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-rw-r--r--Build/source/libs/mpfr/mpfr-src/src/digamma.c402
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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
deleted file mode 100644
index e9e8389b115..00000000000
--- a/Build/source/libs/mpfr/mpfr-src/src/digamma.c
+++ /dev/null
@@ -1,402 +0,0 @@
-/* mpfr_digamma -- digamma function of a floating-point number
-
-Copyright 2009-2020 Free Software Foundation, Inc.
-Contributed by the AriC and Caramba 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
-https://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"
-
-/* FIXME: Check that MPFR_GET_EXP can only be called on regular values
- (in r14025, this is not the case) and that there cannot be integer
- overflows. */
-
-/* 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;
- unsigned long n;
-
- MPFR_ASSERTN (MPFR_IS_POS (x) && MPFR_GET_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_2ui (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_sqr (invxx, 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 */
- mpfr_set_ui (t, 1, MPFR_RNDN); /* err = 0 */
- for (n = 1;; 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, mpfr_bernoulli_cache(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 rounding error coming from 'u' term */
- }
-
- 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;
- mpfr_t t, u, v;
- mpfr_exp_t e1, expv, expx, q;
- int inex;
- MPFR_ZIV_DECL (loop);
-
- 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));
-
- /* 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) */
- expx = MPFR_GET_EXP (x);
- if (expx < 0)
- q = MPFR_PREC(x) + 1 - expx;
- else if (expx <= MPFR_PREC(x))
- q = MPFR_PREC(x) + 1;
- else
- q = expx;
- MPFR_ASSERTN (q <= MPFR_PREC_MAX);
- mpfr_init2 (u, q);
- MPFR_DBGRES(inex = mpfr_ui_sub (u, 1, x, MPFR_RNDN));
- MPFR_ASSERTN(inex == 0);
-
- /* if x is half an integer, cot(Pi*x) = 0, thus Digamma(x) = Digamma(1-x) */
- mpfr_mul_2ui (u, u, 1, MPFR_RNDN);
- inex = mpfr_integer_p (u);
- mpfr_div_2ui (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_GET_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_GET_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_GET_EXP (v);
- mpfr_sub (v, v, t, MPFR_RNDN);
- if (MPFR_GET_EXP (v) < MPFR_GET_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);
-
- 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));
-
- /* compute a precision q such that x+1 is exact */
- if (MPFR_PREC(x) < MPFR_GET_EXP(x))
- q = MPFR_EXP(x);
- else
- q = MPFR_PREC(x) + 1;
-
- /* for very large x, use |digamma(x) - log(x)| < 1/x < 2^(1-EXP(x)) */
- if (MPFR_PREC(y) + 10 < MPFR_EXP(x))
- {
- /* this ensures EXP(x) >= 3, thus x >= 4, thus log(x) > 1 */
- mpfr_init2 (t, MPFR_PREC(y) + 10);
- mpfr_log (t, x, MPFR_RNDZ);
- if (MPFR_CAN_ROUND (t, MPFR_PREC(y) + 10, MPFR_PREC(y), rnd_mode))
- {
- inex = mpfr_set (y, t, rnd_mode);
- mpfr_clear (t);
- return inex;
- }
- mpfr_clear (t);
- }
-
- 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_GET_EXP (t);
- mpfr_sub (t, t, u, MPFR_RNDN);
- if (MPFR_GET_EXP (t) < expt)
- errt += expt - MPFR_EXP(t);
- /* Warning: if u is zero (which happens when x_plus_j >= min at the
- beginning of the while loop above), EXP(u) is not defined.
- In this case we have no error from u. */
- if (MPFR_NOTZERO(u) && MPFR_GET_EXP (t) < MPFR_GET_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_GET_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);
-}