模除 (Chinese Wikipedia)

Boute, Raymond T. The Euclidean definition of the functions div and mod. ACM Transactions on Programming Languages and Systems (ACM Press (New York, NY, USA)). April 1992, 14 (2): 127–144. doi:10.1145/128861.128862 （英语）.
Let us recall Euclid’s theorem.

THEOREM. For any real numbers $D$ and $d$ with $d\neq 0$ , there exists a unique pair of numbers $q,r$ satisfying the following conditions:

(a) $q\in \mathbb {Z}$ ;

(b) $D=d\cdot q+r$ ;

(c) $0\leq r<|d|$ .
For the particular case of integers, the conditions (a)-(c) correspond to part of the axioms for Euclidean rings [9].
With the notational conventions of the theorem, we define $D\operatorname {div} d=q$ and $D{\bmod {d}}=r$ . Clearly the correspondingly labeled conditions are satisfied.
Properties

${\begin{aligned}D\operatorname {div} \,(-d)&=-(D\operatorname {div} d)\\D{\bmod {\left(-d\right)}}&=D{\bmod {d}}\\(-D)\operatorname {div} d&=-(D\operatorname {div} d)-I\cdot J\\(-D){\bmod {d}}&=-(D{\bmod {d}})+d\cdot I\cdot J\end{aligned}}$
where $J=\operatorname {sign} d$ and $I$ is as defined earlier. ( $I={\text{if }}D/d\in \mathbb {Z} {\text{ then }}0{\text{ else }}1$ )

apple.com

developer.apple.com

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archive.org

ANSI. Programming Languages — Full BASIC. New York: American National Standards Institute. 28 January 1987. § 5.4.4. X modulo Y, i.e., X-Y*INT(X/Y).
ANSI. Programming Languages — Full BASIC. New York: American National Standards Institute. 28 January 1987. § 5.4.4. The remainder function, i.e., X-Y*IP(X/Y).

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Jones, Derek M. The New C Standard: An Economic and Cultural Commentary (PDF). Addison-Wesley. 2003 [2018-07-11]. ISBN 9780201709179. （原始内容 (PDF)存档于2018-07-11）（英语）.
C99 reflects almost universal processor behavior (as does the Fortran Standard). This definition truncates toward zero and the expression (-(a/b) == (-a)/b) && (-(a/b) == a/(-b)) is always true. It also means that the absolute value of the result does not depend on the signs of the operands; for example:

+10 / +3 == +3 +10 % +3 == +1 -10 / +3 == -3 -10 % +3 == -1
+10 / -3 == -3 +10 % -3 == +1 -10 / -3 == +3 -10 % -3 == -1
C90

When integers are divided and the division is inexact, if both operands are positive the result of the / operator is the largest integer less than the algebraic quotient and the result of the % operator is positive. If either operand is negative, whether the result of the / operator is the largest integer less than or equal to the algebraic quotient or the smallest integer greater than or equal to the algebraic quotient is implementation-defined, as is the sign of the result of the % operator.
If either operand is negative, the behavior may differ between C90 and C99, depending on the implementation-defined behavior of the C90 implementation.
```
#include <stdio.h>

int main(void)
{
int x = -1,
y = +3;
if ((x%y > 0) ||
    ((x+y)%y == x%y))
    printf("This is a C90 translator behaving differently than C99\n");
}
```
Quoting from the C9X Revision Proposal, WG14/N613, that proposed this change:

The origin of this practice seems to have been a desire to map C’s division directly to the “natural” behavior of the target instruction set, whatever it may be, without requiring extra code overhead that might be necessary to check for special cases and enforce a particular behavior. However, the argument that Fortran programmers are unpleasantly surprised by this aspect of C and that there would be negligible impact on code efficiency was accepted by WG14, who agreed to require Fortran-like behavior in C99.

coffeescript.org

CoffeeScript operators

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api.dart.dev

operator % method - num class - dart:core library - Dart API. api.dart.dev. [2021-06-01].
remainder method - num class - dart:core library - Dart API. api.dart.dev. [2021-06-01].

dlang.org

Expressions - D Programming Language. dlang.org. [2021-06-01].

doi.org

Boute, Raymond T. The Euclidean definition of the functions div and mod. ACM Transactions on Programming Languages and Systems (ACM Press (New York, NY, USA)). April 1992, 14 (2): 127–144. doi:10.1145/128861.128862 （英语）.
Let us recall Euclid’s theorem.

THEOREM. For any real numbers $D$ and $d$ with $d\neq 0$ , there exists a unique pair of numbers $q,r$ satisfying the following conditions:

(a) $q\in \mathbb {Z}$ ;

(b) $D=d\cdot q+r$ ;

(c) $0\leq r<|d|$ .
For the particular case of integers, the conditions (a)-(c) correspond to part of the axioms for Euclidean rings [9].
With the notational conventions of the theorem, we define $D\operatorname {div} d=q$ and $D{\bmod {d}}=r$ . Clearly the correspondingly labeled conditions are satisfied.
Properties

${\begin{aligned}D\operatorname {div} \,(-d)&=-(D\operatorname {div} d)\\D{\bmod {\left(-d\right)}}&=D{\bmod {d}}\\(-D)\operatorname {div} d&=-(D\operatorname {div} d)-I\cdot J\\(-D){\bmod {d}}&=-(D{\bmod {d}})+d\cdot I\cdot J\end{aligned}}$
where $J=\operatorname {sign} d$ and $I$ is as defined earlier. ( $I={\text{if }}D/d\in \mathbb {Z} {\text{ then }}0{\text{ else }}1$ )

elm-lang.org

package.elm-lang.org

Basics - core 1.0.5. package.elm-lang.org. [2022-03-16].
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The Go Programming Language Specification - The Go Programming Language. go.dev. [2022-02-28].

hansotten.com

pascal.hansotten.com

ISO/IEC 7185, Information Technology—Programming Languages–Pascal. International Standard, 2nd ed. (PDF). ISO/IEC 7185, 1990 (E).
A term of the form i div j shall be an error if j is zero; otherwise, the value of i div j shall be such that

abs(i) - abs(j) < abs((i div j) * j) <= abs(i)
where the value shall be zero if abs(i) < abs(j); otherwise, the sign of the value shall be positive if i and j have the same sign and negative if i and j have different signs.
A term of the form i mod j shall be an error if j is zero or negative; otherwise, the value of i mod j shall be that value of (i - (k * j)) for integral k such that 0 <= i mod j < j.
NOTE 2 — Only for i >= 0 and j > 0 does the relation (i div j) * j + i mod j = i hold.
Kathleen Jensen, Niklaus Wirth. PASCAL User Manual and Report － ISO Pascal Standard (PDF). 1991.

div   divide and truncate (i.e., value is not rounded)

mod   modulus: let Remainder = A - (A div B) * B;
  if Remainder < 0 then A mod B = Remainder + B

  otherwise A mod B = Remainder

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ISO/IEC JTC 1/SC 22/WG 4. ISO/IEC 1989:2023 – Programming language COBOL. ISO. January 2023.
ISO/IEC JTC 1/SC 22. ISO/IEC 23271:2012 — Information technology — Common Language Infrastructure (CLI). ISO. February 2012. §§ III.3.55–56.
ISO 1995

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Mathematics · The Julia Language. docs.julialang.org. [2021-11-20].
Mathematics · The Julia Language. docs.julialang.org. [2021-11-20].

khronos.org

GLSL Language Specification, Version 4.50.7 (PDF). section 5.9 Expressions. If both operands are non-negative, then the remainder is non-negative. Results are undefined if one or both operands are negative.
GLSL Language Specification, Version 4.50.7 (PDF). section 8.3 Common Functions.

kotlinlang.org

rem - Kotlin Programming Language. Kotlin. [2021-05-05] （英语）.
mod - Kotlin Programming Language. Kotlin. [2021-05-05] （英语）.

microsoft.com

docs.microsoft.com

Operators. Microsoft. [2021-07-19]. The % operator is defined only in cases where either both sides are positive or both sides are negative. Unlike C, it also operates on floating-point data types, as well as integers.
QuantumWriter. Expressions. docs.microsoft.com. [2018-07-11] （美国英语）.

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Chapter 3: The NASM Language. NASM - The Netwide Assembler version 2.15.05.

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Shell Command Language. pubs.opengroup.org. [2021-02-05].

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perldoc.perl.org

Perl documentation

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pursuit.purescript.org

EuclideanRing.

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F32 - Rust.

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Horvath, Adam. Faster division and modulo operation - the power of two. July 5, 2012. （原始内容存档于2018-03-05）（英语）.

utexas.edu

cs.utexas.edu

ISO/IEC 7185, Information Technology—Programming Languages–Pascal. International Standard, 2nd ed. (PDF). ISO/IEC 7185, 1990 (E).
A term of the form i div j shall be an error if j is zero; otherwise, the value of i div j shall be such that

abs(i) - abs(j) < abs((i div j) * j) <= abs(i)
where the value shall be zero if abs(i) < abs(j); otherwise, the sign of the value shall be positive if i and j have the same sign and negative if i and j have different signs.
A term of the form i mod j shall be an error if j is zero or negative; otherwise, the value of i mod j shall be that value of (i - (k * j)) for integral k such that 0 <= i mod j < j.
NOTE 2 — Only for i >= 0 and j > 0 does the relation (i div j) * j + i mod j = i hold.
Kathleen Jensen, Niklaus Wirth. PASCAL User Manual and Report － ISO Pascal Standard (PDF). 1991.

div   divide and truncate (i.e., value is not rounded)

mod   modulus: let Remainder = A - (A div B) * B;
  if Remainder < 0 then A mod B = Remainder + B

  otherwise A mod B = Remainder

w3.org

Rossberg, Andreas (编). WebAssembly Core Specification: Version 2.0. World Wide Web Consortium. § 4.3.2 Integer Operations. 19 April 2022.

web.archive.org

Jones, Derek M. The New C Standard: An Economic and Cultural Commentary (PDF). Addison-Wesley. 2003 [2018-07-11]. ISBN 9780201709179. （原始内容 (PDF)存档于2018-07-11）（英语）.
C99 reflects almost universal processor behavior (as does the Fortran Standard). This definition truncates toward zero and the expression (-(a/b) == (-a)/b) && (-(a/b) == a/(-b)) is always true. It also means that the absolute value of the result does not depend on the signs of the operands; for example:

+10 / +3 == +3 +10 % +3 == +1 -10 / +3 == -3 -10 % +3 == -1
+10 / -3 == -3 +10 % -3 == +1 -10 / -3 == +3 -10 % -3 == -1
C90

When integers are divided and the division is inexact, if both operands are positive the result of the / operator is the largest integer less than the algebraic quotient and the result of the % operator is positive. If either operand is negative, whether the result of the / operator is the largest integer less than or equal to the algebraic quotient or the smallest integer greater than or equal to the algebraic quotient is implementation-defined, as is the sign of the result of the % operator.
If either operand is negative, the behavior may differ between C90 and C99, depending on the implementation-defined behavior of the C90 implementation.
```
#include <stdio.h>

int main(void)
{
int x = -1,
y = +3;
if ((x%y > 0) ||
    ((x+y)%y == x%y))
    printf("This is a C90 translator behaving differently than C99\n");
}
```
Quoting from the C9X Revision Proposal, WG14/N613, that proposed this change:

The origin of this practice seems to have been a desire to map C’s division directly to the “natural” behavior of the target instruction set, whatever it may be, without requiring extra code overhead that might be necessary to check for special cases and enforce a particular behavior. However, the argument that Fortran programmers are unpleasantly surprised by this aspect of C and that there would be negligible impact on code efficiency was accepted by WG14, who agreed to require Fortran-like behavior in C99.
Horvath, Adam. Faster division and modulo operation - the power of two. July 5, 2012. （原始内容存档于2018-03-05）（英语）.

xs4all.nl

jmvdveer.home.xs4all.nl

A. van Wijngaarden, B. J. Mailloux, J. E. L. Peck, C. H. A. Koster, M. Sintzoff, C. H. Lindsey, L. G. L.T. Meertens and R. G. Fisker. Revised Report on the Algorithmic Language Algol 68. IFIP W.G. 2.1.
10.2.3.3. Operations on integral operands
……

m) OP «÷, %, OVER» = (L INT a, b) L INT:
      IF b ≠ L 0
      THEN L INT q := L 0, r := ABS a;
         WHILE (r := r - ABS b) ≥ L 0 DO q := q + L 1 OD;
         (a < L 0 AND b > L 0 OR a ≥ L 0 AND b < L 0 | -q | q)
      FI;
n) OP «÷×, ÷*, %*, MOD» = (L INT a, b) L INT:
      (L INT r = a - a ÷ b × b; r < 0 | r + ABS b | r);

ziglang.org

Zig Documentation. Zig Programming Language. [2022-12-18].

`div`	divide and truncate (i.e., value is not rounded)
`mod`	modulus: let `Remainder = A - (A div B) * B`; if `Remainder < 0` then `A mod B = Remainder + B` otherwise `A mod B = Remainder`