| /* 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 */ |
| } |
