/* mpfr_log1p -- Compute log(1+x)
Copyright 2001-2018 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"
/* Put in y an approximation of log(1+x) for x small.
We assume |x| < 1, in which case:
|x/2| <= |log(1+x)| = |x - x^2/2 + x^3/3 - x^4/4 + ...| <= |x|.
Return k such that the error is bounded by 2^k*ulp(y).
*/
static int
mpfr_log1p_small (mpfr_ptr y, mpfr_srcptr x)
{
mpfr_prec_t p = MPFR_PREC(y), err;
mpfr_t t, u;
unsigned long i;
int k;
MPFR_ASSERTD(MPFR_GET_EXP (x) <= 0); /* ensures |x| < 1 */
/* in the following, theta represents a value with |theta| <= 2^(1-p)
(might be a different value each time) */
mpfr_init2 (t, p);
mpfr_init2 (u, p);
mpfr_set (t, x, MPFR_RNDF); /* t = x * (1 + theta) */
mpfr_set (y, t, MPFR_RNDF); /* exact */
for (i = 2; ; i++)
{
mpfr_mul (t, t, x, MPFR_RNDF); /* t = x^i * (1 + theta)^i */
mpfr_div_ui (u, t, i, MPFR_RNDF); /* u = x^i/i * (1 + theta)^(i+1) */
if (MPFR_GET_EXP (u) <= MPFR_GET_EXP (y) - p) /* |u| < ulp(y) */
break;
if (i & 1)
mpfr_add (y, y, u, MPFR_RNDF); /* error <= ulp(y) */
else
mpfr_sub (y, y, u, MPFR_RNDF); /* error <= ulp(y) */
}
/* We assume |(1 + theta)^(i+1)| <= 2.
The neglected part is at most |u| + |u|/2 + ... <= 2|u| < 2 ulp(y)
which has to be multiplied by |(1 + theta)^(i+1)| <= 2, thus at most
4 ulp(y).
The rounding error on y is bounded by:
* for the (i-2) add/sub, each error is bounded by ulp(y),
and since |y| <= |x|, this yields (i-2)*ulp(x)
* from Lemma 3.1 from [Higham02] (see algorithms.tex),
the relative error on u at step i is bounded by:
(i+1)*epsilon/(1-(i+1)*epsilon) where epsilon = 2^(1-p).
If (i+1)*epsilon <= 1/2, then the relative error on u at
step i is bounded by 2*(i+1)*epsilon, and since |u| <= 1/2^(i+1)
at step i, this gives an absolute error bound of;
2*epsilon*x*(3/2^3 + 4/2^4 + 5/2^5 + ...) <= 2*2^(1-p)*x =
4*2^(-p)*x <= 4*ulp(x).
If (i+1)*epsilon <= 1/2, then the relative error on u at step i
is bounded by (i+1)*epsilon/(1-(i+1)*epsilon) <= 1, thus it follows
|(1 + theta)^(i+1)| <= 2.
Finally the total error is bounded by 4*ulp(y) + (i-2)*ulp(x) + 4*ulp(x)
= 4*ulp(y) + (i+2)*ulp(x).
Since x/2 <= y, we have ulp(x) <= 2*ulp(y), thus the error is bounded by:
(2*i+8)*ulp(y).
*/
err = 2 * i + 8;
k = __gmpfr_int_ceil_log2 (err);
MPFR_ASSERTN(k < p);
/* if k < p, since k = ceil(log2(err)), we have err <= 2^k <= 2^(p-1),
thus i+4 = err/2 <= 2^(p-2), thus (i+4)*epsilon <= 1/2, which implies
our assumption (i+1)*epsilon <= 1/2. */
mpfr_clear (t);
mpfr_clear (u);
return k;
}
/* The computation of log1p is done by
log1p(x) = log(1+x)
except when x is very small, in which case log1p(x) = x + tiny error,
or when x is small, where we use directly the Taylor expansion.
*/
int
mpfr_log1p (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
int comp, inexact;
mpfr_exp_t ex;
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,
inexact));
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
if (MPFR_IS_NAN (x))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
/* check for inf or -inf (result is not defined) */
else if (MPFR_IS_INF (x))
{
if (MPFR_IS_POS (x))
{
MPFR_SET_INF (y);
MPFR_SET_POS (y);
MPFR_RET (0);
}
else
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
}
else /* x is zero */
{
MPFR_ASSERTD (MPFR_IS_ZERO (x));
MPFR_SET_ZERO (y); /* log1p(+/- 0) = +/- 0 */
MPFR_SET_SAME_SIGN (y, x);
MPFR_RET (0);
}
}
ex = MPFR_GET_EXP (x);
if (ex < 0) /* -0.5 < x < 0.5 */
{
/* For x > 0, abs(log(1+x)-x) < x^2/2.
For x > -0.5, abs(log(1+x)-x) < x^2. */
if (MPFR_IS_POS (x))
MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, - ex - 1, 0, 0, rnd_mode, {});
else
MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, - ex, 0, 1, rnd_mode, {});
}
comp = mpfr_cmp_si (x, -1);
/* log1p(x) is undefined for x < -1 */
if (MPFR_UNLIKELY(comp <= 0))
{
if (comp == 0)
/* x=0: log1p(-1)=-inf (divide-by-zero exception) */
{
MPFR_SET_INF (y);
MPFR_SET_NEG (y);
MPFR_SET_DIVBY0 ();
MPFR_RET (0);
}
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
MPFR_SAVE_EXPO_MARK (expo);
/* General case */
{
/* Declaration of the intermediary variable */
mpfr_t t;
/* Declaration of the size variable */
mpfr_prec_t Ny = MPFR_PREC(y); /* target precision */
mpfr_prec_t Nt; /* working precision */
mpfr_exp_t err; /* error */
MPFR_ZIV_DECL (loop);
/* compute the precision of intermediary variable */
/* the optimal number of bits : see algorithms.tex */
Nt = Ny + MPFR_INT_CEIL_LOG2 (Ny) + 6;
/* if |x| is smaller than 2^(-e), we will loose about e bits
in log(1+x) */
if (MPFR_EXP(x) < 0)
Nt += -MPFR_EXP(x);
/* initialize of intermediary variable */
mpfr_init2 (t, Nt);
/* First computation of log1p */
MPFR_ZIV_INIT (loop, Nt);
for (;;)
{
int k;
/* small case: assuming the AGM algorithm used by mpfr_log uses
log2(p) steps for a precision of p bits, we try the special
variant whenever EXP(x) <= -p/log2(p). */
k = 1 + __gmpfr_int_ceil_log2 (Ny); /* the +1 avoids a division by 0
when Ny=1 */
if (MPFR_GET_EXP (x) <= - (mpfr_exp_t) (Ny / k))
/* this implies EXP(x) <= 0 thus x < 1 */
err = Nt - mpfr_log1p_small (t, x);
else
{
/* compute log1p */
inexact = mpfr_add_ui (t, x, 1, MPFR_RNDN); /* 1+x */
/* if inexact = 0, then t = x+1, and the result is simply log(t) */
if (inexact == 0)
{
inexact = mpfr_log (y, t, rnd_mode);
goto end;
}
mpfr_log (t, t, MPFR_RNDN); /* log(1+x) */
/* the error is bounded by (1/2+2^(1-EXP(t))*ulp(t)
(cf algorithms.tex)
if EXP(t)>=2, then error <= ulp(t)
if EXP(t)<=1, then error <= 2^(2-EXP(t))*ulp(t) */
err = Nt - MAX (0, 2 - MPFR_GET_EXP (t));
}
if (MPFR_LIKELY (MPFR_CAN_ROUND (t, err, Ny, rnd_mode)))
break;
/* increase the precision */
MPFR_ZIV_NEXT (loop, Nt);
mpfr_set_prec (t, Nt);
}
inexact = mpfr_set (y, t, rnd_mode);
end:
MPFR_ZIV_FREE (loop);
mpfr_clear (t);
}
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}