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Diffstat (limited to 'src/ai.c')
| -rw-r--r-- | src/ai.c | 664 |
1 files changed, 664 insertions, 0 deletions
diff --git a/src/ai.c b/src/ai.c new file mode 100644 index 0000000..19d9eb5 --- /dev/null +++ b/src/ai.c @@ -0,0 +1,664 @@ +/* mpfr_ai -- Airy function Ai + +Copyright 2010, 2011, 2012, 2013 Free Software Foundation, Inc. +Contributed by the AriC and Caramel 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. */ + +#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 */ +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) ); + + /* 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); + } + + + /* Save current exponents range */ + MPFR_SAVE_EXPO_MARK (expo); + + if (MPFR_UNLIKELY (MPFR_IS_ZERO (x))) + { + mpfr_t y1, y2; + 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); + } + + /* 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_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_IS_ZERO (x)) || (MPFR_GET_EXP (x) <= 0) ) + cond = 0; + else + cond = mpfr_get_ui (tmp2_sp, MPFR_RNDU) - (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_ZERO (x)) + assumed_exponent = 2; + else + { + if (MPFR_IS_POS (x)) + { + if (MPFR_GET_EXP (x) <= 0) + assumed_exponent = 3; + else + assumed_exponent = (2 + (MPFR_GET_EXP (x)/4 + 1) + + mpfr_get_ui (tmp2_sp, MPFR_RNDU)); + } + /* We do not know Ai (x) yet */ + /* We cover the case when EXP (Ai (x))>=-10 */ + else + assumed_exponent = 10; + } + + wprec = prec + MPFR_INT_CEIL_LOG2 (prec) + 5 + cond + assumed_exponent; + + 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; + 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 (k) + cond + assumed_exponent; + } + else + { + if (correct_bits < prec) + { /* The precision was badly chosen */ + MPFR_LOG_MSG (("Bad assumption on the exponent of Ai(x)", 0)); + MPFR_LOG_MSG ((" (E=%ld)\n", (long) 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 */ +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)); + + /* 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); + } + + /* 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_IS_ZERO (x)) || (MPFR_GET_EXP (x) <= 0) ) + cond = 0; + else + cond = mpfr_get_ui (tmp2_sp, MPFR_RNDU) - (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^{-assumedExp}. */ + if (MPFR_IS_ZERO (x)) + assumed_exponent = 2; + else + { + if (MPFR_IS_POS (x)) + { + if (MPFR_GET_EXP (x) <= 0) + assumed_exponent = 3; + else + assumed_exponent = (2 + (MPFR_GET_EXP (x)/4 + 1) + + mpfr_get_ui (tmp2_sp, MPFR_RNDU)); + } + /* We do not know Ai (x) yet */ + /* We cover the case when EXP (Ai (x))>=-10 */ + else + assumed_exponent = 10; + } + + wprec = prec + MPFR_INT_CEIL_LOG2 (prec) + 6 + cond + assumed_exponent; + + /* 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 *) (*__gmp_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]); + (*__gmp_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 *) (*__gmp_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)", 0)); + MPFR_LOG_MSG ((" (E=%ld)\n", (long) (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); + MPFR_SAVE_EXPO_FREE (expo); + + r = mpfr_set (y, result, rnd); + + mpfr_clear (tmp_sp); + mpfr_clear (tmp2_sp); + for (j=0; j<=L; j++) + mpfr_clear (z[j]); + (*__gmp_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); + + return r; +} + +/* 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); + + /* 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. */ + + return use_ai2 ? mpfr_ai2 (y, x, rnd) : mpfr_ai1 (y, x, rnd); +} |