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/* mpfr_compound_si --- compound(x,n) = (1+x)^n
Copyright 2021 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
https://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 /* needed for MPFR_INT_CEIL_LOG2 */
#include "mpfr-impl.h"
/* assuming |(1+x)^n - 1| < 1/4*ulp(1), return correct rounding,
where s is the sign of n*log2(1+x) */
static int
mpfr_compound_near_one (mpfr_ptr y, int s, mpfr_rnd_t rnd_mode)
{
mpfr_set_ui (y, 1, rnd_mode); /* exact */
if (rnd_mode == MPFR_RNDN || rnd_mode == MPFR_RNDF
|| (s > 0 && (rnd_mode == MPFR_RNDZ || rnd_mode == MPFR_RNDD))
|| (s < 0 && (rnd_mode == MPFR_RNDA || rnd_mode == MPFR_RNDU)))
{
/* round toward 1 */
return -s;
}
else if (s > 0) /* necessarily RNDA or RNDU */
{
/* round toward +Inf */
mpfr_nextabove (y);
return +1;
}
else /* necessarily s < 0 and RNDZ or RNDD */
{
/* round toward 0 */
mpfr_nextbelow (y);
return -1;
}
}
/* put in y the correctly rounded value of (1+x)^n */
int
mpfr_compound_si (mpfr_ptr y, mpfr_srcptr x, long n, mpfr_rnd_t rnd_mode)
{
int inexact, compared, k, nloop;
mpfr_t t;
mpfr_exp_t e;
mpfr_prec_t prec;
MPFR_ZIV_DECL (loop);
MPFR_SAVE_EXPO_DECL (expo);
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg n=%ld rnd=%d",
mpfr_get_prec(x), mpfr_log_prec, x, n, rnd_mode),
("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y, inexact));
/* Special cases */
if (MPFR_IS_SINGULAR (x))
{
if (MPFR_IS_INF (x) && MPFR_IS_NEG (x))
{
/* compound(-Inf,n) is NaN */
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else if (n == 0 || MPFR_IS_ZERO (x))
{
/* compound(x,0) = 1 for x >= -1 or NaN (the only special value
of x that is not concerned is -Inf, already handled);
compound(0,n) = 1 */
return mpfr_set_ui (y, 1, rnd_mode);
}
else if (MPFR_IS_NAN (x))
{
/* compound(NaN,n) is NaN, except for n = 0, already handled. */
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else if (MPFR_IS_INF (x)) /* x = +Inf */
{
MPFR_ASSERTD (MPFR_IS_POS (x));
if (n < 0) /* (1+Inf)^n = +0 for n < 0 */
MPFR_SET_ZERO (y);
else /* n > 0: (1+Inf)^n = +Inf */
MPFR_SET_INF (y);
MPFR_SET_POS (y);
MPFR_RET (0); /* exact 0 or infinity */
}
}
/* (1+x)^n = NaN for x < -1 */
compared = mpfr_cmp_si (x, -1);
if (compared < 0)
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
/* compound(x,0) gives 1 for x >= 1 */
if (n == 0)
return mpfr_set_ui (y, 1, rnd_mode);
if (compared == 0)
{
if (n < 0)
{
/* compound(-1,n) = +Inf with divide-by-zero exception */
MPFR_SET_INF (y);
MPFR_SET_POS (y);
MPFR_SET_DIVBY0 ();
MPFR_RET (0);
}
else
{
/* compound(-1,n) = +0 */
MPFR_SET_ZERO (y);
MPFR_SET_POS (y);
MPFR_RET (0);
}
}
if (n == 1)
return mpfr_add_ui (y, x, 1, rnd_mode);
MPFR_SAVE_EXPO_MARK (expo);
prec = MPFR_PREC(y);
prec += MPFR_INT_CEIL_LOG2 (prec) + 6;
mpfr_init2 (t, prec);
k = MPFR_INT_CEIL_LOG2(SAFE_ABS (unsigned long, n)); /* thus |n| <= 2^k */
MPFR_ZIV_INIT (loop, prec);
for (nloop = 0; ; nloop++)
{
/* we compute (1+x)^n as 2^(n*log2p1(x)) */
inexact = mpfr_log2p1 (t, x, MPFR_RNDN) != 0;
e = MPFR_GET_EXP(t);
/* |t - log2(1+x)| <= 1/2*ulp(t) = 2^(e-prec-1) */
inexact |= mpfr_mul_si (t, t, n, MPFR_RNDN) != 0;
/* |t - n*log2(1+x)| <= 2^(e2-prec-1) + |n|*2^(e-prec-1)
<= 2^(e2-prec-1) + 2^(e+k-prec-1) <= 2^(e+k-prec)
where |n| <= 2^k, and e2 is the new exponent of t. */
MPFR_ASSERTD(MPFR_GET_EXP(t) <= e + k);
e += k;
/* |t - n*log2(1+x)| <= 2^(e-prec) */
/* detect overflow */
if (nloop == 0 && mpfr_cmp_si (t, __gmpfr_emax) >= 0)
{
MPFR_ZIV_FREE (loop);
mpfr_clear (t);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_overflow (y, rnd_mode, 1);
}
/* detect underflow */
if (nloop == 0 && mpfr_cmp_si (t, __gmpfr_emin - 1) <= 0)
{
MPFR_ZIV_FREE (loop);
mpfr_clear (t);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_underflow (y,
(rnd_mode == MPFR_RNDN) ? MPFR_RNDZ : rnd_mode, 1);
}
/* Detect cases where result is 1 or 1+ulp(1) or 1-1/2*ulp(1):
|2^t - 1| = |exp(t*log(2)) - 1| <= |t|*log(2) < |t| */
if (nloop == 0 && MPFR_GET_EXP(t) < - (mpfr_exp_t) MPFR_PREC(y))
{
/* since ulp(1) = 2^(1-PREC(y)), we have |t| < 1/4*ulp(1) */
/* mpfr_compound_near_one must be called in the extended
exponent range, so that 1 is representable. */
inexact = mpfr_compound_near_one (y, MPFR_SIGN (t), rnd_mode);
goto end;
}
inexact |= mpfr_exp2 (t, t, MPFR_RNDA) != 0;
/* |t - (1+x)^n| <= ulp(t) + |t|*log(2)*2^(e-prec)
< 2^(EXP(t)-prec) + 2^(EXP(t)+e-prec) */
e = (e >= 0) ? e + 1 : 1;
/* now |t - (1+x)^n| < 2^(EXP(t)+e-prec) */
if (MPFR_LIKELY (inexact == 0 ||
MPFR_CAN_ROUND (t, prec - e, MPFR_PREC(y), rnd_mode)))
break;
/* Exact cases like compound(0.5,2) = 9/4 must be detected, since
except for 1+x power of 2, the log2p1 above will be inexact,
so that in the Ziv test, inexact != 0 and MPFR_CAN_ROUND will
fail (even for RNDN, as the ternary value cannot be determined),
yielding an infinite loop.
For an exact case in precision prec(y), 1+x will necessarily
be exact in precision prec(y), thus also in prec(t), where
prec(t) >= prec(y), and we can use mpfr_pow_si under this
condition (which will also evaluate some non-exact cases). */
if (mpfr_add_ui (t, x, 1, MPFR_RNDZ) == 0)
{
inexact = mpfr_pow_si (y, t, n, rnd_mode);
goto end;
}
MPFR_ZIV_NEXT (loop, prec);
mpfr_set_prec (t, prec);
}
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);
}
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