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Diffstat (limited to 'Build/source/libs/mpfr/mpfr-src/src/ai.c')
-rw-r--r-- | Build/source/libs/mpfr/mpfr-src/src/ai.c | 690 |
1 files changed, 0 insertions, 690 deletions
diff --git a/Build/source/libs/mpfr/mpfr-src/src/ai.c b/Build/source/libs/mpfr/mpfr-src/src/ai.c deleted file mode 100644 index 66650dddd16..00000000000 --- a/Build/source/libs/mpfr/mpfr-src/src/ai.c +++ /dev/null @@ -1,690 +0,0 @@ -/* mpfr_ai -- Airy function Ai - -Copyright 2010-2020 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. */ - -#define MPFR_NEED_LONGLONG_H -#include "mpfr-impl.h" - -/* Reminder and notations: - ----------------------- - - Ai is the solution of: - / y'' - x*y = 0 - { Ai(0) = 1/ ( 9^(1/3)*Gamma(2/3) ) - \ Ai'(0) = -1/ ( 3^(1/3)*Gamma(1/3) ) - - Series development: - Ai(x) = sum (a_i*x^i) - = sum (t_i) - - Recurrences: - a_(i+3) = a_i / ((i+2)*(i+3)) - t_(i+3) = t_i * x^3 / ((i+2)*(i+3)) - - Values: - a_0 = Ai(0) ~ 0.355 - a_1 = Ai'(0) ~ -0.259 -*/ - - -/* Airy function Ai evaluated by the most naive algorithm. - Assume that x is a finite number. */ -static int -mpfr_ai1 (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd) -{ - MPFR_ZIV_DECL (loop); - MPFR_SAVE_EXPO_DECL (expo); - mpfr_prec_t wprec; /* working precision */ - mpfr_prec_t prec; /* target precision */ - mpfr_prec_t err; /* used to estimate the evaluation error */ - mpfr_prec_t correct_bits; /* estimates the number of correct bits*/ - unsigned long int k; - unsigned long int cond; /* condition number of the series */ - unsigned long int assumed_exponent; /* used as a lowerbound of |EXP(Ai(x))| */ - int r; - mpfr_t s; /* used to store the partial sum */ - mpfr_t ti, tip1; /* used to store successive values of t_i */ - mpfr_t x3; /* used to store x^3 */ - mpfr_t tmp_sp, tmp2_sp; /* small precision variables */ - unsigned long int x3u; /* used to store ceil(x^3) */ - mpfr_t temp1, temp2; - int test1, test2; - - /* Logging */ - MPFR_LOG_FUNC ( - ("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd), - ("y[%Pu]=%.*Rg", mpfr_get_prec (y), mpfr_log_prec, y) ); - - /* Save current exponents range */ - MPFR_SAVE_EXPO_MARK (expo); - - if (MPFR_UNLIKELY (MPFR_IS_ZERO (x))) - { - mpfr_t y1, y2; - prec = MPFR_ADD_PREC (MPFR_PREC (y), 3); - mpfr_init2 (y1, prec); - mpfr_init2 (y2, prec); - MPFR_ZIV_INIT (loop, prec); - - /* ZIV loop */ - for (;;) - { - mpfr_gamma_one_and_two_third (y1, y2, prec); /* y2 = Gamma(2/3)(1 + delta1), |delta1| <= 2^{1-prec}. */ - - r = mpfr_set_ui (y1, 9, MPFR_RNDN); - MPFR_ASSERTD (r == 0); - mpfr_cbrt (y1, y1, MPFR_RNDN); /* y1 = cbrt(9)(1 + delta2), |delta2| <= 2^{-prec}. */ - mpfr_mul (y1, y1, y2, MPFR_RNDN); - mpfr_ui_div (y1, 1, y1, MPFR_RNDN); - if (MPFR_LIKELY (MPFR_CAN_ROUND (y1, prec - 3, MPFR_PREC (y), rnd))) - break; - MPFR_ZIV_NEXT (loop, prec); - } - r = mpfr_set (y, y1, rnd); - MPFR_ZIV_FREE (loop); - MPFR_SAVE_EXPO_FREE (expo); - mpfr_clear (y1); - mpfr_clear (y2); - return mpfr_check_range (y, r, rnd); - } - - /* now x is not zero */ - MPFR_ASSERTD(!MPFR_IS_ZERO(x)); - - /* FIXME: underflow for large values of |x| ? */ - - - /* Set initial precision */ - /* If we compute sum(i=0, N-1, t_i), the relative error is bounded by */ - /* 2*(4N)*2^(1-wprec)*C(|x|)/Ai(x) */ - /* where C(|x|) = 1 if 0<=x<=1 */ - /* and C(|x|) = (1/2)*x^(-1/4)*exp(2/3 x^(3/2)) if x >= 1 */ - - /* A priori, we do not know N, so we estimate it to ~ prec */ - /* If 0<=x<=1, we estimate Ai(x) ~ 1/8 */ - /* if 1<=x, we estimate Ai(x) ~ (1/4)*x^(-1/4)*exp(-2/3 * x^(3/2)) */ - /* if x<=0, ????? */ - - /* We begin with 11 guard bits */ - prec = MPFR_ADD_PREC (MPFR_PREC (y), 11); - MPFR_ZIV_INIT (loop, prec); - - /* The working precision is heuristically chosen in order to obtain */ - /* approximately prec correct bits in the sum. To sum up: the sum */ - /* is stopped when the *exact* sum gives ~ prec correct bit. And */ - /* when it is stopped, the accuracy of the computed sum, with respect*/ - /* to the exact one should be ~prec bits. */ - mpfr_init2 (tmp_sp, MPFR_SMALL_PRECISION); - mpfr_init2 (tmp2_sp, MPFR_SMALL_PRECISION); - mpfr_abs (tmp_sp, x, MPFR_RNDU); - mpfr_pow_ui (tmp_sp, tmp_sp, 3, MPFR_RNDU); - mpfr_sqrt (tmp_sp, tmp_sp, MPFR_RNDU); /* tmp_sp ~ |x|^(3/2) */ - - /* 0.96179669392597567 >~ 2/3 * log2(e). See algorithms.tex */ - mpfr_set_str (tmp2_sp, "0.96179669392597567", 10, MPFR_RNDU); - mpfr_mul (tmp2_sp, tmp_sp, tmp2_sp, MPFR_RNDU); - - /* cond represents the number of lost bits in the evaluation of the sum */ - if (MPFR_GET_EXP (x) <= 0) - cond = 0; - else - { - MPFR_BLOCK_DECL (flags); - - MPFR_BLOCK (flags, cond = mpfr_get_ui (tmp2_sp, MPFR_RNDU)); - MPFR_ASSERTN (! MPFR_ERANGEFLAG (flags)); - cond -= (MPFR_GET_EXP (x) - 1) / 4 + 1; - } - - /* The variable assumed_exponent is used to store the maximal assumed */ - /* exponent of Ai(x). More precisely, we assume that |Ai(x)| will be */ - /* greater than 2^{-assumed_exponent}. */ - if (MPFR_IS_POS (x)) - { - if (MPFR_GET_EXP (x) <= 0) - assumed_exponent = 3; - else - { - unsigned long int t; - MPFR_BLOCK_DECL (flags); - - MPFR_BLOCK (flags, t = mpfr_get_ui (tmp2_sp, MPFR_RNDU)); - MPFR_ASSERTN (! MPFR_ERANGEFLAG (flags)); - assumed_exponent = t + 2 + (MPFR_GET_EXP (x) / 4 + 1); - MPFR_ASSERTN (assumed_exponent > t); - } - } - /* We do not know Ai (x) yet */ - /* We cover the case when EXP (Ai (x))>=-10 */ - else - assumed_exponent = 10; - - { - unsigned long int t, u; - - t = assumed_exponent + cond; - MPFR_ASSERTN (t >= cond); - u = MPFR_INT_CEIL_LOG2 (prec) + 5; - t += u; - MPFR_ASSERTN (t >= u); - wprec = MPFR_ADD_PREC (prec, t); - } - - mpfr_init (ti); - mpfr_init (tip1); - mpfr_init (temp1); - mpfr_init (temp2); - mpfr_init (x3); - mpfr_init (s); - - /* ZIV loop */ - for (;;) - { - MPFR_LOG_MSG (("Working precision: %Pu\n", wprec)); - mpfr_set_prec (ti, wprec); - mpfr_set_prec (tip1, wprec); - mpfr_set_prec (x3, wprec); - mpfr_set_prec (s, wprec); - - mpfr_sqr (x3, x, MPFR_RNDU); - mpfr_mul (x3, x3, x, (MPFR_IS_POS (x)?MPFR_RNDU:MPFR_RNDD)); /* x3=x^3 */ - if (MPFR_IS_NEG (x)) - MPFR_CHANGE_SIGN (x3); - x3u = mpfr_get_ui (x3, MPFR_RNDU); /* x3u >= ceil(x^3) */ - if (MPFR_IS_NEG (x)) - MPFR_CHANGE_SIGN (x3); - - mpfr_gamma_one_and_two_third (temp1, temp2, wprec); - mpfr_set_ui (ti, 9, MPFR_RNDN); - mpfr_cbrt (ti, ti, MPFR_RNDN); - mpfr_mul (ti, ti, temp2, MPFR_RNDN); - mpfr_ui_div (ti, 1, ti , MPFR_RNDN); /* ti = 1/( Gamma (2/3)*9^(1/3) ) */ - - mpfr_set_ui (tip1, 3, MPFR_RNDN); - mpfr_cbrt (tip1, tip1, MPFR_RNDN); - mpfr_mul (tip1, tip1, temp1, MPFR_RNDN); - mpfr_neg (tip1, tip1, MPFR_RNDN); - mpfr_div (tip1, x, tip1, MPFR_RNDN); /* tip1 = -x/(Gamma (1/3)*3^(1/3)) */ - - mpfr_add (s, ti, tip1, MPFR_RNDN); - - - /* Evaluation of the series */ - k = 2; - for (;;) - { - mpfr_mul (ti, ti, x3, MPFR_RNDN); - mpfr_mul (tip1, tip1, x3, MPFR_RNDN); - - mpfr_div_ui2 (ti, ti, k, (k+1), MPFR_RNDN); - mpfr_div_ui2 (tip1, tip1, (k+1), (k+2), MPFR_RNDN); - - k += 3; - mpfr_add (s, s, ti, MPFR_RNDN); - mpfr_add (s, s, tip1, MPFR_RNDN); - - /* FIXME: if s==0 */ - test1 = MPFR_IS_ZERO (ti) - || (MPFR_GET_EXP (ti) + (mpfr_exp_t)prec + 3 <= MPFR_GET_EXP (s)); - test2 = MPFR_IS_ZERO (tip1) - || (MPFR_GET_EXP (tip1) + (mpfr_exp_t)prec + 3 <= MPFR_GET_EXP (s)); - - if ( test1 && test2 && (x3u <= k*(k+1)/2) ) - break; /* FIXME: if k*(k+1) overflows */ - } - - MPFR_LOG_MSG (("Truncation rank: %lu\n", k)); - - err = 4 + MPFR_INT_CEIL_LOG2 (k) + cond - MPFR_GET_EXP (s); - - /* err is the number of bits lost due to the evaluation error */ - /* wprec-(prec+1): number of bits lost due to the approximation error */ - MPFR_LOG_MSG (("Roundoff error: %Pu\n", err)); - MPFR_LOG_MSG (("Approxim error: %Pu\n", wprec-prec-1)); - - if (wprec < err + 1) - correct_bits = 0; - else if (wprec < err + prec +1) - correct_bits = wprec - err - 1; /* since wprec > err + 1, - correct_bits > 0 */ - else - correct_bits = prec; - - if (MPFR_LIKELY (MPFR_CAN_ROUND (s, correct_bits, MPFR_PREC (y), rnd))) - break; - - if (correct_bits == 0) - { - assumed_exponent *= 2; - MPFR_LOG_MSG (("Not a single bit correct (assumed_exponent=%lu)\n", - assumed_exponent)); - wprec = prec + 5 + MPFR_INT_CEIL_LOG2 (prec) + cond + - assumed_exponent; - } - else if (correct_bits < prec) - { /* The precision was badly chosen */ - MPFR_LOG_MSG (("Bad assumption on the exponent of Ai(x)" - " (E=%" MPFR_EXP_FSPEC "d)\n", - (mpfr_eexp_t) MPFR_GET_EXP (s))); - wprec = prec + err + 1; - } - else - { /* We are really in a bad case of the TMD */ - MPFR_ZIV_NEXT (loop, prec); - - /* We update wprec */ - /* We assume that K will not be multiplied by more than 4 */ - wprec = prec + (MPFR_INT_CEIL_LOG2 (k) + 2) + 5 + cond - - MPFR_GET_EXP (s); - } - - } /* End of ZIV loop */ - - MPFR_ZIV_FREE (loop); - - r = mpfr_set (y, s, rnd); - - mpfr_clear (ti); - mpfr_clear (tip1); - mpfr_clear (temp1); - mpfr_clear (temp2); - mpfr_clear (x3); - mpfr_clear (s); - mpfr_clear (tmp_sp); - mpfr_clear (tmp2_sp); - - MPFR_SAVE_EXPO_FREE (expo); - return mpfr_check_range (y, r, rnd); -} - - -/* Airy function Ai evaluated by Smith algorithm. - Assume that x is a finite non-zero number. */ -static int -mpfr_ai2 (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd) -{ - MPFR_ZIV_DECL (loop); - MPFR_SAVE_EXPO_DECL (expo); - mpfr_prec_t wprec; /* working precision */ - mpfr_prec_t prec; /* target precision */ - mpfr_prec_t err; /* used to estimate the evaluation error */ - mpfr_prec_t correctBits; /* estimates the number of correct bits*/ - unsigned long int i, j, L, t; - unsigned long int cond; /* condition number of the series */ - unsigned long int assumed_exponent; /* used as a lowerbound of |EXP(Ai(x))| */ - int r; /* returned ternary value */ - mpfr_t s; /* used to store the partial sum */ - mpfr_t u0, u1; - mpfr_t *z; /* used to store the (x^3j) */ - mpfr_t result; - mpfr_t tmp_sp, tmp2_sp; /* small precision variables */ - unsigned long int x3u; /* used to store ceil (x^3) */ - mpfr_t temp1, temp2; - int test0, test1; - - /* Logging */ - MPFR_LOG_FUNC ( - ("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd), - ("y[%Pu]=%.*Rg", mpfr_get_prec (y), mpfr_log_prec, y)); - - /* Save current exponents range */ - MPFR_SAVE_EXPO_MARK (expo); - - /* FIXME: underflow for large values of |x| */ - - /* Set initial precision */ - /* See the analysis for the naive evaluation */ - - /* We begin with 11 guard bits */ - prec = MPFR_PREC (y) + 11; - MPFR_ZIV_INIT (loop, prec); - - mpfr_init2 (tmp_sp, MPFR_SMALL_PRECISION); - mpfr_init2 (tmp2_sp, MPFR_SMALL_PRECISION); - mpfr_abs (tmp_sp, x, MPFR_RNDU); - mpfr_pow_ui (tmp_sp, tmp_sp, 3, MPFR_RNDU); - mpfr_sqrt (tmp_sp, tmp_sp, MPFR_RNDU); /* tmp_sp ~ |x|^(3/2) */ - - /* 0.96179669392597567 >~ 2/3 * log2(e). See algorithms.tex */ - mpfr_set_str (tmp2_sp, "0.96179669392597567", 10, MPFR_RNDU); - mpfr_mul (tmp2_sp, tmp_sp, tmp2_sp, MPFR_RNDU); - - /* cond represents the number of lost bits in the evaluation of the sum */ - if (MPFR_GET_EXP (x) <= 0) - cond = 0; - else - { - MPFR_BLOCK_DECL (flags); - - MPFR_BLOCK (flags, cond = mpfr_get_ui (tmp2_sp, MPFR_RNDU)); - MPFR_ASSERTN (! MPFR_ERANGEFLAG (flags)); - cond -= (MPFR_GET_EXP (x) - 1) / 4 + 1; - } - - /* This variable is used to store the maximal assumed exponent of */ - /* Ai(x). More precisely, we assume that |Ai(x)| will be greater than */ - /* 2^{-assumed_exponent}. */ - if (MPFR_IS_POS (x)) - { - if (MPFR_GET_EXP (x) <= 0) - assumed_exponent = 3; - else - { - unsigned long int t; - MPFR_BLOCK_DECL (flags); - - MPFR_BLOCK (flags, t = mpfr_get_ui (tmp2_sp, MPFR_RNDU)); - MPFR_ASSERTN (! MPFR_ERANGEFLAG (flags)); - assumed_exponent = t + 2 + (MPFR_GET_EXP (x) / 4 + 1); - MPFR_ASSERTN (assumed_exponent > t); - } - } - /* We do not know Ai(x) yet */ - /* We cover the case when EXP(Ai(x))>=-10 */ - else - assumed_exponent = 10; - - { - unsigned long int t, u; - - t = assumed_exponent + cond; - MPFR_ASSERTN (t >= cond); - u = MPFR_INT_CEIL_LOG2 (prec) + 6; - t += u; - MPFR_ASSERTN (t >= u); - wprec = MPFR_ADD_PREC (prec, t); - } - - /* We assume that the truncation rank will be ~ prec */ - L = __gmpfr_isqrt (prec); - MPFR_LOG_MSG (("size of blocks L = %lu\n", L)); - - z = (mpfr_t *) mpfr_allocate_func ( (L + 1) * sizeof (mpfr_t) ); - MPFR_ASSERTN (z != NULL); - for (j=0; j<=L; j++) - mpfr_init (z[j]); - - mpfr_init (s); - mpfr_init (u0); mpfr_init (u1); - mpfr_init (result); - mpfr_init (temp1); - mpfr_init (temp2); - - /* ZIV loop */ - for (;;) - { - MPFR_LOG_MSG (("working precision: %Pu\n", wprec)); - - for (j=0; j<=L; j++) - mpfr_set_prec (z[j], wprec); - mpfr_set_prec (s, wprec); - mpfr_set_prec (u0, wprec); mpfr_set_prec (u1, wprec); - mpfr_set_prec (result, wprec); - - mpfr_set_ui (u0, 1, MPFR_RNDN); - mpfr_set (u1, x, MPFR_RNDN); - - mpfr_set_ui (z[0], 1, MPFR_RNDU); - mpfr_sqr (z[1], u1, MPFR_RNDU); - mpfr_mul (z[1], z[1], x, (MPFR_IS_POS (x) ? MPFR_RNDU : MPFR_RNDD) ); - - if (MPFR_IS_NEG (x)) - MPFR_CHANGE_SIGN (z[1]); - x3u = mpfr_get_ui (z[1], MPFR_RNDU); /* x3u >= ceil (x^3) */ - if (MPFR_IS_NEG (x)) - MPFR_CHANGE_SIGN (z[1]); - - for (j=2; j<=L ;j++) - { - if (j%2 == 0) - mpfr_sqr (z[j], z[j/2], MPFR_RNDN); - else - mpfr_mul (z[j], z[j-1], z[1], MPFR_RNDN); - } - - mpfr_gamma_one_and_two_third (temp1, temp2, wprec); - mpfr_set_ui (u0, 9, MPFR_RNDN); - mpfr_cbrt (u0, u0, MPFR_RNDN); - mpfr_mul (u0, u0, temp2, MPFR_RNDN); - mpfr_ui_div (u0, 1, u0 , MPFR_RNDN); /* u0 = 1/( Gamma (2/3)*9^(1/3) ) */ - - mpfr_set_ui (u1, 3, MPFR_RNDN); - mpfr_cbrt (u1, u1, MPFR_RNDN); - mpfr_mul (u1, u1, temp1, MPFR_RNDN); - mpfr_neg (u1, u1, MPFR_RNDN); - mpfr_div (u1, x, u1, MPFR_RNDN); /* u1 = -x/(Gamma (1/3)*3^(1/3)) */ - - mpfr_set_ui (result, 0, MPFR_RNDN); - t = 0; - - /* Evaluation of the series by Smith' method */ - for (i=0; ; i++) - { - t += 3 * L; - - /* k = 0 */ - t -= 3; - mpfr_set (s, z[L-1], MPFR_RNDN); - for (j=L-2; ; j--) - { - t -= 3; - mpfr_div_ui2 (s, s, (t+2), (t+3), MPFR_RNDN); - mpfr_add (s, s, z[j], MPFR_RNDN); - if (j==0) - break; - } - mpfr_mul (s, s, u0, MPFR_RNDN); - mpfr_add (result, result, s, MPFR_RNDN); - - mpfr_mul (u0, u0, z[L], MPFR_RNDN); - for (j=0; j<=L-1; j++) - { - mpfr_div_ui2 (u0, u0, (t + 2), (t + 3), MPFR_RNDN); - t += 3; - } - - t++; - - /* k = 1 */ - t -= 3; - mpfr_set (s, z[L-1], MPFR_RNDN); - for (j=L-2; ; j--) - { - t -= 3; - mpfr_div_ui2 (s, s, (t + 2), (t + 3), MPFR_RNDN); - mpfr_add (s, s, z[j], MPFR_RNDN); - if (j==0) - break; - } - mpfr_mul (s, s, u1, MPFR_RNDN); - mpfr_add (result, result, s, MPFR_RNDN); - - mpfr_mul (u1, u1, z[L], MPFR_RNDN); - for (j=0; j<=L-1; j++) - { - mpfr_div_ui2 (u1, u1, (t + 2), (t + 3), MPFR_RNDN); - t += 3; - } - - t++; - - /* k = 2 */ - t++; - - /* End of the loop over k */ - t -= 3; - - test0 = MPFR_IS_ZERO (u0) || - MPFR_GET_EXP (u0) + (mpfr_exp_t)prec + 4 <= MPFR_GET_EXP (result); - test1 = MPFR_IS_ZERO (u1) || - MPFR_GET_EXP (u1) + (mpfr_exp_t)prec + 4 <= MPFR_GET_EXP (result); - - if ( test0 && test1 && (x3u <= (t + 2) * (t + 3) / 2) ) - break; - } - - MPFR_LOG_MSG (("Truncation rank: %lu\n", t)); - - err = (5 + MPFR_INT_CEIL_LOG2 (L+1) + MPFR_INT_CEIL_LOG2 (i+1) - + cond - MPFR_GET_EXP (result)); - - /* err is the number of bits lost due to the evaluation error */ - /* wprec-(prec+1): number of bits lost due to the approximation error */ - MPFR_LOG_MSG (("Roundoff error: %Pu\n", err)); - MPFR_LOG_MSG (("Approxim error: %Pu\n", wprec - prec - 1)); - - if (wprec < err+1) - correctBits = 0; - else - { - if (wprec < err+prec+1) - correctBits = wprec - err - 1; - else - correctBits = prec; - } - - if (MPFR_LIKELY (MPFR_CAN_ROUND (result, correctBits, - MPFR_PREC (y), rnd))) - break; - - for (j=0; j<=L; j++) - mpfr_clear (z[j]); - mpfr_free_func (z, (L + 1) * sizeof (mpfr_t)); - L = __gmpfr_isqrt (t); - MPFR_LOG_MSG (("size of blocks L = %lu\n", L)); - z = (mpfr_t *) mpfr_allocate_func ( (L + 1) * sizeof (mpfr_t)); - MPFR_ASSERTN (z != NULL); - for (j=0; j<=L; j++) - mpfr_init (z[j]); - - if (correctBits == 0) - { - assumed_exponent *= 2; - MPFR_LOG_MSG (("Not a single bit correct (assumed_exponent=%lu)\n", - assumed_exponent)); - wprec = prec + 6 + MPFR_INT_CEIL_LOG2 (t) + cond + assumed_exponent; - } - else - { - if (correctBits < prec) - { /* The precision was badly chosen */ - MPFR_LOG_MSG (("Bad assumption on the exponent of Ai(x)" - " (E=%" MPFR_EXP_FSPEC "d)\n", - (mpfr_eexp_t) MPFR_GET_EXP (result))); - wprec = prec + err + 1; - } - else - { /* We are really in a bad case of the TMD */ - MPFR_ZIV_NEXT (loop, prec); - - /* We update wprec */ - /* We assume that t will not be multiplied by more than 4 */ - wprec = (prec + (MPFR_INT_CEIL_LOG2 (t) + 2) + 6 + cond - - MPFR_GET_EXP (result)); - } - } - } /* End of ZIV loop */ - - MPFR_ZIV_FREE (loop); - - r = mpfr_set (y, result, rnd); - - mpfr_clear (tmp_sp); - mpfr_clear (tmp2_sp); - for (j=0; j<=L; j++) - mpfr_clear (z[j]); - mpfr_free_func (z, (L + 1) * sizeof (mpfr_t)); - - mpfr_clear (s); - mpfr_clear (u0); mpfr_clear (u1); - mpfr_clear (result); - mpfr_clear (temp1); - mpfr_clear (temp2); - - MPFR_SAVE_EXPO_FREE (expo); - return mpfr_check_range (y, r, rnd); -} - -/* We consider that the boundary between the area where the naive method - should preferably be used and the area where Smith' method should preferably - be used has the following form: - it is a triangle defined by two lines (one for the negative values of x, and - one for the positive values of x) crossing at x=0. - - More precisely, - - * If x<0 and MPFR_AI_THRESHOLD1*x + MPFR_AI_THRESHOLD2*prec > MPFR_AI_SCALE, - use Smith' algorithm; - * If x>0 and MPFR_AI_THRESHOLD3*x + MPFR_AI_THRESHOLD2*prec > MPFR_AI_SCALE, - use Smith' algorithm; - * otherwise, use the naive method. -*/ - -#define MPFR_AI_SCALE 1048576 - -int -mpfr_ai (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd) -{ - mpfr_t temp1, temp2; - int use_ai2; - MPFR_SAVE_EXPO_DECL (expo); - - /* Special cases */ - 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)) - return mpfr_set_ui (y, 0, rnd); - /* the cases x = +0 or -0 will be treated below */ - } - - /* The exponent range must be large enough for the computation of temp1. */ - MPFR_SAVE_EXPO_MARK (expo); - - mpfr_init2 (temp1, MPFR_SMALL_PRECISION); - mpfr_init2 (temp2, MPFR_SMALL_PRECISION); - - mpfr_set (temp1, x, MPFR_RNDN); - mpfr_set_si (temp2, MPFR_AI_THRESHOLD2, MPFR_RNDN); - mpfr_mul_ui (temp2, temp2, MPFR_PREC (y) > ULONG_MAX ? - ULONG_MAX : (unsigned long) MPFR_PREC (y), MPFR_RNDN); - - if (MPFR_IS_NEG (x)) - mpfr_mul_si (temp1, temp1, MPFR_AI_THRESHOLD1, MPFR_RNDN); - else - mpfr_mul_si (temp1, temp1, MPFR_AI_THRESHOLD3, MPFR_RNDN); - - mpfr_add (temp1, temp1, temp2, MPFR_RNDN); - mpfr_clear (temp2); - - use_ai2 = mpfr_cmp_si (temp1, MPFR_AI_SCALE) > 0; - mpfr_clear (temp1); - - MPFR_SAVE_EXPO_FREE (expo); /* Ignore all previous exceptions. */ - - /* we use ai2 if |x|*AI_THRESHOLD1/3 + PREC(y)*AI_THRESHOLD2 > AI_SCALE, - which means x cannot be zero in mpfr_ai2 */ - return use_ai2 ? mpfr_ai2 (y, x, rnd) : mpfr_ai1 (y, x, rnd); -} |