Exponentiation operator: Difference between revisions
No edit summary |
|||
| Line 1,536: | Line 1,536: | ||
\\ in the signature and in the call |
\\ in the signature and in the call |
||
function pow(a, b as long) { |
function pow(a, b as long) { |
||
p=a-a ' make p same |
p=a-a ' make p same type as a |
||
p++ |
p++ |
||
if b>0 then for i=1& to b {p*=a} |
if b>0 then for i=1& to b {p*=a} |
||
| Line 1,570: | Line 1,570: | ||
</pre > |
</pre > |
||
=={{header|M4}}== |
=={{header|M4}}== |
||
M4 lacks floating point computation and operator definition. |
M4 lacks floating point computation and operator definition. |
||
Revision as of 15:28, 29 July 2019
You are encouraged to solve this task according to the task description, using any language you may know.
Most programming languages have a built-in implementation of exponentiation.
- Task
Re-implement integer exponentiation for both intint and floatint as both a procedure, and an operator (if your language supports operator definition).
If the language supports operator (or procedure) overloading, then an overloaded form should be provided for both intint and floatint variants.
Ada
First we declare the specifications of the two procedures and the two corresponding operators (written as functions with quoted operators as their names): <lang ada>package Integer_Exponentiation is
-- int^int
procedure Exponentiate (Argument : in Integer;
Exponent : in Natural;
Result : out Integer);
function "**" (Left : Integer;
Right : Natural) return Integer;
-- real^int
procedure Exponentiate (Argument : in Float;
Exponent : in Integer;
Result : out Float);
function "**" (Left : Float;
Right : Integer) return Float;
end Integer_Exponentiation;</lang>
Now we can create a test program:
<lang ada>with Ada.Float_Text_IO, Ada.Integer_Text_IO, Ada.Text_IO; with Integer_Exponentiation;
procedure Test_Integer_Exponentiation is
use Ada.Float_Text_IO, Ada.Integer_Text_IO, Ada.Text_IO; use Integer_Exponentiation; R : Float; I : Integer;
begin
Exponentiate (Argument => 2.5, Exponent => 3, Result => R);
Put ("2.5 ^ 3 = ");
Put (R, Fore => 2, Aft => 4, Exp => 0);
New_Line;
Exponentiate (Argument => -12, Exponent => 3, Result => I);
Put ("-12 ^ 3 = ");
Put (I, Width => 7);
New_Line;
end Test_Integer_Exponentiation;</lang>
Finally we can implement the procedures and operations:
<lang ada>package body Integer_Exponentiation is
-- int^int
procedure Exponentiate (Argument : in Integer;
Exponent : in Natural;
Result : out Integer) is
begin
Result := 1;
for Counter in 1 .. Exponent loop
Result := Result * Argument;
end loop;
end Exponentiate;
function "**" (Left : Integer;
Right : Natural) return Integer is
Result : Integer;
begin
Exponentiate (Argument => Left,
Exponent => Right,
Result => Result);
return Result;
end "**";
-- real^int
procedure Exponentiate (Argument : in Float;
Exponent : in Integer;
Result : out Float) is
begin
Result := 1.0;
if Exponent < 0 then
for Counter in Exponent .. -1 loop
Result := Result / Argument;
end loop;
else
for Counter in 1 .. Exponent loop
Result := Result * Argument;
end loop;
end if;
end Exponentiate;
function "**" (Left : Float;
Right : Integer) return Float is
Result : Float;
begin
Exponentiate (Argument => Left,
Exponent => Right,
Result => Result);
return Result;
end "**";
end Integer_Exponentiation;</lang>
ALGOL 68
<lang algol68>main:(
INT two=2, thirty=30; # test constants # PROC VOID undefined;
- First implement exponentiation using a rather slow but sure FOR loop #
PROC int pow = (INT base, exponent)INT: ( # PROC cannot be over loaded # IF exponent<0 THEN undefined FI; INT out:=( exponent=0 | 1 | base ); FROM 2 TO exponent DO out*:=base OD; out );
printf(($" One Gibi-unit is: int pow("g(0)","g(0)")="g(0)" - (cost: "g(0)
" INT multiplications)"l$,two, thirty, int pow(two,thirty),thirty-1));
- implement exponentiation using a faster binary technique and WHILE LOOP #
OP ** = (INT base, exponent)INT: ( BITS binary exponent:=BIN exponent ; # do exponent arithmetic in binary # INT out := IF bits width ELEM binary exponent THEN base ELSE 1 FI; INT sq := IF exponent < 0 THEN undefined; ~ ELSE base FI;
WHILE
binary exponent := binary exponent SHR 1;
binary exponent /= BIN 0
DO
sq *:= sq;
IF bits width ELEM binary exponent THEN out *:= sq FI
OD;
out
);
printf(($" One Gibi-unit is: "g(0)"**"g(0)"="g(0)" - (cost: "g(0)
" INT multiplications)"l$,two, thirty, two ** thirty,8));
OP ** = (REAL in base, INT in exponent)REAL: ( # ** INT Operator can be overloaded # REAL base := ( in exponent<0 | 1/in base | in base); INT exponent := ABS in exponent; BITS binary exponent:=BIN exponent ; # do exponent arithmetic in binary # REAL out := IF bits width ELEM binary exponent THEN base ELSE 1 FI; REAL sq := base;
WHILE
binary exponent := binary exponent SHR 1;
binary exponent /= BIN 0
DO
sq *:= sq;
IF bits width ELEM binary exponent THEN out *:= sq FI
OD;
out
);
printf(($" One Gibi-unit is: "g(0,1)"**"g(0)"="g(0,1)" - (cost: "g(0)
" REAL multiplications)"l$, 2.0, thirty, 2.0 ** thirty,8));
OP ** = (REAL base, REAL exponent)REAL: ( # ** REAL Operator can be overloaded # exp(ln(base)*exponent) );
printf(($" One Gibi-unit is: "g(0,1)"**"g(0,1)"="g(0,1)" - (cost: "
"depends on precision)"l$, 2.0, 30.0, 2.0 ** 30.0))
)</lang>
- Output:
One Gibi-unit is: int pow(2,30)=1073741824 - (cost: 29 INT multiplications) One Gibi-unit is: 2**30=1073741824 - (cost: 8 INT multiplications) One Gibi-unit is: 2.0**30=1073741824.0 - (cost: 8 REAL multiplications) One Gibi-unit is: 2.0**30.0=1073741824.0 - (cost: depends on precision)
Recursive operator calls
<lang algol68>main:(
INT two=2, thirty=30; # test constants # PROC VOID undefined;
- First implement exponentiation using a rather slow but sure FOR loop #
PROC int pow = (INT base, exponent)INT: ( # PROC cannot be over loaded #
IF exponent<0 THEN undefined FI;
INT out:=( exponent=0 | 1 | base );
FROM 2 TO exponent DO out*:=base OD;
out
);
printf(($" One Gibi-unit is: int pow("g(0)","g(0)")="g(0)" - (cost: "g(0)
" INT multiplications)"l$,two, thirty, int pow(two,thirty),thirty-1));
- implement exponentiation using a faster binary technique and WHILE LOOP #
OP ** = (INT base, exponent)INT:
IF base = 0 THEN 0 ELIF base = 1 THEN 1
ELIF exponent = 0 THEN 1 ELIF exponent = 1 THEN base
ELIF ODD exponent THEN
(base*base) ** (exponent OVER 2) * base
ELSE
(base*base) ** (exponent OVER 2)
FI;
printf(($" One Gibi-unit is: "g(0)"**"g(0)"="g(0)" - (cost: "g(0)
" INT multiplications)"l$,two, thirty, two ** thirty,8));
OP ** = (REAL in base, INT in exponent)REAL: ( # ** INT Operator can be overloaded #
REAL base := ( in exponent<0 | 1/in base | in base);
INT exponent := ABS in exponent;
IF base = 0 THEN 0 ELIF base = 1 THEN 1
ELIF exponent = 0 THEN 1 ELIF exponent = 1 THEN base
ELIF ODD exponent THEN
(base*base) ** (exponent OVER 2) * base
ELSE
(base*base) ** (exponent OVER 2)
FI
);
printf(($" One Gibi-unit is: "g(0,1)"**"g(0)"="g(0,1)" - (cost: "g(0)
" REAL multiplications)"l$, 2.0, thirty, 2.0 ** thirty,8));
OP ** = (REAL base, REAL exponent)REAL: ( # ** REAL Operator can be overloaded #
exp(ln(base)*exponent)
);
printf(($" One Gibi-unit is: "g(0,1)"**"g(0,1)"="g(0,1)" - (cost: "
"depends on precision)"l$, 2.0, 30.0, 2.0 ** 30.0))
)</lang>
- Output:
One Gibi-unit is: int pow(2,30)=1073741824 - (cost: 29 INT multiplications) One Gibi-unit is: 2**30=1073741824 - (cost: 8 INT multiplications) One Gibi-unit is: 2.0**30=1073741824.0 - (cost: 8 REAL multiplications) One Gibi-unit is: 2.0**30.0=1073741824.0 - (cost: depends on precision)
AutoHotkey
<lang AutoHotkey>MsgBox % Pow(5,3) MsgBox % Pow(2.5,4)
Pow(x, n){ r:=1 loop %n% r *= x return r }</lang>
AWK
Traditional awk implementations do not provide an exponent operator, so we define a function to calculate the exponent. This one-liner reads base and exponent from stdin, one pair per line, and writes the result to stdout: <lang awk>$ awk 'function pow(x,n){r=1;for(i=0;i<n;i++)r=r*x;return r}{print pow($1,$2)}' </lang>
- Output:
2.5 2 6.25 10 6 1000000 3 0 1 But this last exponentation is wrong : 10 140 100000000000000048235962126657397336628942202864391877882749784196997612045996878213993935631944257381261260379071613194067266765009513873408 This is because traditionnal awk treat number internaly by finite precision.
<lang awk>If you want to use arbitrary precision number with (more recent) awk, you have to use -M option : $ gawk -M '{ printf("%f\n",$1^$2) }' </lang>
- Output:
10 140 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000,000000
<lang awk>And if you want to use locales for decimal separator, you have tu use -N option : $ gawk -N '{ printf("%f\n",$1^$2) }' </lang>
- Output:
2,5 2 6,250000
BASIC
The vast majority of BASIC implementations don't support defining custom operators, or overloading of any kind.
<lang qbasic>DECLARE FUNCTION powL& (x AS INTEGER, y AS INTEGER) DECLARE FUNCTION powS# (x AS SINGLE, y AS INTEGER)
DIM x AS INTEGER, y AS INTEGER DIM a AS SINGLE
RANDOMIZE TIMER a = RND * 10 x = INT(RND * 10) y = INT(RND * 10) PRINT x, y, powL&(x, y) PRINT a, y, powS#(a, y)
FUNCTION powL& (x AS INTEGER, y AS INTEGER)
DIM n AS INTEGER, m AS LONG
IF x <> 0 THEN
m = 1
IF SGN(y) > 0 THEN
FOR n = 1 TO y
m = m * x
NEXT
END IF
END IF
powL& = m
END FUNCTION
FUNCTION powS# (x AS SINGLE, y AS INTEGER)
DIM n AS INTEGER, m AS DOUBLE
IF x <> 0 THEN
m = 1
IF y <> 0 THEN
FOR n = 1 TO y
m = m * x
NEXT
IF y < 0 THEN m = 1# / m
END IF
END IF
powS# = m
END FUNCTION</lang>
- Output:
0 8 0 7.768213 8 13260781.61887441 1 9 1 2.707636 9 7821.90151734948 8 2 64 9.712946 2 94.34131879665438
BBC BASIC
<lang bbcbasic> PRINT "11^5 = " ; FNipow(11, 5)
PRINT "PI^3 = " ; FNfpow(PI, 3)
END
DEF FNipow(A%, B%)
LOCAL I%, P%
P% = 1
FOR I% = 1 TO 32
P% *= P%
IF B% < 0 THEN P% *= A%
B% = B% << 1
NEXT
= P%
DEF FNfpow(A, B%)
LOCAL I%, P
P = 1
FOR I% = 1 TO 32
P *= P
IF B% < 0 THEN P *= A
B% = B% << 1
NEXT
= P</lang>
- Output:
11^5 = 161051 PI^3 = 31.0062767
IS-BASIC
<lang IS-BASIC>100 DEF POW(X,Y) 110 IF X=0 THEN LET POW=0:EXIT DEF 120 LET POW=EXP(Y*LOG(X)) 130 END DEF 140 PRINT POW(PI,3) 150 PRINT PI^3</lang>
Befunge
Note: Only works for integer bases and powers. <lang befunge>v v \< >&:32p&1-\>32g*\1-:|
$
.
@</lang>
Brat
<lang brat>#Procedure exp = { base, exp |
1.to(exp).reduce 1, { m, n | m = m * base }
}
- Numbers are weird
1.parent.^ = { rhs |
num = my
1.to(rhs).reduce 1 { m, n | m = m * num }
}
p exp 2 5 #Prints 32 p 2 ^ 5 #Prints 32</lang>
C
Two versions are given - one for integer bases, the other for floating point. The integer version returns 0 when the abs(base) is != 1 and the exponent is negative. <lang c>#include <stdio.h>
- include <assert.h>
int ipow(int base, int exp) {
int pow = base;
int v = 1;
if (exp < 0) {
assert (base != 0); /* divide by zero */
return (base*base != 1)? 0: (exp&1)? base : 1;
}
while(exp > 0 )
{
if (exp & 1) v *= pow;
pow *= pow;
exp >>= 1;
}
return v;
}
double dpow(double base, int exp) {
double v=1.0; double pow = (exp <0)? 1.0/base : base; if (exp < 0) exp = - exp;
while(exp > 0 )
{
if (exp & 1) v *= pow;
pow *= pow;
exp >>= 1;
}
return v;
}
int main() {
printf("2^6 = %d\n", ipow(2,6));
printf("2^-6 = %d\n", ipow(2,-6));
printf("2.71^6 = %lf\n", dpow(2.71,6));
printf("2.71^-6 = %lf\n", dpow(2.71,-6));
}</lang>
The C11 standard features type-generic expressions via the _Generic keyword. We can add to the above example to use this feature.
<lang c>
- define generic_pow(base, exp)\
_Generic((base),\
double: dpow,\
int: ipow)\
(base, exp)
int main() {
printf("2^6 = %d\n", generic_pow(2,6));
printf("2^-6 = %d\n", generic_pow(2,-6));
printf("2.71^6 = %lf\n", generic_pow(2.71,6));
printf("2.71^-6 = %lf\n", generic_pow(2.71,-6));
} </lang>
C#
In C# it is possible to overload operators (+, -, *, etc..), but to do so requires the overload to implement at least one argument as the calling type.
What this means, is that if we have the class, A, to do an overload of + - we must set one of the arguments as the type "A". This is because in C#, overloads are defined on a class basis - so when doing an operator, .Net looks at the class to find the operators. In this manner, one of the arguments must be of the class, else .Net would be looking there in vain.
This again means, that a direct overloading of the ^-character between two integers / double and integer is not possible.
However - coming to think of it, one could overload the "int" class, and enter the operator there. --LordMike 17:45, 5 May 2010 (UTC)
<lang csharp> static void Main(string[] args) { Console.WriteLine("5^5 = " + Expon(5, 5)); Console.WriteLine("5.5^5 = " + Expon(5.5, 5)); Console.ReadLine(); }
static double Expon(int Val, int Pow) { return Math.Pow(Val, Pow); } static double Expon(double Val, int Pow) { return Math.Pow(Val, Pow); } </lang>
- Output:
5^5 = 3125 5.5^5 = 5032,84375
C++
While C++ does allow operator overloading, it does not have an exponentiation operator, therefore only a function definition is given. For non-negative exponents the integer and floating point versions are exactly the same, for obvious reasons. For negative exponents, the integer exponentiation would not give integer results; therefore there are several possibilities:
- Use floating point results even for integer exponents.
- Use integer results for integer exponents and give an error for negative exponents.
- Use integer results for integer exponents and return just the integer part (i.e. return 0 if the base is larger than one and the exponent is negative).
The third option somewhat resembles the integer division rules, and has the nice property that it can use the exact same algorithm as the floating point version. Therefore this option is chosen here. Actually the template can be used with any type which supports multiplication, division and explicit initialization from int. Note that there are several aspects about int which are not portably defined; most notably it is not guaranteed
- that the negative of a valid int is again a valid int; indeed for most implementations, the minimal value doesn't have a positive counterpart,
- whether the result of a%b is positive or negative if a is negative, and in which direction the corresponding division is rounded (however, it is guaranteed that (a/b)*b + a%b == a)
The code below tries to avoid those platform dependencies. Note that bitwise operations wouldn't help here either, because the representation of negative numbers can vary as well. <lang cpp>template<typename Number>
Number power(Number base, int exponent)
{
int zerodir;
Number factor;
if (exponent < 0)
{
zerodir = 1;
factor = Number(1)/base;
}
else
{
zerodir = -1;
factor = base;
}
Number result(1);
while (exponent != 0)
{
if (exponent % 2 != 0)
{
result *= factor;
exponent += zerodir;
}
else
{
factor *= factor;
exponent /= 2;
}
}
return result;
}</lang>
Chef
See Basic integer arithmetic#Chef.
Clojure
Operators in Clojure are functions, so this satisfies both requirements. Also, this is polymorphic- it will work with integers, floats, etc, even ratios. (Since operators are implemented as functions they are used in prefix notation) <lang lisp>(defn ** [x n] (reduce * (repeat n x)))</lang> Usage:
(** 2 3) ; 8 (** 7.2 2.1) ; 373.24800000000005 (** 7/2 3) ; 343/8
Common Lisp
Common Lisp has a few forms of iteration. One of the more general is the do loop. Using the do loop, one definition is given below: <lang lisp>(defun my-expt-do (a b)
(do ((x 1 (* x a))
(y 0 (+ y 1)))
((= y b) x)))</lang>
do takes three forms. The first is a list of variable initializers and incrementers. In this case, x, the eventual return value, is initialized to 1, and every iteration of the do loop replaces the value of x with x * a. Similarly, y is initialized to 0 and is replaced with y + 1. The second is a list of conditions and return values. In this case, when y = b, the loop stops, and the current value of x is returned. Common Lisp has no explicit return keyword, so x ends up being the return value for the function. The last form is the body of the loop, and usually consists of some action to perform (that has some side-effect). In this case, all the work is being done by the first and second forms, so there are no extra actions.
Of course, Lisp programmers often prefer recursive solutions. <lang lisp>(defun my-expt-rec (a b)
(cond ((= b 0) 1) (t (* a (my-expt-rec a (- b 1))))))</lang>
This solution uses the fact that a^0 = 1 and that a^b = a * a^{b-1}. cond is essentially a generalized if-statement. It takes a list of forms of the form (cond result). For instance, in this case, if b = 0, then function returns 1. t is the truth constant in Common Lisp and is often used as a default condition (similar to the default keyword in C/C++/Java or the else block in many languages).
Common Lisp has much more lenient rules for identifiers. In particular, ^ is a valid CL identifier. Since it is not already defined in the standard library, we can simply use it as a function name, just like any other function. <lang lisp>(defun ^ (a b)
(do ((x 1 (* x a))
(y 0 (+ y 1)))
((= y b) x)))</lang>
D
D has a built-in exponentiation operator: ^^ <lang d>import std.stdio, std.conv;
struct Number(T) {
T x; // base
alias x this;
string toString() const { return text(x); }
Number opBinary(string op)(in int exponent)
const pure nothrow @nogc if (op == "^^") in {
if (exponent < 0)
assert (x != 0, "Division by zero");
} body {
debug puts("opBinary ^^");
int zerodir;
T factor;
if (exponent < 0) {
zerodir = +1;
factor = T(1) / x;
} else {
zerodir = -1;
factor = x;
}
T result = 1;
int e = exponent;
while (e != 0)
if (e % 2 != 0) {
result *= factor;
e += zerodir;
} else {
factor *= factor;
e /= 2;
}
return Number(result); }
}
void main() {
alias Double = Number!double; writeln(Double(2.5) ^^ 5);
alias Int = Number!int; writeln(Int(3) ^^ 3); writeln(Int(0) ^^ -2); // Division by zero.
}</lang>
- Output:
(Compiled in debug mode, stack trace removed)
core.exception.AssertError@exponentiation_operator.d(11): Division by zero opBinary ^^ 97.6563 opBinary ^^ 27 opBinary
E
Simple, unoptimized implementation which will accept any kind of number for the base. If the base is an int, then the result will be of type float64 if the exponent is negative, and int otherwise.
<lang e>def power(base, exponent :int) {
var r := base
if (exponent < 0) {
for _ in exponent..0 { r /= base }
} else if (exponent <=> 0) {
return 1
} else {
for _ in 2..exponent { r *= base }
}
return r
}</lang>
EchoLisp
<lang lisp>
- this exponentiation function handles integer, rational or float x.
- n is a positive or negative integer.
(define (** x n) (cond
((zero? n) 1) ((< n 0) (/ (** x (- n)))) ;; x**-n = 1 / x**n ((= n 1) x) ((= n 0) 1) ((odd? n) (* x (** x (1- n)))) ;; x**(2p+1) = x * x**2p (else (let ((m (** x (/ n 2)))) (* m m))))) ;; x**2p = (x**p) * (x**p)
(** 3 0) → 1 (** 3 4) → 81 (** 3 5) → 243 (** 10 10) → 10000000000 (** 1.3 10) → 13.785849184900007
(** -3 5) → -243 (** 3 -4) → 1/81 (** 3.7 -4) → 0.005335720890574502 (** 2/3 7) → 128/2187
(lib 'bigint) (** 666 42) → 38540524895511613165266748863173814985473295063157418576769816295283207864908351682948692085553606681763707358759878656
</lang>
Ela
Ela standard prelude already defines an exponentiation operator (**) but we will implement it by ourselves anyway:
<lang ela>open number
_ ^ 0 = 1 x ^ n | n > 0 = f x (n - 1) x
|else = fail "Negative exponent"
where f _ 0 y = y
f a d y = g a d
where g b i | even i = g (b * b) (i `quot` 2)
| else = f b (i - 1) (b * y)
(12 ^ 4, 12 ** 4)</lang>
- Output:
(20736,20736)
Ela supports generic arithmetic functions and generic numeric literals. This is how we can change an implementation of a (^) function and make it generic:
<lang ela>open number
//Function quot from number module is defined only for //integral numbers. We can use this as an universal quot. uquot x y | x is Integral = x `quot` y
| else = x / y
//Changing implementation by using generic numeric literals //(e.g. 2u) and elimitating all comparisons with 0. !x ^ n | n ~= 0u = 1u
| n > 0u = f x (n - 1u) x
| else = fail "Negative exponent"
where f a d y
| d ~= 0u = y
| else = g a d
where g b i | even i = g (b * b) (i `uquot` 2u)
| else = f b (i - 1u) (b * y)
(12 ^ 4, 12.34 ^ 4.04)</lang>
- Output:
(20736,286138.2f)
We have a case of true polymorphism here and no overloading is required. However Ela supports overloading using classes (somewhat similar to Haskell type classes) so we can show how the same implementation could work with overloading (less preferable in this case because of more redundant code but still possible):
<lang ela>open number
//A class that defines our overloadable function class Exponent a where
(^) a->a->_
//Implementation for integers instance Exponent Int where
_ ^ 0 = 1
x ^ n | n > 0 = f x (n - 1) x
|else = fail "Negative exponent"
where f _ 0 y = y
f a d y = g a d
where g b i | even i = g (b * b) (i `quot` 2)
| else = f b (i - 1) (b * y)
//Implementation for floats instance Exponent Single where
x ^ n | n < 0.001 = 1
| n > 0 = f x (n - 1) x
| else = fail "Negative exponent"
where f a d y
| d < 0.001 = y
| else = g a d
where g b i | even i = g (b * b) (i / 2)
| else = f b (i - 1) (b * y)
(12 ^ 4, 12.34 ^ 4.04)</lang>
- Output:
(20736,286138.2f)
Elixir
<lang elixir>defmodule My do
def exp(x,y) when is_integer(x) and is_integer(y) and y>=0 do
IO.write("int> ") # debug test
exp_int(x,y)
end
def exp(x,y) when is_integer(y) do
IO.write("float> ") # debug test
exp_float(x,y)
end
def exp(x,y), do: (IO.write(" "); :math.pow(x,y))
defp exp_int(_,0), do: 1
defp exp_int(x,y), do: Enum.reduce(1..y, 1, fn _,acc -> x * acc end)
defp exp_float(_,y) when y==0, do: 1.0
defp exp_float(x,y) when y<0, do: 1/exp_float(x,-y)
defp exp_float(x,y), do: Enum.reduce(1..y, 1, fn _,acc -> x * acc end)
end
list = [{2,0}, {2,3}, {2,-2},
{2.0,0}, {2.0,3}, {2.0,-2},
{0.5,0}, {0.5,3}, {0.5,-2},
{-2,2}, {-2,3}, {-2.0,2}, {-2.0,3},
]
IO.puts " ___My.exp___ __:math.pow_" Enum.each(list, fn {x,y} ->
sxy = "#{x} ** #{y}"
sexp = inspect My.exp(x,y)
spow = inspect :math.pow(x,y) # For the comparison
:io.fwrite("~10s = ~12s, ~12s~n", [sxy, sexp, spow])
end)</lang>
- Output:
___My.exp___ __:math.pow_ int> 2 ** 0 = 1, 1.0 int> 2 ** 3 = 8, 8.0 float> 2 ** -2 = 0.25, 0.25 float> 2.0 ** 0 = 1.0, 1.0 float> 2.0 ** 3 = 8.0, 8.0 float> 2.0 ** -2 = 0.25, 0.25 float> 0.5 ** 0 = 1.0, 1.0 float> 0.5 ** 3 = 0.125, 0.125 float> 0.5 ** -2 = 4.0, 4.0 int> -2 ** 2 = 4, 4.0 int> -2 ** 3 = -8, -8.0 float> -2.0 ** 2 = 4.0, 4.0 float> -2.0 ** 3 = -8.0, -8.0
Erlang
pow(number, integer) -> number <lang erlang> pow(X, Y) when Y < 0 ->
1/pow(X, -Y);
pow(X, Y) when is_integer(Y) ->
pow(X, Y, 1).
pow(_, 0, B) ->
B;
pow(X, Y, B) ->
B2 = if Y rem 2 =:= 0 -> B; true -> X * B end, pow(X * X, Y div 2, B2).
</lang>
Tail call optimised version which works for both integers and float bases.
ERRE
ERRE does not permit operator overloading, so we can use a procedure only. The procedure below handles *integer powers*: for floating point exponent you must use EXP and LOG predefined functions. <lang ERRE>PROGRAM POWER
PROCEDURE POWER(A,B->POW) ! this routine handles only *INTEGER* powers
LOCAL FLAG% IF B<0 THEN B=-B FLAG%=TRUE POW=1 FOR X=1 TO B DO POW=POW*A END FOR IF FLAG% THEN POW=1/POW
END PROCEDURE
BEGIN
POWER(11,-2->POW) PRINT(POW) POWER(π,3->POW) PRINT(POW)
END PROGRAM</lang>
- Output:
8.264463E-03 31.00628
F#
<lang fsharp> //Integer Exponentiation, more interesting anyway than repeated multiplication. Nigel Galloway, October 12th., 2018 let rec myExp n g=match g with
|0 ->1
|g when g%2=1 ->n*(myExp n (g-1))
|_ ->let p=myExp n (g/2) in p*p
printfn "%d" (myExp 3 15) </lang>
- Output:
14348907
Factor
Simple, unoptimized implementation which accepts a positive or negative exponent: <lang factor>: pow ( f n -- f' )
dup 0 < [ abs pow recip ] [ [ 1 ] 2dip swap [ * ] curry times ] if ;</lang>
Here is a recursive implementation which splits the exponent in two: <lang factor>: pow ( f n -- f' )
{
{ [ dup 0 < ] [ abs pow recip ] }
{ [ dup 0 = ] [ 2drop 1 ] }
[ [ 2 mod 1 = swap 1 ? ] [ [ sq ] [ 2 /i ] bi* pow ] 2bi * ]
} cond ;</lang>
This implementation recurses only when an odd factor is found: <lang factor>USING: combinators kernel math ; IN: test
- (pow) ( f n -- f' )
[ dup even? ] [ [ sq ] [ 2 /i ] bi* ] while dup 1 = [ drop ] [ dupd 1 - (pow) * ] if ;
- pow ( f n -- f' )
{
{ [ dup 0 < ] [ abs (pow) recip ] }
{ [ dup 0 = ] [ 2drop 1 ] }
[ (pow) ]
} cond ;</lang>
A non-recursive version of (pow) can be written as: <lang factor>: (pow) ( f n -- f' )
[ 1 ] 2dip
[ dup 1 = ] [
dup even? [ [ sq ] [ 2 /i ] bi* ] [ [ [ * ] keep ] dip 1 - ] if
] until
drop * ;</lang>
Forth
<lang forth>: ** ( n m -- n^m )
1 swap 0 ?do over * loop nip ;</lang>
<lang forth>: f**n ( f n -- f^n )
dup 0= if drop fdrop 1e else dup 1 and if 1- fdup recurse f* else 2/ fdup f* recurse then then ;</lang>
Fortran
<lang fortran>MODULE Exp_Mod IMPLICIT NONE
INTERFACE OPERATOR (.pow.) ! Using ** instead would overload the standard exponentiation operator
MODULE PROCEDURE Intexp, Realexp
END INTERFACE
CONTAINS
FUNCTION Intexp (base, exponent) INTEGER :: Intexp INTEGER, INTENT(IN) :: base, exponent INTEGER :: i
IF (exponent < 0) THEN
IF (base == 1) THEN
Intexp = 1
ELSE
Intexp = 0
END IF
RETURN
END IF
Intexp = 1
DO i = 1, exponent
Intexp = Intexp * base
END DO
END FUNCTION IntExp
FUNCTION Realexp (base, exponent)
REAL :: Realexp
REAL, INTENT(IN) :: base
INTEGER, INTENT(IN) :: exponent
INTEGER :: i
Realexp = 1.0
IF (exponent < 0) THEN
DO i = exponent, -1
Realexp = Realexp / base
END DO
ELSE
DO i = 1, exponent
Realexp = Realexp * base
END DO
END IF
END FUNCTION RealExp
END MODULE Exp_Mod
PROGRAM EXAMPLE USE Exp_Mod
WRITE(*,*) 2.pow.30, 2.0.pow.30
END PROGRAM EXAMPLE</lang>
- Output:
1073741824 1.073742E+09
FreeBASIC
<lang freebasic>' FB 1.05.0
' Note that 'base' is a keyword in FB, so we use 'base_' instead as a parameter
Function Pow Overload (base_ As Double, exponent As Integer) As Double
If exponent = 0.0 Then Return 1.0
If exponent = 1.0 Then Return base_
If exponent < 0.0 Then Return 1.0 / Pow(base_, -exponent)
Dim power As Double = base_
For i As Integer = 2 To exponent
power *= base_
Next
Return power
End Function
Function Pow Overload(base_ As Integer, exponent As Integer) As Double
Return Pow(CDbl(base_), exponent)
End Function
' check results of these functions using FB's built in '^' operator Print "Pow(2, 2) = "; Pow(2, 2) Print "Pow(2.5, 2) = "; Pow(2.5, 2) Print "Pow(2, -3) = "; Pow(2, -3) Print "Pow(1.78, 3) = "; Pow(1.78, 3) Print Print "2 ^ 2 = "; 2 ^ 2 Print "2.5 ^ 2 = "; 2.5 ^ 2 Print "2 ^ -3 = "; 2 ^ -3 Print "1.78 ^ 3 = "; 1.78 ^ 3 Print Print "Press any key to quit" Sleep</lang>
- Output:
Pow(2, 2) = 4 Pow(2.5, 2) = 6.25 Pow(2, -3) = 0.125 Pow(1.78, 3) = 5.639752000000001 2 ^ 2 = 4 2.5 ^ 2 = 6.25 2 ^ -3 = 0.125 1.78 ^ 3 = 5.639752000000001
GAP
<lang gap>expon := function(a, n, one, mul) local p; p := one; while n > 0 do if IsOddInt(n) then p := mul(a, p); fi; a := mul(a, a); n := QuoInt(n, 2); od; return p; end;
expon(2, 10, 1, \*);
- 1024
- a more creative use of exponentiation
List([0 .. 31], n -> (1 - expon(0, n, 1, \-))/2);
- [ 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0,
- 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1 ]</lang>
Go
Go doesn't support operator defintion. Other notes: While I left the integer algorithm simple, I used the shift and square trick for the float algorithm, just to show an alternative. <lang go>package main
import (
"errors" "fmt"
)
func expI(b, p int) (int, error) {
if p < 0 {
return 0, errors.New("negative power not allowed")
}
r := 1
for i := 1; i <= p; i++ {
r *= b
}
return r, nil
}
func expF(b float32, p int) float32 {
var neg bool
if p < 0 {
neg = true
p = -p
}
r := float32(1)
for pow := b; p > 0; pow *= pow {
if p&1 == 1 {
r *= pow
}
p >>= 1
}
if neg {
r = 1 / r
}
return r
}
func main() {
ti := func(b, p int) {
fmt.Printf("%d^%d: ", b, p)
e, err := expI(b, p)
if err != nil {
fmt.Println(err)
} else {
fmt.Println(e)
}
}
fmt.Println("expI tests")
ti(2, 10)
ti(2, -10)
ti(-2, 10)
ti(-2, 11)
ti(11, 0)
fmt.Println("overflow undetected")
ti(10, 10)
tf := func(b float32, p int) {
fmt.Printf("%g^%d: %g\n", b, p, expF(b, p))
}
fmt.Println("\nexpF tests:")
tf(2, 10)
tf(2, -10)
tf(-2, 10)
tf(-2, 11)
tf(11, 0)
fmt.Println("disallowed in expI, allowed here")
tf(0, -1)
fmt.Println("other interesting cases for 32 bit float type")
tf(10, 39)
tf(10, -39)
tf(-10, 39)
}</lang>
- Output:
expI tests 2^10: 1024 2^-10: negative power not allowed -2^10: 1024 -2^11: -2048 11^0: 1 overflow undetected 10^10: 1410065408 expF tests: 2^10: 1024 2^-10: 0.0009765625 -2^10: 1024 -2^11: -2048 11^0: 1 disallowed in expI, allowed here 0^-1: +Inf other interesting cases for 32 bit float type 10^39: +Inf 10^-39: 0 -10^39: -Inf
Haskell
Here's the exponentiation operator from the Prelude: <lang haskell>(^) :: (Num a, Integral b) => a -> b -> a _ ^ 0 = 1 x ^ n | n > 0 = f x (n-1) x where
f _ 0 y = y
f a d y = g a d where
g b i | even i = g (b*b) (i `quot` 2)
| otherwise = f b (i-1) (b*y)
_ ^ _ = error "Prelude.^: negative exponent"</lang>
There's no difference in Haskell between a procedure (or function) and an operator, other than the infix notation. This routine is overloaded for any integral exponent (which includes the arbitrarily large Integer type) and any numeric type for the bases (including, for example, Complex). It uses the fast "binary" exponentiation algorithm. For a negative exponent, the type of the base must support division (and hence reciprocals):
<lang haskell>(^^) :: (Fractional a, Integral b) => a -> b -> a x ^^ n = if n >= 0 then x^n else recip (x^(negate n))</lang>
This rules out e.g. the integer types as base values in this case. Haskell also has a third exponentiation operator,
<lang haskell>(**) :: Floating a => a -> a -> a x ** y = exp (log x * y)</lang> which is used for floating point arithmetic.
HicEst
<lang hicest>WRITE(Clipboard) pow(5, 3) ! 125 WRITE(ClipBoard) pow(5.5, 7) ! 152243.5234
FUNCTION pow(x, n)
pow = 1
DO i = 1, n
pow = pow * x
ENDDO
END </lang>
Icon and Unicon
The procedure below will take an integer or real base and integer exponent and return base ^ exponent. If exponent is negative, base is coerced to real so as not to return 0. Operator overloading is not supported and this is not an efficient implementation. <lang Icon>procedure main() bases := [5,5.] numbers := [0,2,2.,-1,3] every write("expon(",b := !bases,", ",x := !numbers,")=",(expon(b,x) | "failed") \ 1) end
procedure expon(base,power) local op,res
base := numeric(base) | runerror(102,base) power := power = integer(power) | runerr(101,power)
if power = 0 then return 1 else op := if power < 1 then
(base := real(base)) & "/" # force real base
else "*"
res := 1 every 1 to abs(power) do
res := op(res,base)
return res end</lang>
J
J is concretely specified, which makes it easy to define primitives in terms of other primitives (this is especially true of mathematical primitives, given the language's mathematical emphasis).
So we have any number of options. Here's the simplest, equivalent to the for each number, product = product * number of other languages. The base may be any number, and the exponent may be any non-negative integer (including zero):
<lang j> exp =: */@:#~
10 exp 3
1000
10 exp 0
1</lang>
We can make this more general by allowing the exponent to be any integer (including negatives), at the cost of a slight increase in complexity:
<lang j> exp =: *@:] %: */@:(#~|)
10 exp _3
0.001</lang>
Or, we can define exponentiation as repeated multiplication (as opposed to multiplying a given number of copies of the base)
<lang J> exp =: dyad def 'x *^:y 1'
10 exp 3
1000
10 exp _3
0.001</lang>
Here, when we specify a negative number of repetitions, multiplication's inverse is used that many times.
J's calculus of functions permits us to define exponentiation in its full generality, as the inverse of log (i.e. exp = log-1):
<lang j> exp =: ^.^:_1
81 exp 0.5
9</lang>
Note that the definition above does not use the primitive exponentiation function ^ . The carets in it represent different (but related) things . The function is composed of three parts: ^. ^: _1 . The first part, ^., is the primitive logarithm operator (e.g. 3 = 10^.1000) .
The second part, ^: , is interesting: it is a "meta operator". It takes two arguments: a function f on its left, and a number N on its right. It produces a new function, which, when given an argument, applies f to that argument N times. For example, if we had a function increment, then increment^:3 X would increment X three times, so the result would be X+3.
In the case of ^. ^: _1 , f is ^. (i.e. logarithm) and N is -1. Therefore we apply log negative one times or the inverse of log once (precisely as in log-1).
Similarly, we can define exponentiation as the reverse of the inverse of root. That is, x pow y = y root-1 x:
<lang j> exp =: %:^:_1~
81 exp 0.5
9</lang>
Compare this with the previous definition: it is the same, except that %: , root, has been substituted for ^. , logarithm, and the arguments have been reversed (or reflected) with ~.
That is, J is telling us that power is the same as the reflex of the inverse of root, exactly as we'd expect.
One last note: we said these definitions are the same as ^ in its full generality. What is meant by that? Well, in the context of this puzzle, it means both the base and exponent may be any real number. But J goes further than that: it also permits complex numbers.
Let's use Euler's famous formula, epi*i = -1 as an example:
<lang j> pi =: 3.14159265358979323846
e =: 2.71828182845904523536 i =: 2 %: _1 NB. Square root of -1 e^(pi*i)
_1</lang>
And, as stated, our redefinition is equivalent:
<lang j> exp =: %:^:_1~
e exp (pi*i)
_1</lang>
Java
Java does not support operator definition. This example is unoptimized, but will handle negative exponents as well. It is unnecessary to show intint since an int in Java will be cast as a double. <lang java>public class Exp{
public static void main(String[] args){
System.out.println(pow(2,30));
System.out.println(pow(2.0,30)); //tests
System.out.println(pow(2.0,-2));
}
public static double pow(double base, int exp){
if(exp < 0) return 1 / pow(base, -exp);
double ans = 1.0;
for(;exp > 0;--exp) ans *= base;
return ans;
}
}</lang>
- Output:
1.073741824E9 1.073741824E9 0.25
JavaScript
<lang javascript>function pow(base, exp) {
if (exp != Math.floor(exp))
throw "exponent must be an integer";
if (exp < 0)
return 1 / pow(base, -exp);
var ans = 1;
while (exp > 0) {
ans *= base;
exp--;
}
return ans;
}</lang>
jq
<lang jq># 0^0 => 1
- NOTE: jq converts very large integers to floats.
- This implementation uses reduce to avoid deep recursion
def power_int(n):
if n == 0 then 1
elif . == 0 then 0
elif n < 0 then 1/power_int(-n)
elif ((n | floor) == n) then
( (n % 2) | if . == 0 then 1 else -1 end ) as $sign
| if (. == -1) then $sign
elif . < 0 then (( -(.) | power_int(n) ) * $sign)
else . as $in | reduce range(1;n) as $i ($in; . * $in)
end
else error("This is a toy implementation that requires n be integral")
end ;</lang>
Demonstration:<lang jq>def demo(x;y):
x | [ power_int(y), (log*y|exp) ] ;
demo(2; 3), demo(2; 64), demo(1.1; 1024), demo(1.1; -1024)
- Output:
[8, 7.999999999999998] [18446744073709552000, 18446744073709525000] [2.4328178969536854e+42, 2.4328178969536693e+42] [4.1104597317052596e-43, 4.1104597317052874e-43] </lang>
Julia
<lang julia> function pow(base::Number, exp::Integer)
r = one(base) for i = 1:exp r *= base end return r
end </lang>
- Output:
julia> println("5 ^ 3 ^ 2 = ", 5 ^ 3 ^ 2)
5 ^ 3 ^ 2 = 1953125
julia> println("(5 ^ 3) ^ 2 = ", (5 ^ 3) ^ 2)
(5 ^ 3) ^ 2 = 15625
julia> println("5 ^ (3 ^ 2) = ", 5 ^ (3 ^ 2))
5 ^ (3 ^ 2) = 1953125
Kotlin
Kotlin does not have a dedicated exponentiation operator (we would normally use Java's Math.pow method instead) but it's possible to implement integer and floating power exponentiation (with integer exponents) using infix extension functions which look like non-symbolic operators for these actions: <lang scala>// version 1.0.6
infix fun Int.ipow(exp: Int): Int =
when {
this == 1 -> 1
this == -1 -> if (exp % 2 == 0) 1 else -1
exp < 0 -> throw IllegalArgumentException("invalid exponent")
exp == 0 -> 1
else -> {
var ans = 1
var base = this
var e = exp
while (e > 0) {
if (e and 1 == 1) ans *= base
e = e shr 1
base *= base
}
ans
}
}
infix fun Double.dpow(exp: Int): Double {
var ans = 1.0
var e = exp
var base = if (e < 0) 1.0 / this else this
if (e < 0) e = -e
while (e > 0) {
if (e and 1 == 1) ans *= base
e = e shr 1
base *= base
}
return ans
}
fun main(args: Array<String>) {
println("2 ^ 3 = ${2 ipow 3}")
println("1 ^ -10 = ${1 ipow -10}")
println("-1 ^ -3 = ${-1 ipow -3}")
println()
println("2.0 ^ -3 = ${2.0 dpow -3}")
println("1.5 ^ 0 = ${1.5 dpow 0}")
println("4.5 ^ 2 = ${4.5 dpow 2}")
}</lang>
- Output:
2 ^ 3 = 8 1 ^ -10 = 1 -1 ^ -3 = -1 2.0 ^ -3 = 0.125 1.5 ^ 0 = 1.0 4.5 ^ 2 = 20.25
Liberty BASIC
<lang lb>
print " 11^5 = ", floatPow( 11, 5 ) print " (-11)^5 = ", floatPow( -11, 5 ) print " 11^( -5) = ", floatPow( 11, -5 ) print " 3.1416^3 = ", floatPow( 3.1416, 3 ) print " 0^2 = ", floatPow( 0, 2 ) print " 2^0 = ", floatPow( 2, 0 ) print " -2^0 = ", floatPow( -2, 0 )
end
function floatPow( a, b)
if a <>0 then
m =1
if b =abs( b) then
for n =1 to b
m =m *a
next n
else
m =1 /floatPow( a, 0 - b) ' LB has no unitary minus operator.
end if
else
m =0
end if
floatPow =m
end function
</lang>
Lingo
Lingo doesn't support user-defined operators. <lang lingo>-- As for built-in power() function: -- base can be either integer or float; returns float. on pow (base, exp)
if exp=0 then return 1.0 else if exp<0 then exp = -exp base = 1.0/base end if res = float(base) repeat with i = 2 to exp res = res*base end repeat return res
end</lang>
Logo
<lang logo>to int_power :n :m
if equal? 0 :m [output 1] if equal? 0 modulo :m 2 [output int_power :n*:n :m/2] output :n * int_power :n :m-1
end</lang>
Lua
All numbers in Lua are floating point numbers (thus, there are no real integers). Operator overloading is supported for tables only. <lang lua>number = {}
function number.pow( a, b )
local ret = 1
if b >= 0 then
for i = 1, b do
ret = ret * a.val
end
else
for i = b, -1 do
ret = ret / a.val
end
end
return ret
end
function number.New( v )
local num = { val = v }
local mt = { __pow = number.pow }
setmetatable( num, mt )
return num
end
x = number.New( 5 ) print( x^2 ) --> 25 print( number.pow( x, -4 ) ) --> 0.016</lang>
Lucid
Some misconceptions about Lucid
<lang lucid>pow(n,x)
k = n fby k div 2; p = x fby p*p; y =1 fby if even(k) then y else y*p; result y asa k eq 0;
end</lang>
M2000 Interpreter
<lang M2000 Interpreter> Module Exponentiation { \\ a variable can be any type except a string (no $ in name) \\ variable b is long type. \\ by default we pass by value arguments to a function \\ to pass by reference we have to use & before name, \\ in the signature and in the call function pow(a, b as long) { p=a-a ' make p same type as a p++ if b>0 then for i=1& to b {p*=a} =p } const fst$="{0::-32} {1}" Document exp$ k= pow(11&, 5) exp$=format$(fst$, k, type$(k)="Long")+{ } l=pow(11, 5) exp$=format$(fst$, l, type$(l)="Double")+{ } m=pow(pi, 3) exp$=format$(fst$, m, type$(m)="Decimal")+{ } \\ send to clipboard clipboard exp$ \\ send monospaced type text to console using cr char to change lines Print #-2, exp$ Rem Report exp$ ' send to console using proportional spacing and justification } Exponentiation
</lang>
- Output:
161051 True
161051 True
31.006276680299820175476315064 True
M4
M4 lacks floating point computation and operator definition. <lang M4>define(`power',`ifelse($2,0,1,`eval($1*$0($1,decr($2)))')') power(2,10)</lang>
- Output:
1024
Mathematica / Wolfram Language
Define a function and an infix operator \[CirclePlus] with the same definition: <lang Mathematica>exponentiation[x_,y_Integer]:=Which[y>0,Times@@ConstantArray[x,y],y==0,1,y<0,1/exponentiation[x,-y]] CirclePlus[x_,y_Integer]:=exponentiation[x,y]</lang> Examples: <lang Mathematica>exponentiation[1.23,3] exponentiation[4,0] exponentiation[2.5,-2] 1.23\[CirclePlus]3 4\[CirclePlus]0 2.5\[CirclePlus]-2</lang> gives back: <lang Mathematica>1.86087 1 0.16 1.86087 1 0.16</lang> Note that \[CirclePlus] shows up as a special character in Mathematica namely a circle divided in 4 pieces. Note also that this function supports negative and positive exponents.
Maxima
<lang maxima>"^^^"(a, n) := block(
[p: 1],
while n > 0 do (
if oddp(n) then p: p * a,
a: a * a,
n: quotient(n, 2)
),
p
)$
infix("^^^")$
2 ^^^ 10; 1024
2.5 ^^^ 10; 9536.7431640625</lang>
МК-61/52
<lang>С/П x^y С/П</lang>
Modula-2
Whilst some implementations or dialects of Modula-2 may permit definition or overloading of operators, neither is permitted in N.Wirth's classic language definition and the ISO Modula-2 standard. The operations are therefore given as library functions. <lang modula2> (* Library Interface *) DEFINITION MODULE Exponentiation;
PROCEDURE IntExp(base, exp : INTEGER) : INTEGER;
(* Raises base to the power of exp and returns the result both base and exp must be of type INTEGER *)
PROCEDURE RealExp(base : REAL; exp : INTEGER) : REAL;
(* Raises base to the power of exp and returns the result base must be of type REAL, exp of type INTEGER *)
END Exponentiation.
(* Library Implementation *) IMPLEMENTATION MODULE Exponentiation;
PROCEDURE IntExp(base, exp : INTEGER) : INTEGER;
VAR
i, res : INTEGER;
BEGIN
res := 1;
FOR i := 1 TO exp DO
res := res * base;
END;
RETURN res;
END IntExp;
PROCEDURE RealExp(base: REAL; exp: INTEGER) : REAL;
VAR
i : INTEGER;
res : REAL;
BEGIN
res := 1.0;
IF exp < 0 THEN
FOR i := exp TO -1 DO
res := res / base;
END;
ELSE (* exp >= 0 *)
FOR i := 1 TO exp DO
res := res * base;
END;
END;
RETURN res;
END RealExp;
END Exponentiation. </lang>
Modula-3
<lang modula3>MODULE Expt EXPORTS Main;
IMPORT IO, Fmt;
PROCEDURE IntExpt(arg, exp: INTEGER): INTEGER =
VAR result := 1;
BEGIN
FOR i := 1 TO exp DO
result := result * arg;
END;
RETURN result;
END IntExpt;
PROCEDURE RealExpt(arg: REAL; exp: INTEGER): REAL =
VAR result := 1.0;
BEGIN
IF exp < 0 THEN
FOR i := exp TO -1 DO
result := result / arg;
END;
ELSE
FOR i := 1 TO exp DO
result := result * arg;
END;
END;
RETURN result;
END RealExpt;
BEGIN
IO.Put("2 ^ 4 = " & Fmt.Int(IntExpt(2, 4)) & "\n");
IO.Put("2.5 ^ 4 = " & Fmt.Real(RealExpt(2.5, 4)) & "\n");
END Expt.</lang>
- Output:
2 ^ 4 = 16 2.5 ^ 4 = 39.0625
Nemerle
Macros can be used to define a new operator: <lang Nemerle>using System;
macro @^ (val, pow : int) {
<[ Math.Pow($val, $pow) ]>
}</lang> The file with the macro needs to be compiled as a library, and the resulting assembly must be referenced when compiling source files which use the operator. <lang Nemerle>using System; using System.Console; using Nemerle.Assertions;
module Expon {
Expon(val : int, pow : int) : int // demonstrates simple/naive method
requires pow > 0 otherwise throw ArgumentOutOfRangeException("Negative powers not allowed, will not return int.")
{
mutable result = 1;
repeat(pow) {
result *= val
}
result
}
Expon(val : double, pow : int) : double // demonstrates shift and square method
{
mutable neg = false;
mutable p = pow;
when (pow < 0) {neg = true; p = -pow};
mutable v = val;
mutable result = 1d;
while (p > 0) {
when (p & 1 == 1) result *= v;
v *= v;
p >>= 1;
}
if (neg) 1d/result else result
}
Main() : void
{
def eight = 2^3;
// def oops = 2^1.5; // compilation error as operator is defined for integer exponentiation
def four = Expon(2, 2);
def four_d = Expon(2.0, 2);
WriteLine($"$eight, $four, $four_d");
}
}</lang>
Nim
<lang nim>proc `^`[T: float|int](base: T; exp: int): T =
var (base, exp) = (base, exp) result = 1
if exp < 0:
when T is int:
if base * base != 1: return 0
elif (exp and 1) == 0: return 1
else: return base
else:
base = 1.0 / base
exp = -exp
while exp != 0:
if (exp and 1) != 0:
result *= base
exp = exp shr 1
base *= base
echo "2^6 = ", 2^6 echo "2^-6 = ", 2 ^ -6 echo "2.71^6 = ", 2.71^6 echo "2.71^-6 = ", 2.71 ^ -6</lang>
Objeck
<lang objeck>class Exp {
function : Main(args : String[]) ~ Nil {
Pow(2,30)->PrintLine();
Pow(2.0,30)->PrintLine();
Pow(2.0,-2)->PrintLine();
}
function : native : Pow(base : Float, exp : Int) ~ Float {
if(exp < 0) {
return 1 / base->Power(exp * -1.0);
};
ans := 1.0;
while(exp > 0) {
ans *= base;
exp -= 1;
};
return ans;
}
}</lang>
1.07374182e+009 1.07374182e+009 0.25
OCaml
It is possible to create a generic exponential. For this, one must know the multiplication function, and the unit value. Here, the usual fast algorithm is used:
<lang ocaml>let pow one mul a n =
let rec g p x = function
| 0 -> x
| i ->
g (mul p p) (if i mod 2 = 1 then mul p x else x) (i/2)
in
g a one n
pow 1 ( * ) 2 16;; (* 65536 *) pow 1.0 ( *. ) 2.0 16;; (* 65536. *)
(* pow is not limited to exponentiation *) pow 0 ( + ) 2 16;; (* 32 *) pow "" ( ^ ) "abc " 10;; (* "abc abc abc abc abc abc abc abc abc abc " *) pow [ ] ( @ ) [ 1; 2 ] 10;; (* [1; 2; 1; 2; 1; 2; 1; 2; 1; 2; 1; 2; 1; 2; 1; 2; 1; 2; 1; 2] *)
(* Thue-Morse sequence *) Array.init 32 (fun n -> (1 - pow 1 ( - ) 0 n) lsr 1);;
(* [|0; 1; 1; 0; 1; 0; 0; 1; 1; 0; 0; 1; 0; 1; 1; 0;
1; 0; 0; 1; 0; 1; 1; 0; 0; 1; 1; 0; 1; 0; 0; 1|]
See http://en.wikipedia.org/wiki/Thue-Morse_sequence
- )</lang>
See also Matrix-exponentiation operator#OCaml for a matrix usage.
Oforth
This function works either for int or floats :
<lang Oforth>: powint(r, n) | i |
1 n abs loop: i [ r * ] n isNegative ifTrue: [ inv ] ;
2 3 powint println 2 powint(3) println 1.2 4 powint println 1.2 powint(4) println</lang>
- Output:
8 8 2.0736 2.0736
PARI/GP
This version works for integer and floating-point bases (as well as intmod bases, ...). <lang parigp>ex(a, b)={
my(c = 1); while(b > 1, if(b % 2, c *= a); a = a^2; b >>= 1 ); a * c
};</lang>
PARI/GP also has a built-in operator that works for any type of numerical exponent: <lang parigp>ex2(a, b) = a ^ b;</lang>
Pascal
<lang pascal>Program ExponentiationOperator(output);
function intexp (base, exponent: integer): longint;
var i: integer;
begin
if (exponent < 0) then
if (base = 1) then
intexp := 1
else
intexp := 0
else
begin
intexp := 1;
for i := 1 to exponent do
intexp := intexp * base;
end;
end;
function realexp (base: real; exponent: integer): real;
var i: integer;
begin
realexp := 1.0;
if (exponent < 0) then
for i := exponent to -1 do
realexp := realexp / base
else
for i := 1 to exponent do
realexp := realexp * base;
end;
begin
writeln('2^30: ', intexp(2, 30));
writeln('2.0^30: ', realexp(2.0, 30));
end.</lang>
- Output:
% ./ExponentiationOperator 2^30: 1073741824 2.0^30: 1.07374182400000E+009
With overload
Pascal functions can be overloaded. This means that the two functions can have the same name and the particular function executed will depend on the data types of the arguments.
Program ExponentiationOperator(output);
function newpower (base, exponent: integer): longint;
var
i: integer;
begin
if (exponent < 0) then
if (base = 1) then
newpower := 1
else
newpower := 0
else
begin
newpower := 1;
for i := 1 to exponent do
newpower := newpower * base;
end;
end;
function newpower (base: real; exponent: integer): real;
var
i: integer;
begin
newpower := 1.0;
if (exponent < 0) then
for i := exponent to -1 do
newpower := newpower / base
else
for i := 1 to exponent do
newpower := newpower * base;
end;
begin
writeln('2^30: ', newpower(2, 30));
writeln('2.0^30: ', newpower(2.0, 30));
readln;
end.
Output is as before.
Perl
<lang Perl>#!/usr/bin/perl -w use strict ;
sub expon {
my ( $base , $expo ) = @_ ;
if ( $expo == 0 ) {
return 1 ;
}
elsif ( $expo == 1 ) {
return $base ;
}
elsif ( $expo > 1 ) {
my $prod = 1 ;
foreach my $n ( 0..($expo - 1) ) {
$prod *= $base ;
}
return $prod ;
}
elsif ( $expo < 0 ) {
return 1 / ( expon ( $base , -$expo ) ) ;
}
} print "3 to the power of 10 as a function is " . expon( 3 , 10 ) . " !\n" ; print "3 to the power of 10 as a builtin is " . 3**10 . " !\n" ; print "5.5 to the power of -3 as a function is " . expon( 5.5 , -3 ) . " !\n" ; print "5.5 to the power of -3 as a builtin is " . 5.5**-3 . " !\n" ; </lang>
- Output:
3 to the power of 10 as a function is 59049 ! 3 to the power of 10 as a builtin is 59049 ! 5.5 to the power of -3 as a function is 0.00601051840721262 ! 5.5 to the power of -3 as a builtin is 0.00601051840721262 !
The following version is simpler and much faster for large exponents, since it uses exponentiation by squaring. <lang Perl>sub ex {
my($base,$exp) = @_;
die "Exponent '$exp' must be an integer!" if $exp != int($exp);
return 1 if $exp == 0;
($base, $exp) = (1/$base, -$exp) if $exp < 0;
my $c = 1;
while ($exp > 1) {
$c *= $base if $exp % 2;
$base *= $base;
$exp >>= 1;
}
$base * $c;
}</lang>
Perl 6
<lang perl6>subset Natural of Int where { $^n >= 0 }
multi pow (0, 0) { fail '0**0 is undefined' } multi pow ($base, Natural $exp) { [*] $base xx $exp } multi pow ($base, Int $exp) { 1 / pow $base, -$exp }
sub infix:<***> ($a, $b) { pow $a, $b }
- Testing
say pow .75, -5; say .75 *** -5;</lang>
Phix
The builtin power function handles atoms and integers for both arguments, whereas this deliberately restricts the exponent to an integer.
There is no operator overloading in Phix, or for that matter any builtin overriding.
<lang Phix>function powi(atom b, integer i)
atom v=1
b = iff(i<0 ? 1/b : b)
i = abs(i)
while i>0 do
if and_bits(i,1) then v *= b end if
b *= b
i = floor(i/2)
end while
return v
end function ?powi(-3,-5) ?power(-3,-5)</lang>
- Output:
-0.004115226337 -0.004115226337
PicoLisp
This uses Knuth's algorithm (The Art of Computer Programming, Vol. 2, page 442) <lang PicoLisp>(de ** (X N) # N th power of X
(if (ge0 N)
(let Y 1
(loop
(when (bit? 1 N)
(setq Y (* Y X)) )
(T (=0 (setq N (>> 1 N)))
Y )
(setq X (* X X)) ) )
0 ) )</lang>
PL/I
<lang pli>declare exp generic
(iexp when (fixed, fixed), fexp when (float, fixed) );
iexp: procedure (m, n) returns (fixed binary (31));
declare (m, n) fixed binary (31) nonassignable;
declare exp fixed binary (31) initial (m), i fixed binary;
if m = 0 & n = 0 then signal error;
if n = 0 then return (1);
do i = 2 to n;
exp = exp * m;
end;
return (exp);
end iexp; fexp: procedure (a, n) returns (float (15));
declare (a float, n fixed binary (31)) nonassignable;
declare exp float initial (a), i fixed binary;
if a = 0 & n = 0 then signal error;
if n = 0 then return (1);
do i = 2 to n;
exp = exp * a;
end;
return (exp);
end fexp;</lang>
PowerShell
<lang powershell>function pow($a, [int]$b) {
if ($b -eq -1) { return 1/$a }
if ($b -eq 0) { return 1 }
if ($b -eq 1) { return $a }
if ($b -lt 0) {
$rec = $true # reciprocal needed
$b = -$b
}
$result = $a
2..$b | ForEach-Object {
$result *= $a
}
if ($rec) {
return 1/$result
} else {
return $result
}
}</lang> The function works for both integers and floating-point values as first argument.
PowerShell does not support operator overloading directly (and there wouldn't be an exponentiation operator to overload).
- Output:
PS> pow 2 15 32768 PS> pow 2.71 -4 0,018540559532257 PS> pow (-1.35) 3 −2,460375
The negative first argument needs to be put in parentheses because it would otherwise be passed as string.
This can be circumvented by declaring the first argument to the function as double, but then the return type would be always double while currently pow 2 3 returns an int.
PureBasic
PureBasic does not allow an operator to be redefined or operator overloading. <lang PureBasic>Procedure powI(base, exponent)
Protected i, result.d
If exponent < 0
If base = 1
result = 1
EndIf
ProcedureReturn result
EndIf
result = 1
For i = 1 To exponent
result * base
Next
ProcedureReturn result
EndProcedure
Procedure.f powF(base.f, exponent)
Protected i, magExponent = Abs(exponent), result.d
If base <> 0
result = 1.0
If exponent <> 0
For i = 1 To magExponent
result * base
Next
If exponent < 0
result = 1.0 / result
EndIf
EndIf
EndIf
ProcedureReturn result
EndProcedure
If OpenConsole()
Define x, a.f, exp
x = Random(10) - 5
a = Random(10000) / 10000 * 10
For exp = -3 To 3
PrintN(Str(x) + " ^ " + Str(exp) + " = " + Str(powI(x, exp)))
PrintN(StrF(a) + " ^ " + Str(exp) + " = " + StrF(powF(a, exp)))
PrintN("--------------")
Next
Print(#CRLF$ + #CRLF$ + "Press ENTER to exit")
Input()
CloseConsole()
EndIf</lang>
- Output:
-3 ^ -3 = 0 6.997000 ^ -3 = 0.002919 -------------- -3 ^ -2 = 0 6.997000 ^ -2 = 0.020426 -------------- -3 ^ -1 = 0 6.997000 ^ -1 = 0.142918 -------------- -3 ^ 0 = 1 6.997000 ^ 0 = 1.000000 -------------- -3 ^ 1 = -3 6.997000 ^ 1 = 6.997000 -------------- -3 ^ 2 = 9 6.997000 ^ 2 = 48.958012 -------------- -3 ^ 3 = -27 6.997000 ^ 3 = 342.559235 -------------
Prolog
Declaring an Operator as an Arithmetic Function
In Prolog, we define predicates rather than functions. Still, functions and predicates are related: going one way, we can think of an n-place predicate as a function from its arguments to a member of the set {true, false}; going the other way, we can think of functions as predicates with a hidden ultimate argument, called a "return value". Following the latter approach, Prolog sometimes uses macro expansion to provide functional syntax by
1. catching terms fitting a certain pattern (viz. Base ^^ Exp, which is the same as '^^'(N, 3)),
2. calling the term with an extra argument (viz. call('^^'(Base, Exp), Power)),
3. replacing the occurrence of the term with the value instantiated in the extra argument (viz. Power).
The predicate is/2 supports functional syntax in its second argument: e.g., X is sqrt(2) + 1. New arithmetic functions can be added with the `arithmetic_function/1` directive, wherein the arity attributed to the function is one less than the arity of the predicate which will be called during term expansion and evaluation. The following directives establish ^^/2 as, first, an arithmetic function, and then as a right-associative binary operator (so that X is 2^^2^^2 == X = 2^(2^2)):
<lang prolog>:- arithmetic_function((^^)/2).
- - op(200, xfy, user:(^^)).</lang>
When ^^/2 occurs in an expression in the second argument of is/2, Prolog calls the subsequently defined predicate ^^/3, and obtains the operators replacement value from the predicate's third argument.
Higher-order Predicate:
This solution employs the higher-order predicate foldl/4 from the standard SWI-Prolog library(apply), in conjunction with an auxiliary "folding predicate" (note, the definition uses the ^^ operator as an arithmetic function):
<lang prolog>%% ^^/3 % % True if Power is Base ^ Exp.
^^(Base, Exp, Power) :-
( Exp < 0 -> Power is 1 / (Base ^^ (Exp * -1)) % If exponent is negative, then ...
; Exp > 0 -> length(Powers, Exp), % If exponent is positive, then
foldl( exp_folder(Base), Powers, 1, Power ) % Powers is a list of free variables with length Exp
% and Power is Powers folded with exp_folder/4
; Power = 1 % otherwise Exp must be 0, so
).
%% exp_folder/4 % % True when Power is the product of Base and Powers. % % This predicate is designed to work with foldl and a list of free variables. % It passes the result of each evaluation to the next application through its % fourth argument, instantiating the elements of Powers to each successive Power of the Base.
exp_folder(Base, Power, Powers, Power) :-
Power is Base * Powers.</lang>
Example usage:
<lang prolog>?- X is 2 ^^ 3. X = 8.
?- X is 2 ^^ -3. X = 0.125.
?- X is 2.5 ^^ -3. X = 0.064.
?- X is 2.5 ^^ 3. X = 15.625.</lang>
Recursive Predicate
An implementation of exponentiation using recursion and no control predicates.
<lang prolog>exp_recursive(Base, NegExp, NegPower) :-
NegExp < 0, Exp is NegExp * -1, exp_recursive_(Base, Exp, Base, Power), NegPower is 1 / Power.
exp_recursive(Base, Exp, Power) :-
Exp > 0, exp_recursive_(Base, Exp, Base, Power).
exp_recursive(_, 0, 1).
exp_recursive_(_, 1, Power, Power). exp_recursive_(Base, Exp, Acc, Power) :-
Exp > 1, NewAcc is Base * Acc, NewExp is Exp - 1, exp_recursive_(Base, NewExp, NewAcc, Power).</lang>
Python
<lang python>MULTIPLY = lambda x, y: x*y
class num(float):
# the following method has complexity O(b)
# rather than O(log b) via the rapid exponentiation
def __pow__(self, b):
return reduce(MULTIPLY, [self]*b, 1)
- works with ints as function or operator
print num(2).__pow__(3) print num(2) ** 3
- works with floats as function or operator
print num(2.3).__pow__(8) print num(2.3) ** 8</lang>
R
<lang r># Method pow <- function(x, y) {
x <- as.numeric(x) y <- as.integer(y) prod(rep(x, y))
}
- Operator
"%pow%" <- function(x,y) pow(x,y)
pow(3, 4) # 81 2.5 %pow% 2 # 6.25</lang>
Racket
<lang Racket>#lang racket (define (^ base expt)
(for/fold ((acum 1)) ((i (in-range expt))) (* acum base)))
(^ 5 2) ; 25 (^ 5.0 2) ; 25.0</lang>
Retro
Retro has no floating point support in the standard VM.
From the math' vocabulary:
<lang Retro>: pow ( bp-n ) 1 swap [ over * ] times nip ;</lang>
And in use:
<lang Retro>2 5 ^math'pow</lang>
The fast exponentiation algorithm can be coded as follows:
<lang Retro>: pow ( n m -- n^m ) 1 2rot [ dup 1 and 0 <>
[ [ tuck * swap ] dip ] ifTrue [ dup * ] dip 1 >> dup 0 <>
] while drop drop ;</lang>
REXX
The iPow function doesn't care what kind of number is to be raised to a power,
it can be an integer or floating point number.
Extra error checking was added to verify that the invocation is syntactically correct. <lang rexx>/*REXX program computes and displays various (integer) exponentiations. */
say center('digits='digits(), 79, "─")
say '17**65 is:' say 17**65 say
numeric digits 100; say center('digits='digits(), 79, "─") say '17**65 is:' say 17**65 say
numeric digits 10; say center('digits='digits(), 79, "─") say '2 ** -10 is:' say 2 ** -10 say
numeric digits 30; say center('digits='digits(), 79, "─") say '-3.1415926535897932384626433 ** 3 is:' say -3.1415926535897932384626433 ** 3 say
numeric digits 1000; say center('digits='digits(), 79, "─") say '2 ** 1000 is:' say 2 ** 1000 say
numeric digits 60; say center('digits='digits(), 79, "─") say 'iPow(5, 70) is:' say iPow(5, 70) exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ errMsg: say; say '***error***'; say; say arg(1); say; say; exit 13 /*──────────────────────────────────────────────────────────────────────────────────────*/ iPow: procedure; parse arg x 1 _,p
if arg()<2 then call errMsg "not enough arguments specified"
if arg()>2 then call errMsg "too many arguments specified"
if \datatype(x,'N') then call errMsg "1st arg isn't numeric:" x
if \datatype(p,'W') then call errMsg "2nd arg isn't an integer:" p
if p=0 then return 1
do abs(p) - 1; _=_*x; end /*abs(p)-1*/
if p<0 then _=1/_
return _</lang>
- output
───────────────────────────────────digits=9──────────────────────────────────── 17**65 is: 9.53190909E+79 ──────────────────────────────────digits=100─────────────────────────────────── 17**65 is: 95319090450218007303742536355848761234066170796000792973413605849481890760893457 ───────────────────────────────────digits=10─────────────────────────────────── 2 ** -10 is: 0.0009765625 ───────────────────────────────────digits=30─────────────────────────────────── -3.1415926535897932384626433 ** 3 is: -31.0062766802998201754763126013 ──────────────────────────────────digits=1000────────────────────────────────── 2 ** 1000 is: 10715086071862673209484250490600018105614048117055336074437503883703510511249361224931983788156958581275946729175531468251871452856923140435984577574698574803934567774824230985421074605062371141877954182153046474983581941267398767559165543946077062914571196477686542167660429831652624386837205668069376 ───────────────────────────────────digits=60─────────────────────────────────── ipow(5,70) is: 8470329472543003390683225006796419620513916015625
Ring
<lang ring> see "11^5 = " + ipow(11, 5) + nl see "pi^3 = " + fpow(3.14, 3) + nl
func ipow a, b
p2 = 1
for i = 1 to 32
p2 *= p2
if b < 0 p2 *= a ok
b = b << 1
next
return p2
func fpow a, b
p = 1
for i = 1 to 32
p *= p
if b < 0 p *= a ok
b = b << 1
next
return p
</lang> Output :
11^5 = 161051 pi^3 = 30.96
Ruby
We add a pow method to Numeric objects. To calculate 5.pow 3, this method fills an array [5, 5, 5] and then multiplies together the elements.
<lang ruby>class Numeric
def pow(m)
raise TypeError, "exponent must be an integer: #{m}" unless m.is_a? Integer
puts "pow!!"
Array.new(m, self).reduce(1, :*)
end
end
p 5.pow(3) p 5.5.pow(3) p 5.pow(3.1)</lang>
- Output:
pow!!
125
pow!!
166.375
pow.rb:3:in `pow': exponent must be an integer: 3.1 (TypeError)
from pow.rb:16:in `<main>'
To overload the ** exponentiation operator, this might work, but doesn't: <lang ruby>class Numeric
def **(m) pow(m) end
end</lang> It doesn't work because the ** method is defined independently for Numeric subclasses Fixnum, Bignum and Float. One must: <lang ruby>class Fixnum
def **(m) print "Fixnum " pow(m) end
end class Bignum
def **(m) print "Bignum " pow(m) end
end class Float
def **(m) print "Float " pow(m) end
end
p i=2**64 p i ** 2 p 2.2 ** 3</lang>
- Output:
Fixnum pow!! 18446744073709551616 Bignum pow!! 340282366920938463463374607431768211456 Float pow!! 10.648
Run BASIC
<lang runbasic>print " 11^5 = ";11^5 print " (-11)^5 = ";-11^5 print " 11^( -5) = ";11^-5 print " 3.1416^3 = ";3.1416^3 print " 0^2 = ";0^2 print " 2^0 = ";2^0 print " -2^0 = ";-2^0</lang>
- Output:
11^5 = 161051 (-11)^5 = -161051 11^( -5) = 6.20921325e-6 3.1416^3 = 31.0064942 0^2 = 0 2^0 = 1 -2^0 = 1
Rust
The num crate is the de-facto Rust library for numerical generics and it provides the One trait which allows for an exponentiation function that is generic over both integral and floating point types. The library provides this generic exponentiation function, the implementation of which is the pow function below.
<lang rust>extern crate num;
use num::traits::One;
use std::ops::Mul;
fn pow<T>(mut base: T, mut exp: usize) -> T
where T: Clone + One + Mul<T, Output=T>
{
if exp == 0 { return T::one() }
while exp & 1 == 0 {
base = base.clone() * base;
exp >>= 1;
}
if exp == 1 { return base }
let mut acc = base.clone();
while exp > 1 {
exp >>= 1;
base = base.clone() * base;
if exp & 1 == 1 {
acc = acc * base.clone();
}
}
acc
}</lang>
Scala
There's no distinction between an operator and a method in Scala. Alas, there is no way of adding methods to a class, but one can make it look like a method has been added, through a method commonly known as Pimp My Library. Therefore, we show below how that can beaccomplished. We define the operator ↑ (unicode's uparrow), which is written as \u2191 below, to make cut & paste easier.
To use it, one has to import the implicit from the appropriate object. ExponentI will work for any integral type (Int, BigInt, etc), ExponentF will work for any fractional type (Double, BigDecimal, etc). Importing both at the same time won't work. In this case, it might be better to define implicits for the actual types being used, such as was done in Exponents. <lang scala>object Exponentiation {
import scala.annotation.tailrec
@tailrec def powI[N](n: N, exponent: Int)(implicit num: Integral[N]): N = {
import num._
exponent match {
case 0 => one
case _ if exponent % 2 == 0 => powI((n * n), (exponent / 2))
case _ => powI(n, (exponent - 1)) * n
}
}
@tailrec def powF[N](n: N, exponent: Int)(implicit num: Fractional[N]): N = {
import num._
exponent match {
case 0 => one
case _ if exponent < 0 => one / powF(n, exponent.abs)
case _ if exponent % 2 == 0 => powF((n * n), (exponent / 2))
case _ => powF(n, (exponent - 1)) * n
}
}
class ExponentI[N : Integral](n: N) {
def \u2191(exponent: Int): N = powI(n, exponent)
}
class ExponentF[N : Fractional](n: N) {
def \u2191(exponent: Int): N = powF(n, exponent)
}
object ExponentI {
implicit def toExponentI[N : Integral](n: N): ExponentI[N] = new ExponentI(n)
}
object ExponentF {
implicit def toExponentF[N : Fractional](n: N): ExponentF[N] = new ExponentF(n)
}
object Exponents {
implicit def toExponent(n: Int): ExponentI[Int] = new ExponentI(n)
implicit def toExponent(n: Double): ExponentF[Double] = new ExponentF(n)
}
}</lang>Functions powI and powF above are not tail recursive, since the result of the recursive call is multiplied by n. A tail recursive version of powI would be: <lang scala> @tailrec def powI[N](n: N, exponent: Int, acc:Int=1)(implicit num: Integral[N]): N = {
exponent match {
case 0 => acc
case _ if exponent % 2 == 0 => powI(n * n, exponent / 2, acc)
case _ => powI(n, (exponent - 1), acc*n)
}
}</lang>
Scheme
This definition of the exponentiation procedure ^ operates on bases of all numerical types that the multiplication procedure * operates on, i. e. integer, rational, real, and complex. The notion of an operator does not exist in Scheme. Application of a procedure to its arguments is always expressed with a prefix notation.
<lang scheme>(define (^ base exponent)
(define (*^ exponent acc)
(if (= exponent 0)
acc
(*^ (- exponent 1) (* acc base))))
(*^ exponent 1))
(display (^ 2 3)) (newline) (display (^ (/ 1 2) 3)) (newline) (display (^ 0.5 3)) (newline) (display (^ 2+i 3)) (newline)</lang>
- Output:
8 1/8 0.125 2+11i
Seed7
In Seed7 the ** operator is overloaded for both integerinteger and floatinteger (additionally there is a ** operator for floatfloat). The following re-implementation of both functions does not use another exponentiation function to do the computation. Instead the exponentiation-by-squaring algorithm is used.
<lang seed7>const func integer: intPow (in var integer: base, in var integer: exponent) is func
result
var integer: result is 0;
begin
if exponent < 0 then
raise(NUMERIC_ERROR);
else
if odd(exponent) then
result := base;
else
result := 1;
end if;
exponent := exponent div 2;
while exponent <> 0 do
base *:= base;
if odd(exponent) then
result *:= base;
end if;
exponent := exponent div 2;
end while;
end if;
end func;</lang>
Original source: [1]
<lang seed7>const func float: fltIPow (in var float: base, in var integer: exponent) is func
result
var float: power is 1.0;
local
var integer: stop is 0;
begin
if base = 0.0 then
if exponent < 0 then
power := Infinity;
elsif exponent > 0 then
power := 0.0;
end if;
else
if exponent < 0 then
stop := -1;
end if;
if odd(exponent) then
power := base;
end if;
exponent >>:= 1;
while exponent <> stop do
base *:= base;
if odd(exponent) then
power *:= base;
end if;
exponent >>:= 1;
end while;
if stop = -1 then
power := 1.0 / power;
end if;
end if;
end func;</lang>
Original source: [2]
Since Seed7 supports operator and function overloading a new exponentiation operator like ^* can be defined for integer and float bases:
<lang seed7>$ syntax expr: .(). ^* .() is <- 4;
const func integer: (in var integer: base) ^* (in var integer: exponent) is
return intPow(base, exponent);
const func float: (in var float: base) ^* (in var integer: exponent) is
return fltIPow(base, exponent);</lang>
Sidef
Function definition: <lang ruby>func expon(_, {.is_zero}) { 1 }
func expon(base, exp {.is_neg}) {
expon(1/base, -exp)
}
func expon(base, exp {.is_int}) {
var c = 1
while (exp > 1) {
c *= base if exp.is_odd
base *= base
exp >>= 1
}
return (base * c)
}
say expon(3, 10) say expon(5.5, -3)</lang>
Operator definition: <lang ruby>class Number {
method ⊙(exp) {
expon(self, exp)
}
}
say (3 ⊙ 10) say (5.5 ⊙ -3)</lang>
- Output:
59049 0.006010518407212622088655146506386175807661607813673929376408715251690458302028550142749812171299774605559729526671675432
Slate
This code is from the current slate implementation: <lang slate>x@(Number traits) raisedTo: y@(Integer traits) [
y isZero ifTrue: [^ x unit].
x isZero \/ [y = 1] ifTrue: [^ x].
y isPositive
ifTrue:
"(x * x raisedTo: y // 2) * (x raisedTo: y \\ 2)"
[| count result |
count: 1.
[(count: (count bitShift: 1)) < y] whileTrue.
result: x unit.
[count isPositive]
whileTrue: [result: result squared. (y bitAnd: count) isZero ifFalse: [result: result * x]. count: (count bitShift: -1)].
result] ifFalse: [(x raisedTo: y negated) reciprocal]
].</lang>
For floating numbers:
<lang slate>x@(Float traits) raisedTo: y@(Float traits) "Implements floating-point exponentiation in terms of the natural logarithm and exponential primitives - this is generally faster than the naive method." [
y isZero ifTrue: [^ x unit]. x isZero \/ [y isUnit] ifTrue: [^ x]. (x ln * y) exp
].</lang>
Smalltalk
Extending the class Number, we provide the operator for integers, floating points, rationals numbers (and any other derived class) <lang smalltalk>Number extend [
** anInt [
| r |
( anInt isInteger )
ifFalse:
[ '** works fine only for integer powers'
displayOn: stderr . Character nl displayOn: stderr ].
r := 1.
1 to: anInt do: [ :i | r := ( r * self ) ].
^r
]
].
( 2.5 ** 3 ) displayNl. ( 2 ** 10 ) displayNl. ( 3/7 ** 3 ) displayNl.</lang>
- Output:
15.625 1024 27/343
Standard ML
The following operators only take nonnegative integer exponents. <lang sml>fun expt_int (a, b) = let
fun aux (x, i) = if i = b then x else aux (x * a, i + 1)
in
aux (1, 0)
end
fun expt_real (a, b) = let
fun aux (x, i) = if i = b then x else aux (x * a, i + 1)
in
aux (1.0, 0)
end
val op ** = expt_int infix 6 ** val op *** = expt_real infix 6 ***</lang>
<lang sml>- 2 ** 3; val it = 8 : int - 2.4 *** 3; val it = 13.824 : real</lang>
Stata
<lang stata>mata function pow(a, n) { x = a for(p=1; n>0; n=floor(n/2)) { if(mod(n,2)==1) p = p*x x = x*x } return(p) } end</lang>
Swift
Defines generic function raise(_:to:) and operator ** that will work with all bases conforming to protocol Numeric, including Float and Int. <lang Swift>func raise<T: Numeric>(_ base: T, to exponent: Int) -> T {
precondition(exponent >= 0, "Exponent has to be nonnegative") return Array(repeating: base, count: exponent).reduce(1, *)
}
infix operator **: MultiplicationPrecedence
func **<T: Numeric>(lhs: T, rhs: Int) -> T {
return raise(lhs, to: rhs)
}
let someFloat: Float = 2 let someInt: Int = 10
assert(raise(someFloat, to: someInt) == 1024) assert(someFloat ** someInt == 1024) assert(raise(someInt, to: someInt) == 10000000000) assert(someInt ** someInt == 10000000000)</lang>
Tcl
Tcl already has both an exponentiation function (set x [expr {pow(2.4, 3.5)}]) and operator (set x [expr {2.4 ** 3.5}]). The operator cannot be overridden. The function may be overridden by a procedure in the tcl::mathfunc namespace, relative to
the calling namespace.
This solution does not consider negative exponents. <lang tcl>package require Tcl 8.5 proc tcl::mathfunc::mypow {a b} {
if { ! [string is int -strict $b]} {error "exponent must be an integer"}
set res 1
for {set i 1} {$i <= $b} {incr i} {set res [expr {$res * $a}]}
return $res
} expr {mypow(3, 3)} ;# ==> 27 expr {mypow(3.5, 3)} ;# ==> 42.875 expr {mypow(3.5, 3.2)} ;# ==> exponent must be an integer</lang>
Ursa
<lang ursa># these implementations ignore negative exponents def intpow (int m, int n) if (< n 1) return 1 end if decl int ret set ret 1 for () (> n 0) (dec n) set ret (* ret m) end for return ret end intpow
def floatpow (double m, int n) if (or (< n 1) (and (= m 0) (= n 0))) return 1 end if decl int ret set ret 1 for () (> n 0) (dec n) set ret (* ret m) end for return ret end floatpow</lang>
VBA
<lang vb>Public Function exp(ByVal base As Variant, ByVal exponent As Long) As Variant
Dim result As Variant
If TypeName(base) = "Integer" Or TypeName(base) = "Long" Then
'integer exponentiation
result = 1
If exponent < 0 Then
result = IIf(Abs(base) <> 1, CVErr(2019), IIf(exponent Mod 2 = -1, base, 1))
End If
Do While exponent > 0
If exponent Mod 2 = 1 Then result = result * base
base = base * base
exponent = exponent \ 2
Loop
Else
Debug.Assert IsNumeric(base)
'float exponentiation
If base = 0 Then
If exponent < 0 Then result = CVErr(11)
Else
If exponent < 0 Then
base = 1# / base
exponent = -exponent
End If
result = 1
Do While exponent > 0
If exponent Mod 2 = 1 Then result = result * base
base = base * base
exponent = exponent \ 2
Loop
End If
End If
exp = result
End Function Public Sub main()
Debug.Print "Integer exponentiation" Debug.Print "10^7=", exp(10, 7) Debug.Print "10^4=", exp(10, 4) Debug.Print "(-3)^3=", exp(-3, 3) Debug.Print "(-1)^(-5)=", exp(-1, -5) Debug.Print "10^(-1)=", exp(10, -1) Debug.Print "0^2=", exp(0, 2) Debug.Print "Float exponentiation" Debug.Print "10.0^(-3)=", exp(10#, -3) Debug.Print "10.0^(-4)=", exp(10#, -4) Debug.Print "(-3.0)^(-5)=", exp(-3#, -5) Debug.Print "(-3.0)^(-4)=", exp(-3#, -4) Debug.Print "0.0^(-4)=", exp(0#, -4)
End Sub</lang>
- Output:
Integer exponentiation 10^7= 10000000 10^4= 10000 (-3)^3= -27 (-1)^(-5)= -1 10^(-1)= Fout 2019 0^2= 0 Float exponentiation 10.0^(-3)= 0,001 10.0^(-4)= 0,0001 (-3.0)^(-5)= -4,11522633744856E-03 (-3.0)^(-4)= 1,23456790123457E-02 0.0^(-4)= Fout 11
VBScript
<lang vb> Function pow(x,y) pow = 1 If y < 0 Then For i = 1 To Abs(y) pow = pow * (1/x) Next Else For i = 1 To y pow = pow * x Next End If End Function
WScript.StdOut.Write "2 ^ 0 = " & pow(2,0) WScript.StdOut.WriteLine WScript.StdOut.Write "7 ^ 6 = " & pow(7,6) WScript.StdOut.WriteLine WScript.StdOut.Write "3.14159265359 ^ 9 = " & pow(3.14159265359,9) WScript.StdOut.WriteLine WScript.StdOut.Write "4 ^ -6 = " & pow(4,-6) WScript.StdOut.WriteLine WScript.StdOut.Write "-3 ^ 5 = " & pow(-3,5) WScript.StdOut.WriteLine </lang>
- Output:
2 ^ 0 = 1 7 ^ 6 = 117649 3.14159265359 ^ 9 = 29809.0993334639 4 ^ -6 = 0.000244140625 -3 ^ 5 = -243
XPL0
To create an exponent operator you need to modify the compiler code, which is open source. <lang XPL0>include c:\cxpl\codes; \intrinsic 'code' declarations
func real Power(X, Y); \X raised to the Y power; (X > 0.0) real X; int Y; return Exp(float(Y) * Ln(X));
func IPower(X, Y); \X raised to the Y power int X, Y; int P; [P:= 1; while Y do
[if Y&1 then P:= P*X; X:= X*X; Y:= Y>>1; ];
return P; ];
int X, Y; [Format(9, 0); for X:= 1 to 10 do
[for Y:= 0 to 7 do
RlOut(0, Power(float(X), Y));
CrLf(0);
];
CrLf(0); for X:= 1 to 10 do
[for Y:= 0 to 7 do
[ChOut(0, 9); IntOut(0, IPower(X, Y))];
CrLf(0);
];
]</lang>
- Output:
1 1 1 1 1 1 1 1
1 2 4 8 16 32 64 128
1 3 9 27 81 243 729 2187
1 4 16 64 256 1024 4096 16384
1 5 25 125 625 3125 15625 78125
1 6 36 216 1296 7776 46656 279936
1 7 49 343 2401 16807 117649 823543
1 8 64 512 4096 32768 262144 2097152
1 9 81 729 6561 59049 531441 4782969
1 10 100 1000 10000 100000 1000000 10000000
1 1 1 1 1 1 1 1
1 2 4 8 16 32 64 128
1 3 9 27 81 243 729 2187
1 4 16 64 256 1024 4096 16384
1 5 25 125 625 3125 15625 78125
1 6 36 216 1296 7776 46656 279936
1 7 49 343 2401 16807 117649 823543
1 8 64 512 4096 32768 262144 2097152
1 9 81 729 6561 59049 531441 4782969
1 10 100 1000 10000 100000 1000000 10000000
zkl
Int and Float have pow methods and zkl doesn't allow you to add operators, classes can implement existing ones.
<lang zkl>fcn pow(n,exp){
reg v;
if(n.isType(1)){ // Int
if (exp<0) return(if(n*n!=1) 0 else (if(exp.isOdd) n else 1));
v=1;
}else{
if(exp<0){ n=1.0/n; exp=-exp; }
v=1.0;
}
while(exp>0){
if(exp.isOdd) v*=n;
n*=n;
exp/=2;
}
v
}</lang> <lang zkl>println("2^6 = %d".fmt(pow(2,6))); println("2^-6 = %d".fmt(pow(2,-6))); println("2.71^6 = %f".fmt(pow(2.71,6))); println("2.71^-6 = %f".fmt(pow(2.71,-6)));</lang>
- Output:
2^6 = 64 2^-6 = 0 2.71^6 = 396.109944 2.71^-6 = 0.002525
ZX Spectrum Basic
ZX Spectrum Basic does not support custom operators or integer datatypes, but here we implement exponentation using a function. The function itself makes use of the inbuilt exponentiation operator, which is kind of cheating, but hey this provides a working implementation.
<lang zxbasic>10 PRINT e(3,2): REM 3 ^ 2 20 PRINT e(1.5,2.7): REM 1.5 ^ 2.7 30 STOP 9950 DEF FN e(a,b)=a^b</lang>
- Programming Tasks
- Arithmetic operations
- Arithmetic
- Ada
- ALGOL 68
- AutoHotkey
- AWK
- BASIC
- BBC BASIC
- IS-BASIC
- Befunge
- Brat
- C
- C sharp
- C++
- Chef
- Clojure
- Common Lisp
- D
- E
- EchoLisp
- Ela
- Elixir
- Erlang
- ERRE
- F Sharp
- Factor
- Forth
- Fortran
- FreeBASIC
- GAP
- Go
- Haskell
- HicEst
- Icon
- Unicon
- J
- Java
- JavaScript
- Jq
- Julia
- Kotlin
- Liberty BASIC
- Lingo
- Logo
- Lua
- Lucid
- M2000 Interpreter
- M4
- Mathematica
- Wolfram Language
- Maxima
- МК-61/52
- Modula-2
- Modula-3
- Nemerle
- Nim
- Objeck
- OCaml
- Oforth
- PARI/GP
- Pascal
- Perl
- Perl 6
- Phix
- PicoLisp
- PL/I
- PowerShell
- PureBasic
- Prolog
- Python
- R
- Racket
- Retro
- REXX
- Ring
- Ruby
- Run BASIC
- Rust
- Scala
- Scala examples needing attention
- Examples needing attention
- Scheme
- Seed7
- Sidef
- Slate
- Smalltalk
- Standard ML
- Stata
- Swift
- Tcl
- Ursa
- VBA
- VBScript
- XPL0
- Zkl
- ZX Spectrum Basic