| /* mpfr_digamma -- digamma function of a floating-point number |
| |
| Copyright 2009-2017 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 |
| http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc., |
| 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */ |
| |
| #include "third_party/mpfr/v3_1_6/src/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); |
| } |
