/* mpfr_ai -- Airy function Ai Copyright 2010-2017 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. */ #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); } /* 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_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; } { mpfr_prec_t incr = MPFR_INT_CEIL_LOG2 (prec) + 5 + cond + assumed_exponent; wprec = MPFR_ADD_PREC (prec, incr); } 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)" " (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 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_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)" " (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); 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); /* 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 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); }