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SRFI 143: Fixnums

Abstract

This SRFI describes arithmetic procedures applicable to a limited range of exact integers only. These procedures are semantically similar to the corresponding generic-arithmetic procedures, but allow more efficient implementations.

For more information see: SRFI 143: Fixnums

Rationale

It is common for Schemes that support arbitrarily large exact integers to have two different representations: one for smaller integers (in absolute value) and one for the rest. These are colloquially known as fixnums and bignums respectively. Because the maximum size of a fixnum is typically smaller than the size of a machine word, fixnums can be represented more compactly and operated on more efficiently than bignums.

Specific procedures for fixnum arithmetic are already supported by many Scheme systems. 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 fixnum operations, but makes the range of fixnums implementation-dependent.

This SRFI is a modest extension of the R6RS (rnrs arithmetic fixnum) library, with some procedures renamed for consistency with R7RS-small. New procedures include fxneg, fxabs, fxsquare, and fxsqrt.

Existing implementations employ different implementation strategies for fixnum procedures. Some implement the model specified by R6RS (overflows cause exceptions), some implement modular arithmetic (overflows "wrap around"), and others fail catastrophically. In programs that use fixnums instead of generic arithmetic, overflows are typically programming mistakes.

Specification

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.

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. Consequently, when this SRFI says things like "fx+ is semantically equivalent to +", the phrase "except for the effects of the fixnum rule" is to be understood.

This SRFI uses i, j, k as parameter names for fixnum arguments. Except as noted, the names of fixnum procedures begin with the letters fx. In most cases they correspond to an R7RS-small or SRFI 151 operation on general integers.

Constants

fx-widthconstant

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

fx-greatestconstant

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

fx-leastconstant

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

Predicates

fixnum? objprocedure

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

fx=? i ...procedure

Semantically equivalent to =.

fx<? i ...procedure

Semantically equivalent to <.

fx>? i ...procedure

Semantically equivalent to >.

fx<=? i ...procedure

Semantically equivalent to <=.

fx>=? i ...procedure

Semantically equivalent to >=.

fxzero? iprocedure

Semantically equivalent to zero?.

fxpositive? iprocedure

Semantically equivalent to positive?.

fxnegative? iprocedure

Semantically equivalent to negative?.

fxodd? iprocedure

Semantically equivalent to odd?.

fxeven? iprocedure

Semantically equivalent to even?.

fxmax i j ...procedure

Semantically equivalent to max.

fxmin i j ...procedure

Semantically equivalent to min.

Basic arithmetic

fx+ i jprocedure

Semantically equivalent to +, but accepts exactly two arguments.

fx- i jprocedure

Semantically equivalent to -, but accepts exactly two arguments.

fxneg iprocedure

Semantically equivalent to -, but accepts exactly one argument.

fx* i jprocedure

Semantically equivalent to *, but accepts exactly two arguments.

fxquotient i jprocedure

Semantically equivalent to quotient.

fxremainder i jprocedure

Semantically equivalent to remainder.

fxabs iprocedure

Semantically equivalent to abs. In accordance with the fixnum rule, has undefined results when applied to fx-least.

fxsquare iprocedure

Semantically equivalent to square.

fxsqrt iprocedure

Semantically equivalent to exact-integer-sqrt (not sqrt).

Arithmetic with carry

fx+/carry i j kprocedure

Returns the two fixnum results of the following computation:

(let*-values (((s) (+ i j k))
       ((q r) (balanced/ s (expt 2 fx-width))))
  (values r q))
fx-/carry i j kprocedure

Returns the two fixnum results of the following computation:

(let*-values (((d) (- i j k))
       ((q r) (balanced/ d (expt 2 fx-width))))
  (values r q))
fx*/carry i j kprocedure

Returns the two fixnum results of the following computation:

(let*-values (((s) (+ (* i j) k))
       ((q r) (balanced/ s (expt 2 fx-width))))
  (values r q))

The balanced/ procedure is available in SRFI 141, and also in the R6RS base library under the name of div0-and-mod0.

Bitwise operations

The following procedures are the fixnum counterparts of certain bitwise operations from SRFI 151 and the R6RS (rnrs arithmetic fixnums) library. In case of disagreement, SRFI 151 is preferred. The prefixes bitwise- and integer- are dropped for brevity and compatibility.

fxnot iprocedure

Semantically equivalent to bitwise-not.

fxand i ...procedure

Semantically equivalent to bitwise-and.

fxior i ...procedure

Semantically equivalent to bitwise-ior.

fxxor i ...procedure

Semantically equivalent to bitwise-xor.

fxarithmetic-shift i countprocedure

Semantically equivalent to arithmetic-shift, except that it is an error for the absolute value of count to exceed w-1.

fxarithmetic-shift-left i countprocedure

The same as fxarithmetic-shift except that a negative value of count is an error. This is provided for additional efficiency.

fxarithmetic-shift-right i countprocedure

The same as fxarithmetic-shift except that a non-negative value of count specifies the number of bits to shift right, and a negative value is an error. This is provided for additional efficiency.

fxbit-count iprocedure

Semantically equivalent to SRFI 151 bit-count.

fxlength iprocedure

Semantically equivalent to integer-length.

fxif mask i jprocedure

Semantically equivalent to bitwise-if. It can be implemented as (fxior (fxand mask i) (fxand (fxnot mask) j))).

fxbit-set? index iprocedure

Semantically equivalent to SRFI 151 bit-set?, except that it is an error for index to be larger than or equal to fx-width.

fxcopy-bit index i booleanprocedure

Semantically equivalent to SRFI 151 copy-bit, except that it is an error for index to be larger than or equal to fx-width.

fxfirst-set-bit iprocedure

Semantically equivalent to first-set-bit.

fxbit-field i start endprocedure

Semantically equivalent to bit-field.

(fxbit-field-rotate {{i}} count start end)procedure

Semantically equivalent to SRFI 151 bit-field-rotate.

fxbit-field-reverse i start endprocedure

Semantically equivalent to bit-field-reverse.

Acknowledgements

This SRFI would not be possible without the efforts of Olin Shivers, Audrey Jaffer, Taylor Campbell, and the R6RS editors.

Author

Copyright

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.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

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