sysdeps/ieee754/dbl-64/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 /* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25,
  13    for performance improvement on pipelined processors.
  14  */
  15
  16 /* double log1p(double x)
  17  *
  18  * Method :
  19  *   1. Argument Reduction: find k and f such that
  20  *                      1+x = 2^k * (1+f),
  21  *         where  sqrt(2)/2 < 1+f < sqrt(2) .
  22  *
  23  *      Note. If k=0, then f=x is exact. However, if k!=0, then f
  24  *      may not be representable exactly. In that case, a correction
  25  *      term is need. Let u=1+x rounded. Let c = (1+x)-u, then
  26  *      log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
  27  *      and add back the correction term c/u.
  28  *      (Note: when x > 2**53, one can simply return log(x))
  29  *
  30  *   2. Approximation of log1p(f).
  31  *      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
  32  *               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
  33  *               = 2s + s*R
  34  *      We use a special Reme algorithm on [0,0.1716] to generate
  35  *      a polynomial of degree 14 to approximate R The maximum error
  36  *      of this polynomial approximation is bounded by 2**-58.45. In
  37  *      other words,
  38  *                      2      4      6      8      10      12      14
  39  *          R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s  +Lp6*s  +Lp7*s
  40  *      (the values of Lp1 to Lp7 are listed in the program)
  41  *      and
  42  *          |      2          14          |     -58.45
  43  *          | Lp1*s +...+Lp7*s    -  R(z) | <= 2
  44  *          |                             |
  45  *      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
  46  *      In order to guarantee error in log below 1ulp, we compute log
  47  *      by
  48  *              log1p(f) = f - (hfsq - s*(hfsq+R)).
  49  *
  50  *      3. Finally, log1p(x) = k*ln2 + log1p(f).
  51  *                           = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
  52  *         Here ln2 is split into two floating point number:
  53  *                      ln2_hi + ln2_lo,
  54  *         where n*ln2_hi is always exact for |n| < 2000.
  55  *
  56  * Special cases:
  57  *      log1p(x) is NaN with signal if x < -1 (including -INF) ;
  58  *      log1p(+INF) is +INF; log1p(-1) is -INF with signal;
  59  *      log1p(NaN) is that NaN with no signal.
  60  *
  61  * Accuracy:
  62  *      according to an error analysis, the error is always less than
  63  *      1 ulp (unit in the last place).
  64  *
  65  * Constants:
  66  * The hexadecimal values are the intended ones for the following
  67  * constants. The decimal values may be used, provided that the
  68  * compiler will convert from decimal to binary accurately enough
  69  * to produce the hexadecimal values shown.
  70  *
  71  * Note: Assuming log() return accurate answer, the following
  72  *       algorithm can be used to compute log1p(x) to within a few ULP:
  73  *
  74  *              u = 1+x;
  75  *              if(u==1.0) return x ; else
  76  *                         return log(u)*(x/(u-1.0));
  77  *
  78  *       See HP-15C Advanced Functions Handbook, p.193.
  79  */
  80
  81 #include <math.h>
  82 #include <math_private.h>
  83
  84 static const double
  85   ln2_hi = 6.93147180369123816490e-01,  /* 3fe62e42 fee00000 */
  86   ln2_lo = 1.90821492927058770002e-10,  /* 3dea39ef 35793c76 */
  87   two54 = 1.80143985094819840000e+16,   /* 43500000 00000000 */
  88   Lp[] = { 0.0, 6.666666666666735130e-01, /* 3FE55555 55555593 */
  89            3.999999999940941908e-01, /* 3FD99999 9997FA04 */
  90            2.857142874366239149e-01, /* 3FD24924 94229359 */
  91            2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
  92            1.818357216161805012e-01, /* 3FC74664 96CB03DE */
  93            1.531383769920937332e-01, /* 3FC39A09 D078C69F */
  94            1.479819860511658591e-01 }; /* 3FC2F112 DF3E5244 */
  95
  96 static const double zero = 0.0;
  97
  98 double
  99 __log1p (double x)
 100 {
 101   double hfsq, f, c, s, z, R, u, z2, z4, z6, R1, R2, R3, R4;
 102   int32_t k, hx, hu, ax;
 103
 104   GET_HIGH_WORD (hx, x);
 105   ax = hx & 0x7fffffff;
 106
 107   k = 1;
 108   if (hx < 0x3FDA827A)                          /* x < 0.41422  */
 109     {
 110       if (__builtin_expect (ax >= 0x3ff00000, 0))     /* x <= -1.0 */
 111         {
 112           if (x == -1.0)
 113             return -two54 / (x - x);            /* log1p(-1)=+inf */
 114           else
 115             return (x - x) / (x - x);           /* log1p(x<-1)=NaN */
 116         }
 117       if (__builtin_expect (ax < 0x3e200000, 0))     /* |x| < 2**-29 */
 118         {
 119           math_force_eval (two54 + x);          /* raise inexact */
 120           if (ax < 0x3c900000)                  /* |x| < 2**-54 */
 121             return x;
 122           else
 123             return x - x * x * 0.5;
 124         }
 125       if (hx > 0 || hx <= ((int32_t) 0xbfd2bec3))
 126         {
 127           k = 0; f = x; hu = 1;
 128         }                       /* -0.2929<x<0.41422 */
 129     }
 130   else if (__builtin_expect (hx >= 0x7ff00000, 0))
 131     return x + x;
 132   if (k != 0)
 133     {
 134       if (hx < 0x43400000)
 135         {
 136           u = 1.0 + x;
 137           GET_HIGH_WORD (hu, u);
 138           k = (hu >> 20) - 1023;
 139           c = (k > 0) ? 1.0 - (u - x) : x - (u - 1.0); /* correction term */
 140           c /= u;
 141         }
 142       else
 143         {
 144           u = x;
 145           GET_HIGH_WORD (hu, u);
 146           k = (hu >> 20) - 1023;
 147           c = 0;
 148         }
 149       hu &= 0x000fffff;
 150       if (hu < 0x6a09e)
 151         {
 152           SET_HIGH_WORD (u, hu | 0x3ff00000);   /* normalize u */
 153         }
 154       else
 155         {
 156           k += 1;
 157           SET_HIGH_WORD (u, hu | 0x3fe00000);   /* normalize u/2 */
 158           hu = (0x00100000 - hu) >> 2;
 159         }
 160       f = u - 1.0;
 161     }
 162   hfsq = 0.5 * f * f;
 163   if (hu == 0)          /* |f| < 2**-20 */
 164     {
 165       if (f == zero)
 166         {
 167           if (k == 0)
 168             return zero;
 169           else
 170             {
 171               c += k * ln2_lo; return k * ln2_hi + c;
 172             }
 173         }
 174       R = hfsq * (1.0 - 0.66666666666666666 * f);
 175       if (k == 0)
 176         return f - R;
 177       else
 178         return k * ln2_hi - ((R - (k * ln2_lo + c)) - f);
 179     }
 180   s = f / (2.0 + f);
 181   z = s * s;
 182   R1 = z * Lp[1]; z2 = z * z;
 183   R2 = Lp[2] + z * Lp[3]; z4 = z2 * z2;
 184   R3 = Lp[4] + z * Lp[5]; z6 = z4 * z2;
 185   R4 = Lp[6] + z * Lp[7];
 186   R = R1 + z2 * R2 + z4 * R3 + z6 * R4;
 187   if (k == 0)
 188     return f - (hfsq - s * (hfsq + R));
 189   else
 190     return k * ln2_hi - ((hfsq - (s * (hfsq + R) + (k * ln2_lo + c))) - f);
 191 }
 192 weak_alias (__log1p, log1p)
 193 #ifdef NO_LONG_DOUBLE
 194 strong_alias (__log1p, __log1pl)
 195 weak_alias (__log1p, log1pl)
 196 #endif