newlib-cygwin/newlib/libm/math/e_exp.c

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2000-02-18 03:39:52 +08:00
/* @(#)e_exp.c 5.1 93/09/24 */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* __ieee754_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 Reme 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.
*/
#include "fdlibm.h"
#ifndef _DOUBLE_IS_32BITS
#ifdef __STDC__
static const double
#else
static double
#endif
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 */
#ifdef __STDC__
double __ieee754_exp(double x) /* default IEEE double exp */
#else
double __ieee754_exp(x) /* default IEEE double exp */
double x;
#endif
{
double y,hi,lo,c,t;
Throughout, run newlib with -Wall -Werror option and fix bugs and compiler warnings found this way. * libc/stdio/freopen.c (_freopen_r): Fix bug setting _flags. * libc/include/stdio.h (_rename): Define when building newlib. * libc/include/sys/signal.h (_kill): Ditto. * libc/include/sys/stat.h (_mkdir): Ditto. * libc/include/sys/time.h (_gettimeofday): Ditto. * libc/include/sys/times.h (_times): Ditto. * libc/include/sys/wait.h (_wait): Ditto. * libc/locale/lmessages.c (empty): Don't define for Cygwin. * libc/locale/lmonetary.c (cnv): Ditto. * libc/locale/nl_langinfo.c (nl_langinfo): Ditto for variable s. * libc/posix/collate.c: Throughout cast to avoid compiler warning. * libc/posix/engine.c (matcher): Initialize dp to avoid compiler warning. * libc/posix/glob.c: Disable on Cygwin. Explain why. * libc/posix/regcomp.c: Fix "uninitialized" compiler warnings. (dissect): Deliberately silence gcc compiler warning. Add comment to explain why. * libc/posix/wordexp.c (wordexp): Remove num_bytes variable since result is never used. * libc/posix/popen.c (popen): Ditto for variable last. * libc/reent/mkdirr.c: Include sys/stat.h. * libc/reent/renamer.c: Include stdio.h. * libc/search/hash.c: Throughout use underscored variants of the stat function family. (init_hash): Add missing definition for the __USE_INTERNAL_STAT64 case. * libc/search/hash_bigkey.c (__big_insert): Add parenthesis to avoid compiler warning. * libc/search/hash_page.c (overflow_page): Initalize freep to NULL to avoid compiler warning. * libc/stdio/asiprintf.c (_asiprintf_r): Cast unsigned char * to char * to avoid compiler warning. (asiprintf): Ditto. * libc/stdio/asprintf.c (_asprintf_r): Ditto. (asprintf): Ditto. * libc/stdio/vasiprintf.c (_vasiprintf_r): Ditto. * libc/stdio/vasprintf.c (_vasprintf_r): Ditto. * libc/stdio/mktemp.c (_gettemp): Cast to unsigned char in call to isdigit to avoid compiler warning. * libc/stdio/vfprintf.c (_VFPRINTF_R): Initialize variables used for grouping to avoid compiler warning. Only define and set nseps and nrepeats if they are really used. * libc/stdio/vfwprintf.c (_VFWPRINTF_R): Ditto. Only define state if it is really used. * libc/stdio/vfscanf.c (u_char): Revert to be defined as unsigned char. (__SVFSCANF_R): Cast fmt in call to __mbtowc. * libc/stdlib/mbtowc_r.c (JIS_state_table): Disable when building Cygwin. (JIS_action_table): Ditto. * libc/stdlib/wctomb_r.c (__utf8_wctomb): Add parenthesis to avoid compiler warning. * libc/string/strcasestr.c: Deliberately silence gcc compiler warning. Add comment to explain why. * libc/time/strptime.c (strptime): Cast to unsigned char in calls to isspace to avoid compiler warning. * libm/math/e_atan2.c (__ieee754_atan2): Add parenthesis to avoid compiler warning. * libm/math/e_exp.c (__ieee754_exp): Initialize k to 0 to avoid compiler warning. Drop setting it to 0 later. * libm/math/ef_exp.c (__ieee754_expf): Ditto. * libm/math/e_pow.c (__ieee754_pow): Add braces to avoid compiler warning. * libm/math/ef_pow.c (__ieee754_powf): Ditto. * libm/math/er_lgamma.c (__ieee754_lgamma_r): Initialize nadj to 0 to avoid compiler warning. * libm/math/erf_lgamma.c (__ieee754_lgammaf_r): Ditto. * libm/math/e_rem_pio2.c (__ieee754_rem_pio2): Ditto for variable z. * libm/common/sf_round.c (roundf): Remove signbit variable since result is never used.
2012-08-08 19:04:18 +08:00
__int32_t k = 0,xsb;
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__uint32_t hx;
GET_HIGH_WORD(hx,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) {
__uint32_t 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 = 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 */
}
/* 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) {
__uint32_t hy;
GET_HIGH_WORD(hy,y);
SET_HIGH_WORD(y,hy+(k<<20)); /* add k to y's exponent */
return y;
} else {
__uint32_t hy;
GET_HIGH_WORD(hy,y);
SET_HIGH_WORD(y,hy+((k+1000)<<20)); /* add k to y's exponent */
return y*twom1000;
}
}
#endif /* defined(_DOUBLE_IS_32BITS) */