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The following five operations are central in defining integer operations.
evenp ( x ) oddp ( x ) half ( x ) small ( x ) double ( b , i )
The first is true if x is even, the second is true if x is odd, the third computes 
, the fourth is true if x is zero or minus one, and the fifth computes 
if b is false and 
if b is true. If one applies the halving operation repeatedly to an integer, one eventually ends up with a 'small' integer (zero or minus one). The other way round, all integers can be reached by applying the double operation repeatedly, starting with zero or minus one.
The following operations (name as well as semantics) is taken from the Common Lisp standard.
lognot ( x ) Bitwise complement logior ( x , y ) Bitwise inclusive or logxor ( x , y ) Bitwise exclusive or logand ( x , y ) Bitwise and logeqv ( x , y ) Bitwise exclusive nor lognand ( x , y ) Bitwise nand lognor ( x , y ) Bitwise nor logandc1 ( x , y ) Bitwise and of y and complement of x logandc2 ( x , y ) Bitwise and of x and complement of y logorc1 ( x , y ) Bitwise or of y and complement of x logorc2 ( x , y ) Bitwise or of x and complement of y logtest ( x , y ) True if bitwise and is nonzero ash ( i , c ) Arithmetic shift logbitp ( x , i ) True if bit x of i is one logcount ( x ) Return number of one/zero bits integer-length ( x ) Return size of integer
For the sake of bit operations, integers are treated as infinite series of bits. Positive integers have an infinite number of zero bits and a finite number of one bits. The opposite holds for negative numbers.
For the arithmetic shift, a positive value of c causes i to be multiplied by 
. For a negative value of c, i is halved -c times.
The logcount function returns the number of one-bits in x for positive x and the number of zero bits in x for negative x.
The integer-length of small integers (zero and minus one) is zero. In general, the integer-length indicates the number of times one must halve the integer to get a small integer.
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