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### 8.7 Binary Number Functions

The commands in this chapter all use two-letter sequences beginning with the `b` prefix.

The “binary” operations actually work regardless of the currently displayed radix, although their results make the most sense in a radix like 2, 8, or 16 (as obtained by the `d 2`, `d 8`, or `d 6` commands, respectively). You may also wish to enable display of leading zeros with `d z`. See Radix Modes.

The Calculator maintains a current word size ‘`w`’, an arbitrary positive or negative integer. For a positive word size, all of the binary operations described here operate modulo ‘`2^w`’. In particular, negative arguments are converted to positive integers modulo ‘`2^w`’ by all binary functions.

If the word size is negative, binary operations produce twos-complement integers from ‘`-(2^(-w-1))`’ to ‘`2^(-w-1)-1`’ inclusive. Either mode accepts inputs in any range; the sign of ‘`w`’ affects only the results produced.

The `b c` (`calc-clip`) [`clip`] command can be used to clip a number by reducing it modulo ‘`2^w`’. The commands described in this chapter automatically clip their results to the current word size. Note that other operations like addition do not use the current word size, since integer addition generally is not “binary.” (However, see Simplification Modes, `calc-bin-simplify-mode`.) For example, with a word size of 8 bits `b c` converts a number to the range 0 to 255; with a word size of -8 `b c` converts to the range -128 to 127.

The default word size is 32 bits. All operations except the shifts and rotates allow you to specify a different word size for that one operation by giving a numeric prefix argument: `C-u 8 b c` clips the top of stack to the range 0 to 255 regardless of the current word size. To set the word size permanently, use `b w` (`calc-word-size`). This command displays a prompt with the current word size; press <RET> immediately to keep this word size, or type a new word size at the prompt.

When the binary operations are written in symbolic form, they take an optional second (or third) word-size parameter. When a formula like ‘`and(a,b)`’ is finally evaluated, the word size current at that time will be used, but when ‘`and(a,b,-8)`’ is evaluated, a word size of -8 will always be used. A symbolic binary function will be left in symbolic form unless the all of its argument(s) are integers or integer-valued floats.

If either or both arguments are modulo forms for which ‘`M`’ is a power of two, that power of two is taken as the word size unless a numeric prefix argument overrides it. The current word size is never consulted when modulo-power-of-two forms are involved.

The `b a` (`calc-and`) [`and`] command computes the bitwise AND of the two numbers on the top of the stack. In other words, for each of the ‘`w`’ binary digits of the two numbers (pairwise), the corresponding bit of the result is 1 if and only if both input bits are 1: ‘`and(2#1100, 2#1010) = 2#1000`’.

The `b o` (`calc-or`) [`or`] command computes the bitwise inclusive OR of two numbers. A bit is 1 if either of the input bits, or both, are 1: ‘`or(2#1100, 2#1010) = 2#1110`’.

The `b x` (`calc-xor`) [`xor`] command computes the bitwise exclusive OR of two numbers. A bit is 1 if exactly one of the input bits is 1: ‘`xor(2#1100, 2#1010) = 2#0110`’.

The `b d` (`calc-diff`) [`diff`] command computes the bitwise difference of two numbers; this is defined by ‘`diff(a,b) = and(a,not(b))`’, so that ‘`diff(2#1100, 2#1010) = 2#0100`’.

The `b n` (`calc-not`) [`not`] command computes the bitwise NOT of a number. A bit is 1 if the input bit is 0 and vice-versa.

The `b l` (`calc-lshift-binary`) [`lsh`] command shifts a number left by one bit, or by the number of bits specified in the numeric prefix argument. A negative prefix argument performs a logical right shift, in which zeros are shifted in on the left. In symbolic form, ‘`lsh(a)`’ is short for ‘`lsh(a,1)`’, which in turn is short for ‘`lsh(a,n,w)`’. Bits shifted “off the end,” according to the current word size, are lost.

The `H b l` command also does a left shift, but it takes two arguments from the stack (the value to shift, and, at top-of-stack, the number of bits to shift). This version interprets the prefix argument just like the regular binary operations, i.e., as a word size. The Hyperbolic flag has a similar effect on the rest of the binary shift and rotate commands.

The `b r` (`calc-rshift-binary`) [`rsh`] command shifts a number right by one bit, or by the number of bits specified in the numeric prefix argument: ‘`rsh(a,n) = lsh(a,-n)`’.

The `b L` (`calc-lshift-arith`) [`ash`] command shifts a number left. It is analogous to `lsh`, except that if the shift is rightward (the prefix argument is negative), an arithmetic shift is performed as described below.

The `b R` (`calc-rshift-arith`) [`rash`] command performs an “arithmetic” shift to the right, in which the leftmost bit (according to the current word size) is duplicated rather than shifting in zeros. This corresponds to dividing by a power of two where the input is interpreted as a signed, twos-complement number. (The distinction between the ‘`rsh`’ and ‘`rash`’ operations is totally independent from whether the word size is positive or negative.) With a negative prefix argument, this performs a standard left shift.

The `b t` (`calc-rotate-binary`) [`rot`] command rotates a number one bit to the left. The leftmost bit (according to the current word size) is dropped off the left and shifted in on the right. With a numeric prefix argument, the number is rotated that many bits to the left or right.

See Set Operations, for the `b p` and `b u` commands that pack and unpack binary integers into sets. (For example, `b u` unpacks the number ‘`2#11001`’ to the set of bit-numbers ‘`[0, 3, 4]`’.) Type `b u V #` to count the number of “1” bits in a binary integer.

Another interesting use of the set representation of binary integers is to reverse the bits in, say, a 32-bit integer. Type `b u` to unpack; type `31 <TAB> -` to replace each bit-number in the set with 31 minus that bit-number; type `b p` to pack the set back into a binary integer.