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/* mpfr_erf -- error function of a floating-point number
Copyright 2001, 2003-2016 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"
#define EXP1 2.71828182845904523536 /* exp(1) */
static int mpfr_erf_0 (mpfr_ptr, mpfr_srcptr, double, mpfr_rnd_t);
int
mpfr_erf (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
mpfr_t xf;
int inex, large;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode),
("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;
}
else if (MPFR_IS_INF (x)) /* erf(+inf) = +1, erf(-inf) = -1 */
return mpfr_set_si (y, MPFR_INT_SIGN (x), MPFR_RNDN);
else /* erf(+0) = +0, erf(-0) = -0 */
{
MPFR_ASSERTD (MPFR_IS_ZERO (x));
return mpfr_set (y, x, MPFR_RNDN); /* should keep the sign of x */
}
}
/* now x is neither NaN, Inf nor 0 */
/* first try expansion at x=0 when x is small, or asymptotic expansion
where x is large */
MPFR_SAVE_EXPO_MARK (expo);
/* around x=0, we have erf(x) = 2x/sqrt(Pi) (1 - x^2/3 + ...),
with 1 - x^2/3 <= sqrt(Pi)*erf(x)/2/x <= 1 for x >= 0. This means that
if x^2/3 < 2^(-PREC(y)-1) we can decide of the correct rounding,
unless we have a worst-case for 2x/sqrt(Pi). */
if (MPFR_EXP(x) < - (mpfr_exp_t) (MPFR_PREC(y) / 2))
{
/* we use 2x/sqrt(Pi) (1 - x^2/3) <= erf(x) <= 2x/sqrt(Pi) for x > 0
and 2x/sqrt(Pi) <= erf(x) <= 2x/sqrt(Pi) (1 - x^2/3) for x < 0.
In both cases |2x/sqrt(Pi) (1 - x^2/3)| <= |erf(x)| <= |2x/sqrt(Pi)|.
We will compute l and h such that l <= |2x/sqrt(Pi) (1 - x^2/3)|
and |2x/sqrt(Pi)| <= h. If l and h round to the same value to
precision PREC(y) and rounding rnd_mode, then we are done. */
mpfr_t l, h; /* lower and upper bounds for erf(x) */
int ok, inex2;
mpfr_init2 (l, MPFR_PREC(y) + 17);
mpfr_init2 (h, MPFR_PREC(y) + 17);
/* first compute l */
mpfr_mul (l, x, x, MPFR_RNDU);
mpfr_div_ui (l, l, 3, MPFR_RNDU); /* upper bound on x^2/3 */
mpfr_ui_sub (l, 1, l, MPFR_RNDZ); /* lower bound on 1 - x^2/3 */
mpfr_const_pi (h, MPFR_RNDU); /* upper bound of Pi */
mpfr_sqrt (h, h, MPFR_RNDU); /* upper bound on sqrt(Pi) */
mpfr_div (l, l, h, MPFR_RNDZ); /* lower bound on 1/sqrt(Pi) (1 - x^2/3) */
mpfr_mul_2ui (l, l, 1, MPFR_RNDZ); /* 2/sqrt(Pi) (1 - x^2/3) */
mpfr_mul (l, l, x, MPFR_RNDZ); /* |l| is a lower bound on
|2x/sqrt(Pi) (1 - x^2/3)| */
/* now compute h */
mpfr_const_pi (h, MPFR_RNDD); /* lower bound on Pi */
mpfr_sqrt (h, h, MPFR_RNDD); /* lower bound on sqrt(Pi) */
mpfr_div_2ui (h, h, 1, MPFR_RNDD); /* lower bound on sqrt(Pi)/2 */
/* since sqrt(Pi)/2 < 1, the following should not underflow */
mpfr_div (h, x, h, MPFR_IS_POS(x) ? MPFR_RNDU : MPFR_RNDD);
/* round l and h to precision PREC(y) */
inex = mpfr_prec_round (l, MPFR_PREC(y), rnd_mode);
inex2 = mpfr_prec_round (h, MPFR_PREC(y), rnd_mode);
/* Caution: we also need inex=inex2 (inex might be 0). */
ok = SAME_SIGN (inex, inex2) && mpfr_cmp (l, h) == 0;
if (ok)
mpfr_set (y, h, rnd_mode);
mpfr_clear (l);
mpfr_clear (h);
if (ok)
goto end;
/* this test can still fail for small precision, for example
for x=-0.100E-2 with a target precision of 3 bits, since
the error term x^2/3 is not that small. */
}
mpfr_init2 (xf, 53);
mpfr_const_log2 (xf, MPFR_RNDU);
mpfr_div (xf, x, xf, MPFR_RNDZ); /* round to zero ensures we get a lower
bound of |x/log(2)| */
mpfr_mul (xf, xf, x, MPFR_RNDZ);
large = mpfr_cmp_ui (xf, MPFR_PREC (y) + 1) > 0;
mpfr_clear (xf);
/* when x goes to infinity, we have erf(x) = 1 - 1/sqrt(Pi)/exp(x^2)/x + ...
and |erf(x) - 1| <= exp(-x^2) is true for any x >= 0, thus if
exp(-x^2) < 2^(-PREC(y)-1) the result is 1 or 1-epsilon.
This rewrites as x^2/log(2) > p+1. */
if (MPFR_UNLIKELY (large))
/* |erf x| = 1 or 1- */
{
mpfr_rnd_t rnd2 = MPFR_IS_POS (x) ? rnd_mode : MPFR_INVERT_RND(rnd_mode);
if (rnd2 == MPFR_RNDN || rnd2 == MPFR_RNDU || rnd2 == MPFR_RNDA)
{
inex = MPFR_INT_SIGN (x);
mpfr_set_si (y, inex, rnd2);
}
else /* round to zero */
{
inex = -MPFR_INT_SIGN (x);
mpfr_setmax (y, 0); /* warning: setmax keeps the old sign of y */
MPFR_SET_SAME_SIGN (y, x);
}
}
else /* use Taylor */
{
double xf2;
/* FIXME: get rid of doubles/mpfr_get_d here */
xf2 = mpfr_get_d (x, MPFR_RNDN);
xf2 = xf2 * xf2; /* xf2 ~ x^2 */
inex = mpfr_erf_0 (y, x, xf2, rnd_mode);
}
end:
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inex, rnd_mode);
}
/* return x*2^e */
static double
mul_2exp (double x, mpfr_exp_t e)
{
if (e > 0)
{
while (e--)
x *= 2.0;
}
else
{
while (e++)
x /= 2.0;
}
return x;
}
/* evaluates erf(x) using the expansion at x=0:
erf(x) = 2/sqrt(Pi) * sum((-1)^k*x^(2k+1)/k!/(2k+1), k=0..infinity)
Assumes x is neither NaN nor infinite nor zero.
Assumes also that e*x^2 <= n (target precision).
*/
static int
mpfr_erf_0 (mpfr_ptr res, mpfr_srcptr x, double xf2, mpfr_rnd_t rnd_mode)
{
mpfr_prec_t n, m;
mpfr_exp_t nuk, sigmak;
double tauk;
mpfr_t y, s, t, u;
unsigned int k;
int log2tauk;
int inex;
MPFR_ZIV_DECL (loop);
n = MPFR_PREC (res); /* target precision */
/* initial working precision */
m = n + (mpfr_prec_t) (xf2 / LOG2) + 8 + MPFR_INT_CEIL_LOG2 (n);
mpfr_init2 (y, m);
mpfr_init2 (s, m);
mpfr_init2 (t, m);
mpfr_init2 (u, m);
MPFR_ZIV_INIT (loop, m);
for (;;)
{
mpfr_mul (y, x, x, MPFR_RNDU); /* err <= 1 ulp */
mpfr_set_ui (s, 1, MPFR_RNDN);
mpfr_set_ui (t, 1, MPFR_RNDN);
tauk = 0.0;
for (k = 1; ; k++)
{
mpfr_mul (t, y, t, MPFR_RNDU);
mpfr_div_ui (t, t, k, MPFR_RNDU);
mpfr_div_ui (u, t, 2 * k + 1, MPFR_RNDU);
sigmak = MPFR_GET_EXP (s);
if (k % 2)
mpfr_sub (s, s, u, MPFR_RNDN);
else
mpfr_add (s, s, u, MPFR_RNDN);
sigmak -= MPFR_GET_EXP(s);
nuk = MPFR_GET_EXP(u) - MPFR_GET_EXP(s);
if ((nuk < - (mpfr_exp_t) m) && ((double) k >= xf2))
break;
/* tauk <- 1/2 + tauk * 2^sigmak + (1+8k)*2^nuk */
tauk = 0.5 + mul_2exp (tauk, sigmak)
+ mul_2exp (1.0 + 8.0 * (double) k, nuk);
}
mpfr_mul (s, x, s, MPFR_RNDU);
MPFR_SET_EXP (s, MPFR_GET_EXP (s) + 1);
mpfr_const_pi (t, MPFR_RNDZ);
mpfr_sqrt (t, t, MPFR_RNDZ);
mpfr_div (s, s, t, MPFR_RNDN);
tauk = 4.0 * tauk + 11.0; /* final ulp-error on s */
log2tauk = __gmpfr_ceil_log2 (tauk);
if (MPFR_LIKELY (MPFR_CAN_ROUND (s, m - log2tauk, n, rnd_mode)))
break;
/* Actualisation of the precision */
MPFR_ZIV_NEXT (loop, m);
mpfr_set_prec (y, m);
mpfr_set_prec (s, m);
mpfr_set_prec (t, m);
mpfr_set_prec (u, m);
}
MPFR_ZIV_FREE (loop);
inex = mpfr_set (res, s, rnd_mode);
mpfr_clear (y);
mpfr_clear (t);
mpfr_clear (u);
mpfr_clear (s);
return inex;
}
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