(input);
  }

  @SuppressWarnings("unchecked")
  public default BiFunction<INPUTS, INPUTS, OUTPUT> toBiFunction() {
    return (input, input2) -> apply(input, input2);
  }

  @SuppressWarnings("unchecked")
  public default <PRE_INPUTS> VarargsFunction<PRE_INPUTS, OUTPUT> transformArguments(Function<PRE_INPUTS, INPUTS> transformer) {
    return inputs -> apply((INPUTS[]) Arrays.stream(inputs).parallel().map(transformer).toArray());
  }
}

import java.math.BigDecimal;
import java.math.BigInteger;
import java.util.Arrays;
import java.util.HashMap;
import java.util.Map;
import java.util.function.Function;
import java.util.function.UnaryOperator;
import java.util.stream.Collectors;
import java.util.stream.LongStream;

@FunctionalInterface
public interface Y<FUNCTION> extends SelfApplicable<FixedPoint<FUNCTION>> {
  public static void main(String... arguments) {
    BigInteger TWO = BigInteger.ONE.add(BigInteger.ONE);

    Function<Number, Long> toLong = Number::longValue;
    Function<Number, BigInteger> toBigInteger = toLong.andThen(BigInteger::valueOf);

    /* Based on https://gist.github.com/aruld/3965968/#comment-604392 */
    Y<VarargsFunction<Number, Number>> combinator = y -> f -> x -> f.apply(y.selfApply().apply(f)).apply(x);
    FixedPoint<VarargsFunction<Number, Number>> fixedPoint = combinator.selfApply();

    VarargsFunction<Number, Number> fibonacci = fixedPoint.apply(
      f -> VarargsFunction.upgrade(
        toBigInteger.andThen(
          n -> (n.compareTo(TWO) <= 0)
            ? 1
            : new BigInteger(f.apply(n.subtract(BigInteger.ONE)).toString())
              .add(new BigInteger(f.apply(n.subtract(TWO)).toString()))
        )
      )
    );

    VarargsFunction<Number, Number> factorial = fixedPoint.apply(
      f -> VarargsFunction.upgrade(
        toBigInteger.andThen(
          n -> (n.compareTo(BigInteger.ONE) <= 0)
            ? 1
            : n.multiply(new BigInteger(f.apply(n.subtract(BigInteger.ONE)).toString()))
        )
      )
    );

    VarargsFunction<Number, Number> ackermann = fixedPoint.apply(
      f -> VarargsFunction.upgrade(
        (BigInteger m, BigInteger n) -> m.equals(BigInteger.ZERO)
          ? n.add(BigInteger.ONE)
          : f.apply(
              m.subtract(BigInteger.ONE),
              n.equals(BigInteger.ZERO)
                ? BigInteger.ONE
                  : f.apply(m, n.subtract(BigInteger.ONE))
            )
      ).transformArguments(toBigInteger)
    );

    Map<String, VarargsFunction<Number, Number>> functions = new HashMap<>();
    functions.put("fibonacci", fibonacci);
    functions.put("factorial", factorial);
    functions.put("ackermann", ackermann);

    Map<VarargsFunction<Number, Number>, Number[]> parameters = new HashMap<>();
    parameters.put(functions.get("fibonacci"), new Number[]{20});
    parameters.put(functions.get("factorial"), new Number[]{10});
    parameters.put(functions.get("ackermann"), new Number[]{3, 2});

    functions.entrySet().stream().parallel().map(
      entry -> entry.getKey()
        + Arrays.toString(parameters.get(entry.getValue()))
        + " = "
        + entry.getValue().apply(parameters.get(entry.getValue()))
    ).forEach(System.out::println);
  }
}

Output:

(may depend on which function gets processed first)

factorial[10] = 3628800
ackermann[3, 2] = 29
fibonacci[20] = 6765

JavaScript

The standard version of the Y combinator does not use lexically bound local variables (or any local variables at all), which necessitates adding a wrapper function and some code duplication - the remaining locale variables are only there to make the relationship to the previous implementation more explicit:

function Y(f) {
    var g = f((function(h) {
        return function() {
            var g = f(h(h));
            return g.apply(this, arguments);
        }
    })(function(h) {
        return function() {
            var g = f(h(h));
            return g.apply(this, arguments);
        }
    }));
    return g;
}

var fac = Y(function(f) {
    return function (n) {
        return n > 1 ? n * f(n - 1) : 1;
    };
});

var fib = Y(function(f) {
    return function(n) {
        return n > 1 ? f(n - 1) + f(n - 2) : n;
    };
});

Changing the order of function application (i.e. the place where f gets called) and making use of the fact that we're generating a fixed-point, this can be reduced to

function Y(f) {
    return (function(h) {
        return h(h);
    })(function(h) {
        return f(function() {
            return h(h).apply(this, arguments);
        });
    });
}

A functionally equivalent version using the implicit this parameter is also possible:

function pseudoY(f) {
    return (function(h) {
        return h(h);
    })(function(h) {
        return f.bind(function() {
            return h(h).apply(null, arguments);
        });
    });
}

var fac = pseudoY(function(n) {
    return n > 1 ? n * this(n - 1) : 1;
});

var fib = pseudoY(function(n) {
    return n > 1 ? this(n - 1) + this(n - 2) : n;
});

However, pseudoY() is not a fixed-point combinator.

The usual version using recursion, disallowed by the task:

function Y(f) {
    return function() {
    	return f(Y(f)).apply(this, arguments);
    };
}

Another version which is disallowed because it uses arguments.callee for a function to get itself recursively:

function Y(f) {
    return function() {
    	return f(arguments.callee).apply(this, arguments);
    };
}

ECMAScript 2015 (ES6) variants

Since ECMAScript 2015 (ES6) just reached final draft, there are new ways to encode the applicative order Y combinator. These use the new fat arrow function expression syntax, and are made to allow functions of more than one argument through the use of new rest parameters syntax and the corresponding new spread operator syntax. Also showcases new default parameter value syntax:

let
    Y= // Except for the η-abstraction necessary for applicative order languages, this is the formal Y combinator.
        f=>((g=>(f((...x)=>g(g)(...x))))
            (g=>(f((...x)=>g(g)(...x))))),
    Y2= // Using β-abstraction to eliminate code repetition.
        f=>((f=>f(f))
            (g=>(f((...x)=>g(g)(...x))))),
    Y3= // Using β-abstraction to separate out the self application combinator δ.
        ((δ=>f=>δ(g=>(f((...x)=>g(g)(...x)))))
         ((f=>f(f)))),
    fix= // β/η-equivalent fix point combinator. Easier to convert to memoise than the Y combinator.
        (((f)=>(g)=>(h)=>(f(h)(g(h)))) // The Substitute combinator out of SKI calculus
         ((f)=>(g)=>(...x)=>(f(g(g)))(...x)) // S((S(KS)K)S(S(KS)K))(KI)
         ((f)=>(g)=>(...x)=>(f(g(g)))(...x))),
    fix2= // β/η-converted form of fix above into a more compact form
        f=>(f=>f(f))(g=>(...x)=>f(g(g))(...x)),
    opentailfact= // Open version of the tail call variant of the factorial function
        fact=>(n,m=1)=>n<2?m:fact(n-1,n*m);
    tailfact= // Tail call version of factorial function
        Y(opentailfact);

ECMAScript 2015 (ES6) also permits a really compact polyvariadic variant for mutually recursive functions:

let
    polyfix= // A version that takes an array instead of multiple arguments would simply use l instead of (...l) for parameter
        (...l)=>(
            (f=>f(f))
            (g=>l.map(f=>(...x)=>f(...g(g))(...x)))),
    [even,odd]= // The new destructive assignment syntax for arrays
        polyfix(
            (even,odd)=>n=>(n===0)||odd(n-1),
            (even,odd)=>n=>(n!==0)&&even(n-1));

A minimalist version:

var Y = f => (x => x(x))(y => f(x => y(y)(x)));
var fac = Y(f => n => n > 1 ? n * f(n-1) : 1);

Joy

DEFINE y == [dup cons] swap concat dup cons i;
     fac == [[pop null] [pop succ] [[dup pred] dip i *] ifte] y.

Julia

               _
   _       _ _(_)_     |  Documentation: https://docs.julialang.org
  (_)     | (_) (_)    |
   _ _   _| |_  __ _   |  Type "?" for help, "]?" for Pkg help.
  | | | | | | |/ _` |  |
  | | |_| | | | (_| |  |  Version 1.6.3 (2021-09-23)
 _/ |\__'_|_|_|\__'_|  |  Official https://julialang.org/ release
|__/                   |

julia> using Markdown

julia> @doc md"""
       # Y Combinator

       $λf. (λx. f (x x)) (λx. f (x x))$
       """ ->
       Y = f -> (x -> x(x))(y -> f((t...) -> y(y)(t...)))
Y

Usage:

julia> fac = f -> (n -> n < 2 ? 1 : n * f(n - 1))
#9 (generic function with 1 method)

julia> fib = f -> (n -> n == 0 ? 0 : (n == 1 ? 1 : f(n - 1) + f(n - 2)))
#13 (generic function with 1 method)

julia> Y(fac).(1:10)
10-element Vector{Int64}:
       1
       2
       6
      24
     120
     720
    5040
   40320
  362880
 3628800

julia> Y(fib).(1:10)
10-element Vector{Int64}:
  1
  1
  2
  3
  5
  8
 13
 21
 34
 55

Kitten

define y<S..., T...> (S..., (S..., (S... -> T...) -> T...) -> T...):
  -> f; { f y } f call

define fac (Int32, (Int32 -> Int32) -> Int32):
  -> x, rec;
  if (x <= 1) { 1 } else { (x - 1) rec call * x }

define fib (Int32, (Int32 -> Int32) -> Int32):
  -> x, rec;
  if (x <= 2):
    1
  else:
    (x - 1) rec call -> a;
    (x - 2) rec call -> b;
    a + b

5  \fac y say  // 120
10 \fib y say  // 55

Klingphix

:fac
    dup 1 great [dup 1 sub fac mult] if
;

 
:fib
    dup 1 great [dup 1 sub fib swap 2 sub fib add] if
;

 
:test
    print ": " print
    10 [over exec print " " print] for
    nl
;

 
@fib "fib" test
@fac "fac" test

"End " input

Output:

fib: 1 1 2 3 5 8 13 21 34 55
fac: 1 2 6 24 120 720 5040 40320 362880 3628800
End

Kotlin

// version 1.1.2

typealias Func<T, R> = (T) -> R

class RecursiveFunc<T, R>(val p: (RecursiveFunc<T, R>) -> Func<T, R>) 

fun <T, R> y(f: (Func<T, R>) -> Func<T, R>): Func<T, R> {
    val rec = RecursiveFunc<T, R> { r -> f { r.p(r)(it) } }
    return rec.p(rec)
}

fun fac(f: Func<Int, Int>) = { x: Int -> if (x <= 1) 1 else x * f(x - 1) }

fun fib(f: Func<Int, Int>) = { x: Int -> if (x <= 2) 1 else f(x - 1) + f(x - 2) }

fun main(args: Array<String>) {
    print("Factorial(1..10)   : ")
    for (i in 1..10) print("${y(::fac)(i)}  ") 
    print("\nFibonacci(1..10)   : ")   
    for (i in 1..10) print("${y(::fib)(i)}  ")
    println()
}

Output:

Factorial(1..10)   : 1  2  6  24  120  720  5040  40320  362880  3628800  
Fibonacci(1..10)   : 1  1  2  3  5  8  13  21  34  55  

Lambdatalk

Tested in http://lambdaway.free.fr/lambdawalks/?view=Ycombinator

1) defining the Ycombinator
{def Y {lambda {:f} {:f :f}}}

2) defining non recursive functions
2.1) factorial 
{def almost-fac 
 {lambda {:f :n}
  {if {= :n 1}
   then 1
   else {* :n {:f :f {- :n 1}}}}}}

2.2) fibonacci
{def almost-fibo
 {lambda {:f :n}
  {if {<   :n 2}
   then 1
   else {+ {:f :f {- :n 1}} {:f :f {- :n 2}}}}}}

3) testing
{{Y almost-fac} 6}  
-> 720
{{Y almost-fibo} 8} 
-> 34

Lang

Y combinator function:

# Disable warning for shadowing of predefined function
lang.errorOutput = -1

fp.combY = (fp.f) -> {
	# fp.f must be provided by the function with a partially called combinator function, because fp.f will not be available in the callee scope
	fp.func = (fp.f, fp.x) -> {
		fp.callFunc = (fp.f, fp.x, &args...) -> return fp.f(fp.x(fp.x))(&args...)
		
		return fn.combAN(fp.callFunc, fp.f, fp.x)
	}
	
	return fn.combM(fn.combA2(fp.func, fp.f))
}

# Re-enable warning output
lang.errorOutput = 1

Usage (Factorial):

fp.fac = (fp.func) -> {
	fp.retFunc = (fp.func, $n) -> {
		return parser.op($n < 2?1:$n * fp.func($n - 1))
	}
	
	return fn.combAN(fp.retFunc, fp.func)
}

# Apply Y combinator
fp.facY = fp.combY(fp.fac)

# Use function
fn.println(fp.facY(10))

Usage (Fibonacci):

fp.fib = (fp.func) -> {
	fp.retFunc = (fp.func, $x) -> {
		return parser.op($x < 2?1:fp.func($x - 2) + fp.func($x - 1))
	}
	
	return fn.combAN(fp.retFunc, fp.func)
}

fp.fibY = fp.combY(fp.fib)

fn.println(fp.fibY(10))

Lua

Y = function (f)
   return function(...)
      return (function(x) return x(x) end)(function(x) return f(function(y) return x(x)(y) end) end)(...)
   end
end

Usage:

almostfactorial = function(f) return function(n) return n > 0 and n * f(n-1) or 1 end end
almostfibs = function(f) return function(n) return n < 2 and n or f(n-1) + f(n-2) end end
factorial, fibs = Y(almostfactorial), Y(almostfibs)
print(factorial(7))

M2000 Interpreter

Lambda functions in M2000 are value types. They have a list of closures, but closures are copies, except for those closures which are reference types. Lambdas can keep state in closures (they are mutable). But here we didn't do that. Y combinator is a lambda which return a lambda with a closure as f function. This function called passing as first argument itself by value.

Module Ycombinator {
	\\ y() return value. no use of closure
	y=lambda (g, x)->g(g, x)
	Print y(lambda (g, n as decimal)->if(n=0->1, n*g(g, n-1)), 10)=3628800 ' true
	Print y(lambda (g, n)->if(n<=1->n,g(g, n-1)+g(g, n-2)), 10)=55 ' true
	
	\\ Using closure in y, y() return function
	y=lambda (g)->lambda g (x) -> g(g, x)
	fact=y((lambda (g, n as decimal)-> if(n=0->1, n*g(g, n-1))))
	Print fact(6)=720, fact(24)=620448401733239439360000@
	fib=y(lambda (g, n)->if(n<=1->n, g(g, n-1)+g(g, n-2)))
	Print  fib(10)=55
	}
Ycombinator

Module Checkit {
	Rem {
		all lambda arguments passed by value in this example
		There is no recursion in these lambdas
		Y combinator make  argument f as closure, as a copy of f
		m(m, argument) pass as first argument a copy of m
		so never a function, here, call itself, only call a copy who get it as argument before the call.
	}
	Y=lambda (f)-> {
		=lambda f (x)->f(f,x)
	}
	fac_step=lambda (m, n)-> {
		if n<2 then =1 else =n*m(m, n-1)
	}
	fac=Y(fac_step)
	fib_step=lambda (m, n)-> {
		if n<=1 then =n else =m(m, n-1)+m(m, n-2)
	}
	fib=Y(fib_step)
	For i=1 to 10 {
		Print fib(i), fac(i)
	}
}
Checkit
Module CheckRecursion {
	fac=lambda (n) -> {
		if n<2 then =1 else =n*Lambda(n-1)
	}
	fib=lambda (n) -> {
		if n<=1 then =n else =lambda(n-1)+lambda(n-2)
	}
	For i=1 to 10:Print fib(i), fac(i):Next
}
CheckRecursion

MANOOL

Here one additional technique is demonstrated: the Y combinator is applied to a function during compilation due to the $ operator, which is optional:

{ {extern "manool.org.18/std/0.3/all"} in
: let { Y = {proc {F} as {proc {X} as X[X]}[{proc {X} with {F} as F[{proc {Y} with {X} as X[X][Y]}]}]} } in
  { for { N = Range[10] } do
  : (WriteLine) Out; N "! = "
    {Y: proc {Rec} as {proc {N} with {Rec} as: if N == 0 then 1 else N * Rec[N - 1]}}$[N]
  }
  { for { N = Range[10] } do
  : (WriteLine) Out; "Fib " N " = "
    {Y: proc {Rec} as {proc {N} with {Rec} as: if N == 0 then 0 else: if N == 1 then 1 else Rec[N - 2] + Rec[N - 1]}}$[N]
  }
}

Using less syntactic sugar:

{ {extern "manool.org.18/std/0.3/all"} in
: let { Y = {proc {F} as {proc {X} as X[X]}[{proc {F; X} as F[{proc {X; Y} as X[X][Y]}.Bind[X]]}.Bind[F]]} } in
  { for { N = Range[10] } do
  : (WriteLine) Out; N "! = "
    {Y: proc {Rec} as {proc {Rec; N} as: if N == 0 then 1 else N * Rec[N - 1]}.Bind[Rec]}$[N]
  }
  { for { N = Range[10] } do
  : (WriteLine) Out; "Fib " N " = "
    {Y: proc {Rec} as {proc {Rec; N} as: if N == 0 then 0 else: if N == 1 then 1 else Rec[N - 2] + Rec[N - 1]}.Bind[Rec]}$[N]
  }
}

Output:

0! = 1
1! = 1
2! = 2
3! = 6
4! = 24
5! = 120
6! = 720
7! = 5040
8! = 40320
9! = 362880
Fib 0 = 0
Fib 1 = 1
Fib 2 = 1
Fib 3 = 2
Fib 4 = 3
Fib 5 = 5
Fib 6 = 8
Fib 7 = 13
Fib 8 = 21
Fib 9 = 34

Maple

> Y:=f->(x->x(x))(g->f((()->g(g)(args)))):
> Yfac:=Y(f->(x->`if`(x<2,1,x*f(x-1)))):
> seq( Yfac( i ), i = 1 .. 10 );
          1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800
> Yfib:=Y(f->(x->`if`(x<2,x,f(x-1)+f(x-2)))):
> seq( Yfib( i ), i = 1 .. 10 );
                    1, 1, 2, 3, 5, 8, 13, 21, 34, 55

Mathematica / Wolfram Language

Y = Function[f, #[#] &[Function[g, f[g[g][##] &]]]];
factorial = Y[Function[f, If[# < 1, 1, # f[# - 1]] &]];
fibonacci = Y[Function[f, If[# < 2, #, f[# - 1] + f[# - 2]] &]];

Moonscript

Z = (f using nil) -> ((x) -> x x) (x) -> f (...) -> (x x) ...
factorial = Z (f using nil) -> (n) -> if n == 0 then 1 else n * f n - 1

Nim

# The following is implemented for a strict language as a Z-Combinator;
# Z-combinators differ from Y-combinators in lacking one Beta reduction of
# the extra `T` argument to the function to be recursed...

import sugar

proc fixz[T, TResult](f: ((T) -> TResult) -> ((T) -> TResult)): (T) -> TResult =
  type RecursiveFunc = object # any entity that wraps the recursion!
    recfnc: ((RecursiveFunc) -> ((T) -> TResult))
  let g = (x: RecursiveFunc) => f ((a: T) => x.recfnc(x)(a))
  g(RecursiveFunc(recfnc: g))

let facz = fixz((f: (int) -> int) =>
  ((n: int) => (if n <= 1: 1 else: n * f(n - 1))))
let fibz = fixz((f: (int) -> int) =>
  ((n: int) => (if n < 2: n else: f(n - 2) + f(n - 1))))

echo facz(10)
echo fibz(10)

# by adding some laziness, we can get a true Y-Combinator...
# note that there is no specified parmater(s) - truly fix point!...

#[
proc fixy[T](f: () -> T -> T): T =
  type RecursiveFunc = object # any entity that wraps the recursion!
    recfnc: ((RecursiveFunc) -> T)
  let g = ((x: RecursiveFunc) => f(() => x.recfnc(x)))
  g(RecursiveFunc(recfnc: g))
# ]#

# same thing using direct recursion as Nim has...
# note that this version of fix uses function recursion in its own definition;
# thus its use just means that the recursion has been "pulled" into the "fix" function,
# instead of the function that uses it...
proc fixy[T](f: () -> T -> T): T = f(() => (fixy(f)))

# these are dreadfully inefficient as they becursively build stack!...
let facy = fixy((f: () -> (int -> int)) =>
  ((n: int) => (if n <= 1: 1 else: n * f()(n - 1))))

let fiby = fixy((f: () -> (int -> int)) =>
  ((n: int) => (if n < 2: n else: f()(n - 2) + f()(n - 1))))

echo facy 10
echo fiby 10

# something that can be done with the Y-Combinator that con't be done with the Z...
# given the following Co-Inductive Stream (CIS) definition...
type CIS[T] = object
  head: T
  tail: () -> CIS[T]

# Using a double Y-Combinator recursion...
# defines a continuous stream of Fibonacci numbers; there are other simpler ways,
# this way implements recursion by using the Y-combinator, although it is
# much slower than other ways due to the many additional function calls,
# it demonstrates something that can't be done with the Z-combinator...
iterator fibsy: int {.closure.} = # two recursions...
  let fbsfnc: (CIS[(int, int)] -> CIS[(int, int)]) = # first one...
    fixy((fnc: () -> (CIS[(int,int)] -> CIS[(int,int)])) =>
      ((cis: CIS[(int,int)]) => (
        let (f,s) = cis.head;
        CIS[(int,int)](head: (s, f + s), tail: () => fnc()(cis.tail())))))
  var fbsgen: CIS[(int, int)] = # second recursion
    fixy((cis: () -> CIS[(int,int)]) => # cis is a lazy thunk used directly below!
      fbsfnc(CIS[(int,int)](head: (1,0), tail: cis)))
  while true: yield fbsgen.head[0]; fbsgen = fbsgen.tail()

let fibs = fibsy
for _ in 1 .. 20: stdout.write fibs(), " "
echo()

Output:

3628800
55
3628800
55
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181

At least this last example version building a sequence of Fibonacci numbers doesn't build stack as it the use of CIS's means that it is a type of continuation passing/trampolining style.

Note that these would likely never be practically used in Nim as the language offers both direct variable binding recursion and recursion on proc's as well as other forms of recursion so it would never normally be necessary. Also note that these implementations not using recursive bindings on variables are "non-sharing" fix point combinators, whereas sharing is sometimes desired/required and thus recursion on variable bindings is required.

Objective-C

Works with: Mac OS X version 10.6+

Works with: iOS version 4.0+

#import <Foundation/Foundation.h>

typedef int (^Func)(int);
typedef Func (^FuncFunc)(Func);
typedef Func (^RecursiveFunc)(id); // hide recursive typing behind dynamic typing

Func Y(FuncFunc f) {
  RecursiveFunc r =
  ^(id y) {
    RecursiveFunc w = y; // cast value back into desired type
    return f(^(int x) {
      return w(w)(x);
    });
  };
  return r(r);
}

int main (int argc, const char *argv[]) {
  @autoreleasepool {

    Func fib = Y(^Func(Func f) {
      return ^(int n) {
        if (n <= 2) return 1;
        return  f(n - 1) + f(n - 2);
      };
    });
    Func fac = Y(^Func(Func f) {
      return ^(int n) {
        if (n <= 1) return 1;
        return n * f(n - 1);
      };
    });

    Func fib = fix(almost_fib);
    Func fac = fix(almost_fac);
    NSLog(@"fib(10) = %d", fib(10));
    NSLog(@"fac(10) = %d", fac(10));

  }
  return 0;
}

The usual version using recursion, disallowed by the task:

Func Y(FuncFunc f) {
  return ^(int x) {
    return f(Y(f))(x);
  };
}

OCaml

The Y-combinator over functions may be written directly in OCaml provided rectypes are enabled:

let fix f g = (fun x a -> f (x x) a) (fun x a -> f (x x) a) g

Polymorphic variants are the simplest workaround in the absence of rectypes:

let fix f = (fun (`X x) -> f(x (`X x))) (`X(fun (`X x) y -> f(x (`X x)) y));;

Otherwise, an ordinary variant can be defined and used:

type 'a mu = Roll of ('a mu -> 'a);;

let unroll (Roll x) = x;;

let fix f = (fun x a -> f (unroll x x) a) (Roll (fun x a -> f (unroll x x) a));;

let fac f = function
    0 -> 1
  | n -> n * f (n-1)
;;

let fib f = function
    0 -> 0
  | 1 -> 1
  | n -> f (n-1) + f (n-2)
;;

(* val unroll : 'a mu -> 'a mu -> 'a = <fun>
val fix : (('a -> 'b) -> 'a -> 'b) -> 'a -> 'b = <fun>
val fac : (int -> int) -> int -> int = <fun>
val fib : (int -> int) -> int -> int = <fun> *)

fix fac 5;;
(* - : int = 120 *)

fix fib 8;;
(* - : int = 21 *)

The usual version using recursion, disallowed by the task:

let rec fix f x = f (fix f) x;;

Oforth

These combinators work for any number of parameters (see Ackermann usage)

With recursion into Y definition (so non stateless Y) :

: Y(f)   #[ f Y f perform ] ;

Without recursion into Y definition (stateless Y).

: X(me, f)   #[ me f me perform f perform ] ;
: Y(f)       #X f X ;

Usage :

: almost-fact(n, f)   n ifZero: [ 1 ] else: [ n n 1 - f perform * ] ;
#almost-fact Y => fact

: almost-fib(n, f)   n 1 <= ifTrue: [ n ] else: [ n 1 - f perform n 2 - f perform + ] ;
#almost-fib Y => fib

: almost-Ackermann(m, n, f)
   m 0 == ifTrue: [ n 1 + return ]
   n 0 == ifTrue: [ 1 m 1 - f perform return ]
   n 1 - m f perform m 1 - f perform ;
#almost-Ackermann Y => Ackermann

Order

#include <order/interpreter.h>

#define ORDER_PP_DEF_8y                                             \
ORDER_PP_FN(8fn(8F,                                                 \
            8let((8R, 8fn(8G,                                       \
                          8ap(8F, 8fn(8A, 8ap(8ap(8G, 8G), 8A))))), \
                 8ap(8R, 8R))))

#define ORDER_PP_DEF_8fac \
ORDER_PP_FN(8fn(8F, 8X,   \
                8if(8less_eq(8X, 0), 1, 8times(8X, 8ap(8F, 8minus(8X, 1))))))

#define ORDER_PP_DEF_8fib                                           \
ORDER_PP_FN(8fn(8F, 8X,                                             \
                8if(8less(8X, 2), 8X, 8plus(8ap(8F, 8minus(8X, 1)), \
                                            8ap(8F, 8minus(8X, 2))))))

ORDER_PP(8to_lit(8ap(8y(8fac), 10)))    // 3628800
ORDER_PP(8ap(8y(8fib), 10))             // 55

Oz

declare
  Y = fun {$ F}
         {fun {$ X} {X X} end
          fun {$ X} {F fun {$ Z} {{X X} Z} end} end}
      end

  Fac = {Y fun {$ F}
              fun {$ N}
                 if N == 0 then 1 else N*{F N-1} end
              end
           end}

  Fib = {Y fun {$ F}
              fun {$ N}
                 case N of 0 then 0
                 [] 1 then 1
                 else {F N-1} + {F N-2}
                 end
              end
           end}
in
  {Show {Fac 5}}
  {Show {Fib 8}}

PARI/GP

As of 2.8.0, GP cannot make general self-references in closures declared inline, so the Y combinator is required to implement these functions recursively in that environment, e.g., for use in parallel processing.

Y(f)=x->f(f,x);
fact=Y((f,n)->if(n,n*f(f,n-1),1));
fib=Y((f,n)->if(n>1,f(f,n-1)+f(f,n-2),n));
apply(fact, [1..10])
apply(fib, [1..10])

Output:

%1 = [1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800]
%2 = [1, 1, 2, 3, 5, 8, 13, 21, 34, 55]

Perl

sub Y { my $f = shift;                                # λf.
    sub { my $x = shift; $x->($x) }->(                #   (λx.x x)
	sub {my $y = shift; $f->(sub {$y->($y)(@_)})} #   λy.f λz.y y z
    )
}
my $fac = sub {my $f = shift;
    sub {my $n = shift; $n < 2 ? 1 : $n * $f->($n-1)}
};
my $fib = sub {my $f = shift;
    sub {my $n = shift; $n == 0 ? 0 : $n == 1 ? 1 : $f->($n-1) + $f->($n-2)}
};
for my $f ($fac, $fib) {
    print join(' ', map Y($f)->($_), 0..9), "\n";
}

Output:

1 1 2 6 24 120 720 5040 40320 362880
0 1 1 2 3 5 8 13 21 34

The usual version using recursion, disallowed by the task:

sub Y { my $f = shift;
    sub {$f->(Y($f))->(@_)}
}

Phix

Translation of: C

After (over) simplifying things, the Y function has become a bit of a joke, but at least the recursion has been shifted out of fib/fac

Before saying anything too derogatory about Y(f)=f, it is clearly a fixed-point combinator, and I feel compelled to quote from the Mike Vanier link above:
"It doesn't matter whether you use cos or (lambda (x) (cos x)) as your cosine function; they will both do the same thing."
Anyone thinking they can do better may find some inspiration at Currying, Closures/Value_capture, Partial_function_application, and/or Function_composition

with javascript_semantics
function call_fn(integer f, n)
    return call_func(f,{f,n})
end function
 
function Y(integer f)
    return f
end function
 
function fac(integer self, integer n)
    return iff(n>1?n*call_fn(self,n-1):1)
end function
 
function fib(integer self, integer n)
    return iff(n>1?call_fn(self,n-1)+call_fn(self,n-2):n)
end function
 
procedure test(string name, integer rid=routine_id(name))
    integer f = Y(rid)
    printf(1,"%s: ",{name})
    for i=1 to 10 do
        printf(1," %d",call_fn(f,i))
    end for
    printf(1,"\n");
end procedure
test("fac")
test("fib")

Output:

fac:  1 2 6 24 120 720 5040 40320 362880 3628800
fib:  1 1 2 3 5 8 13 21 34 55

Phixmonti

0 var subr

def fac
    dup 1 > if
        dup 1 - subr exec *
    endif
enddef
 
def fib
    dup 1 > if
        dup 1 - subr exec swap 2 - subr exec +
    endif
enddef
 
def test
    print ": " print
    var subr
    10 for
        subr exec print " " print
    endfor
    nl
enddef

getid fac "fac" test
getid fib "fib" test

PHP

Works with: PHP version 5.3+

<?php
function Y($f) {
  $g = function($w) use($f) {
    return $f(function() use($w) {
      return call_user_func_array($w($w), func_get_args());
    });
  };
  return $g($g);
}

$fibonacci = Y(function($f) {
  return function($i) use($f) { return ($i <= 1) ? $i : ($f($i-1) + $f($i-2)); };
});

echo $fibonacci(10), "\n";

$factorial = Y(function($f) {
  return function($i) use($f) { return ($i <= 1) ? 1 : ($f($i - 1) * $i); };
});

echo $factorial(10), "\n";
?>

The usual version using recursion, disallowed by the task:

function Y($f) {
  return function() use($f) {
    return call_user_func_array($f(Y($f)), func_get_args());
  };
}

Works with: PHP version pre-5.3 and 5.3+

with create_function instead of real closures. A little far-fetched, but...

<?php
function Y($f) {
  $g = create_function('$w', '$f = '.var_export($f,true).';
    return $f(create_function(\'\', \'$w = \'.var_export($w,true).\';
      return call_user_func_array($w($w), func_get_args());
    \'));
  ');
  return $g($g);
}

function almost_fib($f) {
  return create_function('$i', '$f = '.var_export($f,true).';
    return ($i <= 1) ? $i : ($f($i-1) + $f($i-2));
  ');
};
$fibonacci = Y('almost_fib');
echo $fibonacci(10), "\n";

function almost_fac($f) {
  return create_function('$i', '$f = '.var_export($f,true).';
    return ($i <= 1) ? 1 : ($f($i - 1) * $i);
  ');
};
$factorial = Y('almost_fac');
echo $factorial(10), "\n";
?>

A functionally equivalent version using the $this parameter in closures is also possible:

Works with: PHP version 5.4+

<?php
function pseudoY($f) {
    $g = function($w) use ($f) {
        return $f->bindTo(function() use ($w) {
            return call_user_func_array($w($w), func_get_args());
        });
    };
    return $g($g);
}

$factorial = pseudoY(function($n) {
    return $n > 1 ? $n * $this($n - 1) : 1;
});
echo $factorial(10), "\n";

$fibonacci = pseudoY(function($n) {
    return $n > 1 ? $this($n - 1) + $this($n - 2) : $n;
});
echo $fibonacci(10), "\n";
?>

However, pseudoY() is not a fixed-point combinator.

PicoLisp

Translation of: Common Lisp

(de Y (F)
   (let X (curry (F) (Y) (F (curry (Y) @ (pass (Y Y)))))
      (X X) ) )

Factorial

# Factorial
(de fact (F)
   (curry (F) (N)
      (if (=0 N)
         1
         (* N (F (dec N))) ) ) )

: ((Y fact) 6)
-> 720

Fibonacci sequence

# Fibonacci
(de fibo (F)
   (curry (F) (N)
      (if (> 2 N)
         1
         (+ (F (dec N)) (F (- N 2))) ) ) )

: ((Y fibo) 22)
-> 28657

Ackermann function

# Ackermann
(de ack (F)
   (curry (F) (X Y)
      (cond
         ((=0 X) (inc Y))
         ((=0 Y) (F (dec X) 1))
         (T (F (dec X) (F X (dec Y)))) ) ) )

: ((Y ack) 3 4)
-> 125

Pop11

define Y(f);
    procedure (x); x(x) endprocedure(
        procedure (y);
            f(procedure(z); (y(y))(z) endprocedure)
        endprocedure
    )
enddefine;

define fac(h);
    procedure (n);
       if n = 0 then 1 else n * h(n - 1) endif
    endprocedure
enddefine;

define fib(h);
    procedure (n);
        if n < 2 then 1 else h(n - 1) + h(n - 2) endif
    endprocedure
enddefine;

Y(fac)(5) =>
Y(fib)(5) =>

Output:

** 120
** 8

PostScript

Translation of: Joy

Library: initlib

y {
    {dup cons} exch concat dup cons i
}.

/fac {
    { {pop zero?} {pop succ} {{dup pred} dip i *} ifte }
    y
}.

PowerShell

Translation of: Python

PowerShell Doesn't have true closure, in order to fake it, the script-block is converted to text and inserted whole into the next function using variable expansion in double-quoted strings. For simple translation of lambda calculus, $lambda$ translates as param inside of a ScriptBlock, $(\ldots )$ translates as Invoke-Expression "{}", invocation (written as a space) translates to InvokeReturnAsIs. ${\begin{array}{lcl}fac&:=&\lambda f.(\lambda n.{\mbox{if }}n\leq 0{\mbox{ then }}1{\mbox{ else }}n*(f\ n-1))\\fib&:=&\lambda f.(\lambda n.{\mbox{if }}n=0{\mbox{ or }}n=1{\mbox{ then }}1{\mbox{ else }}(f\ n-1)+(f\ n-2))\\Z&:=&\lambda f.(\lambda x.f\ (\lambda y.x\ x\ y))\ (\lambda x.f\ (\lambda y.x\ x\ y))\\\end{array}}$

$fac = {
    	param([ScriptBlock] $f)
    	invoke-expression @"
    	{
    		param([int] `$n)
    		if (`$n -le 0) {1}
    		else {`$n * {$f}.InvokeReturnAsIs(`$n - 1)}
    	}
"@
    }

$fib = {
	param([ScriptBlock] $f)
	invoke-expression @"
	{
		param([int] `$n)
		switch (`$n)
        {
        0 {1}
        1 {1}
        default {{$f}.InvokeReturnAsIs(`$n-1)+{$f}.InvokeReturnAsIs(`$n-2)}
        }
	}
"@
}

$Z = {
    param([ScriptBlock] $f)
    invoke-expression @"
    {
        param([ScriptBlock] `$x)
        {$f}.InvokeReturnAsIs(`$(invoke-expression @`"
        {
            param(```$y)
            {`$x}.InvokeReturnAsIs({`$x}).InvokeReturnAsIs(```$y)
        }
`"@))
    }.InvokeReturnAsIs({
        param([ScriptBlock] `$x)
        {$f}.InvokeReturnAsIs(`$(invoke-expression @`"
        {
            param(```$y)
            {`$x}.InvokeReturnAsIs({`$x}).InvokeReturnAsIs(```$y)
        }
`"@))
    })
"@
}

$Z.InvokeReturnAsIs($fac).InvokeReturnAsIs(5)
$Z.InvokeReturnAsIs($fib).InvokeReturnAsIs(5)

GetNewClosure() was added in Powershell 2, allowing for an implementation without metaprogramming. The following was tested with Powershell 4.

$Y = {
    param ($f)

    {
        param ($x)
        
        $f.InvokeReturnAsIs({
            param ($y)

            $x.InvokeReturnAsIs($x).InvokeReturnAsIs($y)
        }.GetNewClosure())
        
    }.InvokeReturnAsIs({
        param ($x)

        $f.InvokeReturnAsIs({
            param ($y)

            $x.InvokeReturnAsIs($x).InvokeReturnAsIs($y)
        }.GetNewClosure())

    }.GetNewClosure())
}

$fact = {
    param ($f)

    {
        param ($n)
        
        if ($n -eq 0) { 1 } else { $n * $f.InvokeReturnAsIs($n - 1) }

    }.GetNewClosure()
}

$fib = {
    param ($f)

    {
        param ($n)

        if ($n -lt 2) { 1 } else { $f.InvokeReturnAsIs($n - 1) + $f.InvokeReturnAsIs($n - 2) }

    }.GetNewClosure()
}

$Y.invoke($fact).invoke(5)
$Y.invoke($fib).invoke(5)

Prolog

Works with SWI-Prolog and module lambda, written by Ulrich Neumerkel found there http://www.complang.tuwien.ac.at/ulrich/Prolog-inedit/lambda.pl.

The code is inspired from this page : http://www.complang.tuwien.ac.at/ulrich/Prolog-inedit/ISO-Hiord#Hiord (p 106).
Original code is from Hermenegildo and al : Hiord: A Type-Free Higher-Order Logic Programming Language with Predicate Abstraction, pdf accessible here http://www.stups.uni-duesseldorf.de/asap/?id=129.

:- use_module(lambda).

% The Y combinator
y(P, Arg, R) :-
	Pred = P +\Nb2^F2^call(P,Nb2,F2,P),
	call(Pred, Arg, R).


test_y_combinator :-
    % code for Fibonacci function
    Fib   = \NFib^RFib^RFibr1^(NFib < 2 ->
			         RFib = NFib
			      ;
			         NFib1 is NFib - 1,
			         NFib2 is NFib - 2,
			         call(RFibr1,NFib1,RFib1,RFibr1),
			         call(RFibr1,NFib2,RFib2,RFibr1),
			         RFib is RFib1 + RFib2
			      ),

    y(Fib, 10, FR), format('Fib(~w) = ~w~n', [10, FR]),

    % code for Factorial function
    Fact =  \NFact^RFact^RFactr1^(NFact = 1 ->
			            RFact = NFact
                                 ;
			            NFact1 is NFact - 1,
			            call(RFactr1,NFact1,RFact1,RFactr1),
			            RFact is NFact * RFact1
			         ),

    y(Fact, 10, FF), format('Fact(~w) = ~w~n', [10, FF]).

Output:

 ?- test_y_combinator.
Fib(10) = 55
Fact(10) = 3628800
true.

Python

>>> Y = lambda f: (lambda x: x(x))(lambda y: f(lambda *args: y(y)(*args)))
>>> fac = lambda f: lambda n: (1 if n<2 else n*f(n-1))
>>> [ Y(fac)(i) for i in range(10) ]
[1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
>>> fib = lambda f: lambda n: 0 if n == 0 else (1 if n == 1 else f(n-1) + f(n-2))
>>> [ Y(fib)(i) for i in range(10) ]
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]

The usual version using recursion, disallowed by the task:

Y = lambda f: lambda *args: f(Y(f))(*args)

Y = lambda b: ((lambda f: b(lambda *x: f(f)(*x)))((lambda f: b(lambda *x: f(f)(*x)))))

Q

> Y: {{x x} {({y {(x x) y} x} y) x} x}
> fac: {{$[y<2; 1; y*x y-1]} x}
> (Y fac) 6
720j
> fib: {{$[y<2; 1;  (x y-1) + (x y-2)]} x}
> (Y fib) each til 20
1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765

Quackery

From the Wikipedia article Fixed-point combinator:

The Y combinator is a particular implementation of a fixed-point combinator in lambda calculus. Its structure is determined by the limitations of lambda calculus. It is not necessary or helpful to use this structure in implementing the fixed-point combinator in other languages.

recursive is a stateless fixed-point combinator which takes a stateless nest (named or nameless) and returns a recursive version of the nest.

As per the task it is used here to compute factorial and Fibonacci numbers.

Without the restriction on self referencing, recursive could be defined as [ ' [ this ] swap nested join ].

  [ ' stack nested nested
    ' share nested join
    swap nested join
    dup dup 0 peek put ]   is recursive (   x --> x )

  [ over 2 < iff
      [ 2drop 1 ] done
    dip [ dup 1 - ] do * ] is factorial ( n x --> n )

  [ over 2 < iff drop done
    swap 1 - tuck 1 -
    over do dip do + ]     is fibonacci ( n x --> n )

  say "8 factorial = " 8 ' factorial recursive do echo cr
  say "8 fibonacci = " 8 ' fibonacci recursive do echo cr

Output:

8 factorial = 40320
8 fibonacci = 21

R

#' Y = λf.(λs.ss)(λx.f(xx))
#' Z = λf.(λs.ss)(λx.f(λz.(xx)z))
#' 

fixp.Y <- \ (f) (\ (x) (x) (x)) (\ (y) (f) ((y) (y))) # y-combinator
fixp.Z <- \ (f) (\ (x) (x) (x)) (\ (y) (f) (\ (...) (y) (y) (...))) # z-combinator

Y-combinator test:

fac.y <- fixp.Y (\ (f) \ (n) if (n<2) 1 else n*f(n-1))
fac.y(9) # [1] 362880

fib.y <- fixp.Y (\ (f) \ (n) if (n <= 1) n else f(n-1) + f(n-2))
fib.y(9) # [1] 34

Z-combinator test:

fac.z <- fixp.Z (\ (f) \ (n) if (n<2) 1 else n*f(n-1))
fac.z(9) # [1] 362880

fib.z <- fixp.Z (\ (f) \ (n) if (n <= 1) n else f(n-1) + f(n-2))
fib.z(9) # [1] 34

You can verify these codes by here

Racket

The lazy implementation

#lang lazy

(define Y (λ (f) ((λ (x) (f (x x))) (λ (x) (f (x x))))))

(define Fact
  (Y (λ (fact) (λ (n) (if (zero? n) 1 (* n (fact (- n 1))))))))
(define Fib
  (Y (λ (fib) (λ (n) (if (<= n 1) n (+ (fib (- n 1)) (fib (- n 2))))))))

Output:

> (!! (map Fact '(1 2 4 8 16)))
'(1 2 24 40320 20922789888000)
> (!! (map Fib '(1 2 4 8 16)))
'(0 1 2 13 610)

Strict realization:

#lang racket
(define Y (λ (b) ((λ (f) (b (λ (x) ((f f) x))))
                  (λ (f) (b (λ (x) ((f f) x)))))))

Definitions of Fact and Fib functions will be the same as in Lazy Racket.

Finally, a definition in Typed Racket is a little difficult as in other statically typed languages:

#lang typed/racket

(: make-recursive : (All (S T) ((S -> T) -> (S -> T)) -> (S -> T)))
(define-type Tau (All (S T) (Rec this (this -> (S -> T)))))
(define (make-recursive f)
  ((lambda: ([x : (Tau S T)]) (f (lambda (z) ((x x) z))))
   (lambda: ([x : (Tau S T)]) (f (lambda (z) ((x x) z))))))

(: fact : Number -> Number)
(define fact (make-recursive
              (lambda: ([fact : (Number -> Number)])
                (lambda: ([n : Number])
                  (if (zero? n)
                    1
                    (* n (fact (- n 1))))))))

(fact 5)

Raku

(formerly Perl 6)

sub Y (&f) { sub (&x) { x(&x) }( sub (&y) { f(sub ($x) { y(&y)($x) }) } ) }
sub fac (&f) { sub ($n) { $n < 2 ?? 1 !! $n * f($n - 1) } }
sub fib (&f) { sub ($n) { $n < 2 ?? $n !! f($n - 1) + f($n - 2) } }
say map Y($_), ^10 for &fac, &fib;

Output:

(1 1 2 6 24 120 720 5040 40320 362880)
(0 1 1 2 3 5 8 13 21 34)

Note that Raku doesn't actually need a Y combinator because you can name anonymous functions from the inside:

say .(10) given sub (Int $x) { $x < 2 ?? 1 !! $x * &?ROUTINE($x - 1); }

REBOL

Y: closure [g] [do func [f] [f :f] closure [f] [g func [x] [do f :f :x]]]

usage example

fact*: closure [h] [func [n] [either n <= 1 [1] [n * h n - 1]]]
fact: Y :fact*

REXX

Programming note:   length,   reverse,   sign,   trunc,   b2x,   d2x,   and   x2d   are REXX BIFs   (Built In Functions).

/*REXX program implements and displays  a  stateless   Y   combinator.                  */
numeric digits 1000                                               /*allow big numbers.  */
say '    fib'   Y(fib      (50) )                                 /*Fibonacci series.   */
say '    fib'   Y(fib      (12 11 10 9 8 7 6 5 4 3 2 1 0)  )      /*Fibonacci series.   */
say '   fact'   Y(fact     (60) )                                 /*single    factorial.*/
say '   fact'   Y(fact     (0 1 2 3 4 5 6 7 8 9 10 11) )          /*single    factorial.*/
say '  Dfact'   Y(dfact    (4 5 6 7 8 9 10 11 12 13)   )          /*double    factorial.*/
say '  Tfact'   Y(tfact    (4 5 6 7 8 9 10 11 12 13)   )          /*triple    factorial.*/
say '  Qfact'   Y(qfact    (4 5 6 7 8 40) )                       /*quadruple factorial.*/
say ' length'   Y(length   (when for to where whenceforth) )      /*lengths   of words. */
say 'reverse'   Y(reverse  (123 66188 3007 45.54 MAS I MA) )      /*reverses  strings.  */
say '   sign'   Y(sign     (-8 0 8) )                             /*sign of the numbers.*/
say '  trunc'   Y(trunc    (-7.0005 12 3.14159 6.4 78.999) )      /*truncates numbers.  */
say '    b2x'   Y(b2x      (1 10 11 100 1000 10000 11111 ) )      /*converts BIN──►HEX. */
say '    d2x'   Y(d2x      (8 9 10 11 12 88 89 90 91 6789) )      /*converts DEC──►HEX. */
say '    x2d'   Y(x2d      (8 9 10 11 12 88 89 90 91 6789) )      /*converts HEX──►DEC. */
exit 0                                           /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
Y: parse arg Y _; $=; do j=1 for words(_); interpret '$=$' Y"("word(_,j)')'; end; return $
/*──────────────────────────────────────────────────────────────────────────────────────*/
fib:   procedure; parse arg x;  if x<2  then return x;  s= 0;  a= 0;   b= 1
                                    do j=2  to x;  s= a+b;  a= b;  b= s;  end;    return s
/*──────────────────────────────────────────────────────────────────────────────────────*/
dfact: procedure; parse arg x;  != 1;    do j=x  to 2  by -2;   != !*j;   end;    return !
tfact: procedure; parse arg x;  != 1;    do j=x  to 2  by -3;   != !*j;   end;    return !
qfact: procedure; parse arg x;  != 1;    do j=x  to 2  by -4;   != !*j;   end;    return !
 fact: procedure; parse arg x;  != 1;    do j=2  to x       ;   != !*j;   end;    return !

output   when using the internal default input:

    fib  12586269025
    fib  144 89 55 34 21 13 8 5 3 2 1 1 0
   fact  8320987112741390144276341183223364380754172606361245952449277696409600000000000000
   fact  1 1 2 6 24 120 720 5040 40320 362880 3628800 39916800
  Dfact  8 15 48 105 384 945 3840 10395 46080 135135
  Tfact  4 10 18 28 80 162 280 880 1944 3640
  Qfact  4 5 12 21 32 3805072588800
 length  4 3 2 5 11
reverse  321 88166 7003 45.54 SAM I AM
   sign  -1 0 1
  trunc  -7 12 3 6 78
    b2x  1 2 3 4 8 10 1F
    d2x  8 9 A B C 58 59 5A 5B 1A85
    x2d  8 9 16 17 18 136 137 144 145 26505

Ruby

Using a lambda:

y = lambda do |f|
  lambda {|g| g[g]}[lambda do |g|
      f[lambda {|*args| g[g][*args]}]
    end]
end

fac = lambda{|f| lambda{|n| n < 2 ? 1 : n * f[n-1]}}
p Array.new(10) {|i| y[fac][i]}   #=> [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]

fib = lambda{|f| lambda{|n| n < 2 ? n : f[n-1] + f[n-2]}}
p Array.new(10) {|i| y[fib][i]}   #=> [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]

Same as the above, using the new short lambda syntax:

Works with: Ruby version 1.9

y = ->(f) {->(g) {g.(g)}.(->(g) { f.(->(*args) {g.(g).(*args)})})}
             
fac = ->(f) { ->(n) { n < 2 ? 1 : n * f.(n-1) } }

p 10.times.map {|i| y.(fac).(i)}

fib = ->(f) { ->(n) { n < 2 ? n : f.(n-2) + f.(n-1) } }

p 10.times.map {|i| y.(fib).(i)}

Using a method:

Works with: Ruby version 1.9

def y(&f)
  lambda do |g|
    f.call {|*args| g[g][*args]}
  end.tap {|g| break g[g]}
end

fac = y {|&f| lambda {|n| n < 2 ? 1 : n * f[n - 1]}}
fib = y {|&f| lambda {|n| n < 2 ? n : f[n - 1] + f[n - 2]}}

p Array.new(10) {|i| fac[i]}
# => [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
p Array.new(10) {|i| fib[i]}
# => [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]

The usual version using recursion, disallowed by the task:

y = lambda do |f|
  lambda {|*args| f[y[f]][*args]}
end

Rust

Works with: Rust version 1.44.1 stable

//! A simple implementation of the Y Combinator:
//! λf.(λx.xx)(λx.f(xx))
//! <=> λf.(λx.f(xx))(λx.f(xx))

/// A function type that takes its own type as an input is an infinite recursive type.
/// We introduce the "Apply" trait, which will allow us to have an input with the same type as self, and break the recursion.
/// The input is going to be a trait object that implements the desired function in the interface.
trait Apply<T, R> {
    fn apply(&self, f: &dyn Apply<T, R>, t: T) -> R;
}

/// If we were to pass in self as f, we get:
/// λf.λt.sft
/// => λs.λt.sst [s/f]
/// => λs.ss
impl<T, R, F> Apply<T, R> for F where F: Fn(&dyn Apply<T, R>, T) -> R {
    fn apply(&self, f: &dyn Apply<T, R>, t: T) -> R {
        self(f, t)
    }
}

/// (λt(λx.(λy.xxy))(λx.(λy.f(λz.xxz)y)))t
/// => (λx.xx)(λx.f(xx))
/// => Yf
fn y<T, R>(f: impl Fn(&dyn Fn(T) -> R, T) -> R) -> impl Fn(T) -> R {
    move |t| (&|x: &dyn Apply<T, R>, y| x.apply(x, y))
             (&|x: &dyn Apply<T, R>, y| f(&|z| x.apply(x, z), y), t)
}

/// Factorial of n.
fn fac(n: usize) -> usize {
    let almost_fac = |f: &dyn Fn(usize) -> usize, x| if x == 0 { 1 } else { x * f(x - 1) };
    y(almost_fac)(n)
}

/// nth Fibonacci number.
fn fib(n: usize) -> usize {
    let almost_fib = |f: &dyn Fn((usize, usize, usize)) -> usize, (a0, a1, x)|
        match x {
            0 => a0,
            1 => a1,
            _ => f((a1, a0 + a1, x - 1)),
        };

    y(almost_fib)((1, 1, n))
}

/// Driver function.
fn main() {
    let n = 10;
    println!("fac({}) = {}", n, fac(n));
    println!("fib({}) = {}", n, fib(n));
}

Output:

fac(10) = 3628800
fib(10) = 89

Scala

Credit goes to the thread in scala blog

def Y[A, B](f: (A => B) => (A => B)): A => B = {
  case class W(wf: W => (A => B)) {
    def apply(w: W): A => B = wf(w)
  }
  val g: W => (A => B) = w => f(w(w))(_)
  g(W(g))
}

Example

val fac: Int => Int = Y[Int, Int](f => i => if (i <= 0) 1 else f(i - 1) * i)
fac(6)  //> res0: Int = 720

val fib: Int => Int = Y[Int, Int](f => i => if (i < 2) i else f(i - 1) + f(i - 2))
fib(6)  //> res1: Int = 8

Scheme

(define Y                 ; (Y f) = (g g) where
  (lambda (f)             ;         (g g) = (f  (lambda a (apply (g g) a)))
    ((lambda (g) (g g))   ; (Y f) ==        (f  (lambda a (apply (Y f) a)))
     (lambda (g)       
       (f  (lambda a (apply (g g) a))))))) 

;; head-recursive factorial
(define fac                ; fac = (Y f) = (f      (lambda a (apply (Y f) a))) 
  (Y (lambda (r)           ;     = (lambda (x) ... (r     (- x 1)) ... )
       (lambda (x)         ;        where   r    = (lambda a (apply (Y f) a))
         (if (< x 2)       ;               (r ... ) == ((Y f) ... )
             1             ;     == (lambda (x) ... (fac  (- x 1)) ... )
             (* x (r (- x 1))))))))

;; tail-recursive factorial
(define fac2
  (lambda (x)            
    ((Y (lambda (r)        ;       (Y f) == (f     (lambda a (apply (Y f) a))) 
          (lambda (x acc)  ;          r         == (lambda a (apply (Y f) a))
            (if (< x 2)    ;         (r ... )   == ((Y f) ... )
                acc
                (r (- x 1) (* x acc))))))
     x 1)))

; double-recursive Fibonacci
(define fib
  (Y (lambda (f)
       (lambda (x)
         (if (< x 2)
             x
             (+ (f (- x 1)) (f (- x 2))))))))

; tail-recursive Fibonacci
(define fib2
  (lambda (x)
    ((Y (lambda (f)
          (lambda (x a b)
            (if (< x 1)
                a
                (f (- x 1) b (+ a b))))))
     x 0 1)))

(display (fac 6))
(newline)

(display (fib2 134))
(newline)

Output:

720
4517090495650391871408712937

If we were allowed to use recursion (with Y referring to itself by name in its body) we could define the equivalent to the above as

(define Yr        ; (Y f) == (f  (lambda a (apply (Y f) a)))
  (lambda (f)
    (f  (lambda a (apply (Yr f) a)))))

And another way is:

(define Y2r
  (lambda (f)
    (lambda a (apply (f (Y2r f)) a))))

Which, non-recursively, is

(define Y2                ; (Y2 f) = (g g) where
  (lambda (f)             ;          (g g) = (lambda a (apply (f (g g)) a))
    ((lambda (g) (g g))   ; (Y2 f) ==       (lambda a (apply (f (Y2 f)) a))
     (lambda (g)
       (lambda a (apply (f (g g)) a))))))

Shen

(define y
  F -> ((/. X (X X))
        (/. X (F (/. Z ((X X) Z))))))

(let Fac (y (/. F N (if (= 0 N)
                      1
                      (* N (F (- N 1))))))
  (output "~A~%~A~%~A~%"
    (Fac 0)
    (Fac 5)
    (Fac 10)))

Output:

1
120
3628800

Sidef

var y = ->(f) {->(g) {g(g)}(->(g) { f(->(*args) {g(g)(args...)})})}

var fac = ->(f) { ->(n) { n < 2 ? 1 : (n * f(n-1)) } }
say 10.of { |i| y(fac)(i) }

var fib = ->(f) { ->(n) { n < 2 ? n : (f(n-2) + f(n-1)) } }
say 10.of { |i| y(fib)(i) }

Output:

[1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]

Slate

The Y combinator is already defined in slate as:

Method traits define: #Y &builder:
  [[| :f | [| :x | f applyWith: (x applyWith: x)]
	   applyWith: [| :x | f applyWith: (x applyWith: x)]]].

Smalltalk

Works with: GNU Smalltalk

Y := [:f| [:x| x value: x] value: [:g| f value: [:x| (g value: g) value: x] ] ].

fib := Y value: [:f| [:i| i <= 1 ifTrue: [i] ifFalse: [(f value: i-1) + (f value: i-2)] ] ].

(fib value: 10) displayNl.

fact := Y value: [:f| [:i| i = 0 ifTrue: [1] ifFalse: [(f value: i-1) * i] ] ].

(fact value: 10) displayNl.

Output:

55
3628800

The usual version using recursion, disallowed by the task:

Y := [:f| [:x| (f value: (Y value: f)) value: x] ].

Standard ML

- datatype 'a mu = Roll of ('a mu -> 'a)
  fun unroll (Roll x) = x

  fun fix f = (fn x => fn a => f (unroll x x) a) (Roll (fn x => fn a => f (unroll x x) a))

  fun fac f 0 = 1
    | fac f n = n * f (n-1)

  fun fib f 0 = 0
    | fib f 1 = 1
    | fib f n = f (n-1) + f (n-2)
;
datatype 'a mu = Roll of 'a mu -> 'a
val unroll = fn : 'a mu -> 'a mu -> 'a
val fix = fn : (('a -> 'b) -> 'a -> 'b) -> 'a -> 'b
val fac = fn : (int -> int) -> int -> int
val fib = fn : (int -> int) -> int -> int
- List.tabulate (10, fix fac);
val it = [1,1,2,6,24,120,720,5040,40320,362880] : int list
- List.tabulate (10, fix fib);
val it = [0,1,1,2,3,5,8,13,21,34] : int list

The usual version using recursion, disallowed by the task:

fun fix f x = f (fix f) x

SuperCollider

The direct implementation will not work, because SuperCollider evaluates x.(x) before calling f.

y = { |f| { |x| f.(x.(x)) }.({ |x| f.(x.(x)) }) };

For lazy evaluation, this call needs to be postponed by passing a function to f that makes this call (this is what is called the z-combinator):

// z-combinator

z = { |f| { |x| f.({ |args| x.(x).(args) }) }.({ |x| f.({ |args| x.(x).(args) }) }) };

// this can be also factored differently
(
y = { |f|
	{ |r| r.(r) }.(
		{ |x| f.({ |args| x.(x).(args) }) }
	)
};
)

// the same in a reduced form

(
r = { |x| x.(x) };
z = { |f| r.({ |y| f.(r.(y).(_)) }) };
)


// factorial
k = { |f| { |x| if(x < 2, 1, { x * f.(x - 1) }) } };

g = z.(k);

g.(5) // 120

(1..10).collect(g) // [ 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800 ]



// fibonacci

k = { |f| { |x| if(x <= 2, 1, { f.(x - 1) + f.(x - 2) }) } };

g = z.(k);

g.(3)

(1..10).collect(g) // [ 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 ]

Swift

Using a recursive type:

struct RecursiveFunc<F> {
  let o : RecursiveFunc<F> -> F
}

func Y<A, B>(f: (A -> B) -> A -> B) -> A -> B {
  let r = RecursiveFunc<A -> B> { w in f { w.o(w)($0) } }
  return r.o(r)
}

let fac = Y { (f: Int -> Int) in
  { $0 <= 1 ? 1 : $0 * f($0-1) }
}
let fib = Y { (f: Int -> Int) in
  { $0 <= 2 ? 1 : f($0-1)+f($0-2) }
}
println("fac(5) = \(fac(5))")
println("fib(9) = \(fib(9))")

Output:

fac(5) = 120
fib(9) = 34

Without a recursive type, and instead using Any to erase the type:

Works with: Swift version 1.2+

(for Swift 1.1 replace as! with as)

func Y<A, B>(f: (A -> B) -> A -> B) -> A -> B {
  typealias RecursiveFunc = Any -> A -> B
  let r : RecursiveFunc = { (z: Any) in let w = z as! RecursiveFunc; return f { w(w)($0) } }
  return r(r)
}

The usual version using recursion, disallowed by the task:

func Y<In, Out>( f: (In->Out) -> (In->Out) ) -> (In->Out) {
    return { x in f(Y(f))(x) }
}

Tailspin

// YCombinator is not needed since tailspin supports recursion readily, but this demonstrates passing functions as parameters
 
templates combinator&{stepper:}
  templates makeStep&{rec:}
    $ -> stepper&{next: rec&{rec: rec}} !
  end makeStep
  $ -> makeStep&{rec: makeStep} !
end combinator
 
templates factorial
  templates seed&{next:}
    <=0> 1 !
    <>
      $ * ($ - 1 -> next) !
  end seed
  $ -> combinator&{stepper: seed} !
end factorial
 
5 -> factorial -> 'factorial 5: $;
' -> !OUT::write
 
templates fibonacci
  templates seed&{next:}
    <..1> $ !
    <>
      ($ - 2 -> next) + ($ - 1 -> next) !
  end seed
  $ -> combinator&{stepper: seed} !
end fibonacci
 
5 -> fibonacci -> 'fibonacci 5: $;
' -> !OUT::write

Output:

factorial 5: 120
fibonacci 5: 5

Tcl

Y combinator is derived in great detail here.

TXR

This prints out 24, the factorial of 4:

;; The Y combinator:
(defun y (f) 
  [(op @1 @1)
   (op f (op [@@1 @@1]))])

;; The Y-combinator-based factorial:
(defun fac (f) 
  (do if (zerop @1) 
         1 
         (* @1 [f (- @1 1)])))

;; Test:
(format t "~s\n" [[y fac] 4])

Both the op and do operators are a syntactic sugar for currying, in two different flavors. The forms within do that are symbols are evaluated in the normal Lisp-2 style and the first symbol can be an operator. Under op, any forms that are symbols are evaluated in the Lisp-2 style, and the first form is expected to evaluate to a function. The name do stems from the fact that the operator is used for currying over special forms like if in the above example, where there is evaluation control. Operators can have side effects: they can "do" something. Consider (do set a @1) which yields a function of one argument which assigns that argument to a.

The compounded @@... notation allows for inner functions to refer to outer parameters, when the notation is nested. Consider

(op foo @1 (op bar @2 @@2))

. Here the @2 refers to the second argument of the anonymous function denoted by the inner op. The @@2 refers to the second argument of the outer op.

Ursala

The standard y combinator doesn't work in Ursala due to eager evaluation, but an alternative is easily defined as shown

(r "f") "x" = "f"("f","x")
my_fix "h"  = r ("f","x"). ("h" r "f") "x"

or by this shorter expression for the same thing in point free form.

my_fix = //~&R+ ^|H\~&+ ; //~&R

Normally you'd like to define a function recursively by writing $f=h(f)$, where $h(f)$ is just the body of the function with recursive calls to $f$ in it. With a fixed point combinator such as my_fix as defined above, you do almost the same thing, except it's $f=$my_fix "f". $h$("f"), where the dot represents lambda abstraction and the quotes signify a dummy variable. Using this method, the definition of the factorial function becomes

#import nat

fact = my_fix "f". ~&?\1! product^/~& "f"+ predecessor

To make it easier, the compiler has a directive to let you install your own fixed point combinator for it to use, which looks like this,

#fix my_fix

with your choice of function to be used in place of my_fix. Having done that, you may express recursive functions per convention by circular definitions, as in this example of a Fibonacci function.

fib = {0,1}?</1! sum+ fib~~+ predecessor^~/~& predecessor

Note that this way is only syntactic sugar for the for explicit way shown above. Without a fixed point combinator given in the #fix directive, this definition of fib would not have compiled. (Ursala allows user defined fixed point combinators because they're good for other things besides functions.) To confirm that all this works, here is a test program applying both of the functions defined above to the numbers from 1 to 8.

#cast %nLW

examples = (fact* <1,2,3,4,5,6,7,8>,fib* <1,2,3,4,5,6,7,8>)

Output:

(
   <1,2,6,24,120,720,5040,40320>,
   <1,2,3,5,8,13,21,34>)

The fixed point combinator defined above is theoretically correct but inefficient and limited to first order functions, whereas the standard distribution includes a library (sol) providing a hierarchy of fixed point combinators suitable for production use and with higher order functions. A more efficient alternative implementation of my_fix would be general_function_fixer 0 (with 0 signifying the lowest order of fixed point combinators), or if that's too easy, then by this definition.

#import sol

#fix general_function_fixer 1

my_fix "h" = "h" my_fix "h"

Note that this equation is solved using the next fixed point combinator in the hierarchy.

Verbexx

/////// Y-combinator function (for single-argument lambdas) ///////

y @FN [f]
{ @( x -> { @f (z -> {@(@x x) z}) } )   // output of this expression is treated as a verb, due to outer @(  )
   ( x -> { @f (z -> {@(@x x) z}) } )   // this is the argument supplied to the above verb expression
};


/////// Function to generate an anonymous factorial function as the return value -- (not tail-recursive) ///////

fact_gen @FN [f]
{ n -> { (n<=0) ? {1} {n * (@f n-1)}
       }
};


/////// Function to generate an anonymous fibonacci function as the return value -- (not tail-recursive) ///////

fib_gen @FN [f]
{ n -> { (n<=0) ? { 0                                    }
                  { (n<=2) ? {1} { (@f n-1) + (@f n-2) } } 
       }
};
                

/////// loops to test the above functions ///////

@VAR factorial = @y fact_gen;
@VAR fibonacci = @y fib_gen;

@LOOP init:{@VAR i = -1} while:(i <= 20) next:{i++}
{ @SAY  i "factorial =" (@factorial i) };

@LOOP init:{     i = -1} while:(i <= 16) next:{i++}
{ @SAY "fibonacci<" i "> =" (@fibonacci i) };

Vim Script

There is no lambda in Vim (yet?), so here is a way to fake it using a Dictionary. This also provides garbage collection.

" Translated from Python.  Works with: Vim 7.0

func! Lambx(sig, expr, dict)
    let fanon = {'d': a:dict}
    exec printf("
	\func fanon.f(%s) dict\n
	\  return %s\n
	\endfunc",
	\ a:sig, a:expr)
    return fanon
endfunc

func! Callx(fanon, arglist)
    return call(a:fanon.f, a:arglist, a:fanon.d)
endfunc

let g:Y = Lambx('f', 'Callx(Lambx("x", "Callx(a:x, [a:x])", {}), [Lambx("y", ''Callx(self.f, [Lambx("...", "Callx(Callx(self.y, [self.y]), a:000)", {"y": a:y})])'', {"f": a:f})])', {})

let g:fac = Lambx('f', 'Lambx("n", "a:n<2 ? 1 : a:n * Callx(self.f, [a:n-1])", {"f": a:f})', {})

echo Callx(Callx(g:Y, [g:fac]), [5])
echo map(range(10), 'Callx(Callx(Y, [fac]), [v:val])')

Update: since Vim 7.4.2044 (or so...), the following can be used (the feature check was added with 7.4.2121):

if !has("lambda")
    echoerr 'Lambda feature required'
    finish
endif
let Y = {f -> {x -> x(x)}({y -> f({... -> call(y(y), a:000)})})}
let Fac = {f -> {n -> n<2 ? 1 : n * f(n-1)}}

echo Y(Fac)(5)
echo map(range(10), 'Y(Fac)(v:val)')

Output:

120
[1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]

Wart

# Better names due to Jim Weirich: http://vimeo.com/45140590
def (Y improver)
  ((fn(gen) gen.gen)
   (fn(gen)
     (fn(n)
       ((improver gen.gen) n))))

factorial <- (Y (fn(f)
                  (fn(n)
                    (if zero?.n
                      1
                      (n * (f n-1))))))

prn factorial.5

Wren

Translation of: Go

var y = Fn.new { |f|
    var g = Fn.new { |r| f.call { |x| r.call(r).call(x) } }
    return g.call(g)
}

var almostFac = Fn.new { |f| Fn.new { |x| x <= 1 ? 1 : x * f.call(x-1) } }

var almostFib = Fn.new { |f| Fn.new { |x| x <= 2 ? 1 : f.call(x-1) + f.call(x-2) } }

var fac = y.call(almostFac)
var fib = y.call(almostFib)

System.print("fac(10) = %(fac.call(10))")
System.print("fib(10) = %(fib.call(10))")

Output:

fac(10) = 3628800
fib(10) = 55

XQuery

Version 3.0 of the XPath and XQuery specifications added support for function items.

let $Y := function($f) {  
    (function($x) { ($x)($x) })( function($g) { $f( (function($a) { $g($g) ($a)})  ) } )
  }
let $fac := $Y(function($f) { function($n) { if($n <  2) then 1  else $n * $f($n - 1) } })
let $fib := $Y(function($f) { function($n) { if($n <= 1) then $n else $f($n - 1) + $f($n - 2) } })
return (
    $fac(6),
    $fib(6)
)

Output:

720 8

zkl

fcn Y(f){ fcn(g){ g(g) }( 'wrap(h){ f( 'wrap(a){ h(h)(a) }) }) }

Functions don't get to look outside of their scope so data in enclosing scopes needs to be bound to a function, the fp (function application/cheap currying) method does this. 'wrap is syntactic sugar for fp.

fcn almost_factorial(f){ fcn(n,f){ if(n<=1) 1 else n*f(n-1) }.fp1(f) }
Y(almost_factorial)(6).println();
[0..10].apply(Y(almost_factorial)).println();

Output:

720
L(1,1,2,6,24,120,720,5040,40320,362880,3628800)

fcn almost_fib(f){ fcn(n,f){ if(n<2) 1 else f(n-1)+f(n-2) }.fp1(f) }
Y(almost_fib)(9).println();
[0..10].apply(Y(almost_fib)).println();

Output:

55
L(1,1,2,3,5,8,13,21,34,55,89)