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 ```/* 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 */ ``` ``` x *= 0x1p1023; ``` ``` 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; ``` ``` 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); ``` ```} ``` ``` ```