mirror of https://gitlab.com/rnger/amath
198 lines
6.9 KiB
C
198 lines
6.9 KiB
C
/*-
|
|
* Copyright (c) 2014-2017 Carsten Sonne Larsen <cs@innolan.net>
|
|
* 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.
|
|
*
|
|
* Project homepage:
|
|
* https://amath.innolan.net
|
|
*
|
|
* The original source code can be obtained from:
|
|
* http://www.netlib.org/fdlibm/s_log1p.c
|
|
*
|
|
* =================================================================
|
|
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
|
*
|
|
* 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.
|
|
* =================================================================
|
|
*/
|
|
|
|
#include "prim.h"
|
|
|
|
/* double log1p(double x)
|
|
*
|
|
* Method :
|
|
* 1. Argument Reduction: find k and f such that
|
|
* 1+x = 2^k * (1+f),
|
|
* where sqrt(2)/2 < 1+f < sqrt(2) .
|
|
*
|
|
* Note. If k=0, then f=x is exact. However, if k!=0, then f
|
|
* may not be representable exactly. In that case, a correction
|
|
* term is need. Let u=1+x rounded. Let c = (1+x)-u, then
|
|
* log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
|
|
* and add back the correction term c/u.
|
|
* (Note: when x > 2**53, one can simply return log(x))
|
|
*
|
|
* 2. Approximation of log1p(f).
|
|
* Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
|
|
* = 2s + 2/3 s**3 + 2/5 s**5 + .....,
|
|
* = 2s + s*R
|
|
* We use a special Remes algorithm on [0,0.1716] to generate
|
|
* a polynomial of degree 14 to approximate R The maximum error
|
|
* of this polynomial approximation is bounded by 2**-58.45. In
|
|
* other words,
|
|
* 2 4 6 8 10 12 14
|
|
* R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s
|
|
* (the values of Lp1 to Lp7 are listed in the program)
|
|
* and
|
|
* | 2 14 | -58.45
|
|
* | Lp1*s +...+Lp7*s - R(z) | <= 2
|
|
* | |
|
|
* Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
|
|
* In order to guarantee error in log below 1ulp, we compute log
|
|
* by
|
|
* log1p(f) = f - (hfsq - s*(hfsq+R)).
|
|
*
|
|
* 3. Finally, log1p(x) = k*ln2 + log1p(f).
|
|
* = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
|
|
* Here ln2 is split into two floating point number:
|
|
* ln2_hi + ln2_lo,
|
|
* where n*ln2_hi is always exact for |n| < 2000.
|
|
*
|
|
* Special cases:
|
|
* log1p(x) is NaN with signal if x < -1 (including -INF) ;
|
|
* log1p(+INF) is +INF; log1p(-1) is -INF with signal;
|
|
* log1p(NaN) is that NaN with no signal.
|
|
*
|
|
* Accuracy:
|
|
* according to an error analysis, the error is always less than
|
|
* 1 ulp (unit in the last place).
|
|
*
|
|
* 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.
|
|
*
|
|
* Note: Assuming log() return accurate answer, the following
|
|
* algorithm can be used to compute log1p(x) to within a few ULP:
|
|
*
|
|
* u = 1+x;
|
|
* if(u==1.0) return x ; else
|
|
* return log(u)*(x/(u-1.0));
|
|
*
|
|
* See HP-15C Advanced Functions Handbook, p.193.
|
|
*/
|
|
|
|
static const double
|
|
ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
|
|
ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
|
|
two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
|
|
Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
|
|
Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
|
|
Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
|
|
Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
|
|
Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
|
|
Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
|
|
Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
|
|
|
|
static double zero = 0.0;
|
|
|
|
double log1p(double x)
|
|
{
|
|
double hfsq,f,c,s,z,R,u;
|
|
int32_t k,hx,hu,ax;
|
|
|
|
f = 0.0;
|
|
c = 0.0;
|
|
hu = 0;
|
|
|
|
GET_HIGH_WORD(hx,x); /* high word of x */
|
|
ax = hx&0x7fffffff;
|
|
|
|
k = 1;
|
|
if (hx < 0x3FDA827A) { /* x < 0.41422 */
|
|
if(ax>=0x3ff00000) { /* x <= -1.0 */
|
|
if(x==-1.0) return -two54/zero; /* log1p(-1)=+inf */
|
|
else return (x-x)/(x-x); /* log1p(x<-1)=NaN */
|
|
}
|
|
if(ax<0x3e200000) { /* |x| < 2**-29 */
|
|
if(two54+x>zero /* raise inexact */
|
|
&&ax<0x3c900000) /* |x| < 2**-54 */
|
|
return x;
|
|
else
|
|
return x - x*x*0.5;
|
|
}
|
|
if(hx>0||hx<=((int)0xbfd2bec3)) {
|
|
k=0;
|
|
f=x;
|
|
hu=1;
|
|
} /* -0.2929<x<0.41422 */
|
|
}
|
|
if (hx >= 0x7ff00000) return x+x;
|
|
if(k!=0) {
|
|
if(hx<0x43400000) {
|
|
u = 1.0+x;
|
|
GET_HIGH_WORD(hu,u); /* high word of u */
|
|
k = (hu>>20)-1023;
|
|
c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */
|
|
c /= u;
|
|
} else {
|
|
u = x;
|
|
GET_HIGH_WORD(hu,u); /* high word of u */
|
|
k = (hu>>20)-1023;
|
|
c = 0;
|
|
}
|
|
hu &= 0x000fffff;
|
|
if(hu<0x6a09e) {
|
|
SET_HIGH_WORD(u, hu|0x3ff00000); /* normalize u */
|
|
} else {
|
|
k += 1;
|
|
SET_HIGH_WORD(u, hu|0x3fe00000); /* normalize u/2 */
|
|
hu = (0x00100000-hu)>>2;
|
|
}
|
|
f = u-1.0;
|
|
}
|
|
hfsq=0.5*f*f;
|
|
if(hu==0) { /* |f| < 2**-20 */
|
|
if(f==zero) {
|
|
if(k==0) return zero;
|
|
else {
|
|
c += k*ln2_lo;
|
|
return k*ln2_hi+c;
|
|
}
|
|
}
|
|
R = hfsq*(1.0-0.66666666666666666*f);
|
|
if(k==0) return f-R;
|
|
else
|
|
return k*ln2_hi-((R-(k*ln2_lo+c))-f);
|
|
}
|
|
s = f/(2.0+f);
|
|
z = s*s;
|
|
R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
|
|
if(k==0) return f-(hfsq-s*(hfsq+R));
|
|
else
|
|
return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
|
|
}
|