Gray code
You are encouraged to solve this task according to the task description, using any language you may know.
Gray code is a form of binary encoding where transitions between consecutive numbers differ by only one bit. This is a useful encoding for reducing hardware data hazards with values that change rapidly and/or connect to slower hardware as inputs. It is also useful for generating inputs for Karnaugh maps in order from left to right or top to bottom.
Create functions to encode a number to and decode a number from Gray code.
Display the normal binary representations, Gray code representations, and decoded Gray code values for all 5-bit binary numbers (0-31 inclusive, leading 0's not necessary).
There are many possible Gray codes. The following encodes what is called "binary reflected Gray code."
Encoding (MSB is bit 0, b is binary, g is Gray code):
if b[i-1] = 1 g[i] = not b[i] else g[i] = b[i]
Or:
g = b xor (b logically right shifted 1 time)
Decoding (MSB is bit 0, b is binary, g is Gray code):
b[0] = g[0] for other bits: b[i] = g[i] xor b[i-1]
- Reference
- Converting Between Gray and Binary Codes. It includes step-by-step animations.
11l
<lang 11l>F gray_encode(n)
R n (+) n >> 1
F gray_decode(=n)
V m = n >> 1
L m != 0
n (+)= m
m >>= 1
R n
print(‘DEC, BIN => GRAY => DEC’) L(i) 32
V gray = gray_encode(i) V dec = gray_decode(gray) print(‘ #2, #. => #. => #2’.format(i, bin(i).zfill(5), bin(gray).zfill(5), dec))</lang>
- Output:
DEC, BIN => GRAY => DEC 0, 00000 => 00000 => 0 1, 00001 => 00001 => 1 2, 00010 => 00011 => 2 3, 00011 => 00010 => 3 4, 00100 => 00110 => 4 5, 00101 => 00111 => 5 6, 00110 => 00101 => 6 7, 00111 => 00100 => 7 8, 01000 => 01100 => 8 9, 01001 => 01101 => 9 10, 01010 => 01111 => 10 11, 01011 => 01110 => 11 12, 01100 => 01010 => 12 13, 01101 => 01011 => 13 14, 01110 => 01001 => 14 15, 01111 => 01000 => 15 16, 10000 => 11000 => 16 17, 10001 => 11001 => 17 18, 10010 => 11011 => 18 19, 10011 => 11010 => 19 20, 10100 => 11110 => 20 21, 10101 => 11111 => 21 22, 10110 => 11101 => 22 23, 10111 => 11100 => 23 24, 11000 => 10100 => 24 25, 11001 => 10101 => 25 26, 11010 => 10111 => 26 27, 11011 => 10110 => 27 28, 11100 => 10010 => 28 29, 11101 => 10011 => 29 30, 11110 => 10001 => 30 31, 11111 => 10000 => 31
8080 Assembly
<lang 8080asm> org 100h xra a ; set A=0 loop: push psw ; print number as decimal call decout call padding ; print padding pop psw push psw call binout ; print number as binary call padding pop psw ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; mov b,a ; gray encode ana a ; clear carry rar ; shift right xra b ; xor the original ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; push psw call binout ; print gray number as binary call padding pop psw ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; mov b,a ; gray decode decode: ana a ; clear carry jz done ; when no more bits are left, stop rar ; shift right mov c,a ; keep that value xra b ; xor into output value mov b,a ; that is the output value mov a,c ; restore intermediate jmp decode ; do next bit done: mov a,b ; give output value ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; push psw call binout ; print decoded number as binary call padding pop psw push psw call decout ; print decoded number as decimal lxi d,nl call strout pop psw inr a ; next number ani 1fh ; are we there yet? jnz loop ; if not, do next number ret ;; Print A as two-digit number decout: mvi c,10 call dgtout mvi c,1 dgtout: mvi e,'0' - 1 dgtloop: inr e sub c jnc dgtloop add c push psw mvi c,2 call 5 pop psw ret ;; Print A as five-bit binary number binout: ani 1fh ral ral ral mvi c,5 binloop: ral push psw push b mvi c,2 mvi a,0 aci '0' mov e,a call 5 pop b pop psw dcr c jnz binloop ret ;; Print padding padding: lxi d,arrow strout: mvi c,9 jmp 5 arrow: db ' ==> $' nl: db 13,10,'$'</lang>
- Output:
00 ==> 00000 ==> 00000 ==> 00000 ==> 00 01 ==> 00001 ==> 00001 ==> 00001 ==> 01 02 ==> 00010 ==> 00011 ==> 00010 ==> 02 03 ==> 00011 ==> 00010 ==> 00011 ==> 03 04 ==> 00100 ==> 00110 ==> 00100 ==> 04 05 ==> 00101 ==> 00111 ==> 00101 ==> 05 06 ==> 00110 ==> 00101 ==> 00110 ==> 06 07 ==> 00111 ==> 00100 ==> 00111 ==> 07 08 ==> 01000 ==> 01100 ==> 01000 ==> 08 09 ==> 01001 ==> 01101 ==> 01001 ==> 09 10 ==> 01010 ==> 01111 ==> 01010 ==> 10 11 ==> 01011 ==> 01110 ==> 01011 ==> 11 12 ==> 01100 ==> 01010 ==> 01100 ==> 12 13 ==> 01101 ==> 01011 ==> 01101 ==> 13 14 ==> 01110 ==> 01001 ==> 01110 ==> 14 15 ==> 01111 ==> 01000 ==> 01111 ==> 15 16 ==> 10000 ==> 11000 ==> 10000 ==> 16 17 ==> 10001 ==> 11001 ==> 10001 ==> 17 18 ==> 10010 ==> 11011 ==> 10010 ==> 18 19 ==> 10011 ==> 11010 ==> 10011 ==> 19 20 ==> 10100 ==> 11110 ==> 10100 ==> 20 21 ==> 10101 ==> 11111 ==> 10101 ==> 21 22 ==> 10110 ==> 11101 ==> 10110 ==> 22 23 ==> 10111 ==> 11100 ==> 10111 ==> 23 24 ==> 11000 ==> 10100 ==> 11000 ==> 24 25 ==> 11001 ==> 10101 ==> 11001 ==> 25 26 ==> 11010 ==> 10111 ==> 11010 ==> 26 27 ==> 11011 ==> 10110 ==> 11011 ==> 27 28 ==> 11100 ==> 10010 ==> 11100 ==> 28 29 ==> 11101 ==> 10011 ==> 11101 ==> 29 30 ==> 11110 ==> 10001 ==> 11110 ==> 30 31 ==> 11111 ==> 10000 ==> 11111 ==> 31
Action!
<lang Action!>PROC ToBinaryStr(BYTE n CHAR ARRAY s)
BYTE i
s(0)=8 i=8 SetBlock(s+1,8,'0) WHILE n DO s(i)=(n&1)+'0 n==RSH 1 i==-1 OD
RETURN
PROC PrintB2(BYTE n)
IF n<10 THEN Put(32) FI PrintB(n)
RETURN
PROC PrintBin5(BYTE n)
CHAR ARRAY s(9),sub(6)
ToBinaryStr(n,s) SCopyS(sub,s,4,s(0)) Print(sub)
RETURN
BYTE FUNC Encode(BYTE n) RETURN (n XOR (n RSH 1))
BYTE FUNC Decode(BYTE n)
BYTE res
res=n
DO
n==RSH 1
IF n THEN
res==XOR n
ELSE
EXIT
FI
OD
RETURN (res)
PROC Main()
BYTE i,g,b CHAR ARRAY sep=" -> "
FOR i=0 TO 31 DO PrintB2(i) Print(sep) PrintBin5(i) Print(sep) g=Encode(i) PrintBin5(g) Print(sep) b=Decode(g) PrintBin5(b) Print(sep) PrintB2(b) PutE() OD
RETURN</lang>
- Output:
Screenshot from Atari 8-bit computer
0 -> 00000 -> 00000 -> 00000 -> 0 1 -> 00001 -> 00001 -> 00001 -> 1 2 -> 00010 -> 00011 -> 00010 -> 2 3 -> 00011 -> 00010 -> 00011 -> 3 4 -> 00100 -> 00110 -> 00100 -> 4 5 -> 00101 -> 00111 -> 00101 -> 5 6 -> 00110 -> 00101 -> 00110 -> 6 7 -> 00111 -> 00100 -> 00111 -> 7 8 -> 01000 -> 01100 -> 01000 -> 8 9 -> 01001 -> 01101 -> 01001 -> 9 10 -> 01010 -> 01111 -> 01010 -> 10 11 -> 01011 -> 01110 -> 01011 -> 11 12 -> 01100 -> 01010 -> 01100 -> 12 13 -> 01101 -> 01011 -> 01101 -> 13 14 -> 01110 -> 01001 -> 01110 -> 14 15 -> 01111 -> 01000 -> 01111 -> 15 16 -> 10000 -> 11000 -> 10000 -> 16 17 -> 10001 -> 11001 -> 10001 -> 17 18 -> 10010 -> 11011 -> 10010 -> 18 19 -> 10011 -> 11010 -> 10011 -> 19 20 -> 10100 -> 11110 -> 10100 -> 20 21 -> 10101 -> 11111 -> 10101 -> 21 22 -> 10110 -> 11101 -> 10110 -> 22 23 -> 10111 -> 11100 -> 10111 -> 23 24 -> 11000 -> 10100 -> 11000 -> 24 25 -> 11001 -> 10101 -> 11001 -> 25 26 -> 11010 -> 10111 -> 11010 -> 26 27 -> 11011 -> 10110 -> 11011 -> 27 28 -> 11100 -> 10010 -> 11100 -> 28 29 -> 11101 -> 10011 -> 11101 -> 29 30 -> 11110 -> 10001 -> 11110 -> 30 31 -> 11111 -> 10000 -> 11111 -> 31
Ada
Demonstrates the use of shift operators. Code scalable to 6, 7 or 8 bits. Values are implemented with 8 bits according to representation clause of Unsigned_8 (check package Interfaces). <lang Ada>with Ada.Text_IO, Interfaces; use Ada.Text_IO, Interfaces;
procedure Gray is
Bits : constant := 5; -- Change only this line for 6 or 7-bit encodings subtype Values is Unsigned_8 range 0 .. 2 ** Bits - 1; package Values_Io is new Ada.Text_IO.Modular_IO (Values);
function Encode (Binary : Values) return Values is
begin
return Binary xor Shift_Right (Binary, 1);
end Encode;
pragma Inline (Encode);
function Decode (Gray : Values) return Values is
Binary, Bit : Values;
Mask : Values := 2 ** (Bits - 1);
begin
Bit := Gray and Mask;
Binary := Bit;
for I in 2 .. Bits loop
Bit := Shift_Right (Bit, 1);
Mask := Shift_Right (Mask, 1);
Bit := (Gray and Mask) xor Bit;
Binary := Binary + Bit;
end loop;
return Binary;
end Decode;
pragma Inline (Decode);
HT : constant Character := Character'Val (9); J : Values;
begin
Put_Line ("Num" & HT & "Binary" & HT & HT & "Gray" & HT & HT & "decoded");
for I in Values'Range loop
J := Encode (I);
Values_Io.Put (I, 4);
Put (": " & HT);
Values_Io.Put (I, Bits + 2, 2);
Put (" =>" & HT);
Values_Io.Put (J, Bits + 2, 2);
Put (" => " & HT);
Values_Io.Put (Decode (J), 4);
New_Line;
end loop;
end Gray;</lang>
Check compactness of assembly code generated by GNAT :http://pastebin.com/qtNjeQk9
- Output:
Num Binary Gray decoded 0: 2#0# => 2#0# => 0 1: 2#1# => 2#1# => 1 2: 2#10# => 2#11# => 2 3: 2#11# => 2#10# => 3 4: 2#100# => 2#110# => 4 5: 2#101# => 2#111# => 5 6: 2#110# => 2#101# => 6 7: 2#111# => 2#100# => 7 8: 2#1000# => 2#1100# => 8 9: 2#1001# => 2#1101# => 9 10: 2#1010# => 2#1111# => 10 11: 2#1011# => 2#1110# => 11 12: 2#1100# => 2#1010# => 12 13: 2#1101# => 2#1011# => 13 14: 2#1110# => 2#1001# => 14 15: 2#1111# => 2#1000# => 15 16: 2#10000# => 2#11000# => 16 17: 2#10001# => 2#11001# => 17 18: 2#10010# => 2#11011# => 18 19: 2#10011# => 2#11010# => 19 20: 2#10100# => 2#11110# => 20 21: 2#10101# => 2#11111# => 21 22: 2#10110# => 2#11101# => 22 23: 2#10111# => 2#11100# => 23 24: 2#11000# => 2#10100# => 24 25: 2#11001# => 2#10101# => 25 26: 2#11010# => 2#10111# => 26 27: 2#11011# => 2#10110# => 27 28: 2#11100# => 2#10010# => 28 29: 2#11101# => 2#10011# => 29 30: 2#11110# => 2#10001# => 30 31: 2#11111# => 2#10000# => 31
Aime
<lang aime>integer gray_encode(integer n) {
n ^ (n >> 1);
}
integer gray_decode(integer n) {
integer p;
p = n;
while (n >>= 1) {
p ^= n;
}
p;
}</lang> Demonstration code: <lang aime>integer main(void) {
integer i, g, b;
i = 0;
while (i < 32) {
g = gray_encode(i);
b = gray_decode(g);
o_winteger(2, i);
o_text(": ");
o_fxinteger(5, 2, i);
o_text(" => ");
o_fxinteger(5, 2, g);
o_text(" => ");
o_fxinteger(5, 2, b);
o_text(": ");
o_winteger(2, b);
o_byte('\n');
i += 1;
}
return 0;
}</lang>
- Output:
0: 00000 => 00000 => 00000: 0 1: 00001 => 00001 => 00001: 1 2: 00010 => 00011 => 00010: 2 3: 00011 => 00010 => 00011: 3 4: 00100 => 00110 => 00100: 4 5: 00101 => 00111 => 00101: 5 6: 00110 => 00101 => 00110: 6 7: 00111 => 00100 => 00111: 7 8: 01000 => 01100 => 01000: 8 9: 01001 => 01101 => 01001: 9 10: 01010 => 01111 => 01010: 10 11: 01011 => 01110 => 01011: 11 12: 01100 => 01010 => 01100: 12 13: 01101 => 01011 => 01101: 13 14: 01110 => 01001 => 01110: 14 15: 01111 => 01000 => 01111: 15 16: 10000 => 11000 => 10000: 16 17: 10001 => 11001 => 10001: 17 18: 10010 => 11011 => 10010: 18 19: 10011 => 11010 => 10011: 19 20: 10100 => 11110 => 10100: 20 21: 10101 => 11111 => 10101: 21 22: 10110 => 11101 => 10110: 22 23: 10111 => 11100 => 10111: 23 24: 11000 => 10100 => 11000: 24 25: 11001 => 10101 => 11001: 25 26: 11010 => 10111 => 11010: 26 27: 11011 => 10110 => 11011: 27 28: 11100 => 10010 => 11100: 28 29: 11101 => 10011 => 11101: 29 30: 11110 => 10001 => 11110: 30 31: 11111 => 10000 => 11111: 31
ALGOL 68
In Algol 68 the BITS mode is specifically designed for manipulating machine words as a row of bits so it is natural to treat Gray encoded integers as values of MODE BITS. The standard operator BIN (INT) : BITS converts an INT value to a BITS value. The ABS (BITS) : INT operator performs the reverse conversion, though it has not been needed for this task. It is also natural in the language for simple operations on values to be implemented as operators, rather than as functions, as in the program below. <lang algol68>BEGIN
OP GRAY = (BITS b) BITS : b XOR (b SHR 1); CO Convert to Gray code CO
OP YARG = (BITS g) BITS : CO Convert from Gray code CO
BEGIN
BITS b := g, mask := g SHR 1;
WHILE mask /= 2r0 DO b := b XOR mask; mask := mask SHR 1 OD;
b
END;
FOR i FROM 0 TO 31 DO
printf (($zd, ": ", 2(2r5d, " >= "), 2r5dl$, i, BIN i, GRAY BIN i, YARG GRAY BIN i))
OD
END</lang>
- Output:
0: 00000 >= 00000 >= 00000 1: 00001 >= 00001 >= 00001 2: 00010 >= 00011 >= 00010 3: 00011 >= 00010 >= 00011 4: 00100 >= 00110 >= 00100 5: 00101 >= 00111 >= 00101 6: 00110 >= 00101 >= 00110 7: 00111 >= 00100 >= 00111 8: 01000 >= 01100 >= 01000 9: 01001 >= 01101 >= 01001 10: 01010 >= 01111 >= 01010 11: 01011 >= 01110 >= 01011 12: 01100 >= 01010 >= 01100 13: 01101 >= 01011 >= 01101 14: 01110 >= 01001 >= 01110 15: 01111 >= 01000 >= 01111 16: 10000 >= 11000 >= 10000 17: 10001 >= 11001 >= 10001 18: 10010 >= 11011 >= 10010 19: 10011 >= 11010 >= 10011 20: 10100 >= 11110 >= 10100 21: 10101 >= 11111 >= 10101 22: 10110 >= 11101 >= 10110 23: 10111 >= 11100 >= 10111 24: 11000 >= 10100 >= 11000 25: 11001 >= 10101 >= 11001 26: 11010 >= 10111 >= 11010 27: 11011 >= 10110 >= 11011 28: 11100 >= 10010 >= 11100 29: 11101 >= 10011 >= 11101 30: 11110 >= 10001 >= 11110 31: 11111 >= 10000 >= 11111
APL
Generate the complete N-bit Gray sequence as a matrix:run <lang apl>N←5 ({(0,⍵)⍪1,⊖⍵}⍣N)(1 0⍴⍬)</lang>
- Output:
0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1 1 0 0 0 1 1 1 0 0 1 0 1 0 0 1 0 0 0 1 1 0 0 0 1 1 0 1 0 1 1 1 1 0 1 1 1 0 0 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 0 0 0 1 1 0 0 0 1 1 0 0 1 1 1 0 1 1 1 1 0 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 1 0 1 0 0 1 0 1 0 1 1 0 1 1 1 1 0 1 1 0 1 0 0 1 0 1 0 0 1 1 1 0 0 0 1 1 0 0 0 0
Encode and decode an individual integer:run <lang apl>N←5 grayEncode←{a≠N↑(0,a←(N⍴2)⊤⍵)} grayDecode←{2⊥≠⌿N N↑N(2×N)⍴⍵,0,N⍴0}
grayEncode 19</lang>
- Output:
1 1 0 1 0
Arturo
<lang rebol>toGray: function [n]-> xor n shr n 1 fromGray: function [n][
p: n
while [n > 0][
n: shr n 1
p: xor p n
]
return p
]
loop 0..31 'num [
encoded: toGray num decoded: fromGray encoded
print [
pad to :string num 2 ":"
pad as.binary num 5 "=>"
pad as.binary encoded 5 "=>"
pad as.binary decoded 5 ":"
pad to :string decoded 2
]
]</lang>
- Output:
0 : 0 => 0 => 0 : 0 1 : 1 => 1 => 1 : 1 2 : 10 => 11 => 10 : 2 3 : 11 => 10 => 11 : 3 4 : 100 => 110 => 100 : 4 5 : 101 => 111 => 101 : 5 6 : 110 => 101 => 110 : 6 7 : 111 => 100 => 111 : 7 8 : 1000 => 1100 => 1000 : 8 9 : 1001 => 1101 => 1001 : 9 10 : 1010 => 1111 => 1010 : 10 11 : 1011 => 1110 => 1011 : 11 12 : 1100 => 1010 => 1100 : 12 13 : 1101 => 1011 => 1101 : 13 14 : 1110 => 1001 => 1110 : 14 15 : 1111 => 1000 => 1111 : 15 16 : 10000 => 11000 => 10000 : 16 17 : 10001 => 11001 => 10001 : 17 18 : 10010 => 11011 => 10010 : 18 19 : 10011 => 11010 => 10011 : 19 20 : 10100 => 11110 => 10100 : 20 21 : 10101 => 11111 => 10101 : 21 22 : 10110 => 11101 => 10110 : 22 23 : 10111 => 11100 => 10111 : 23 24 : 11000 => 10100 => 11000 : 24 25 : 11001 => 10101 => 11001 : 25 26 : 11010 => 10111 => 11010 : 26 27 : 11011 => 10110 => 11011 : 27 28 : 11100 => 10010 => 11100 : 28 29 : 11101 => 10011 => 11101 : 29 30 : 11110 => 10001 => 11110 : 30 31 : 11111 => 10000 => 11111 : 31
AutoHotkey
<lang AHK>gray_encode(n){ return n ^ (n >> 1) }
gray_decode(n){ p := n while (n >>= 1) p ^= n return p }
BinString(n){ Loop 5 If ( n & ( 1 << (A_Index-1) ) ) o := "1" . o else o := "0" . o return o }
Loop 32 n:=A_Index-1, out .= n " : " BinString(n) " => " BinString(e:=gray_encode(n)) . " => " BinString(gray_decode(e)) " => " BinString(n) "`n" MsgBox % clipboard := out</lang>
- Output:
0 : 00000 => 00000 => 00000 => 00000 1 : 00001 => 00001 => 00001 => 00001 2 : 00010 => 00011 => 00010 => 00010 3 : 00011 => 00010 => 00011 => 00011 4 : 00100 => 00110 => 00100 => 00100 5 : 00101 => 00111 => 00101 => 00101 6 : 00110 => 00101 => 00110 => 00110 7 : 00111 => 00100 => 00111 => 00111 8 : 01000 => 01100 => 01000 => 01000 9 : 01001 => 01101 => 01001 => 01001 10 : 01010 => 01111 => 01010 => 01010 11 : 01011 => 01110 => 01011 => 01011 12 : 01100 => 01010 => 01100 => 01100 13 : 01101 => 01011 => 01101 => 01101 14 : 01110 => 01001 => 01110 => 01110 15 : 01111 => 01000 => 01111 => 01111 16 : 10000 => 11000 => 10000 => 10000 17 : 10001 => 11001 => 10001 => 10001 18 : 10010 => 11011 => 10010 => 10010 19 : 10011 => 11010 => 10011 => 10011 20 : 10100 => 11110 => 10100 => 10100 21 : 10101 => 11111 => 10101 => 10101 22 : 10110 => 11101 => 10110 => 10110 23 : 10111 => 11100 => 10111 => 10111 24 : 11000 => 10100 => 11000 => 11000 25 : 11001 => 10101 => 11001 => 11001 26 : 11010 => 10111 => 11010 => 11010 27 : 11011 => 10110 => 11011 => 11011 28 : 11100 => 10010 => 11100 => 11100 29 : 11101 => 10011 => 11101 => 11101 30 : 11110 => 10001 => 11110 => 11110 31 : 11111 => 10000 => 11111 => 11111
AWK
<lang awk># Tested using GAWK
function bits2str(bits, data, mask) {
# Source: https://www.gnu.org/software/gawk/manual/html_node/Bitwise-Functions.html if (bits == 0) return "0"
mask = 1
for (; bits != 0; bits = rshift(bits, 1))
data = (and(bits, mask) ? "1" : "0") data
while ((length(data) % 8) != 0)
data = "0" data
return data
}
function gray_encode(n){
# Source: https://en.wikipedia.org/wiki/Gray_code#Converting_to_and_from_Gray_code return xor(n,rshift(n,1))
}
function gray_decode(n){
# Source: https://en.wikipedia.org/wiki/Gray_code#Converting_to_and_from_Gray_code mask = rshift(n,1) while(mask != 0){ n = xor(n,mask) mask = rshift(mask,1) } return n
}
BEGIN{
for (i=0; i < 32; i++)
printf "%-3s => %05d => %05d => %05d\n",i, bits2str(i),bits2str(gray_encode(i)), bits2str(gray_decode(gray_encode(i)))
}</lang>
- Output:
0 => 00000 => 00000 => 00000 1 => 00001 => 00001 => 00001 2 => 00010 => 00011 => 00010 3 => 00011 => 00010 => 00011 4 => 00100 => 00110 => 00100 5 => 00101 => 00111 => 00101 6 => 00110 => 00101 => 00110 7 => 00111 => 00100 => 00111 8 => 01000 => 01100 => 01000 9 => 01001 => 01101 => 01001 10 => 01010 => 01111 => 01010 11 => 01011 => 01110 => 01011 12 => 01100 => 01010 => 01100 13 => 01101 => 01011 => 01101 14 => 01110 => 01001 => 01110 15 => 01111 => 01000 => 01111 16 => 10000 => 11000 => 10000 17 => 10001 => 11001 => 10001 18 => 10010 => 11011 => 10010 19 => 10011 => 11010 => 10011 20 => 10100 => 11110 => 10100 21 => 10101 => 11111 => 10101 22 => 10110 => 11101 => 10110 23 => 10111 => 11100 => 10111 24 => 11000 => 10100 => 11000 25 => 11001 => 10101 => 11001 26 => 11010 => 10111 => 11010 27 => 11011 => 10110 => 11011 28 => 11100 => 10010 => 11100 29 => 11101 => 10011 => 11101 30 => 11110 => 10001 => 11110 31 => 11111 => 10000 => 11111
Batch File
<lang dos>:: Gray Code Task from Rosetta Code
- Batch File Implementation
@echo off rem -------------- define batch file macros with parameters appended rem more info: https://www.dostips.com/forum/viewtopic.php?f=3&t=2518 setlocal disabledelayedexpansion % == required for macro ==% (set \n=^^^ %== this creates escaped line feed for macro ==% )
rem convert to binary (unsigned) rem argument: natnum bitlength outputvar rem note: if natnum is negative, then !outputvar! is empty set tobinary=for %%# in (1 2) do if %%#==2 ( %\n%
for /f "tokens=1,2,3" %%a in ("!args!") do ( %\n%
set "natnum=%%a"^&set "bitlength=%%b"^&set "outputvar=%%c") %\n%
set "!outputvar!=" %\n%
if !natnum! geq 0 ( %\n%
set "currnum=!natnum!" %\n%
for /l %%m in (1,1,!bitlength!) do ( %\n%
set /a "bit=!currnum!%%2" %\n%
for %%v in (!outputvar!) do set "!outputvar!=!bit!!%%v!" %\n%
set /a "currnum/=2" %\n%
) %\n%
) %\n%
) else set args=
goto :main-thing %== jump to the main thing ==% rem -------------- usual "call" sections rem the sad disadvantage of using these is that they are slow (TnT)
rem gray code encoder rem argument: natnum outputvar
- encoder
set /a "%~2=%~1^(%~1>>1)" goto :eof
rem gray code decoder rem argument: natnum outputvar
- decoder
set "inp=%~1" & set "%~2=0"
- while-loop-1
if %inp% gtr 0 (
set /a "%~2^=%inp%, inp>>=1" goto while-loop-1
) goto :eof
rem -------------- main thing
- main-thing
setlocal enabledelayedexpansion echo(# -^> bin -^> enc -^> dec for /l %%n in (0,1,31) do (
%tobinary% %%n 5 bin call :encoder "%%n" "enc" %tobinary% !enc! 5 gray call :decoder "!enc!" "dec" %tobinary% !dec! 5 rebin echo(%%n -^> !bin! -^> !gray! -^> !rebin!
) exit /b 0</lang>
- Output:
# -> bin -> enc -> dec 0 -> 00000 -> 00000 -> 00000 1 -> 00001 -> 00001 -> 00001 2 -> 00010 -> 00011 -> 00010 3 -> 00011 -> 00010 -> 00011 4 -> 00100 -> 00110 -> 00100 5 -> 00101 -> 00111 -> 00101 6 -> 00110 -> 00101 -> 00110 7 -> 00111 -> 00100 -> 00111 8 -> 01000 -> 01100 -> 01000 9 -> 01001 -> 01101 -> 01001 10 -> 01010 -> 01111 -> 01010 11 -> 01011 -> 01110 -> 01011 12 -> 01100 -> 01010 -> 01100 13 -> 01101 -> 01011 -> 01101 14 -> 01110 -> 01001 -> 01110 15 -> 01111 -> 01000 -> 01111 16 -> 10000 -> 11000 -> 10000 17 -> 10001 -> 11001 -> 10001 18 -> 10010 -> 11011 -> 10010 19 -> 10011 -> 11010 -> 10011 20 -> 10100 -> 11110 -> 10100 21 -> 10101 -> 11111 -> 10101 22 -> 10110 -> 11101 -> 10110 23 -> 10111 -> 11100 -> 10111 24 -> 11000 -> 10100 -> 11000 25 -> 11001 -> 10101 -> 11001 26 -> 11010 -> 10111 -> 11010 27 -> 11011 -> 10110 -> 11011 28 -> 11100 -> 10010 -> 11100 29 -> 11101 -> 10011 -> 11101 30 -> 11110 -> 10001 -> 11110 31 -> 11111 -> 10000 -> 11111
BBC BASIC
<lang bbcbasic> INSTALL @lib$+"STRINGLIB"
PRINT " Decimal Binary Gray Decoded"
FOR number% = 0 TO 31
gray% = FNgrayencode(number%)
PRINT number% " " FN_tobase(number%, 2, 5) ;
PRINT " " FN_tobase(gray%, 2, 5) FNgraydecode(gray%)
NEXT
END
DEF FNgrayencode(B%) = B% EOR (B% >>> 1)
DEF FNgraydecode(G%) : LOCAL B%
REPEAT B% EOR= G% : G% = G% >>> 1 : UNTIL G% = 0
= B%</lang>
bc
This language has no bitwise logic. We must repeat, with each bit, the exclusive-or calculation. This solution uses h % 2 and i % 2 to grab the low bits, and repeats if (h % 2 != i % 2) to check if the exclusive-or is one. Our encoding and decoding functions are identical except that h always comes from the decoded integer.
<lang bc>scale = 0 /* to use integer division */
/* encode Gray code */ define e(i) { auto h, r
if (i <= 0) return 0 h = i / 2 r = e(h) * 2 /* recurse */ if (h % 2 != i % 2) r += 1 /* xor low bits of h, i */ return r }
/* decode Gray code */ define d(i) { auto h, r
if (i <= 0) return 0 h = d(i / 2) /* recurse */ r = h * 2 if (h % 2 != i % 2) r += 1 /* xor low bits of h, i */ return r }
/* print i as 5 binary digits */
define p(i) {
auto d, d[]
for (d = 0; d <= 4; d++) { d[d] = i % 2 i /= 2 } for (d = 4; d >= 0; d--) { if(d[d] == 0) "0" if(d[d] == 1) "1" } }
for (i = 0; i < 32; i++) { /* original */ t = p(i); " => " /* encoded */ e = e(i); t = p(e); " => " /* decoded */ d = d(e); t = p(d); " " } quit</lang>
- Output:
00000 => 00000 => 00000 00001 => 00001 => 00001 00010 => 00011 => 00010 00011 => 00010 => 00011 00100 => 00110 => 00100 00101 => 00111 => 00101 00110 => 00101 => 00110 00111 => 00100 => 00111 01000 => 01100 => 01000 01001 => 01101 => 01001 01010 => 01111 => 01010 01011 => 01110 => 01011 01100 => 01010 => 01100 01101 => 01011 => 01101 01110 => 01001 => 01110 01111 => 01000 => 01111 10000 => 11000 => 10000 10001 => 11001 => 10001 10010 => 11011 => 10010 10011 => 11010 => 10011 10100 => 11110 => 10100 10101 => 11111 => 10101 10110 => 11101 => 10110 10111 => 11100 => 10111 11000 => 10100 => 11000 11001 => 10101 => 11001 11010 => 10111 => 11010 11011 => 10110 => 11011 11100 => 10010 => 11100 11101 => 10011 => 11101 11110 => 10001 => 11110 11111 => 10000 => 11111
C
<lang c>int gray_encode(int n) {
return n ^ (n >> 1);
}
int gray_decode(int n) {
int p = n; while (n >>= 1) p ^= n; return p;
}</lang> Demonstration code: <lang c>#include <stdio.h>
/* Simple bool formatter, only good on range 0..31 */ void fmtbool(int n, char *buf) {
char *b = buf + 5;
*b=0;
do {
*--b = '0' + (n & 1); n >>= 1;
} while (b != buf);
}
int main(int argc, char **argv) {
int i,g,b; char bi[6],bg[6],bb[6];
for (i=0 ; i<32 ; i++) {
g = gray_encode(i); b = gray_decode(g); fmtbool(i,bi); fmtbool(g,bg); fmtbool(b,bb); printf("%2d : %5s => %5s => %5s : %2d\n", i, bi, bg, bb, b);
} return 0;
}</lang>
- Output:
0 : 00000 => 00000 => 00000 : 0 1 : 00001 => 00001 => 00001 : 1 2 : 00010 => 00011 => 00010 : 2 3 : 00011 => 00010 => 00011 : 3 4 : 00100 => 00110 => 00100 : 4 5 : 00101 => 00111 => 00101 : 5 6 : 00110 => 00101 => 00110 : 6 7 : 00111 => 00100 => 00111 : 7 8 : 01000 => 01100 => 01000 : 8 9 : 01001 => 01101 => 01001 : 9 10 : 01010 => 01111 => 01010 : 10 11 : 01011 => 01110 => 01011 : 11 12 : 01100 => 01010 => 01100 : 12 13 : 01101 => 01011 => 01101 : 13 14 : 01110 => 01001 => 01110 : 14 15 : 01111 => 01000 => 01111 : 15 16 : 10000 => 11000 => 10000 : 16 17 : 10001 => 11001 => 10001 : 17 18 : 10010 => 11011 => 10010 : 18 19 : 10011 => 11010 => 10011 : 19 20 : 10100 => 11110 => 10100 : 20 21 : 10101 => 11111 => 10101 : 21 22 : 10110 => 11101 => 10110 : 22 23 : 10111 => 11100 => 10111 : 23 24 : 11000 => 10100 => 11000 : 24 25 : 11001 => 10101 => 11001 : 25 26 : 11010 => 10111 => 11010 : 26 27 : 11011 => 10110 => 11011 : 27 28 : 11100 => 10010 => 11100 : 28 29 : 11101 => 10011 => 11101 : 29 30 : 11110 => 10001 => 11110 : 30 31 : 11111 => 10000 => 11111 : 31
C#
<lang c sharp>using System;
public class Gray {
public static ulong grayEncode(ulong n) {
return n^(n>>1);
}
public static ulong grayDecode(ulong n) {
ulong i=1<<8*64-2; //long is 64-bit
ulong p, b=p=n&i;
while((i>>=1)>0)
b|=p=n&i^p>>1;
return b;
}
public static void Main(string[] args) {
Console.WriteLine("Number\tBinary\tGray\tDecoded");
for(ulong i=0;i<32;i++) {
Console.WriteLine(string.Format("{0}\t{1}\t{2}\t{3}", i, Convert.ToString((long)i, 2), Convert.ToString((long)grayEncode(i), 2), grayDecode(grayEncode(i))));
}
}
}</lang>
- Output:
Number Binary Gray Decoded 0 0 0 0 1 1 1 1 2 10 11 2 3 11 10 3 4 100 110 4 5 101 111 5 6 110 101 6 7 111 100 7 8 1000 1100 8 9 1001 1101 9 10 1010 1111 10 11 1011 1110 11 12 1100 1010 12 13 1101 1011 13 14 1110 1001 14 15 1111 1000 15 16 10000 11000 16 17 10001 11001 17 18 10010 11011 18 19 10011 11010 19 20 10100 11110 20 21 10101 11111 21 22 10110 11101 22 23 10111 11100 23 24 11000 10100 24 25 11001 10101 25 26 11010 10111 26 27 11011 10110 27 28 11100 10010 28 29 11101 10011 29 30 11110 10001 30 31 11111 10000 31
C++
<lang cpp>
- include <bitset>
- include <iostream>
- include <string>
- include <assert.h>
uint32_t gray_encode(uint32_t b) {
return b ^ (b >> 1);
}
uint32_t gray_decode(uint32_t g) {
for (uint32_t bit = 1U << 31; bit > 1; bit >>= 1)
{
if (g & bit) g ^= bit >> 1;
}
return g;
}
std::string to_binary(int value) // utility function {
const std::bitset<32> bs(value);
const std::string str(bs.to_string());
const size_t pos(str.find('1'));
return pos == std::string::npos ? "0" : str.substr(pos);
}
int main() {
std::cout << "Number\tBinary\tGray\tDecoded\n";
for (uint32_t n = 0; n < 32; ++n)
{
uint32_t g = gray_encode(n);
assert(gray_decode(g) == n);
std::cout << n << "\t" << to_binary(n) << "\t" << to_binary(g) << "\t" << g << "\n"; }
}</lang>
- Output:
Number Binary Gray Decoded 0 0 0 0 1 1 1 1 2 10 11 3 3 11 10 2 4 100 110 6 5 101 111 7 6 110 101 5 7 111 100 4 8 1000 1100 12 9 1001 1101 13 10 1010 1111 15 11 1011 1110 14 12 1100 1010 10 13 1101 1011 11 14 1110 1001 9 15 1111 1000 8 16 10000 11000 24 17 10001 11001 25 18 10010 11011 27 19 10011 11010 26 20 10100 11110 30 21 10101 11111 31 22 10110 11101 29 23 10111 11100 28 24 11000 10100 20 25 11001 10101 21 26 11010 10111 23 27 11011 10110 22 28 11100 10010 18 29 11101 10011 19 30 11110 10001 17 31 11111 10000 16
CoffeeScript
<lang coffeescript> gray_encode = (n) ->
n ^ (n >> 1)
gray_decode = (g) ->
n = g n ^= g while g >>= 1 n
for i in [0..32]
console.log gray_decode gray_encode(i)
</lang>
Common Lisp
<lang lisp>(defun gray-encode (n)
(logxor n (ash n -1)))
(defun gray-decode (n)
(do ((p n (logxor p n))) ((zerop n) p) (setf n (ash n -1))))
(loop for i to 31 do
(let* ((g (gray-encode i)) (b (gray-decode g)))
(format t "~2d:~6b =>~6b =>~6b :~2d~%" i i g b b)))</lang>
- Output:
0: 0 => 0 => 0 : 0 1: 1 => 1 => 1 : 1 2: 10 => 11 => 10 : 2 3: 11 => 10 => 11 : 3 4: 100 => 110 => 100 : 4 5: 101 => 111 => 101 : 5 6: 110 => 101 => 110 : 6 7: 111 => 100 => 111 : 7 8: 1000 => 1100 => 1000 : 8 9: 1001 => 1101 => 1001 : 9 10: 1010 => 1111 => 1010 :10 11: 1011 => 1110 => 1011 :11 12: 1100 => 1010 => 1100 :12 13: 1101 => 1011 => 1101 :13 14: 1110 => 1001 => 1110 :14 15: 1111 => 1000 => 1111 :15 16: 10000 => 11000 => 10000 :16 17: 10001 => 11001 => 10001 :17 18: 10010 => 11011 => 10010 :18 19: 10011 => 11010 => 10011 :19 20: 10100 => 11110 => 10100 :20 21: 10101 => 11111 => 10101 :21 22: 10110 => 11101 => 10110 :22 23: 10111 => 11100 => 10111 :23 24: 11000 => 10100 => 11000 :24 25: 11001 => 10101 => 11001 :25 26: 11010 => 10111 => 11010 :26 27: 11011 => 10110 => 11011 :27 28: 11100 => 10010 => 11100 :28 29: 11101 => 10011 => 11101 :29 30: 11110 => 10001 => 11110 :30 31: 11111 => 10000 => 11111 :31
Component Pascal
BlackBox Component Builder <lang oberon2> MODULE GrayCodes; IMPORT StdLog,SYSTEM;
PROCEDURE Encode*(i: INTEGER; OUT x: INTEGER); VAR j: INTEGER; s,r: SET; BEGIN s := BITS(i);j := MAX(SET); WHILE (j >= 0) & ~(j IN s) DO DEC(j) END; r := {};IF j >= 0 THEN INCL(r,j) END; WHILE j > 0 DO IF ((j IN s) & ~(j - 1 IN s)) OR (~(j IN s) & (j - 1 IN s)) THEN INCL(r,j-1) END; DEC(j) END; x := SYSTEM.VAL(INTEGER,r) END Encode;
PROCEDURE Decode*(x: INTEGER; OUT i: INTEGER); VAR j: INTEGER; s,r: SET; BEGIN s := BITS(x);r:={};j := MAX(SET); WHILE (j >= 0) & ~(j IN s) DO DEC(j) END; IF j >= 0 THEN INCL(r,j) END; WHILE j > 0 DO IF ((j IN r) & ~(j - 1 IN s)) OR (~(j IN r) & (j - 1 IN s)) THEN INCL(r,j-1) END; DEC(j) END; i := SYSTEM.VAL(INTEGER,r); END Decode;
PROCEDURE Do*;
VAR
grayCode,binCode: INTEGER;
i: INTEGER;
BEGIN
StdLog.String(" i ");StdLog.String(" bin code ");StdLog.String(" gray code ");StdLog.Ln;
StdLog.String("---");StdLog.String(" ----------------");StdLog.String(" ---------------");StdLog.Ln;
FOR i := 0 TO 32 DO;
Encode(i,grayCode);Decode(grayCode,binCode);
StdLog.IntForm(i,10,3,' ',FALSE);
StdLog.IntForm(binCode,2,16,' ',TRUE);
StdLog.IntForm(grayCode,2,16,' ',TRUE);
StdLog.Ln;
END
END Do;
END GrayCodes.
</lang>
Execute: ^QGrayCodes.Do
- Output:
i bin code gray code --- ---------------- --------------- 0 0%2 0%2 1 1%2 1%2 2 10%2 11%2 3 11%2 10%2 4 100%2 110%2 5 101%2 111%2 6 110%2 101%2 7 111%2 100%2 8 1000%2 1100%2 9 1001%2 1101%2 10 1010%2 1111%2 11 1011%2 1110%2 12 1100%2 1010%2 13 1101%2 1011%2 14 1110%2 1001%2 15 1111%2 1000%2 16 10000%2 11000%2 17 10001%2 11001%2 18 10010%2 11011%2 19 10011%2 11010%2 20 10100%2 11110%2 21 10101%2 11111%2 22 10110%2 11101%2 23 10111%2 11100%2 24 11000%2 10100%2 25 11001%2 10101%2 26 11010%2 10111%2 27 11011%2 10110%2 28 11100%2 10010%2 29 11101%2 10011%2 30 11110%2 10001%2 31 11111%2 10000%2 32 100000%2 110000%2
Crystal
<lang ruby> def gray_encode(bin)
bin ^ (bin >> 1)
end
def gray_decode(gray)
bin = gray while gray > 0 gray >>= 1 bin ^= gray end bin
end </lang> Demonstration code: <lang ruby> (0..31).each do |n|
gr = gray_encode n bin = gray_decode gr printf "%2d : %05b => %05b => %05b : %2d\n", n, n, gr, bin, bin
end </lang>
- Output:
0 : 00000 => 00000 => 00000 : 0 1 : 00001 => 00001 => 00001 : 1 2 : 00010 => 00011 => 00010 : 2 3 : 00011 => 00010 => 00011 : 3 4 : 00100 => 00110 => 00100 : 4 5 : 00101 => 00111 => 00101 : 5 6 : 00110 => 00101 => 00110 : 6 7 : 00111 => 00100 => 00111 : 7 8 : 01000 => 01100 => 01000 : 8 9 : 01001 => 01101 => 01001 : 9 10 : 01010 => 01111 => 01010 : 10 11 : 01011 => 01110 => 01011 : 11 12 : 01100 => 01010 => 01100 : 12 13 : 01101 => 01011 => 01101 : 13 14 : 01110 => 01001 => 01110 : 14 15 : 01111 => 01000 => 01111 : 15 16 : 10000 => 11000 => 10000 : 16 17 : 10001 => 11001 => 10001 : 17 18 : 10010 => 11011 => 10010 : 18 19 : 10011 => 11010 => 10011 : 19 20 : 10100 => 11110 => 10100 : 20 21 : 10101 => 11111 => 10101 : 21 22 : 10110 => 11101 => 10110 : 22 23 : 10111 => 11100 => 10111 : 23 24 : 11000 => 10100 => 11000 : 24 25 : 11001 => 10101 => 11001 : 25 26 : 11010 => 10111 => 11010 : 26 27 : 11011 => 10110 => 11011 : 27 28 : 11100 => 10010 => 11100 : 28 29 : 11101 => 10011 => 11101 : 29 30 : 11110 => 10001 => 11110 : 30 31 : 11111 => 10000 => 11111 : 31
D
<lang d>uint grayEncode(in uint n) pure nothrow @nogc {
return n ^ (n >> 1);
}
uint grayDecode(uint n) pure nothrow @nogc {
auto p = n;
while (n >>= 1)
p ^= n;
return p;
}
void main() {
import std.stdio;
" N N2 enc dec2 dec".writeln;
foreach (immutable n; 0 .. 32) {
immutable g = n.grayEncode;
immutable d = g.grayDecode;
writefln("%2d: %5b => %5b => %5b: %2d", n, n, g, d, d);
assert(d == n);
}
}</lang>
- Output:
N N2 enc dec2 dec 0: 0 => 0 => 0: 0 1: 1 => 1 => 1: 1 2: 10 => 11 => 10: 2 3: 11 => 10 => 11: 3 4: 100 => 110 => 100: 4 5: 101 => 111 => 101: 5 6: 110 => 101 => 110: 6 7: 111 => 100 => 111: 7 8: 1000 => 1100 => 1000: 8 9: 1001 => 1101 => 1001: 9 10: 1010 => 1111 => 1010: 10 11: 1011 => 1110 => 1011: 11 12: 1100 => 1010 => 1100: 12 13: 1101 => 1011 => 1101: 13 14: 1110 => 1001 => 1110: 14 15: 1111 => 1000 => 1111: 15 16: 10000 => 11000 => 10000: 16 17: 10001 => 11001 => 10001: 17 18: 10010 => 11011 => 10010: 18 19: 10011 => 11010 => 10011: 19 20: 10100 => 11110 => 10100: 20 21: 10101 => 11111 => 10101: 21 22: 10110 => 11101 => 10110: 22 23: 10111 => 11100 => 10111: 23 24: 11000 => 10100 => 11000: 24 25: 11001 => 10101 => 11001: 25 26: 11010 => 10111 => 11010: 26 27: 11011 => 10110 => 11011: 27 28: 11100 => 10010 => 11100: 28 29: 11101 => 10011 => 11101: 29 30: 11110 => 10001 => 11110: 30 31: 11111 => 10000 => 11111: 31
Compile-Time version
This version uses a compile time generated translation table, if maximum bit width of the numbers is a constant. The encoding table is generated recursively, then the decode table is calculated and appended. Same output. <lang d>import std.stdio, std.algorithm;
T[] gray(int N : 1, T)() pure nothrow {
return [T(0), 1];
}
/// Recursively generate gray encoding mapping table. T[] gray(int N, T)() pure nothrow if (N <= T.sizeof * 8) {
enum T M = T(2) ^^ (N - 1);
T[] g = gray!(N - 1, T)();
foreach (immutable i; 0 .. M)
g ~= M + g[M - i - 1];
return g;
}
T[][] grayDict(int N, T)() pure nothrow {
T[][] dict = [gray!(N, T)(), [0]];
// Append inversed gray encoding mapping.
foreach (immutable i; 1 .. dict[0].length)
dict[1] ~= cast(T)countUntil(dict[0], i);
return dict;
}
enum M { Encode = 0, Decode = 1 }
T gray(int N, T)(in T n, in int mode=M.Encode) pure nothrow {
// Generated at compile time. enum dict = grayDict!(N, T)(); return dict[mode][n];
}
void main() {
foreach (immutable i; 0 .. 32) {
immutable encoded = gray!(5)(i, M.Encode);
immutable decoded = gray!(5)(encoded, M.Decode);
writefln("%2d: %5b => %5b : %2d", i, i, encoded, decoded);
}
}</lang>
Short Functional-Style Generator
<lang d>import std.stdio, std.algorithm, std.range;
string[] g(in uint n) pure nothrow {
return n ? g(n - 1).map!q{'0' ~ a}.array ~
g(n - 1).retro.map!q{'1' ~ a}.array
: [""];
}
void main() {
4.g.writeln;
}</lang>
- Output:
["0000", "0001", "0011", "0010", "0110", "0111", "0101", "0100", "1100", "1101", "1111", "1110", "1010", "1011", "1001", "1000"]
Delphi
<lang delphi>program GrayCode;
{$APPTYPE CONSOLE}
uses SysUtils;
function Encode(v: Integer): Integer; begin
Result := v xor (v shr 1);
end;
function Decode(v: Integer): Integer; begin
Result := 0; while v > 0 do begin Result := Result xor v; v := v shr 1; end;
end;
function IntToBin(aValue: LongInt; aDigits: Integer): string; begin
Result := StringOfChar('0', aDigits);
while aValue > 0 do
begin
if (aValue and 1) = 1 then
Result[aDigits] := '1';
Dec(aDigits);
aValue := aValue shr 1;
end;
end;
var
i, g, d: Integer;
begin
Writeln('decimal binary gray decoded');
for i := 0 to 31 do
begin
g := Encode(i);
d := Decode(g);
Writeln(Format(' %2d %s %s %s %2d', [i, IntToBin(i, 5), IntToBin(g, 5), IntToBin(d, 5), d]));
end;
end.</lang>
- Output:
decimal binary gray decoded 0 00000 00000 00000 0 1 00001 00001 00001 1 2 00010 00011 00010 2 3 00011 00010 00011 3 4 00100 00110 00100 4 5 00101 00111 00101 5 6 00110 00101 00110 6 7 00111 00100 00111 7 8 01000 01100 01000 8 9 01001 01101 01001 9 10 01010 01111 01010 10 11 01011 01110 01011 11 12 01100 01010 01100 12 13 01101 01011 01101 13 14 01110 01001 01110 14 15 01111 01000 01111 15 16 10000 11000 10000 16 17 10001 11001 10001 17 18 10010 11011 10010 18 19 10011 11010 10011 19 20 10100 11110 10100 20 21 10101 11111 10101 21 22 10110 11101 10110 22 23 10111 11100 10111 23 24 11000 10100 11000 24 25 11001 10101 11001 25 26 11010 10111 11010 26 27 11011 10110 11011 27 28 11100 10010 11100 28 29 11101 10011 11101 29 30 11110 10001 11110 30 31 11111 10000 11111 31
DWScript
<lang delphi>function Encode(v : Integer) : Integer; begin
Result := v xor (v shr 1);
end;
function Decode(v : Integer) : Integer; begin
Result := 0;
while v>0 do begin
Result := Result xor v;
v := v shr 1;
end;
end;
PrintLn('decimal binary gray decoded');
var i : Integer; for i:=0 to 31 do begin
var g := Encode(i);
var d := Decode(g);
PrintLn(Format(' %2d %s %s %s %2d',
[i, IntToBin(i, 5), IntToBin(g, 5), IntToBin(d, 5), d]));
end;</lang>
Elixir
<lang elixir>defmodule Gray_code do
use Bitwise def encode(n), do: bxor(n, bsr(n,1)) def decode(g), do: decode(g,0) def decode(0,n), do: n def decode(g,n), do: decode(bsr(g,1), bxor(g,n))
end
Enum.each(0..31, fn(n) ->
g = Gray_code.encode(n)
d = Gray_code.decode(g)
:io.fwrite("~2B : ~5.2.0B : ~5.2.0B : ~5.2.0B : ~2B~n", [n, n, g, d, d])
end)</lang> output is the same as "Erlang".
Erlang
<lang erlang>-module(gray). -export([encode/1, decode/1]).
encode(N) -> N bxor (N bsr 1).
decode(G) -> decode(G,0).
decode(0,N) -> N; decode(G,N) -> decode(G bsr 1, G bxor N). </lang>
Demonstration code: <lang erlang>-module(testgray).
test_encode(N) ->
G = gray:encode(N),
D = gray:decode(G),
io:fwrite("~2B : ~5.2.0B : ~5.2.0B : ~5.2.0B : ~2B~n", [N, N, G, D, D]).
test_encode(N, N) -> []; test_encode(I, N) when I < N -> test_encode(I), test_encode(I+1, N).
main(_) -> test_encode(0,32).</lang>
- Output:
0 : 00000 : 00000 : 00000 : 0 1 : 00001 : 00001 : 00001 : 1 2 : 00010 : 00011 : 00010 : 2 3 : 00011 : 00010 : 00011 : 3 4 : 00100 : 00110 : 00100 : 4 5 : 00101 : 00111 : 00101 : 5 6 : 00110 : 00101 : 00110 : 6 7 : 00111 : 00100 : 00111 : 7 8 : 01000 : 01100 : 01000 : 8 9 : 01001 : 01101 : 01001 : 9 10 : 01010 : 01111 : 01010 : 10 11 : 01011 : 01110 : 01011 : 11 12 : 01100 : 01010 : 01100 : 12 13 : 01101 : 01011 : 01101 : 13 14 : 01110 : 01001 : 01110 : 14 15 : 01111 : 01000 : 01111 : 15 16 : 10000 : 11000 : 10000 : 16 17 : 10001 : 11001 : 10001 : 17 18 : 10010 : 11011 : 10010 : 18 19 : 10011 : 11010 : 10011 : 19 20 : 10100 : 11110 : 10100 : 20 21 : 10101 : 11111 : 10101 : 21 22 : 10110 : 11101 : 10110 : 22 23 : 10111 : 11100 : 10111 : 23 24 : 11000 : 10100 : 11000 : 24 25 : 11001 : 10101 : 11001 : 25 26 : 11010 : 10111 : 11010 : 26 27 : 11011 : 10110 : 11011 : 27 28 : 11100 : 10010 : 11100 : 28 29 : 11101 : 10011 : 11101 : 29 30 : 11110 : 10001 : 11110 : 30 31 : 11111 : 10000 : 11111 : 31
Euphoria
<lang euphoria>function gray_encode(integer n)
return xor_bits(n,floor(n/2))
end function
function gray_decode(integer n)
integer g
g = 0
while n > 0 do
g = xor_bits(g,n)
n = floor(n/2)
end while
return g
end function
function dcb(integer n)
atom d,m
d = 0
m = 1
while n do
d += remainder(n,2)*m
n = floor(n/2)
m *= 10
end while
return d
end function
integer j for i = #0 to #1F do
printf(1,"%05d => ",dcb(i)) j = gray_encode(i) printf(1,"%05d => ",dcb(j)) j = gray_decode(j) printf(1,"%05d\n",dcb(j))
end for</lang>
- Output:
00000 => 00000 => 00000 00001 => 00001 => 00001 00010 => 00011 => 00010 00011 => 00010 => 00011 00100 => 00110 => 00100 00101 => 00111 => 00101 00110 => 00101 => 00110 00111 => 00100 => 00111 01000 => 01100 => 01000 01001 => 01101 => 01001 01010 => 01111 => 01010 01011 => 01110 => 01011 01100 => 01010 => 01100 01101 => 01011 => 01101 01110 => 01001 => 01110 01111 => 01000 => 01111 10000 => 11000 => 10000 10001 => 11001 => 10001 10010 => 11011 => 10010 10011 => 11010 => 10011 10100 => 11110 => 10100 10101 => 11111 => 10101 10110 => 11101 => 10110 10111 => 11100 => 10111 11000 => 10100 => 11000 11001 => 10101 => 11001 11010 => 10111 => 11010 11011 => 10110 => 11011 11100 => 10010 => 11100 11101 => 10011 => 11101 11110 => 10001 => 11110 11111 => 10000 => 11111
F#
The Function
<lang fsharp> // Functıons to translate bınary to grey code and vv. Nigel Galloway: December 7th., 2018 let grayCode,invGrayCode=let fN g (n:uint8)=n^^^(n>>>g) in ((fN 1),(fN 1>>fN 2>>fN 4)) </lang>
The Task
<lang fsharp> [0uy..31uy]|>List.iter(fun n->let g=grayCode n in printfn "%2d -> %5s (%2d) -> %2d" n (System.Convert.ToString(g,2)) g (invGrayCode g))</lang>
- Output:
0 -> 0 ( 0) -> 0 1 -> 1 ( 1) -> 1 2 -> 11 ( 3) -> 2 3 -> 10 ( 2) -> 3 4 -> 110 ( 6) -> 4 5 -> 111 ( 7) -> 5 6 -> 101 ( 5) -> 6 7 -> 100 ( 4) -> 7 8 -> 1100 (12) -> 8 9 -> 1101 (13) -> 9 10 -> 1111 (15) -> 10 11 -> 1110 (14) -> 11 12 -> 1010 (10) -> 12 13 -> 1011 (11) -> 13 14 -> 1001 ( 9) -> 14 15 -> 1000 ( 8) -> 15 16 -> 11000 (24) -> 16 17 -> 11001 (25) -> 17 18 -> 11011 (27) -> 18 19 -> 11010 (26) -> 19 20 -> 11110 (30) -> 20 21 -> 11111 (31) -> 21 22 -> 11101 (29) -> 22 23 -> 11100 (28) -> 23 24 -> 10100 (20) -> 24 25 -> 10101 (21) -> 25 26 -> 10111 (23) -> 26 27 -> 10110 (22) -> 27 28 -> 10010 (18) -> 28 29 -> 10011 (19) -> 29 30 -> 10001 (17) -> 30 31 -> 10000 (16) -> 31
Factor
Translation of C. <lang factor>USING: math.ranges locals ; IN: rosetta-gray
- gray-encode ( n -- n' ) dup -1 shift bitxor ;
- gray-decode ( n! -- n' )
n :> p!
[ n -1 shift dup n! 0 = not ] [
p n bitxor p!
] while
p ;
- main ( -- )
-1 32 [a,b] [ dup [ >bin ] [ gray-encode ] bi [ >bin ] [ gray-decode ] bi 4array . ] each ;
MAIN: main </lang> Running above code prints: <lang factor>{ -1 "-1" "0" 0 } { 0 "0" "0" 0 } { 1 "1" "1" 1 } { 2 "10" "11" 2 } { 3 "11" "10" 3 } { 4 "100" "110" 4 } { 5 "101" "111" 5 } { 6 "110" "101" 6 } { 7 "111" "100" 7 } { 8 "1000" "1100" 8 } { 9 "1001" "1101" 9 } { 10 "1010" "1111" 10 } { 11 "1011" "1110" 11 } { 12 "1100" "1010" 12 } { 13 "1101" "1011" 13 } { 14 "1110" "1001" 14 } { 15 "1111" "1000" 15 } { 16 "10000" "11000" 16 } { 17 "10001" "11001" 17 } { 18 "10010" "11011" 18 } { 19 "10011" "11010" 19 } { 20 "10100" "11110" 20 } { 21 "10101" "11111" 21 } { 22 "10110" "11101" 22 } { 23 "10111" "11100" 23 } { 24 "11000" "10100" 24 } { 25 "11001" "10101" 25 } { 26 "11010" "10111" 26 } { 27 "11011" "10110" 27 } { 28 "11100" "10010" 28 } { 29 "11101" "10011" 29 } { 30 "11110" "10001" 30 } { 31 "11111" "10000" 31 } { 32 "100000" "110000" 32 }</lang>
Forth
As a low level language Forth provides efficient bit manipulation operators. These functions take input parameters from the stack and return the result on the stack.
<lang forth>: >gray ( n -- n' ) dup 2/ xor ; \ n' = n xor (n logically right shifted 1 time)
\ 2/ is Forth divide by 2, ie: shift right 1
- gray> ( n -- n )
0 1 31 lshift ( -- g b mask )
begin
>r \ save a copy of mask on return stack
2dup 2/ xor
r@ and or
r> 1 rshift
dup 0=
until
drop nip ; \ clean the parameter stack leaving result only
- test
2 base ! \ set system number base to 2. ie: Binary 32 0 do cr I dup 5 .r ." ==> " \ print numbers (binary) right justified 5 places >gray dup 5 .r ." ==> " gray> 5 .r loop decimal ; \ revert to BASE 10
</lang>
- Output:
FORTH> test
0 ==> 0 ==> 0
1 ==> 1 ==> 1
10 ==> 11 ==> 10
11 ==> 10 ==> 11
100 ==> 110 ==> 100
101 ==> 111 ==> 101
110 ==> 101 ==> 110
111 ==> 100 ==> 111
1000 ==> 1100 ==> 1000
1001 ==> 1101 ==> 1001
1010 ==> 1111 ==> 1010
1011 ==> 1110 ==> 1011
1100 ==> 1010 ==> 1100
1101 ==> 1011 ==> 1101
1110 ==> 1001 ==> 1110
1111 ==> 1000 ==> 1111
10000 ==> 11000 ==> 10000
10001 ==> 11001 ==> 10001
10010 ==> 11011 ==> 10010
10011 ==> 11010 ==> 10011
10100 ==> 11110 ==> 10100
10101 ==> 11111 ==> 10101
10110 ==> 11101 ==> 10110
10111 ==> 11100 ==> 10111
11000 ==> 10100 ==> 11000
11001 ==> 10101 ==> 11001
11010 ==> 10111 ==> 11010
11011 ==> 10110 ==> 11011
11100 ==> 10010 ==> 11100
11101 ==> 10011 ==> 11101
11110 ==> 10001 ==> 11110
11111 ==> 10000 ==> 11111 ok
Fortran
Using MIL-STD-1753 extensions in Fortran 77, and formulas found at OEIS for direct and inverse Gray code : <lang fortran> PROGRAM GRAY
IMPLICIT NONE
INTEGER IGRAY,I,J,K
CHARACTER*5 A,B,C
DO 10 I=0,31
J=IGRAY(I,1)
K=IGRAY(J,-1)
CALL BINARY(A,I,5)
CALL BINARY(B,J,5)
CALL BINARY(C,K,5)
PRINT 99,I,A,B,C,K
10 CONTINUE
99 FORMAT(I2,3H : ,A5,4H => ,A5,4H => ,A5,3H : ,I2)
END
FUNCTION IGRAY(N,D)
IMPLICIT NONE
INTEGER D,K,N,IGRAY
IF(D.LT.0) GO TO 10
IGRAY=IEOR(N,ISHFT(N,-1))
RETURN
10 K=N
IGRAY=0
20 IGRAY=IEOR(IGRAY,K)
K=K/2
IF(K.NE.0) GO TO 20
END
SUBROUTINE BINARY(S,N,K)
IMPLICIT NONE
INTEGER I,K,L,N
CHARACTER*(*) S
L=LEN(S)
DO 10 I=0,K-1
C The following line may replace the next block-if, C on machines using ASCII code : C S(L-I:L-I)=CHAR(48+IAND(1,ISHFT(N,-I))) C On EBCDIC machines, use 240 instead of 48.
IF(BTEST(N,I)) THEN
S(L-I:L-I)='1'
ELSE
S(L-I:L-I)='0'
END IF
10 CONTINUE
S(1:L-K)=
END</lang>
0 : 00000 => 00000 => 00000 : 0 1 : 00001 => 00001 => 00001 : 1 2 : 00010 => 00011 => 00010 : 2 3 : 00011 => 00010 => 00011 : 3 4 : 00100 => 00110 => 00100 : 4 5 : 00101 => 00111 => 00101 : 5 6 : 00110 => 00101 => 00110 : 6 7 : 00111 => 00100 => 00111 : 7 8 : 01000 => 01100 => 01000 : 8 9 : 01001 => 01101 => 01001 : 9 10 : 01010 => 01111 => 01010 : 10 11 : 01011 => 01110 => 01011 : 11 12 : 01100 => 01010 => 01100 : 12 13 : 01101 => 01011 => 01101 : 13 14 : 01110 => 01001 => 01110 : 14 15 : 01111 => 01000 => 01111 : 15 16 : 10000 => 11000 => 10000 : 16 17 : 10001 => 11001 => 10001 : 17 18 : 10010 => 11011 => 10010 : 18 19 : 10011 => 11010 => 10011 : 19 20 : 10100 => 11110 => 10100 : 20 21 : 10101 => 11111 => 10101 : 21 22 : 10110 => 11101 => 10110 : 22 23 : 10111 => 11100 => 10111 : 23 24 : 11000 => 10100 => 11000 : 24 25 : 11001 => 10101 => 11001 : 25 26 : 11010 => 10111 => 11010 : 26 27 : 11011 => 10110 => 11011 : 27 28 : 11100 => 10010 => 11100 : 28 29 : 11101 => 10011 => 11101 : 29 30 : 11110 => 10001 => 11110 : 30 31 : 11111 => 10000 => 11111 : 31
FreeBASIC
<lang freebasic>' version 18-01-2017 ' compile with: fbc -s console
Function gray2bin(g As UInteger) As UInteger
Dim As UInteger b = g
While g
g Shr= 1
b Xor= g
Wend
Return b
End Function
Function bin2gray(b As UInteger) As UInteger
Return b Xor (b Shr 1)
End Function
' ------=< MAIN >=------
Dim As UInteger i
Print " i binary gray gra2bin" Print String(32,"=")
For i = 0 To 31
Print Using "## --> "; i; print Bin(i,5); " --> "; Print Bin(bin2gray(i),5); " --> "; Print Bin(gray2bin(bin2gray(i)),5)
Next
' empty keyboard buffer While Inkey <> "" : Wend Print : Print "hit any key to end program" Sleep End</lang>
- Output:
i binary gray gra2bin ================================ 0 --> 00000 --> 00000 --> 00000 1 --> 00001 --> 00001 --> 00001 2 --> 00010 --> 00011 --> 00010 3 --> 00011 --> 00010 --> 00011 4 --> 00100 --> 00110 --> 00100 5 --> 00101 --> 00111 --> 00101 6 --> 00110 --> 00101 --> 00110 7 --> 00111 --> 00100 --> 00111 8 --> 01000 --> 01100 --> 01000 9 --> 01001 --> 01101 --> 01001 10 --> 01010 --> 01111 --> 01010 11 --> 01011 --> 01110 --> 01011 12 --> 01100 --> 01010 --> 01100 13 --> 01101 --> 01011 --> 01101 14 --> 01110 --> 01001 --> 01110 15 --> 01111 --> 01000 --> 01111 16 --> 10000 --> 11000 --> 10000 17 --> 10001 --> 11001 --> 10001 18 --> 10010 --> 11011 --> 10010 19 --> 10011 --> 11010 --> 10011 20 --> 10100 --> 11110 --> 10100 21 --> 10101 --> 11111 --> 10101 22 --> 10110 --> 11101 --> 10110 23 --> 10111 --> 11100 --> 10111 24 --> 11000 --> 10100 --> 11000 25 --> 11001 --> 10101 --> 11001 26 --> 11010 --> 10111 --> 11010 27 --> 11011 --> 10110 --> 11011 28 --> 11100 --> 10010 --> 11100 29 --> 11101 --> 10011 --> 11101 30 --> 11110 --> 10001 --> 11110 31 --> 11111 --> 10000 --> 11111
Frink
Frink has built-in functions to convert to and from binary reflected Gray code. <lang frink> for i=0 to 31 {
gray = binaryToGray[i] back = grayToBinary[gray] println[(i->binary) + "\t" + (gray->binary) + "\t" + (back->binary)]
} </lang>
Go
Binary reflected, as described in the task. Reading down through the solutions, the Euphoria decode algorithm caught my eye as being concise and easy to read. <lang go>package main
import "fmt"
func enc(b int) int {
return b ^ b>>1
}
func dec(g int) (b int) {
for ; g != 0; g >>= 1 {
b ^= g
}
return
}
func main() {
fmt.Println("decimal binary gray decoded")
for b := 0; b < 32; b++ {
g := enc(b)
d := dec(g)
fmt.Printf(" %2d %05b %05b %05b %2d\n", b, b, g, d, d)
}
}</lang>
- Output:
decimal binary gray decoded 0 00000 00000 00000 0 1 00001 00001 00001 1 2 00010 00011 00010 2 3 00011 00010 00011 3 4 00100 00110 00100 4 5 00101 00111 00101 5 6 00110 00101 00110 6 7 00111 00100 00111 7 8 01000 01100 01000 8 9 01001 01101 01001 9 10 01010 01111 01010 10 11 01011 01110 01011 11 12 01100 01010 01100 12 13 01101 01011 01101 13 14 01110 01001 01110 14 15 01111 01000 01111 15 16 10000 11000 10000 16 17 10001 11001 10001 17 18 10010 11011 10010 18 19 10011 11010 10011 19 20 10100 11110 10100 20 21 10101 11111 10101 21 22 10110 11101 10110 22 23 10111 11100 10111 23 24 11000 10100 11000 24 25 11001 10101 11001 25 26 11010 10111 11010 26 27 11011 10110 11011 27 28 11100 10010 11100 28 29 11101 10011 11101 29 30 11110 10001 11110 30 31 11111 10000 11111 31
Groovy
Solution: <lang groovy>def grayEncode = { i ->
i ^ (i >>> 1)
}
def grayDecode; grayDecode = { int code ->
if(code <= 0) return 0 def h = grayDecode(code >>> 1) return (h << 1) + ((code ^ h) & 1)
}</lang>
Test: <lang groovy>def binary = { i, minBits = 1 ->
def remainder = i
def bin = []
while (remainder > 0 || bin.size() <= minBits) {
bin << (remainder & 1)
remainder >>>= 1
}
bin
}
println "number binary gray code decode" println "====== ====== ========= ======" (0..31).each {
def code = grayEncode(it)
def decode = grayDecode(code)
def iB = binary(it, 5)
def cB = binary(code, 5)
printf(" %2d %1d%1d%1d%1d%1d %1d%1d%1d%1d%1d %2d",
it, iB[4],iB[3],iB[2],iB[1],iB[0], cB[4],cB[3],cB[2],cB[1],cB[0], decode)
println()
}</lang>
Results:
number binary gray code decode
====== ====== ========= ======
0 00000 00000 0
1 00001 00001 1
2 00010 00011 2
3 00011 00010 3
4 00100 00110 4
5 00101 00111 5
6 00110 00101 6
7 00111 00100 7
8 01000 01100 8
9 01001 01101 9
10 01010 01111 10
11 01011 01110 11
12 01100 01010 12
13 01101 01011 13
14 01110 01001 14
15 01111 01000 15
16 10000 11000 16
17 10001 11001 17
18 10010 11011 18
19 10011 11010 19
20 10100 11110 20
21 10101 11111 21
22 10110 11101 22
23 10111 11100 23
24 11000 10100 24
25 11001 10101 25
26 11010 10111 26
27 11011 10110 27
28 11100 10010 28
29 11101 10011 29
30 11110 10001 30
31 11111 10000 31
Haskell
For zero padding, replace the %5s specifiers in the format string with %05s.
<lang Haskell>import Data.Bits import Data.Char import Numeric import Control.Monad import Text.Printf
grayToBin :: (Integral t, Bits t) => t -> t grayToBin 0 = 0 grayToBin g = g `xor` (grayToBin $ g `shiftR` 1)
binToGray :: (Integral t, Bits t) => t -> t binToGray b = b `xor` (b `shiftR` 1)
showBinary :: (Integral t, Show t) => t -> String showBinary n = showIntAtBase 2 intToDigit n ""
showGrayCode :: (Integral t, Bits t, PrintfArg t, Show t) => t -> IO () showGrayCode num = do
let bin = showBinary num let gray = showBinary (binToGray num) printf "int: %2d -> bin: %5s -> gray: %5s\n" num bin gray
main = forM_ [0..31::Int] showGrayCode</lang>
Icon and Unicon
The following works in both languages: <lang unicon>link bitint
procedure main()
every write(right(i := 0 to 10,4),":",right(int2bit(i),10)," -> ",
right(g := gEncode(i),10)," -> ",
right(b := gDecode(g),10)," -> ",
right(bit2int(b),10))
end
procedure gEncode(b)
return int2bit(ixor(b, ishift(b,-1)))
end
procedure gDecode(g)
b := g[1] every i := 2 to *g do b ||:= if g[i] == b[i-1] then "0" else "1" return b
end</lang>
Sample run:
->gc 0: 0 -> 0 -> 0 -> 0 1: 1 -> 1 -> 1 -> 1 2: 10 -> 11 -> 10 -> 2 3: 11 -> 10 -> 11 -> 3 4: 100 -> 110 -> 100 -> 4 5: 101 -> 111 -> 101 -> 5 6: 110 -> 101 -> 110 -> 6 7: 111 -> 100 -> 111 -> 7 8: 1000 -> 1100 -> 1000 -> 8 9: 1001 -> 1101 -> 1001 -> 9 10: 1010 -> 1111 -> 1010 -> 10 ->
J
G2B is an invertible function which will translate Gray code to Binary:
<lang j>G2B=: ~:/\&.|:</lang>
Thus G2B inv will translate binary to Gray code.
Required example:
<lang j> n=:i.32
G2B=: ~:/\&.|: (,: ,.@".&.>) 'n';'#:n';'G2B inv#:n';'#.G2B G2B inv#:n'
+--+---------+----------+----------------+ |n |#:n |G2B inv#:n|#.G2B G2B inv#:n| +--+---------+----------+----------------+ | 0|0 0 0 0 0|0 0 0 0 0 | 0 | | 1|0 0 0 0 1|0 0 0 0 1 | 1 | | 2|0 0 0 1 0|0 0 0 1 1 | 2 | | 3|0 0 0 1 1|0 0 0 1 0 | 3 | | 4|0 0 1 0 0|0 0 1 1 0 | 4 | | 5|0 0 1 0 1|0 0 1 1 1 | 5 | | 6|0 0 1 1 0|0 0 1 0 1 | 6 | | 7|0 0 1 1 1|0 0 1 0 0 | 7 | | 8|0 1 0 0 0|0 1 1 0 0 | 8 | | 9|0 1 0 0 1|0 1 1 0 1 | 9 | |10|0 1 0 1 0|0 1 1 1 1 |10 | |11|0 1 0 1 1|0 1 1 1 0 |11 | |12|0 1 1 0 0|0 1 0 1 0 |12 | |13|0 1 1 0 1|0 1 0 1 1 |13 | |14|0 1 1 1 0|0 1 0 0 1 |14 | |15|0 1 1 1 1|0 1 0 0 0 |15 | |16|1 0 0 0 0|1 1 0 0 0 |16 | |17|1 0 0 0 1|1 1 0 0 1 |17 | |18|1 0 0 1 0|1 1 0 1 1 |18 | |19|1 0 0 1 1|1 1 0 1 0 |19 | |20|1 0 1 0 0|1 1 1 1 0 |20 | |21|1 0 1 0 1|1 1 1 1 1 |21 | |22|1 0 1 1 0|1 1 1 0 1 |22 | |23|1 0 1 1 1|1 1 1 0 0 |23 | |24|1 1 0 0 0|1 0 1 0 0 |24 | |25|1 1 0 0 1|1 0 1 0 1 |25 | |26|1 1 0 1 0|1 0 1 1 1 |26 | |27|1 1 0 1 1|1 0 1 1 0 |27 | |28|1 1 1 0 0|1 0 0 1 0 |28 | |29|1 1 1 0 1|1 0 0 1 1 |29 | |30|1 1 1 1 0|1 0 0 0 1 |30 | |31|1 1 1 1 1|1 0 0 0 0 |31 | +--+---------+----------+----------------+</lang>
Java
<lang java> public class Gray { public static long grayEncode(long n){ return n ^ (n >>> 1); }
public static long grayDecode(long n) { long p = n; while ((n >>>= 1) != 0) p ^= n; return p; } public static void main(String[] args){ System.out.println("i\tBinary\tGray\tDecoded"); for(int i = -1; i < 32;i++){ System.out.print(i +"\t"); System.out.print(Integer.toBinaryString(i) + "\t"); System.out.print(Long.toBinaryString(grayEncode(i))+ "\t"); System.out.println(grayDecode(grayEncode(i))); } } } </lang>
- Output:
i Binary Gray Decoded -1 11111111111111111111111111111111 1000000000000000000000000000000000000000000000000000000000000000 -1 0 0 0 0 1 1 1 1 2 10 11 2 3 11 10 3 4 100 110 4 5 101 111 5 6 110 101 6 7 111 100 7 8 1000 1100 8 9 1001 1101 9 10 1010 1111 10 11 1011 1110 11 12 1100 1010 12 13 1101 1011 13 14 1110 1001 14 15 1111 1000 15 16 10000 11000 16 17 10001 11001 17 18 10010 11011 18 19 10011 11010 19 20 10100 11110 20 21 10101 11111 21 22 10110 11101 22 23 10111 11100 23 24 11000 10100 24 25 11001 10101 25 26 11010 10111 26 27 11011 10110 27 28 11100 10010 28 29 11101 10011 29 30 11110 10001 30 31 11111 10000 31
Here is an example encoding function that does it in a bit-by-bit, more human way: <lang java>public static long grayEncode(long n){ long result = 0; for(int exp = 0; n > 0; n /= 2, exp++){ long nextHighestBit = (n >> 1) & 1; if(nextHighestBit == 1){ result += ((n & 1) == 0) ? (1 << exp) : 0; //flip the bit }else{ result += (n & 1) * (1 << exp); //"n & 1" is "this bit", don't flip it } } return result; }</lang> This decoding function should work for gray codes of any size:
<lang java>public static BigInteger grayDecode(BigInteger n){
String nBits = n.toString(2);
String result = nBits.substring(0, 1);
for(int i = 1; i < nBits.length(); i++){
//bin[i] = gray[i] ^ bin[i-1]
//XOR with characters result += nBits.charAt(i) != result.charAt(i - 1) ? "1" : "0"; } return new BigInteger(result, 2); }</lang>
JavaScript
The following code is ES2015.
Module gray-code.js
<lang javascript>export function encode (number) {
return number ^ (number >> 1)
}
export function decode (encodedNumber) {
let number = encodedNumber
while (encodedNumber >>= 1) {
number ^= encodedNumber
}
return number
}</lang> Test <lang javascript>import printf from 'printf' // Module must be installed with npm first import * as gray from './gray-code.js'
console.log(
'Number\t' + 'Binary\t' + 'Gray Code\t' + 'Decoded Gray Code'
)
for (let number = 0; number < 32; number++) {
const grayCode = gray.encode(number) const decodedGrayCode = gray.decode(grayCode)
console.log(printf(
'%2d\t%05d\t%05d\t\t%2d',
number,
number.toString(2),
grayCode.toString(2),
decodedGrayCode
))
}</lang>
- Output:
Number Binary Gray Code Decoded Gray Code 0 00000 00000 0 1 00001 00001 1 2 00010 00011 2 3 00011 00010 3 4 00100 00110 4 5 00101 00111 5 6 00110 00101 6 7 00111 00100 7 8 01000 01100 8 9 01001 01101 9 10 01010 01111 10 11 01011 01110 11 12 01100 01010 12 13 01101 01011 13 14 01110 01001 14 15 01111 01000 15 16 10000 11000 16 17 10001 11001 17 18 10010 11011 18 19 10011 11010 19 20 10100 11110 20 21 10101 11111 21 22 10110 11101 22 23 10111 11100 23 24 11000 10100 24 25 11001 10101 25 26 11010 10111 26 27 11011 10110 27 28 11100 10010 28 29 11101 10011 29 30 11110 10001 30 31 11111 10000 31
Julia
<lang julia>grayencode(n::Integer) = n ⊻ (n >> 1) function graydecode(n::Integer)
r = n
while (n >>= 1) != 0
r ⊻= n
end
return r
end</lang>
Note that these functions work for any integer type, including arbitrary-precision integers (the built-in BigInt type).
K
Binary to Gray code
<lang K> xor: {~x=y}
gray:{x[0],xor':x}
/ variant: using shift
gray1:{(x[0],xor[1_ x;-1_ x])}
/ variant: iterative
gray2:{x[0],{:[x[y-1]=1;~x[y];x[y]]}[x]'1+!(#x)-1}</lang>
Gray code to binary
"Accumulated xor" <lang K> g2b:xor\</lang>
An alternative is to find the inverse of the gray code by tracing until fixpoint. Here we find that 1 1 1 1 1 is the inverse of 1 0 0 0 0 <lang K> gray\1 0 0 0 0 (1 0 0 0 0
1 1 0 0 0 1 0 1 0 0 1 1 1 1 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 1 1 1 1 1 1)
</lang>
As a function (*| takes the last result) <lang K> g2b1:*|{gray x}\</lang>
Iterative version with "do" <lang K> g2b2:{c:#x;b:c#0;b[0]:x[0];i:1;do[#x;b[i]:xor[x[i];b[i-1]];i+:1];b}</lang>
Presentation
<lang K> gray:{x[0],xor':x}
g2b:xor\
/ using allcomb instead of 2_vs'!32 for nicer presentation
allcomb:{+(x#y)_vs!_ y^x}
a:(+allcomb . 5 2)
`0:,/{n:2_sv x;gg:gray x;gb:g2b gg;n2:2_sv gb;
,/$((2$n)," : ",$x," -> ",$gg," -> ",$gb," : ",(2$n2),"\n") }'a</lang>
- Output:
0 : 00000 -> 00000 -> 00000 : 0 1 : 00001 -> 00001 -> 00001 : 1 2 : 00010 -> 00011 -> 00010 : 2 3 : 00011 -> 00010 -> 00011 : 3 4 : 00100 -> 00110 -> 00100 : 4 5 : 00101 -> 00111 -> 00101 : 5 6 : 00110 -> 00101 -> 00110 : 6 7 : 00111 -> 00100 -> 00111 : 7 8 : 01000 -> 01100 -> 01000 : 8 9 : 01001 -> 01101 -> 01001 : 9 10 : 01010 -> 01111 -> 01010 : 10 11 : 01011 -> 01110 -> 01011 : 11 12 : 01100 -> 01010 -> 01100 : 12 13 : 01101 -> 01011 -> 01101 : 13 14 : 01110 -> 01001 -> 01110 : 14 15 : 01111 -> 01000 -> 01111 : 15 16 : 10000 -> 11000 -> 10000 : 16 17 : 10001 -> 11001 -> 10001 : 17 18 : 10010 -> 11011 -> 10010 : 18 19 : 10011 -> 11010 -> 10011 : 19 20 : 10100 -> 11110 -> 10100 : 20 21 : 10101 -> 11111 -> 10101 : 21 22 : 10110 -> 11101 -> 10110 : 22 23 : 10111 -> 11100 -> 10111 : 23 24 : 11000 -> 10100 -> 11000 : 24 25 : 11001 -> 10101 -> 11001 : 25 26 : 11010 -> 10111 -> 11010 : 26 27 : 11011 -> 10110 -> 11011 : 27 28 : 11100 -> 10010 -> 11100 : 28 29 : 11101 -> 10011 -> 11101 : 29 30 : 11110 -> 10001 -> 11110 : 30 31 : 11111 -> 10000 -> 11111 : 31
Kotlin
<lang scala>// version 1.0.6
object Gray {
fun encode(n: Int) = n xor (n shr 1)
fun decode(n: Int): Int {
var p = n
var nn = n
while (nn != 0) {
nn = nn shr 1
p = p xor nn
}
return p
}
}
fun main(args: Array<String>) {
println("Number\tBinary\tGray\tDecoded")
for (i in 0..31) {
print("$i\t${Integer.toBinaryString(i)}\t")
val g = Gray.encode(i)
println("${Integer.toBinaryString(g)}\t${Gray.decode(g)}")
}
}</lang>
- Output:
Number Binary Gray Decoded 0 0 0 0 1 1 1 1 2 10 11 2 3 11 10 3 4 100 110 4 5 101 111 5 6 110 101 6 7 111 100 7 8 1000 1100 8 9 1001 1101 9 10 1010 1111 10 11 1011 1110 11 12 1100 1010 12 13 1101 1011 13 14 1110 1001 14 15 1111 1000 15 16 10000 11000 16 17 10001 11001 17 18 10010 11011 18 19 10011 11010 19 20 10100 11110 20 21 10101 11111 21 22 10110 11101 22 23 10111 11100 23 24 11000 10100 24 25 11001 10101 25 26 11010 10111 26 27 11011 10110 27 28 11100 10010 28 29 11101 10011 29 30 11110 10001 30 31 11111 10000 31
Liberty BASIC
<lang lb>
for r =0 to 31
print " Decimal "; using( "###", r); " is ";
B$ =dec2Bin$( r)
print " binary "; B$; ". Binary "; B$;
G$ =Bin2Gray$( dec2Bin$( r))
print " is "; G$; " in Gray code, or ";
B$ =Gray2Bin$( G$)
print B$; " in pure binary."
next r
end
function Bin2Gray$( bin$) ' Given a binary number as a string, returns Gray code as a string.
g$ =left$( bin$, 1)
for i =2 to len( bin$)
bitA =val( mid$( bin$, i -1, 1))
bitB =val( mid$( bin$, i, 1))
AXorB =bitA xor bitB
g$ =g$ +str$( AXorB)
next i
Bin2Gray$ =g$
end function
function Gray2Bin$( g$) ' Given a Gray code as a string, returns equivalent binary num.
' as a string
gl =len( g$)
b$ =left$( g$, 1)
for i =2 to len( g$)
bitA =val( mid$( b$, i -1, 1))
bitB =val( mid$( g$, i, 1))
AXorB =bitA xor bitB
b$ =b$ +str$( AXorB)
next i
Gray2Bin$ =right$( b$, gl)
end function
function dec2Bin$( num) ' Given an integer decimal, returns binary equivalent as a string
n =num
dec2Bin$ =""
while ( num >0)
dec2Bin$ =str$( num mod 2) +dec2Bin$
num =int( num /2)
wend
if ( n >255) then nBits =16 else nBits =8
dec2Bin$ =right$( "0000000000000000" +dec2Bin$, nBits) ' Pad to 8 bit or 16 bit
end function
function bin2Dec( b$) ' Given a binary number as a string, returns decimal equivalent num.
t =0
d =len( b$)
for k =d to 1 step -1
t =t +val( mid$( b$, k, 1)) *2^( d -k)
next k
bin2Dec =t
end function
</lang>
- Output:
Decimal 0 is binary 00000000. Binary 00000000 is 00000000 in Gray code, or 00000000 in pure binary. Decimal 1 is binary 00000001. Binary 00000001 is 00000001 in Gray code, or 00000001 in pure binary. Decimal 2 is binary 00000010. Binary 00000010 is 00000011 in Gray code, or 00000010 in pure binary. Decimal 3 is binary 00000011. Binary 00000011 is 00000010 in Gray code, or 00000011 in pure binary. Decimal 4 is binary 00000100. Binary 00000100 is 00000110 in Gray code, or 00000100 in pure binary. Decimal 5 is binary 00000101. Binary 00000101 is 00000111 in Gray code, or 00000101 in pure binary. Decimal 6 is binary 00000110. Binary 00000110 is 00000101 in Gray code, or 00000110 in pure binary. Decimal 7 is binary 00000111. Binary 00000111 is 00000100 in Gray code, or 00000111 in pure binary. Decimal 8 is binary 00001000. Binary 00001000 is 00001100 in Gray code, or 00001000 in pure binary. Decimal 9 is binary 00001001. Binary 00001001 is 00001101 in Gray code, or 00001001 in pure binary. Decimal 10 is binary 00001010. Binary 00001010 is 00001111 in Gray code, or 00001010 in pure binary. Decimal 11 is binary 00001011. Binary 00001011 is 00001110 in Gray code, or 00001011 in pure binary. Decimal 12 is binary 00001100. Binary 00001100 is 00001010 in Gray code, or 00001100 in pure binary. Decimal 13 is binary 00001101. Binary 00001101 is 00001011 in Gray code, or 00001101 in pure binary. Decimal 14 is binary 00001110. Binary 00001110 is 00001001 in Gray code, or 00001110 in pure binary. Decimal 15 is binary 00001111. Binary 00001111 is 00001000 in Gray code, or 00001111 in pure binary. Decimal 16 is binary 00010000. Binary 00010000 is 00011000 in Gray code, or 00010000 in pure binary. Decimal 17 is binary 00010001. Binary 00010001 is 00011001 in Gray code, or 00010001 in pure binary. Decimal 18 is binary 00010010. Binary 00010010 is 00011011 in Gray code, or 00010010 in pure binary. Decimal 19 is binary 00010011. Binary 00010011 is 00011010 in Gray code, or 00010011 in pure binary. Decimal 20 is binary 00010100. Binary 00010100 is 00011110 in Gray code, or 00010100 in pure binary. Decimal 21 is binary 00010101. Binary 00010101 is 00011111 in Gray code, or 00010101 in pure binary. Decimal 22 is binary 00010110. Binary 00010110 is 00011101 in Gray code, or 00010110 in pure binary. Decimal 23 is binary 00010111. Binary 00010111 is 00011100 in Gray code, or 00010111 in pure binary. Decimal 24 is binary 00011000. Binary 00011000 is 00010100 in Gray code, or 00011000 in pure binary. Decimal 25 is binary 00011001. Binary 00011001 is 00010101 in Gray code, or 00011001 in pure binary. Decimal 26 is binary 00011010. Binary 00011010 is 00010111 in Gray code, or 00011010 in pure binary. Decimal 27 is binary 00011011. Binary 00011011 is 00010110 in Gray code, or 00011011 in pure binary. Decimal 28 is binary 00011100. Binary 00011100 is 00010010 in Gray code, or 00011100 in pure binary. Decimal 29 is binary 00011101. Binary 00011101 is 00010011 in Gray code, or 00011101 in pure binary. Decimal 30 is binary 00011110. Binary 00011110 is 00010001 in Gray code, or 00011110 in pure binary. Decimal 31 is binary 00011111. Binary 00011111 is 00010000 in Gray code, or 00011111 in pure binary.
Limbo
<lang limbo>implement Gray;
include "sys.m"; sys: Sys; print: import sys; include "draw.m";
Gray: module { init: fn(nil: ref Draw->Context, args: list of string); # Export gray and grayinv so that this module can be used as either a # standalone program or as a library: gray: fn(n: int): int; grayinv: fn(n: int): int; };
init(nil: ref Draw->Context, args: list of string) { sys = load Sys Sys->PATH; for(i := 0; i < 32; i++) { g := gray(i); f := grayinv(g); print("%2d %5s %2d %5s %5s %2d\n", i, binstr(i), g, binstr(g), binstr(f), f); } }
gray(n: int): int { return n ^ (n >> 1); }
grayinv(n: int): int { r := 0; while(n) { r ^= n; n >>= 1; } return r; }
binstr(n: int): string { if(!n) return "0"; s := ""; while(n) { s = (string (n&1)) + s; n >>= 1; } return s; }</lang>
- Output:
0 0 0 0 0 0
1 1 1 1 1 1
2 10 3 11 10 2
3 11 2 10 11 3
4 100 6 110 100 4
5 101 7 111 101 5
6 110 5 101 110 6
7 111 4 100 111 7
8 1000 12 1100 1000 8
9 1001 13 1101 1001 9
10 1010 15 1111 1010 10
11 1011 14 1110 1011 11
12 1100 10 1010 1100 12
13 1101 11 1011 1101 13
14 1110 9 1001 1110 14
15 1111 8 1000 1111 15
16 10000 24 11000 10000 16
17 10001 25 11001 10001 17
18 10010 27 11011 10010 18
19 10011 26 11010 10011 19
20 10100 30 11110 10100 20
21 10101 31 11111 10101 21
22 10110 29 11101 10110 22
23 10111 28 11100 10111 23
24 11000 20 10100 11000 24
25 11001 21 10101 11001 25
26 11010 23 10111 11010 26
27 11011 22 10110 11011 27
28 11100 18 10010 11100 28
29 11101 19 10011 11101 29
30 11110 17 10001 11110 30
31 11111 16 10000 11111 31
Lobster
<lang Lobster>def grey_encode(n) -> int:
return n ^ (n >> 1)
def grey_decode(n) -> int:
var p = n
n = n >> 1
while n != 0:
p = p ^ n
n = n >> 1
return p
for(32) i:
let g = grey_encode(i)
let b = grey_decode(g)
print(number_to_string(i, 10, 2) + " : " +
number_to_string(i, 2, 5) + " ⇾ " +
number_to_string(g, 2, 5) + " ⇾ " +
number_to_string(b, 2, 5) + " : " +
number_to_string(b, 10, 2))</lang>
- Output:
00 : 00000 ⇾ 00000 ⇾ 00000 : 00 01 : 00001 ⇾ 00001 ⇾ 00001 : 01 02 : 00010 ⇾ 00011 ⇾ 00010 : 02 03 : 00011 ⇾ 00010 ⇾ 00011 : 03 04 : 00100 ⇾ 00110 ⇾ 00100 : 04 05 : 00101 ⇾ 00111 ⇾ 00101 : 05 06 : 00110 ⇾ 00101 ⇾ 00110 : 06 07 : 00111 ⇾ 00100 ⇾ 00111 : 07 08 : 01000 ⇾ 01100 ⇾ 01000 : 08 09 : 01001 ⇾ 01101 ⇾ 01001 : 09 10 : 01010 ⇾ 01111 ⇾ 01010 : 10 11 : 01011 ⇾ 01110 ⇾ 01011 : 11 12 : 01100 ⇾ 01010 ⇾ 01100 : 12 13 : 01101 ⇾ 01011 ⇾ 01101 : 13 14 : 01110 ⇾ 01001 ⇾ 01110 : 14 15 : 01111 ⇾ 01000 ⇾ 01111 : 15 16 : 10000 ⇾ 11000 ⇾ 10000 : 16 17 : 10001 ⇾ 11001 ⇾ 10001 : 17 18 : 10010 ⇾ 11011 ⇾ 10010 : 18 19 : 10011 ⇾ 11010 ⇾ 10011 : 19 20 : 10100 ⇾ 11110 ⇾ 10100 : 20 21 : 10101 ⇾ 11111 ⇾ 10101 : 21 22 : 10110 ⇾ 11101 ⇾ 10110 : 22 23 : 10111 ⇾ 11100 ⇾ 10111 : 23 24 : 11000 ⇾ 10100 ⇾ 11000 : 24 25 : 11001 ⇾ 10101 ⇾ 11001 : 25 26 : 11010 ⇾ 10111 ⇾ 11010 : 26 27 : 11011 ⇾ 10110 ⇾ 11011 : 27 28 : 11100 ⇾ 10010 ⇾ 11100 : 28 29 : 11101 ⇾ 10011 ⇾ 11101 : 29 30 : 11110 ⇾ 10001 ⇾ 11110 : 30 31 : 11111 ⇾ 10000 ⇾ 11111 : 31
Logo
<lang logo>to gray_encode :number
output bitxor :number lshift :number -1
end
to gray_decode :code
local "value make "value 0 while [:code > 0] [ make "value bitxor :code :value make "code lshift :code -1 ] output :value
end</lang>
Demonstration code, including formatters: <lang logo>to format :str :width [pad (char 32)]
while [(count :str) < :width] [ make "str word :pad :str ] output :str
end
- Output binary representation of a number
to binary :number [:width 1]
local "bits
ifelse [:number = 0] [
make "bits 0
] [
make "bits "
while [:number > 0] [
make "bits word (bitand :number 1) :bits
make "number lshift :number -1
]
]
output (format :bits :width 0)
end
repeat 32 [
make "num repcount - 1
make "gray gray_encode :num
make "decoded gray_decode :gray
print (sentence (format :num 2) ": (binary :num 5) ": (binary :gray 5) ":
(binary :decoded 5) ": (format :decoded 2)) ]
bye</lang>
- Output:
0 : 00000 : 00000 : 00000 : 0 1 : 00001 : 00001 : 00001 : 1 2 : 00010 : 00011 : 00010 : 2 3 : 00011 : 00010 : 00011 : 3 4 : 00100 : 00110 : 00100 : 4 5 : 00101 : 00111 : 00101 : 5 6 : 00110 : 00101 : 00110 : 6 7 : 00111 : 00100 : 00111 : 7 8 : 01000 : 01100 : 01000 : 8 9 : 01001 : 01101 : 01001 : 9 10 : 01010 : 01111 : 01010 : 10 11 : 01011 : 01110 : 01011 : 11 12 : 01100 : 01010 : 01100 : 12 13 : 01101 : 01011 : 01101 : 13 14 : 01110 : 01001 : 01110 : 14 15 : 01111 : 01000 : 01111 : 15 16 : 10000 : 11000 : 10000 : 16 17 : 10001 : 11001 : 10001 : 17 18 : 10010 : 11011 : 10010 : 18 19 : 10011 : 11010 : 10011 : 19 20 : 10100 : 11110 : 10100 : 20 21 : 10101 : 11111 : 10101 : 21 22 : 10110 : 11101 : 10110 : 22 23 : 10111 : 11100 : 10111 : 23 24 : 11000 : 10100 : 11000 : 24 25 : 11001 : 10101 : 11001 : 25 26 : 11010 : 10111 : 11010 : 26 27 : 11011 : 10110 : 11011 : 27 28 : 11100 : 10010 : 11100 : 28 29 : 11101 : 10011 : 11101 : 29 30 : 11110 : 10001 : 11110 : 30 31 : 11111 : 10000 : 11111 : 31
Lua
This code uses the Lua BitOp module. Designed to be a module named gray.lua. <lang lua>local _M = {}
local bit = require('bit') local math = require('math')
_M.encode = function(number)
return bit.bxor(number, bit.rshift(number, 1));
end
_M.decode = function(gray_code)
local value = 0 while gray_code > 0 do gray_code, value = bit.rshift(gray_code, 1), bit.bxor(gray_code, value) end return value
end
return _M</lang>
Demonstration code: <lang lua>local bit = require 'bit' local gray = require 'gray'
-- simple binary string formatter local function to_bit_string(n, width)
width = width or 1 local output = "" while n > 0 do output = bit.band(n,1) .. output n = bit.rshift(n,1) end while #output < width do output = '0' .. output end return output
end
for i = 0,31 do
g = gray.encode(i);
gd = gray.decode(g);
print(string.format("%2d : %s => %s => %s : %2d", i,
to_bit_string(i,5), to_bit_string(g, 5),
to_bit_string(gd,5), gd))
end</lang>
- Output:
0 : 00000 => 00000 => 00000 : 0 1 : 00001 => 00001 => 00001 : 1 2 : 00010 => 00011 => 00010 : 2 3 : 00011 => 00010 => 00011 : 3 4 : 00100 => 00110 => 00100 : 4 5 : 00101 => 00111 => 00101 : 5 6 : 00110 => 00101 => 00110 : 6 7 : 00111 => 00100 => 00111 : 7 8 : 01000 => 01100 => 01000 : 8 9 : 01001 => 01101 => 01001 : 9 10 : 01010 => 01111 => 01010 : 10 11 : 01011 => 01110 => 01011 : 11 12 : 01100 => 01010 => 01100 : 12 13 : 01101 => 01011 => 01101 : 13 14 : 01110 => 01001 => 01110 : 14 15 : 01111 => 01000 => 01111 : 15 16 : 10000 => 11000 => 10000 : 16 17 : 10001 => 11001 => 10001 : 17 18 : 10010 => 11011 => 10010 : 18 19 : 10011 => 11010 => 10011 : 19 20 : 10100 => 11110 => 10100 : 20 21 : 10101 => 11111 => 10101 : 21 22 : 10110 => 11101 => 10110 : 22 23 : 10111 => 11100 => 10111 : 23 24 : 11000 => 10100 => 11000 : 24 25 : 11001 => 10101 => 11001 : 25 26 : 11010 => 10111 => 11010 : 26 27 : 11011 => 10110 => 11011 : 27 28 : 11100 => 10010 => 11100 : 28 29 : 11101 => 10011 => 11101 : 29 30 : 11110 => 10001 => 11110 : 30 31 : 11111 => 10000 => 11111 : 31
M2000 Interpreter
Additions to showing the modules/functions replacement mechanism of M2000
<lang M2000 Interpreter> Module Code32 (&code(), &decode()){
Const d$="{0::-2} {1:-6} {2:-6} {3:-6} {4::-2}"
For i=0 to 32
g=code(i)
b=decode(g)
Print format$(d$, i, @bin$(i), @bin$(g), @bin$(b), b)
Next
// static function
Function bin$(a)
a$=""
Do n= a mod 2 : a$=if$(n=1->"1", "0")+a$ : a|div 2 : Until a==0
=a$
End Function
} Module GrayCode {
Module doit (&a(), &b()) { }
Function GrayEncode(a) {
=binary.xor(a, binary.shift(a,-1))
}
Function GrayDecode(a) {
b=0
Do b=binary.xor(a, b) : a=binary.shift(a,-1) : Until a==0
=b
}
// pass 2 functions to Code32
doit &GrayEncode(), &GrayDecode()
} // pass Code32 to GrayCode in place of doit GrayCode ; doit as Code32 </lang>
- Output:
0 0 0 0 0 1 1 1 1 1 2 10 11 10 2 3 11 10 11 3 4 100 110 100 4 5 101 111 101 5 6 110 101 110 6 7 111 100 111 7 8 1000 1100 1000 8 9 1001 1101 1001 9 10 1010 1111 1010 10 11 1011 1110 1011 11 12 1100 1010 1100 12 13 1101 1011 1101 13 14 1110 1001 1110 14 15 1111 1000 1111 15 16 10000 11000 10000 16 17 10001 11001 10001 17 18 10010 11011 10010 18 19 10011 11010 10011 19 20 10100 11110 10100 20 21 10101 11111 10101 21 22 10110 11101 10110 22 23 10111 11100 10111 23 24 11000 10100 11000 24 25 11001 10101 11001 25 26 11010 10111 11010 26 27 11011 10110 11011 27 28 11100 10010 11100 28 29 11101 10011 11101 29 30 11110 10001 11110 30 31 11111 10000 11111 31 32 100000 110000 100000 32
Mathematica / Wolfram Language
<lang Mathematica>graycode[n_]:=BitXor[n,BitShiftRight[n]] graydecode[n_]:=Fold[BitXor,0,FixedPointList[BitShiftRight,n]]</lang>
- Output:
Required example:
Grid[{# ,IntegerDigits[#,2],IntegerDigits[graycode@#,2], IntegerDigits[graydecode@graycode@#,2]}&/@Range[32]]
1 {1} {1} {1}
2 {1,0} {1,1} {1,0}
3 {1,1} {1,0} {1,1}
...
15 {1,1,1,1} {1,0,0,0} {1,1,1,1}
...
30 {1,1,1,1,0} {1,0,0,0,1} {1,1,1,1,0}
31 {1,1,1,1,1} {1,0,0,0,0} {1,1,1,1,1}
32 {1,0,0,0,0,0} {1,1,0,0,0,0} {1,0,0,0,0,0}
MATLAB
<lang MATLAB> %% Gray Code Generator % this script generates gray codes of n bits % total 2^n -1 continuous gray codes will be generated. % this code follows a recursive approach. therefore, % it can be slow for large n
clear all; clc;
bits = input('Enter the number of bits: '); if (bits<1)
disp('Sorry, number of bits should be positive');
elseif (mod(bits,1)~=0)
disp('Sorry, number of bits can only be positive integers');
else
initial_container = [0;1];
if bits == 1
result = initial_container;
else
previous_container = initial_container;
for i=2:bits
new_gray_container = zeros(2^i,i);
new_gray_container(1:(2^i)/2,1) = 0;
new_gray_container(((2^i)/2)+1:end,1) = 1;
for j = 1:(2^i)/2
new_gray_container(j,2:end) = previous_container(j,:);
end
for j = ((2^i)/2)+1:2^i
new_gray_container(j,2:end) = previous_container((2^i)+1-j,:);
end
previous_container = new_gray_container;
end
result = previous_container;
end
fprintf('Gray code of %d bits',bits);
disp(' ');
disp(result);
end </lang>
- Output:
Enter the number of bits: 5
Gray code of 5 bits
0 0 0 0 0
0 0 0 0 1
0 0 0 1 1
0 0 0 1 0
0 0 1 1 0
0 0 1 1 1
0 0 1 0 1
0 0 1 0 0
0 1 1 0 0
0 1 1 0 1
0 1 1 1 1
0 1 1 1 0
0 1 0 1 0
0 1 0 1 1
0 1 0 0 1
0 1 0 0 0
1 1 0 0 0
1 1 0 0 1
1 1 0 1 1
1 1 0 1 0
1 1 1 1 0
1 1 1 1 1
1 1 1 0 1
1 1 1 0 0
1 0 1 0 0
1 0 1 0 1
1 0 1 1 1
1 0 1 1 0
1 0 0 1 0
1 0 0 1 1
1 0 0 0 1
1 0 0 0 0
Mercury
The following is a full implementation of Gray encoding and decoding. It publicly exposes the gray type along with the following value conversion functions:
- gray.from_int/1
- gray.to_int/1
The from_int/1 and to_int/1 functions are value conversion functions. from_int/1 converts an int value into the enclosing gray type. to_int/1 converts a gray value back into a regular int type.
The additional gray.coerce/2 predicate converts the representation underlying a gray value into an int value or vice versa (it is moded in both directions). For type safety reasons we do not wish to generally expose the underlying representation, but for some purposes, most notably I/O or storage or their ilk we have to break the type safety. The coerce/2 predicate is used for this purpose.
<lang mercury>:- module gray.
- - interface.
- - import_module int.
- - type gray.
% VALUE conversion functions
- - func gray.from_int(int) = gray.
- - func gray.to_int(gray) = int.
% REPRESENTATION conversion predicate
- - pred gray.coerce(gray, int).
- - mode gray.coerce(in, out) is det.
- - mode gray.coerce(out, in) is det.
- - implementation.
- - import_module list.
- - type gray
---> gray(int).
gray.from_int(X) = gray(X `xor` (X >> 1)).
gray.to_int(gray(G)) = (G > 0 -> G `xor` gray.to_int(gray(G >> 1))
; G).
gray.coerce(gray(I), I).
- - end_module gray.</lang>
The following program tests the above code:
<lang mercury>:- module gray_test.
- - interface.
- - import_module io.
- - pred main(io::di, io::uo) is det.
- - implementation.
- - import_module gray.
- - import_module int, list, string.
- - pred check_conversion(list(int)::in, list(gray)::out) is semidet.
- - pred display_lists(list(int)::in, list(gray)::in, io::di, io::uo) is det.
- - pred display_record(int::in, gray::in, io::di, io::uo) is det.
main(!IO) :-
Numbers = 0..31,
( check_conversion(Numbers, Grays) ->
io.format("%8s %8s %8s\n", [s("Number"), s("Binary"), s("Gray")], !IO),
io.format("%8s %8s %8s\n", [s("------"), s("------"), s("----")], !IO),
display_lists(Numbers, Grays, !IO)
; io.write("Either conversion or back-conversion failed.\n", !IO)).
check_conversion(Numbers, Grays) :-
Grays = list.map(gray.from_int, Numbers), Numbers = list.map(gray.to_int, Grays).
display_lists(Numbers, Grays, !IO) :-
list.foldl_corresponding(display_record, Numbers, Grays, !IO).
display_record(Number, Gray, !IO) :-
gray.coerce(Gray, GrayRep),
NumBin = string.int_to_base_string(Number, 2),
GrayBin = string.int_to_base_string(GrayRep, 2),
io.format("%8d %8s %8s\n", [i(Number), s(NumBin), s(GrayBin)], !IO).
- - end_module gray_test.</lang>
The main/2 predicate generates a list of numbers from 0 to 31 inclusive and then checks that conversion is working properly. It does so by calling the check_conversion/2 predicate with the list of numbers as an input and the list of Gray-encoded numbers as an output. Note the absence of the usual kinds of testing you'd see in most programming languages. gray.from_int/1 is mapped over the Numbers (input) list and placed into the Grays (output) list. Then gray.to_int is mapped over the Grays list and placed into the Numbers (input) list. Or so it would seem to those used to imperative or functional languages.
In reality what's happening is unification. Since the Grays list is not yet populated, unification is very similar notionally to assignment in other languages. Numbers, however, is instantiated and thus unification is more like testing for equality.
If the conversions check out, main/2 prints off some headers and then displays the lists. Here we're cluttering up the namespace of the gray_test module a little by providing a one-line predicate. While it is true that we could just take the contents of that predicate and place it inline, we've chosen not to do that because the name display_lists communicates more effectively what we intend. The compiler is smart enough to automatically inline that predicate call so there's no efficiency reason not to do it.
However we choose to do that, the result is the same: repeated calls to display_record/4. In that predicate the aforementioned coerce/2 predicate extracts, in this case, the Gray value's representation. This value and the corresponding int value are then converted into a string showing the base-2 representation of their values. io.format/4 then prints them off in a nice format.
The output of the program looks like this:
Number Binary Gray
------ ------ ----
0 0 0
1 1 1
2 10 11
3 11 10
4 100 110
5 101 111
6 110 101
7 111 100
8 1000 1100
9 1001 1101
10 1010 1111
11 1011 1110
12 1100 1010
13 1101 1011
14 1110 1001
15 1111 1000
16 10000 11000
17 10001 11001
18 10010 11011
19 10011 11010
20 10100 11110
21 10101 11111
22 10110 11101
23 10111 11100
24 11000 10100
25 11001 10101
26 11010 10111
27 11011 10110
28 11100 10010
29 11101 10011
30 11110 10001
31 11111 10000
Nim
<lang nim>proc grayEncode(n: int): int =
n xor (n shr 1)
proc grayDecode(n: int): int =
result = n var t = n while t > 0: t = t shr 1 result = result xor t</lang>
Demonstration code: <lang nim>import strutils, strformat
for i in 0 .. 32:
echo &"{i:>2} => {toBin(grayEncode(i), 6)} => {grayDecode(grayEncode(i)):>2}"</lang>
- Output:
0 => 000000 => 0 1 => 000001 => 1 2 => 000011 => 2 3 => 000010 => 3 4 => 000110 => 4 5 => 000111 => 5 6 => 000101 => 6 7 => 000100 => 7 8 => 001100 => 8 9 => 001101 => 9 10 => 001111 => 10 11 => 001110 => 11 12 => 001010 => 12 13 => 001011 => 13 14 => 001001 => 14 15 => 001000 => 15 16 => 011000 => 16 17 => 011001 => 17 18 => 011011 => 18 19 => 011010 => 19 20 => 011110 => 20 21 => 011111 => 21 22 => 011101 => 22 23 => 011100 => 23 24 => 010100 => 24 25 => 010101 => 25 26 => 010111 => 26 27 => 010110 => 27 28 => 010010 => 28 29 => 010011 => 29 30 => 010001 => 30 31 => 010000 => 31 32 => 110000 => 32
NOWUT
adapted from C <lang NOWUT>; link with PIOxxx.OBJ
sectiondata
output: db " : " inbinary: db "00000 => " graybinary: db "00000 => " outbinary: db "00000"
db 13,10,0 ; carriage return and null terminator
sectioncode
start!
gosub initplatform
beginfunc
localvar i.d,g.d,b.d
i=0
whileless i,32
callex g,gray_encode,i
callex b,gray_decode,g
callex ,bin2string,i,inbinary,5 ; 5 = number of binary digits
callex ,bin2string,g,graybinary,5
callex ,bin2string,b,outbinary,5
callex ,printhex8,i ; display hex value
; because there is no PIO routine for decimals...
callex ,printnt,output.a
i=_+1
wend
endfunc
end
gray_encode:
beginfunc n.d
n=_ xor (n shr 1)
endfunc n
returnex 4 ; clean off 1 parameter from the stack
gray_decode:
beginfunc n.d
localvar p.d
p=n
whilegreater n,1
n=_ shr 1 > p=_ xor n
wend
endfunc p
returnex 4 ; clean off 1 parameter from the stack
bin2string:
beginfunc digits.d,straddr.d,value.d
whilegreater digits,0
digits=_-1
[straddr].b=value shr digits and 1+$30 ; write an ASCII '0' or '1'
straddr=_+1 ; increment the pointer
wend
endfunc
returnex $0C ; clean off 3 parameters from the stack
</lang>
- Output:
00 : 00000 => 00000 => 00000 01 : 00001 => 00001 => 00001 02 : 00010 => 00011 => 00010 03 : 00011 => 00010 => 00011 04 : 00100 => 00110 => 00100 05 : 00101 => 00111 => 00101 06 : 00110 => 00101 => 00110 07 : 00111 => 00100 => 00111 08 : 01000 => 01100 => 01000 09 : 01001 => 01101 => 01001 0A : 01010 => 01111 => 01010 0B : 01011 => 01110 => 01011 0C : 01100 => 01010 => 01100 0D : 01101 => 01011 => 01101 0E : 01110 => 01001 => 01110 0F : 01111 => 01000 => 01111 10 : 10000 => 11000 => 10000 11 : 10001 => 11001 => 10001 12 : 10010 => 11011 => 10010 13 : 10011 => 11010 => 10011 14 : 10100 => 11110 => 10100 15 : 10101 => 11111 => 10101 16 : 10110 => 11101 => 10110 17 : 10111 => 11100 => 10111 18 : 11000 => 10100 => 11000 19 : 11001 => 10101 => 11001 1A : 11010 => 10111 => 11010 1B : 11011 => 10110 => 11011 1C : 11100 => 10010 => 11100 1D : 11101 => 10011 => 11101 1E : 11110 => 10001 => 11110 1F : 11111 => 10000 => 11111
OCaml
<lang ocaml>let gray_encode b =
b lxor (b lsr 1)
let gray_decode n =
let rec aux p n = if n = 0 then p else aux (p lxor n) (n lsr 1) in aux n (n lsr 1)
let bool_string len n =
let s = String.make len '0' in let rec aux i n = if n land 1 = 1 then s.[i] <- '1'; if i <= 0 then s else aux (pred i) (n lsr 1) in aux (pred len) n
let () =
let s = bool_string 5 in for i = 0 to pred 32 do let g = gray_encode i in let b = gray_decode g in Printf.printf "%2d : %s => %s => %s : %2d\n" i (s i) (s g) (s b) b done</lang>
PARI/GP
This code may have exposed a bug in PARI 2.4.4: apply(Str, 1) fails.
As a workaround I used a closure: apply(k->Str(k), 1).
<lang parigp>toGray(n)=bitxor(n,n>>1);
fromGray(n)=my(k=1,m=n);while(m>>k,n=bitxor(n,n>>k);k+=k);n;
bin(n)=concat(apply(k->Str(k),binary(n)))
for(n=0,31,print(n"\t"bin(n)"\t"bin(g=toGray(n))"\t"fromGray(g)))</lang>
- Output:
0 0 0 0 1 1 1 1 2 10 11 2 3 11 10 3 4 100 110 4 5 101 111 5 6 110 101 6 7 111 100 7 8 1000 1100 8 9 1001 1101 9 10 1010 1111 10 11 1011 1110 11 12 1100 1010 12 13 1101 1011 13 14 1110 1001 14 15 1111 1000 15 16 10000 11000 16 17 10001 11001 17 18 10010 11011 18 19 10011 11010 19 20 10100 11110 20 21 10101 11111 21 22 10110 11101 22 23 10111 11100 23 24 11000 10100 24 25 11001 10101 25 26 11010 10111 26 27 11011 10110 27 28 11100 10010 28 29 11101 10011 29 30 11110 10001 30 31 11111 10000 31
Pascal
See Delphi
Perl
<lang perl>sub bin2gray {
return $_[0] ^ ($_[0] >> 1);
}
sub gray2bin {
my ($num)= @_;
my $bin= $num;
while( $num >>= 1 ) {
# a bit ends up flipped iff an odd number of bits to its left is set.
$bin ^= $num; # different from the suggested algorithm;
} # avoids using bit mask and explicit bittery
return $bin;
}
for (0..31) {
my $gr= bin2gray($_); printf "%d\t%b\t%b\t%b\n", $_, $_, $gr, gray2bin($gr);
}</lang>
Phix
(turned out to be almost the same as Euphoria)
with javascript_semantics function gray_encode(integer n) return xor_bits(n,floor(n/2)) end function function gray_decode(integer n) integer r = 0 while n>0 do r = xor_bits(r,n) n = floor(n/2) end while return r end function integer e,d puts(1," N Binary Gray Decoded\n"& "== ===== ===== =======\n") for i=0 to 31 do e = gray_encode(i) d = gray_decode(e) printf(1,"%2d %05b %05b %2d\n",{i,i,e,d}) end for
- Output:
N Binary Gray Decoded == ===== ===== ======= 0 00000 00000 0 1 00001 00001 1 2 00010 00011 2 3 00011 00010 3 4 00100 00110 4 5 00101 00111 5 6 00110 00101 6 7 00111 00100 7 8 01000 01100 8 9 01001 01101 9 10 01010 01111 10 11 01011 01110 11 12 01100 01010 12 13 01101 01011 13 14 01110 01001 14 15 01111 01000 15 16 10000 11000 16 17 10001 11001 17 18 10010 11011 18 19 10011 11010 19 20 10100 11110 20 21 10101 11111 21 22 10110 11101 22 23 10111 11100 23 24 11000 10100 24 25 11001 10101 25 26 11010 10111 26 27 11011 10110 27 28 11100 10010 28 29 11101 10011 29 30 11110 10001 30 31 11111 10000 31
PHP
<lang php> <?php
/**
* @author Elad Yosifon */
/**
* @param int $binary * @return int */
function gray_encode($binary){ return $binary ^ ($binary >> 1); }
/**
* @param int $gray * @return int */
function gray_decode($gray){ $binary = $gray; while($gray >>= 1) $binary ^= $gray; return $binary; }
for($i=0;$i<32;$i++){ $gray_encoded = gray_encode($i); printf("%2d : %05b => %05b => %05b : %2d \n",$i, $i, $gray_encoded, $gray_encoded, gray_decode($gray_encoded)); } </lang>
- Output:
0 : 00000 => 00000 => 00000 : 0 1 : 00001 => 00001 => 00001 : 1 2 : 00010 => 00011 => 00011 : 2 3 : 00011 => 00010 => 00010 : 3 4 : 00100 => 00110 => 00110 : 4 5 : 00101 => 00111 => 00111 : 5 6 : 00110 => 00101 => 00101 : 6 7 : 00111 => 00100 => 00100 : 7 8 : 01000 => 01100 => 01100 : 8 9 : 01001 => 01101 => 01101 : 9 10 : 01010 => 01111 => 01111 : 10 11 : 01011 => 01110 => 01110 : 11 12 : 01100 => 01010 => 01010 : 12 13 : 01101 => 01011 => 01011 : 13 14 : 01110 => 01001 => 01001 : 14 15 : 01111 => 01000 => 01000 : 15 16 : 10000 => 11000 => 11000 : 16 17 : 10001 => 11001 => 11001 : 17 18 : 10010 => 11011 => 11011 : 18 19 : 10011 => 11010 => 11010 : 19 20 : 10100 => 11110 => 11110 : 20 21 : 10101 => 11111 => 11111 : 21 22 : 10110 => 11101 => 11101 : 22 23 : 10111 => 11100 => 11100 : 23 24 : 11000 => 10100 => 10100 : 24 25 : 11001 => 10101 => 10101 : 25 26 : 11010 => 10111 => 10111 : 26 27 : 11011 => 10110 => 10110 : 27 28 : 11100 => 10010 => 10010 : 28 29 : 11101 => 10011 => 10011 : 29 30 : 11110 => 10001 => 10001 : 30 31 : 11111 => 10000 => 10000 : 31
PicoLisp
<lang PicoLisp>(de grayEncode (N)
(bin (x| N (>> 1 N))) )
(de grayDecode (G)
(bin
(pack
(let X 0
(mapcar
'((C) (setq X (x| X (format C))))
(chop G) ) ) ) ) )</lang>
Test: <lang PicoLisp>(prinl " Binary Gray Decoded") (for I (range 0 31)
(let G (grayEncode I)
(tab (4 9 9 9) I (bin I) G (grayDecode G)) ) )</lang>
- Output:
Binary Gray Decoded 0 0 0 0 1 1 1 1 2 10 11 2 3 11 10 3 4 100 110 4 5 101 111 5 6 110 101 6 7 111 100 7 8 1000 1100 8 9 1001 1101 9 10 1010 1111 10 11 1011 1110 11 12 1100 1010 12 13 1101 1011 13 14 1110 1001 14 15 1111 1000 15 16 10000 11000 16 17 10001 11001 17 18 10010 11011 18 19 10011 11010 19 20 10100 11110 20 21 10101 11111 21 22 10110 11101 22 23 10111 11100 23 24 11000 10100 24 25 11001 10101 25 26 11010 10111 26 27 11011 10110 27 28 11100 10010 28 29 11101 10011 29 30 11110 10001 30 31 11111 10000 31
PL/I
<lang PL/I>(stringrange, stringsize): Gray_code: procedure options (main); /* 15 November 2013 */
declare (bin(0:31), g(0:31), b2(0:31)) bit (5); declare (c, carry) bit (1); declare (i, j) fixed binary (7);
bin(0) = '00000'b;
do i = 0 to 31;
if i > 0 then
do;
carry = '1'b;
bin(i) = bin(i-1);
do j = 5 to 1 by -1;
c = substr(bin(i), j, 1) & carry;
substr(bin(i), j, 1) = substr(bin(i), j, 1) ^ carry;
carry = c;
end;
end;
g(i) = bin(i) ^ '0'b || substr(bin(i), 1, 4);
end;
do i = 0 to 31;
substr(b2(i), 1, 1) = substr(g(i), 1, 1);
do j = 2 to 5;
substr(b2(i), j, 1) = substr(g(i), j, 1) ^ substr(bin(i), j-1, 1);
end;
end;
do i = 0 to 31;
put skip edit (i, bin(i), g(i), b2(i)) (f(2), 3(x(1), b));
end;
end Gray_code;</lang>
0 00000 00000 00000 1 00001 00001 00001 2 00010 00011 00010 3 00011 00010 00011 4 00100 00110 00100 5 00101 00111 00101 6 00110 00101 00110 7 00111 00100 00111 8 01000 01100 01000 9 01001 01101 01001 10 01010 01111 01010 11 01011 01110 01011 12 01100 01010 01100 13 01101 01011 01101 14 01110 01001 01110 15 01111 01000 01111 16 10000 11000 10000 17 10001 11001 10001 18 10010 11011 10010 19 10011 11010 10011 20 10100 11110 10100 21 10101 11111 10101 22 10110 11101 10110 23 10111 11100 10111 24 11000 10100 11000 25 11001 10101 11001 26 11010 10111 11010 27 11011 10110 11011 28 11100 10010 11100 29 11101 10011 11101 30 11110 10001 11110 31 11111 10000 11111
PowerBASIC
<lang powerbasic>function gray%(byval n%)
gray%=n% xor (n%\2)
end function
function igray%(byval n%)
r%=0 while n%>0 r%=r% xor n% shift right n%,1 wend igray%=r%
end function
print " N GRAY INV" for n%=0 to 31
g%=gray%(n%) print bin$(n%);" ";bin$(g%);" ";bin$(igray%(g%))
next</lang>
Prolog
Codecs
The encoding and decoding predicates are simple and will work with any Prolog that supports bitwise integer operations.
<lang Prolog>to_gray(N, G) :-
N0 is N >> 1, G is N xor N0.
from_gray(G, N) :-
( G > 0
-> S is G >> 1,
from_gray(S, N0),
N is G xor N0
; N is G ).</lang>
Test Code
A quick driver around this to test it will prove the point. (This test script uses features not available in every Prolog implementation.)
<lang Prolog>:- use_module(library(apply)).
to_gray(N, G) :-
N0 is N >> 1, G is N xor N0.
from_gray(G, N) :-
( G > 0
-> S is G >> 1,
from_gray(S, N0),
N is G xor N0
; N is G ).
make_num(In, Out) :-
atom_to_term(In, Out, _), integer(Out).
write_record(Number, Gray, Decoded) :-
format('~w~10|~2r~10+~2r~10+~2r~10+~w~n',
[Number, Number, Gray, Decoded, Decoded]).
go :-
setof(N, between(0, 31, N), Numbers),
maplist(to_gray, Numbers, Grays),
maplist(from_gray, Grays, Decodeds),
format('~w~10|~w~10+~w~10+~w~10+~w~n',
['Number', 'Binary', 'Gray', 'Decoded', 'Number']),
format('~w~10|~w~10+~w~10+~w~10+~w~n',
['------', '------', '----', '-------', '------']),
maplist(write_record, Numbers, Grays, Decodeds).
go :- halt(1). </lang>
- Output:
Putting all of this in a file, we execute it, getting the following results:
% swipl -q -t go -f gray.pl # OR: yap -q -z go,halt -f gray.pl Number Binary Gray Decoded Number ------ ------ ---- ------- ------ 0 0 0 0 0 1 1 1 1 1 2 10 11 10 2 3 11 10 11 3 4 100 110 100 4 5 101 111 101 5 6 110 101 110 6 7 111 100 111 7 8 1000 1100 1000 8 9 1001 1101 1001 9 10 1010 1111 1010 10 11 1011 1110 1011 11 12 1100 1010 1100 12 13 1101 1011 1101 13 14 1110 1001 1110 14 15 1111 1000 1111 15 16 10000 11000 10000 16 17 10001 11001 10001 17 18 10010 11011 10010 18 19 10011 11010 10011 19 20 10100 11110 10100 20 21 10101 11111 10101 21 22 10110 11101 10110 22 23 10111 11100 10111 23 24 11000 10100 11000 24 25 11001 10101 11001 25 26 11010 10111 11010 26 27 11011 10110 11011 27 28 11100 10010 11100 28 29 11101 10011 11101 29 30 11110 10001 11110 30 31 11111 10000 11111 31
PureBasic
<lang PureBasic>Procedure.i gray_encode(n)
ProcedureReturn n ! (n >> 1)
EndProcedure
Procedure.i gray_decode(g)
Protected bit = 1 << (8 * SizeOf(Integer) - 2) Protected b = g & bit, p = b >> 1 While bit > 1 bit >> 1 b | (p ! (g & bit)) p = (b & bit) >> 1 Wend ProcedureReturn b
EndProcedure
If OpenConsole()
PrintN("Number Gray Binary Decoded")
Define i, n
For i = 0 To 31
g = gray_encode(i)
Print(RSet(Str(i), 2, "0") + Space(5))
Print(RSet(Bin(g, #PB_Byte), 5, "0") + Space(2))
n = gray_decode(g)
Print(RSet(Bin(n, #PB_Byte), 5, "0") + Space(3))
PrintN(RSet(Str(n), 2, "0"))
Next
Print(#CRLF$ + #CRLF$ + "Press ENTER to exit"): Input()
CloseConsole()
EndIf</lang>
- Output:
Number Gray Binary Decoded 00 00000 00000 00 01 00001 00001 01 02 00011 00010 02 03 00010 00011 03 04 00110 00100 04 05 00111 00101 05 06 00101 00110 06 07 00100 00111 07 08 01100 01000 08 09 01101 01001 09 10 01111 01010 10 11 01110 01011 11 12 01010 01100 12 13 01011 01101 13 14 01001 01110 14 15 01000 01111 15 16 11000 10000 16 17 11001 10001 17 18 11011 10010 18 19 11010 10011 19 20 11110 10100 20 21 11111 10101 21 22 11101 10110 22 23 11100 10111 23 24 10100 11000 24 25 10101 11001 25 26 10111 11010 26 27 10110 11011 27 28 10010 11100 28 29 10011 11101 29 30 10001 11110 30 31 10000 11111 31
Python
Python: on integers
Works with python integers <lang>def gray_encode(n):
return n ^ n >> 1
def gray_decode(n):
m = n >> 1
while m:
n ^= m
m >>= 1
return n
if __name__ == '__main__':
print("DEC, BIN => GRAY => DEC")
for i in range(32):
gray = gray_encode(i)
dec = gray_decode(gray)
print(f" {i:>2d}, {i:>05b} => {gray:>05b} => {dec:>2d}")</lang>
- Output:
DEC, BIN => GRAY => DEC 0, 00000 => 00000 => 0 1, 00001 => 00001 => 1 2, 00010 => 00011 => 2 3, 00011 => 00010 => 3 4, 00100 => 00110 => 4 5, 00101 => 00111 => 5 6, 00110 => 00101 => 6 7, 00111 => 00100 => 7 8, 01000 => 01100 => 8 9, 01001 => 01101 => 9 10, 01010 => 01111 => 10 11, 01011 => 01110 => 11 12, 01100 => 01010 => 12 13, 01101 => 01011 => 13 14, 01110 => 01001 => 14 15, 01111 => 01000 => 15 16, 10000 => 11000 => 16 17, 10001 => 11001 => 17 18, 10010 => 11011 => 18 19, 10011 => 11010 => 19 20, 10100 => 11110 => 20 21, 10101 => 11111 => 21 22, 10110 => 11101 => 22 23, 10111 => 11100 => 23 24, 11000 => 10100 => 24 25, 11001 => 10101 => 25 26, 11010 => 10111 => 26 27, 11011 => 10110 => 27 28, 11100 => 10010 => 28 29, 11101 => 10011 => 29 30, 11110 => 10001 => 30 31, 11111 => 10000 => 31
Python: on lists of bits
This example works with lists of discrete binary digits.
- First some int<>bin conversion routines
<lang python>>>> def int2bin(n): 'From positive integer to list of binary bits, msb at index 0' if n: bits = [] while n: n,remainder = divmod(n, 2) bits.insert(0, remainder) return bits else: return [0]
>>> def bin2int(bits):
'From binary bits, msb at index 0 to integer'
i = 0
for bit in bits:
i = i * 2 + bit
return i</lang>
- Now the bin<>gray converters.
These follow closely the methods in the animation seen here: Converting Between Gray and Binary Codes. <lang python>>>> def bin2gray(bits): return bits[:1] + [i ^ ishift for i, ishift in zip(bits[:-1], bits[1:])]
>>> def gray2bin(bits): b = [bits[0]] for nextb in bits[1:]: b.append(b[-1] ^ nextb) return b</lang>
- Sample output
<lang python>>>> for i in range(16): print('int:%2i -> bin:%12r -> gray:%12r -> bin:%12r -> int:%2i' % ( i, int2bin(i), bin2gray(int2bin(i)), gray2bin(bin2gray(int2bin(i))), bin2int(gray2bin(bin2gray(int2bin(i)))) ))
int: 0 -> bin: [0] -> gray: [0] -> bin: [0] -> int: 0
int: 1 -> bin: [1] -> gray: [1] -> bin: [1] -> int: 1
int: 2 -> bin: [1, 0] -> gray: [1, 1] -> bin: [1, 0] -> int: 2
int: 3 -> bin: [1, 1] -> gray: [1, 0] -> bin: [1, 1] -> int: 3
int: 4 -> bin: [1, 0, 0] -> gray: [1, 1, 0] -> bin: [1, 0, 0] -> int: 4
int: 5 -> bin: [1, 0, 1] -> gray: [1, 1, 1] -> bin: [1, 0, 1] -> int: 5
int: 6 -> bin: [1, 1, 0] -> gray: [1, 0, 1] -> bin: [1, 1, 0] -> int: 6
int: 7 -> bin: [1, 1, 1] -> gray: [1, 0, 0] -> bin: [1, 1, 1] -> int: 7
int: 8 -> bin:[1, 0, 0, 0] -> gray:[1, 1, 0, 0] -> bin:[1, 0, 0, 0] -> int: 8
int: 9 -> bin:[1, 0, 0, 1] -> gray:[1, 1, 0, 1] -> bin:[1, 0, 0, 1] -> int: 9
int:10 -> bin:[1, 0, 1, 0] -> gray:[1, 1, 1, 1] -> bin:[1, 0, 1, 0] -> int:10
int:11 -> bin:[1, 0, 1, 1] -> gray:[1, 1, 1, 0] -> bin:[1, 0, 1, 1] -> int:11
int:12 -> bin:[1, 1, 0, 0] -> gray:[1, 0, 1, 0] -> bin:[1, 1, 0, 0] -> int:12
int:13 -> bin:[1, 1, 0, 1] -> gray:[1, 0, 1, 1] -> bin:[1, 1, 0, 1] -> int:13
int:14 -> bin:[1, 1, 1, 0] -> gray:[1, 0, 0, 1] -> bin:[1, 1, 1, 0] -> int:14
int:15 -> bin:[1, 1, 1, 1] -> gray:[1, 0, 0, 0] -> bin:[1, 1, 1, 1] -> int:15
>>> </lang>
Quackery
<lang Quackery> [ dup 1 >> ^ ] is encodegray ( n --> n )
[ dup
[ dip [ 1 >> ]
over ^
over 0 = until ]
nip ] is decodegray ( n --> n )
[ [] unrot times
[ 2 /mod char 0 +
rot join swap ]
drop echo$ ] is echobin ( n n --> )
say "number encoded decoded" cr
say "------ ------- -------" cr
32 times
[ sp i^ 5 echobin
say " -> "
i^ encodegray dup 5 echobin
say " -> "
decodegray 5 echobin cr ]</lang>
- Output:
number encoded decoded ------ ------- ------- 00000 -> 00000 -> 00000 00001 -> 00001 -> 00001 00010 -> 00011 -> 00010 00011 -> 00010 -> 00011 00100 -> 00110 -> 00100 00101 -> 00111 -> 00101 00110 -> 00101 -> 00110 00111 -> 00100 -> 00111 01000 -> 01100 -> 01000 01001 -> 01101 -> 01001 01010 -> 01111 -> 01010 01011 -> 01110 -> 01011 01100 -> 01010 -> 01100 01101 -> 01011 -> 01101 01110 -> 01001 -> 01110 01111 -> 01000 -> 01111 10000 -> 11000 -> 10000 10001 -> 11001 -> 10001 10010 -> 11011 -> 10010 10011 -> 11010 -> 10011 10100 -> 11110 -> 10100 10101 -> 11111 -> 10101 10110 -> 11101 -> 10110 10111 -> 11100 -> 10111 11000 -> 10100 -> 11000 11001 -> 10101 -> 11001 11010 -> 10111 -> 11010 11011 -> 10110 -> 11011 11100 -> 10010 -> 11100 11101 -> 10011 -> 11101 11110 -> 10001 -> 11110 11111 -> 10000 -> 11111
R
<lang r> GrayEncode <- function(binary) { gray <- substr(binary,1,1) repeat { if (substr(binary,1,1) != substr(binary,2,2)) gray <- paste(gray,"1",sep="") else gray <- paste(gray,"0",sep="") binary <- substr(binary,2,nchar(binary)) if (nchar(binary) <=1) { break } } return (gray) } GrayDecode <- function(gray) { binary <- substr(gray,1,1) repeat { if (substr(binary,nchar(binary),nchar(binary)) != substr(gray,2,2)) binary <- paste(binary ,"1",sep="") else binary <- paste(binary ,"0",sep="") gray <- substr(gray,2,nchar(gray))
if (nchar(gray) <=1) { break } } return (binary) } </lang>
Racket
<lang racket>
- lang racket
(define (gray-encode n) (bitwise-xor n (arithmetic-shift n -1)))
(define (gray-decode n)
(letrec ([loop (lambda(g bits)
(if (> bits 0)
(loop (bitwise-xor g bits) (arithmetic-shift bits -1))
g))])
(loop 0 n)))
(define (to-bin n) (format "~b" n)) (define (show-table)
(for ([i (in-range 1 32)])
(printf "~a | ~a | ~a ~n"
(~r i #:min-width 2 #:pad-string "0")
(~a (to-bin(gray-encode i)) #:width 5 #:align 'right #:pad-string "0")
(~a (to-bin (gray-decode(gray-encode i))) #:width 5 #:align 'right #:pad-string "0"))))
</lang>
- Output:
> (show-table) 01 | 00001 | 00001 02 | 00011 | 00010 03 | 00010 | 00011 04 | 00110 | 00100 05 | 00111 | 00101 06 | 00101 | 00110 07 | 00100 | 00111 08 | 01100 | 01000 09 | 01101 | 01001 10 | 01111 | 01010 11 | 01110 | 01011 12 | 01010 | 01100 13 | 01011 | 01101 14 | 01001 | 01110 15 | 01000 | 01111 16 | 11000 | 10000 17 | 11001 | 10001 18 | 11011 | 10010 19 | 11010 | 10011 20 | 11110 | 10100 21 | 11111 | 10101 22 | 11101 | 10110 23 | 11100 | 10111 24 | 10100 | 11000 25 | 10101 | 11001 26 | 10111 | 11010 27 | 10110 | 11011 28 | 10010 | 11100 29 | 10011 | 11101 30 | 10001 | 11110 31 | 10000 | 11111
Raku
(formerly Perl 6) <lang perl6>sub gray_encode ( Int $n --> Int ) {
return $n +^ ( $n +> 1 );
}
sub gray_decode ( Int $n is copy --> Int ) {
my $mask = 1 +< (32-2); $n +^= $mask +> 1 if $n +& $mask while $mask +>= 1; return $n;
}
for ^32 -> $n {
my $g = gray_encode($n); my $d = gray_decode($g); printf "%2d: %5b => %5b => %5b: %2d\n", $n, $n, $g, $d, $d; die if $d != $n;
}</lang>
- Output:
0: 0 => 0 => 0: 0 1: 1 => 1 => 1: 1 2: 10 => 11 => 10: 2 3: 11 => 10 => 11: 3 4: 100 => 110 => 100: 4 5: 101 => 111 => 101: 5 6: 110 => 101 => 110: 6 7: 111 => 100 => 111: 7 8: 1000 => 1100 => 1000: 8 9: 1001 => 1101 => 1001: 9 10: 1010 => 1111 => 1010: 10 11: 1011 => 1110 => 1011: 11 12: 1100 => 1010 => 1100: 12 13: 1101 => 1011 => 1101: 13 14: 1110 => 1001 => 1110: 14 15: 1111 => 1000 => 1111: 15 16: 10000 => 11000 => 10000: 16 17: 10001 => 11001 => 10001: 17 18: 10010 => 11011 => 10010: 18 19: 10011 => 11010 => 10011: 19 20: 10100 => 11110 => 10100: 20 21: 10101 => 11111 => 10101: 21 22: 10110 => 11101 => 10110: 22 23: 10111 => 11100 => 10111: 23 24: 11000 => 10100 => 11000: 24 25: 11001 => 10101 => 11001: 25 26: 11010 => 10111 => 11010: 26 27: 11011 => 10110 => 11011: 27 28: 11100 => 10010 => 11100: 28 29: 11101 => 10011 => 11101: 29 30: 11110 => 10001 => 11110: 30 31: 11111 => 10000 => 11111: 31
Raku distinguishes numeric bitwise operators with a leading + sign, so +< and +> are left and right shift, while +& is a bitwise AND, while +^ is bitwise XOR (here used as part of an assignment metaoperator).
REXX
The leading zeroes for the binary numbers and the gray code could've easily been elided. <lang rexx>/*REXX program converts decimal number ───► binary ───► gray code ───► binary.*/ parse arg N . /*get the optional argument from the CL*/ if N== | N=="," then N=31 /*Not specified? Then use the default.*/ L=max(1,length(strip(x2b(d2x(N)),'L',0))) /*find the max binary length of N.*/ w=14 /*used for the formatting of cell width*/ _=center('binary',w,'─') /*the 2nd and 4th part of the header.*/ say center('decimal', w, "─")'►' _"►" center('gray code', w, '─')"►" _
/* [+] the output header*/
do j=0 to N; b=right(x2b(d2x(j)),L,0) /*process 0 ──► N. */
g=b2gray(b) /*convert binary number to gray code. */
a=gray2b(g) /*convert the gray code to binary. */
say center(j,w+1) center(b,w+1) center(g,w+1) center(a,w+1)
end /*j*/
exit /*stick a fork in it, we're all done. */ /*────────────────────────────────────────────────────────────────────────────*/ b2gray: procedure; parse arg x 1 $ 2; do b=2 to length(x)
$=$||(substr(x,b-1,1) && substr(x,b,1))
end /*b*/
return $
/*────────────────────────────────────────────────────────────────────────────*/ gray2b: procedure; parse arg x 1 $ 2; do g=2 to length(x)
$=$ || (right($,1) && substr(x,g,1))
end /*g*/ /* ↑ */
/* │ */
return $ /*this is an eXclusive OR ►─────────┘ */</lang>
output when using the default input:
───decimal────► ────binary────► ──gray code───► ────binary────
0 00000 00000 00000
1 00001 00001 00001
2 00010 00011 00010
3 00011 00010 00011
4 00100 00110 00100
5 00101 00111 00101
6 00110 00101 00110
7 00111 00100 00111
8 01000 01100 01000
9 01001 01101 01001
10 01010 01111 01010
11 01011 01110 01011
12 01100 01010 01100
13 01101 01011 01101
14 01110 01001 01110
15 01111 01000 01111
16 10000 11000 10000
17 10001 11001 10001
18 10010 11011 10010
19 10011 11010 10011
20 10100 11110 10100
21 10101 11111 10101
22 10110 11101 10110
23 10111 11100 10111
24 11000 10100 11000
25 11001 10101 11001
26 11010 10111 11010
27 11011 10110 11011
28 11100 10010 11100
29 11101 10011 11101
30 11110 10001 11110
31 11111 10000 11111
Ring
<lang ring>
- Project : Gray code
pos = 5 see "0 : 00000 => 00000 => 00000" + nl for n = 1 to 31
res1 = tobase(n, 2, pos)
res2 = tobase(grayencode(n), 2, pos)
res3 = tobase(graydecode(n), 2, pos)
see "" + n + " : " + res1 + " => " + res2 + " => " + res3 + nl
next
func grayencode(n)
return n ^ (n >> 1)
func graydecode(n)
p = n
while (n = n >> 1)
p = p ^ n
end
return p
func tobase(nr, base, pos)
binary = 0
i = 1
while(nr != 0)
remainder = nr % base
nr = floor(nr/base)
binary= binary + (remainder*i)
i = i*10
end
result = ""
for nr = 1 to pos - len(string(binary))
result = result + "0"
next
result = result + string(binary)
return result
</lang> Output:
0 : 00000 => 00000 => 00000 1 : 00001 => 00001 => 00001 2 : 00010 => 00011 => 00010 3 : 00011 => 00010 => 00011 4 : 00100 => 00110 => 00100 5 : 00101 => 00111 => 00101 6 : 00110 => 00101 => 00110 7 : 00111 => 00100 => 00111 8 : 01000 => 01100 => 01000 9 : 01001 => 01101 => 01001 10 : 01010 => 01111 => 01010 11 : 01011 => 01110 => 01011 12 : 01100 => 01010 => 01100 13 : 01101 => 01011 => 01101 14 : 01110 => 01001 => 01110 15 : 01111 => 01000 => 01111 16 : 10000 => 11000 => 10000 17 : 10001 => 11001 => 10001 18 : 10010 => 11011 => 10010 19 : 10011 => 11010 => 10011 20 : 10100 => 11110 => 10100 21 : 10101 => 11111 => 10101 22 : 10110 => 11101 => 10110 23 : 10111 => 11100 => 10111 24 : 11000 => 10100 => 11000 25 : 11001 => 10101 => 11001 26 : 11010 => 10111 => 11010 27 : 11011 => 10110 => 11011 28 : 11100 => 10010 => 11100 29 : 11101 => 10011 => 11101 30 : 11110 => 10001 => 11110 31 : 11111 => 10000 => 11111
Ruby
Integer#from_gray has recursion so it can use each bit of the answer to compute the next bit.
<lang ruby>class Integer
# Converts a normal integer to a Gray code.
def to_gray
raise Math::DomainError, "integer is negative" if self < 0
self ^ (self >> 1)
end
# Converts a Gray code to a normal integer.
def from_gray
raise Math::DomainError, "integer is negative" if self < 0
recurse = proc do |i|
next 0 if i == 0
o = recurse[i >> 1] << 1
o | (i[0] ^ o[1])
end
recurse[self]
end
end
(0..31).each do |number|
encoded = number.to_gray
decoded = encoded.from_gray
printf "%2d : %5b => %5b => %5b : %2d\n",
number, number, encoded, decoded, decoded
end</lang>
- Output:
0 : 0 => 0 => 0 : 0 1 : 1 => 1 => 1 : 1 2 : 10 => 11 => 10 : 2 3 : 11 => 10 => 11 : 3 4 : 100 => 110 => 100 : 4 5 : 101 => 111 => 101 : 5 6 : 110 => 101 => 110 : 6 7 : 111 => 100 => 111 : 7 8 : 1000 => 1100 => 1000 : 8 9 : 1001 => 1101 => 1001 : 9 10 : 1010 => 1111 => 1010 : 10 11 : 1011 => 1110 => 1011 : 11 12 : 1100 => 1010 => 1100 : 12 13 : 1101 => 1011 => 1101 : 13 14 : 1110 => 1001 => 1110 : 14 15 : 1111 => 1000 => 1111 : 15 16 : 10000 => 11000 => 10000 : 16 17 : 10001 => 11001 => 10001 : 17 18 : 10010 => 11011 => 10010 : 18 19 : 10011 => 11010 => 10011 : 19 20 : 10100 => 11110 => 10100 : 20 21 : 10101 => 11111 => 10101 : 21 22 : 10110 => 11101 => 10110 : 22 23 : 10111 => 11100 => 10111 : 23 24 : 11000 => 10100 => 11000 : 24 25 : 11001 => 10101 => 11001 : 25 26 : 11010 => 10111 => 11010 : 26 27 : 11011 => 10110 => 11011 : 27 28 : 11100 => 10010 => 11100 : 28 29 : 11101 => 10011 => 11101 : 29 30 : 11110 => 10001 => 11110 : 30 31 : 11111 => 10000 => 11111 : 31
Rust
<lang rust>fn gray_encode(integer: u64) -> u64 {
(integer >> 1) ^ integer
}
fn gray_decode(integer: u64) -> u64 {
match integer {
0 => 0,
_ => integer ^ gray_decode(integer >> 1)
}
}
fn main() {
for i in 0..32 {
println!("{:2} {:0>5b} {:0>5b} {:2}", i, i, gray_encode(i),
gray_decode(i));
}
}</lang>
Scala
Functional style: the Gray code is encoded to, and decoded from a String.
The scanLeft function takes a sequence (here, of characters) and produces a collection containing cumulative results of applying an operator going left to right.
Here the operator is exclusive-or, "^", and we can use "_" placeholders to represent the arguments to the left and right. tail removes the "0" we added as the initial accumulator value, and mkString turns the collection back into a String, that we can parse into an integer (Integer.parseInt is directly from the java.lang package).
<lang scala>def encode(n: Int) = (n ^ (n >>> 1)).toBinaryString
def decode(s: String) = Integer.parseInt( s.scanLeft(0)(_ ^ _.asDigit).tail.mkString , 2)
println("decimal binary gray decoded") for (i <- 0 to 31; g = encode(i))
println("%7d %6s %5s %7s".format(i, i.toBinaryString, g, decode(g)))
</lang>
- Output:
decimal binary gray decoded
0 0 0 0
1 1 1 1
2 10 11 2
3 11 10 3
4 100 110 4
5 101 111 5
6 110 101 6
7 111 100 7
8 1000 1100 8
9 1001 1101 9
10 1010 1111 10
11 1011 1110 11
12 1100 1010 12
13 1101 1011 13
14 1110 1001 14
15 1111 1000 15
16 10000 11000 16
17 10001 11001 17
18 10010 11011 18
19 10011 11010 19
20 10100 11110 20
21 10101 11111 21
22 10110 11101 22
23 10111 11100 23
24 11000 10100 24
25 11001 10101 25
26 11010 10111 26
27 11011 10110 27
28 11100 10010 28
29 11101 10011 29
30 11110 10001 30
31 11111 10000 31
Alternatively, more imperative style: <lang scala>def encode(n: Long) = n ^ (n >>> 1)
def decode(n: Long) = {
var g = 0L
var bits = n
while (bits > 0) {
g ^= bits
bits >>= 1
}
g
}
def toBin(n: Long) = ("0000" + n.toBinaryString) takeRight 5
println("decimal binary gray decoded") for (i <- 0 until 32) {
val g = encode(i)
println("%7d %6s %5s %7s".format(i, toBin(i), toBin(g), decode(g)))
}</lang> Improved version of decode using functional style (recursion+local method). No vars and mutations. <lang scala>def decode(n:Long)={
def calc(g:Long,bits:Long):Long=if (bits>0) calc(g^bits, bits>>1) else g calc(0, n)
}</lang>
- Output:
decimal binary gray decoded
0 00000 00000 0
1 00001 00001 1
2 00010 00011 2
3 00011 00010 3
4 00100 00110 4
5 00101 00111 5
6 00110 00101 6
7 00111 00100 7
8 01000 01100 8
9 01001 01101 9
10 01010 01111 10
11 01011 01110 11
12 01100 01010 12
13 01101 01011 13
14 01110 01001 14
15 01111 01000 15
16 10000 11000 16
17 10001 11001 17
18 10010 11011 18
19 10011 11010 19
20 10100 11110 20
21 10101 11111 21
22 10110 11101 22
23 10111 11100 23
24 11000 10100 24
25 11001 10101 25
26 11010 10111 26
27 11011 10110 27
28 11100 10010 28
29 11101 10011 29
30 11110 10001 30
31 11111 10000 31
Scratch
![[1]](http://i.imgur.com/0sw5D4T.png){kind=link}
Seed7
The type bin32 is intended for bit operations that are not defined for integer values. Bin32 is used for the exclusive or (><) operation. <lang seed7>$ include "seed7_05.s7i";
include "bin32.s7i";
const func integer: grayEncode (in integer: n) is
return ord(bin32(n) >< bin32(n >> 1));
const func integer: grayDecode (in var integer: n) is func
result
var integer: decoded is 0;
begin
decoded := n;
while n > 1 do
n >>:= 1;
decoded := ord(bin32(decoded) >< bin32(n));
end while;
end func;
const proc: main is func
local
var integer: i is 0;
begin
for i range 0 to 32 do
writeln(i <& " => " <& grayEncode(i) radix 2 lpad0 6 <& " => " <& grayDecode(grayEncode(i)));
end for;
end func;</lang>
- Output:
0 => 000000 => 0 1 => 000001 => 1 2 => 000011 => 2 3 => 000010 => 3 4 => 000110 => 4 5 => 000111 => 5 6 => 000101 => 6 7 => 000100 => 7 8 => 001100 => 8 9 => 001101 => 9 10 => 001111 => 10 11 => 001110 => 11 12 => 001010 => 12 13 => 001011 => 13 14 => 001001 => 14 15 => 001000 => 15 16 => 011000 => 16 17 => 011001 => 17 18 => 011011 => 18 19 => 011010 => 19 20 => 011110 => 20 21 => 011111 => 21 22 => 011101 => 22 23 => 011100 => 23 24 => 010100 => 24 25 => 010101 => 25 26 => 010111 => 26 27 => 010110 => 27 28 => 010010 => 28 29 => 010011 => 29 30 => 010001 => 30 31 => 010000 => 31 32 => 110000 => 32
SenseTalk
Note: Inputs and outputs as strings <lang sensetalk> function BinaryToGray param1 set theResult to "" repeat for each character in param1 if the counter is equal to 1 put it after theResult else if it is equal to previousCharacter put "0" after theResult else put "1" after theResult end if end if set previousCharacter to it end repeat return theResult end BinaryToGray
function GrayToBinary param1 set theResult to param1 repeat for each character in param1 if the counter is equal to 1 next repeat end if set currentChar to it set lastCharInd to the counter - 1 repeat for lastCharInd down to 1 if currentChar is equal to character it of param1 set currentChar to "0" else set currentChar to "1" end if end repeat set character the counter of theResult to currentChar end repeat
return theResult end GrayToBinary </lang>
- Output:
binary => gray => decoded 00000 => 00000 => 00000 00001 => 00001 => 00001 00010 => 00011 => 00010 00011 => 00010 => 00011 00100 => 00110 => 00100 00101 => 00111 => 00101 00110 => 00101 => 00110 00111 => 00100 => 00111 01000 => 01100 => 01000 01001 => 01101 => 01001 01010 => 01111 => 01010 01011 => 01110 => 01011 01100 => 01010 => 01100 01101 => 01011 => 01101 01110 => 01001 => 01110 01111 => 01000 => 01111 10000 => 11000 => 10000 10001 => 11001 => 10001 10010 => 11011 => 10010 10011 => 11010 => 10011 10100 => 11110 => 10100 10101 => 11111 => 10101 10110 => 11101 => 10110 10111 => 11100 => 10111 11000 => 10100 => 11000 11001 => 10101 => 11001 11010 => 10111 => 11010 11011 => 10110 => 11011 11100 => 10010 => 11100 11101 => 10011 => 11101 11110 => 10001 => 11110 11111 => 10000 => 11111 101001110101111 => 111101001111000 => 101001110101111 101001110110000 => 111101001101000 => 101001110110000 101001110110001 => 111101001101001 => 101001110110001 101001110110010 => 111101001101011 => 101001110110010
Sidef
<lang ruby>func bin2gray(n) {
n ^ (n >> 1)
}
func gray2bin(num) {
var bin = num
while (num >>= 1) { bin ^= num }
return bin
}
{ |i|
var gr = bin2gray(i)
printf("%d\t%b\t%b\t%b\n", i, i, gr, gray2bin(gr))
} << ^32</lang>
- Output:
0 0 0 0 1 1 1 1 2 10 11 10 3 11 10 11 4 100 110 100 5 101 111 101 6 110 101 110 7 111 100 111 8 1000 1100 1000 9 1001 1101 1001 10 1010 1111 1010 11 1011 1110 1011 12 1100 1010 1100 13 1101 1011 1101 14 1110 1001 1110 15 1111 1000 1111 16 10000 11000 10000 17 10001 11001 10001 18 10010 11011 10010 19 10011 11010 10011 20 10100 11110 10100 21 10101 11111 10101 22 10110 11101 10110 23 10111 11100 10111 24 11000 10100 11000 25 11001 10101 11001 26 11010 10111 11010 27 11011 10110 11011 28 11100 10010 11100 29 11101 10011 11101 30 11110 10001 11110 31 11111 10000 11111
SQL
<lang sql> DECLARE @binary AS NVARCHAR(MAX) = '001010111' DECLARE @gray AS NVARCHAR(MAX) =
--Encoder SET @gray = LEFT(@binary, 1)
WHILE LEN(@binary) > 1
BEGIN
IF LEFT(@binary, 1) != SUBSTRING(@binary, 2, 1)
SET @gray = @gray + '1'
ELSE
SET @gray = @gray + '0'
SET @binary = RIGHT(@binary, LEN(@binary) - 1) END
SELECT @gray
--Decoder SET @binary = LEFT(@gray, 1)
WHILE LEN(@gray) > 1
BEGIN
IF RIGHT(@binary, 1) != SUBSTRING(@gray, 2, 1)
SET @binary = @binary + '1'
ELSE
SET @binary = @binary + '0'
SET @gray = RIGHT(@gray, LEN(@gray) - 1) END
SELECT @binary </lang>
Standard ML
<lang sml>fun gray_encode b =
Word.xorb (b, Word.>> (b, 0w1))
fun gray_decode n =
let
fun aux (p, n) =
if n = 0w0 then p
else aux (Word.xorb (p, n), Word.>> (n, 0w1))
in
aux (n, Word.>> (n, 0w1))
end;
val s = Word.fmt StringCvt.BIN; fun aux i =
if i = 0w32 then
()
else
let
val g = gray_encode i
val b = gray_decode g
in
print (Word.toString i ^ " :\t" ^ s i ^ " => " ^ s g ^ " => " ^ s b ^ "\t: " ^ Word.toString b ^ "\n");
aux (i + 0w1)
end;
aux 0w0</lang>
Swift
<lang swift>func grayEncode(_ i: Int) -> Int {
return (i >> 1) ^ i
}
func grayDecode(_ i: Int) -> Int {
switch i {
case 0:
return 0
case _:
return i ^ grayDecode(i >> 1)
}
}
for i in 0..<32 {
let iStr = String(i, radix: 2) let encode = grayEncode(i) let encodeStr = String(encode, radix: 2) let decode = grayDecode(encode) let decodeStr = String(decode, radix: 2)
print("\(i) (\(iStr)) => \(encode) (\(encodeStr)) => \(decode) (\(decodeStr))")
}</lang>
- Output:
0 (0) => 0 (0) => 0 (0) 1 (1) => 1 (1) => 1 (1) 2 (10) => 3 (11) => 2 (10) 3 (11) => 2 (10) => 3 (11) 4 (100) => 6 (110) => 4 (100) 5 (101) => 7 (111) => 5 (101) 6 (110) => 5 (101) => 6 (110) 7 (111) => 4 (100) => 7 (111) 8 (1000) => 12 (1100) => 8 (1000) 9 (1001) => 13 (1101) => 9 (1001) 10 (1010) => 15 (1111) => 10 (1010) 11 (1011) => 14 (1110) => 11 (1011) 12 (1100) => 10 (1010) => 12 (1100) 13 (1101) => 11 (1011) => 13 (1101) 14 (1110) => 9 (1001) => 14 (1110) 15 (1111) => 8 (1000) => 15 (1111) 16 (10000) => 24 (11000) => 16 (10000) 17 (10001) => 25 (11001) => 17 (10001) 18 (10010) => 27 (11011) => 18 (10010) 19 (10011) => 26 (11010) => 19 (10011) 20 (10100) => 30 (11110) => 20 (10100) 21 (10101) => 31 (11111) => 21 (10101) 22 (10110) => 29 (11101) => 22 (10110) 23 (10111) => 28 (11100) => 23 (10111) 24 (11000) => 20 (10100) => 24 (11000) 25 (11001) => 21 (10101) => 25 (11001) 26 (11010) => 23 (10111) => 26 (11010) 27 (11011) => 22 (10110) => 27 (11011) 28 (11100) => 18 (10010) => 28 (11100) 29 (11101) => 19 (10011) => 29 (11101) 30 (11110) => 17 (10001) => 30 (11110) 31 (11111) => 16 (10000) => 31 (11111)
Tcl
<lang tcl>namespace eval gray {
proc encode n {
expr {$n ^ $n >> 1}
}
proc decode n {
# Compute some bit at least as large as MSB set i [expr {2**int(ceil(log($n+1)/log(2)))}] set b [set bprev [expr {$n & $i}]] while {[set i [expr {$i >> 1}]]} { set b [expr {$b | [set bprev [expr {$n & $i ^ $bprev >> 1}]]}] } return $b
}
}</lang> Demonstrating: <lang tcl>package require Tcl 8.6; # Just for %b format specifier for {set i 0} {$i < 32} {incr i} {
set g [gray::encode $i] set b [gray::decode $g] puts [format "%2d: %05b => %05b => %05b : %2d" $i $i $g $b $b]
}</lang>
- Output:
0: 00000 => 00000 => 00000 : 0 1: 00001 => 00001 => 00001 : 1 2: 00010 => 00011 => 00010 : 2 3: 00011 => 00010 => 00011 : 3 4: 00100 => 00110 => 00100 : 4 5: 00101 => 00111 => 00101 : 5 6: 00110 => 00101 => 00110 : 6 7: 00111 => 00100 => 00111 : 7 8: 01000 => 01100 => 01000 : 8 9: 01001 => 01101 => 01001 : 9 10: 01010 => 01111 => 01010 : 10 11: 01011 => 01110 => 01011 : 11 12: 01100 => 01010 => 01100 : 12 13: 01101 => 01011 => 01101 : 13 14: 01110 => 01001 => 01110 : 14 15: 01111 => 01000 => 01111 : 15 16: 10000 => 11000 => 10000 : 16 17: 10001 => 11001 => 10001 : 17 18: 10010 => 11011 => 10010 : 18 19: 10011 => 11010 => 10011 : 19 20: 10100 => 11110 => 10100 : 20 21: 10101 => 11111 => 10101 : 21 22: 10110 => 11101 => 10110 : 22 23: 10111 => 11100 => 10111 : 23 24: 11000 => 10100 => 11000 : 24 25: 11001 => 10101 => 11001 : 25 26: 11010 => 10111 => 11010 : 26 27: 11011 => 10110 => 11011 : 27 28: 11100 => 10010 => 11100 : 28 29: 11101 => 10011 => 11101 : 29 30: 11110 => 10001 => 11110 : 30 31: 11111 => 10000 => 11111 : 31
Ursala
<lang Ursala>#import std
- import nat
xor = ~&Y&& not ~&B # either and not both
btog = xor*+ zipp0@iitBX # map xor over the argument zipped with its shift
gtob = ~&y+ =><0> ^C/xor@lrhPX ~&r # fold xor over the next input with previous output
- show+
test = mat` * 2-$'01'***K7xSS pad0*K7 <.~&,btog,gtob+ btog>* iota32</lang>
- Output:
00000 00000 00000 00001 00001 00001 00010 00011 00010 00011 00010 00011 00100 00110 00100 00101 00111 00101 00110 00101 00110 00111 00100 00111 01000 01100 01000 01001 01101 01001 01010 01111 01010 01011 01110 01011 01100 01010 01100 01101 01011 01101 01110 01001 01110 01111 01000 01111 10000 11000 10000 10001 11001 10001 10010 11011 10010 10011 11010 10011 10100 11110 10100 10101 11111 10101 10110 11101 10110 10111 11100 10111 11000 10100 11000 11001 10101 11001 11010 10111 11010 11011 10110 11011 11100 10010 11100 11101 10011 11101 11110 10001 11110 11111 10000 11111
VBScript
<lang vb>Function Encoder(ByVal n)
Encoder = n Xor (n \ 2)
End Function
Function Decoder(ByVal n)
Dim g : g = 0
Do While n > 0
g = g Xor n
n = n \ 2
Loop
Decoder = g
End Function
' Decimal to Binary Function Dec2bin(ByVal n, ByVal length)
Dim i, strbin : strbin = ""
For i = 1 to 5
strbin = (n Mod 2) & strbin
n = n \ 2
Next
Dec2Bin = strbin
End Function
WScript.StdOut.WriteLine("Binary -> Gray Code -> Binary") For i = 0 to 31
encoded = Encoder(i) decoded = Decoder(encoded) WScript.StdOut.WriteLine(Dec2Bin(i, 5) & " -> " & Dec2Bin(encoded, 5) & " -> " & Dec2Bin(decoded, 5))
Next</lang>
- Output:
Binary -> Gray Code -> Binary 00000 -> 00000 -> 00000 00001 -> 00001 -> 00001 00010 -> 00011 -> 00010 00011 -> 00010 -> 00011 00100 -> 00110 -> 00100 00101 -> 00111 -> 00101 00110 -> 00101 -> 00110 00111 -> 00100 -> 00111 01000 -> 01100 -> 01000 01001 -> 01101 -> 01001 01010 -> 01111 -> 01010 01011 -> 01110 -> 01011 01100 -> 01010 -> 01100 01101 -> 01011 -> 01101 01110 -> 01001 -> 01110 01111 -> 01000 -> 01111 10000 -> 11000 -> 10000 10001 -> 11001 -> 10001 10010 -> 11011 -> 10010 10011 -> 11010 -> 10011 10100 -> 11110 -> 10100 10101 -> 11111 -> 10101 10110 -> 11101 -> 10110 10111 -> 11100 -> 10111 11000 -> 10100 -> 11000 11001 -> 10101 -> 11001 11010 -> 10111 -> 11010 11011 -> 10110 -> 11011 11100 -> 10010 -> 11100 11101 -> 10011 -> 11101 11110 -> 10001 -> 11110 11111 -> 10000 -> 11111
Verilog
Function Based Approach:
<lang Verilog> `timescale 1ns/10ps `default_nettype wire
module graytestbench;
localparam aw = 8;
function [aw:0] binn_to_gray;
input [aw:0] binn;
begin :b2g
binn_to_gray = binn ^ (binn >> 1);
end endfunction
function [aw:0] gray_to_binn;
input [aw:0] gray;
begin :g2b
reg [aw:0] binn;
integer i;
for(i=0; i <= aw; i = i+1) begin
binn[i] = ^(gray >> i);
end
gray_to_binn = binn;
end endfunction
initial begin :test_graycode
integer ii;
reg[aw:0] gray;
reg[aw:0] binn;
for(ii=0; ii < 10; ii=ii+1) begin
gray = binn_to_gray(ii[aw:0]);
binn = gray_to_binn(gray);
$display("test_graycode: i:%x gray:%x:%b binn:%x", ii[aw:0], gray, gray, binn);
end
$stop;
end
endmodule
`default_nettype none
</lang>
Module Based Approach:
<lang Verilog> `timescale 1ns/10ps `default_nettype none
module gray_counter #(
parameter SIZE=4
) (
input wire i_clk, input wire i_rst_n,
input wire i_inc,
output wire [SIZE-1:0] o_count_gray, output wire [SIZE-1:0] o_count_binn
);
reg [SIZE-1:0] state_gray; reg [SIZE-1:0] state_binn; reg [SIZE-1:0] logic_gray; reg [SIZE-1:0] logic_binn;
always @(posedge i_clk or negedge i_rst_n) begin
if (!i_rst_n) begin
state_gray <= 0;
state_binn <= 0;
end
else begin
state_gray <= logic_gray;
state_binn <= logic_binn;
end
end
always @* begin
logic_binn = state_binn + i_inc; logic_gray = (logic_binn>>1) ^ logic_binn;
end
assign o_count_gray = state_gray; assign o_count_binn = state_binn;
endmodule
`default_nettype none
</lang>
VHDL
Combinatorial encoder: <lang VHDL>LIBRARY ieee; USE ieee.std_logic_1164.all;
entity b2g is
port( bin : in std_logic_vector (4 downto 0);
gray : out std_logic_vector (4 downto 0)
);
end b2g ;
architecture rtl of b2g is
constant N : integer := bin'high;
begin
gray <= bin(n) & ( bin(N-1 downto 0) xor bin(N downto 1));
end architecture rtl;</lang>
Combinatorial decoder: <lang VHDL>LIBRARY ieee; USE ieee.std_logic_1164.all;
entity g2b is
port( gray : in std_logic_vector (4 downto 0);
bin : buffer std_logic_vector (4 downto 0)
);
end g2b ;
architecture rtl of g2b is
constant N : integer := bin'high;
begin
bin(N) <= gray(N); gen_xor: for i in N-1 downto 0 generate bin(i) <= gray(i) xor bin(i+1); end generate;
end architecture rtl;</lang>
Wren
<lang ecmascript>import "/fmt" for Fmt
var toGray = Fn.new { |n| n ^ (n>>1) }
var fromGray = Fn.new { |g|
var b = 0
while (g != 0) {
b = b ^ g
g = g >> 1
}
return b
}
System.print("decimal binary gray decoded") for (b in 0..31) {
System.write(" %(Fmt.d(2, b)) %(Fmt.bz(5, b))")
var g = toGray.call(b)
System.write(" %(Fmt.bz(5, g))")
System.print(" %(Fmt.bz(5, fromGray.call(g)))")
}</lang>
- Output:
decimal binary gray decoded 0 00000 00000 00000 1 00001 00001 00001 2 00010 00011 00010 3 00011 00010 00011 4 00100 00110 00100 5 00101 00111 00101 6 00110 00101 00110 7 00111 00100 00111 8 01000 01100 01000 9 01001 01101 01001 10 01010 01111 01010 11 01011 01110 01011 12 01100 01010 01100 13 01101 01011 01101 14 01110 01001 01110 15 01111 01000 01111 16 10000 11000 10000 17 10001 11001 10001 18 10010 11011 10010 19 10011 11010 10011 20 10100 11110 10100 21 10101 11111 10101 22 10110 11101 10110 23 10111 11100 10111 24 11000 10100 11000 25 11001 10101 11001 26 11010 10111 11010 27 11011 10110 11011 28 11100 10010 11100 29 11101 10011 11101 30 11110 10001 11110 31 11111 10000 11111
XBasic
Intrinsic function BIN$ has been used.
<lang xbasic> PROGRAM "graycode" VERSION "0.0001"
DECLARE FUNCTION Entry() INTERNAL FUNCTION Encode(v&) INTERNAL FUNCTION Decode(v&)
FUNCTION Entry()
PRINT "decimal binary gray decoded"
FOR i& = 0 TO 31
g& = Encode(i&)
d& = Decode(g&)
PRINT FORMAT$(" ##", i&); " "; BIN$(i&, 5); " "; BIN$(g&, 5);
PRINT " "; BIN$(d&, 5); FORMAT$(" ##", d&)
NEXT i&
END FUNCTION
FUNCTION Encode(v&) END FUNCTION v& ^ (v& >> 1)
FUNCTION Decode(v&)
result& = 0 DO WHILE v& > 0 result& = result& ^ v& v& = v& >> 1 LOOP
END FUNCTION result&
END PROGRAM </lang>
- Output:
decimal binary gray decoded 0 00000 00000 00000 0 1 00001 00001 00001 1 2 00010 00011 00010 2 3 00011 00010 00011 3 4 00100 00110 00100 4 5 00101 00111 00101 5 6 00110 00101 00110 6 7 00111 00100 00111 7 8 01000 01100 01000 8 9 01001 01101 01001 9 10 01010 01111 01010 10 11 01011 01110 01011 11 12 01100 01010 01100 12 13 01101 01011 01101 13 14 01110 01001 01110 14 15 01111 01000 01111 15 16 10000 11000 10000 16 17 10001 11001 10001 17 18 10010 11011 10010 18 19 10011 11010 10011 19 20 10100 11110 10100 20 21 10101 11111 10101 21 22 10110 11101 10110 22 23 10111 11100 10111 23 24 11000 10100 11000 24 25 11001 10101 11001 25 26 11010 10111 11010 26 27 11011 10110 11011 27 28 11100 10010 11100 28 29 11101 10011 11101 29 30 11110 10001 11110 30 31 11111 10000 11111 31
XPL0
<lang XPL0>include c:\cxpl\codes; \intrinsic 'code' declarations
func Gray2Bin(N); \Convert N from Gray code to binary int N; int S; [S:= 1; repeat N:= N>>S | N;
S:= S<<1;
until S=32; return N; ]; \Gray2Bin
func Bin2Gray(N); \Convert N from binary to Gray code
int N;
return N>>1 | N;
proc BinOut(N); \Output N in binary
int N;
int R;
[R:= N&1;
N:= N>>1;
if N then BinOut(N);
ChOut(0, R+^0);
]; \BinOut
int N, G;
[for N:= 0 to 31 do
[BinOut(N); ChOut(0, 9\tab\);
G:= Bin2Gray(N);
BinOut(G); ChOut(0, 9\tab\);
BinOut(Gray2Bin(G)); CrLf(0);
];
]</lang>
- Output:
0 0 0 1 1 1 10 11 10 11 10 11 100 110 100 101 111 101 110 101 110 111 100 111 1000 1100 1000 1001 1101 1001 1010 1111 1010 1011 1110 1011 1100 1010 1100 1101 1011 1101 1110 1001 1110 1111 1000 1111 10000 11000 10000 10001 11001 10001 10010 11011 10010 10011 11010 10011 10100 11110 10100 10101 11111 10101 10110 11101 10110 10111 11100 10111 11000 10100 11000 11001 10101 11001 11010 10111 11010 11011 10110 11011 11100 10010 11100 11101 10011 11101 11110 10001 11110 11111 10000 11111
zkl
<lang zkl>fcn grayEncode(n){ n.bitXor(n.shiftRight(1)) } fcn grayDecode(g){ b:=g; while(g/=2){ b=b.bitXor(g) } b }</lang> <lang zkl>foreach n in ([0..31]){
g:=grayEncode(n); b:=grayDecode(g);
println("%2d(%05.2B) --> %2d(%05.2B) --> %2d(%05.2B)".fmt(n,n,g,g,b,b));
}</lang>
- Output:
0(00000) --> 0(00000) --> 0(00000) 1(00001) --> 1(00001) --> 1(00001) 2(00010) --> 3(00011) --> 2(00010) 3(00011) --> 2(00010) --> 3(00011) 4(00100) --> 6(00110) --> 4(00100) 5(00101) --> 7(00111) --> 5(00101) 6(00110) --> 5(00101) --> 6(00110) 7(00111) --> 4(00100) --> 7(00111) 8(01000) --> 12(01100) --> 8(01000) 9(01001) --> 13(01101) --> 9(01001) 10(01010) --> 15(01111) --> 10(01010) 11(01011) --> 14(01110) --> 11(01011) 12(01100) --> 10(01010) --> 12(01100) 13(01101) --> 11(01011) --> 13(01101) 14(01110) --> 9(01001) --> 14(01110) 15(01111) --> 8(01000) --> 15(01111) 16(10000) --> 24(11000) --> 16(10000) 17(10001) --> 25(11001) --> 17(10001) 18(10010) --> 27(11011) --> 18(10010) 19(10011) --> 26(11010) --> 19(10011) 20(10100) --> 30(11110) --> 20(10100) 21(10101) --> 31(11111) --> 21(10101) 22(10110) --> 29(11101) --> 22(10110) 23(10111) --> 28(11100) --> 23(10111) 24(11000) --> 20(10100) --> 24(11000) 25(11001) --> 21(10101) --> 25(11001) 26(11010) --> 23(10111) --> 26(11010) 27(11011) --> 22(10110) --> 27(11011) 28(11100) --> 18(10010) --> 28(11100) 29(11101) --> 19(10011) --> 29(11101) 30(11110) --> 17(10001) --> 30(11110) 31(11111) --> 16(10000) --> 31(11111)
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