25b040c287
updates #13745 Multiprecision squaring can be done in a straightforward manner with about half the multiplications of a basic multiplication due to the symmetry of the operands. This change implements basic squaring for nat types and uses it for Int multiplication when the same variable is supplied to both arguments of z.Mul(x, x). This has some overhead to allocate a temporary variable to hold the cross products, shift them to double and add them to the diagonal terms. There is a speed benefit in the intermediate range when the overhead is neglible and the asymptotic performance of karatsuba multiplication has not been reached. basicSqrThreshold = 20 karatsubaSqrThreshold = 400 Were set by running calibrate_test.go to measure timing differences between the algorithms. Benchmarks for squaring: name old time/op new time/op delta IntSqr/1-4 51.5ns ±25% 25.1ns ± 7% -51.38% (p=0.008 n=5+5) IntSqr/2-4 79.1ns ± 4% 72.4ns ± 2% -8.47% (p=0.008 n=5+5) IntSqr/3-4 102ns ± 4% 97ns ± 5% ~ (p=0.056 n=5+5) IntSqr/5-4 161ns ± 4% 163ns ± 7% ~ (p=0.952 n=5+5) IntSqr/8-4 277ns ± 5% 267ns ± 6% ~ (p=0.087 n=5+5) IntSqr/10-4 358ns ± 3% 360ns ± 4% ~ (p=0.730 n=5+5) IntSqr/20-4 1.07µs ± 3% 1.01µs ± 6% ~ (p=0.056 n=5+5) IntSqr/30-4 2.36µs ± 4% 1.72µs ± 2% -27.03% (p=0.008 n=5+5) IntSqr/50-4 5.19µs ± 3% 3.88µs ± 4% -25.37% (p=0.008 n=5+5) IntSqr/80-4 11.3µs ± 4% 8.6µs ± 3% -23.78% (p=0.008 n=5+5) IntSqr/100-4 16.2µs ± 4% 12.8µs ± 3% -21.49% (p=0.008 n=5+5) IntSqr/200-4 50.1µs ± 5% 44.7µs ± 3% -10.65% (p=0.008 n=5+5) IntSqr/300-4 105µs ±11% 95µs ± 3% -9.50% (p=0.008 n=5+5) IntSqr/500-4 231µs ± 5% 227µs ± 2% ~ (p=0.310 n=5+5) IntSqr/800-4 496µs ± 9% 459µs ± 3% -7.40% (p=0.016 n=5+5) IntSqr/1000-4 700µs ± 3% 710µs ± 5% ~ (p=0.841 n=5+5) Show a speed up of 10-25% in the range where basicSqr is optimal, improved single word squaring and no significant difference when the fallback to standard multiplication is used. Change-Id: Iae2c82ca91cf890823f91e5c83bbe9a2c534b72b Reviewed-on: https://go-review.googlesource.com/53638 Reviewed-by: Robert Griesemer <gri@golang.org> Run-TryBot: Robert Griesemer <gri@golang.org> TryBot-Result: Gobot Gobot <gobot@golang.org> |
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