/* 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 */
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);
}