compiler/mlib/s_log1p.c

   1 /* @(#)s_log1p.c 5.1 93/09/24 */
   2 /*
   3  * ====================================================
   4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   5  *
   6  * Developed at SunPro, a Sun Microsystems, Inc. business.
   7  * Permission to use, copy, modify, and distribute this
   8  * software is freely granted, provided that this notice
   9  * is preserved.
  10  * ====================================================
  11  */
  12
  13 #ifndef lint
  14 static char rcsid[] = "$FreeBSD: src/lib/msun/src/s_log1p.c,v 1.8 2005/12/04 12:28:33 bde Exp $";
  15 #endif
  16
  17 /* double log1p(double x)
  18  *
  19  * Method :
  20  *   1. Argument Reduction: find k and f such that
  21  *                      1+x = 2^k * (1+f),
  22  *         where  sqrt(2)/2 < 1+f < sqrt(2) .
  23  *
  24  *      Note. If k=0, then f=x is exact. However, if k!=0, then f
  25  *      may not be representable exactly. In that case, a correction
  26  *      term is need. Let u=1+x rounded. Let c = (1+x)-u, then
  27  *      log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
  28  *      and add back the correction term c/u.
  29  *      (Note: when x > 2**53, one can simply return log(x))
  30  *
  31  *   2. Approximation of log1p(f).
  32  *      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
  33  *               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
  34  *               = 2s + s*R
  35  *      We use a special Reme algorithm on [0,0.1716] to generate
  36  *      a polynomial of degree 14 to approximate R The maximum error
  37  *      of this polynomial approximation is bounded by 2**-58.45. In
  38  *      other words,
  39  *                      2      4      6      8      10      12      14
  40  *          R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s  +Lp6*s  +Lp7*s
  41  *      (the values of Lp1 to Lp7 are listed in the program)
  42  *      and
  43  *          |      2          14          |     -58.45
  44  *          | Lp1*s +...+Lp7*s    -  R(z) | <= 2
  45  *          |                             |
  46  *      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
  47  *      In order to guarantee error in log below 1ulp, we compute log
  48  *      by
  49  *              log1p(f) = f - (hfsq - s*(hfsq+R)).
  50  *
  51  *      3. Finally, log1p(x) = k*ln2 + log1p(f).
  52  *                           = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
  53  *         Here ln2 is split into two floating point number:
  54  *                      ln2_hi + ln2_lo,
  55  *         where n*ln2_hi is always exact for |n| < 2000.
  56  *
  57  * Special cases:
  58  *      log1p(x) is NaN with signal if x < -1 (including -INF) ;
  59  *      log1p(+INF) is +INF; log1p(-1) is -INF with signal;
  60  *      log1p(NaN) is that NaN with no signal.
  61  *
  62  * Accuracy:
  63  *      according to an error analysis, the error is always less than
  64  *      1 ulp (unit in the last place).
  65  *
  66  * Constants:
  67  * The hexadecimal values are the intended ones for the following
  68  * constants. The decimal values may be used, provided that the
  69  * compiler will convert from decimal to binary accurately enough
  70  * to produce the hexadecimal values shown.
  71  *
  72  * Note: Assuming log() return accurate answer, the following
  73  *       algorithm can be used to compute log1p(x) to within a few ULP:
  74  *
  75  *              u = 1+x;
  76  *              if(u==1.0) return x ; else
  77  *                         return log(u)*(x/(u-1.0));
  78  *
  79  *       See HP-15C Advanced Functions Handbook, p.193.
  80  */
  81
  82 #include "math.h"
  83 #include "math_private.h"
  84
  85 static const double
  86 ln2_hi  =  6.93147180369123816490e-01,  /* 3fe62e42 fee00000 */
  87 ln2_lo  =  1.90821492927058770002e-10,  /* 3dea39ef 35793c76 */
  88 two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
  89 Lp1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
  90 Lp2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
  91 Lp3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
  92 Lp4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
  93 Lp5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
  94 Lp6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
  95 Lp7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
  96
  97 static const double zero = 0.0;
  98
  99 double
 100 log1p(double x)
 101 {
 102         double hfsq,f,c,s,z,R,u;
 103         int32_t k,hx,hu,ax;
 104
 105         GET_HIGH_WORD(hx,x);
 106         ax = hx&0x7fffffff;
 107
 108         k = 1;
 109         if (hx < 0x3FDA827A) {                  /* 1+x < sqrt(2)+ */
 110             if(ax>=0x3ff00000) {                /* x <= -1.0 */
 111                 if(x==-1.0) return -two54/zero; /* log1p(-1)=+inf */
 112                 else return (x-x)/(x-x);        /* log1p(x<-1)=NaN */
 113             }
 114             if(ax<0x3e200000) {                 /* |x| < 2**-29 */
 115                 if(two54+x>zero                 /* raise inexact */
 116                     &&ax<0x3c900000)            /* |x| < 2**-54 */
 117                     return x;
 118                 else
 119                     return x - x*x*0.5;
 120             }
 121             if(hx>0||hx<=((int32_t)0xbfd2bec4)) {
 122                 k=0;f=x;hu=1;}          /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
 123         }
 124         if (hx >= 0x7ff00000) return x+x;
 125         if(k!=0) {
 126             if(hx<0x43400000) {
 127                 u  = 1.0+x;
 128                 GET_HIGH_WORD(hu,u);
 129                 k  = (hu>>20)-1023;
 130                 c  = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */
 131                 c /= u;
 132             } else {
 133                 u  = x;
 134                 GET_HIGH_WORD(hu,u);
 135                 k  = (hu>>20)-1023;
 136                 c  = 0;
 137             }
 138             hu &= 0x000fffff;
 139             /*
 140              * The approximation to sqrt(2) used in thresholds is not
 141              * critical.  However, the ones used above must give less
 142              * strict bounds than the one here so that the k==0 case is
 143              * never reached from here, since here we have committed to
 144              * using the correction term but don't use it if k==0.
 145              */
 146             if(hu<0x6a09e) {                    /* u ~< sqrt(2) */
 147                 SET_HIGH_WORD(u,hu|0x3ff00000); /* normalize u */
 148             } else {
 149                 k += 1;
 150                 SET_HIGH_WORD(u,hu|0x3fe00000); /* normalize u/2 */
 151                 hu = (0x00100000-hu)>>2;
 152             }
 153             f = u-1.0;
 154         }
 155         hfsq=0.5*f*f;
 156         if(hu==0) {     /* |f| < 2**-20 */
 157             if(f==zero) if(k==0) return zero;
 158                         else {c += k*ln2_lo; return k*ln2_hi+c;}
 159             R = hfsq*(1.0-0.66666666666666666*f);
 160             if(k==0) return f-R; else
 161                      return k*ln2_hi-((R-(k*ln2_lo+c))-f);
 162         }
 163         s = f/(2.0+f);
 164         z = s*s;
 165         R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
 166         if(k==0) return f-(hfsq-s*(hfsq+R)); else
 167                  return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
 168 }