1/* mpc_log -- Take the logarithm of a complex number. 2 3Copyright (C) 2008, 2009, 2010, 2011, 2012 INRIA 4 5This file is part of GNU MPC. 6 7GNU MPC is free software; you can redistribute it and/or modify it under 8the terms of the GNU Lesser General Public License as published by the 9Free Software Foundation; either version 3 of the License, or (at your 10option) any later version. 11 12GNU MPC is distributed in the hope that it will be useful, but WITHOUT ANY 13WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS 14FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for 15more details. 16 17You should have received a copy of the GNU Lesser General Public License 18along with this program. If not, see http://www.gnu.org/licenses/ . 19*/ 20 21#include <stdio.h> /* for MPC_ASSERT */ 22#include "mpc-impl.h" 23 24int 25mpc_log (mpc_ptr rop, mpc_srcptr op, mpc_rnd_t rnd){ 26 int ok, underflow = 0; 27 mpfr_srcptr x, y; 28 mpfr_t v, w; 29 mpfr_prec_t prec; 30 int loops; 31 int re_cmp, im_cmp; 32 int inex_re, inex_im; 33 int err; 34 mpfr_exp_t expw; 35 int sgnw; 36 37 /* special values: NaN and infinities */ 38 if (!mpc_fin_p (op)) { 39 if (mpfr_nan_p (mpc_realref (op))) { 40 if (mpfr_inf_p (mpc_imagref (op))) 41 mpfr_set_inf (mpc_realref (rop), +1); 42 else 43 mpfr_set_nan (mpc_realref (rop)); 44 mpfr_set_nan (mpc_imagref (rop)); 45 inex_im = 0; /* Inf/NaN is exact */ 46 } 47 else if (mpfr_nan_p (mpc_imagref (op))) { 48 if (mpfr_inf_p (mpc_realref (op))) 49 mpfr_set_inf (mpc_realref (rop), +1); 50 else 51 mpfr_set_nan (mpc_realref (rop)); 52 mpfr_set_nan (mpc_imagref (rop)); 53 inex_im = 0; /* Inf/NaN is exact */ 54 } 55 else /* We have an infinity in at least one part. */ { 56 inex_im = mpfr_atan2 (mpc_imagref (rop), mpc_imagref (op), mpc_realref (op), 57 MPC_RND_IM (rnd)); 58 mpfr_set_inf (mpc_realref (rop), +1); 59 } 60 return MPC_INEX(0, inex_im); 61 } 62 63 /* special cases: real and purely imaginary numbers */ 64 re_cmp = mpfr_cmp_ui (mpc_realref (op), 0); 65 im_cmp = mpfr_cmp_ui (mpc_imagref (op), 0); 66 if (im_cmp == 0) { 67 if (re_cmp == 0) { 68 inex_im = mpfr_atan2 (mpc_imagref (rop), mpc_imagref (op), mpc_realref (op), 69 MPC_RND_IM (rnd)); 70 mpfr_set_inf (mpc_realref (rop), -1); 71 inex_re = 0; /* -Inf is exact */ 72 } 73 else if (re_cmp > 0) { 74 inex_re = mpfr_log (mpc_realref (rop), mpc_realref (op), MPC_RND_RE (rnd)); 75 inex_im = mpfr_set (mpc_imagref (rop), mpc_imagref (op), MPC_RND_IM (rnd)); 76 } 77 else { 78 /* op = x + 0*y; let w = -x = |x| */ 79 int negative_zero; 80 mpfr_rnd_t rnd_im; 81 82 negative_zero = mpfr_signbit (mpc_imagref (op)); 83 if (negative_zero) 84 rnd_im = INV_RND (MPC_RND_IM (rnd)); 85 else 86 rnd_im = MPC_RND_IM (rnd); 87 w [0] = *mpc_realref (op); 88 MPFR_CHANGE_SIGN (w); 89 inex_re = mpfr_log (mpc_realref (rop), w, MPC_RND_RE (rnd)); 90 inex_im = mpfr_const_pi (mpc_imagref (rop), rnd_im); 91 if (negative_zero) { 92 mpc_conj (rop, rop, MPC_RNDNN); 93 inex_im = -inex_im; 94 } 95 } 96 return MPC_INEX(inex_re, inex_im); 97 } 98 else if (re_cmp == 0) { 99 if (im_cmp > 0) { 100 inex_re = mpfr_log (mpc_realref (rop), mpc_imagref (op), MPC_RND_RE (rnd)); 101 inex_im = mpfr_const_pi (mpc_imagref (rop), MPC_RND_IM (rnd)); 102 /* division by 2 does not change the ternary flag */ 103 mpfr_div_2ui (mpc_imagref (rop), mpc_imagref (rop), 1, GMP_RNDN); 104 } 105 else { 106 w [0] = *mpc_imagref (op); 107 MPFR_CHANGE_SIGN (w); 108 inex_re = mpfr_log (mpc_realref (rop), w, MPC_RND_RE (rnd)); 109 inex_im = mpfr_const_pi (mpc_imagref (rop), INV_RND (MPC_RND_IM (rnd))); 110 /* division by 2 does not change the ternary flag */ 111 mpfr_div_2ui (mpc_imagref (rop), mpc_imagref (rop), 1, GMP_RNDN); 112 mpfr_neg (mpc_imagref (rop), mpc_imagref (rop), GMP_RNDN); 113 inex_im = -inex_im; /* negate the ternary flag */ 114 } 115 return MPC_INEX(inex_re, inex_im); 116 } 117 118 prec = MPC_PREC_RE(rop); 119 mpfr_init2 (w, 2); 120 /* let op = x + iy; log = 1/2 log (x^2 + y^2) + i atan2 (y, x) */ 121 /* loop for the real part: 1/2 log (x^2 + y^2), fast, but unsafe */ 122 /* implementation */ 123 ok = 0; 124 for (loops = 1; !ok && loops <= 2; loops++) { 125 prec += mpc_ceil_log2 (prec) + 4; 126 mpfr_set_prec (w, prec); 127 128 mpc_abs (w, op, GMP_RNDN); 129 /* error 0.5 ulp */ 130 if (mpfr_inf_p (w)) 131 /* intermediate overflow; the logarithm may be representable. 132 Intermediate underflow is impossible. */ 133 break; 134 135 mpfr_log (w, w, GMP_RNDN); 136 /* generic error of log: (2^(- exp(w)) + 0.5) ulp */ 137 138 if (mpfr_zero_p (w)) 139 /* impossible to round, switch to second algorithm */ 140 break; 141 142 err = MPC_MAX (-mpfr_get_exp (w), 0) + 1; 143 /* number of lost digits */ 144 ok = mpfr_can_round (w, prec - err, GMP_RNDN, GMP_RNDZ, 145 mpfr_get_prec (mpc_realref (rop)) + (MPC_RND_RE (rnd) == GMP_RNDN)); 146 } 147 148 if (!ok) { 149 prec = MPC_PREC_RE(rop); 150 mpfr_init2 (v, 2); 151 /* compute 1/2 log (x^2 + y^2) = log |x| + 1/2 * log (1 + (y/x)^2) 152 if |x| >= |y|; otherwise, exchange x and y */ 153 if (mpfr_cmpabs (mpc_realref (op), mpc_imagref (op)) >= 0) { 154 x = mpc_realref (op); 155 y = mpc_imagref (op); 156 } 157 else { 158 x = mpc_imagref (op); 159 y = mpc_realref (op); 160 } 161 162 do { 163 prec += mpc_ceil_log2 (prec) + 4; 164 mpfr_set_prec (v, prec); 165 mpfr_set_prec (w, prec); 166 167 mpfr_div (v, y, x, GMP_RNDD); /* error 1 ulp */ 168 mpfr_sqr (v, v, GMP_RNDD); 169 /* generic error of multiplication: 170 1 + 2*1*(2+1*2^(1-prec)) <= 5.0625 since prec >= 6 */ 171 mpfr_log1p (v, v, GMP_RNDD); 172 /* error 1 + 4*5.0625 = 21.25 , see algorithms.tex */ 173 mpfr_div_2ui (v, v, 1, GMP_RNDD); 174 /* If the result is 0, then there has been an underflow somewhere. */ 175 176 mpfr_abs (w, x, GMP_RNDN); /* exact */ 177 mpfr_log (w, w, GMP_RNDN); /* error 0.5 ulp */ 178 expw = mpfr_get_exp (w); 179 sgnw = mpfr_signbit (w); 180 181 mpfr_add (w, w, v, GMP_RNDN); 182 if (!sgnw) /* v is positive, so no cancellation; 183 error 22.25 ulp; error counts lost bits */ 184 err = 5; 185 else 186 err = MPC_MAX (5 + mpfr_get_exp (v), 187 /* 21.25 ulp (v) rewritten in ulp (result, now in w) */ 188 -1 + expw - mpfr_get_exp (w) 189 /* 0.5 ulp (previous w), rewritten in ulp (result) */ 190 ) + 2; 191 192 /* handle one special case: |x|=1, and (y/x)^2 underflows; 193 then 1/2*log(x^2+y^2) \approx 1/2*y^2 also underflows. */ 194 if ( (mpfr_cmp_si (x, -1) == 0 || mpfr_cmp_ui (x, 1) == 0) 195 && mpfr_zero_p (w)) 196 underflow = 1; 197 198 } while (!underflow && 199 !mpfr_can_round (w, prec - err, GMP_RNDN, GMP_RNDZ, 200 mpfr_get_prec (mpc_realref (rop)) + (MPC_RND_RE (rnd) == GMP_RNDN))); 201 mpfr_clear (v); 202 } 203 204 /* imaginary part */ 205 inex_im = mpfr_atan2 (mpc_imagref (rop), mpc_imagref (op), mpc_realref (op), 206 MPC_RND_IM (rnd)); 207 208 /* set the real part; cannot be done before if rop==op */ 209 if (underflow) 210 /* create underflow in result */ 211 inex_re = mpfr_set_ui_2exp (mpc_realref (rop), 1, 212 mpfr_get_emin_min () - 2, MPC_RND_RE (rnd)); 213 else 214 inex_re = mpfr_set (mpc_realref (rop), w, MPC_RND_RE (rnd)); 215 mpfr_clear (w); 216 return MPC_INEX(inex_re, inex_im); 217} 218