1/* origin: FreeBSD /usr/src/lib/msun/src/e_exp.c */ 2/* 3 * ==================================================== 4 * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved. 5 * 6 * Permission to use, copy, modify, and distribute this 7 * software is freely granted, provided that this notice 8 * is preserved. 9 * ==================================================== 10 */ 11/* exp(x) 12 * Returns the exponential of x. 13 * 14 * Method 15 * 1. Argument reduction: 16 * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. 17 * Given x, find r and integer k such that 18 * 19 * x = k*ln2 + r, |r| <= 0.5*ln2. 20 * 21 * Here r will be represented as r = hi-lo for better 22 * accuracy. 23 * 24 * 2. Approximation of exp(r) by a special rational function on 25 * the interval [0,0.34658]: 26 * Write 27 * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... 28 * We use a special Remez algorithm on [0,0.34658] to generate 29 * a polynomial of degree 5 to approximate R. The maximum error 30 * of this polynomial approximation is bounded by 2**-59. In 31 * other words, 32 * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 33 * (where z=r*r, and the values of P1 to P5 are listed below) 34 * and 35 * | 5 | -59 36 * | 2.0+P1*z+...+P5*z - R(z) | <= 2 37 * | | 38 * The computation of exp(r) thus becomes 39 * 2*r 40 * exp(r) = 1 + ---------- 41 * R(r) - r 42 * r*c(r) 43 * = 1 + r + ----------- (for better accuracy) 44 * 2 - c(r) 45 * where 46 * 2 4 10 47 * c(r) = r - (P1*r + P2*r + ... + P5*r ). 48 * 49 * 3. Scale back to obtain exp(x): 50 * From step 1, we have 51 * exp(x) = 2^k * exp(r) 52 * 53 * Special cases: 54 * exp(INF) is INF, exp(NaN) is NaN; 55 * exp(-INF) is 0, and 56 * for finite argument, only exp(0)=1 is exact. 57 * 58 * Accuracy: 59 * according to an error analysis, the error is always less than 60 * 1 ulp (unit in the last place). 61 * 62 * Misc. info. 63 * For IEEE double 64 * if x > 709.782712893383973096 then exp(x) overflows 65 * if x < -745.133219101941108420 then exp(x) underflows 66 */ 67 68#include "libm.h" 69 70static const double 71half[2] = {0.5,-0.5}, 72ln2hi = 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ 73ln2lo = 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ 74invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */ 75P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ 76P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ 77P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ 78P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ 79P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ 80 81double exp(double x) 82{ 83 double_t hi, lo, c, xx, y; 84 int k, sign; 85 uint32_t hx; 86 87 GET_HIGH_WORD(hx, x); 88 sign = hx>>31; 89 hx &= 0x7fffffff; /* high word of |x| */ 90 91 /* special cases */ 92 if (hx >= 0x4086232b) { /* if |x| >= 708.39... */ 93 if (isnan(x)) 94 return x; 95 if (x > 709.782712893383973096) { 96 /* overflow if x!=inf */ 97 x *= 0x1p1023; 98 return x; 99 } 100 if (x < -708.39641853226410622) { 101 /* underflow if x!=-inf */ 102 FORCE_EVAL((float)(-0x1p-149/x)); 103 if (x < -745.13321910194110842) 104 return 0; 105 } 106 } 107 108 /* argument reduction */ 109 if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ 110 if (hx >= 0x3ff0a2b2) /* if |x| >= 1.5 ln2 */ 111 k = (int)(invln2*x + half[sign]); 112 else 113 k = 1 - sign - sign; 114 hi = x - k*ln2hi; /* k*ln2hi is exact here */ 115 lo = k*ln2lo; 116 x = hi - lo; 117 } else if (hx > 0x3e300000) { /* if |x| > 2**-28 */ 118 k = 0; 119 hi = x; 120 lo = 0; 121 } else { 122 /* inexact if x!=0 */ 123 FORCE_EVAL(0x1p1023 + x); 124 return 1 + x; 125 } 126 127 /* x is now in primary range */ 128 xx = x*x; 129 c = x - xx*(P1+xx*(P2+xx*(P3+xx*(P4+xx*P5)))); 130 y = 1 + (x*c/(2-c) - lo + hi); 131 if (k == 0) 132 return y; 133 return scalbn(y, k); 134} 135