1 /* mpfr_erf -- error function of a floating-point number 2 3 Copyright 2001, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc. 4 Contributed by the AriC and Caramel projects, INRIA. 5 6 This file is part of the GNU MPFR Library. 7 8 The GNU MPFR Library is free software; you can redistribute it and/or modify 9 it under the terms of the GNU Lesser General Public License as published by 10 the Free Software Foundation; either version 3 of the License, or (at your 11 option) any later version. 12 13 The GNU MPFR Library is distributed in the hope that it will be useful, but 14 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY 15 or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public 16 License for more details. 17 18 You should have received a copy of the GNU Lesser General Public License 19 along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see 20 http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc., 21 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */ 22 23 #define MPFR_NEED_LONGLONG_H 24 #include "mpfr-impl.h" 25 26 #define EXP1 2.71828182845904523536 /* exp(1) */ 27 28 static int mpfr_erf_0 (mpfr_ptr, mpfr_srcptr, double, mpfr_rnd_t); 29 30 int 31 mpfr_erf (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode) 32 { 33 mpfr_t xf; 34 int inex, large; 35 MPFR_SAVE_EXPO_DECL (expo); 36 37 MPFR_LOG_FUNC 38 (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode), 39 ("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y, inex)); 40 41 if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x))) 42 { 43 if (MPFR_IS_NAN (x)) 44 { 45 MPFR_SET_NAN (y); 46 MPFR_RET_NAN; 47 } 48 else if (MPFR_IS_INF (x)) /* erf(+inf) = +1, erf(-inf) = -1 */ 49 return mpfr_set_si (y, MPFR_INT_SIGN (x), MPFR_RNDN); 50 else /* erf(+0) = +0, erf(-0) = -0 */ 51 { 52 MPFR_ASSERTD (MPFR_IS_ZERO (x)); 53 return mpfr_set (y, x, MPFR_RNDN); /* should keep the sign of x */ 54 } 55 } 56 57 /* now x is neither NaN, Inf nor 0 */ 58 59 /* first try expansion at x=0 when x is small, or asymptotic expansion 60 where x is large */ 61 62 MPFR_SAVE_EXPO_MARK (expo); 63 64 /* around x=0, we have erf(x) = 2x/sqrt(Pi) (1 - x^2/3 + ...), 65 with 1 - x^2/3 <= sqrt(Pi)*erf(x)/2/x <= 1 for x >= 0. This means that 66 if x^2/3 < 2^(-PREC(y)-1) we can decide of the correct rounding, 67 unless we have a worst-case for 2x/sqrt(Pi). */ 68 if (MPFR_EXP(x) < - (mpfr_exp_t) (MPFR_PREC(y) / 2)) 69 { 70 /* we use 2x/sqrt(Pi) (1 - x^2/3) <= erf(x) <= 2x/sqrt(Pi) for x > 0 71 and 2x/sqrt(Pi) <= erf(x) <= 2x/sqrt(Pi) (1 - x^2/3) for x < 0. 72 In both cases |2x/sqrt(Pi) (1 - x^2/3)| <= |erf(x)| <= |2x/sqrt(Pi)|. 73 We will compute l and h such that l <= |2x/sqrt(Pi) (1 - x^2/3)| 74 and |2x/sqrt(Pi)| <= h. If l and h round to the same value to 75 precision PREC(y) and rounding rnd_mode, then we are done. */ 76 mpfr_t l, h; /* lower and upper bounds for erf(x) */ 77 int ok, inex2; 78 79 mpfr_init2 (l, MPFR_PREC(y) + 17); 80 mpfr_init2 (h, MPFR_PREC(y) + 17); 81 /* first compute l */ 82 mpfr_mul (l, x, x, MPFR_RNDU); 83 mpfr_div_ui (l, l, 3, MPFR_RNDU); /* upper bound on x^2/3 */ 84 mpfr_ui_sub (l, 1, l, MPFR_RNDZ); /* lower bound on 1 - x^2/3 */ 85 mpfr_const_pi (h, MPFR_RNDU); /* upper bound of Pi */ 86 mpfr_sqrt (h, h, MPFR_RNDU); /* upper bound on sqrt(Pi) */ 87 mpfr_div (l, l, h, MPFR_RNDZ); /* lower bound on 1/sqrt(Pi) (1 - x^2/3) */ 88 mpfr_mul_2ui (l, l, 1, MPFR_RNDZ); /* 2/sqrt(Pi) (1 - x^2/3) */ 89 mpfr_mul (l, l, x, MPFR_RNDZ); /* |l| is a lower bound on 90 |2x/sqrt(Pi) (1 - x^2/3)| */ 91 /* now compute h */ 92 mpfr_const_pi (h, MPFR_RNDD); /* lower bound on Pi */ 93 mpfr_sqrt (h, h, MPFR_RNDD); /* lower bound on sqrt(Pi) */ 94 mpfr_div_2ui (h, h, 1, MPFR_RNDD); /* lower bound on sqrt(Pi)/2 */ 95 /* since sqrt(Pi)/2 < 1, the following should not underflow */ 96 mpfr_div (h, x, h, MPFR_IS_POS(x) ? MPFR_RNDU : MPFR_RNDD); 97 /* round l and h to precision PREC(y) */ 98 inex = mpfr_prec_round (l, MPFR_PREC(y), rnd_mode); 99 inex2 = mpfr_prec_round (h, MPFR_PREC(y), rnd_mode); 100 /* Caution: we also need inex=inex2 (inex might be 0). */ 101 ok = SAME_SIGN (inex, inex2) && mpfr_cmp (l, h) == 0; 102 if (ok) 103 mpfr_set (y, h, rnd_mode); 104 mpfr_clear (l); 105 mpfr_clear (h); 106 if (ok) 107 goto end; 108 /* this test can still fail for small precision, for example 109 for x=-0.100E-2 with a target precision of 3 bits, since 110 the error term x^2/3 is not that small. */ 111 } 112 113 mpfr_init2 (xf, 53); 114 mpfr_const_log2 (xf, MPFR_RNDU); 115 mpfr_div (xf, x, xf, MPFR_RNDZ); /* round to zero ensures we get a lower 116 bound of |x/log(2)| */ 117 mpfr_mul (xf, xf, x, MPFR_RNDZ); 118 large = mpfr_cmp_ui (xf, MPFR_PREC (y) + 1) > 0; 119 mpfr_clear (xf); 120 121 /* when x goes to infinity, we have erf(x) = 1 - 1/sqrt(Pi)/exp(x^2)/x + ... 122 and |erf(x) - 1| <= exp(-x^2) is true for any x >= 0, thus if 123 exp(-x^2) < 2^(-PREC(y)-1) the result is 1 or 1-epsilon. 124 This rewrites as x^2/log(2) > p+1. */ 125 if (MPFR_UNLIKELY (large)) 126 /* |erf x| = 1 or 1- */ 127 { 128 mpfr_rnd_t rnd2 = MPFR_IS_POS (x) ? rnd_mode : MPFR_INVERT_RND(rnd_mode); 129 if (rnd2 == MPFR_RNDN || rnd2 == MPFR_RNDU || rnd2 == MPFR_RNDA) 130 { 131 inex = MPFR_INT_SIGN (x); 132 mpfr_set_si (y, inex, rnd2); 133 } 134 else /* round to zero */ 135 { 136 inex = -MPFR_INT_SIGN (x); 137 mpfr_setmax (y, 0); /* warning: setmax keeps the old sign of y */ 138 MPFR_SET_SAME_SIGN (y, x); 139 } 140 } 141 else /* use Taylor */ 142 { 143 double xf2; 144 145 /* FIXME: get rid of doubles/mpfr_get_d here */ 146 xf2 = mpfr_get_d (x, MPFR_RNDN); 147 xf2 = xf2 * xf2; /* xf2 ~ x^2 */ 148 inex = mpfr_erf_0 (y, x, xf2, rnd_mode); 149 } 150 151 end: 152 MPFR_SAVE_EXPO_FREE (expo); 153 return mpfr_check_range (y, inex, rnd_mode); 154 } 155 156 /* return x*2^e */ 157 static double 158 mul_2exp (double x, mpfr_exp_t e) 159 { 160 if (e > 0) 161 { 162 while (e--) 163 x *= 2.0; 164 } 165 else 166 { 167 while (e++) 168 x /= 2.0; 169 } 170 171 return x; 172 } 173 174 /* evaluates erf(x) using the expansion at x=0: 175 176 erf(x) = 2/sqrt(Pi) * sum((-1)^k*x^(2k+1)/k!/(2k+1), k=0..infinity) 177 178 Assumes x is neither NaN nor infinite nor zero. 179 Assumes also that e*x^2 <= n (target precision). 180 */ 181 static int 182 mpfr_erf_0 (mpfr_ptr res, mpfr_srcptr x, double xf2, mpfr_rnd_t rnd_mode) 183 { 184 mpfr_prec_t n, m; 185 mpfr_exp_t nuk, sigmak; 186 double tauk; 187 mpfr_t y, s, t, u; 188 unsigned int k; 189 int log2tauk; 190 int inex; 191 MPFR_ZIV_DECL (loop); 192 193 n = MPFR_PREC (res); /* target precision */ 194 195 /* initial working precision */ 196 m = n + (mpfr_prec_t) (xf2 / LOG2) + 8 + MPFR_INT_CEIL_LOG2 (n); 197 198 mpfr_init2 (y, m); 199 mpfr_init2 (s, m); 200 mpfr_init2 (t, m); 201 mpfr_init2 (u, m); 202 203 MPFR_ZIV_INIT (loop, m); 204 for (;;) 205 { 206 mpfr_mul (y, x, x, MPFR_RNDU); /* err <= 1 ulp */ 207 mpfr_set_ui (s, 1, MPFR_RNDN); 208 mpfr_set_ui (t, 1, MPFR_RNDN); 209 tauk = 0.0; 210 211 for (k = 1; ; k++) 212 { 213 mpfr_mul (t, y, t, MPFR_RNDU); 214 mpfr_div_ui (t, t, k, MPFR_RNDU); 215 mpfr_div_ui (u, t, 2 * k + 1, MPFR_RNDU); 216 sigmak = MPFR_GET_EXP (s); 217 if (k % 2) 218 mpfr_sub (s, s, u, MPFR_RNDN); 219 else 220 mpfr_add (s, s, u, MPFR_RNDN); 221 sigmak -= MPFR_GET_EXP(s); 222 nuk = MPFR_GET_EXP(u) - MPFR_GET_EXP(s); 223 224 if ((nuk < - (mpfr_exp_t) m) && ((double) k >= xf2)) 225 break; 226 227 /* tauk <- 1/2 + tauk * 2^sigmak + (1+8k)*2^nuk */ 228 tauk = 0.5 + mul_2exp (tauk, sigmak) 229 + mul_2exp (1.0 + 8.0 * (double) k, nuk); 230 } 231 232 mpfr_mul (s, x, s, MPFR_RNDU); 233 MPFR_SET_EXP (s, MPFR_GET_EXP (s) + 1); 234 235 mpfr_const_pi (t, MPFR_RNDZ); 236 mpfr_sqrt (t, t, MPFR_RNDZ); 237 mpfr_div (s, s, t, MPFR_RNDN); 238 tauk = 4.0 * tauk + 11.0; /* final ulp-error on s */ 239 log2tauk = __gmpfr_ceil_log2 (tauk); 240 241 if (MPFR_LIKELY (MPFR_CAN_ROUND (s, m - log2tauk, n, rnd_mode))) 242 break; 243 244 /* Actualisation of the precision */ 245 MPFR_ZIV_NEXT (loop, m); 246 mpfr_set_prec (y, m); 247 mpfr_set_prec (s, m); 248 mpfr_set_prec (t, m); 249 mpfr_set_prec (u, m); 250 251 } 252 MPFR_ZIV_FREE (loop); 253 254 inex = mpfr_set (res, s, rnd_mode); 255 256 mpfr_clear (y); 257 mpfr_clear (t); 258 mpfr_clear (u); 259 mpfr_clear (s); 260 261 return inex; 262 } 263