xref: /dragonfly/contrib/mpfr/src/erfc.c (revision ab6d115f)

/* mpfr_erfc -- The Complementary Error Function of a floating-point number

Copyright 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc.
Contributed by the AriC and Caramel 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 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);
}