path: root/eint.c

/* mpfr_eint, mpfr_eint1 -- the exponential integral

Copyright 2005, 2006, 2007, 2008, 2009, 2010 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 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"

/* eint1(x) = -gamma - log(x) - sum((-1)^k*z^k/k/k!, k=1..infinity) for x > 0
            = - eint(-x) for x < 0
   where
   eint (x) = gamma + log(x) + sum(z^k/k/k!, k=1..infinity) for x > 0
   eint (x) is undefined for x < 0.
*/

/* compute in y an approximation of sum(x^k/k/k!, k=1..infinity),
   and return e such that the absolute error is bound by 2^e ulp(y) */
static mp_exp_t
mpfr_eint_aux (mpfr_t y, mpfr_srcptr x)
{
  mpfr_t eps; /* dynamic (absolute) error bound on t */
  mpfr_t erru, errs;
  mpz_t m, s, t, u;
  mp_exp_t e, sizeinbase;
  mpfr_prec_t w = MPFR_PREC(y);
  unsigned long k;
  MPFR_GROUP_DECL (group);

  /* for |x| <= 1, we have S := sum(x^k/k/k!, k=1..infinity) = x + R(x)
     where |R(x)| <= (x/2)^2/(1-x/2) <= 2*(x/2)^2
     thus |R(x)/x| <= |x|/2
     thus if |x| <= 2^(-PREC(y)) we have |S - o(x)| <= ulp(y) */

  if (MPFR_GET_EXP(x) <= - (mp_exp_t) w)
    {
      mpfr_set (y, x, MPFR_RNDN);
      return 0;
    }

  mpz_init (s); /* initializes to 0 */
  mpz_init (t);
  mpz_init (u);
  mpz_init (m);
  MPFR_GROUP_INIT_3 (group, 31, eps, erru, errs);
  e = mpfr_get_z_2exp (m, x); /* x = m * 2^e */
  MPFR_ASSERTD (mpz_sizeinbase (m, 2) == MPFR_PREC (x));
  if (MPFR_PREC (x) > w)
    {
      e += MPFR_PREC (x) - w;
      mpz_tdiv_q_2exp (m, m, MPFR_PREC (x) - w);
    }
  /* remove trailing zeroes from m: this will speed up much cases where
     x is a small integer divided by a power of 2 */
  k = mpz_scan1 (m, 0);
  mpz_tdiv_q_2exp (m, m, k);
  e += k;
  /* initialize t to 2^w */
  mpz_set_ui (t, 1);
  mpz_mul_2exp (t, t, w);
  mpfr_set_ui (eps, 0, MPFR_RNDN); /* eps[0] = 0 */
  mpfr_set_ui (errs, 0, MPFR_RNDN);
  for (k = 1;; k++)
    {
      /* let eps[k] be the absolute error on t[k]:
         since t[k] = trunc(t[k-1]*m*2^e/k), we have
         eps[k+1] <= 1 + eps[k-1]*m*2^e/k + t[k-1]*m*2^(1-w)*2^e/k
                  =  1 + (eps[k-1] + t[k-1]*2^(1-w))*m*2^e/k
                  = 1 + (eps[k-1]*2^(w-1) + t[k-1])*2^(1-w)*m*2^e/k */
      mpfr_mul_2ui (eps, eps, w - 1, MPFR_RNDU);
      mpfr_add_z (eps, eps, t, MPFR_RNDU);
      MPFR_MPZ_SIZEINBASE2 (sizeinbase, m);
      mpfr_mul_2si (eps, eps, sizeinbase - (w - 1) + e, MPFR_RNDU);
      mpfr_div_ui (eps, eps, k, MPFR_RNDU);
      mpfr_add_ui (eps, eps, 1, MPFR_RNDU);
      mpz_mul (t, t, m);
      if (e < 0)
        mpz_tdiv_q_2exp (t, t, -e);
      else
        mpz_mul_2exp (t, t, e);
      mpz_tdiv_q_ui (t, t, k);
      mpz_tdiv_q_ui (u, t, k);
      mpz_add (s, s, u);
      /* the absolute error on u is <= 1 + eps[k]/k */
      mpfr_div_ui (erru, eps, k, MPFR_RNDU);
      mpfr_add_ui (erru, erru, 1, MPFR_RNDU);
      /* and that on s is the sum of all errors on u */
      mpfr_add (errs, errs, erru, MPFR_RNDU);
      /* we are done when t is smaller than errs */
      if (mpz_sgn (t) == 0)
        sizeinbase = 0;
      else
        MPFR_MPZ_SIZEINBASE2 (sizeinbase, t);
      if (sizeinbase < MPFR_GET_EXP (errs))
        break;
    }
  /* the truncation error is bounded by (|t|+eps)/k*(|x|/k + |x|^2/k^2 + ...)
     <= (|t|+eps)/k*|x|/(k-|x|) */
  mpz_abs (t, t);
  mpfr_add_z (eps, eps, t, MPFR_RNDU);
  mpfr_div_ui (eps, eps, k, MPFR_RNDU);
  mpfr_abs (erru, x, MPFR_RNDU); /* |x| */
  mpfr_mul (eps, eps, erru, MPFR_RNDU);
  mpfr_ui_sub (erru, k, erru, MPFR_RNDD);
  if (MPFR_IS_NEG (erru))
    {
      /* the truncated series does not converge, return fail */
      e = w;
    }
  else
    {
      mpfr_div (eps, eps, erru, MPFR_RNDU);
      mpfr_add (errs, errs, eps, MPFR_RNDU);
      mpfr_set_z (y, s, MPFR_RNDN);
      mpfr_div_2ui (y, y, w, MPFR_RNDN);
      /* errs was an absolute error bound on s. We must convert it to an error
         in terms of ulp(y). Since ulp(y) = 2^(EXP(y)-PREC(y)), we must
         divide the error by 2^(EXP(y)-PREC(y)), but since we divided also
         y by 2^w = 2^PREC(y), we must simply divide by 2^EXP(y). */
      e = MPFR_GET_EXP (errs) - MPFR_GET_EXP (y);
    }
  MPFR_GROUP_CLEAR (group);
  mpz_clear (s);
  mpz_clear (t);
  mpz_clear (u);
  mpz_clear (m);
  return e;
}

/* Return in y an approximation of Ei(x) using the asymptotic expansion:
   Ei(x) = exp(x)/x * (1 + 1/x + 2/x^2 + ... + k!/x^k + ...)
   Assumes x >= PREC(y) * log(2).
   Returns the error bound in terms of ulp(y).
*/
static mp_exp_t
mpfr_eint_asympt (mpfr_ptr y, mpfr_srcptr x)
{
  mpfr_prec_t p = MPFR_PREC(y);
  mpfr_t invx, t, err;
  unsigned long k;
  mp_exp_t err_exp;

  mpfr_init2 (t, p);
  mpfr_init2 (invx, p);
  mpfr_init2 (err, 31); /* error in ulps on y */
  mpfr_ui_div (invx, 1, x, MPFR_RNDN); /* invx = 1/x*(1+u) with |u|<=2^(1-p) */
  mpfr_set_ui (t, 1, MPFR_RNDN); /* exact */
  mpfr_set (y, t, MPFR_RNDN);
  mpfr_set_ui (err, 0, MPFR_RNDN);
  for (k = 1; MPFR_GET_EXP(t) + (mp_exp_t) p > MPFR_GET_EXP(y); k++)
    {
      mpfr_mul (t, t, invx, MPFR_RNDN); /* 2 more roundings */
      mpfr_mul_ui (t, t, k, MPFR_RNDN); /* 1 more rounding: t = k!/x^k*(1+u)^e
                                          with u=2^{-p} and |e| <= 3*k */
      /* we use the fact that |(1+u)^n-1| <= 2*|n*u| for |n*u| <= 1, thus
         the error on t is less than 6*k*2^{-p}*t <= 6*k*ulp(t) */
      /* err is in terms of ulp(y): transform it in terms of ulp(t) */
      mpfr_mul_2si (err, err, MPFR_GET_EXP(y) - MPFR_GET_EXP(t), MPFR_RNDU);
      mpfr_add_ui (err, err, 6 * k, MPFR_RNDU);
      /* transform back in terms of ulp(y) */
      mpfr_div_2si (err, err, MPFR_GET_EXP(y) - MPFR_GET_EXP(t), MPFR_RNDU);
      mpfr_add (y, y, t, MPFR_RNDN);
    }
  /* add the truncation error bounded by ulp(y): 1 ulp */
  mpfr_mul (y, y, invx, MPFR_RNDN); /* err <= 2*err + 3/2 */
  mpfr_exp (t, x, MPFR_RNDN); /* err(t) <= 1/2*ulp(t) */
  mpfr_mul (y, y, t, MPFR_RNDN); /* again: err <= 2*err + 3/2 */
  mpfr_mul_2ui (err, err, 2, MPFR_RNDU);
  mpfr_add_ui (err, err, 8, MPFR_RNDU);
  err_exp = MPFR_GET_EXP(err);
  mpfr_clear (t);
  mpfr_clear (invx);
  mpfr_clear (err);
  return err_exp;
}

int
mpfr_eint (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd)
{
  int inex;
  mpfr_t tmp, ump;
  mp_exp_t err, te;
  mpfr_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)))
    {
      /* exp(NaN) = exp(-Inf) = NaN */
      if (MPFR_IS_NAN (x) || (MPFR_IS_INF (x) && MPFR_IS_NEG(x)))
        {
          MPFR_SET_NAN (y);
          MPFR_RET_NAN;
        }
      /* eint(+inf) = +inf */
      else if (MPFR_IS_INF (x))
        {
          MPFR_SET_INF(y);
          MPFR_SET_POS(y);
          MPFR_RET(0);
        }
      else /* eint(+/-0) = -Inf */
        {
          MPFR_SET_INF(y);
          MPFR_SET_NEG(y);
          MPFR_RET(0);
        }
    }

  /* eint(x) = NaN for x < 0 */
  if (MPFR_IS_NEG(x))
    {
      MPFR_SET_NAN (y);
      MPFR_RET_NAN;
    }

  MPFR_SAVE_EXPO_MARK (expo);

  /* Since eint(x) >= exp(x)/x, we have log2(eint(x)) >= (x-log(x))/log(2).
     Let's compute k <= (x-log(x))/log(2) in a low precision. If k >= emax,
     then log2(eint(x)) >= emax, and eint(x) >= 2^emax, i.e. it overflows. */
  mpfr_init2 (tmp, 64);
  mpfr_init2 (ump, 64);
  mpfr_log (tmp, x, MPFR_RNDU);
  mpfr_sub (ump, x, tmp, MPFR_RNDD);
  mpfr_const_log2 (tmp, MPFR_RNDU);
  mpfr_div (ump, ump, tmp, MPFR_RNDD);
  /* FIXME: We really need mpfr_set_exp_t and mpfr_cmp_exp_t functions. */
  MPFR_ASSERTN (MPFR_EMAX_MAX <= LONG_MAX);
  if (mpfr_cmp_ui (ump, __gmpfr_emax) >= 0)
    {
      mpfr_clear (tmp);
      mpfr_clear (ump);
      MPFR_SAVE_EXPO_FREE (expo);
      return mpfr_overflow (y, rnd, 1);
    }

  /* Init stuff */
  prec = MPFR_PREC (y) + 2 * MPFR_INT_CEIL_LOG2 (MPFR_PREC (y)) + 6;

  /* eint() has a root 0.37250741078136663446..., so if x is near,
     already take more bits */
  if (MPFR_GET_EXP(x) == -1) /* 1/4 <= x < 1/2 */
    {
      double d;
      d = mpfr_get_d (x, MPFR_RNDN) - 0.37250741078136663;
      d = (d == 0.0) ? -53 : __gmpfr_ceil_log2 (d);
      prec += -d;
    }

  mpfr_set_prec (tmp, prec);
  mpfr_set_prec (ump, prec);

  MPFR_ZIV_INIT (loop, prec);            /* Initialize the ZivLoop controler */
  for (;;)                               /* Infinite loop */
    {
      /* We need that the smallest value of k!/x^k is smaller than 2^(-p).
         The minimum is obtained for x=k, and it is smaller than e*sqrt(x)/e^x
         for x>=1. */
      if (MPFR_GET_EXP (x) > 0 && mpfr_cmp_d (x, ((double) prec +
                            0.5 * (double) MPFR_GET_EXP (x)) * LOG2 + 1.0) > 0)
        err = mpfr_eint_asympt (tmp, x);
      else
        {
          err = mpfr_eint_aux (tmp, x); /* error <= 2^err ulp(tmp) */
          te = MPFR_GET_EXP(tmp);
          mpfr_const_euler (ump, MPFR_RNDN); /* 0.577 -> EXP(ump)=0 */
          mpfr_add (tmp, tmp, ump, MPFR_RNDN);
          /* error <= 1/2 + 1/2*2^(EXP(ump)-EXP(tmp)) + 2^(te-EXP(tmp)+err)
             <= 1/2 + 2^(MAX(EXP(ump), te+err+1) - EXP(tmp))
             <= 2^(MAX(0, 1 + MAX(EXP(ump), te+err+1) - EXP(tmp))) */
          err = MAX(1, te + err + 2) - MPFR_GET_EXP(tmp);
          err = MAX(0, err);
          te = MPFR_GET_EXP(tmp);
          mpfr_log (ump, x, MPFR_RNDN);
          mpfr_add (tmp, tmp, ump, MPFR_RNDN);
          /* same formula as above, except now EXP(ump) is not 0 */
          err += te + 1;
          if (MPFR_LIKELY (!MPFR_IS_ZERO (ump)))
            err = MAX (MPFR_GET_EXP (ump), err);
          err = MAX(0, err - MPFR_GET_EXP (tmp));
        }
      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_set_prec (ump, prec);
    }
  MPFR_ZIV_FREE (loop);                  /* Free the ZivLoop Controler */

  inex = mpfr_set (y, tmp, rnd);    /* Set y to the computed value */
  mpfr_clear (tmp);
  mpfr_clear (ump);

  MPFR_SAVE_EXPO_FREE (expo);
  return mpfr_check_range (y, inex, rnd);
}