--- /dev/null
+/* mpfr_erfc -- The Complementary Error Function of a floating-point number
+
+Copyright 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc.
+Contributed by the Arenaire and Cacao 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 2.1 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.LIB. If not, 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 mp_exp_t
+mpfr_erfc_asympt (mpfr_ptr y, mpfr_srcptr x)
+{
+ mpfr_t t, xx, err;
+ unsigned long k;
+ mp_prec_t prec = MPFR_PREC(y);
+ mp_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, GMP_RNDD); /* err <= 1 */
+ mpfr_ui_div (xx, 1, xx, GMP_RNDU); /* upper bound for 1/(2x^2), err <= 2 */
+ mpfr_div_2ui (xx, xx, 1, GMP_RNDU); /* exact */
+ mpfr_set_ui (t, 1, GMP_RNDN); /* current term, exact */
+ mpfr_set (y, t, GMP_RNDN); /* current sum */
+ mpfr_set_ui (err, 0, GMP_RNDN);
+ for (k = 1; ; k++)
+ {
+ mpfr_mul_ui (t, t, 2 * k - 1, GMP_RNDU); /* err <= 4k-3 */
+ mpfr_mul (t, t, xx, GMP_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), GMP_RNDU);
+ mpfr_add_ui (err, err, 14 * k, GMP_RNDU); /* 2^(1-p) * t <= 2 ulp(t) */
+ mpfr_div_2si (err, err, MPFR_GET_EXP (y) - MPFR_GET_EXP (t), GMP_RNDU);
+ if (MPFR_GET_EXP (t) + (mp_exp_t) prec <= MPFR_GET_EXP (y))
+ {
+ /* the truncation error is bounded by |t| < ulp(y) */
+ mpfr_add_ui (err, err, 1, GMP_RNDU);
+ break;
+ }
+ if (k & 1)
+ mpfr_sub (y, y, t, GMP_RNDN);
+ else
+ mpfr_add (y, y, t, GMP_RNDN);
+ }
+ /* the error on y is bounded by err*ulp(y) */
+ mpfr_mul (t, x, x, GMP_RNDU); /* rel. err <= 2^(1-p) */
+ mpfr_div_2ui (err, err, 3, GMP_RNDU); /* err/8 */
+ mpfr_add (err, err, t, GMP_RNDU); /* err/8 + xx */
+ mpfr_mul_2ui (err, err, 3, GMP_RNDU); /* err + 8*xx */
+ mpfr_exp (t, t, GMP_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, GMP_RNDN); /* err <= 1/2*ulp(t) + (4*|x*x|+1)*ulp(t)
+ <= (4*|x*x|+3/2)*ulp(t) */
+ mpfr_const_pi (xx, GMP_RNDZ); /* err <= ulp(Pi) */
+ mpfr_sqrt (xx, xx, GMP_RNDN); /* err <= 1/2*ulp(xx) + ulp(Pi)/2/sqrt(Pi)
+ <= 3/2*ulp(xx) */
+ mpfr_mul (t, t, xx, GMP_RNDN); /* err <= (8 |xx| + 13/2) * ulp(t) */
+ mpfr_div (y, y, t, GMP_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, GMP_RNDD); /* t <= x^2 */
+ mpfr_neg (t, t, GMP_RNDU); /* -x^2 <= t */
+ mpfr_exp (t, t, GMP_RNDU); /* exp(-x^2) <= t */
+ mpfr_const_pi (xx, GMP_RNDD); /* xx <= sqrt(Pi), cached */
+ mpfr_mul (xx, xx, x, GMP_RNDD); /* xx <= sqrt(Pi)*x */
+ mpfr_div (y, t, xx, GMP_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, GMP_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, mp_rnd_t rnd)
+{
+ int inex;
+ mpfr_t tmp;
+ mp_exp_t te, err;
+ mp_prec_t prec;
+ MPFR_SAVE_EXPO_DECL (expo);
+ MPFR_ZIV_DECL (loop);
+
+ MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", x, x, rnd),
+ ("y[%#R]=%R inexact=%d", y, 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)
+ {
+ /* for x >= 27282, erfc(x) < 2^(-2^30-1) */
+ if (mpfr_cmp_ui (x, 27282) >= 0)
+ return mpfr_underflow (y, (rnd == GMP_RNDN) ? GMP_RNDZ : rnd, 1);
+ }
+
+ if (MPFR_SIGN (x) < 0)
+ {
+ mp_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, GMP_RNDN);
+ mpfr_set_inexflag ();
+ if (rnd == GMP_RNDZ || rnd == GMP_RNDD)
+ {
+ mpfr_nextbelow (y);
+ return -1;
+ }
+ else
+ return 1;
+ }
+ 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, GMP_RNDZ);
+ mpfr_mul (t, t, u, GMP_RNDZ); /* t <= x^2/log(2) */
+ mpfr_neg (u, x, GMP_RNDZ); /* 0 <= u <= |x| */
+ mpfr_log2 (u, u, GMP_RNDZ); /* u <= log2(|x|) */
+ mpfr_add (t, t, u, GMP_RNDZ); /* t <= log2|x| + x^2 / log(2) */
+ near_2 = mpfr_cmp_ui (t, MPFR_PREC(y)) >= 0;
+ mpfr_clear (t);
+ mpfr_clear (u);
+ if (near_2)
+ goto near_two;
+ }
+ }
+
+ /* Init stuff */
+ MPFR_SAVE_EXPO_MARK (expo);
+
+ /* 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 == GMP_RNDN) ? GMP_RNDZ : rnd, 1);
+ }
+ }
+ else
+ {
+ mpfr_erf (tmp, x, GMP_RNDN);
+ MPFR_ASSERTD (!MPFR_IS_SINGULAR (tmp)); /* FIXME: 0 only for x=0 ? */
+ te = MPFR_GET_EXP (tmp);
+ mpfr_ui_sub (tmp, 1, tmp, GMP_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 Controler */
+
+ 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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