# Integral Division and Remainder Operators

## Abstract

This SRFI provides a fairly complete set of integral division and remainder operators.

## Rationale

Most programming languages provide at least one operation for division,
and sometimes related operations for computing integral quotients and
remainders. (There is at least one (fortunately today defunct)
programming language that provides addition, subtraction, and division,
with multiplication notably absent, being expressible as division.)
Everyone agrees that a pair of operators for computing integral
quotients *q* and remainders *r* from division of dividend *n* by divisor *d*,
should satisfy the relations

*n*=*dq*+*r*,- |
*r*| < |*d*|, and *q*is an integer.

Such a pair of operators will be called a division operator pair. Many programming languages provide only one division operator pair. Some, such as C, leave the semantics unspecified when either or each of the dividend and the divisor is negative. If the dividend and divisor are both integers, then the remainder will also be an integer.

To describe the semantics of a division operator pair, it suffices to
define the integer *q*, from which *r* can be uniquely derived by the
relation

r=n-dq,

provided that this choice of *q* induced an *r* satisfying |*r*| < |*d*|. For
an extensive discussion of five of the six division operator pairs proposed
here, and some broken but standardized operator pairs that fail to
satisfy properties (1)-(3), see Raymond T. Boute, [http://dl.acm.org/citation.cfm?id=128862
"The Euclidean Definition of the Functions DIV and MOD"], ACM TOPLAS 14(2), April 1992,
pp. 127-144.

Unfortunately, most programming languages give nondescript names such as DIV(IDE), QUOT(IENT), MOD(ULO), and REM(AINDER) to these operations. The language should make clear to programmers what division operations their programs are performing, especially when negative dividends and divisors can arise, but perhaps may not often be tested.

## Specification

For each of six division operator pairs — floor, ceiling, truncate,
round, Euclidean and balanced — there is a family of three procedures: one, named
`<operator>/`, to compute the division and to return both quotient and
remainder as multiple return values; one, named `<operator>-quotient`,
to compute the quotient; and one, named `<operator>-remainder`, to
compute the remainder. Each division operator pair is specified by
defining the quotient *q* in terms of the dividend *a* and the divisor *n*.
Tacitly the remainder *r* is as above: *r* = *n* - *dq*.

It is an error if any of the arguments are not integers (exact or inexact). It is also an error to supply zero as a divisor to any of these procedures. If any argument is inexact, the result is inexact, unless the implementation can prove that the inexactness cannot affect the result, as in the case of dividing an exact zero by an inexact number.

`(floor/ `*dividend*` `*divisor*`)`

`(floor-quotient `*dividend*` `*divisor*`)`

`(floor-remainder `*dividend*` `*divisor*`)`

q= floor(n/d)

Thus *r* is negative iff *d* is negative.

`(ceiling/ `*dividend*` `*divisor*`)`

`(ceiling-quotient `*dividend*` `*divisor*`)`

`(ceiling-remainder `*dividend*` `*divisor*`)`

q= ceiling(n/d)

Thus *r* is negative iff *d* is non-negative.

If *divisor* is the number of units in a block, and <dividend> is
some number of units, then `(ceiling-quotient `*dividend*` `*divisor*`)`
gives the number of blocks needed to cover *dividend* units. For
example, *divisor* might be the number of bytes in a disk sector, and
*dividend* the number of bytes in a file; then the quotient is the
number of disk sectors needed to store the contents of the file. For
another example, *divisor* might be the number of octets in the
output of a cryptographic hash function, and *dividend* the number of
octets desired in a key for a symmetric cipher, to be derived using
the cryptographic hash function; then the quotient is the number of
hash values needed to concatenate to make a key.

`(truncate/ `*dividend*` `*divisor*`)`

`(truncate-quotient `*dividend*` `*divisor*`)`

`(truncate-remainder `*dividend*` `*divisor*`)`

q= truncate(n/d)

Thus *r* is negative iff *n* is negative. However, by any divisor, the
quotient of +1, 0, or -1 is 0; that is, three contiguous dividends by
a common divisor share a common quotient. Of the other division
operator pairs, only the round pair exhibits this property.

`(round/ `*dividend*` `*divisor*`)`

`(round-quotient `*dividend*` `*divisor*`)`

`(round-remainder `*dividend*` `*divisor*`)`

q= round(n/d)

The round function rounds to the nearest integer, breaking ties by choosing the nearest even integer. Nothing general can be said about the sign of r. Like the truncate operator pair, the quotient of +1, 0, or -1 by any divisor is 0, so that three contiguous dividends by a common divisor share a common quotient.

`(euclidean/ `*dividend*` `*divisor*`)`

`(euclidean-quotient `*dividend*` `*divisor*`)`

`(euclidean-remainder `*dividend*` `*divisor*`)`

If

d> 0,q= floor(n/d); ifd< 0,q= ceiling(n/d).

This division operator pair satisfies the stronger property

0 <=

r< |d|,

used often in mathematics. Thus, for example,
`(euclidean-remainder `*dividend*` `*divisor*`)`
is always a valid index into a vector whose
length is at least the absolute value of *divisor*.
This division operator pair is so
named because it is the subject of the Euclidean division algorithm.

`(balanced/ `*dividend*` `*divisor*`)`

`(balanced-quotient `*dividend*` `*divisor*`)`

`(balanced-remainder `*dividend*` `*divisor*`)`

This division operator pair satisfies the property

-|d/2| <= r < |d/2|.

When *d* is a power of 2, say 2^{k} for some *k*, this reduces to

-2

^{(k - 1)}<=r< 2^{(k - 1)}.

Computer scientists will immediately recognize this as the interval of
integers representable in two's-complement with (*k* - 1) bits.

## Related Standards

### R5RS

The R5RS gives the names `quotient` and `remainder` to the truncating
division operator pair, and the name `modulo` to the remainder half of
the flooring division operator pair. For all these three procedures in
the R5RS, the dividend may be any integer, and the divisor may be any
nonzero integer.

### R6RS

The R6RS gives the names `div`, `mod`, and `div-and-mod` to the Euclidean division operator
family, and the names `div0`, `mod0`, and `div0-and-mod0` to the balanced operator family.
For all six of these procedures, the dividend may be
any real number, and the divisor may be any nonzero real number. The R5RS procedures
are also available in the R5RS compatibility library.

### R7RS

The truncate and floor families are part of R7RS-small. Three of them are also available under their R5RS names, for backward compatibility. The dividend may be any integer, and the divisor may be any nonzero integer.

### Common Lisp

Common Lisp provides four integral division functions, `floor`, `ceiling`,
`truncate`, and `round`; and two remainder functions, `mod` and `rem`. The
division functions comprise both the quotient and remainder of a
division operator pair, and return them as two values, of which the
latter, the remainder, may be implicitly ignored in Common Lisp. The
divisor argument is optional in Common Lisp's integral division
functions; if omitted, it is taken to be 1. `mod` is the remainder half
of the flooring division operator pair; `rem` is the remainder half of
the truncating division operator pair. Common Lisp does not provide
any part of the Euclidean division operator pair.

For all six of these functions in Common Lisp, the dividend may be any
real number, and the divisor may be any nonzero real number. Common
Lisp also provides four extra functions `ffloor`, `fceiling`, `ftruncate`,
and `fround`, which differ from their `f`-less variants only in
floating-point contagion rules.

## Copying

Copyright (c) 2009, 2010, Taylor R. Campbell.

Verbatim copying and distribution of this entire article are permitted worldwide, without royalty, in any medium, provided this notice, and the copyright notice, are preserved.

This is a draft. If you wish to derive a work from this article, contact the author. [This has been done. —John Cowan]

## Implementation

Code: http://git.savannah.gnu.org/cgit/mit-scheme.git/plain/src/runtime/division.scm Tests: http://git.savannah.gnu.org/cgit/mit-scheme.git/plain/tests/runtime/test-division.scm

The implementation accepts any real numbers as arguments. It returns inexact values if any argument is inexact, unless the dividend is an exact zero, in which case the quotient and remainder are exact zeros.