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 zero = 0.0L;
 121
 122 long double
 123 __log1pl (long double xm1)
 124 {
 125   long double x, y, z, r, s;
 126   ieee854_long_double_shape_type u;
 127   int32_t hx;
 128   int e;
 129
 130   /* Test for NaN or infinity input. */
 131   u.value = xm1;
 132   hx = u.parts32.w0;
 133   if (hx >= 0x7fff0000)
 134     return xm1;
 135
 136   /* log1p(+- 0) = +- 0.  */
 137   if (((hx & 0x7fffffff) == 0)
 138       && (u.parts32.w1 | u.parts32.w2 | u.parts32.w3) == 0)
 139     return xm1;
 140
 141   x = xm1 + 1.0L;
 142
 143   /* log1p(-1) = -inf */
 144   if (x <= 0.0L)
 145     {
 146       if (x == 0.0L)
 147         return (-1.0L / (x - x));
 148       else
 149         return (zero / (x - x));
 150     }
 151
 152   /* Separate mantissa from exponent.  */
 153
 154   /* Use frexp used so that denormal numbers will be handled properly.  */
 155   x = __frexpl (x, &e);
 156
 157   /* Logarithm using log(x) = z + z^3 P(z^2)/Q(z^2),
 158      where z = 2(x-1)/x+1).  */
 159   if ((e > 2) || (e < -2))
 160     {
 161       if (x < sqrth)
 162         {                       /* 2( 2x-1 )/( 2x+1 ) */
 163           e -= 1;
 164           z = x - 0.5L;
 165           y = 0.5L * z + 0.5L;
 166         }
 167       else
 168         {                       /*  2 (x-1)/(x+1)   */
 169           z = x - 0.5L;
 170           z -= 0.5L;
 171           y = 0.5L * x + 0.5L;
 172         }
 173       x = z / y;
 174       z = x * x;
 175       r = ((((R5 * z
 176               + R4) * z
 177              + R3) * z
 178             + R2) * z
 179            + R1) * z
 180         + R0;
 181       s = (((((z
 182                + S5) * z
 183               + S4) * z
 184              + S3) * z
 185             + S2) * z
 186            + S1) * z
 187         + S0;
 188       z = x * (z * r / s);
 189       z = z + e * C2;
 190       z = z + x;
 191       z = z + e * C1;
 192       return (z);
 193     }
 194
 195
 196   /* Logarithm using log(1+x) = x - .5x^2 + x^3 P(x)/Q(x). */
 197
 198   if (x < sqrth)
 199     {
 200       e -= 1;
 201       if (e != 0)
 202         x = 2.0L * x - 1.0L;    /*  2x - 1  */
 203       else
 204         x = xm1;
 205     }
 206   else
 207     {
 208       if (e != 0)
 209         x = x - 1.0L;
 210       else
 211         x = xm1;
 212     }
 213   z = x * x;
 214   r = (((((((((((P12 * x
 215                  + P11) * x
 216                 + P10) * x
 217                + P9) * x
 218               + P8) * x
 219              + P7) * x
 220             + P6) * x
 221            + P5) * x
 222           + P4) * x
 223          + P3) * x
 224         + P2) * x
 225        + P1) * x
 226     + P0;
 227   s = (((((((((((x
 228                  + Q11) * x
 229                 + Q10) * x
 230                + Q9) * x
 231               + Q8) * x
 232              + Q7) * x
 233             + Q6) * x
 234            + Q5) * x
 235           + Q4) * x
 236          + Q3) * x
 237         + Q2) * x
 238        + Q1) * x
 239     + Q0;
 240   y = x * (z * r / s);
 241   y = y + e * C2;
 242   z = y - 0.5L * z;
 243   z = z + x;
 244   z = z + e * C1;
 245   return (z);
 246 }
 247
 248 weak_alias (__log1pl, log1pl)