1 // Copyright 2009 The Go Authors. All rights reserved. 2 // Use of this source code is governed by a BSD-style 3 // license that can be found in the LICENSE file. 4 5 package math 6 7 func isOddInt(x float64) bool { 8 xi, xf := Modf(x) 9 return xf == 0 && int64(xi)&1 == 1 10 } 11 12 // Special cases taken from FreeBSD's /usr/src/lib/msun/src/e_pow.c 13 // updated by IEEE Std. 754-2008 "Section 9.2.1 Special values". 14 15 // Pow returns x**y, the base-x exponential of y. 16 // 17 // Special cases are (in order): 18 // Pow(x, 0) = 1 for any x 19 // Pow(1, y) = 1 for any y 20 // Pow(x, 1) = x for any x 21 // Pow(NaN, y) = NaN 22 // Pow(x, NaN) = NaN 23 // Pow(0, y) = Inf for y an odd integer < 0 24 // Pow(0, -Inf) = +Inf 25 // Pow(0, +Inf) = +0 26 // Pow(0, y) = +Inf for finite y < 0 and not an odd integer 27 // Pow(0, y) = 0 for y an odd integer > 0 28 // Pow(0, y) = +0 for finite y > 0 and not an odd integer 29 // Pow(-1, Inf) = 1 30 // Pow(x, +Inf) = +Inf for |x| > 1 31 // Pow(x, -Inf) = +0 for |x| > 1 32 // Pow(x, +Inf) = +0 for |x| < 1 33 // Pow(x, -Inf) = +Inf for |x| < 1 34 // Pow(+Inf, y) = +Inf for y > 0 35 // Pow(+Inf, y) = +0 for y < 0 36 // Pow(-Inf, y) = Pow(-0, -y) 37 // Pow(x, y) = NaN for finite x < 0 and finite non-integer y 38 func Pow(x, y float64) float64 39 40 func pow(x, y float64) float64 { 41 switch { 42 case y == 0 || x == 1: 43 return 1 44 case y == 1: 45 return x 46 case IsNaN(x) || IsNaN(y): 47 return NaN() 48 case x == 0: 49 switch { 50 case y < 0: 51 if isOddInt(y) { 52 return Copysign(Inf(1), x) 53 } 54 return Inf(1) 55 case y > 0: 56 if isOddInt(y) { 57 return x 58 } 59 return 0 60 } 61 case IsInf(y, 0): 62 switch { 63 case x == -1: 64 return 1 65 case (Abs(x) < 1) == IsInf(y, 1): 66 return 0 67 default: 68 return Inf(1) 69 } 70 case IsInf(x, 0): 71 if IsInf(x, -1) { 72 return Pow(1/x, -y) // Pow(-0, -y) 73 } 74 switch { 75 case y < 0: 76 return 0 77 case y > 0: 78 return Inf(1) 79 } 80 case y == 0.5: 81 return Sqrt(x) 82 case y == -0.5: 83 return 1 / Sqrt(x) 84 } 85 86 absy := y 87 flip := false 88 if absy < 0 { 89 absy = -absy 90 flip = true 91 } 92 yi, yf := Modf(absy) 93 if yf != 0 && x < 0 { 94 return NaN() 95 } 96 if yi >= 1<<63 { 97 // yi is a large even int that will lead to overflow (or underflow to 0) 98 // for all x except -1 (x == 1 was handled earlier) 99 switch { 100 case x == -1: 101 return 1 102 case (Abs(x) < 1) == (y > 0): 103 return 0 104 default: 105 return Inf(1) 106 } 107 } 108 109 // ans = a1 * 2**ae (= 1 for now). 110 a1 := 1.0 111 ae := 0 112 113 // ans *= x**yf 114 if yf != 0 { 115 if yf > 0.5 { 116 yf-- 117 yi++ 118 } 119 a1 = Exp(yf * Log(x)) 120 } 121 122 // ans *= x**yi 123 // by multiplying in successive squarings 124 // of x according to bits of yi. 125 // accumulate powers of two into exp. 126 x1, xe := Frexp(x) 127 for i := int64(yi); i != 0; i >>= 1 { 128 if xe < -1<<12 || 1<<12 < xe { 129 // catch xe before it overflows the left shift below 130 // Since i !=0 it has at least one bit still set, so ae will accumulate xe 131 // on at least one more iteration, ae += xe is a lower bound on ae 132 // the lower bound on ae exceeds the size of a float64 exp 133 // so the final call to Ldexp will produce under/overflow (0/Inf) 134 ae += xe 135 break 136 } 137 if i&1 == 1 { 138 a1 *= x1 139 ae += xe 140 } 141 x1 *= x1 142 xe <<= 1 143 if x1 < .5 { 144 x1 += x1 145 xe-- 146 } 147 } 148 149 // ans = a1*2**ae 150 // if flip { ans = 1 / ans } 151 // but in the opposite order 152 if flip { 153 a1 = 1 / a1 154 ae = -ae 155 } 156 return Ldexp(a1, ae) 157 } 158
