Efficient Algorithms for Integer Division by Constants Using Multiplication

Abstract

We present a complete analysis of the integer division of a single unsigned dividend word by a single unsigned divisor word based on double-word multiplication of the dividend by an inverse of the divisor. The well-known advantage of this method yields run-time efficiency, if the inverse of the divisor can be calculated at compile time, since multiplication is much faster than division in arithmetic units. Our analysis leads to the discovery of a limit to the straightforward application of this method in the form of a critical dividend, which fortunately associates with a minority of the possible divisors (20%) and defines only a small upper part of the available dividend space. We present two algorithms for ascertaining whether a critical dividend exists and, if so, its value along with a circumvention of this limit. For completeness, we include an algorithm for integer division of a unsigned double-word dividend by an unsigned single-word divisor in which the quotient is not limited to a single word and the remainder is an intrinsic part of the result.

Euclidean Affine Functions and Applications to Calendar Algorithms

“…For α = 1, Theorems 3.2 and 3.4 set α ′ to 2 k • δ −1 and 2 k • δ −1 , respectively. The former is the choice taken by [1,5,10] and the latter by [16]. Finally, [22] and an appendix to [5], consider both and, in this particular case, rigourously justify the heuristics we have suggested for choosing between the two approaches.…”

“…This section covers optimisations for (α • r + β)/δ. The particular case α = 1 and β = 0 has been considered by many authors [1,5,10,16,22], who have derived strength reductions whereby r/δ is reduced to a multiplication and cheaper operations. Major compilers implement the algorithms of [10].…”

“…Divisions by constants appear in some of the most common tasks performed by software systems, including printing decimal numbers (division by powers of 10) and working with times (division by 24, 60 and 3600). Since division is the slowest of the four basic arithmetical operations, various authors [1,5,10,16,22] have proposed strength reduction optimisations (i.e., the replacement of instructions with alternatives that are mathematically equivalent but faster) for integer divisions when divisors are constants known by the compiler. The algorithms proposed by Granlund and Montgomery [10] have been implemented by major compilers.…”

Integer division by constants: optimal bounds

“…Their approach was later refined by Robison [7]. Similarly, Granlund and Montgomery's approach [5] (without an intermediate addition) was refined by Cavagnino and Werbrouck [13], and later by Warren [11]. As remarked by Robison [7], the two approaches (with and without an intermediate addition) are complementary: we can choose one or the other depending on the divisor.…”

Faster remainder by direct computation: Applications to compilers and software libraries

“…Given any nonzero 32‐bit divisor known at compile time, the optimizing compiler can (and usually will) replace the division by a multiplication followed by a shift. Following the work of Cavagnino and Werbrouck, Warren finds that Granlund and Montgomery choose a slightly suboptimal number of fractional bits for some divisors. Warren's slightly better approach is found in LLVM's Clang compiler (see Algorithm 1).…”