mirror of
https://gitlab.com/rnger/amath
synced 2026-10-21 04:59:34 +00:00
200 lines
6.9 KiB
C
200 lines
6.9 KiB
C
/* @(#)e_exp.c 1.6 04/04/22 */
|
|
|
|
/*
|
|
* Copyright (c) 2015 Carsten Larsen
|
|
* All rights reserved.
|
|
*
|
|
* Redistribution and use in source and binary forms, with or without
|
|
* modification, are permitted provided that the following conditions
|
|
* are met:
|
|
* 1. Redistributions of source code must retain the above copyright
|
|
* notice, this list of conditions and the following disclaimer.
|
|
* 2. Redistributions in binary form must reproduce the above copyright
|
|
* notice, this list of conditions and the following disclaimer in the
|
|
* documentation and/or other materials provided with the distribution.
|
|
*
|
|
* THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
|
|
* IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
|
|
* OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
|
|
* IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
|
|
* INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
|
|
* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
|
|
* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
|
|
* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
|
|
* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
|
|
* THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
|
|
*
|
|
* The origin source code can be obtained from:
|
|
* http://www.netlib.org/fdlibm/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.
|
|
* ====================================================
|
|
*/
|
|
|
|
#include "prim.h"
|
|
#include "math.h"
|
|
|
|
static const double
|
|
one = 1.0,
|
|
halF[2] = {0.5,-0.5,},
|
|
huge = 1.0e+300,
|
|
twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/
|
|
o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
|
|
u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
|
|
ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
|
|
-6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
|
|
ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
|
|
-1.90821492927058770002e-10,},/* 0xbdea39ef, 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 */
|
|
|
|
/**
|
|
* @brief Returns the exponential of x.
|
|
* @version 1.6
|
|
* @date 04/04/22
|
|
* @details
|
|
* <pre>
|
|
* 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 Remes 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*R1(r)
|
|
* = 1 + r + ----------- (for better accuracy)
|
|
* 2 - R1(r)
|
|
* where
|
|
* 2 4 10
|
|
* R1(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 > 7.09782712893383973096e+02 then exp(x) overflow
|
|
* if x < -7.45133219101941108420e+02 then exp(x) underflow
|
|
*
|
|
* Constants:
|
|
* The hexadecimal values are the intended ones for the following
|
|
* constants. The decimal values may be used, provided that the
|
|
* compiler will convert from decimal to binary accurately enough
|
|
* to produce the hexadecimal values shown.
|
|
* </pre>
|
|
* @copyright Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
|
* @license Developed at SunSoft, a Sun Microsystems, Inc. business. Permission
|
|
* to use, copy, modify, and distribute this software is freely granted,
|
|
* provided that this notice is preserved.
|
|
*/
|
|
|
|
double exp(double x) /* default IEEE double exp */
|
|
{
|
|
double y,hi,lo,c,t;
|
|
sword k,xsb;
|
|
uword hx;
|
|
|
|
lo = 0.0;
|
|
hi = 0.0;
|
|
|
|
GET_HIGH_WORD(hx,x); /* high word of x */
|
|
xsb = (hx>>31)&1; /* sign bit of x */
|
|
hx &= 0x7fffffff; /* high word of |x| */
|
|
|
|
/* filter out non-finite argument */
|
|
if(hx >= 0x40862E42) { /* if |x|>=709.78... */
|
|
if(hx>=0x7ff00000) {
|
|
uword lx;
|
|
GET_LOW_WORD(lx,x);
|
|
if(((hx&0xfffff)|lx)!=0)
|
|
return x+x; /* NaN */
|
|
else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */
|
|
}
|
|
if(x > o_threshold) return huge*huge; /* overflow */
|
|
if(x < u_threshold) return twom1000*twom1000; /* underflow */
|
|
}
|
|
|
|
/* argument reduction */
|
|
if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
|
|
if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
|
|
hi = x-ln2HI[xsb];
|
|
lo=ln2LO[xsb];
|
|
k = 1-xsb-xsb;
|
|
} else {
|
|
k = (sword)(invln2*x+halF[xsb]);
|
|
t = k;
|
|
hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
|
|
lo = t*ln2LO[0];
|
|
}
|
|
x = hi - lo;
|
|
}
|
|
else if(hx < 0x3e300000) { /* when |x|<2**-28 */
|
|
if(huge+x>one) {
|
|
return one+x; /* trigger inexact */
|
|
} else {
|
|
k = 0;
|
|
}
|
|
}
|
|
else k = 0;
|
|
|
|
/* x is now in primary range */
|
|
t = x*x;
|
|
c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
|
|
if(k==0) return one-((x*c)/(c-2.0)-x);
|
|
else y = one-((lo-(x*c)/(2.0-c))-hi);
|
|
if(k >= -1021) {
|
|
uword hy;
|
|
GET_HIGH_WORD(hy, y);
|
|
SET_HIGH_WORD(y, hy + (k<<20)); /* add k to y's exponent */
|
|
return y;
|
|
} else {
|
|
uword hy;
|
|
GET_HIGH_WORD(hy, y);
|
|
SET_HIGH_WORD(y, hy + ((k+1000)<<20));/* add k to y's exponent */
|
|
return y*twom1000;
|
|
}
|
|
}
|