Fixnums­Cowan

Abstract

Fixnums are an implementation-defined subset of the exact integers. Every implementation of this SRFI must define its fixnum range as a closed interval [-2^w-1, 2^w-1-1], where w is an integer greater than or equal to 24. Every mathematical integer within an implementation's fixnum range must correspond to an exact integer that is representable within the implementation. A fixnum is an exact integer whose value lies within this fixnum range.

Rationale

Fixnum arithmetic is already supported by many systems, mainly for efficiency. Standardizing fixnum arithmetic increases the portability of code that uses it. Standardizing the range of fixnums would make fixnum operations inefficient on some systems, which would defeat their purpose. Therefore, this SRFI specifies some of the semantics of fixnums, but makes the range implementation-dependent.

Existing implementations employ different implementation strategies for fixnums: Some implement the model specified by R6RS (overflows cause exceptions), some implement modular arithmetic (overflows “wrap around”), and others do not handle arithmetic overflows at all. In programs that use fixnums instead of generic arithmetic, overflows are typically programming mistakes.

Specification

Fixnum operations perform integer arithmetic on their fixnum arguments. If any argument is not a fixnum, or if the mathematical result is not representable as a fixnum, it is an error: this is known as the fixnum rule. In particular, this means that fixnum operations may return a mathematically incorrect fixnum in these situations without raising an error. Exceptions to the fixnum rule are noted below.

This SRFI uses fx, fx₁, fx₂, etc., as parameter names for fixnum arguments. Except as noted, the name of fixnum procedures begin with the letters fx. In most cases they correspond to an R7RS-small or DivisionRiastradh or BitwiseCowan operation on general integers.

Constants

fx-width

Bound to the value w that specifies the implementation-defined range. (R6RS fixnum-width is a procedure that always returns this value.)

fx-greatest

Bound to the value 2^w-1-1, the largest representable fixnum. (R6RS greatest-fixnum is a procedure that always returns this value.)

fx-least

Bound to the value -2^w-1, the smallest representable fixnum. (R6RS least-fixnum is a procedure that always returns this value.)

Predicate

(fixnum? obj)

Returns #t if obj is an exact integer within the fixnum range, and #f otherwise.

Basic arithmetic

The following procedures are the fixnum counterparts of procedures from R7RS-small:

fxzero? fxpositive? fxnegative? fxodd? fxeven? fx= fx< fx> fx<= fx>= fxmax fxmin fx+ fx- fx* fxabs fxsquare fxsqrt fxexpt

Except for the effects of the fixnum rule, the fx versions have the same arguments and semantics as their generic counterparts, with the following additional modifications:

• The procedures fx+ and fx* accept exactly two arguments.

• The procedure fx- accepts either one or two arguments.

• The fxsqrt procedure is the counterpart of exact-integer-sqrt rather than sqrt.

Note that in accordance with the fixnum rule the procedure fxabs has undefined results when applied to fx-least.

Arithmetic with carry

(fx+/carry fx₁ fx₂ fx₃)‌‌

Returns the two fixnum results of the following computation:

(let* ((s (+ fx1 fx2 fx3)) (s0 (balanced-remainder s (expt 2 (fixnum-width)))) (s1 (balanced-quotient s (expt 2 (fixnum-width))))) (values s0 s1))

(fx-/carry fx₁ fx₂ fx₃)‌‌

Returns the two fixnum results of the following computation:

(let* ((d (- fx1 fx2 fx3)) (d0 (balanced-remainder d (expt 2 (fixnum-width)))) (d1 (balanced-quotient d (expt 2 (fixnum-width))))) (values d0 d1))

(fx*/carry fx₁ fx₂ fx₃)‌‌

Returns the two fixnum results of the following computation:

(let* ((s (+ (* fx1, fx1)) fx3)) (s0 (balanced-remainder s (expt 2 (fixnum-width)))) (s1 (balanced-quotient s (expt 2 (fixnum-width))))) (values s0 s1))

Integer division

The following procedures are the fixnum counterparts of procedures from SRFI 141:

fxfloor/ fxfloor-quotient fxfloor-remainder fxceiling/ fxceiling-quotient fxceiling-remainder fxtruncate/ fxtruncate-quotient fxtruncate-remainder fxround/ fxround-quotient fxround-remainder fxeuclidean/ fxeuclidean-quotient fxeuclidean-remainder fxbalanced/ fxbalanced-quotient fxbalanced-remainder

Except for the effects of the fixnum rule, the fx versions have the same arguments and semantics as their generic counterparts.

Bitwise operations

The following procedures are the fixnum counterparts of procedures from SRFI 141:

fxnot fxand fxior fxxor fxeqv fxnand fxnor fxandc1 fxandc2 fxorc1 fxorc2 farithmetic-shift fxbit-count fxinteger-length fxif fxbit-set? fxcopy-bit fxbit-swap fxany-bit-set? fxevery-bit-set? fxfirst-set-bit fxbit-field fxbit-field-any? fxbit-field-every? fxbit-field-clear fxbit-field-set fxbit-field-replace fbit-field-replace-same fxbit-field-rotate fxbit-field-reverse fxbit-field-append fixnum->list list->fixnum fixnum->vector vector->fixnum fxbits fxfold fxfor-each fxunfold

Except for the effects of the fixnum rule, the fx versions have the same arguments and semantics as their generic counterparts, with the following additional modifications:

• The prefix bitwise- in the SRFI 141 functions is dropped for brevity and compatibility.

• Despite the fixnum rule, the fxarithmetic-shift procedure produces a defined result on all fixnums by discarding any higher-order bits that do not fit into the fixnum width.

• The fixnum->list and list->fixnum procedures correspond to the integer->list and list->integer procedures respectively, and the same for their vector analogues.

The following additional bitwise procedure is provided:

(fxlogical-shift i count)

When left shifting (count > 0), returns the same result as fxarithmetic-shift. When right shifting, always inserts 0 bits at the most significant end rather than copies of the sign bit.

The result of a logical shift depends on the value of fx-width. This means that if fx-width were 8 (which this SRFI does not permit), (fxlogical-shift -8 -1) would be #x74, or 116, rather than -4.