usr/src/lib/libm/common/C/log1p.c

   1 /*
   2  * CDDL HEADER START
   3  *
   4  * The contents of this file are subject to the terms of the
   5  * Common Development and Distribution License (the "License").
   6  * You may not use this file except in compliance with the License.
   7  *
   8  * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
   9  * or http://www.opensolaris.org/os/licensing.
  10  * See the License for the specific language governing permissions
  11  * and limitations under the License.
  12  *
  13  * When distributing Covered Code, include this CDDL HEADER in each
  14  * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
  15  * If applicable, add the following below this CDDL HEADER, with the
  16  * fields enclosed by brackets "[]" replaced with your own identifying
  17  * information: Portions Copyright [yyyy] [name of copyright owner]
  18  *
  19  * CDDL HEADER END
  20  */
  21 /*
  22  * Copyright 2011 Nexenta Systems, Inc.  All rights reserved.
  23  */
  24 /*
  25  * Copyright 2005 Sun Microsystems, Inc.  All rights reserved.
  26  * Use is subject to license terms.
  27  */
  28
  29 #pragma weak __log1p = log1p
  30
  31 /* INDENT OFF */
  32 /*
  33  * Method :
  34  *   1. Argument Reduction: find k and f such that
  35  *                      1+x = 2^k * (1+f),
  36  *         where  sqrt(2)/2 < 1+f < sqrt(2) .
  37  *
  38  *      Note. If k=0, then f=x is exact. However, if k != 0, then f
  39  *      may not be representable exactly. In that case, a correction
  40  *      term is need. Let u=1+x rounded. Let c = (1+x)-u, then
  41  *      log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
  42  *      and add back the correction term c/u.
  43  *      (Note: when x > 2**53, one can simply return log(x))
  44  *
  45  *   2. Approximation of log1p(f).
  46  *      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
  47  *               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
  48  *               = 2s + s*R
  49  *      We use a special Reme algorithm on [0,0.1716] to generate
  50  *      a polynomial of degree 14 to approximate R The maximum error
  51  *      of this polynomial approximation is bounded by 2**-58.45. In
  52  *      other words,
  53  *                      2      4      6      8      10      12      14
  54  *          R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s  +Lp6*s  +Lp7*s
  55  *      (the values of Lp1 to Lp7 are listed in the program)
  56  *      and
  57  *          |      2          14          |     -58.45
  58  *          | Lp1*s +...+Lp7*s    -  R(z) | <= 2
  59  *          |                             |
  60  *      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
  61  *      In order to guarantee error in log below 1ulp, we compute log
  62  *      by
  63  *              log1p(f) = f - (hfsq - s*(hfsq+R)).
  64  *
  65  *      3. Finally, log1p(x) = k*ln2 + log1p(f).
  66  *                           = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
  67  *         Here ln2 is splitted into two floating point number:
  68  *                      ln2_hi + ln2_lo,
  69  *         where n*ln2_hi is always exact for |n| < 2000.
  70  *
  71  * Special cases:
  72  *      log1p(x) is NaN with signal if x < -1 (including -INF) ;
  73  *      log1p(+INF) is +INF; log1p(-1) is -INF with signal;
  74  *      log1p(NaN) is that NaN with no signal.
  75  *
  76  * Accuracy:
  77  *      according to an error analysis, the error is always less than
  78  *      1 ulp (unit in the last place).
  79  *
  80  * Constants:
  81  * The hexadecimal values are the intended ones for the following
  82  * constants. The decimal values may be used, provided that the
  83  * compiler will convert from decimal to binary accurately enough
  84  * to produce the hexadecimal values shown.
  85  *
  86  * Note: Assuming log() return accurate answer, the following
  87  *       algorithm can be used to compute log1p(x) to within a few ULP:
  88  *
  89  *              u = 1+x;
  90  *              if (u == 1.0) return x ; else
  91  *                         return log(u)*(x/(u-1.0));
  92  *
  93  *       See HP-15C Advanced Functions Handbook, p.193.
  94  */
  95 /* INDENT ON */
  96
  97 #include "libm.h"
  98
  99 static const double xxx[] = {
 100 /* ln2_hi */    6.93147180369123816490e-01,     /* 3fe62e42 fee00000 */
 101 /* ln2_lo */    1.90821492927058770002e-10,     /* 3dea39ef 35793c76 */
 102 /* two54 */     1.80143985094819840000e+16,     /* 43500000 00000000 */
 103 /* Lp1 */       6.666666666666735130e-01,       /* 3FE55555 55555593 */
 104 /* Lp2 */       3.999999999940941908e-01,       /* 3FD99999 9997FA04 */
 105 /* Lp3 */       2.857142874366239149e-01,       /* 3FD24924 94229359 */
 106 /* Lp4 */       2.222219843214978396e-01,       /* 3FCC71C5 1D8E78AF */
 107 /* Lp5 */       1.818357216161805012e-01,       /* 3FC74664 96CB03DE */
 108 /* Lp6 */       1.531383769920937332e-01,       /* 3FC39A09 D078C69F */
 109 /* Lp7 */       1.479819860511658591e-01,       /* 3FC2F112 DF3E5244 */
 110 /* zero */      0.0
 111 };
 112 #define ln2_hi  xxx[0]
 113 #define ln2_lo  xxx[1]
 114 #define two54   xxx[2]
 115 #define Lp1     xxx[3]
 116 #define Lp2     xxx[4]
 117 #define Lp3     xxx[5]
 118 #define Lp4     xxx[6]
 119 #define Lp5     xxx[7]
 120 #define Lp6     xxx[8]
 121 #define Lp7     xxx[9]
 122 #define zero    xxx[10]
 123
 124 double
 125 log1p(double x) {
 126         double  hfsq, f, c = 0.0, s, z, R, u;
 127         int     k, hx, hu, ax;
 128
 129         hx = ((int *)&x)[HIWORD];               /* high word of x */
 130         ax = hx & 0x7fffffff;
 131
 132         if (ax >= 0x7ff00000) { /* x is inf or nan */
 133                 if (((hx - 0xfff00000) | ((int *)&x)[LOWORD]) == 0) /* -inf */
 134                         return (_SVID_libm_err(x, x, 44));
 135                 return (x * x);
 136         }
 137
 138         k = 1;
 139         if (hx < 0x3FDA827A) {  /* x < 0.41422  */
 140                 if (ax >= 0x3ff00000)   /* x <= -1.0 */
 141                         return (_SVID_libm_err(x, x, x == -1.0 ? 43 : 44));
 142                 if (ax < 0x3e200000) {  /* |x| < 2**-29 */
 143                         if (two54 + x > zero && /* raise inexact */
 144                             ax < 0x3c900000)    /* |x| < 2**-54 */
 145                                 return (x);
 146                         else
 147                                 return (x - x * x * 0.5);
 148                 }
 149                 if (hx > 0 || hx <= (int)0xbfd2bec3) {  /* -0.2929<x<0.41422 */
 150                         k = 0;
 151                         f = x;
 152                         hu = 1;
 153                 }
 154         }
 155         /* We will initialize 'c' here. */
 156         if (k != 0) {
 157                 if (hx < 0x43400000) {
 158                         u = 1.0 + x;
 159                         hu = ((int *)&u)[HIWORD];       /* high word of u */
 160                         k = (hu >> 20) - 1023;
 161                         /*
 162                          * correction term
 163                          */
 164                         c = k > 0 ? 1.0 - (u - x) : x - (u - 1.0);
 165                         c /= u;
 166                 } else {
 167                         u = x;
 168                         hu = ((int *)&u)[HIWORD];       /* high word of u */
 169                         k = (hu >> 20) - 1023;
 170                         c = 0;
 171                 }
 172                 hu &= 0x000fffff;
 173                 if (hu < 0x6a09e) {     /* normalize u */
 174                         ((int *)&u)[HIWORD] = hu | 0x3ff00000;
 175                 } else {                        /* normalize u/2 */
 176                         k += 1;
 177                         ((int *)&u)[HIWORD] = hu | 0x3fe00000;
 178                         hu = (0x00100000 - hu) >> 2;
 179                 }
 180                 f = u - 1.0;
 181         }
 182         hfsq = 0.5 * f * f;
 183         if (hu == 0) {          /* |f| < 2**-20 */
 184                 if (f == zero) {
 185                         if (k == 0)
 186                                 return (zero);
 187                         /* We already initialized 'c' before, when (k != 0) */
 188                         c += k * ln2_lo;
 189                         return (k * ln2_hi + c);
 190                 }
 191                 R = hfsq * (1.0 - 0.66666666666666666 * f);
 192                 if (k == 0)
 193                         return (f - R);
 194                 return (k * ln2_hi - ((R - (k * ln2_lo + c)) - f));
 195         }
 196         s = f / (2.0 + f);
 197         z = s * s;
 198         R = z * (Lp1 + z * (Lp2 + z * (Lp3 + z * (Lp4 + z * (Lp5 +
 199                 z * (Lp6 + z * Lp7))))));
 200         if (k == 0)
 201                 return (f - (hfsq - s * (hfsq + R)));
 202         return (k * ln2_hi - ((hfsq - (s * (hfsq + R) +
 203                 (k * ln2_lo + c))) - f));
 204 }