Two's complement

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

<lang 6502asm>LDA #%01010101 EOR #255 CLC ADC #1 ;result: #%10101011</lang>

16-bit

<lang 6502asm>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+1</lang>

8086 Assembly

<lang asm>mov al,17 neg al ;8-bit mov bx,4C00h neg bx ;16-bit</lang>

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. <lang algol68>BEGIN

   INT a := 3;
   print( ( -a, " ", 1 + ABS NOT BIN a, newline ) )

END</lang>

Output:

-3 -3

ALGOL W

Translation of: ALGOL 68

<lang algolw>begin

   integer a;
   a := 3;
   write( i_w := 1, s_w := 1, -a, 1 + number( not bitstring( a ) ) )

end.</lang>

Output:

-3 -3

C

<lang C>int a = 3; a = -a;</lang>

J

J uses twos complement natively: <lang J> -3 _3</lang>

We can see this by extracting bits representing the number. In this example, we limit ourselves to 8 bits:

<lang J> (8#2)#:3 0 0 0 0 0 0 1 1

  (8#2)#:-3

1 1 1 1 1 1 0 1</lang>

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.

PL/M

Works with: 8080 PL/M Compiler

... under CP/M (or an emulator)

Even though the original PL/M 8080 compiler only handles unsigned integers, -A two's complements A. <lang pli>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</lang>

Output:

01 FF

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.

<lang perl6>use FixedInt;

1. 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</lang>

Output:

144114427045277101
0b111111111111111110100111011001111000010101110110110101101
761030578771
0b000000000000000001011000100110000111101010001001001010011

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: <lang ecmascript>var a = 0 a = -a System.print(a) // -0</lang> 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:

<lang ecmascript>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)")

}</lang>

Output:

-2147483648 -> 2147483648
-2147483647 -> 2147483647
-2 -> 2
-1 -> 1
0 -> 0
1 -> -1
2 -> -2
2147483646 -> -2147483646
2147483647 -> -2147483647

Z80 Assembly

8-Bit

Zilog Z80

<lang z80>ld a,%00001111 neg ;returns %11110001 in a</lang>

Game Boy <lang z80>ld a,%00001111 cpl ;game boy doesn't have NEG but it has CPL which flips all the bits. inc a ;returns %11110001 in a</lang>

16 Bit

NEG and CPL only work on the accumulator A. The following can be written to work with BC, DE, HL, IX, or IY. <lang z80>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</lang>