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...achine (e.g., in cross-compilation). CASH’s implementation uses native arithmetic to carry the constant folding. Strength reduction The code for constant multiplication, constant division and modulu=-=s [GM94]-=- has been adapted from gcc. Induction variable Multiplications of induction variables with constants are turned into repeated strength reduction additions. Algebraic optimizations False predicates See...

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...ments of the group of quadratic forms of discriminant −D and the equivalent forms given by GAMMA2 and the corresponding values of γ2: Q ˜ Q γ2(τ ˜Q ) [1, 1, 6] [1, 3, 8] −151.73142930462826 [2, 1, 3] =-=[2, 9, 26]-=- −1.6342853476858265 − 12.303828997932955 i [2, −1, 3] [2, 3, 4] −1.6342853476858265 + 12.303828997932955 i From this, the minimal polynomial of g2 is: Factoring we find that H23[g2](X) =X 3 + 155 X 2...

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...ication operations are simply integer arithmetic operations mod q. The division instruction is always particularly time-consuming, so we replace it with multiplication using the following proposition =-=[24]-=-: Proposition 1. If M satis es 2n+ℓ ≤ Md ≤ 2n+ℓ + 2ℓ for 2ℓ−1 < d < 2ℓ, then ⌊ ⌋ X d = ⌊ 2−ℓ ⌊ XM 2n ⌋⌋ ⌊ = 2−ℓ (⌊ X(M−2n ) 2n ⌋ )⌋ + X for 0 ≤ X < 2n. Note that the second equality, since often M > 2...

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...mit directssupport for integer division, and instead rely onssoftware implementation.sA case of particular interestsis where a divisor is a compile-time constant [1] [7], orsa run-time loop-invariant =-=[6]-=-.sPrevious work hassshown that in such situations, an unsigned integersdivision x/d can be profitably computed ass(ax+b)/2s, where integer a approximates the scaledsreciprocal 2s/d, integer b comp...

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...it complements locality improving array transformations [7]. Other previous work on eliminating division and modulo operations have focused on the case when the denominator is a compile-time constant =-=[1, 12, 14]-=-. In [12], a division with constant denominator is turned into a load-constant, a multiplication, one or two shifts, and two add/subtracts; a modulo with constant denominator is converted into those i...

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...sion 4.3 of the GMP library. When applied to the problem of dividing a large integer by a single word, the new algorithm gives a speedup of roughly 30%, benchmarked on AMD and Intel processors in the x86 64 family. I. INTRODUCTION Integer division instructions are either not present at all in current microprocessors, or if they are present, they are considerably slower than the corresponding multiplication instructions. Multiplication instructions in turn are at least a few times slower than addition instructions, both in terms of throughput and latency. The situation was similar a decade ago [1], and the trend has continued so that division latency is now typically 5-15 times higher than multiplication latency, and division throughput is up to 50 times worse than multiplication throughput. Another trend is that branches cost gradually more, except for branches that the hardware can predict correctly. But some branches are inherently unpredictable. Division can be implemented using multiplication, by first computing an approximate reciprocal, e.g., by Newton iteration, followed by a multiplication that results in a candidate quotient. Finally, the remainder corresponding to this candi...