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 // Exp returns e**x, the base-e exponential of x. 8 // 9 // Special cases are: 10 // Exp(+Inf) = +Inf 11 // Exp(NaN) = NaN 12 // Very large values overflow to 0 or +Inf. 13 // Very small values underflow to 1. 14 func Exp(x float64) float64 15 16 // The original C code, the long comment, and the constants 17 // below are from FreeBSD's /usr/src/lib/msun/src/e_exp.c 18 // and came with this notice. The go code is a simplified 19 // version of the original C. 20 // 21 // ==================================================== 22 // Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved. 23 // 24 // Permission to use, copy, modify, and distribute this 25 // software is freely granted, provided that this notice 26 // is preserved. 27 // ==================================================== 28 // 29 // 30 // exp(x) 31 // Returns the exponential of x. 32 // 33 // Method 34 // 1. Argument reduction: 35 // Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. 36 // Given x, find r and integer k such that 37 // 38 // x = k*ln2 + r, |r| <= 0.5*ln2. 39 // 40 // Here r will be represented as r = hi-lo for better 41 // accuracy. 42 // 43 // 2. Approximation of exp(r) by a special rational function on 44 // the interval [0,0.34658]: 45 // Write 46 // R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... 47 // We use a special Remez algorithm on [0,0.34658] to generate 48 // a polynomial of degree 5 to approximate R. The maximum error 49 // of this polynomial approximation is bounded by 2**-59. In 50 // other words, 51 // R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 52 // (where z=r*r, and the values of P1 to P5 are listed below) 53 // and 54 // | 5 | -59 55 // | 2.0+P1*z+...+P5*z - R(z) | <= 2 56 // | | 57 // The computation of exp(r) thus becomes 58 // 2*r 59 // exp(r) = 1 + ------- 60 // R - r 61 // r*R1(r) 62 // = 1 + r + ----------- (for better accuracy) 63 // 2 - R1(r) 64 // where 65 // 2 4 10 66 // R1(r) = r - (P1*r + P2*r + ... + P5*r ). 67 // 68 // 3. Scale back to obtain exp(x): 69 // From step 1, we have 70 // exp(x) = 2**k * exp(r) 71 // 72 // Special cases: 73 // exp(INF) is INF, exp(NaN) is NaN; 74 // exp(-INF) is 0, and 75 // for finite argument, only exp(0)=1 is exact. 76 // 77 // Accuracy: 78 // according to an error analysis, the error is always less than 79 // 1 ulp (unit in the last place). 80 // 81 // Misc. info. 82 // For IEEE double 83 // if x > 7.09782712893383973096e+02 then exp(x) overflow 84 // if x < -7.45133219101941108420e+02 then exp(x) underflow 85 // 86 // Constants: 87 // The hexadecimal values are the intended ones for the following 88 // constants. The decimal values may be used, provided that the 89 // compiler will convert from decimal to binary accurately enough 90 // to produce the hexadecimal values shown. 91 92 func exp(x float64) float64 { 93 const ( 94 Ln2Hi = 6.93147180369123816490e-01 95 Ln2Lo = 1.90821492927058770002e-10 96 Log2e = 1.44269504088896338700e+00 97 98 Overflow = 7.09782712893383973096e+02 99 Underflow = -7.45133219101941108420e+02 100 NearZero = 1.0 / (1 << 28) // 2**-28 101 ) 102 103 // special cases 104 switch { 105 case IsNaN(x) || IsInf(x, 1): 106 return x 107 case IsInf(x, -1): 108 return 0 109 case x > Overflow: 110 return Inf(1) 111 case x < Underflow: 112 return 0 113 case -NearZero < x && x < NearZero: 114 return 1 + x 115 } 116 117 // reduce; computed as r = hi - lo for extra precision. 118 var k int 119 switch { 120 case x < 0: 121 k = int(Log2e*x - 0.5) 122 case x > 0: 123 k = int(Log2e*x + 0.5) 124 } 125 hi := x - float64(k)*Ln2Hi 126 lo := float64(k) * Ln2Lo 127 128 // compute 129 return expmulti(hi, lo, k) 130 } 131 132 // Exp2 returns 2**x, the base-2 exponential of x. 133 // 134 // Special cases are the same as Exp. 135 func Exp2(x float64) float64 136 137 func exp2(x float64) float64 { 138 const ( 139 Ln2Hi = 6.93147180369123816490e-01 140 Ln2Lo = 1.90821492927058770002e-10 141 142 Overflow = 1.0239999999999999e+03 143 Underflow = -1.0740e+03 144 ) 145 146 // special cases 147 switch { 148 case IsNaN(x) || IsInf(x, 1): 149 return x 150 case IsInf(x, -1): 151 return 0 152 case x > Overflow: 153 return Inf(1) 154 case x < Underflow: 155 return 0 156 } 157 158 // argument reduction; x = rlg(e) + k with |r| ln(2)/2. 159 // computed as r = hi - lo for extra precision. 160 var k int 161 switch { 162 case x > 0: 163 k = int(x + 0.5) 164 case x < 0: 165 k = int(x - 0.5) 166 } 167 t := x - float64(k) 168 hi := t * Ln2Hi 169 lo := -t * Ln2Lo 170 171 // compute 172 return expmulti(hi, lo, k) 173 } 174 175 // exp1 returns e**r 2**k where r = hi - lo and |r| ln(2)/2. 176 func expmulti(hi, lo float64, k int) float64 { 177 const ( 178 P1 = 1.66666666666666657415e-01 /* 0x3FC55555; 0x55555555 */ 179 P2 = -2.77777777770155933842e-03 /* 0xBF66C16C; 0x16BEBD93 */ 180 P3 = 6.61375632143793436117e-05 /* 0x3F11566A; 0xAF25DE2C */ 181 P4 = -1.65339022054652515390e-06 /* 0xBEBBBD41; 0xC5D26BF1 */ 182 P5 = 4.13813679705723846039e-08 /* 0x3E663769; 0x72BEA4D0 */ 183 ) 184 185 r := hi - lo 186 t := r * r 187 c := r - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))) 188 y := 1 - ((lo - (r*c)/(2-c)) - hi) 189 // TODO(rsc): make sure Ldexp can handle boundary k 190 return Ldexp(y, k) 191 } 192
