libm/s_log1p.c

   1 /*
   2  * ====================================================
   3  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   4  *
   5  * Developed at SunPro, a Sun Microsystems, Inc. business.
   6  * Permission to use, copy, modify, and distribute this
   7  * software is freely granted, provided that this notice
   8  * is preserved.
   9  * ====================================================
  10  */
  11
  12 /* double log1p(double x)
  13  *
  14  * Method :
  15  *   1. Argument Reduction: find k and f such that
  16  *                      1+x = 2^k * (1+f),
  17  *         where  sqrt(2)/2 < 1+f < sqrt(2) .
  18  *
  19  *      Note. If k=0, then f=x is exact. However, if k!=0, then f
  20  *      may not be representable exactly. In that case, a correction
  21  *      term is need. Let u=1+x rounded. Let c = (1+x)-u, then
  22  *      log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
  23  *      and add back the correction term c/u.
  24  *      (Note: when x > 2**53, one can simply return log(x))
  25  *
  26  *   2. Approximation of log1p(f).
  27  *      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
  28  *               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
  29  *               = 2s + s*R
  30  *      We use a special Reme algorithm on [0,0.1716] to generate
  31  *      a polynomial of degree 14 to approximate R The maximum error
  32  *      of this polynomial approximation is bounded by 2**-58.45. In
  33  *      other words,
  34  *                      2      4      6      8      10      12      14
  35  *          R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s  +Lp6*s  +Lp7*s
  36  *      (the values of Lp1 to Lp7 are listed in the program)
  37  *      and
  38  *          |      2          14          |     -58.45
  39  *          | Lp1*s +...+Lp7*s    -  R(z) | <= 2
  40  *          |                             |
  41  *      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
  42  *      In order to guarantee error in log below 1ulp, we compute log
  43  *      by
  44  *              log1p(f) = f - (hfsq - s*(hfsq+R)).
  45  *
  46  *      3. Finally, log1p(x) = k*ln2 + log1p(f).
  47  *                           = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
  48  *         Here ln2 is split into two floating point number:
  49  *                      ln2_hi + ln2_lo,
  50  *         where n*ln2_hi is always exact for |n| < 2000.
  51  *
  52  * Special cases:
  53  *      log1p(x) is NaN with signal if x < -1 (including -INF) ;
  54  *      log1p(+INF) is +INF; log1p(-1) is -INF with signal;
  55  *      log1p(NaN) is that NaN with no signal.
  56  *
  57  * Accuracy:
  58  *      according to an error analysis, the error is always less than
  59  *      1 ulp (unit in the last place).
  60  *
  61  * Constants:
  62  * The hexadecimal values are the intended ones for the following
  63  * constants. The decimal values may be used, provided that the
  64  * compiler will convert from decimal to binary accurately enough
  65  * to produce the hexadecimal values shown.
  66  *
  67  * Note: Assuming log() return accurate answer, the following
  68  *       algorithm can be used to compute log1p(x) to within a few ULP:
  69  *
  70  *              u = 1+x;
  71  *              if(u==1.0) return x ; else
  72  *                         return log(u)*(x/(u-1.0));
  73  *
  74  *       See HP-15C Advanced Functions Handbook, p.193.
  75  */
  76
  77 #include "math.h"
  78 #include "math_private.h"
  79
  80 static const double
  81 ln2_hi  =  6.93147180369123816490e-01,  /* 3fe62e42 fee00000 */
  82 ln2_lo  =  1.90821492927058770002e-10,  /* 3dea39ef 35793c76 */
  83 two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
  84 Lp1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
  85 Lp2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
  86 Lp3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
  87 Lp4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
  88 Lp5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
  89 Lp6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
  90 Lp7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
  91
  92 static const double zero = 0.0;
  93
  94 double log1p(double x)
  95 {
  96         double hfsq,f=0,c=0,s,z,R,u;
  97         int32_t k,hx,hu=0,ax;
  98
  99         GET_HIGH_WORD(hx,x);
 100         ax = hx&0x7fffffff;
 101
 102         k = 1;
 103         if (hx < 0x3FDA827A) {                  /* x < 0.41422  */
 104             if(ax>=0x3ff00000) {                /* x <= -1.0 */
 105                 if(x==-1.0) return -two54/zero; /* log1p(-1)=+inf */
 106                 else return (x-x)/(x-x);        /* log1p(x<-1)=NaN */
 107             }
 108             if(ax<0x3e200000) {                 /* |x| < 2**-29 */
 109                 if(two54+x>zero                 /* raise inexact */
 110                     &&ax<0x3c900000)            /* |x| < 2**-54 */
 111                     return x;
 112                 else
 113                     return x - x*x*0.5;
 114             }
 115             if(hx>0||hx<=((int32_t)0xbfd2bec3)) {
 116                 k=0;f=x;hu=1;}  /* -0.2929<x<0.41422 */
 117         }
 118         if (hx >= 0x7ff00000) return x+x;
 119         if(k!=0) {
 120             if(hx<0x43400000) {
 121                 u  = 1.0+x;
 122                 GET_HIGH_WORD(hu,u);
 123                 k  = (hu>>20)-1023;
 124                 c  = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */
 125                 c /= u;
 126             } else {
 127                 u  = x;
 128                 GET_HIGH_WORD(hu,u);
 129                 k  = (hu>>20)-1023;
 130                 c  = 0;
 131             }
 132             hu &= 0x000fffff;
 133             if(hu<0x6a09e) {
 134                 SET_HIGH_WORD(u,hu|0x3ff00000); /* normalize u */
 135             } else {
 136                 k += 1;
 137                 SET_HIGH_WORD(u,hu|0x3fe00000); /* normalize u/2 */
 138                 hu = (0x00100000-hu)>>2;
 139             }
 140             f = u-1.0;
 141         }
 142         hfsq=0.5*f*f;
 143         if(hu==0) {     /* |f| < 2**-20 */
 144             if(f==zero) {if(k==0) return zero;
 145                         else {c += k*ln2_lo; return k*ln2_hi+c;}
 146             }
 147             R = hfsq*(1.0-0.66666666666666666*f);
 148             if(k==0) return f-R; else
 149                      return k*ln2_hi-((R-(k*ln2_lo+c))-f);
 150         }
 151         s = f/(2.0+f);
 152         z = s*s;
 153         R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
 154         if(k==0) return f-(hfsq-s*(hfsq+R)); else
 155                  return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
 156 }
 157 libm_hidden_def(log1p)