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/* mpfr_mpn_exp -- auxiliary function for mpfr_get_str and mpfr_set_str
Copyright 1999-2020 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
#include "mpfr-impl.h"
/* this function computes an approximation of b^e in {a, n}, with exponent
stored in exp_r. The computed value is rounded toward zero (truncated).
It returns an integer f such that the final error is bounded by 2^f ulps,
that is:
a*2^exp_r <= b^e <= 2^exp_r (a + 2^f),
where a represents {a, n}, i.e. the integer
a[0] + a[1]*B + ... + a[n-1]*B^(n-1) where B=2^GMP_NUMB_BITS
Return -1 is the result is exact.
Return -2 if an overflow occurred in the computation of exp_r.
*/
long
mpfr_mpn_exp (mp_limb_t *a, mpfr_exp_t *exp_r, int b, mpfr_exp_t e, size_t n)
{
mp_limb_t *c, B;
mpfr_exp_t f, h;
int i;
unsigned long t; /* number of bits in e */
unsigned long bits;
size_t n1;
unsigned int error; /* (number - 1) of loop a^2b inexact */
/* error == t means no error */
int err_s_a2 = 0;
int err_s_ab = 0; /* number of error when shift A^2, AB */
MPFR_TMP_DECL(marker);
MPFR_ASSERTN(e > 0);
MPFR_ASSERTN((2 <= b) && (b <= 62));
MPFR_TMP_MARK(marker);
/* initialization of a, b, f, h */
/* normalize the base */
B = (mp_limb_t) b;
count_leading_zeros (h, B);
bits = GMP_NUMB_BITS - h;
B = B << h;
h = - h;
/* allocate space for A and set it to B */
c = MPFR_TMP_LIMBS_ALLOC (2 * n);
a [n - 1] = B;
MPN_ZERO (a, n - 1);
/* initial exponent for A: invariant is A = {a, n} * 2^f */
f = h - (n - 1) * GMP_NUMB_BITS;
/* determine number of bits in e */
count_leading_zeros (t, (mp_limb_t) e);
t = GMP_NUMB_BITS - t; /* number of bits of exponent e */
error = t; /* error <= GMP_NUMB_BITS */
MPN_ZERO (c, 2 * n);
for (i = t - 2; i >= 0; i--)
{
/* determine precision needed */
bits = n * GMP_NUMB_BITS - mpn_scan1 (a, 0);
n1 = (n * GMP_NUMB_BITS - bits) / GMP_NUMB_BITS;
/* square of A : {c+2n1, 2(n-n1)} = {a+n1, n-n1}^2 */
/* TODO: we should use a short square here, but this needs to redo
the error analysis */
mpn_sqr (c + 2 * n1, a + n1, n - n1);
/* set {c+n, 2n1-n} to 0 : {c, n} = {a, n}^2*K^n */
/* check overflow on f */
if (MPFR_UNLIKELY(f < MPFR_EXP_MIN/2 || f > MPFR_EXP_MAX/2))
{
overflow:
MPFR_TMP_FREE(marker);
return -2;
}
/* FIXME: Could f = 2*f + n * GMP_NUMB_BITS be used? */
f = 2*f;
MPFR_SADD_OVERFLOW (f, f, n * GMP_NUMB_BITS,
mpfr_exp_t, mpfr_uexp_t,
MPFR_EXP_MIN, MPFR_EXP_MAX,
goto overflow, goto overflow);
if ((c[2*n - 1] & MPFR_LIMB_HIGHBIT) == 0)
{
/* shift A by one bit to the left */
mpn_lshift (a, c + n, n, 1);
a[0] |= mpn_lshift (c + n - 1, c + n - 1, 1, 1);
f --;
if (error != t)
err_s_a2 ++;
}
else
MPN_COPY (a, c + n, n);
if ((error == t) && (2 * n1 <= n) &&
(mpn_scan1 (c + 2 * n1, 0) < (n - 2 * n1) * GMP_NUMB_BITS))
error = i;
if (e & ((mpfr_exp_t) 1 << i))
{
/* multiply A by B */
c[2 * n - 1] = mpn_mul_1 (c + n - 1, a, n, B);
f += h + GMP_NUMB_BITS;
if ((c[2 * n - 1] & MPFR_LIMB_HIGHBIT) == 0)
{ /* shift A by one bit to the left */
mpn_lshift (a, c + n, n, 1);
a[0] |= mpn_lshift (c + n - 1, c + n - 1, 1, 1);
f --;
}
else
{
MPN_COPY (a, c + n, n);
if (error != t)
err_s_ab ++;
}
if ((error == t) && (c[n - 1] != 0))
error = i;
}
}
MPFR_TMP_FREE(marker);
*exp_r = f;
if (error == t)
return -1; /* result is exact */
else /* error <= t-2 <= GMP_NUMB_BITS-2
err_s_ab, err_s_a2 <= t-1 */
{
/* if there are p loops after the first inexact result, with
j shifts in a^2 and l shifts in a*b, then the final error is
at most 2^(p+ceil((j+1)/2)+l+1)*ulp(res).
This is bounded by 2^(5/2*t-1/2) where t is the number of bits of e.
*/
error = error + err_s_ab + err_s_a2 / 2 + 3; /* <= 5t/2-1/2 */
#if 0
if ((error - 1) >= ((n * GMP_NUMB_BITS - 1) / 2))
error = n * GMP_NUMB_BITS; /* result is completely wrong:
this is very unlikely since error is
at most 5/2*log_2(e), and
n * GMP_NUMB_BITS is at least
3*log_2(e) */
#endif
return error;
}
}
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