sysdeps/ieee754/ldbl-128/s_log1pl.c

   1 /*                                                      log1pl.c
   2  *
   3  *      Relative error logarithm
   4  *      Natural logarithm of 1+x, 128-bit long double precision
   5  *
   6  *
   7  *
   8  * SYNOPSIS:
   9  *
  10  * long double x, y, log1pl();
  11  *
  12  * y = log1pl( x );
  13  *
  14  *
  15  *
  16  * DESCRIPTION:
  17  *
  18  * Returns the base e (2.718...) logarithm of 1+x.
  19  *
  20  * The argument 1+x is separated into its exponent and fractional
  21  * parts.  If the exponent is between -1 and +1, the logarithm
  22  * of the fraction is approximated by
  23  *
  24  *     log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x).
  25  *
  26  * Otherwise, setting  z = 2(w-1)/(w+1),
  27  *
  28  *     log(w) = z + z^3 P(z)/Q(z).
  29  *
  30  *
  31  *
  32  * ACCURACY:
  33  *
  34  *                      Relative error:
  35  * arithmetic   domain     # trials      peak         rms
  36  *    IEEE      -1, 8       100000      1.9e-34     4.3e-35
  37  */
  38
  39 /* Copyright 2001 by Stephen L. Moshier
  40
  41     This library is free software; you can redistribute it and/or
  42     modify it under the terms of the GNU Lesser General Public
  43     License as published by the Free Software Foundation; either
  44     version 2.1 of the License, or (at your option) any later version.
  45
  46     This library is distributed in the hope that it will be useful,
  47     but WITHOUT ANY WARRANTY; without even the implied warranty of
  48     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
  49     Lesser General Public License for more details.
  50
  51     You should have received a copy of the GNU Lesser General Public
  52     License along with this library; if not, write to the Free Software
  53     Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307  USA */
  54
  55
  56 #include "math.h"
  57 #include "math_private.h"
  58
  59 /* Coefficients for log(1+x) = x - x^2 / 2 + x^3 P(x)/Q(x)
  60  * 1/sqrt(2) <= 1+x < sqrt(2)
  61  * Theoretical peak relative error = 5.3e-37,
  62  * relative peak error spread = 2.3e-14
  63  */
  64 static const long double
  65   P12 = 1.538612243596254322971797716843006400388E-6L,
  66   P11 = 4.998469661968096229986658302195402690910E-1L,
  67   P10 = 2.321125933898420063925789532045674660756E1L,
  68   P9 = 4.114517881637811823002128927449878962058E2L,
  69   P8 = 3.824952356185897735160588078446136783779E3L,
  70   P7 = 2.128857716871515081352991964243375186031E4L,
  71   P6 = 7.594356839258970405033155585486712125861E4L,
  72   P5 = 1.797628303815655343403735250238293741397E5L,
  73   P4 = 2.854829159639697837788887080758954924001E5L,
  74   P3 = 3.007007295140399532324943111654767187848E5L,
  75   P2 = 2.014652742082537582487669938141683759923E5L,
  76   P1 = 7.771154681358524243729929227226708890930E4L,
  77   P0 = 1.313572404063446165910279910527789794488E4L,
  78   /* Q12 = 1.000000000000000000000000000000000000000E0L, */
  79   Q11 = 4.839208193348159620282142911143429644326E1L,
  80   Q10 = 9.104928120962988414618126155557301584078E2L,
  81   Q9 = 9.147150349299596453976674231612674085381E3L,
  82   Q8 = 5.605842085972455027590989944010492125825E4L,
  83   Q7 = 2.248234257620569139969141618556349415120E5L,
  84   Q6 = 6.132189329546557743179177159925690841200E5L,
  85   Q5 = 1.158019977462989115839826904108208787040E6L,
  86   Q4 = 1.514882452993549494932585972882995548426E6L,
  87   Q3 = 1.347518538384329112529391120390701166528E6L,
  88   Q2 = 7.777690340007566932935753241556479363645E5L,
  89   Q1 = 2.626900195321832660448791748036714883242E5L,
  90   Q0 = 3.940717212190338497730839731583397586124E4L;
  91
  92 /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
  93  * where z = 2(x-1)/(x+1)
  94  * 1/sqrt(2) <= x < sqrt(2)
  95  * Theoretical peak relative error = 1.1e-35,
  96  * relative peak error spread 1.1e-9
  97  */
  98 static const long double
  99   R5 = -8.828896441624934385266096344596648080902E-1L,
 100   R4 = 8.057002716646055371965756206836056074715E1L,
 101   R3 = -2.024301798136027039250415126250455056397E3L,
 102   R2 = 2.048819892795278657810231591630928516206E4L,
 103   R1 = -8.977257995689735303686582344659576526998E4L,
 104   R0 = 1.418134209872192732479751274970992665513E5L,
 105   /* S6 = 1.000000000000000000000000000000000000000E0L, */
 106   S5 = -1.186359407982897997337150403816839480438E2L,
 107   S4 = 3.998526750980007367835804959888064681098E3L,
 108   S3 = -5.748542087379434595104154610899551484314E4L,
 109   S2 = 4.001557694070773974936904547424676279307E5L,
 110   S1 = -1.332535117259762928288745111081235577029E6L,
 111   S0 = 1.701761051846631278975701529965589676574E6L;
 112
 113 /* C1 + C2 = ln 2 */
 114 static const long double C1 = 6.93145751953125E-1L;
 115 static const long double C2 = 1.428606820309417232121458176568075500134E-6L;
 116
 117 static const long double sqrth = 0.7071067811865475244008443621048490392848L;
 118 /* ln (2^16384 * (1 - 2^-113)) */
 119 static const long double maxlog = 1.1356523406294143949491931077970764891253E4L;
 120 static const long double big = 2e4932L;
 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   ieee854_long_double_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.w0;
 134   if (hx >= 0x7fff0000)
 135     return xm1;
 136
 137   /* log1p(+- 0) = +- 0.  */
 138   if (((hx & 0x7fffffff) == 0)
 139       && (u.parts32.w1 | u.parts32.w2 | u.parts32.w3) == 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 }
 248
 249 weak_alias (__log1pl, log1pl)