contrib/mpc/src/log10.c

   1 /* mpc_log10 -- Take the base-10 logarithm of a complex number.
   2
   3 Copyright (C) 2012 INRIA
   4
   5 This file is part of GNU MPC.
   6
   7 GNU MPC is free software; you can redistribute it and/or modify it under
   8 the terms of the GNU Lesser General Public License as published by the
   9 Free Software Foundation; either version 3 of the License, or (at your
  10 option) any later version.
  11
  12 GNU MPC is distributed in the hope that it will be useful, but WITHOUT ANY
  13 WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
  14 FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for
  15 more details.
  16
  17 You should have received a copy of the GNU Lesser General Public License
  18 along with this program. If not, see http://www.gnu.org/licenses/ .
  19 */
  20
  21 #include <limits.h> /* for CHAR_BIT */
  22 #include "mpc-impl.h"
  23
  24 /* Auxiliary functions which implement Ziv's strategy for special cases.
  25    if flag = 0: compute only real part
  26    if flag = 1: compute only imaginary
  27    Exact cases should be dealt with separately. */
  28 static int
  29 mpc_log10_aux (mpc_ptr rop, mpc_srcptr op, mpc_rnd_t rnd, int flag, int nb)
  30 {
  31   mp_prec_t prec = (MPFR_PREC_MIN > 4) ? MPFR_PREC_MIN : 4;
  32   mpc_t tmp;
  33   mpfr_t log10;
  34   int ok = 0, ret;
  35
  36   prec = mpfr_get_prec ((flag == 0) ? mpc_realref (rop) : mpc_imagref (rop));
  37   prec += 10;
  38   mpc_init2 (tmp, prec);
  39   mpfr_init2 (log10, prec);
  40   while (ok == 0)
  41     {
  42       mpfr_set_ui (log10, 10, GMP_RNDN); /* exact since prec >= 4 */
  43       mpfr_log (log10, log10, GMP_RNDN);
  44       /* In each case we have two roundings, thus the final value is
  45          x * (1+u)^2 where x is the exact value, and |u| <= 2^(-prec-1).
  46          Thus the error is always less than 3 ulps. */
  47       switch (nb)
  48         {
  49         case 0: /* imag <- atan2(y/x) */
  50           mpfr_atan2 (mpc_imagref (tmp), mpc_imagref (op), mpc_realref (op),
  51                       MPC_RND_IM (rnd));
  52           mpfr_div (mpc_imagref (tmp), mpc_imagref (tmp), log10, GMP_RNDN);
  53           ok = mpfr_can_round (mpc_imagref (tmp), prec - 2, GMP_RNDN,
  54                                GMP_RNDZ, MPC_PREC_IM(rop) +
  55                                (MPC_RND_IM (rnd) == GMP_RNDN));
  56           if (ok)
  57             ret = mpfr_set (mpc_imagref (rop), mpc_imagref (tmp),
  58                             MPC_RND_IM (rnd));
  59           break;
  60         case 1: /* real <- log(x) */
  61           mpfr_log (mpc_realref (tmp), mpc_realref (op), MPC_RND_RE (rnd));
  62           mpfr_div (mpc_realref (tmp), mpc_realref (tmp), log10, GMP_RNDN);
  63           ok = mpfr_can_round (mpc_realref (tmp), prec - 2, GMP_RNDN,
  64                                GMP_RNDZ, MPC_PREC_RE(rop) +
  65                                (MPC_RND_RE (rnd) == GMP_RNDN));
  66           if (ok)
  67             ret = mpfr_set (mpc_realref (rop), mpc_realref (tmp),
  68                             MPC_RND_RE (rnd));
  69           break;
  70         case 2: /* imag <- pi */
  71           mpfr_const_pi (mpc_imagref (tmp), MPC_RND_IM (rnd));
  72           mpfr_div (mpc_imagref (tmp), mpc_imagref (tmp), log10, GMP_RNDN);
  73           ok = mpfr_can_round (mpc_imagref (tmp), prec - 2, GMP_RNDN,
  74                                GMP_RNDZ, MPC_PREC_IM(rop) +
  75                                (MPC_RND_IM (rnd) == GMP_RNDN));
  76           if (ok)
  77             ret = mpfr_set (mpc_imagref (rop), mpc_imagref (tmp),
  78                             MPC_RND_IM (rnd));
  79           break;
  80         }
  81       prec += prec / 2;
  82       mpc_set_prec (tmp, prec);
  83       mpfr_set_prec (log10, prec);
  84     }
  85   mpc_clear (tmp);
  86   mpfr_clear (log10);
  87   return ret;
  88 }
  89
  90 int
  91 mpc_log10 (mpc_ptr rop, mpc_srcptr op, mpc_rnd_t rnd)
  92 {
  93   int ok = 0, loops = 0, re_cmp, im_cmp, inex_re, inex_im, negative_zero;
  94   mpfr_t w;
  95   mpfr_prec_t prec;
  96   mpfr_rnd_t rnd_im;
  97   mpc_t ww;
  98   mpc_rnd_t invrnd;
  99
 100   /* special values: NaN and infinities: same as mpc_log */
 101   if (!mpc_fin_p (op)) /* real or imaginary parts are NaN or Inf */
 102     {
 103       if (mpfr_nan_p (mpc_realref (op)))
 104         {
 105           if (mpfr_inf_p (mpc_imagref (op)))
 106             /* (NaN, Inf) -> (+Inf, NaN) */
 107             mpfr_set_inf (mpc_realref (rop), +1);
 108           else
 109             /* (NaN, xxx) -> (NaN, NaN) */
 110             mpfr_set_nan (mpc_realref (rop));
 111          mpfr_set_nan (mpc_imagref (rop));
 112          inex_im = 0; /* Inf/NaN is exact */
 113         }
 114       else if (mpfr_nan_p (mpc_imagref (op)))
 115         {
 116           if (mpfr_inf_p (mpc_realref (op)))
 117             /* (Inf, NaN) -> (+Inf, NaN) */
 118             mpfr_set_inf (mpc_realref (rop), +1);
 119           else
 120             /* (xxx, NaN) -> (NaN, NaN) */
 121             mpfr_set_nan (mpc_realref (rop));
 122           mpfr_set_nan (mpc_imagref (rop));
 123           inex_im = 0; /* Inf/NaN is exact */
 124         }
 125       else /* We have an infinity in at least one part. */
 126         {
 127           /* (+Inf, y) -> (+Inf, 0) for finite positive-signed y */
 128           if (mpfr_inf_p (mpc_realref (op)) && mpfr_signbit (mpc_realref (op))
 129               == 0 && mpfr_number_p (mpc_imagref (op)))
 130             inex_im = mpfr_atan2 (mpc_imagref (rop), mpc_imagref (op),
 131                                   mpc_realref (op), MPC_RND_IM (rnd));
 132           else
 133             /* (xxx, Inf) -> (+Inf, atan2(Inf/xxx))
 134                (Inf, yyy) -> (+Inf, atan2(yyy/Inf)) */
 135             inex_im = mpc_log10_aux (rop, op, rnd, 1, 0);
 136           mpfr_set_inf (mpc_realref (rop), +1);
 137         }
 138       return MPC_INEX(0, inex_im);
 139     }
 140
 141    /* special cases: real and purely imaginary numbers */
 142   re_cmp = mpfr_cmp_ui (mpc_realref (op), 0);
 143   im_cmp = mpfr_cmp_ui (mpc_imagref (op), 0);
 144   if (im_cmp == 0) /* Im(op) = 0 */
 145     {
 146       if (re_cmp == 0) /* Re(op) = 0 */
 147         {
 148           if (mpfr_signbit (mpc_realref (op)) == 0)
 149             inex_im = mpfr_atan2 (mpc_imagref (rop), mpc_imagref (op),
 150                                   mpc_realref (op), MPC_RND_IM (rnd));
 151           else
 152             inex_im = mpc_log10_aux (rop, op, rnd, 1, 0);
 153           mpfr_set_inf (mpc_realref (rop), -1);
 154           inex_re = 0; /* -Inf is exact */
 155         }
 156       else if (re_cmp > 0)
 157         {
 158           inex_re = mpfr_log10 (mpc_realref (rop), mpc_realref (op),
 159                                 MPC_RND_RE (rnd));
 160           inex_im = mpfr_set (mpc_imagref (rop), mpc_imagref (op),
 161                               MPC_RND_IM (rnd));
 162         }
 163       else /* log10(x + 0*i) for negative x */
 164         { /* op = x + 0*i; let w = -x = |x| */
 165           negative_zero = mpfr_signbit (mpc_imagref (op));
 166           if (negative_zero)
 167             rnd_im = INV_RND (MPC_RND_IM (rnd));
 168           else
 169             rnd_im = MPC_RND_IM (rnd);
 170           ww->re[0] = *mpc_realref (op);
 171           MPFR_CHANGE_SIGN (ww->re);
 172           ww->im[0] = *mpc_imagref (op);
 173           if (mpfr_cmp_ui (ww->re, 1) == 0)
 174             inex_re = mpfr_set_ui (mpc_realref (rop), 0, MPC_RND_RE (rnd));
 175           else
 176             inex_re = mpc_log10_aux (rop, ww, rnd, 0, 1);
 177           inex_im = mpc_log10_aux (rop, op, MPC_RND (0,rnd_im), 1, 2);
 178           if (negative_zero)
 179             {
 180               mpc_conj (rop, rop, MPC_RNDNN);
 181               inex_im = -inex_im;
 182             }
 183         }
 184       return MPC_INEX(inex_re, inex_im);
 185     }
 186    else if (re_cmp == 0)
 187      {
 188        if (im_cmp > 0)
 189          {
 190            inex_re = mpfr_log10 (mpc_realref (rop), mpc_imagref (op), MPC_RND_RE (rnd));
 191            inex_im = mpc_log10_aux (rop, op, rnd, 1, 2);
 192            /* division by 2 does not change the ternary flag */
 193            mpfr_div_2ui (mpc_imagref (rop), mpc_imagref (rop), 1, GMP_RNDN);
 194          }
 195        else
 196          {
 197            w [0] = *mpc_imagref (op);
 198            MPFR_CHANGE_SIGN (w);
 199            inex_re = mpfr_log10 (mpc_realref (rop), w, MPC_RND_RE (rnd));
 200            invrnd = MPC_RND (0, INV_RND (MPC_RND_IM (rnd)));
 201            inex_im = mpc_log10_aux (rop, op, invrnd, 1, 2);
 202            /* division by 2 does not change the ternary flag */
 203            mpfr_div_2ui (mpc_imagref (rop), mpc_imagref (rop), 1, GMP_RNDN);
 204            mpfr_neg (mpc_imagref (rop), mpc_imagref (rop), GMP_RNDN);
 205            inex_im = -inex_im; /* negate the ternary flag */
 206          }
 207        return MPC_INEX(inex_re, inex_im);
 208      }
 209
 210   /* generic case: neither Re(op) nor Im(op) is NaN, Inf or zero */
 211   prec = MPC_PREC_RE(rop);
 212   mpfr_init2 (w, prec);
 213   mpc_init2 (ww, prec);
 214   /* let op = x + iy; compute log(op)/log(10) */
 215   while (ok == 0)
 216     {
 217       loops ++;
 218       prec += (loops <= 2) ? mpc_ceil_log2 (prec) + 4 : prec / 2;
 219       mpfr_set_prec (w, prec);
 220       mpc_set_prec (ww, prec);
 221
 222       mpc_log (ww, op, MPC_RNDNN);
 223       mpfr_set_ui (w, 10, GMP_RNDN); /* exact since prec >= 4 */
 224       mpfr_log (w, w, GMP_RNDN);
 225       mpc_div_fr (ww, ww, w, MPC_RNDNN);
 226
 227       ok = mpfr_can_round (mpc_realref (ww), prec - 2, GMP_RNDN, GMP_RNDZ,
 228                            MPC_PREC_RE(rop) + (MPC_RND_RE (rnd) == GMP_RNDN));
 229
 230       /* Special code to deal with cases where the real part of log10(x+i*y)
 231          is exact, like x=3 and y=1. Since Re(log10(x+i*y)) = log10(x^2+y^2)/2
 232          this happens whenever x^2+y^2 is a nonnegative power of 10.
 233          Indeed x^2+y^2 cannot equal 10^(a/2^b) for a, b integers, a odd, b>0,
 234          since x^2+y^2 is rational, and 10^(a/2^b) is irrational.
 235          Similarly, for b=0, x^2+y^2 cannot equal 10^a for a < 0 since x^2+y^2
 236          is a rational with denominator a power of 2.
 237          Now let x^2+y^2 = 10^s. Without loss of generality we can assume
 238          x = u/2^e and y = v/2^e with u, v, e integers: u^2+v^2 = 10^s*2^(2e)
 239          thus u^2+v^2 = 0 mod 2^(2e). By recurrence on e, necessarily
 240          u = v = 0 mod 2^e, thus x and y are necessarily integers.
 241       */
 242       if ((ok == 0) && (loops == 1) && mpfr_integer_p (mpc_realref (op)) &&
 243           mpfr_integer_p (mpc_imagref (op)))
 244         {
 245           mpz_t x, y;
 246           unsigned long s, v;
 247
 248           mpz_init (x);
 249           mpz_init (y);
 250           mpfr_get_z (x, mpc_realref (op), GMP_RNDN); /* exact */
 251           mpfr_get_z (y, mpc_imagref (op), GMP_RNDN); /* exact */
 252           mpz_mul (x, x, x);
 253           mpz_mul (y, y, y);
 254           mpz_add (x, x, y); /* x^2+y^2 */
 255           v = mpz_scan1 (x, 0);
 256           /* if x = 10^s then necessarily s = v */
 257           s = mpz_sizeinbase (x, 10);
 258           /* since s is either the number of digits of x or one more,
 259              then x = 10^(s-1) or 10^(s-2) */
 260           if (s == v + 1 || s == v + 2)
 261             {
 262               mpz_div_2exp (x, x, v);
 263               mpz_ui_pow_ui (y, 5, v);
 264               if (mpz_cmp (y, x) == 0) /* Re(log10(x+i*y)) is exactly v/2 */
 265                 {
 266                   /* we reset the precision of Re(ww) so that v can be
 267                      represented exactly */
 268                   mpfr_set_prec (mpc_realref (ww), sizeof(unsigned long)*CHAR_BIT);
 269                   mpfr_set_ui_2exp (mpc_realref (ww), v, -1, GMP_RNDN); /* exact */
 270                   ok = 1;
 271                 }
 272             }
 273           mpz_clear (x);
 274           mpz_clear (y);
 275         }
 276
 277       ok = ok && mpfr_can_round (mpc_imagref (ww), prec-2, GMP_RNDN, GMP_RNDZ,
 278                             MPC_PREC_IM(rop) + (MPC_RND_IM (rnd) == GMP_RNDN));
 279     }
 280
 281   inex_re = mpfr_set (mpc_realref(rop), mpc_realref (ww), MPC_RND_RE (rnd));
 282   inex_im = mpfr_set (mpc_imagref(rop), mpc_imagref (ww), MPC_RND_IM (rnd));
 283   mpfr_clear (w);
 284   mpc_clear (ww);
 285   return MPC_INEX(inex_re, inex_im);
 286 }