compiler/stdc/math/ld128/s_log1pl.c

   1 /*      $OpenBSD: s_log1pl.c,v 1.1 2011/07/06 00:02:42 martynas Exp $   */
   2
   3 /*
   4  * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
   5  *
   6  * Permission to use, copy, modify, and distribute this software for any
   7  * purpose with or without fee is hereby granted, provided that the above
   8  * copyright notice and this permission notice appear in all copies.
   9  *
  10  * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
  11  * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
  12  * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
  13  * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
  14  * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
  15  * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
  16  * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
  17  */
  18
  19 /*                                                      log1pl.c
  20  *
  21  *      Relative error logarithm
  22  *      Natural logarithm of 1+x, 128-bit long double precision
  23  *
  24  *
  25  *
  26  * SYNOPSIS:
  27  *
  28  * long double x, y, log1pl();
  29  *
  30  * y = log1pl( x );
  31  *
  32  *
  33  *
  34  * DESCRIPTION:
  35  *
  36  * Returns the base e (2.718...) logarithm of 1+x.
  37  *
  38  * The argument 1+x is separated into its exponent and fractional
  39  * parts.  If the exponent is between -1 and +1, the logarithm
  40  * of the fraction is approximated by
  41  *
  42  *     log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x).
  43  *
  44  * Otherwise, setting  z = 2(w-1)/(w+1),
  45  *
  46  *     log(w) = z + z^3 P(z)/Q(z).
  47  *
  48  *
  49  *
  50  * ACCURACY:
  51  *
  52  *                      Relative error:
  53  * arithmetic   domain     # trials      peak         rms
  54  *    IEEE      -1, 8       100000      1.9e-34     4.3e-35
  55  */
  56
  57 #include "math.h"
  58
  59 #include "math_private.h"
  60
  61 /* Coefficients for log(1+x) = x - x^2 / 2 + x^3 P(x)/Q(x)
  62  * 1/sqrt(2) <= 1+x < sqrt(2)
  63  * Theoretical peak relative error = 5.3e-37,
  64  * relative peak error spread = 2.3e-14
  65  */
  66 static const long double
  67   P12 = 1.538612243596254322971797716843006400388E-6L,
  68   P11 = 4.998469661968096229986658302195402690910E-1L,
  69   P10 = 2.321125933898420063925789532045674660756E1L,
  70   P9 = 4.114517881637811823002128927449878962058E2L,
  71   P8 = 3.824952356185897735160588078446136783779E3L,
  72   P7 = 2.128857716871515081352991964243375186031E4L,
  73   P6 = 7.594356839258970405033155585486712125861E4L,
  74   P5 = 1.797628303815655343403735250238293741397E5L,
  75   P4 = 2.854829159639697837788887080758954924001E5L,
  76   P3 = 3.007007295140399532324943111654767187848E5L,
  77   P2 = 2.014652742082537582487669938141683759923E5L,
  78   P1 = 7.771154681358524243729929227226708890930E4L,
  79   P0 = 1.313572404063446165910279910527789794488E4L,
  80   /* Q12 = 1.000000000000000000000000000000000000000E0L, */
  81   Q11 = 4.839208193348159620282142911143429644326E1L,
  82   Q10 = 9.104928120962988414618126155557301584078E2L,
  83   Q9 = 9.147150349299596453976674231612674085381E3L,
  84   Q8 = 5.605842085972455027590989944010492125825E4L,
  85   Q7 = 2.248234257620569139969141618556349415120E5L,
  86   Q6 = 6.132189329546557743179177159925690841200E5L,
  87   Q5 = 1.158019977462989115839826904108208787040E6L,
  88   Q4 = 1.514882452993549494932585972882995548426E6L,
  89   Q3 = 1.347518538384329112529391120390701166528E6L,
  90   Q2 = 7.777690340007566932935753241556479363645E5L,
  91   Q1 = 2.626900195321832660448791748036714883242E5L,
  92   Q0 = 3.940717212190338497730839731583397586124E4L;
  93
  94 /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
  95  * where z = 2(x-1)/(x+1)
  96  * 1/sqrt(2) <= x < sqrt(2)
  97  * Theoretical peak relative error = 1.1e-35,
  98  * relative peak error spread 1.1e-9
  99  */
 100 static const long double
 101   R5 = -8.828896441624934385266096344596648080902E-1L,
 102   R4 = 8.057002716646055371965756206836056074715E1L,
 103   R3 = -2.024301798136027039250415126250455056397E3L,
 104   R2 = 2.048819892795278657810231591630928516206E4L,
 105   R1 = -8.977257995689735303686582344659576526998E4L,
 106   R0 = 1.418134209872192732479751274970992665513E5L,
 107   /* S6 = 1.000000000000000000000000000000000000000E0L, */
 108   S5 = -1.186359407982897997337150403816839480438E2L,
 109   S4 = 3.998526750980007367835804959888064681098E3L,
 110   S3 = -5.748542087379434595104154610899551484314E4L,
 111   S2 = 4.001557694070773974936904547424676279307E5L,
 112   S1 = -1.332535117259762928288745111081235577029E6L,
 113   S0 = 1.701761051846631278975701529965589676574E6L;
 114
 115 /* C1 + C2 = ln 2 */
 116 static const long double C1 = 6.93145751953125E-1L;
 117 static const long double C2 = 1.428606820309417232121458176568075500134E-6L;
 118
 119 static const long double sqrth = 0.7071067811865475244008443621048490392848L;
 120 /* ln (2^16384 * (1 - 2^-113)) */
 121 static const long double zero = 0.0L;
 122
 123 long double
 124 log1pl(long double xm1)
 125 {
 126   long double x, y, z, r, s;
 127   ieee_quad_shape_type u;
 128   int32_t hx;
 129   int e;
 130
 131   /* Test for NaN or infinity input. */
 132   u.value = xm1;
 133   hx = u.parts32.mswhi;
 134   if (hx >= 0x7fff0000)
 135     return xm1;
 136
 137   /* log1p(+- 0) = +- 0.  */
 138   if (((hx & 0x7fffffff) == 0)
 139       && (u.parts32.mswlo | u.parts32.lswhi | u.parts32.lswlo) == 0)
 140     return xm1;
 141
 142   x = xm1 + 1.0L;
 143
 144   /* log1p(-1) = -inf */
 145   if (x <= 0.0L)
 146     {
 147       if (x == 0.0L)
 148         return (-1.0L / (x - x));
 149       else
 150         return (zero / (x - x));
 151     }
 152
 153   /* Separate mantissa from exponent.  */
 154
 155   /* Use frexp used so that denormal numbers will be handled properly.  */
 156   x = frexpl (x, &e);
 157
 158   /* Logarithm using log(x) = z + z^3 P(z^2)/Q(z^2),
 159      where z = 2(x-1)/x+1).  */
 160   if ((e > 2) || (e < -2))
 161     {
 162       if (x < sqrth)
 163         {                       /* 2( 2x-1 )/( 2x+1 ) */
 164           e -= 1;
 165           z = x - 0.5L;
 166           y = 0.5L * z + 0.5L;
 167         }
 168       else
 169         {                       /*  2 (x-1)/(x+1)   */
 170           z = x - 0.5L;
 171           z -= 0.5L;
 172           y = 0.5L * x + 0.5L;
 173         }
 174       x = z / y;
 175       z = x * x;
 176       r = ((((R5 * z
 177               + R4) * z
 178              + R3) * z
 179             + R2) * z
 180            + R1) * z
 181         + R0;
 182       s = (((((z
 183                + S5) * z
 184               + S4) * z
 185              + S3) * z
 186             + S2) * z
 187            + S1) * z
 188         + S0;
 189       z = x * (z * r / s);
 190       z = z + e * C2;
 191       z = z + x;
 192       z = z + e * C1;
 193       return (z);
 194     }
 195
 196
 197   /* Logarithm using log(1+x) = x - .5x^2 + x^3 P(x)/Q(x). */
 198
 199   if (x < sqrth)
 200     {
 201       e -= 1;
 202       if (e != 0)
 203         x = 2.0L * x - 1.0L;    /*  2x - 1  */
 204       else
 205         x = xm1;
 206     }
 207   else
 208     {
 209       if (e != 0)
 210         x = x - 1.0L;
 211       else
 212         x = xm1;
 213     }
 214   z = x * x;
 215   r = (((((((((((P12 * x
 216                  + P11) * x
 217                 + P10) * x
 218                + P9) * x
 219               + P8) * x
 220              + P7) * x
 221             + P6) * x
 222            + P5) * x
 223           + P4) * x
 224          + P3) * x
 225         + P2) * x
 226        + P1) * x
 227     + P0;
 228   s = (((((((((((x
 229                  + Q11) * x
 230                 + Q10) * x
 231                + Q9) * x
 232               + Q8) * x
 233              + Q7) * x
 234             + Q6) * x
 235            + Q5) * x
 236           + Q4) * x
 237          + Q3) * x
 238         + Q2) * x
 239        + Q1) * x
 240     + Q0;
 241   y = x * (z * r / s);
 242   y = y + e * C2;
 243   z = y - 0.5L * z;
 244   z = z + x;
 245   z = z + e * C1;
 246   return (z);
 247 }