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/* mpfr_erfc -- The Complementary Error Function of a floating-point number
Copyright 2005-2017 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"
/* erfc(x) = 1 - erf(x) */
/* Put in y an approximation of erfc(x) for large x, using formulae 7.1.23 and
7.1.24 from Abramowitz and Stegun.
Returns e such that the error is bounded by 2^e ulp(y),
or returns 0 in case of underflow.
*/
static mpfr_exp_t
mpfr_erfc_asympt (mpfr_ptr y, mpfr_srcptr x)
{
mpfr_t t, xx, err;
unsigned long k;
mpfr_prec_t prec = MPFR_PREC(y);
mpfr_exp_t exp_err;
mpfr_init2 (t, prec);
mpfr_init2 (xx, prec);
mpfr_init2 (err, 31);
/* let u = 2^(1-p), and let us represent the error as (1+u)^err
with a bound for err */
mpfr_mul (xx, x, x, MPFR_RNDD); /* err <= 1 */
mpfr_ui_div (xx, 1, xx, MPFR_RNDU); /* upper bound for 1/(2x^2), err <= 2 */
mpfr_div_2ui (xx, xx, 1, MPFR_RNDU); /* exact */
mpfr_set_ui (t, 1, MPFR_RNDN); /* current term, exact */
mpfr_set (y, t, MPFR_RNDN); /* current sum */
mpfr_set_ui (err, 0, MPFR_RNDN);
for (k = 1; ; k++)
{
mpfr_mul_ui (t, t, 2 * k - 1, MPFR_RNDU); /* err <= 4k-3 */
mpfr_mul (t, t, xx, MPFR_RNDU); /* err <= 4k */
/* for -1 < x < 1, and |nx| < 1, we have |(1+x)^n| <= 1+7/4|nx|.
Indeed, for x>=0: log((1+x)^n) = n*log(1+x) <= n*x. Let y=n*x < 1,
then exp(y) <= 1+7/4*y.
For x<=0, let x=-x, we can prove by induction that (1-x)^n >= 1-n*x.*/
mpfr_mul_2si (err, err, MPFR_GET_EXP (y) - MPFR_GET_EXP (t), MPFR_RNDU);
mpfr_add_ui (err, err, 14 * k, MPFR_RNDU); /* 2^(1-p) * t <= 2 ulp(t) */
mpfr_div_2si (err, err, MPFR_GET_EXP (y) - MPFR_GET_EXP (t), MPFR_RNDU);
if (MPFR_GET_EXP (t) + (mpfr_exp_t) prec <= MPFR_GET_EXP (y))
{
/* the truncation error is bounded by |t| < ulp(y) */
mpfr_add_ui (err, err, 1, MPFR_RNDU);
break;
}
if (k & 1)
mpfr_sub (y, y, t, MPFR_RNDN);
else
mpfr_add (y, y, t, MPFR_RNDN);
}
/* the error on y is bounded by err*ulp(y) */
mpfr_mul (t, x, x, MPFR_RNDU); /* rel. err <= 2^(1-p) */
mpfr_div_2ui (err, err, 3, MPFR_RNDU); /* err/8 */
mpfr_add (err, err, t, MPFR_RNDU); /* err/8 + xx */
mpfr_mul_2ui (err, err, 3, MPFR_RNDU); /* err + 8*xx */
mpfr_exp (t, t, MPFR_RNDU); /* err <= 1/2*ulp(t) + err(x*x)*t
<= 1/2*ulp(t)+2*|x*x|*ulp(t)
<= (2*|x*x|+1/2)*ulp(t) */
mpfr_mul (t, t, x, MPFR_RNDN); /* err <= 1/2*ulp(t) + (4*|x*x|+1)*ulp(t)
<= (4*|x*x|+3/2)*ulp(t) */
mpfr_const_pi (xx, MPFR_RNDZ); /* err <= ulp(Pi) */
mpfr_sqrt (xx, xx, MPFR_RNDN); /* err <= 1/2*ulp(xx) + ulp(Pi)/2/sqrt(Pi)
<= 3/2*ulp(xx) */
mpfr_mul (t, t, xx, MPFR_RNDN); /* err <= (8 |xx| + 13/2) * ulp(t) */
mpfr_div (y, y, t, MPFR_RNDN); /* the relative error on input y is bounded
by (1+u)^err with u = 2^(1-p), that on
t is bounded by (1+u)^(8 |xx| + 13/2),
thus that on output y is bounded by
8 |xx| + 7 + err. */
if (MPFR_IS_ZERO(y))
{
/* If y is zero, most probably we have underflow. We check it directly
using the fact that erfc(x) <= exp(-x^2)/sqrt(Pi)/x for x >= 0.
We compute an upper approximation of exp(-x^2)/sqrt(Pi)/x.
*/
mpfr_mul (t, x, x, MPFR_RNDD); /* t <= x^2 */
mpfr_neg (t, t, MPFR_RNDU); /* -x^2 <= t */
mpfr_exp (t, t, MPFR_RNDU); /* exp(-x^2) <= t */
mpfr_const_pi (xx, MPFR_RNDD); /* xx <= sqrt(Pi), cached */
mpfr_mul (xx, xx, x, MPFR_RNDD); /* xx <= sqrt(Pi)*x */
mpfr_div (y, t, xx, MPFR_RNDN); /* if y is zero, this means that the upper
approximation of exp(-x^2)/sqrt(Pi)/x
is nearer from 0 than from 2^(-emin-1),
thus we have underflow. */
exp_err = 0;
}
else
{
mpfr_add_ui (err, err, 7, MPFR_RNDU);
exp_err = MPFR_GET_EXP (err);
}
mpfr_clear (t);
mpfr_clear (xx);
mpfr_clear (err);
return exp_err;
}
int
mpfr_erfc (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd)
{
int inex;
mpfr_t tmp;
mpfr_exp_t te, err;
mpfr_prec_t prec;
mpfr_exp_t emin = mpfr_get_emin ();
MPFR_SAVE_EXPO_DECL (expo);
MPFR_ZIV_DECL (loop);
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd),
("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y, inex));
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
if (MPFR_IS_NAN (x))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
/* erfc(+inf) = 0+, erfc(-inf) = 2 erfc (0) = 1 */
else if (MPFR_IS_INF (x))
return mpfr_set_ui (y, MPFR_IS_POS (x) ? 0 : 2, rnd);
else
return mpfr_set_ui (y, 1, rnd);
}
if (MPFR_SIGN (x) > 0)
{
/* by default, emin = 1-2^30, thus the smallest representable
number is 1/2*2^emin = 2^(-2^30):
for x >= 27282, erfc(x) < 2^(-2^30-1), and
for x >= 1787897414, erfc(x) < 2^(-2^62-1).
*/
if ((emin >= -1073741823 && mpfr_cmp_ui (x, 27282) >= 0) ||
mpfr_cmp_ui (x, 1787897414) >= 0)
{
/* May be incorrect if MPFR_EMAX_MAX >= 2^62. */
MPFR_ASSERTN ((MPFR_EMAX_MAX >> 31) >> 31 == 0);
return mpfr_underflow (y, (rnd == MPFR_RNDN) ? MPFR_RNDZ : rnd, 1);
}
}
/* Init stuff */
MPFR_SAVE_EXPO_MARK (expo);
if (MPFR_SIGN (x) < 0)
{
mpfr_exp_t e = MPFR_EXP(x);
/* For x < 0 going to -infinity, erfc(x) tends to 2 by below.
More precisely, we have 2 + 1/sqrt(Pi)/x/exp(x^2) < erfc(x) < 2.
Thus log2 |2 - erfc(x)| <= -log2|x| - x^2 / log(2).
If |2 - erfc(x)| < 2^(-PREC(y)) then the result is either 2 or
nextbelow(2).
For x <= -27282, -log2|x| - x^2 / log(2) <= -2^30.
*/
if ((MPFR_PREC(y) <= 7 && e >= 2) || /* x <= -2 */
(MPFR_PREC(y) <= 25 && e >= 3) || /* x <= -4 */
(MPFR_PREC(y) <= 120 && mpfr_cmp_si (x, -9) <= 0) ||
mpfr_cmp_si (x, -27282) <= 0)
{
near_two:
mpfr_set_ui (y, 2, MPFR_RNDN);
mpfr_set_inexflag ();
if (rnd == MPFR_RNDZ || rnd == MPFR_RNDD)
{
mpfr_nextbelow (y);
inex = -1;
}
else
inex = 1;
goto end;
}
else if (e >= 3) /* more accurate test */
{
mpfr_t t, u;
int near_2;
mpfr_init2 (t, 32);
mpfr_init2 (u, 32);
/* the following is 1/log(2) rounded to zero on 32 bits */
mpfr_set_str_binary (t, "1.0111000101010100011101100101001");
mpfr_sqr (u, x, MPFR_RNDZ);
mpfr_mul (t, t, u, MPFR_RNDZ); /* t <= x^2/log(2) */
mpfr_neg (u, x, MPFR_RNDZ); /* 0 <= u <= |x| */
mpfr_log2 (u, u, MPFR_RNDZ); /* u <= log2(|x|) */
mpfr_add (t, t, u, MPFR_RNDZ); /* t <= log2|x| + x^2 / log(2) */
/* Taking into account that mpfr_exp_t >= mpfr_prec_t */
mpfr_set_exp_t (u, MPFR_PREC (y), MPFR_RNDU);
near_2 = mpfr_cmp (t, u) >= 0; /* 1 if PREC(y) <= u <= t <= ... */
mpfr_clear (t);
mpfr_clear (u);
if (near_2)
goto near_two;
}
}
/* erfc(x) ~ 1, with error < 2^(EXP(x)+1) */
MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, __gmpfr_one, - MPFR_GET_EXP (x) - 1,
0, MPFR_SIGN(x) < 0,
rnd, inex = _inexact; goto end);
prec = MPFR_PREC (y) + MPFR_INT_CEIL_LOG2 (MPFR_PREC (y)) + 3;
if (MPFR_GET_EXP (x) > 0)
prec += 2 * MPFR_GET_EXP(x);
mpfr_init2 (tmp, prec);
MPFR_ZIV_INIT (loop, prec); /* Initialize the ZivLoop controler */
for (;;) /* Infinite loop */
{
/* use asymptotic formula only whenever x^2 >= p*log(2),
otherwise it will not converge */
if (MPFR_SIGN (x) > 0 &&
2 * MPFR_GET_EXP (x) - 2 >= MPFR_INT_CEIL_LOG2 (prec))
/* we have x^2 >= p in that case */
{
err = mpfr_erfc_asympt (tmp, x);
if (err == 0) /* underflow case */
{
mpfr_clear (tmp);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_underflow (y, (rnd == MPFR_RNDN) ? MPFR_RNDZ : rnd, 1);
}
}
else
{
mpfr_erf (tmp, x, MPFR_RNDN);
MPFR_ASSERTD (!MPFR_IS_SINGULAR (tmp)); /* FIXME: 0 only for x=0 ? */
te = MPFR_GET_EXP (tmp);
mpfr_ui_sub (tmp, 1, tmp, MPFR_RNDN);
/* See error analysis in algorithms.tex for details */
if (MPFR_IS_ZERO (tmp))
{
prec *= 2;
err = prec; /* ensures MPFR_CAN_ROUND fails */
}
else
err = MAX (te - MPFR_GET_EXP (tmp), 0) + 1;
}
if (MPFR_LIKELY (MPFR_CAN_ROUND (tmp, prec - err, MPFR_PREC (y), rnd)))
break;
MPFR_ZIV_NEXT (loop, prec); /* Increase used precision */
mpfr_set_prec (tmp, prec);
}
MPFR_ZIV_FREE (loop); /* Free the ZivLoop Controller */
inex = mpfr_set (y, tmp, rnd); /* Set y to the computed value */
mpfr_clear (tmp);
end:
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inex, rnd);
}
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