Bitwise arithmetic procedures

Abstract

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 worthwhile replacing the core ops 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) and a few procedures from SRFI 60 and the general vector SRFI 133. SRFI 33 was never finalized, but is a reasonably comprehensive proposal. SRFI 60 (based on SLIB) is smaller but has a few procedures of its own; some of its procedures have both native (often CL) and SRFI 33 names. R6RS is a subset of SRFI 60, except that all procedure names begin with a bitwise- prefix.

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 persistant 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.

Procedure index

bitwise-not 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? first-set-bit 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 bit-field-append integer->list list->integer bits bitwise-fold bitwise-for-each bitwise-unfold

Rationale

General design principles

• These operations interpret exact integers using two's-complement representation.

• It is not optional for the associative bitwise ops to be n-ary instead of merely binary. They are required to be n-ary. Programmers can reliably write bitwise-and with 3 arguments, for example.

• The word or is never used by itself, only with modifiers: xor, ior, nor, orc1, or orc2. This is the same rule as Common Lisp.

• Extra and redundant functions such as bitwise-count, bitwise-nor and the bit-field ops have been included. Settling on a standard choice of names makes it easier to read code that uses these sorts of operations. It also means computations can be clearly expressed using the more powerful ops rather than synthesized with a snarled mess of bitwise-ands, bitwise-ors, and bitwise-nots. What we gain is having an agreed-upon set of names by which we can refer to these functions. If you believe in "small is beautiful," then what is your motivation for including anything beyond bitwise-nand?

• The programmer doesn't have to re-implement the redundant functions, and stumble over the boundary cases and error checking. The programmer can express himself using a full palette of building blocks.

• Compilers can directly implement many of these ops for great efficiency gains without requiring any tricky analysis.

• Logical right or left shift operations are excluded because they are not well defined on general integers; they are only defined on integers in some finite range. Remember that, in this library, integers are interpreted as semi-infinite bit strings that have only a finite number of ones or a finite number of zeros. Logical shifting operates on bit strings of some fixed size. If we shift left, then leftmost bits "fall off" the end (and zeros shift in on the right). If we shift right, then zeros shift into the string on the left (and rightmost bits fall off the end). So to define a logical shift operation, we must specify the size of the window. Typically this is the width of the underlying machine's register set (e.g., 32 bits). This is blatantly machine-specific and unportable, and clearly not the right thing for Scheme's more machine-independent general integers.

Common Lisp

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

• "load" and "deposit" are the wrong verbs (e.g., Common Lisp's ldb and dpb ops), since these guys have nothing to do with the store.

• boole has been removed; it is not one with the Way of Scheme. Boolean functions are directly encoded in Scheme as first-class functions.

• The name choices are more in tune with Scheme conventions (hyphenation, using ? to mark a predicate, etc.). Common Lisp's name choices were more historically motivated, for reasons of backward compatibility with Maclisp and Zetalisp.

• The prefix log has been changed to bitwise- (e.g, lognot to bitwise-not), as the prefix bitwise- more accurately reflects what they do.

• The six trivial binary boolean ops that return constants, the left or right arguments, and the bitwise-not of the left or right arguments, do not appear in this SRFI.

SRFI 33

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

• The name bitwise-merge is replaced by bitwise-if, the name used in SRFI 60 and R6RS.

• The name extract-bit-field (bit-field-extract in Olin's revisions) is replaced by bit-field, the name used in SRFI 60 and R6RS.

• The names any-bits-set? and all-bits-set? are replaced by any-bit-set? and every-bit-set?, in accordance with Olin's revisions.

• The name test-bit-field? has been renamed bit-field-any? and supplemented with bit-field-every?, in accordance with Olin's revisions.

• Because copy-bit-field means different things in SRFI 33 and SRFI 60, SRFI 33's name copy-bit-field (bit-field-copy in Olin's revisions) has been changed to bit-field-replace-same.

SRFI 60

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

• The names rotate-bit-field and reverse-bit-field are replaced by bit-field-rotate and bit-field-reverse, in parallel with Olin's revisions.

• The procedures copy-bit, integer->list and list->integer are incorporated into this SRFI unchanged.

• The procedure booleans->integer is a convenient way to specify a bitwise integer in the absence of datum syntax: it accepts an arbitrary number of boolean arguments and returns a non-negative integer. So in this SRFI it has the short name bits, roughly analogous to list, string, and vector.

Other sources

The following procedures are inspired by SRFI 133: bit-swap, bit-field-append, bitwise-fold, bitwise-for-each, bitwise-unfold.

• The procedure bit-field-set is the counterpart of bit-field-clear.

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 to be operated on is placed before the "source" argument that provides the bits to be transferred to it.

• In SRFI 33, bitwise-nand and bitwise-nor accepted an arbitrary number of arguments even though they are not commutative. Olin's late revisions made them dyadic, and so does this SRFI.

• Common Lisp bit-field operations use a byte spec to encapsulate the position and size of the field. SRFI 33 bit-field operations had leading position and size arguments instead. These have been replaced in this SRFI by start (inclusive) and end (exclusive) arguments, the convention used not only in SRFI 60 and R6RS but also in most other subsequence operations in Scheme standards and SRFIs.

Specification

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.

(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 0, then the kth bit of the result is the kth bit of i, otherwise the kth bit of j. This is equivalent to:

(bitwise-ior (bitwise-and (bitwise-not mask) i) (bitwise-and mask j))

Single-bit operations

(bit-set? index i)

Is bit index set in bitstring i (where index is a non-negative exact integer)? As always, the rightmost/least-significant bit in i is bit 0.

(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.

(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))) or (= test-bits` (bitwise-and test-bits i'')))` respectively.

(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 designated bit field from i, shifted down to the least-significant position in the result.

(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-every? i start end)

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

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

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

(bit-field-replace dst src start end)

Returns dst with the designated bit field replaced by the least-significant end-start bits in src.

(bit-field-replace-same dst src start end)

Returns dst with the selected field's bits replaced by the corresponding field's bits in src.

(bit-field-rotate i count start end)

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

(bit-field-reverse i start end)

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

(bit-field-append { i start end })

The number of arguments must be a multiple of three. The field specified by each triple of (i, start, end) arguments is extracted, and the fields are concatenated in left-to-right order and returned as an integer.

Bits as booleans

(integer->list i [ len ])

Returns a list of len booleans corresponding to each bit of the non-negative integer i. #t is returned for each 1; #f for 0. The len argument defaults to (integer-length i).

(list->integer list)

Returns an integer formed from the booleans in list; it is an error if list 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. integer->list and list->integer are inverses in the sense of equal?.

(bits bool ...)

Returns the integer coded by the bool arguments.

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 to (integer-length i), 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 all bits are exhausted, the final accumulated result becomes the result of bitwise-fold.

(bitwise-for-each proc i)

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

(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. Then get a new state by applying successor to the current state, and repeat this algorithm.

(make-bitwise-generator i)

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

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.

Function	CL	SRFI 33	SRFI 33 late revs	SRFI 60	R6RS	This SRFI
Bitwise NOT	lognot	bitwise-not	bitwise-not	lognot, bitwise-not	bitwise-not	bitwise-not
Bitwise AND	logand	bitwise-and	bitwise-and	logand, bitwise-and	bitwise-and	bitwise-and
Bitwise IOR	logior	bitwise-ior	bitwise-ior	logior, bitwise-ior	bitwise-ior	bitwise-ior
Bitwise XOR	logxor	bitwise-xor	bitwise-xor	logxor, bitwise-xor	bitwise-xor	bitwise-xor
Bitwise EQV	logeqv	bitwise-eqv	bitwise-eqv	---	---	bitwise-eqv
Bitwise NAND	lognand	bitwise-nand	bitwise-nand	---	---	bitwise-nand
Bitwise NOR	lognor	bitwise-nor	bitwise-nor	---	---	bitwise-nor
Bitwise AND with NOT of first arg	logandc1	bitwise-andc1	bitwise-andc1	---	---	bitwise-andc1
Bitwise AND with NOT of second arg	logandc2	bitwise-andc2	bitwise-andc2	---	---	bitwise-andc2
Bitwise OR with NOT of first arg	logorc1	bitwise-orc1	bitwise-orc1	---	---	bitwise-orc1
Bitwise OR with NOT of second arg	logorc2	bitwise-orc2	bitwise-orc2	---	---	bitwise-orc2
Arithmetic shift	ash	arithmetic-shift	arithmetic-shift	ash, arithmetic-shift	bitwise-arithmetic-shift	arithmetic-shift
Population count	logcount	bit-count	bit-count	logcount, bit-count	bitwise-bit-count	bit-count
Integer length	integer-length	integer-length	integer-length	integer-length	bitwise-integer-length	integer-length
Mask selects source of bits	---	bitwise-merge	bitwise-merge	bitwise-if, bitwise-merge	bitwise-if	bitwise-if
Test single bit	logbitp	bit-set?	bit-set?	logbit?, bit-set?	bitwise-bit-set?	bit-set?
See if any mask bits set	logtest	any-bits-set?	any-bit-set?	logtest, any-bit-set?	---	any-bit-set
See if all mask bits set	---	all-bits-set?	every-bit-set?	---	---	every-bit-set?
Replace single bit	---	---	copy-bit	bitwise-copy-bit	---	copy-bit
Swap bits	---	---	---	---	---	bit-swap
Find first bit set	---	first-bit-set	first-set-bit	log2-binary-factors, first-set-bit	---	first-set-bit
Extract bit field	ldb	extract-bit-field	extract-bit-field	bit-field	bitwise-bit-field	bit-field
Test bit field (any)	ldb-test	test-bit-field?	bit-field-any?	---	---	bit-field-any?
Test bit field (every)	---	---	bit-field-every?	---	---	bit-field-every?
Clear bit field	mask-field	clear-bit-field	bit-field-clear	---	---	bit-field-clear
Replace bit field	dpb	replace-bit-field	bit-field-replace	copy-bit-field	bitwise-copy-bit-field	bit-field-replace
Replace corresponding bit field	deposit-field	deposit-field	copy-bit-field	---	---	bit-field-copy-same
Fill bit field	---	---	---	---	---	bit-field-fill
Rotate bit field	---	---	---	rotate-bit-field	bitwise-rotate-bit-field	bit-field-rotate
Reverse bit field	---	---	---	reverse-bit-field	bitwise-reverse-bit-field	bit-field-reverse
Append bit fields	---	---	---	---	---	bit-field-append
Integer to boolean list	---	---	---	integer->list	---	integer->list
Boolean list to integer	---	---	---	list->integer	---	list->integer
Booleans to integer	---	---	---	booleans->integer	---	bits

||Bitwise fold||---||---||---||---||---||bitwise-fold|

Bitwise for-each	---	---	---	---	---	bitwise-for-each
Bitwise unfold	---	---	---	---	---	bitwise-unfold