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Bitwise Operations on Integers ============================== In a computer, an integer is represented as a binary number, a sequence of "bits" (digits which are either zero or one). A bitwise operation acts on the individual bits of such a sequence. For example, "shifting" moves the whole sequence left or right one or more places, reproducing the same pattern "moved over". The bitwise operations in Emacs Lisp apply only to integers. - Function: lsh INTEGER1 COUNT `lsh', which is an abbreviation for "logical shift", shifts the bits in INTEGER1 to the left COUNT places, or to the right if COUNT is negative. If COUNT is negative, `lsh' shifts zeros into the most-significant bit, producing a positive result even if INTEGER1 is negative. Contrast this with `ash', below. Thus, the decimal number 5 is the binary number 00000101. Shifted once to the left, with a zero put in the one's place, the number becomes 00001010, decimal 10. Here are two examples of shifting the pattern of bits one place to the left. Since the contents of the rightmost place has been moved one place to the left, a value has to be inserted into the rightmost place. With `lsh', a zero is placed into the rightmost place. (These examples show only the low-order eight bits of the binary pattern; the rest are all zero.) (lsh 5 1) => 10 ;; Decimal 5 becomes decimal 10. 00000101 => 00001010 (lsh 7 1) => 14 ;; Decimal 7 becomes decimal 14. 00000111 => 00001110 As the examples illustrate, shifting the pattern of bits one place to the left produces a number that is twice the value of the previous number. Note, however that functions do not check for overflow, and a returned value may be negative (and in any case, no more than a 24 bit value) when an integer is sufficiently left shifted. For example, left shifting 8,388,607 produces -2: (lsh 8388607 1) ; left shift => -2 In binary, in the 24 bit implementation, the numbers looks like this: ;; Decimal 8,388,607 0111 1111 1111 1111 1111 1111 which becomes the following when left shifted: ;; Decimal -2 1111 1111 1111 1111 1111 1110 Shifting the pattern of bits two places to the left produces results like this (with 8-bit binary numbers): (lsh 3 2) => 12 ;; Decimal 3 becomes decimal 12. 00000011 => 00001100 On the other hand, shifting the pattern of bits one place to the right looks like this: (lsh 6 -1) => 3 ;; Decimal 6 becomes decimal 3. 00000110 => 00000011 (lsh 5 -1) => 2 ;; Decimal 5 becomes decimal 2. 00000101 => 00000010 As the example illustrates, shifting the pattern of bits one place to the right divides the value of the binary number by two, rounding downward. - Function: ash INTEGER1 COUNT `ash' ("arithmetic shift") shifts the bits in INTEGER1 to the left COUNT places, or to the right if COUNT is negative. `ash' gives the same results as `lsh' except when INTEGER1 and COUNT are both negative. In that case, `ash' puts a one in the leftmost position, while `lsh' puts a zero in the leftmost position. Thus, with `ash', shifting the pattern of bits one place to the right looks like this: (ash -6 -1) => -3 ;; Decimal -6 ;; becomes decimal -3. 1111 1111 1111 1111 1111 1010 => 1111 1111 1111 1111 1111 1101 In contrast, shifting the pattern of bits one place to the right with `lsh' looks like this: (lsh -6 -1) => 8388605 ;; Decimal -6 ;; becomes decimal 8,388,605. 1111 1111 1111 1111 1111 1010 => 0111 1111 1111 1111 1111 1101 In this case, the 1 in the leftmost position is shifted one place to the right, and a zero is shifted into the leftmost position. Here are other examples: ; 24-bit binary values (lsh 5 2) ; 5 = 0000 0000 0000 0000 0000 0101 => 20 ; 20 = 0000 0000 0000 0000 0001 0100 (ash 5 2) => 20 (lsh -5 2) ; -5 = 1111 1111 1111 1111 1111 1011 => -20 ; -20 = 1111 1111 1111 1111 1110 1100 (ash -5 2) => -20 (lsh 5 -2) ; 5 = 0000 0000 0000 0000 0000 0101 => 1 ; 1 = 0000 0000 0000 0000 0000 0001 (ash 5 -2) => 1 (lsh -5 -2) ; -5 = 1111 1111 1111 1111 1111 1011 => 4194302 ; 0011 1111 1111 1111 1111 1110 (ash -5 -2) ; -5 = 1111 1111 1111 1111 1111 1011 => -2 ; -2 = 1111 1111 1111 1111 1111 1110 - Function: logand &rest INTS-OR-MARKERS This function returns the "logical and" of the arguments: the Nth bit is set in the result if, and only if, the Nth bit is set in all the arguments. ("Set" means that the value of the bit is 1 rather than 0.) For example, using 4-bit binary numbers, the "logical and" of 13 and 12 is 12: 1101 combined with 1100 produces 1100. In both the binary numbers, the leftmost two bits are set (i.e., they are 1's), so the leftmost two bits of the returned value are set. However, for the rightmost two bits, each is zero in at least one of the arguments, so the rightmost two bits of the returned value are 0's. Therefore, (logand 13 12) => 12 If `logand' is not passed any argument, it returns a value of -1. This number is an identity element for `logand' because its binary representation consists entirely of ones. If `logand' is passed just one argument, it returns that argument. ; 24-bit binary values (logand 14 13) ; 14 = 0000 0000 0000 0000 0000 1110 ; 13 = 0000 0000 0000 0000 0000 1101 => 12 ; 12 = 0000 0000 0000 0000 0000 1100 (logand 14 13 4) ; 14 = 0000 0000 0000 0000 0000 1110 ; 13 = 0000 0000 0000 0000 0000 1101 ; 4 = 0000 0000 0000 0000 0000 0100 => 4 ; 4 = 0000 0000 0000 0000 0000 0100 (logand) => -1 ; -1 = 1111 1111 1111 1111 1111 1111 - Function: logior &rest INTS-OR-MARKERS This function returns the "inclusive or" of its arguments: the Nth bit is set in the result if, and only if, the Nth bit is set in at least one of the arguments. If there are no arguments, the result is zero, which is an identity element for this operation. If `logior' is passed just one argument, it returns that argument. ; 24-bit binary values (logior 12 5) ; 12 = 0000 0000 0000 0000 0000 1100 ; 5 = 0000 0000 0000 0000 0000 0101 => 13 ; 13 = 0000 0000 0000 0000 0000 1101 (logior 12 5 7) ; 12 = 0000 0000 0000 0000 0000 1100 ; 5 = 0000 0000 0000 0000 0000 0101 ; 7 = 0000 0000 0000 0000 0000 0111 => 15 ; 15 = 0000 0000 0000 0000 0000 1111 - Function: logxor &rest INTS-OR-MARKERS This function returns the "exclusive or" of its arguments: the Nth bit is set in the result if, and only if, the Nth bit is set in an odd number of the arguments. If there are no arguments, the result is 0. If `logxor' is passed just one argument, it returns that argument. ; 24-bit binary values (logxor 12 5) ; 12 = 0000 0000 0000 0000 0000 1100 ; 5 = 0000 0000 0000 0000 0000 0101 => 9 ; 9 = 0000 0000 0000 0000 0000 1001 (logxor 12 5 7) ; 12 = 0000 0000 0000 0000 0000 1100 ; 5 = 0000 0000 0000 0000 0000 0101 ; 7 = 0000 0000 0000 0000 0000 0111 => 14 ; 14 = 0000 0000 0000 0000 0000 1110 - Function: lognot INTEGER This function returns the logical complement of its argument: the Nth bit is one in the result if, and only if, the Nth bit is zero in INTEGER, and vice-versa. ;; 5 = 0000 0000 0000 0000 0000 0101 ;; becomes ;; -6 = 1111 1111 1111 1111 1111 1010 (lognot 5) => -6