Quelle s_log1p.cpp Sprache: C

/* @(#)s_log1p.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.
* ====================================================
*/

//#include <sys/cdefs.h>
//__FBSDID("$FreeBSD$");

/* 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 Reme 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.
*/

#include <float.h>

#include "math_private.h"

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 const double zero = 0.0;
static volatile double vzero = 0.0;

double
log1p(double x)
{
double hfsq,f,c,s,z,R,u;
int32_t k,hx,hu,ax;

GET_HIGH_WORD(hx,x);
ax = hx&0x7fffffff;

k = 1;
if (hx < 0x3FDA827A) {   /* 1+x < sqrt(2)+ */
     if(ax>=0x3ff00000) {  /* x <= -1.0 */
  if(x==-1.0) return -two54/vzero; /* 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<=((int32_t)0xbfd2bec4)) {
  k=0;f=x;hu=1;}  /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
}
if (hx >= 0x7ff00000) return x+x;
if(k!=0) {
     if(hx<0x43400000) {
  STRICT_ASSIGN(double,u,1.0+x);
  GET_HIGH_WORD(hu,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);
         k  = (hu>>20)-1023;
  c  = 0;
     }
     hu &= 0x000fffff;
     /*
     * The approximation to sqrt(2) used in thresholds is not
     * critical.  However, the ones used above must give less
     * strict bounds than the one here so that the k==0 case is
     * never reached from here, since here we have committed to
     * using the correction term but don't use it if k==0.
     */
     if(hu<0x6a09e) {   /* u ~< sqrt(2) */
         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);
}

quality100%

¤ Dauer der Verarbeitung: 0.35 Sekunden (vorverarbeitet) ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.