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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[2] = {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);
}
|