--- /dev/null
+/* mpfr_li2 -- Dilogarithm.
+
+Copyright 2007, 2008, 2009 Free Software Foundation, Inc.
+Contributed by the Arenaire and Cacao 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 2.1 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.LIB. If not, write to
+the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston,
+MA 02110-1301, USA. */
+
+#define MPFR_NEED_LONGLONG_H
+#include "mpfr-impl.h"
+
+/* assuming B[0]...B[2(n-1)] are computed, computes and stores B[2n]*(2n+1)!
+
+ t/(exp(t)-1) = sum(B[j]*t^j/j!, j=0..infinity)
+ thus t = (exp(t)-1) * sum(B[j]*t^j/j!, n=0..infinity).
+ Taking the coefficient of degree n+1 > 1, we get:
+ 0 = sum(1/(n+1-k)!*B[k]/k!, k=0..n)
+ which gives:
+ B[n] = -sum(binomial(n+1,k)*B[k], k=0..n-1)/(n+1).
+
+ Let C[n] = B[n]*(n+1)!.
+ Then C[n] = -sum(binomial(n+1,k)*C[k]*n!/(k+1)!, k=0..n-1),
+ which proves that the C[n] are integers.
+*/
+static mpz_t *
+bernoulli (mpz_t * b, unsigned long n)
+{
+ if (n == 0)
+ {
+ b = (mpz_t *) (*__gmp_allocate_func) (sizeof (mpz_t));
+ mpz_init_set_ui (b[0], 1);
+ }
+ else
+ {
+ mpz_t t;
+ unsigned long k;
+
+ b = (mpz_t *) (*__gmp_reallocate_func)
+ (b, n * sizeof (mpz_t), (n + 1) * sizeof (mpz_t));
+ mpz_init (b[n]);
+ /* b[n] = -sum(binomial(2n+1,2k)*C[k]*(2n)!/(2k+1)!, k=0..n-1) */
+ mpz_init_set_ui (t, 2 * n + 1);
+ mpz_mul_ui (t, t, 2 * n - 1);
+ mpz_mul_ui (t, t, 2 * n);
+ mpz_mul_ui (t, t, n);
+ mpz_div_ui (t, t, 3); /* exact: t=binomial(2*n+1,2*k)*(2*n)!/(2*k+1)!
+ for k=n-1 */
+ mpz_mul (b[n], t, b[n - 1]);
+ for (k = n - 1; k-- > 0;)
+ {
+ mpz_mul_ui (t, t, 2 * k + 1);
+ mpz_mul_ui (t, t, 2 * k + 2);
+ mpz_mul_ui (t, t, 2 * k + 2);
+ mpz_mul_ui (t, t, 2 * k + 3);
+ mpz_div_ui (t, t, 2 * (n - k) + 1);
+ mpz_div_ui (t, t, 2 * (n - k));
+ mpz_addmul (b[n], t, b[k]);
+ }
+ /* take into account C[1] */
+ mpz_mul_ui (t, t, 2 * n + 1);
+ mpz_div_2exp (t, t, 1);
+ mpz_sub (b[n], b[n], t);
+ mpz_neg (b[n], b[n]);
+ mpz_clear (t);
+ }
+ return b;
+}
+
+/* Compute the alternating series
+ s = S(z) = \sum_{k=0}^infty B_{2k} (z))^{2k+1} / (2k+1)!
+ with 0 < z <= log(2) to the precision of s rounded in the direction
+ rnd_mode.
+ Return the maximum index of the truncature which is useful
+ for determinating the relative error.
+*/
+static int
+li2_series (mpfr_t sum, mpfr_srcptr z, mpfr_rnd_t rnd_mode)
+{
+ int i, Bm, Bmax;
+ mpfr_t s, u, v, w;
+ mpfr_prec_t sump, p;
+ mp_exp_t se, err;
+ mpz_t *B;
+ MPFR_ZIV_DECL (loop);
+
+ /* The series converges for |z| < 2 pi, but in mpfr_li2 the argument is
+ reduced so that 0 < z <= log(2). Here is additionnal check that z is
+ (nearly) correct */
+ MPFR_ASSERTD (MPFR_IS_STRICTPOS (z));
+ MPFR_ASSERTD (mpfr_cmp_d (z, 0.6953125) <= 0);
+
+ sump = MPFR_PREC (sum); /* target precision */
+ p = sump + MPFR_INT_CEIL_LOG2 (sump) + 4; /* the working precision */
+ mpfr_init2 (s, p);
+ mpfr_init2 (u, p);
+ mpfr_init2 (v, p);
+ mpfr_init2 (w, p);
+
+ B = bernoulli ((mpz_t *) 0, 0);
+ Bm = Bmax = 1;
+
+ MPFR_ZIV_INIT (loop, p);
+ for (;;)
+ {
+ mpfr_sqr (u, z, GMP_RNDU);
+ mpfr_set (v, z, GMP_RNDU);
+ mpfr_set (s, z, GMP_RNDU);
+ se = MPFR_GET_EXP (s);
+ err = 0;
+
+ for (i = 1;; i++)
+ {
+ if (i >= Bmax)
+ B = bernoulli (B, Bmax++); /* B_2i * (2i+1)!, exact */
+
+ mpfr_mul (v, u, v, GMP_RNDU);
+ mpfr_div_ui (v, v, 2 * i, GMP_RNDU);
+ mpfr_div_ui (v, v, 2 * i, GMP_RNDU);
+ mpfr_div_ui (v, v, 2 * i + 1, GMP_RNDU);
+ mpfr_div_ui (v, v, 2 * i + 1, GMP_RNDU);
+ /* here, v_2i = v_{2i-2} / (2i * (2i+1))^2 */
+
+ mpfr_mul_z (w, v, B[i], GMP_RNDN);
+ /* here, w_2i = v_2i * B_2i * (2i+1)! with
+ error(w_2i) < 2^(5 * i + 8) ulp(w_2i) (see algorithms.tex) */
+
+ mpfr_add (s, s, w, GMP_RNDN);
+
+ err = MAX (err + se, 5 * i + 8 + MPFR_GET_EXP (w))
+ - MPFR_GET_EXP (s);
+ err = 2 + MAX (-1, err);
+ se = MPFR_GET_EXP (s);
+ if (MPFR_GET_EXP (w) <= se - (mp_exp_t) p)
+ break;
+ }
+
+ /* the previous value of err is the rounding error,
+ the truncation error is less than EXP(z) - 6 * i - 5
+ (see algorithms.tex) */
+ err = MAX (err, MPFR_GET_EXP (z) - 6 * i - 5) + 1;
+ if (MPFR_CAN_ROUND (s, (mp_exp_t) p - err, sump, rnd_mode))
+ break;
+
+ MPFR_ZIV_NEXT (loop, p);
+ mpfr_set_prec (s, p);
+ mpfr_set_prec (u, p);
+ mpfr_set_prec (v, p);
+ mpfr_set_prec (w, p);
+ }
+ MPFR_ZIV_FREE (loop);
+ mpfr_set (sum, s, rnd_mode);
+
+ Bm = Bmax;
+ while (Bm--)
+ mpz_clear (B[Bm]);
+ (*__gmp_free_func) (B, Bmax * sizeof (mpz_t));
+ mpfr_clears (s, u, v, w, (mpfr_ptr) 0);
+
+ /* Let K be the returned value.
+ 1. As we compute an alternating series, the truncation error has the same
+ sign as the next term w_{K+2} which is positive iff K%4 == 0.
+ 2. Assume that error(z) <= (1+t) z', where z' is the actual value, then
+ error(s) <= 2 * (K+1) * t (see algorithms.tex).
+ */
+ return 2 * i;
+}
+
+/* try asymptotic expansion when x is large and positive:
+ Li2(x) = -log(x)^2/2 + Pi^2/3 - 1/x + O(1/x^2).
+ More precisely for x >= 2 we have for g(x) = -log(x)^2/2 + Pi^2/3:
+ -2 <= x * (Li2(x) - g(x)) <= -1
+ thus |Li2(x) - g(x)| <= 2/x.
+ Assumes x >= 38, which ensures log(x)^2/2 >= 2*Pi^2/3, and g(x) <= -3.3.
+ Return 0 if asymptotic expansion failed (unable to round), otherwise
+ returns correct ternary value.
+*/
+static int
+mpfr_li2_asympt_pos (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
+{
+ mpfr_t g, h;
+ mp_prec_t w = MPFR_PREC (y) + 20;
+ int inex = 0;
+
+ MPFR_ASSERTN (mpfr_cmp_ui (x, 38) >= 0);
+
+ mpfr_init2 (g, w);
+ mpfr_init2 (h, w);
+ mpfr_log (g, x, GMP_RNDN); /* rel. error <= |(1 + theta) - 1| */
+ mpfr_sqr (g, g, GMP_RNDN); /* rel. error <= |(1 + theta)^3 - 1| <= 2^(2-w) */
+ mpfr_div_2ui (g, g, 1, GMP_RNDN); /* rel. error <= 2^(2-w) */
+ mpfr_const_pi (h, GMP_RNDN); /* error <= 2^(1-w) */
+ mpfr_sqr (h, h, GMP_RNDN); /* rel. error <= 2^(2-w) */
+ mpfr_div_ui (h, h, 3, GMP_RNDN); /* rel. error <= |(1 + theta)^4 - 1|
+ <= 5 * 2^(-w) */
+ /* since x is chosen such that log(x)^2/2 >= 2 * (Pi^2/3), we should have
+ g >= 2*h, thus |g-h| >= |h|, and the relative error on g is at most
+ multiplied by 2 in the difference, and that by h is unchanged. */
+ MPFR_ASSERTN (MPFR_EXP (g) > MPFR_EXP (h));
+ mpfr_sub (g, h, g, GMP_RNDN); /* err <= ulp(g)/2 + g*2^(3-w) + g*5*2^(-w)
+ <= ulp(g) * (1/2 + 8 + 5) < 14 ulp(g).
+
+ If in addition 2/x <= 2 ulp(g), i.e.,
+ 1/x <= ulp(g), then the total error is
+ bounded by 16 ulp(g). */
+ if ((MPFR_EXP (x) >= (mp_exp_t) w - MPFR_EXP (g)) &&
+ MPFR_CAN_ROUND (g, w - 4, MPFR_PREC (y), rnd_mode))
+ inex = mpfr_set (y, g, rnd_mode);
+
+ mpfr_clear (g);
+ mpfr_clear (h);
+
+ return inex;
+}
+
+/* try asymptotic expansion when x is large and negative:
+ Li2(x) = -log(-x)^2/2 - Pi^2/6 - 1/x + O(1/x^2).
+ More precisely for x <= -2 we have for g(x) = -log(-x)^2/2 - Pi^2/6:
+ |Li2(x) - g(x)| <= 1/|x|.
+ Assumes x <= -7, which ensures |log(-x)^2/2| >= Pi^2/6, and g(x) <= -3.5.
+ Return 0 if asymptotic expansion failed (unable to round), otherwise
+ returns correct ternary value.
+*/
+static int
+mpfr_li2_asympt_neg (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
+{
+ mpfr_t g, h;
+ mp_prec_t w = MPFR_PREC (y) + 20;
+ int inex = 0;
+
+ MPFR_ASSERTN (mpfr_cmp_si (x, -7) <= 0);
+
+ mpfr_init2 (g, w);
+ mpfr_init2 (h, w);
+ mpfr_neg (g, x, GMP_RNDN);
+ mpfr_log (g, g, GMP_RNDN); /* rel. error <= |(1 + theta) - 1| */
+ mpfr_sqr (g, g, GMP_RNDN); /* rel. error <= |(1 + theta)^3 - 1| <= 2^(2-w) */
+ mpfr_div_2ui (g, g, 1, GMP_RNDN); /* rel. error <= 2^(2-w) */
+ mpfr_const_pi (h, GMP_RNDN); /* error <= 2^(1-w) */
+ mpfr_sqr (h, h, GMP_RNDN); /* rel. error <= 2^(2-w) */
+ mpfr_div_ui (h, h, 6, GMP_RNDN); /* rel. error <= |(1 + theta)^4 - 1|
+ <= 5 * 2^(-w) */
+ MPFR_ASSERTN (MPFR_EXP (g) >= MPFR_EXP (h));
+ mpfr_add (g, g, h, GMP_RNDN); /* err <= ulp(g)/2 + g*2^(2-w) + g*5*2^(-w)
+ <= ulp(g) * (1/2 + 4 + 5) < 10 ulp(g).
+
+ If in addition |1/x| <= 4 ulp(g), then the
+ total error is bounded by 16 ulp(g). */
+ if ((MPFR_EXP (x) >= (mp_exp_t) (w - 2) - MPFR_EXP (g)) &&
+ MPFR_CAN_ROUND (g, w - 4, MPFR_PREC (y), rnd_mode))
+ inex = mpfr_neg (y, g, rnd_mode);
+
+ mpfr_clear (g);
+ mpfr_clear (h);
+
+ return inex;
+}
+
+/* Compute the real part of the dilogarithm defined by
+ Li2(x) = -\Int_{t=0}^x log(1-t)/t dt */
+int
+mpfr_li2 (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
+{
+ int inexact;
+ mp_exp_t err;
+ mpfr_prec_t yp, m;
+ MPFR_ZIV_DECL (loop);
+ MPFR_SAVE_EXPO_DECL (expo);
+
+ MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", x, x, rnd_mode), ("y[%#R]=%R", y));
+
+ 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))
+ {
+ MPFR_SET_NEG (y);
+ MPFR_SET_INF (y);
+ MPFR_RET (0);
+ }
+ else /* x is zero */
+ {
+ MPFR_ASSERTD (MPFR_IS_ZERO (x));
+ MPFR_SET_SAME_SIGN (y, x);
+ MPFR_SET_ZERO (y);
+ MPFR_RET (0);
+ }
+ }
+
+ /* Li2(x) = x + x^2/4 + x^3/9 + ..., more precisely for 0 < x <= 1/2
+ we have |Li2(x) - x| < x^2/2 <= 2^(2EXP(x)-1) and for -1/2 <= x < 0
+ we have |Li2(x) - x| < x^2/4 <= 2^(2EXP(x)-2) */
+ if (MPFR_IS_POS (x))
+ MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, -MPFR_GET_EXP (x), 1, 1, rnd_mode,
+ {});
+ else
+ MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, -MPFR_GET_EXP (x), 2, 0, rnd_mode,
+ {});
+
+ MPFR_SAVE_EXPO_MARK (expo);
+ yp = MPFR_PREC (y);
+ m = yp + MPFR_INT_CEIL_LOG2 (yp) + 13;
+
+ if (MPFR_LIKELY ((mpfr_cmp_ui (x, 0) > 0) && (mpfr_cmp_d (x, 0.5) <= 0)))
+ /* 0 < x <= 1/2: Li2(x) = S(-log(1-x))-log^2(1-x)/4 */
+ {
+ mpfr_t s, u;
+ mp_exp_t expo_l;
+ int k;
+
+ mpfr_init2 (u, m);
+ mpfr_init2 (s, m);
+
+ MPFR_ZIV_INIT (loop, m);
+ for (;;)
+ {
+ mpfr_ui_sub (u, 1, x, GMP_RNDN);
+ mpfr_log (u, u, GMP_RNDU);
+ mpfr_neg (u, u, GMP_RNDN); /* u = -log(1-x) */
+ expo_l = MPFR_GET_EXP (u);
+ k = li2_series (s, u, GMP_RNDU);
+ err = 1 + MPFR_INT_CEIL_LOG2 (k + 1);
+
+ mpfr_sqr (u, u, GMP_RNDU);
+ mpfr_div_2ui (u, u, 2, GMP_RNDU); /* u = log^2(1-x) / 4 */
+ mpfr_sub (s, s, u, GMP_RNDN);
+
+ /* error(s) <= (0.5 + 2^(d-EXP(s))
+ + 2^(3 + MAX(1, - expo_l) - EXP(s))) ulp(s) */
+ err = MAX (err, MAX (1, - expo_l) - 1) - MPFR_GET_EXP (s);
+ err = 2 + MAX (-1, err);
+ if (MPFR_CAN_ROUND (s, (mp_exp_t) m - err, yp, rnd_mode))
+ break;
+
+ MPFR_ZIV_NEXT (loop, m);
+ mpfr_set_prec (u, m);
+ mpfr_set_prec (s, m);
+ }
+ MPFR_ZIV_FREE (loop);
+ inexact = mpfr_set (y, s, rnd_mode);
+
+ mpfr_clear (u);
+ mpfr_clear (s);
+ MPFR_SAVE_EXPO_FREE (expo);
+ return mpfr_check_range (y, inexact, rnd_mode);
+ }
+ else if (!mpfr_cmp_ui (x, 1))
+ /* Li2(1)= pi^2 / 6 */
+ {
+ mpfr_t u;
+ mpfr_init2 (u, m);
+
+ MPFR_ZIV_INIT (loop, m);
+ for (;;)
+ {
+ mpfr_const_pi (u, GMP_RNDU);
+ mpfr_sqr (u, u, GMP_RNDN);
+ mpfr_div_ui (u, u, 6, GMP_RNDN);
+
+ err = m - 4; /* error(u) <= 19/2 ulp(u) */
+ if (MPFR_CAN_ROUND (u, err, yp, rnd_mode))
+ break;
+
+ MPFR_ZIV_NEXT (loop, m);
+ mpfr_set_prec (u, m);
+ }
+ MPFR_ZIV_FREE (loop);
+ inexact = mpfr_set (y, u, rnd_mode);
+
+ mpfr_clear (u);
+ MPFR_SAVE_EXPO_FREE (expo);
+ return mpfr_check_range (y, inexact, rnd_mode);
+ }
+ else if (mpfr_cmp_ui (x, 2) >= 0)
+ /* x >= 2: Li2(x) = -S(-log(1-1/x))-log^2(x)/2+log^2(1-1/x)/4+pi^2/3 */
+ {
+ int k;
+ mp_exp_t expo_l;
+ mpfr_t s, u, xx;
+
+ if (mpfr_cmp_ui (x, 38) >= 0)
+ {
+ inexact = mpfr_li2_asympt_pos (y, x, rnd_mode);
+ if (inexact != 0)
+ goto end_of_case_gt2;
+ }
+
+ mpfr_init2 (u, m);
+ mpfr_init2 (s, m);
+ mpfr_init2 (xx, m);
+
+ MPFR_ZIV_INIT (loop, m);
+ for (;;)
+ {
+ mpfr_ui_div (xx, 1, x, GMP_RNDN);
+ mpfr_neg (xx, xx, GMP_RNDN);
+ mpfr_log1p (u, xx, GMP_RNDD);
+ mpfr_neg (u, u, GMP_RNDU); /* u = -log(1-1/x) */
+ expo_l = MPFR_GET_EXP (u);
+ k = li2_series (s, u, GMP_RNDN);
+ mpfr_neg (s, s, GMP_RNDN);
+ err = MPFR_INT_CEIL_LOG2 (k + 1) + 1; /* error(s) <= 2^err ulp(s) */
+
+ mpfr_sqr (u, u, GMP_RNDN);
+ mpfr_div_2ui (u, u, 2, GMP_RNDN); /* u= log^2(1-1/x)/4 */
+ mpfr_add (s, s, u, GMP_RNDN);
+ err =
+ MAX (err,
+ 3 + MAX (1, -expo_l) + MPFR_GET_EXP (u)) - MPFR_GET_EXP (s);
+ err = 2 + MAX (-1, err); /* error(s) <= 2^err ulp(s) */
+ err += MPFR_GET_EXP (s);
+
+ mpfr_log (u, x, GMP_RNDU);
+ mpfr_sqr (u, u, GMP_RNDN);
+ mpfr_div_2ui (u, u, 1, GMP_RNDN); /* u = log^2(x)/2 */
+ mpfr_sub (s, s, u, GMP_RNDN);
+ err = MAX (err, 3 + MPFR_GET_EXP (u)) - MPFR_GET_EXP (s);
+ err = 2 + MAX (-1, err); /* error(s) <= 2^err ulp(s) */
+ err += MPFR_GET_EXP (s);
+
+ mpfr_const_pi (u, GMP_RNDU);
+ mpfr_sqr (u, u, GMP_RNDN);
+ mpfr_div_ui (u, u, 3, GMP_RNDN); /* u = pi^2/3 */
+ mpfr_add (s, s, u, GMP_RNDN);
+ err = MAX (err, 2) - MPFR_GET_EXP (s);
+ err = 2 + MAX (-1, err); /* error(s) <= 2^err ulp(s) */
+ if (MPFR_CAN_ROUND (s, (mp_exp_t) m - err, yp, rnd_mode))
+ break;
+
+ MPFR_ZIV_NEXT (loop, m);
+ mpfr_set_prec (u, m);
+ mpfr_set_prec (s, m);
+ mpfr_set_prec (xx, m);
+ }
+ MPFR_ZIV_FREE (loop);
+ inexact = mpfr_set (y, s, rnd_mode);
+ mpfr_clears (s, u, xx, (mpfr_ptr) 0);
+
+ end_of_case_gt2:
+ MPFR_SAVE_EXPO_FREE (expo);
+ return mpfr_check_range (y, inexact, rnd_mode);
+ }
+ else if (mpfr_cmp_ui (x, 1) > 0)
+ /* 2 > x > 1: Li2(x) = S(log(x))+log^2(x)/4-log(x)log(x-1)+pi^2/6 */
+ {
+ int k;
+ mp_exp_t e1, e2;
+ mpfr_t s, u, v, xx;
+ mpfr_init2 (s, m);
+ mpfr_init2 (u, m);
+ mpfr_init2 (v, m);
+ mpfr_init2 (xx, m);
+
+ MPFR_ZIV_INIT (loop, m);
+ for (;;)
+ {
+ mpfr_log (v, x, GMP_RNDU);
+ k = li2_series (s, v, GMP_RNDN);
+ e1 = MPFR_GET_EXP (s);
+
+ mpfr_sqr (u, v, GMP_RNDN);
+ mpfr_div_2ui (u, u, 2, GMP_RNDN); /* u = log^2(x)/4 */
+ mpfr_add (s, s, u, GMP_RNDN);
+
+ mpfr_sub_ui (xx, x, 1, GMP_RNDN);
+ mpfr_log (u, xx, GMP_RNDU);
+ e2 = MPFR_GET_EXP (u);
+ mpfr_mul (u, v, u, GMP_RNDN); /* u = log(x) * log(x-1) */
+ mpfr_sub (s, s, u, GMP_RNDN);
+
+ mpfr_const_pi (u, GMP_RNDU);
+ mpfr_sqr (u, u, GMP_RNDN);
+ mpfr_div_ui (u, u, 6, GMP_RNDN); /* u = pi^2/6 */
+ mpfr_add (s, s, u, GMP_RNDN);
+ /* error(s) <= (31 + (k+1) * 2^(1-e1) + 2^(1-e2)) ulp(s)
+ see algorithms.tex */
+ err = MAX (MPFR_INT_CEIL_LOG2 (k + 1) + 1 - e1, 1 - e2);
+ err = 2 + MAX (5, err);
+ if (MPFR_CAN_ROUND (s, (mp_exp_t) m - err, yp, rnd_mode))
+ break;
+
+ MPFR_ZIV_NEXT (loop, m);
+ mpfr_set_prec (s, m);
+ mpfr_set_prec (u, m);
+ mpfr_set_prec (v, m);
+ mpfr_set_prec (xx, m);
+ }
+ MPFR_ZIV_FREE (loop);
+ inexact = mpfr_set (y, s, rnd_mode);
+
+ mpfr_clears (s, u, v, xx, (mpfr_ptr) 0);
+ MPFR_SAVE_EXPO_FREE (expo);
+ return mpfr_check_range (y, inexact, rnd_mode);
+ }
+ else if (mpfr_cmp_ui_2exp (x, 1, -1) > 0) /* 1/2 < x < 1 */
+ /* 1 > x > 1/2: Li2(x) = -S(-log(x))+log^2(x)/4-log(x)log(1-x)+pi^2/6 */
+ {
+ int k;
+ mpfr_t s, u, v, xx;
+ mpfr_init2 (s, m);
+ mpfr_init2 (u, m);
+ mpfr_init2 (v, m);
+ mpfr_init2 (xx, m);
+
+
+ MPFR_ZIV_INIT (loop, m);
+ for (;;)
+ {
+ mpfr_log (u, x, GMP_RNDD);
+ mpfr_neg (u, u, GMP_RNDN);
+ k = li2_series (s, u, GMP_RNDN);
+ mpfr_neg (s, s, GMP_RNDN);
+ err = 1 + MPFR_INT_CEIL_LOG2 (k + 1) - MPFR_GET_EXP (s);
+
+ mpfr_ui_sub (xx, 1, x, GMP_RNDN);
+ mpfr_log (v, xx, GMP_RNDU);
+ mpfr_mul (v, v, u, GMP_RNDN); /* v = - log(x) * log(1-x) */
+ mpfr_add (s, s, v, GMP_RNDN);
+ err = MAX (err, 1 - MPFR_GET_EXP (v));
+ err = 2 + MAX (3, err) - MPFR_GET_EXP (s);
+
+ mpfr_sqr (u, u, GMP_RNDN);
+ mpfr_div_2ui (u, u, 2, GMP_RNDN); /* u = log^2(x)/4 */
+ mpfr_add (s, s, u, GMP_RNDN);
+ err = MAX (err, 2 + MPFR_GET_EXP (u)) - MPFR_GET_EXP (s);
+ err = 2 + MAX (-1, err) + MPFR_GET_EXP (s);
+
+ mpfr_const_pi (u, GMP_RNDU);
+ mpfr_sqr (u, u, GMP_RNDN);
+ mpfr_div_ui (u, u, 6, GMP_RNDN); /* u = pi^2/6 */
+ mpfr_add (s, s, u, GMP_RNDN);
+ err = MAX (err, 3) - MPFR_GET_EXP (s);
+ err = 2 + MAX (-1, err);
+
+ if (MPFR_CAN_ROUND (s, (mp_exp_t) m - err, yp, rnd_mode))
+ break;
+
+ MPFR_ZIV_NEXT (loop, m);
+ mpfr_set_prec (s, m);
+ mpfr_set_prec (u, m);
+ mpfr_set_prec (v, m);
+ mpfr_set_prec (xx, m);
+ }
+ MPFR_ZIV_FREE (loop);
+ inexact = mpfr_set (y, s, rnd_mode);
+
+ mpfr_clears (s, u, v, xx, (mpfr_ptr) 0);
+ MPFR_SAVE_EXPO_FREE (expo);
+ return mpfr_check_range (y, inexact, rnd_mode);
+ }
+ else if (mpfr_cmp_si (x, -1) >= 0)
+ /* 0 > x >= -1: Li2(x) = -S(log(1-x))-log^2(1-x)/4 */
+ {
+ int k;
+ mp_exp_t expo_l;
+ mpfr_t s, u, xx;
+ mpfr_init2 (s, m);
+ mpfr_init2 (u, m);
+ mpfr_init2 (xx, m);
+
+ MPFR_ZIV_INIT (loop, m);
+ for (;;)
+ {
+ mpfr_neg (xx, x, GMP_RNDN);
+ mpfr_log1p (u, xx, GMP_RNDN);
+ k = li2_series (s, u, GMP_RNDN);
+ mpfr_neg (s, s, GMP_RNDN);
+ expo_l = MPFR_GET_EXP (u);
+ err = 1 + MPFR_INT_CEIL_LOG2 (k + 1) - MPFR_GET_EXP (s);
+
+ mpfr_sqr (u, u, GMP_RNDN);
+ mpfr_div_2ui (u, u, 2, GMP_RNDN); /* u = log^2(1-x)/4 */
+ mpfr_sub (s, s, u, GMP_RNDN);
+ err = MAX (err, - expo_l);
+ err = 2 + MAX (err, 3);
+ if (MPFR_CAN_ROUND (s, (mp_exp_t) m - err, yp, rnd_mode))
+ break;
+
+ MPFR_ZIV_NEXT (loop, m);
+ mpfr_set_prec (s, m);
+ mpfr_set_prec (u, m);
+ mpfr_set_prec (xx, m);
+ }
+ MPFR_ZIV_FREE (loop);
+ inexact = mpfr_set (y, s, rnd_mode);
+
+ mpfr_clears (s, u, xx, (mpfr_ptr) 0);
+ MPFR_SAVE_EXPO_FREE (expo);
+ return mpfr_check_range (y, inexact, rnd_mode);
+ }
+ else
+ /* x < -1: Li2(x)
+ = S(log(1-1/x))-log^2(-x)/4-log(1-x)log(-x)/2+log^2(1-x)/4-pi^2/6 */
+ {
+ int k;
+ mpfr_t s, u, v, w, xx;
+
+ if (mpfr_cmp_si (x, -7) <= 0)
+ {
+ inexact = mpfr_li2_asympt_neg (y, x, rnd_mode);
+ if (inexact != 0)
+ goto end_of_case_ltm1;
+ }
+
+ mpfr_init2 (s, m);
+ mpfr_init2 (u, m);
+ mpfr_init2 (v, m);
+ mpfr_init2 (w, m);
+ mpfr_init2 (xx, m);
+
+ MPFR_ZIV_INIT (loop, m);
+ for (;;)
+ {
+ mpfr_ui_div (xx, 1, x, GMP_RNDN);
+ mpfr_neg (xx, xx, GMP_RNDN);
+ mpfr_log1p (u, xx, GMP_RNDN);
+ k = li2_series (s, u, GMP_RNDN);
+
+ mpfr_ui_sub (xx, 1, x, GMP_RNDN);
+ mpfr_log (u, xx, GMP_RNDU);
+ mpfr_neg (xx, x, GMP_RNDN);
+ mpfr_log (v, xx, GMP_RNDU);
+ mpfr_mul (w, v, u, GMP_RNDN);
+ mpfr_div_2ui (w, w, 1, GMP_RNDN); /* w = log(-x) * log(1-x) / 2 */
+ mpfr_sub (s, s, w, GMP_RNDN);
+ err = 1 + MAX (3, MPFR_INT_CEIL_LOG2 (k+1) + 1 - MPFR_GET_EXP (s))
+ + MPFR_GET_EXP (s);
+
+ mpfr_sqr (w, v, GMP_RNDN);
+ mpfr_div_2ui (w, w, 2, GMP_RNDN); /* w = log^2(-x) / 4 */
+ mpfr_sub (s, s, w, GMP_RNDN);
+ err = MAX (err, 3 + MPFR_GET_EXP(w)) - MPFR_GET_EXP (s);
+ err = 2 + MAX (-1, err) + MPFR_GET_EXP (s);
+
+ mpfr_sqr (w, u, GMP_RNDN);
+ mpfr_div_2ui (w, w, 2, GMP_RNDN); /* w = log^2(1-x) / 4 */
+ mpfr_add (s, s, w, GMP_RNDN);
+ err = MAX (err, 3 + MPFR_GET_EXP (w)) - MPFR_GET_EXP (s);
+ err = 2 + MAX (-1, err) + MPFR_GET_EXP (s);
+
+ mpfr_const_pi (w, GMP_RNDU);
+ mpfr_sqr (w, w, GMP_RNDN);
+ mpfr_div_ui (w, w, 6, GMP_RNDN); /* w = pi^2 / 6 */
+ mpfr_sub (s, s, w, GMP_RNDN);
+ err = MAX (err, 3) - MPFR_GET_EXP (s);
+ err = 2 + MAX (-1, err) + MPFR_GET_EXP (s);
+
+ if (MPFR_CAN_ROUND (s, (mp_exp_t) m - err, yp, rnd_mode))
+ break;
+
+ MPFR_ZIV_NEXT (loop, m);
+ mpfr_set_prec (s, m);
+ mpfr_set_prec (u, m);
+ mpfr_set_prec (v, m);
+ mpfr_set_prec (w, m);
+ mpfr_set_prec (xx, m);
+ }
+ MPFR_ZIV_FREE (loop);
+ inexact = mpfr_set (y, s, rnd_mode);
+ mpfr_clears (s, u, v, w, xx, (mpfr_ptr) 0);
+
+ end_of_case_ltm1:
+ MPFR_SAVE_EXPO_FREE (expo);
+ return mpfr_check_range (y, inexact, rnd_mode);
+ }
+
+ MPFR_ASSERTN (0); /* should never reach this point */
+}
projects / msp430-gcc.git / blobdiff