digamma.c revision 1.1.1.4
1/* mpfr_digamma -- digamma function of a floating-point number 2 3Copyright 2009-2020 Free Software Foundation, Inc. 4Contributed by the AriC and Caramba projects, INRIA. 5 6This file is part of the GNU MPFR Library. 7 8The GNU MPFR Library is free software; you can redistribute it and/or modify 9it under the terms of the GNU Lesser General Public License as published by 10the Free Software Foundation; either version 3 of the License, or (at your 11option) any later version. 12 13The GNU MPFR Library is distributed in the hope that it will be useful, but 14WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY 15or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public 16License for more details. 17 18You should have received a copy of the GNU Lesser General Public License 19along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see 20https://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc., 2151 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */ 22 23#include "mpfr-impl.h" 24 25/* FIXME: Check that MPFR_GET_EXP can only be called on regular values 26 (in r14025, this is not the case) and that there cannot be integer 27 overflows. */ 28 29/* Put in s an approximation of digamma(x). 30 Assumes x >= 2. 31 Assumes s does not overlap with x. 32 Returns an integer e such that the error is bounded by 2^e ulps 33 of the result s. 34*/ 35static mpfr_exp_t 36mpfr_digamma_approx (mpfr_ptr s, mpfr_srcptr x) 37{ 38 mpfr_prec_t p = MPFR_PREC (s); 39 mpfr_t t, u, invxx; 40 mpfr_exp_t e, exps, f, expu; 41 unsigned long n; 42 43 MPFR_ASSERTN (MPFR_IS_POS (x) && MPFR_GET_EXP (x) >= 2); 44 45 mpfr_init2 (t, p); 46 mpfr_init2 (u, p); 47 mpfr_init2 (invxx, p); 48 49 mpfr_log (s, x, MPFR_RNDN); /* error <= 1/2 ulp */ 50 mpfr_ui_div (t, 1, x, MPFR_RNDN); /* error <= 1/2 ulp */ 51 mpfr_div_2ui (t, t, 1, MPFR_RNDN); /* exact */ 52 mpfr_sub (s, s, t, MPFR_RNDN); 53 /* error <= 1/2 + 1/2*2^(EXP(olds)-EXP(s)) + 1/2*2^(EXP(t)-EXP(s)). 54 For x >= 2, log(x) >= 2*(1/(2x)), thus olds >= 2t, and olds - t >= olds/2, 55 thus 0 <= EXP(olds)-EXP(s) <= 1, and EXP(t)-EXP(s) <= 0, thus 56 error <= 1/2 + 1/2*2 + 1/2 <= 2 ulps. */ 57 e = 2; /* initial error */ 58 mpfr_sqr (invxx, x, MPFR_RNDZ); /* invxx = x^2 * (1 + theta) 59 for |theta| <= 2^(-p) */ 60 mpfr_ui_div (invxx, 1, invxx, MPFR_RNDU); /* invxx = 1/x^2 * (1 + theta)^2 */ 61 62 /* in the following we note err=xxx when the ratio between the approximation 63 and the exact result can be written (1 + theta)^xxx for |theta| <= 2^(-p), 64 following Higham's method */ 65 mpfr_set_ui (t, 1, MPFR_RNDN); /* err = 0 */ 66 for (n = 1;; n++) 67 { 68 /* The main term is Bernoulli[2n]/(2n)/x^(2n) = B[n]/(2n+1)!(2n)/x^(2n) 69 = B[n]*t[n]/(2n) where t[n]/t[n-1] = 1/(2n)/(2n+1)/x^2. */ 70 mpfr_mul (t, t, invxx, MPFR_RNDU); /* err = err + 3 */ 71 mpfr_div_ui (t, t, 2 * n, MPFR_RNDU); /* err = err + 1 */ 72 mpfr_div_ui (t, t, 2 * n + 1, MPFR_RNDU); /* err = err + 1 */ 73 /* we thus have err = 5n here */ 74 mpfr_div_ui (u, t, 2 * n, MPFR_RNDU); /* err = 5n+1 */ 75 mpfr_mul_z (u, u, mpfr_bernoulli_cache(n), MPFR_RNDU);/* err = 5n+2, and the 76 absolute error is bounded 77 by 10n+4 ulp(u) [Rule 11] */ 78 /* if the terms 'u' are decreasing by a factor two at least, 79 then the error coming from those is bounded by 80 sum((10n+4)/2^n, n=1..infinity) = 24 */ 81 exps = MPFR_GET_EXP (s); 82 expu = MPFR_GET_EXP (u); 83 if (expu < exps - (mpfr_exp_t) p) 84 break; 85 mpfr_sub (s, s, u, MPFR_RNDN); /* error <= 24 + n/2 */ 86 if (MPFR_GET_EXP (s) < exps) 87 e <<= exps - MPFR_GET_EXP (s); 88 e ++; /* error in mpfr_sub */ 89 f = 10 * n + 4; 90 while (expu < exps) 91 { 92 f = (1 + f) / 2; 93 expu ++; 94 } 95 e += f; /* total rounding error coming from 'u' term */ 96 } 97 98 mpfr_clear (t); 99 mpfr_clear (u); 100 mpfr_clear (invxx); 101 102 f = 0; 103 while (e > 1) 104 { 105 f++; 106 e = (e + 1) / 2; 107 /* Invariant: 2^f * e does not decrease */ 108 } 109 return f; 110} 111 112/* Use the reflection formula Digamma(1-x) = Digamma(x) + Pi * cot(Pi*x), 113 i.e., Digamma(x) = Digamma(1-x) - Pi * cot(Pi*x). 114 Assume x < 1/2. */ 115static int 116mpfr_digamma_reflection (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode) 117{ 118 mpfr_prec_t p = MPFR_PREC(y) + 10; 119 mpfr_t t, u, v; 120 mpfr_exp_t e1, expv, expx, q; 121 int inex; 122 MPFR_ZIV_DECL (loop); 123 124 MPFR_LOG_FUNC 125 (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec(x), mpfr_log_prec, x, rnd_mode), 126 ("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec(y), mpfr_log_prec, y, inex)); 127 128 /* we want that 1-x is exact with precision q: if 0 < x < 1/2, then 129 q = PREC(x)-EXP(x) is ok, otherwise if -1 <= x < 0, q = PREC(x)-EXP(x) 130 is ok, otherwise for x < -1, PREC(x) is ok if EXP(x) <= PREC(x), 131 otherwise we need EXP(x) */ 132 expx = MPFR_GET_EXP (x); 133 if (expx < 0) 134 q = MPFR_PREC(x) + 1 - expx; 135 else if (expx <= MPFR_PREC(x)) 136 q = MPFR_PREC(x) + 1; 137 else 138 q = expx; 139 MPFR_ASSERTN (q <= MPFR_PREC_MAX); 140 mpfr_init2 (u, q); 141 MPFR_DBGRES(inex = mpfr_ui_sub (u, 1, x, MPFR_RNDN)); 142 MPFR_ASSERTN(inex == 0); 143 144 /* if x is half an integer, cot(Pi*x) = 0, thus Digamma(x) = Digamma(1-x) */ 145 mpfr_mul_2ui (u, u, 1, MPFR_RNDN); 146 inex = mpfr_integer_p (u); 147 mpfr_div_2ui (u, u, 1, MPFR_RNDN); 148 if (inex) 149 { 150 inex = mpfr_digamma (y, u, rnd_mode); 151 goto end; 152 } 153 154 mpfr_init2 (t, p); 155 mpfr_init2 (v, p); 156 157 MPFR_ZIV_INIT (loop, p); 158 for (;;) 159 { 160 mpfr_const_pi (v, MPFR_RNDN); /* v = Pi*(1+theta) for |theta|<=2^(-p) */ 161 mpfr_mul (t, v, x, MPFR_RNDN); /* (1+theta)^2 */ 162 e1 = MPFR_GET_EXP(t) - (mpfr_exp_t) p + 1; /* bound for t: err(t) <= 2^e1 */ 163 mpfr_cot (t, t, MPFR_RNDN); 164 /* cot(t * (1+h)) = cot(t) - theta * (1 + cot(t)^2) with |theta|<=t*h */ 165 if (MPFR_GET_EXP(t) > 0) 166 e1 = e1 + 2 * MPFR_EXP(t) + 1; 167 else 168 e1 = e1 + 1; 169 /* now theta * (1 + cot(t)^2) <= 2^e1 */ 170 e1 += (mpfr_exp_t) p - MPFR_EXP(t); /* error is now 2^e1 ulps */ 171 mpfr_mul (t, t, v, MPFR_RNDN); 172 e1 ++; 173 mpfr_digamma (v, u, MPFR_RNDN); /* error <= 1/2 ulp */ 174 expv = MPFR_GET_EXP (v); 175 mpfr_sub (v, v, t, MPFR_RNDN); 176 if (MPFR_GET_EXP (v) < MPFR_GET_EXP (t)) 177 e1 += MPFR_EXP(t) - MPFR_EXP(v); /* scale error for t wrt new v */ 178 /* now take into account the 1/2 ulp error for v */ 179 if (expv - MPFR_EXP(v) - 1 > e1) 180 e1 = expv - MPFR_EXP(v) - 1; 181 else 182 e1 ++; 183 e1 ++; /* rounding error for mpfr_sub */ 184 if (MPFR_CAN_ROUND (v, p - e1, MPFR_PREC(y), rnd_mode)) 185 break; 186 MPFR_ZIV_NEXT (loop, p); 187 mpfr_set_prec (t, p); 188 mpfr_set_prec (v, p); 189 } 190 MPFR_ZIV_FREE (loop); 191 192 inex = mpfr_set (y, v, rnd_mode); 193 194 mpfr_clear (t); 195 mpfr_clear (v); 196 end: 197 mpfr_clear (u); 198 199 return inex; 200} 201 202/* we have x >= 1/2 here */ 203static int 204mpfr_digamma_positive (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode) 205{ 206 mpfr_prec_t p = MPFR_PREC(y) + 10, q; 207 mpfr_t t, u, x_plus_j; 208 int inex; 209 mpfr_exp_t errt, erru, expt; 210 unsigned long j = 0, min; 211 MPFR_ZIV_DECL (loop); 212 213 MPFR_LOG_FUNC 214 (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec(x), mpfr_log_prec, x, rnd_mode), 215 ("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec(y), mpfr_log_prec, y, inex)); 216 217 /* compute a precision q such that x+1 is exact */ 218 if (MPFR_PREC(x) < MPFR_GET_EXP(x)) 219 q = MPFR_EXP(x); 220 else 221 q = MPFR_PREC(x) + 1; 222 223 /* for very large x, use |digamma(x) - log(x)| < 1/x < 2^(1-EXP(x)) */ 224 if (MPFR_PREC(y) + 10 < MPFR_EXP(x)) 225 { 226 /* this ensures EXP(x) >= 3, thus x >= 4, thus log(x) > 1 */ 227 mpfr_init2 (t, MPFR_PREC(y) + 10); 228 mpfr_log (t, x, MPFR_RNDZ); 229 if (MPFR_CAN_ROUND (t, MPFR_PREC(y) + 10, MPFR_PREC(y), rnd_mode)) 230 { 231 inex = mpfr_set (y, t, rnd_mode); 232 mpfr_clear (t); 233 return inex; 234 } 235 mpfr_clear (t); 236 } 237 238 mpfr_init2 (x_plus_j, q); 239 240 mpfr_init2 (t, p); 241 mpfr_init2 (u, p); 242 MPFR_ZIV_INIT (loop, p); 243 for(;;) 244 { 245 /* Lower bound for x+j in mpfr_digamma_approx call: since the smallest 246 term of the divergent series for Digamma(x) is about exp(-2*Pi*x), and 247 we want it to be less than 2^(-p), this gives x > p*log(2)/(2*Pi) 248 i.e., x >= 0.1103 p. 249 To be safe, we ensure x >= 0.25 * p. 250 */ 251 min = (p + 3) / 4; 252 if (min < 2) 253 min = 2; 254 255 mpfr_set (x_plus_j, x, MPFR_RNDN); 256 mpfr_set_ui (u, 0, MPFR_RNDN); 257 j = 0; 258 while (mpfr_cmp_ui (x_plus_j, min) < 0) 259 { 260 j ++; 261 mpfr_ui_div (t, 1, x_plus_j, MPFR_RNDN); /* err <= 1/2 ulp */ 262 mpfr_add (u, u, t, MPFR_RNDN); 263 inex = mpfr_add_ui (x_plus_j, x_plus_j, 1, MPFR_RNDZ); 264 if (inex != 0) /* we lost one bit */ 265 { 266 q ++; 267 mpfr_prec_round (x_plus_j, q, MPFR_RNDZ); 268 mpfr_nextabove (x_plus_j); 269 } 270 /* since all terms are positive, the error is bounded by j ulps */ 271 } 272 for (erru = 0; j > 1; erru++, j = (j + 1) / 2); 273 errt = mpfr_digamma_approx (t, x_plus_j); 274 expt = MPFR_GET_EXP (t); 275 mpfr_sub (t, t, u, MPFR_RNDN); 276 if (MPFR_GET_EXP (t) < expt) 277 errt += expt - MPFR_EXP(t); 278 /* Warning: if u is zero (which happens when x_plus_j >= min at the 279 beginning of the while loop above), EXP(u) is not defined. 280 In this case we have no error from u. */ 281 if (MPFR_NOTZERO(u) && MPFR_GET_EXP (t) < MPFR_GET_EXP (u)) 282 erru += MPFR_EXP(u) - MPFR_EXP(t); 283 if (errt > erru) 284 errt = errt + 1; 285 else if (errt == erru) 286 errt = errt + 2; 287 else 288 errt = erru + 1; 289 if (MPFR_CAN_ROUND (t, p - errt, MPFR_PREC(y), rnd_mode)) 290 break; 291 MPFR_ZIV_NEXT (loop, p); 292 mpfr_set_prec (t, p); 293 mpfr_set_prec (u, p); 294 } 295 MPFR_ZIV_FREE (loop); 296 inex = mpfr_set (y, t, rnd_mode); 297 mpfr_clear (t); 298 mpfr_clear (u); 299 mpfr_clear (x_plus_j); 300 return inex; 301} 302 303int 304mpfr_digamma (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode) 305{ 306 int inex; 307 MPFR_SAVE_EXPO_DECL (expo); 308 309 MPFR_LOG_FUNC 310 (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec(x), mpfr_log_prec, x, rnd_mode), 311 ("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec(y), mpfr_log_prec, y, inex)); 312 313 if (MPFR_UNLIKELY(MPFR_IS_SINGULAR(x))) 314 { 315 if (MPFR_IS_NAN(x)) 316 { 317 MPFR_SET_NAN(y); 318 MPFR_RET_NAN; 319 } 320 else if (MPFR_IS_INF(x)) 321 { 322 if (MPFR_IS_POS(x)) /* Digamma(+Inf) = +Inf */ 323 { 324 MPFR_SET_SAME_SIGN(y, x); 325 MPFR_SET_INF(y); 326 MPFR_RET(0); 327 } 328 else /* Digamma(-Inf) = NaN */ 329 { 330 MPFR_SET_NAN(y); 331 MPFR_RET_NAN; 332 } 333 } 334 else /* Zero case */ 335 { 336 /* the following works also in case of overlap */ 337 MPFR_SET_INF(y); 338 MPFR_SET_OPPOSITE_SIGN(y, x); 339 MPFR_SET_DIVBY0 (); 340 MPFR_RET(0); 341 } 342 } 343 344 /* Digamma is undefined for negative integers */ 345 if (MPFR_IS_NEG(x) && mpfr_integer_p (x)) 346 { 347 MPFR_SET_NAN(y); 348 MPFR_RET_NAN; 349 } 350 351 /* now x is a normal number */ 352 353 MPFR_SAVE_EXPO_MARK (expo); 354 /* for x very small, we have Digamma(x) = -1/x - gamma + O(x), more precisely 355 -1 < Digamma(x) + 1/x < 0 for -0.2 < x < 0.2, thus: 356 (i) either x is a power of two, then 1/x is exactly representable, and 357 as long as 1/2*ulp(1/x) > 1, we can conclude; 358 (ii) otherwise assume x has <= n bits, and y has <= n+1 bits, then 359 |y + 1/x| >= 2^(-2n) ufp(y), where ufp means unit in first place. 360 Since |Digamma(x) + 1/x| <= 1, if 2^(-2n) ufp(y) >= 2, then 361 |y - Digamma(x)| >= 2^(-2n-1)ufp(y), and rounding -1/x gives the correct result. 362 If x < 2^E, then y > 2^(-E), thus ufp(y) > 2^(-E-1). 363 A sufficient condition is thus EXP(x) <= -2 MAX(PREC(x),PREC(Y)). */ 364 if (MPFR_GET_EXP (x) < -2) 365 { 366 if (MPFR_EXP(x) <= -2 * (mpfr_exp_t) MAX(MPFR_PREC(x), MPFR_PREC(y))) 367 { 368 int signx = MPFR_SIGN(x); 369 inex = mpfr_si_div (y, -1, x, rnd_mode); 370 if (inex == 0) /* x is a power of two */ 371 { /* result always -1/x, except when rounding down */ 372 if (rnd_mode == MPFR_RNDA) 373 rnd_mode = (signx > 0) ? MPFR_RNDD : MPFR_RNDU; 374 if (rnd_mode == MPFR_RNDZ) 375 rnd_mode = (signx > 0) ? MPFR_RNDU : MPFR_RNDD; 376 if (rnd_mode == MPFR_RNDU) 377 inex = 1; 378 else if (rnd_mode == MPFR_RNDD) 379 { 380 mpfr_nextbelow (y); 381 inex = -1; 382 } 383 else /* nearest */ 384 inex = 1; 385 } 386 MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags); 387 goto end; 388 } 389 } 390 391 if (MPFR_IS_NEG(x)) 392 inex = mpfr_digamma_reflection (y, x, rnd_mode); 393 /* if x < 1/2 we use the reflection formula */ 394 else if (MPFR_EXP(x) < 0) 395 inex = mpfr_digamma_reflection (y, x, rnd_mode); 396 else 397 inex = mpfr_digamma_positive (y, x, rnd_mode); 398 399 end: 400 MPFR_SAVE_EXPO_FREE (expo); 401 return mpfr_check_range (y, inex, rnd_mode); 402} 403