Two's complement
You are encouraged to solve this task according to the task description, using any language you may know.
Two's complement is an important concept in representing negative numbers. To turn a positive integer negative, flip the bits and add one.
- Task
Show how to calculate the two's complement of an integer. (It doesn't necessarily need to be a 32 bit integer.)
6502 Assembly
8-Bit
LDA #%01010101
EOR #255
CLC
ADC #1 ;result: #%1010101116-bit
myVar equ $20
LDA #3
STA myVar
LDA #0
STA myVar+1 ;equivalent C: uint16_t myVar = 3;
negate:
LDA myVar+1
EOR #255
STA myVar+1
LDA myVar
EOR #255
STA myVar
CLC
ADC #1
STA myVar
;this handles the case if we started with something where the low byte was zero.
LDA myVar+1
ADC #0
STA myVar+168000 Assembly
MOVE.L #3,D0
NEG.L D0 ;D0 = #$FFFFFFFD8086 Assembly
mov al,17
neg al ;8-bit
mov bx,4C00h
neg bx ;16-bit
ALGOL 68
Algol 68 uses whatever representation the hardware the program is running on uses, which is almost certainly two's complement. So, as in C and most other languages, -a two's complements a. Using Algol 68's bit manipulation facilities, we can explicitely two's complement a positive integer, as shown in this example.
Note: BIN a converts a to a BITS (bit-string) value, the NOT operator will flip the bits and the ABS operator will convert back to an integer, so 1 + ABS NOT BIN a is a long-winded alternative to -a. Note in Algol 68, the BIN operator cannot be applied to negative integers, so 1 + ABS NOT BIN -3 won't work.
BEGIN
INT a := 3;
print( ( -a, " ", 1 + ABS NOT BIN a, newline ) )
END- Output:
-3 -3
ALGOL W
begin
integer a;
a := 3;
write( i_w := 1, s_w := 1, -a, 1 + number( not bitstring( a ) ) )
end.- Output:
-3 -3
ARM Assembly
MOV R0,#0x0000000F
MOV R1,#1
MVN R0,R0 ;flips the bits of R0, R0 = 0xFFFFFFF0
ADD R0,R0,R1 ;R0 = 0xFFFFFFF1C
int a = 3;
a = -a;
C++
In C++ the two's complement of an integer 'n' is its arithmetic negation '-n'.
The two's complement of an integer 'n' can also be calculated by adding 1 to its bitwise complement '~n'.
#include <cstdint>
#include <iostream>
#include <iomanip>
#include <vector>
#include <bitset>
std::string to_hex(const int32_t number) {
std::stringstream stream;
stream << std::setfill('0') << std::setw(8) << std::hex << number;
return stream.str();
}
std::string to_binary(const int32_t number) {
std::stringstream stream;
stream << std::bitset<16>(number);
return stream.str();
}
int32_t twos_complement(const int32_t number) {
return ~number + 1;
}
std::string to_upper_case(std::string str) {
for ( char& ch : str ) {
ch = toupper(ch);
}
return str;
}
int main() {
std::vector<int32_t> examples = { 0, 1, -1, 42 };
std::cout << std::setw(9) << "decimal" << std::setw(12) << "hex"
<< std::setw(17) << "binary" << std::setw(25) << "two's complement" << std::endl;
std::cout << std::setw(6) << "-----------" << std::setw(12) << "--------"
<< std::setw(20) << "----------------" << std::setw(20) << "----------------" << std::endl;
for ( const int32_t& example : examples ) {
std::cout << std::setw(6) << example << std::setw(17) << to_upper_case(to_hex(example))
<< std::setw(20) << to_binary(example) << std::setw(13) << twos_complement(example) << std::endl;
}
}
- Output:
decimal hex binary two's complement
----------- -------- ---------------- ----------------
0 00000000 0000000000000000 0
1 00000001 0000000000000001 -1
-1 FFFFFFFF 1111111111111111 1
42 0000002A 0000000000101010 -42
Craft Basic
let d = 1234567
dim b[d * -1, d * -1 + 1, -2, -1, 0, 1, 2, d - 2, d - 1]
arraysize s, b
for i = 0 to s - 1
print b[i], " : ", b[i] * -1
next i
end
- Output:
-1234567 : 1234567 -1234566 : 1234566 -2 : 2 -1 : 1 0 : 0 1 : -1 2 : -2 1234565 : -1234565 1234566 : -1234566
Delphi
procedure TwosCompliment(Memo: TMemo);
var N: integer;
begin
N:=123456789;
Memo.Lines.Add(Format('N=%10d $%0.8x',[N,N]));
Memo.Lines.Add('');
Memo.Lines.Add('N:=(N xor $FFFFFFFF)+1');
N:=(N xor $FFFFFFFF)+1;
Memo.Lines.Add('');
Memo.Lines.Add(Format('N=%10d $%0.8x',[N,N]));
end;
- Output:
N= 123456789 $075BCD15 N:=(N xor $FFFFFFFF)+1 N=-123456789 $F8A432EB Elapsed Time: 5.206 ms.
Forth
Forth uses two's complement internally. One can use the 'base' variable to switch from one base to another.
: 2s'complement base @ swap dup negate swap 2 base ! u. u. base ! ;
25 2s'complement
- Output:
11001 1111111111111111111111111111111111111111111111111111111111100111 ok
FreeBASIC
In FreeBASIC as in C, if a number n is any integer type, then -n is the two's complement of n, with type preserved.
Dim As Integer d1 = 2147483648, d2 = 2147483646
Dim As Integer b(1 To ...) = {-d1, -d1+1, -2, -1, 0, 1, 2, d1-2, d1-1}
For i As Integer = 1 To Ubound(b)
Print b(i); " -> "; -b(i)
Next i
Sleep- Output:
0000000000000011 -> 1111111111111101
inline assembly
Dim As Integer a = &b000011
Dim As Integer a2c, l
#ifdef __FB_64BIT__
l = 16
Asm
mov rax, [a]
neg rax
mov [a2c], rax
End Asm
#else
l = 8
Asm
mov eax, [a]
neg eax
mov [a2c], eax
End Asm
#endif
Print Bin(a, l); " -> "; Bin(a2c, l)
Sleep- Output:
-2147483648 -> 2147483648 -2147483647 -> 2147483647 -2 -> 2 -1 -> 1 0 -> 0 1 -> -1 2 -> -2 2147483646 -> -2147483646 2147483647 -> -2147483647
J
J uses twos complement natively:
-3
_3
We can see this by extracting bits representing the number. In this example, we limit ourselves to 8 bits:
(8#2)#:3
0 0 0 0 0 0 1 1
(8#2)#:-3
1 1 1 1 1 1 0 1
Java
In Java the two's complement of an integer 'n' is its arithmetic negation '-n'.
The two's complement of an integer 'n' can also be calculated by adding 1 to its bitwise complement '~n'.
import java.util.List;
public final class TwosComplement {
public static void main(String[] args) {
List<Integer> examples = List.of( 0, 1, -1, 42 );
System.out.println(String.format("%9s%12s%24s%34s", "decimal", "hex", "binary", "two's complement"));
System.out.println(
String.format("%6s%12s%24s%32s", "-----------", "--------", "----------", "----------------"));
for ( int example : examples ) {
System.out.println(String.format("%5d%18s%36s%13d",
example, toHex(example), toBinary(example), twosComplement(example)));
}
}
private static String toHex(int number) {
return String.format("%8s", Integer.toHexString(number).toUpperCase()).replace(" ", "0");
}
private static String toBinary(int number) {
return String.format("%32s", Integer.toBinaryString(number)).replace(" ", "0");
}
private static int twosComplement(int number) {
return ~number + 1;
}
}
- Output:
decimal hex binary two's complement
----------- -------- ---------- ----------------
0 00000000 00000000000000000000000000000000 0
1 00000001 00000000000000000000000000000001 -1
-1 FFFFFFFF 11111111111111111111111111111111 1
42 0000002A 00000000000000000000000000101010 -42
Jakt
fn main() {
let n = 0xabcdeabcdeu64
println("{:064b}", n)
println("{:064b}", ~n + 1)
println("{:064b}", -n) // Same result
}- Output:
0000000000000000000000001010101111001101111010101011110011011110 1111111111111111111111110101010000110010000101010100001100100010 1111111111111111111111110101010000110010000101010100001100100010
Julia
In Julia as in C, if a number n is any integer type, then -n is the two's complement of n, with type preserved. This is true even if n is unsigned.
M2000 Interpreter
Module Complement2{
// we use binary.and to get a number in range of byte 0 to 255
byte k, v
v=random(1, 255) ' there is no two's complement for zero
z=binary.and(binary.not(v)+1, 0xFF)
print v
print z
print z=255-v+1 // z is type of byte always positive
print sint(z+0xFFFFFF00)=-v // using 4bytes, we add unsinged 0xFFFFFF00
}
Complement2
Complement2- Output:
2
254
True
True
208
48
True
True
Nim
We define a function to compute the two’s complement by taking the logical complement and adding one. As Nim uses the native complement of the computer, which is two’s complement, the result should be equal to the arithmetic negation.
import std/[strformat, strutils]
func twosComplement[T: SomeSignedInt](n: T): T =
## Compute the two's complement of "n".
not n + 1
echo &"""{"n":^15}{"2's complement":^15}{"-n":^15}"""
for n in [0i32, 1i32, -1i32]:
echo &"{n.toHex:^15}{twosComplement(n).toHex:^15}{(-n).toHex:^15}"
for n in [-50i8, 50i8]:
echo &"{n.toHex:^15}{twosComplement(n).toHex:^15}{(-n).toHex:^15}"
- Output:
n 2's complement -n
──────── ────────────── ────────
00000000 00000000 00000000
00000001 FFFFFFFF FFFFFFFF
FFFFFFFF 00000001 00000001
CE 32 32
32 CE CE
Perl
use strict;
use warnings;
for ( -2**31, -2**31+1, -2, -1, 0, 1, 2, 2**31-2, 2**31-1 ) {
printf "$_ -> %d\n", $_ == 0 ? 0 : ~$_+1
}
Output is the same as the Wren entry.
Phix
inline assembly
without js integer a = 0b000011, a2c #ilASM{ [32] mov eax,[a] neg eax mov [a2c],eax [64] mov rax,[a] neg rax mov [a2c],rax } printf(1,"%032b -> %032b\n",{a,a2c})
- Output:
00000000000000000000000000000011 -> 11111111111111111111111111111101
normal hll
with javascript_semantics integer a = 0b000011 printf(1,"%032b -> %032b\n",{a,-a})
Same output (naturally the rhs is twice as long under 64 bit, in both cases)
PL/M
... under CP/M (or an emulator)
Even though the original PL/M 8080 compiler only handles unsigned integers, -A two's complements A.
100H: /* TWO'S COMPLEMENT *?
/* CP/M BDOS SYSTEM CALL */
BDOS: PROCEDURE( FN, ARG ); DECLARE FN BYTE, ARG ADDRESS; GOTO 5;END;
/* CONSOLE OUTPUT ROUTINES */
PR$CHAR: PROCEDURE( C ); DECLARE C BYTE; CALL BDOS( 2, C ); END;
PR$NL: PROCEDURE; CALL PR$CHAR( 0DH ); CALL PR$CHAR( 0AH ); END;
PR$HEX: PROCEDURE( B ); /* PRINTS B AS A 2 DIGIT HEX NUMBER */
DECLARE B BYTE;
DECLARE D BYTE;
IF ( D := SHR( B, 4 ) ) > 9 THEN CALL PR$CHAR( ( D - 10 ) + 'A' );
ELSE CALL PR$CHAR( D + '0' );
IF ( D := B AND 0FH ) > 9 THEN CALL PR$CHAR( ( D - 10 ) + 'A' );
ELSE CALL PR$CHAR( D + '0' );
END PR$HEX ;
DECLARE A BYTE;
A = 1;
CALL PR$HEX( A );
CALL PR$CHAR( ' ' );
A = -A;
CALL PR$HEX( A );
CALL PR$NL;
EOF- Output:
01 FF
Python
-n
or
~n+1
Quackery
Ways of calculating the two's complement of an integer in Quackery include negate, ~ 1+, -1 *, and 0 swap -.
These are demonstrated as a dialogue in the Quackery shell.
/O> 37 negate ... Stack: -37 /O> negate ... Stack: 37 /O> ~ 1+ ... Stack: -37 /O> ~ 1+ ... Stack: 37 /O> -1 * ... Stack: -37 /O> -1 * ... Stack: 37 /O> 0 swap - ... Stack: -37 /O> 0 swap - ... Stack: 37 /O>
Raku
By default Rakus integers are arbitrary sized, theoretically of infinite length. You can't really take the twos complement of an infinitely long number; so, we need to specifically use fixed size integers.
There is a module available from the Raku ecosystem that provides fixed size integer support, named (appropriately enough.) FixedInt.
FixedInt supports fixed bit size integers, not only 8 bit, 16 bit, 32 bit or 64 bit, but ANY integer size. 22 bit, 35 bit, 191 bit, whatever.
Here we'll demonstrate twos complement on a 57 bit integer.
use FixedInt;
# Instantiate a new 57(!) bit fixed size integer
my \fixedint = FixedInt.new: :57bits;
fixedint = (2³⁷ / 72 - 5¹⁷); # Set it to a large value
say fixedint; # Echo the value to the console in decimal format
say fixedint.bin; # Echo the value to the console in binary format
fixedint.=C2; # Take the twos complement
say fixedint; # Echo the value to the console in decimal format
say fixedint.bin; # Echo the value to the console in binary format
- Output:
144114427045277101 0b111111111111111110100111011001111000010101110110110101101 761030578771 0b000000000000000001011000100110000111101010001001001010011
RPL
RPL can handle integers whose size can be set from 1 to 64 bits. For this task, a 8-bit size is selected with STWS.
RPL considers 'true' integers (i.e. declared by the prefix #) to be positive, so the two's complement can not be done by the NEG instruction.
8 STWS BIN #3d NOT 1 +
- Output:
1: #11111101b
Ruby
Ruby integers have a built-in 'one-complement'method: ~, which flips all bits. Adding 1 leads to a negative integer:
~42 + 1 # => -42
Wren
Strictly speaking, Wren doesn't have integers. Instead all numbers are 'IEEE 754' 64 bit floating point values (their underlying C type being double) and negative numbers are therefore represented using the offset binary method rather than two's complement.
This is illustrated by running the following code:
var a = 0
a = -a
System.print(a) // -0
which produces 'negative zero' rather than just 'zero'.
However, when using the bitwise operators, Wren's VM emulates the corresponding operation in C by first converting the operands to unsigned 32 bit values, performing the operation and then converting the result back to a double.
We can therefore emulate how two's complement works on signed 32 bit integers by using the bitwise complement operator ~ to flip the bits as follows:
var pow32 = 2.pow(32)
var pow31 = 2.pow(31)
var bs = [-pow31, -pow31+1, -2, -1, 0, 1, 2, pow31-2, pow31-1]
for (b in bs) {
var b2 = ~b + 1
if (b2 > pow31) b2 = b2 - pow32
System.print("%(b) -> %(b2)")
}
- Output:
-2147483648 -> 2147483648 -2147483647 -> 2147483647 -2 -> 2 -1 -> 1 0 -> 0 1 -> -1 2 -> -2 2147483646 -> -2147483646 2147483647 -> -2147483647
XPL0
int I; char C;
[I:= 123;
I:= (~I) + 1;
IntOut(0, I); CrLf(0);
C:= -123;
C:= ~(C-1);
IntOut(0, C); CrLf(0);
]- Output:
-123 123
Z80 Assembly
8-Bit
Zilog Z80
ld a,%00001111
neg ;returns %11110001 in aGame Boy
ld a,%00001111
cpl ;game boy doesn't have NEG but it has CPL which flips all the bits.
inc a ;returns %11110001 in a16 Bit
NEG and CPL only work on the accumulator A.
The following can be written to work with BC, DE, HL, IX, or IY.
xor a ;ld a,0
sub c
ld c,a
sbc a ;loads &FF into A if "sub c" set the carry (borrow) flag. Otherwise, a remains zero.
sub b
ld b,a