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 #include "math.h"
  42 #include "math_private.h"
  43
  44 /* Coefficients for log(1+x) = x - x^2 / 2 + x^3 P(x)/Q(x)
  45  * 1/sqrt(2) <= 1+x < sqrt(2)
  46  * Theoretical peak relative error = 5.3e-37,
  47  * relative peak error spread = 2.3e-14
  48  */
  49 static long double
  50   P12 = 1.538612243596254322971797716843006400388E-6L,
  51   P11 = 4.998469661968096229986658302195402690910E-1L,
  52   P10 = 2.321125933898420063925789532045674660756E1L,
  53   P9 = 4.114517881637811823002128927449878962058E2L,
  54   P8 = 3.824952356185897735160588078446136783779E3L,
  55   P7 = 2.128857716871515081352991964243375186031E4L,
  56   P6 = 7.594356839258970405033155585486712125861E4L,
  57   P5 = 1.797628303815655343403735250238293741397E5L,
  58   P4 = 2.854829159639697837788887080758954924001E5L,
  59   P3 = 3.007007295140399532324943111654767187848E5L,
  60   P2 = 2.014652742082537582487669938141683759923E5L,
  61   P1 = 7.771154681358524243729929227226708890930E4L,
  62   P0 = 1.313572404063446165910279910527789794488E4L,
  63   /* Q12 = 1.000000000000000000000000000000000000000E0L, */
  64   Q11 = 4.839208193348159620282142911143429644326E1L,
  65   Q10 = 9.104928120962988414618126155557301584078E2L,
  66   Q9 = 9.147150349299596453976674231612674085381E3L,
  67   Q8 = 5.605842085972455027590989944010492125825E4L,
  68   Q7 = 2.248234257620569139969141618556349415120E5L,
  69   Q6 = 6.132189329546557743179177159925690841200E5L,
  70   Q5 = 1.158019977462989115839826904108208787040E6L,
  71   Q4 = 1.514882452993549494932585972882995548426E6L,
  72   Q3 = 1.347518538384329112529391120390701166528E6L,
  73   Q2 = 7.777690340007566932935753241556479363645E5L,
  74   Q1 = 2.626900195321832660448791748036714883242E5L,
  75   Q0 = 3.940717212190338497730839731583397586124E4L;
  76
  77 /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
  78  * where z = 2(x-1)/(x+1)
  79  * 1/sqrt(2) <= x < sqrt(2)
  80  * Theoretical peak relative error = 1.1e-35,
  81  * relative peak error spread 1.1e-9
  82  */
  83 static long double
  84   R5 = -8.828896441624934385266096344596648080902E-1L,
  85   R4 = 8.057002716646055371965756206836056074715E1L,
  86   R3 = -2.024301798136027039250415126250455056397E3L,
  87   R2 = 2.048819892795278657810231591630928516206E4L,
  88   R1 = -8.977257995689735303686582344659576526998E4L,
  89   R0 = 1.418134209872192732479751274970992665513E5L,
  90   /* S6 = 1.000000000000000000000000000000000000000E0L, */
  91   S5 = -1.186359407982897997337150403816839480438E2L,
  92   S4 = 3.998526750980007367835804959888064681098E3L,
  93   S3 = -5.748542087379434595104154610899551484314E4L,
  94   S2 = 4.001557694070773974936904547424676279307E5L,
  95   S1 = -1.332535117259762928288745111081235577029E6L,
  96   S0 = 1.701761051846631278975701529965589676574E6L;
  97
  98 /* C1 + C2 = ln 2 */
  99 static long double C1 = 6.93145751953125E-1L;
 100 static long double C2 = 1.428606820309417232121458176568075500134E-6L;
 101
 102 static long double sqrth = 0.7071067811865475244008443621048490392848L;
 103 /* ln (2^16384 * (1 - 2^-113)) */
 104 static long double maxlog = 1.1356523406294143949491931077970764891253E4L;
 105 static long double big = 2e4932L;
 106 static long double zero = 0.0L;
 107
 108 #if 1
 109 /* Make sure these are prototyped.  */
 110 long double frexpl (long double, int *);
 111 long double ldexpl (long double, int);
 112 #endif
 113
 114
 115 long double
 116 __log1pl (long double xm1)
 117 {
 118   long double x, y, z, r, s;
 119   ieee854_long_double_shape_type u;
 120   int32_t ix;
 121   int e;
 122
 123   /* Test for NaN or infinity input. */
 124   u.value = xm1;
 125   ix = u.parts32.w0 & 0x7fffffff;
 126   if (ix >= 0x7fff0000)
 127     return xm1;
 128
 129   /* log1p(+- 0) = +- 0.  */
 130   if ((ix == 0) && (u.parts32.w1 | u.parts32.w2 | u.parts32.w3) == 0)
 131     return xm1;
 132
 133   x = xm1 + 1.0L;
 134
 135   /* log1p(-1) = -inf */
 136   if (x <= 0.0L)
 137     {
 138       if (x == 0.0L)
 139         return (-1.0L / zero);
 140       else
 141         return (zero / zero);
 142     }
 143
 144   /* Separate mantissa from exponent.  */
 145
 146   /* Use frexp used so that denormal numbers will be handled properly.  */
 147   x = frexpl (x, &e);
 148
 149   /* Logarithm using log(x) = z + z^3 P(z^2)/Q(z^2),
 150      where z = 2(x-1)/x+1).  */
 151   if ((e > 2) || (e < -2))
 152     {
 153       if (x < sqrth)
 154         {                       /* 2( 2x-1 )/( 2x+1 ) */
 155           e -= 1;
 156           z = x - 0.5L;
 157           y = 0.5L * z + 0.5L;
 158         }
 159       else
 160         {                       /*  2 (x-1)/(x+1)   */
 161           z = x - 0.5L;
 162           z -= 0.5L;
 163           y = 0.5L * x + 0.5L;
 164         }
 165       x = z / y;
 166       z = x * x;
 167       r = ((((R5 * z
 168               + R4) * z
 169              + R3) * z
 170             + R2) * z
 171            + R1) * z
 172         + R0;
 173       s = (((((z
 174                + S5) * z
 175               + S4) * z
 176              + S3) * z
 177             + S2) * z
 178            + S1) * z
 179         + S0;
 180       z = x * (z * r / s);
 181       z = z + e * C2;
 182       z = z + x;
 183       z = z + e * C1;
 184       return (z);
 185     }
 186
 187
 188   /* Logarithm using log(1+x) = x - .5x^2 + x^3 P(x)/Q(x). */
 189
 190   if (x < sqrth)
 191     {
 192       e -= 1;
 193       if (e != 0)
 194         x = 2.0L * x - 1.0L;    /*  2x - 1  */
 195       else
 196         x = xm1;
 197     }
 198   else
 199     {
 200       if (e != 0)
 201         x = x - 1.0L;
 202       else
 203         x = xm1;
 204     }
 205   z = x * x;
 206   r = (((((((((((P12 * x
 207                  + P11) * x
 208                 + P10) * x
 209                + P9) * x
 210               + P8) * x
 211              + P7) * x
 212             + P6) * x
 213            + P5) * x
 214           + P4) * x
 215          + P3) * x
 216         + P2) * x
 217        + P1) * x
 218     + P0;
 219   s = (((((((((((x
 220                  + Q11) * x
 221                 + Q10) * x
 222                + Q9) * x
 223               + Q8) * x
 224              + Q7) * x
 225             + Q6) * x
 226            + Q5) * x
 227           + Q4) * x
 228          + Q3) * x
 229         + Q2) * x
 230        + Q1) * x
 231     + Q0;
 232   y = x * (z * r / s);
 233   y = y + e * C2;
 234   z = y - 0.5L * z;
 235   z = z + x;
 236   z = z + e * C1;
 237   return (z);
 238 }
 239
 240 weak_alias (__log1pl, log1pl)
 241 #ifdef NO_LONG_DOUBLE
 242 strong_alias (__log1p, __log1pl)
 243 weak_alias (__log1p, log1pl)
 244 #endif