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## 43 bitwise

The package `bitwise` provides functions that allow to manipulate bits of integer constants. As always maxima attempts to simplify the result of the operation if the actual value of a constant isn’t known considering attributes that might be known for the variables, see the `declare` mechanism.

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### 43.1 Functions and Variables for bitwise

Function: bit_not (int)

  Inverts all bits of a signed integer. The result of this action reads `-int - 1`.

      (%i1) load("bitwise")$

      (%i2) bit_not(i);
      (%o2)                      bit_not(i)

      (%i3) bit_not(bit_not(i));
      (%o3)                           i

      (%i4) bit_not(3);
      (%o4)                          - 4

      (%i5) bit_not(100);
      (%o5)                         - 101

      (%i6) bit_not(-101);
      (%o6)                          100

  ·

  Categories: Number theory ·Binary operations ·Package bitwise

Function: bit_and (int1, ...)

  This function calculates a bitwise `and` of two or more signed integers.

      (%i1) load("bitwise")$

      (%i2) bit_and(i,i);
      (%o2)                           i

      (%i3) bit_and(i,i,i);
      (%o3)                           i

      (%i4) bit_and(1,3);
      (%o4)                           1

      (%i5) bit_and(-7,7);
      (%o5)                           1

  If it is known if one of the parameters to `bit_and` is even this information is taken into consideration by the function.

      (%i1) load("bitwise")$

      (%i2) declare(e,even,o,odd);
      (%o2)                         done

      (%i3) bit_and(1,e);
      (%o3)                           0

      (%i4) bit_and(1,o);
      (%o4)                           1

  ·

  Categories: Number theory ·Binary operations ·Package bitwise

Function: bit_or (int1, ...)

  This function calculates a bitwise `or` of two or more signed integers.

      (%i1) load("bitwise")$

      (%i2) bit_or(i,i);
      (%o2)                           i

      (%i3) bit_or(i,i,i);
      (%o3)                           i

      (%i4) bit_or(1,3);
      (%o4)                           3

      (%i5) bit_or(-7,7);
      (%o5)                          - 1

  If it is known if one of the parameters to `bit_or` is even this information is taken into consideration by the function.

      (%i1) load("bitwise")$

      (%i2) declare(e,even,o,odd);
      (%o2)                         done

      (%i3) bit_or(1,e);
      (%o3)                         e + 1

      (%i4) bit_or(1,o);
      (%o4)                           o

  ·

  Categories: Number theory ·Binary operations ·Package bitwise

Function: bit_xor (int1, ...)

  This function calculates a bitwise `or` of two or more signed integers.

      (%i1) load("bitwise")$

      (%i2) bit_xor(i,i);
      (%o2)                           0

      (%i3) bit_xor(i,i,i);
      (%o3)                           i

      (%i4) bit_xor(1,3);
      (%o4)                           2

      (%i5) bit_xor(-7,7);
      (%o5)                          - 2

  If it is known if one of the parameters to `bit_xor` is even this information is taken into consideration by the function.

      (%i1) load("bitwise")$

      (%i2) declare(e,even,o,odd);
      (%o2)                         done

      (%i3) bit_xor(1,e);
      (%o3)                         e + 1

      (%i4) bit_xor(1,o);
      (%o4)                         o - 1

  ·

  Categories: Number theory ·Binary operations ·Package bitwise

Function: bit_lsh (int, nBits)

  This function shifts all bits of the signed integer `int` to the left by `nBits` bits. The width of the integer is extended by `nBits` for this process. The result of `bit_lsh` therefore is `int * 2`.

      (%i1) load("bitwise")$

      (%i2) bit_lsh(0,1);
      (%o2)                           0

      (%i3) bit_lsh(1,0);
      (%o3)                           1

      (%i4) bit_lsh(1,1);
      (%o4)                           2

      (%i5) bit_lsh(1,i);
      (%o5)                     bit_lsh(1, i)

      (%i6) bit_lsh(-3,1);
      (%o6)                          - 6

      (%i7) bit_lsh(-2,1);
      (%o7)                          - 4

  ·

  Categories: Number theory ·Binary operations ·Package bitwise

Function: bit_rsh (int, nBits)

  This function shifts all bits of the signed integer `int` to the right by `nBits` bits. The width of the integer is reduced by `nBits` for this process.

      (%i1) load("bitwise")$

      (%i2) bit_rsh(0,1);
      (%o2)                           0

      (%i3) bit_rsh(2,0);
      (%o3)                           2

      (%i4) bit_rsh(2,1);
      (%o4)                           1

      (%i5) bit_rsh(2,2);
      (%o5)                           0

      (%i6) bit_rsh(-3,1);
      (%o6)                          - 2

      (%i7) bit_rsh(-2,1);
      (%o7)                          - 1

      (%i8) bit_rsh(-2,2);
      (%o8)                          - 1

  ·

  Categories: Number theory ·Binary operations ·Package bitwise

Function: bit_length (int)

  determines how many bits a variable needs to be long in order to store the number `int`. This function only operates on positive numbers.

      (%i1) load("bitwise")$

      (%i2) bit_length(0);
      (%o2)                           0

      (%i3) bit_length(1);
      (%o3)                           1

      (%i4) bit_length(7);
      (%o4)                           3

      (%i5) bit_length(8);
      (%o5)                           4

  ·

  Categories: Number theory ·Binary operations ·Package bitwise

Function: bit_onep (int, nBit)

  determines if bits `nBit` is set in the signed integer `int`.

      (%i1) load("bitwise")$

      (%i2) bit_onep(85,0);
      (%o2)                         true

      (%i3) bit_onep(85,1);
      (%o3)                         false

      (%i4) bit_onep(85,2);
      (%o4)                         true

      (%i5) bit_onep(85,3);
      (%o5)                         false

      (%i6) bit_onep(85,100);
      (%o6)                         false

      (%i7) bit_onep(i,100);
      (%o7)                   bit_onep(i, 100)

  For signed numbers the sign bit is interpreted to be more than `nBit` to the left of the leftmost bit of `int` that reads `1`.

      (%i1) load("bitwise")$

      (%i2) bit_onep(-2,0);
      (%o2)                         false

      (%i3) bit_onep(-2,1);
      (%o3)                         true

      (%i4) bit_onep(-2,2);
      (%o4)                         true

      (%i5) bit_onep(-2,3);
      (%o5)                         true

      (%i6) bit_onep(-2,4);
      (%o6)                         true

  If it is known if the number to be tested is even this information is taken into consideration by the function.

      (%i1) load("bitwise")$

      (%i2) declare(e,even,o,odd);
      (%o2)                         done

      (%i3) bit_onep(e,0);
      (%o3)                         false

      (%i4) bit_onep(o,0);
      (%o4)                         true

  ·

  Categories: Number theory ·Binary operations ·Package bitwise

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