/* mpfr_eint, mpfr_eint1 -- the exponential integral
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"
/* 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 mpfr_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;
mpfr_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) <= - (mpfr_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 mpfr_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;
mpfr_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) + (mpfr_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;
mpfr_exp_t err, te;
mpfr_prec_t prec;
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)))
{
/* eint(NaN) = eint(-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_set_divby0 ();
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_cmpfr_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 */
/* FIXME: do not use native floating-point here. */
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 Controller */
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);
}