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-rw-r--r--Build/source/libs/mpfr/mpfr-src/src/digamma.c378
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diff --git a/Build/source/libs/mpfr/mpfr-src/src/digamma.c b/Build/source/libs/mpfr/mpfr-src/src/digamma.c
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+/* 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);
+}