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88672de7af
As necessary, math functions were structured to use stubs, so that they can
be accelerated with assembly on any platform.
Technique used was minimax polynomial approximation using tables of
polynomial coefficients, with argument range reduction.
Benchmark New Old Speedup
BenchmarkAcos 12.2 47.5 3.89
BenchmarkAcosh 18.5 56.2 3.04
BenchmarkAsin 13.1 40.6 3.10
BenchmarkAsinh 19.4 62.8 3.24
BenchmarkAtan 10.1 23 2.28
BenchmarkAtanh 19.1 53.2 2.79
BenchmarkAtan2 16.5 33.9 2.05
BenchmarkCbrt 14.8 58 3.92
BenchmarkErf 10.8 20.1 1.86
BenchmarkErfc 11.2 23.5 2.10
BenchmarkExp 8.77 53.8 6.13
BenchmarkExpm1 10.1 38.3 3.79
BenchmarkLog 13.1 40.1 3.06
BenchmarkLog1p 12.7 38.3 3.02
BenchmarkPowInt 31.7 40.5 1.28
BenchmarkPowFrac 33.1 141 4.26
BenchmarkTan 11.5 30 2.61
Accuracy was tested against a high precision
reference function to determine maximum error.
Note: ulperr is error in "units in the last place"
max
ulperr
Acos 1.15
Acosh 1.07
Asin 2.22
Asinh 1.72
Atan 1.41
Atanh 3.00
Atan2 1.45
Cbrt 1.18
Erf 1.29
Erfc 4.82
Exp 1.00
Expm1 2.26
Log 0.94
Log1p 2.39
Tan 3.14
Pow will have 99.99% correctly rounded results with reasonable inputs
producing numeric (non Inf or NaN) results
Change-Id: I850e8cf7b70426e8b54ec49d74acd4cddc8c6cb2
Reviewed-on: https://go-review.googlesource.com/38585
Reviewed-by: Michael Munday <munday@ca.ibm.com>
Run-TryBot: Michael Munday <munday@ca.ibm.com>
TryBot-Result: Gobot Gobot <gobot@golang.org>
80 lines
2.2 KiB
Go
80 lines
2.2 KiB
Go
// Copyright 2009 The Go Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package math
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// The go code is a modified version of the original C code from
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// http://www.netlib.org/fdlibm/s_cbrt.c and came with this notice.
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//
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// ====================================================
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// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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//
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// Developed at SunSoft, a Sun Microsystems, Inc. business.
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// Permission to use, copy, modify, and distribute this
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// software is freely granted, provided that this notice
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// is preserved.
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// ====================================================
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// Cbrt returns the cube root of x.
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//
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// Special cases are:
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// Cbrt(±0) = ±0
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// Cbrt(±Inf) = ±Inf
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// Cbrt(NaN) = NaN
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func Cbrt(x float64) float64
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func cbrt(x float64) float64 {
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const (
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B1 = 715094163 // (682-0.03306235651)*2**20
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B2 = 696219795 // (664-0.03306235651)*2**20
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C = 5.42857142857142815906e-01 // 19/35 = 0x3FE15F15F15F15F1
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D = -7.05306122448979611050e-01 // -864/1225 = 0xBFE691DE2532C834
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E = 1.41428571428571436819e+00 // 99/70 = 0x3FF6A0EA0EA0EA0F
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F = 1.60714285714285720630e+00 // 45/28 = 0x3FF9B6DB6DB6DB6E
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G = 3.57142857142857150787e-01 // 5/14 = 0x3FD6DB6DB6DB6DB7
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SmallestNormal = 2.22507385850720138309e-308 // 2**-1022 = 0x0010000000000000
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)
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// special cases
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switch {
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case x == 0 || IsNaN(x) || IsInf(x, 0):
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return x
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}
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sign := false
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if x < 0 {
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x = -x
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sign = true
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}
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// rough cbrt to 5 bits
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t := Float64frombits(Float64bits(x)/3 + B1<<32)
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if x < SmallestNormal {
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// subnormal number
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t = float64(1 << 54) // set t= 2**54
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t *= x
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t = Float64frombits(Float64bits(t)/3 + B2<<32)
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}
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// new cbrt to 23 bits
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r := t * t / x
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s := C + r*t
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t *= G + F/(s+E+D/s)
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// chop to 22 bits, make larger than cbrt(x)
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t = Float64frombits(Float64bits(t)&(0xFFFFFFFFC<<28) + 1<<30)
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// one step newton iteration to 53 bits with error less than 0.667ulps
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s = t * t // t*t is exact
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r = x / s
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w := t + t
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r = (r - t) / (w + r) // r-s is exact
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t = t + t*r
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// restore the sign bit
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if sign {
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t = -t
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}
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return t
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}
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