diff options
Diffstat (limited to 'Build/source/libs/mpfr/mpfr-src/src/digamma.c')
-rw-r--r-- | Build/source/libs/mpfr/mpfr-src/src/digamma.c | 378 |
1 files changed, 378 insertions, 0 deletions
diff --git a/Build/source/libs/mpfr/mpfr-src/src/digamma.c b/Build/source/libs/mpfr/mpfr-src/src/digamma.c new file mode 100644 index 00000000000..1c4e7df4606 --- /dev/null +++ b/Build/source/libs/mpfr/mpfr-src/src/digamma.c @@ -0,0 +1,378 @@ +/* mpfr_digamma -- digamma function of a floating-point number + +Copyright 2009-2015 Free Software Foundation, Inc. +Contributed by the AriC and Caramel projects, INRIA. + +This file is part of the GNU MPFR Library. + +The GNU MPFR Library is free software; you can redistribute it and/or modify +it under the terms of the GNU Lesser General Public License as published by +the Free Software Foundation; either version 3 of the License, or (at your +option) any later version. + +The GNU MPFR Library is distributed in the hope that it will be useful, but +WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY +or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public +License for more details. + +You should have received a copy of the GNU Lesser General Public License +along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see +http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc., +51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */ + +#include "mpfr-impl.h" + +/* Put in s an approximation of digamma(x). + Assumes x >= 2. + Assumes s does not overlap with x. + Returns an integer e such that the error is bounded by 2^e ulps + of the result s. +*/ +static mpfr_exp_t +mpfr_digamma_approx (mpfr_ptr s, mpfr_srcptr x) +{ + mpfr_prec_t p = MPFR_PREC (s); + mpfr_t t, u, invxx; + mpfr_exp_t e, exps, f, expu; + mpz_t *INITIALIZED(B); /* variable B declared as initialized */ + unsigned long n0, n; /* number of allocated B[] */ + + MPFR_ASSERTN(MPFR_IS_POS(x) && (MPFR_EXP(x) >= 2)); + + mpfr_init2 (t, p); + mpfr_init2 (u, p); + mpfr_init2 (invxx, p); + + mpfr_log (s, x, MPFR_RNDN); /* error <= 1/2 ulp */ + mpfr_ui_div (t, 1, x, MPFR_RNDN); /* error <= 1/2 ulp */ + mpfr_div_2exp (t, t, 1, MPFR_RNDN); /* exact */ + mpfr_sub (s, s, t, MPFR_RNDN); + /* error <= 1/2 + 1/2*2^(EXP(olds)-EXP(s)) + 1/2*2^(EXP(t)-EXP(s)). + For x >= 2, log(x) >= 2*(1/(2x)), thus olds >= 2t, and olds - t >= olds/2, + thus 0 <= EXP(olds)-EXP(s) <= 1, and EXP(t)-EXP(s) <= 0, thus + error <= 1/2 + 1/2*2 + 1/2 <= 2 ulps. */ + e = 2; /* initial error */ + mpfr_mul (invxx, x, x, MPFR_RNDZ); /* invxx = x^2 * (1 + theta) + for |theta| <= 2^(-p) */ + mpfr_ui_div (invxx, 1, invxx, MPFR_RNDU); /* invxx = 1/x^2 * (1 + theta)^2 */ + + /* in the following we note err=xxx when the ratio between the approximation + and the exact result can be written (1 + theta)^xxx for |theta| <= 2^(-p), + following Higham's method */ + B = mpfr_bernoulli_internal ((mpz_t *) 0, 0); + mpfr_set_ui (t, 1, MPFR_RNDN); /* err = 0 */ + for (n = 1;; n++) + { + /* compute next Bernoulli number */ + B = mpfr_bernoulli_internal (B, n); + /* The main term is Bernoulli[2n]/(2n)/x^(2n) = B[n]/(2n+1)!(2n)/x^(2n) + = B[n]*t[n]/(2n) where t[n]/t[n-1] = 1/(2n)/(2n+1)/x^2. */ + mpfr_mul (t, t, invxx, MPFR_RNDU); /* err = err + 3 */ + mpfr_div_ui (t, t, 2 * n, MPFR_RNDU); /* err = err + 1 */ + mpfr_div_ui (t, t, 2 * n + 1, MPFR_RNDU); /* err = err + 1 */ + /* we thus have err = 5n here */ + mpfr_div_ui (u, t, 2 * n, MPFR_RNDU); /* err = 5n+1 */ + mpfr_mul_z (u, u, B[n], MPFR_RNDU); /* err = 5n+2, and the + absolute error is bounded + by 10n+4 ulp(u) [Rule 11] */ + /* if the terms 'u' are decreasing by a factor two at least, + then the error coming from those is bounded by + sum((10n+4)/2^n, n=1..infinity) = 24 */ + exps = mpfr_get_exp (s); + expu = mpfr_get_exp (u); + if (expu < exps - (mpfr_exp_t) p) + break; + mpfr_sub (s, s, u, MPFR_RNDN); /* error <= 24 + n/2 */ + if (mpfr_get_exp (s) < exps) + e <<= exps - mpfr_get_exp (s); + e ++; /* error in mpfr_sub */ + f = 10 * n + 4; + while (expu < exps) + { + f = (1 + f) / 2; + expu ++; + } + e += f; /* total rouding error coming from 'u' term */ + } + + n0 = ++n; + while (n--) + mpz_clear (B[n]); + (*__gmp_free_func) (B, n0 * sizeof (mpz_t)); + + mpfr_clear (t); + mpfr_clear (u); + mpfr_clear (invxx); + + f = 0; + while (e > 1) + { + f++; + e = (e + 1) / 2; + /* Invariant: 2^f * e does not decrease */ + } + return f; +} + +/* Use the reflection formula Digamma(1-x) = Digamma(x) + Pi * cot(Pi*x), + i.e., Digamma(x) = Digamma(1-x) - Pi * cot(Pi*x). + Assume x < 1/2. */ +static int +mpfr_digamma_reflection (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode) +{ + mpfr_prec_t p = MPFR_PREC(y) + 10, q; + mpfr_t t, u, v; + mpfr_exp_t e1, expv; + int inex; + MPFR_ZIV_DECL (loop); + + /* we want that 1-x is exact with precision q: if 0 < x < 1/2, then + q = PREC(x)-EXP(x) is ok, otherwise if -1 <= x < 0, q = PREC(x)-EXP(x) + is ok, otherwise for x < -1, PREC(x) is ok if EXP(x) <= PREC(x), + otherwise we need EXP(x) */ + if (MPFR_EXP(x) < 0) + q = MPFR_PREC(x) + 1 - MPFR_EXP(x); + else if (MPFR_EXP(x) <= MPFR_PREC(x)) + q = MPFR_PREC(x) + 1; + else + q = MPFR_EXP(x); + mpfr_init2 (u, q); + MPFR_ASSERTN(mpfr_ui_sub (u, 1, x, MPFR_RNDN) == 0); + + /* if x is half an integer, cot(Pi*x) = 0, thus Digamma(x) = Digamma(1-x) */ + mpfr_mul_2exp (u, u, 1, MPFR_RNDN); + inex = mpfr_integer_p (u); + mpfr_div_2exp (u, u, 1, MPFR_RNDN); + if (inex) + { + inex = mpfr_digamma (y, u, rnd_mode); + goto end; + } + + mpfr_init2 (t, p); + mpfr_init2 (v, p); + + MPFR_ZIV_INIT (loop, p); + for (;;) + { + mpfr_const_pi (v, MPFR_RNDN); /* v = Pi*(1+theta) for |theta|<=2^(-p) */ + mpfr_mul (t, v, x, MPFR_RNDN); /* (1+theta)^2 */ + e1 = MPFR_EXP(t) - (mpfr_exp_t) p + 1; /* bound for t: err(t) <= 2^e1 */ + mpfr_cot (t, t, MPFR_RNDN); + /* cot(t * (1+h)) = cot(t) - theta * (1 + cot(t)^2) with |theta|<=t*h */ + if (MPFR_EXP(t) > 0) + e1 = e1 + 2 * MPFR_EXP(t) + 1; + else + e1 = e1 + 1; + /* now theta * (1 + cot(t)^2) <= 2^e1 */ + e1 += (mpfr_exp_t) p - MPFR_EXP(t); /* error is now 2^e1 ulps */ + mpfr_mul (t, t, v, MPFR_RNDN); + e1 ++; + mpfr_digamma (v, u, MPFR_RNDN); /* error <= 1/2 ulp */ + expv = MPFR_EXP(v); + mpfr_sub (v, v, t, MPFR_RNDN); + if (MPFR_EXP(v) < MPFR_EXP(t)) + e1 += MPFR_EXP(t) - MPFR_EXP(v); /* scale error for t wrt new v */ + /* now take into account the 1/2 ulp error for v */ + if (expv - MPFR_EXP(v) - 1 > e1) + e1 = expv - MPFR_EXP(v) - 1; + else + e1 ++; + e1 ++; /* rounding error for mpfr_sub */ + if (MPFR_CAN_ROUND (v, p - e1, MPFR_PREC(y), rnd_mode)) + break; + MPFR_ZIV_NEXT (loop, p); + mpfr_set_prec (t, p); + mpfr_set_prec (v, p); + } + MPFR_ZIV_FREE (loop); + + inex = mpfr_set (y, v, rnd_mode); + + mpfr_clear (t); + mpfr_clear (v); + end: + mpfr_clear (u); + + return inex; +} + +/* we have x >= 1/2 here */ +static int +mpfr_digamma_positive (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode) +{ + mpfr_prec_t p = MPFR_PREC(y) + 10, q; + mpfr_t t, u, x_plus_j; + int inex; + mpfr_exp_t errt, erru, expt; + unsigned long j = 0, min; + MPFR_ZIV_DECL (loop); + + /* compute a precision q such that x+1 is exact */ + if (MPFR_PREC(x) < MPFR_EXP(x)) + q = MPFR_EXP(x); + else + q = MPFR_PREC(x) + 1; + mpfr_init2 (x_plus_j, q); + + mpfr_init2 (t, p); + mpfr_init2 (u, p); + MPFR_ZIV_INIT (loop, p); + for(;;) + { + /* Lower bound for x+j in mpfr_digamma_approx call: since the smallest + term of the divergent series for Digamma(x) is about exp(-2*Pi*x), and + we want it to be less than 2^(-p), this gives x > p*log(2)/(2*Pi) + i.e., x >= 0.1103 p. + To be safe, we ensure x >= 0.25 * p. + */ + min = (p + 3) / 4; + if (min < 2) + min = 2; + + mpfr_set (x_plus_j, x, MPFR_RNDN); + mpfr_set_ui (u, 0, MPFR_RNDN); + j = 0; + while (mpfr_cmp_ui (x_plus_j, min) < 0) + { + j ++; + mpfr_ui_div (t, 1, x_plus_j, MPFR_RNDN); /* err <= 1/2 ulp */ + mpfr_add (u, u, t, MPFR_RNDN); + inex = mpfr_add_ui (x_plus_j, x_plus_j, 1, MPFR_RNDZ); + if (inex != 0) /* we lost one bit */ + { + q ++; + mpfr_prec_round (x_plus_j, q, MPFR_RNDZ); + mpfr_nextabove (x_plus_j); + } + /* since all terms are positive, the error is bounded by j ulps */ + } + for (erru = 0; j > 1; erru++, j = (j + 1) / 2); + errt = mpfr_digamma_approx (t, x_plus_j); + expt = MPFR_EXP(t); + mpfr_sub (t, t, u, MPFR_RNDN); + if (MPFR_EXP(t) < expt) + errt += expt - MPFR_EXP(t); + if (MPFR_EXP(t) < MPFR_EXP(u)) + erru += MPFR_EXP(u) - MPFR_EXP(t); + if (errt > erru) + errt = errt + 1; + else if (errt == erru) + errt = errt + 2; + else + errt = erru + 1; + if (MPFR_CAN_ROUND (t, p - errt, MPFR_PREC(y), rnd_mode)) + break; + MPFR_ZIV_NEXT (loop, p); + mpfr_set_prec (t, p); + mpfr_set_prec (u, p); + } + MPFR_ZIV_FREE (loop); + inex = mpfr_set (y, t, rnd_mode); + mpfr_clear (t); + mpfr_clear (u); + mpfr_clear (x_plus_j); + return inex; +} + +int +mpfr_digamma (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode) +{ + int inex; + MPFR_SAVE_EXPO_DECL (expo); + + MPFR_LOG_FUNC + (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec(x), mpfr_log_prec, x, rnd_mode), + ("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec(y), mpfr_log_prec, y, inex)); + + + if (MPFR_UNLIKELY(MPFR_IS_SINGULAR(x))) + { + if (MPFR_IS_NAN(x)) + { + MPFR_SET_NAN(y); + MPFR_RET_NAN; + } + else if (MPFR_IS_INF(x)) + { + if (MPFR_IS_POS(x)) /* Digamma(+Inf) = +Inf */ + { + MPFR_SET_SAME_SIGN(y, x); + MPFR_SET_INF(y); + MPFR_RET(0); + } + else /* Digamma(-Inf) = NaN */ + { + MPFR_SET_NAN(y); + MPFR_RET_NAN; + } + } + else /* Zero case */ + { + /* the following works also in case of overlap */ + MPFR_SET_INF(y); + MPFR_SET_OPPOSITE_SIGN(y, x); + mpfr_set_divby0 (); + MPFR_RET(0); + } + } + + /* Digamma is undefined for negative integers */ + if (MPFR_IS_NEG(x) && mpfr_integer_p (x)) + { + MPFR_SET_NAN(y); + MPFR_RET_NAN; + } + + /* now x is a normal number */ + + MPFR_SAVE_EXPO_MARK (expo); + /* for x very small, we have Digamma(x) = -1/x - gamma + O(x), more precisely + -1 < Digamma(x) + 1/x < 0 for -0.2 < x < 0.2, thus: + (i) either x is a power of two, then 1/x is exactly representable, and + as long as 1/2*ulp(1/x) > 1, we can conclude; + (ii) otherwise assume x has <= n bits, and y has <= n+1 bits, then + |y + 1/x| >= 2^(-2n) ufp(y), where ufp means unit in first place. + Since |Digamma(x) + 1/x| <= 1, if 2^(-2n) ufp(y) >= 2, then + |y - Digamma(x)| >= 2^(-2n-1)ufp(y), and rounding -1/x gives the correct result. + If x < 2^E, then y > 2^(-E), thus ufp(y) > 2^(-E-1). + A sufficient condition is thus EXP(x) <= -2 MAX(PREC(x),PREC(Y)). */ + if (MPFR_EXP(x) < -2) + { + if (MPFR_EXP(x) <= -2 * (mpfr_exp_t) MAX(MPFR_PREC(x), MPFR_PREC(y))) + { + int signx = MPFR_SIGN(x); + inex = mpfr_si_div (y, -1, x, rnd_mode); + if (inex == 0) /* x is a power of two */ + { /* result always -1/x, except when rounding down */ + if (rnd_mode == MPFR_RNDA) + rnd_mode = (signx > 0) ? MPFR_RNDD : MPFR_RNDU; + if (rnd_mode == MPFR_RNDZ) + rnd_mode = (signx > 0) ? MPFR_RNDU : MPFR_RNDD; + if (rnd_mode == MPFR_RNDU) + inex = 1; + else if (rnd_mode == MPFR_RNDD) + { + mpfr_nextbelow (y); + inex = -1; + } + else /* nearest */ + inex = 1; + } + MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags); + goto end; + } + } + + if (MPFR_IS_NEG(x)) + inex = mpfr_digamma_reflection (y, x, rnd_mode); + /* if x < 1/2 we use the reflection formula */ + else if (MPFR_EXP(x) < 0) + inex = mpfr_digamma_reflection (y, x, rnd_mode); + else + inex = mpfr_digamma_positive (y, x, rnd_mode); + + end: + MPFR_SAVE_EXPO_FREE (expo); + return mpfr_check_range (y, inex, rnd_mode); +} |