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