1 // Copyright 2010 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 // The original C code, the long comment, and the constants 8 // below are from FreeBSD's /usr/src/lib/msun/src/s_expm1.c 9 // and came with this notice. The go code is a simplified 10 // version of the original C. 11 // 12 // ==================================================== 13 // Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 14 // 15 // Developed at SunPro, a Sun Microsystems, Inc. business. 16 // Permission to use, copy, modify, and distribute this 17 // software is freely granted, provided that this notice 18 // is preserved. 19 // ==================================================== 20 // 21 // expm1(x) 22 // Returns exp(x)-1, the exponential of x minus 1. 23 // 24 // Method 25 // 1. Argument reduction: 26 // Given x, find r and integer k such that 27 // 28 // x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658 29 // 30 // Here a correction term c will be computed to compensate 31 // the error in r when rounded to a floating-point number. 32 // 33 // 2. Approximating expm1(r) by a special rational function on 34 // the interval [0,0.34658]: 35 // Since 36 // r*(exp(r)+1)/(exp(r)-1) = 2+ r**2/6 - r**4/360 + ... 37 // we define R1(r*r) by 38 // r*(exp(r)+1)/(exp(r)-1) = 2+ r**2/6 * R1(r*r) 39 // That is, 40 // R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r) 41 // = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r)) 42 // = 1 - r**2/60 + r**4/2520 - r**6/100800 + ... 43 // We use a special Reme algorithm on [0,0.347] to generate 44 // a polynomial of degree 5 in r*r to approximate R1. The 45 // maximum error of this polynomial approximation is bounded 46 // by 2**-61. In other words, 47 // R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5 48 // where Q1 = -1.6666666666666567384E-2, 49 // Q2 = 3.9682539681370365873E-4, 50 // Q3 = -9.9206344733435987357E-6, 51 // Q4 = 2.5051361420808517002E-7, 52 // Q5 = -6.2843505682382617102E-9; 53 // (where z=r*r, and the values of Q1 to Q5 are listed below) 54 // with error bounded by 55 // | 5 | -61 56 // | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 57 // | | 58 // 59 // expm1(r) = exp(r)-1 is then computed by the following 60 // specific way which minimize the accumulation rounding error: 61 // 2 3 62 // r r [ 3 - (R1 + R1*r/2) ] 63 // expm1(r) = r + --- + --- * [--------------------] 64 // 2 2 [ 6 - r*(3 - R1*r/2) ] 65 // 66 // To compensate the error in the argument reduction, we use 67 // expm1(r+c) = expm1(r) + c + expm1(r)*c 68 // ~ expm1(r) + c + r*c 69 // Thus c+r*c will be added in as the correction terms for 70 // expm1(r+c). Now rearrange the term to avoid optimization 71 // screw up: 72 // ( 2 2 ) 73 // ({ ( r [ R1 - (3 - R1*r/2) ] ) } r ) 74 // expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- ) 75 // ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 ) 76 // ( ) 77 // 78 // = r - E 79 // 3. Scale back to obtain expm1(x): 80 // From step 1, we have 81 // expm1(x) = either 2**k*[expm1(r)+1] - 1 82 // = or 2**k*[expm1(r) + (1-2**-k)] 83 // 4. Implementation notes: 84 // (A). To save one multiplication, we scale the coefficient Qi 85 // to Qi*2**i, and replace z by (x**2)/2. 86 // (B). To achieve maximum accuracy, we compute expm1(x) by 87 // (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf) 88 // (ii) if k=0, return r-E 89 // (iii) if k=-1, return 0.5*(r-E)-0.5 90 // (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E) 91 // else return 1.0+2.0*(r-E); 92 // (v) if (k<-2||k>56) return 2**k(1-(E-r)) - 1 (or exp(x)-1) 93 // (vi) if k <= 20, return 2**k((1-2**-k)-(E-r)), else 94 // (vii) return 2**k(1-((E+2**-k)-r)) 95 // 96 // Special cases: 97 // expm1(INF) is INF, expm1(NaN) is NaN; 98 // expm1(-INF) is -1, and 99 // for finite argument, only expm1(0)=0 is exact. 100 // 101 // Accuracy: 102 // according to an error analysis, the error is always less than 103 // 1 ulp (unit in the last place). 104 // 105 // Misc. info. 106 // For IEEE double 107 // if x > 7.09782712893383973096e+02 then expm1(x) overflow 108 // 109 // Constants: 110 // The hexadecimal values are the intended ones for the following 111 // constants. The decimal values may be used, provided that the 112 // compiler will convert from decimal to binary accurately enough 113 // to produce the hexadecimal values shown. 114 // 115 116 // Expm1 returns e**x - 1, the base-e exponential of x minus 1. 117 // It is more accurate than Exp(x) - 1 when x is near zero. 118 // 119 // Special cases are: 120 // Expm1(+Inf) = +Inf 121 // Expm1(-Inf) = -1 122 // Expm1(NaN) = NaN 123 // Very large values overflow to -1 or +Inf. 124 func Expm1(x float64) float64 125 126 func expm1(x float64) float64 { 127 const ( 128 Othreshold = 7.09782712893383973096e+02 // 0x40862E42FEFA39EF 129 Ln2X56 = 3.88162421113569373274e+01 // 0x4043687a9f1af2b1 130 Ln2HalfX3 = 1.03972077083991796413e+00 // 0x3ff0a2b23f3bab73 131 Ln2Half = 3.46573590279972654709e-01 // 0x3fd62e42fefa39ef 132 Ln2Hi = 6.93147180369123816490e-01 // 0x3fe62e42fee00000 133 Ln2Lo = 1.90821492927058770002e-10 // 0x3dea39ef35793c76 134 InvLn2 = 1.44269504088896338700e+00 // 0x3ff71547652b82fe 135 Tiny = 1.0 / (1 << 54) // 2**-54 = 0x3c90000000000000 136 // scaled coefficients related to expm1 137 Q1 = -3.33333333333331316428e-02 // 0xBFA11111111110F4 138 Q2 = 1.58730158725481460165e-03 // 0x3F5A01A019FE5585 139 Q3 = -7.93650757867487942473e-05 // 0xBF14CE199EAADBB7 140 Q4 = 4.00821782732936239552e-06 // 0x3ED0CFCA86E65239 141 Q5 = -2.01099218183624371326e-07 // 0xBE8AFDB76E09C32D 142 ) 143 144 // special cases 145 switch { 146 case IsInf(x, 1) || IsNaN(x): 147 return x 148 case IsInf(x, -1): 149 return -1 150 } 151 152 absx := x 153 sign := false 154 if x < 0 { 155 absx = -absx 156 sign = true 157 } 158 159 // filter out huge argument 160 if absx >= Ln2X56 { // if |x| >= 56 * ln2 161 if sign { 162 return -1 // x < -56*ln2, return -1 163 } 164 if absx >= Othreshold { // if |x| >= 709.78... 165 return Inf(1) 166 } 167 } 168 169 // argument reduction 170 var c float64 171 var k int 172 if absx > Ln2Half { // if |x| > 0.5 * ln2 173 var hi, lo float64 174 if absx < Ln2HalfX3 { // and |x| < 1.5 * ln2 175 if !sign { 176 hi = x - Ln2Hi 177 lo = Ln2Lo 178 k = 1 179 } else { 180 hi = x + Ln2Hi 181 lo = -Ln2Lo 182 k = -1 183 } 184 } else { 185 if !sign { 186 k = int(InvLn2*x + 0.5) 187 } else { 188 k = int(InvLn2*x - 0.5) 189 } 190 t := float64(k) 191 hi = x - t*Ln2Hi // t * Ln2Hi is exact here 192 lo = t * Ln2Lo 193 } 194 x = hi - lo 195 c = (hi - x) - lo 196 } else if absx < Tiny { // when |x| < 2**-54, return x 197 return x 198 } else { 199 k = 0 200 } 201 202 // x is now in primary range 203 hfx := 0.5 * x 204 hxs := x * hfx 205 r1 := 1 + hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5)))) 206 t := 3 - r1*hfx 207 e := hxs * ((r1 - t) / (6.0 - x*t)) 208 if k == 0 { 209 return x - (x*e - hxs) // c is 0 210 } 211 e = (x*(e-c) - c) 212 e -= hxs 213 switch { 214 case k == -1: 215 return 0.5*(x-e) - 0.5 216 case k == 1: 217 if x < -0.25 { 218 return -2 * (e - (x + 0.5)) 219 } 220 return 1 + 2*(x-e) 221 case k <= -2 || k > 56: // suffice to return exp(x)-1 222 y := 1 - (e - x) 223 y = Float64frombits(Float64bits(y) + uint64(k)<<52) // add k to y's exponent 224 return y - 1 225 } 226 if k < 20 { 227 t := Float64frombits(0x3ff0000000000000 - (0x20000000000000 >> uint(k))) // t=1-2**-k 228 y := t - (e - x) 229 y = Float64frombits(Float64bits(y) + uint64(k)<<52) // add k to y's exponent 230 return y 231 } 232 t = Float64frombits(uint64(0x3ff-k) << 52) // 2**-k 233 y := x - (e + t) 234 y++ 235 y = Float64frombits(Float64bits(y) + uint64(k)<<52) // add k to y's exponent 236 return y 237 } 238
