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134 lines
4.0 KiB
134 lines
4.0 KiB
/* origin: FreeBSD /usr/src/lib/msun/src/e_exp.c */
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/*
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* ====================================================
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* Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
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*
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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/* exp(x)
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* Returns the exponential of x.
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*
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* Method
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* 1. Argument reduction:
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* Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
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* Given x, find r and integer k such that
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*
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* x = k*ln2 + r, |r| <= 0.5*ln2.
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*
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* Here r will be represented as r = hi-lo for better
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* accuracy.
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*
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* 2. Approximation of exp(r) by a special rational function on
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* the interval [0,0.34658]:
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* Write
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* R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
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* We use a special Remez algorithm on [0,0.34658] to generate
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* a polynomial of degree 5 to approximate R. The maximum error
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* of this polynomial approximation is bounded by 2**-59. In
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* other words,
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* R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
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* (where z=r*r, and the values of P1 to P5 are listed below)
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* and
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* | 5 | -59
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* | 2.0+P1*z+...+P5*z - R(z) | <= 2
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* | |
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* The computation of exp(r) thus becomes
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* 2*r
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* exp(r) = 1 + ----------
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* R(r) - r
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* r*c(r)
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* = 1 + r + ----------- (for better accuracy)
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* 2 - c(r)
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* where
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* 2 4 10
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* c(r) = r - (P1*r + P2*r + ... + P5*r ).
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*
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* 3. Scale back to obtain exp(x):
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* From step 1, we have
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* exp(x) = 2^k * exp(r)
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*
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* Special cases:
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* exp(INF) is INF, exp(NaN) is NaN;
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* exp(-INF) is 0, and
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* for finite argument, only exp(0)=1 is exact.
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*
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* Accuracy:
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* according to an error analysis, the error is always less than
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* 1 ulp (unit in the last place).
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*
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* Misc. info.
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* For IEEE double
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* if x > 709.782712893383973096 then exp(x) overflows
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* if x < -745.133219101941108420 then exp(x) underflows
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*/
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#include "libm.h"
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static const double
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half[2] = {0.5,-0.5},
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ln2hi = 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
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ln2lo = 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
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invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
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P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
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P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
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P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
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P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
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P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
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double exp(double x)
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{
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double_t hi, lo, c, xx, y;
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int k, sign;
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uint32_t hx;
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GET_HIGH_WORD(hx, x);
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sign = hx>>31;
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hx &= 0x7fffffff; /* high word of |x| */
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/* special cases */
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if (hx >= 0x4086232b) { /* if |x| >= 708.39... */
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if (isnan(x))
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return x;
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if (x > 709.782712893383973096) {
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/* overflow if x!=inf */
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x *= 0x1p1023;
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return x;
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}
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if (x < -708.39641853226410622) {
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/* underflow if x!=-inf */
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FORCE_EVAL((float)(-0x1p-149/x));
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if (x < -745.13321910194110842)
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return 0;
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}
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}
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/* argument reduction */
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if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
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if (hx >= 0x3ff0a2b2) /* if |x| >= 1.5 ln2 */
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k = (int)(invln2*x + half[sign]);
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else
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k = 1 - sign - sign;
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hi = x - k*ln2hi; /* k*ln2hi is exact here */
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lo = k*ln2lo;
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x = hi - lo;
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} else if (hx > 0x3e300000) { /* if |x| > 2**-28 */
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k = 0;
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hi = x;
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lo = 0;
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} else {
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/* inexact if x!=0 */
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FORCE_EVAL(0x1p1023 + x);
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return 1 + x;
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}
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/* x is now in primary range */
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xx = x*x;
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c = x - xx*(P1+xx*(P2+xx*(P3+xx*(P4+xx*P5))));
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y = 1 + (x*c/(2-c) - lo + hi);
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if (k == 0)
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return y;
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return scalbn(y, k);
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}
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