SRFI 151: Bitwise Operations


John Cowan


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This SRFI proposes a coherent and comprehensive set of procedures for performing bitwise logical operations on integers; it is accompanied by a reference implementation of the spec in terms of a set of seven core operators. The sample implementation is portable, as efficient as practical with pure Scheme arithmetic (it is much more efficient to replace the core operators with C or assembly language if possible), and open source.

The precise semantics of these operators is almost never an issue. A consistent, portable set of names and parameter conventions, however, is. Hence this SRFI, which is based mainly on SRFI 33, with some changes and additions from Olin's late revisions to SRFI 33 (which were never consummated). SRFI 60 (based on SLIB) is smaller but has a few procedures of its own; some of its procedures have both native (often Common Lisp) and SRFI 33 names. They have been incorporated into this SRFI. R6RS is a subset of SRFI 60, except that all procedure names begin with a bitwise- prefix. A few procedures have been added from the general vector SRFI 133.

Among the applications of bitwise operations are: hashing, Galois-field calculations of error-detecting and error-correcting codes, cryptography and ciphers, pseudo-random number generation, register-transfer-level modeling of digital logic designs, Fast-Fourier transforms, packing and unpacking numbers in persistent data structures, space-filling curves with applications to dimension reduction and sparse multi-dimensional database indexes, and generating approximate seed values for root-finders and transcendental function algorithms.

This SRFI differs from SRFI 142 in only two ways:

  1. The bitwise-if function has the argument ordering of SLIB, SRFI 60, and R6RS rather than the ordering of SRFI 33.

  2. The order in which bits are processed by the procedures listed in the "Bits conversion" section has been clarified and some of the procedures' names have been changed. See "Bit processing order" for details.


General design principles

Common Lisp

The core of this design mirrors the structure of Common Lisp's pretty closely. Here are some differences:


This SRFI contains all the procedures of SRFI 33, and retains their original names with these exceptions:


SRFI 60 includes six procedures that do not have SRFI 33 equivalents. They are incorporated into this SRFI as follows:

Other sources

Argument ordering and semantics

In general, these procedures place the bitstring arguments to be operated on first. Where the operation is not commutative, the "destination" argument that provides the background bits to be operated on is placed before the "source" argument that provides the bits to be transferred to it.

Bit processing order

In SLIB and SRFI 60, the the order in which bits were processed by integer->list and list->integer was not clearly specified. When SRFI 142 was written, the specification was clarified to process bits from least significant to most significant, so that (integer->list 6) => (#f #t #t). However, the SLIB and SRFI 60 implementation processed them from the most significant bit to the least-significant bit, so that (integer->list 6) => (#t #t #f). This SRFI retains the little-endian order, but renames the procedures to bits->list and list->bits to avoid a silent breaking change from SLIB and SRFI 60. The same is true of the closely analogous integer->vector, vector->integer, and bits procedures.


Procedure index

bitwise-and   bitwise-ior 
bitwise-xor   bitwise-eqv
bitwise-nand  bitwise-nor 
bitwise-andc1 bitwise-andc2
bitwise-orc1  bitwise-orc2 

arithmetic-shift bit-count 
integer-length bitwise-if 

bit-set? copy-bit bit-swap
any-bit-set? every-bit-set?

bit-field bit-field-any? bit-field-every?
bit-field-clear bit-field-set
bit-field-replace  bit-field-replace-same
bit-field-rotate bit-field-reverse

bits->list list->bits bits->vector vector->bits
bitwise-fold bitwise-for-each bitwise-unfold

In the following procedure specifications all parameters and return values are exact integers unless otherwise indicated (except that procedures with names ending in ? are predicates, as usual). It is an error to pass values of other types as arguments to these functions.

Bitstrings are represented by exact integers, using a two's-complement encoding of the bitstring. Thus every integer represents a semi-infinite bitstring, having either a finite number of zeros (negative integers) or a finite number of ones (non-negative integers). The bits of a bitstring are numbered from the rightmost/least-significant bit: bit #0 is the rightmost or 20 bit, bit #1 is the next or 21 bit, and so forth.

Basic operations

(bitwise-not i)

Returns the bitwise complement of i; that is, all 1 bits are changed to 0 bits and all 0 bits to 1 bits.

  (bitwise-not 10) => -11
  (bitwise-not -37) => 36

The following ten procedures correspond to the useful set of non-trivial two-argument boolean functions. For each such function, the corresponding bitwise operator maps that function across a pair of bitstrings in a bit-wise fashion. The core idea of this group of functions is this bitwise "lifting" of the set of dyadic boolean functions to bitstring parameters.

(bitwise-and i ...)
(bitwise-ior i ...)
(bitwise-xor i ...)
(bitwise-eqv i ...)

These operations are associative. When passed no arguments, the procedures return the identity values -1, 0, 0, and -1 respectively.

The bitwise-eqv procedure produces the complement of the bitwise-xor procedure. When applied to three arguments, it does not produce a 1 bit everywhere that a, b and c all agree. That is, it does not produce

     (bitwise-ior (bitwise-and a b c)
                  (bitwise-and (bitwise-not a)
                               (bitwise-not b)
                               (bitwise-not c)))

Rather, it produces (bitwise-eqv a (bitwise-eqv b c)) or the equivalent (bitwise-eqv (bitwise-eqv a b) c).

      (bitwise-ior 3  10)     =>  11
      (bitwise-and 11 26)     =>  10
      (bitwise-xor 3 10)      =>   9
      (bitwise-eqv 37 12)     => -42
      (bitwise-and 37 12)     =>   4

(bitwise-nand i j)
(bitwise-nor i j)
(bitwise-andc1 i j)
(bitwise-andc2 i j)
(bitwise-orc1 i j)
(bitwise-orc2 i j)

These operations are not associative.

      (bitwise-nand 11 26) =>  -11
      (bitwise-nor  11 26) => -28
      (bitwise-andc1 11 26) => 16
      (bitwise-andc2 11 26) => 1
      (bitwise-orc1 11 26) => -2
      (bitwise-orc2 11 26) => -17

Integer operations

(arithmetic-shift i count)

Returns the arithmetic left shift when count>0; right shift when count<0.

    (arithmetic-shift 8 2) => 32
    (arithmetic-shift 4 0) => 4
    (arithmetic-shift 8 -1) => 4
    (arithmetic-shift -100000000000000000000000000000000 -100) => -79

(bit-count i)

Returns the population count of 1's (i >= 0) or 0's (i < 0). The result is always non-negative.

Compatibility note: The R6RS analogue bitwise-bit-count applies bitwise-not to the population count before returning it if i is negative.

    (bit-count 0) =>  0
    (bit-count -1) =>  0
    (bit-count 7) =>  3
    (bit-count  13) =>  3 ;Two's-complement binary: ...0001101
    (bit-count -13) =>  2 ;Two's-complement binary: ...1110011
    (bit-count  30) =>  4 ;Two's-complement binary: ...0011110
    (bit-count -30) =>  4 ;Two's-complement binary: ...1100010
    (bit-count (expt 2 100)) =>  1
    (bit-count (- (expt 2 100))) =>  100
    (bit-count (- (1+ (expt 2 100)))) =>  1

(integer-length i)

The number of bits needed to represent i, i.e.

	(ceiling (/ (log (if (negative? integer)
			     (- integer)
			     (+ 1 integer)))
		    (log 2)))

The result is always non-negative. For non-negative i, this is the number of bits needed to represent i in an unsigned binary representation. For all i, (+ 1 (integer-length i)) is the number of bits needed to represent i in a signed twos-complement representation.

    (integer-length  0) => 0
    (integer-length  1) => 1
    (integer-length -1) => 0
    (integer-length  7) => 3
    (integer-length -7) => 3
    (integer-length  8) => 4
    (integer-length -8) => 3

(bitwise-if mask i j)

Merge the bitstrings i and j, with bitstring mask determining from which string to take each bit. That is, if the kth bit of mask is 1, then the kth bit of the result is the kth bit of i, otherwise the kth bit of j.

    (bitwise-if 3 1 8) => 9
    (bitwise-if 3 8 1) => 0
    (bitwise-if 1 1 2) => 3
    (bitwise-if #b00111100 #b11110000 #b00001111) => #b00110011

Single-bit operations

As always, the rightmost/least-significant bit in i is bit 0.

(bit-set? index i)

Is bit index set in bitstring i (where index is a non-negative exact integer)?

Compatibility note: The R6RS analogue bitwise-bit-set? accepts its arguments in the opposite order.

    (bit-set? 1 1) =>  false
    (bit-set? 0 1) =>  true
    (bit-set? 3 10) =>  true
    (bit-set? 1000000 -1) =>  true
    (bit-set? 2 6) =>  true
    (bit-set? 0 6) =>  false

(copy-bit index i boolean)

Returns an integer the same as i except in the indexth bit, which is 1 if boolean is #t and 0 if boolean is #f.

Compatibility note: The R6RS analogue bitwise-copy-bit as originally documented has a completely different interface. (bitwise-copy-bit dest index source) replaces the index'th bit of dest with the index'th bit of source. It is equivalent to (bit-field-replace-same dest source index (+ index 1)). However, an erratum made a silent breaking change to interpret the third argument as 0 for a false bit and 1 for a true bit. Some R6RS implementations applied this erratum but others did not.

(copy-bit 0 0 #t) => #b1
(copy-bit 2 0 #t) => #b100
(copy-bit 2 #b1111 #f) => #b1011

(bit-swap index1 index2 i)

Returns an integer the same as i except that the index1th bit and the index2th bit have been exchanged.

(bit-swap 0 2 4) => #b1

(any-bit-set? test-bits i)
(every-bit-set? test-bits i)

Determines if any/all of the bits set in bitstring test-bits are set in bitstring i. I.e., returns (not (zero? (bitwise-and test-bits i))) and (= test-bits (bitwise-and test-bits i))) respectively.

    (any-bit-set? 3 6) => #t
    (any-bit-set? 3 12) => #f
    (every-bit-set? 4 6) => #t
    (every-bit-set? 7 6) => #f

(first-set-bit i)

Return the index of the first (smallest index) 1 bit in bitstring i. Return -1 if i contains no 1 bits (i.e., if i is zero).

    (first-set-bit 1) => 0
    (first-set-bit 2) => 1
    (first-set-bit 0) => -1
    (first-set-bit 40) => 3
    (first-set-bit -28) => 2
    (first-set-bit (expt  2 99)) => 99
    (first-set-bit (expt -2 99)) => 99

Bit field operations

These functions operate on a contiguous field of bits (a "byte," in Common Lisp parlance) in a given bitstring. The start and end arguments, which are not optional, are non-negative exact integers specifying the field: it is the end-start bits running from bit start to bit end-1.

(bit-field i start end)

Returns the field from i, shifted down to the least-significant position in the result.

   (bit-field #b1101101010 0 4) => #b1010
   (bit-field #b1101101010 3 9) => #b101101
   (bit-field #b1101101010 4 9) => #b10110
   (bit-field #b1101101010 4 10) => #b110110
   (bit-field 6 0 1) => 0
   (bit-field 6 1 3) => 3
   (bit-field 6 2 999) => 1
   (bit-field #x100000000000000000000000000000000 128 129) => 1

(bit-field-any? i start end)

Returns true if any of the field's bits are set in bitstring i, and false otherwise.

  (bit-field-any? #b1001001 1 6) => #t
  (bit-field-any? #b1000001 1 6) => #f

(bit-field-every? i start end)

Returns false if any of the field's bits are not set in bitstring i, and true otherwise.

  (bit-field-every? #b1011110 1 5) => #t
  (bit-field-every? #b1011010 1 5) => #f

(bit-field-clear i start end)
(bit-field-set i start end)

Returns i with the field's bits set to all 0s/1s.

   (bit-field-clear #b101010 1 4) => #b100000
   (bit-field-set #b101010 1 4) => #b101110

(bit-field-replace dest source start end)

Returns dest with the field replaced by the least-significant end-start bits in source.

   (bit-field-replace #b101010 #b010 1 4) => #b100100
   (bit-field-replace #b110 1 0 1) => #b111
   (bit-field-replace #b110 1 1 2) => #b110

(bit-field-replace-same dest source start end)

Returns dest with its field replaced by the corresponding field in source.

   (bit-field-replace-same #b1111 #b0000 1 3) => #b1001

(bit-field-rotate i count start end)

Returns i with the field cyclically permuted by count bits towards high-order.

Compatibility note: The R6RS analogue bitwise-rotate-bit-field uses the argument ordering i start end count.

   (bit-field-rotate #b110 0 0 10) => #b110
   (bit-field-rotate #b110 0 0 256) => #b110
   (bit-field-rotate #x100000000000000000000000000000000 1 0 129) => 1
   (bit-field-rotate #b110 1 1 2) => #b110
   (bit-field-rotate #b110 1 2 4) => #b1010
   (bit-field-rotate #b0111 -1 1 4) => #b1011

(bit-field-reverse i start end)

Returns i with the order of the bits in the field reversed.

   (bit-field-reverse 6 1 3) => 6
   (bit-field-reverse 6 1 4) => 12
   (bit-field-reverse 1 0 32) => #x80000000
   (bit-field-reverse 1 0 31) => #x40000000
   (bit-field-reverse 1 0 30) => #x20000000
   (bit-field-reverse #x140000000000000000000000000000000 0 129) => 5

Bits conversion

(bits->list i [ len ])
(bits->vector i [ len ])

Returns a list/vector of len booleans corresponding to each bit of the non-negative integer i, returning bit #0 as the first element, bit #1 as the second, and so on. #t is returned for each 1; #f for 0.

   (bits->list #b1110101) => (#t #f #t #f #t #t #t)
   (bits->list 3 5) => (#t #t #f #f #f)
   (bits->list 6 4) => (#f #t #t #f)

   (bits->vector #b1110101) => #(#t #f #t #f #t #t #t)

(list->bits list)
(vector->bits vector)

Returns an integer formed from the booleans in list/vector, using the first element as bit #0, the second element as bit #1, and so on. It is an error if list/vector contains non-booleans. A 1 bit is coded for each #t; a 0 bit for #f. Note that the result is never a negative integer.

   (list->bits '(#t #f #t #f #t #t #t)) => #b1110101
   (list->bits '(#f #f #t #f #t #f #t #t #t)) => #b111010100
   (list->bits '(#f #t #t)) => 6
   (list->bits '(#f #t #t #f)) => 6
   (list->bits '(#f #f #t #t)) => 12

   (vector->bits '#(#t #f #t #f #t #t #t)) => #b1110101
   (vector->bits '#(#f #f #t #f #t #f #t #t #t)) => #b111010100
   (vector->bits '#(#f #t #t)) => 6
   (vector->bits '#(#f #t #t #f)) => 6
   (vector->bits '#(#f #f #t #t)) => 12

For positive integers, bits->list and list->bits are inverses in the sense of equal?, and so are bits->vector and vector->bits.

(bits bool ...)

Returns the integer coded by the bool arguments. The first argument is bit #0, the second argument is bit #1, and so on. Note that the result is never a negative integer.

  (bits #t #f #t #f #t #t #t) => #b1110101
  (bits #f #f #t #f #t #f #t #t #t) => #b111010100

Fold, unfold, and generate

It is an error if the arguments named proc, stop?, mapper, successor are not procedures. The arguments named seed may be any Scheme object.

(bitwise-fold proc seed i)

For each bit b of i from bit #0 (inclusive) to bit (integer-length i) (exclusive), proc is called as (proc b r), where r is the current accumulated result. The initial value of r is seed, and the value returned by proc becomes the next accumulated result. When the last bit has been processed, the final accumulated result becomes the result of bitwise-fold.

  (bitwise-fold cons '() #b1010111) => (#t #f #t #f #t #t #t)

(bitwise-for-each proc i)

Repeatedly applies proc to the bits of i starting with bit #0 (inclusive) and ending with bit (integer-length i) (exclusive). The values returned by proc are discarded. Returns an unspecified value.

      (let ((count 0))
        (bitwise-for-each (lambda (b) (if b (set! count (+ count 1))))

(bitwise-unfold stop? mapper successor seed)

Generates a non-negative integer bit by bit, starting with bit 0. If the result of applying stop? to the current state (whose initial value is seed) is true, return the currently accumulated bits as an integer. Otherwise, apply mapper to the current state to obtain the next bit of the result by interpreting a true value as a 1 bit and a false value as a 0 bit. Then get a new state by applying successor to the current state, and repeat this algorithm.

  (bitwise-unfold (lambda (i) (= i 10))
                  (lambda (i) (+ i 1))
                  0) => #b101010101

(make-bitwise-generator i)

Returns a SRFI 121 generator that generates all the bits of i starting with bit #0. Note that the generator is infinite.

  (let ((g (make-bitwise-generator #b110)))
    (test #f (g))
    (test #t (g))
    (test #t (g))
    (test #f (g)))


The implementation is in the repository of this SRFI, and includes the following files:

It is very important for performance to provide a C or assembly implementation of the core operators. There is a cond-expand in srfi-151.sld that can be extended to take advantage of this. Currently it contains options for Chibi and Gauche. In addition, if the R6RS library (rnrs arithmetic bitwise) is available, it will be used in place of bitwise-core.scm.

Temporary note: Chibi assumes a C library named bit. The bit.{so,dll} file can be found in $PREFIX/lib/chibi/srfi/{33,142}. Even more temporary note: this is buggy, so the Chibi arm of the conditional is currently commented out.

Comparison of proposals

The following table compares the names of the bitwise (aka logical) functions of Common Lisp, SRFI 33, Olin's revisions, SRFI 60, R6RS, and this SRFI. Italic procedure names indicate that the equivalence is only rough: argument orderings or precise semantics are not the same as in this SRFI.

FunctionCLSRFI 33SRFI 33 late revsSRFI 60R6RSThis SRFI
Bitwise NOTlognotbitwise-notbitwise-notlognot, bitwise-notbitwise-notbitwise-not
Bitwise ANDlogandbitwise-andbitwise-andlogand, bitwise-andbitwise-andbitwise-and
Bitwise IORlogiorbitwise-iorbitwise-iorlogior, bitwise-iorbitwise-iorbitwise-ior
Bitwise XORlogxorbitwise-xorbitwise-xorlogxor, bitwise-xorbitwise-xorbitwise-xor
Bitwise EQVlogeqvbitwise-eqvbitwise-eqv------bitwise-eqv
Bitwise NANDlognandbitwise-nandbitwise-nand------bitwise-nand
Bitwise NORlognorbitwise-norbitwise-nor------bitwise-nor
Bitwise AND with NOT of first arglogandc1bitwise-andc1bitwise-andc1------bitwise-andc1
Bitwise AND with NOT of second arglogandc2bitwise-andc2bitwise-andc2------bitwise-andc2
Bitwise OR with NOT of first arglogorc1bitwise-orc1bitwise-orc1------bitwise-orc1
Bitwise OR with NOT of second arglogorc2bitwise-orc2bitwise-orc2------bitwise-orc2
Arithmetic shiftasharithmetic-shiftarithmetic-shiftash, arithmetic-shiftbitwise-arithmetic-shiftarithmetic-shift
Population countlogcountbit-countbit-countlogcount, bit-countbitwise-bit-countbit-count
Integer lengthinteger-lengthinteger-lengthinteger-lengthinteger-lengthbitwise-lengthinteger-length
Mask selects source of bits---bitwise-mergebitwise-mergebitwise-if, bitwise-mergebitwise-ifbitwise-if
Test single bitlogbitpbit-set?bit-set?logbit?, bit-set?bitwise-bit-set?bit-set?
See if any mask bits setlogtestany-bits-set?any-bit-set?logtest, any-bits-set?---any-bit-set?
See if all mask bits set---all-bits-set?every-bit-set?------every-bit-set?
Replace single bit------copy-bitcopy-bitbitwise-copy-bitcopy-bit
Swap bits---------------bit-swap
Find first bit set---first-set-bitfirst-set-bitlog2-binary-factors, first-set-bitbitwise-first-bit-setfirst-set-bit
Extract bit fieldldbextract-bit-fieldextract-bit-fieldbit-fieldbitwise-bit-fieldbit-field
Test bit field (any)ldb-testtest-bit-field?bit-field-any?------bit-field-any?
Test bit field (every)------bit-field-every?------bit-field-every?
Clear bit fieldmask-fieldclear-bit-fieldbit-field-clear------bit-field-clear
Set bit field---------------bit-field-set
Replace bit fielddpbreplace-bit-fieldbit-field-replacecopy-bit-fieldbitwise-copy-bit-fieldbit-field-replace
Replace corresponding bit fielddeposit-fieldcopy-bit-fieldbit-field-copy------bit-field-replace-same
Rotate bit field---------rotate-bit-fieldbitwise-rotate-bit-fieldbit-field-rotate
Reverse bit field---------reverse-bit-fieldbitwise-reverse-bit-fieldbit-field-reverse
Bits to boolean list---------integer->list---bits->list
Bits to boolean vector---------------bits->vector
Boolean list to bits---------list->integer---list->bits
Boolean vector to bits---------------vector->bits
Booleans to integer---------booleans->integer---bits
Bitwise fold---------------bitwise-fold
Bitwise for-each---------------bitwise-for-each
Bitwise unfold---------------bitwise-unfold
Bit generator---------------make-bitwise-generator


This SRFI would not exist without the efforts of Olin Shivers, Aubrey Jaffer, and Taylor Campbell.


Copyright (C) John Cowan (2016). All Rights Reserved.

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.


Editor: Arthur A. Gleckler