/* mpfr_exp -- exponential of a floating-point number
Copyright 1999, 2001-2017 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 /* for MPFR_MPZ_SIZEINBASE2 */
#include "mpfr-impl.h"
/* y <- exp(p/2^r) within 1 ulp, using 2^m terms from the series
Assume |p/2^r| < 1.
We use the following binary splitting formula:
P(a,b) = p if a+1=b, P(a,c)*P(c,b) otherwise
Q(a,b) = a*2^r if a+1=b [except Q(0,1)=1], Q(a,c)*Q(c,b) otherwise
T(a,b) = P(a,b) if a+1=b, Q(c,b)*T(a,c)+P(a,c)*T(c,b) otherwise
Then exp(p/2^r) ~ T(0,i)/Q(0,i) for i so that (p/2^r)^i/i! is small enough.
Since P(a,b) = p^(b-a), and we consider only values of b-a of the form 2^j,
we don't need to compute P(), we only precompute p^(2^j) in the ptoj[] array
below.
Since Q(a,b) is divisible by 2^(r*(b-a-1)), we don't compute the power of
two part.
*/
static void
mpfr_exp_rational (mpfr_ptr y, mpz_ptr p, long r, int m,
mpz_t *Q, mpfr_prec_t *mult)
{
unsigned long n, i, j;
mpz_t *S, *ptoj;
mpfr_prec_t *log2_nb_terms;
mpfr_exp_t diff, expo;
mpfr_prec_t precy = MPFR_PREC(y), prec_i_have, prec_ptoj;
int k, l;
MPFR_ASSERTN ((size_t) m < sizeof (long) * CHAR_BIT - 1);
S = Q + (m+1);
ptoj = Q + 2*(m+1); /* ptoj[i] = mantissa^(2^i) */
log2_nb_terms = mult + (m+1);
/* Normalize p */
MPFR_ASSERTD (mpz_cmp_ui (p, 0) != 0);
n = mpz_scan1 (p, 0); /* number of trailing zeros in p */
mpz_tdiv_q_2exp (p, p, n);
r -= n; /* since |p/2^r| < 1 and p >= 1, r >= 1 */
/* Set initial var */
mpz_set (ptoj[0], p);
for (k = 1; k < m; k++)
mpz_mul (ptoj[k], ptoj[k-1], ptoj[k-1]); /* ptoj[k] = p^(2^k) */
mpz_set_ui (Q[0], 1);
mpz_set_ui (S[0], 1);
k = 0;
mult[0] = 0; /* the multiplier P[k]/Q[k] for the remaining terms
satisfies P[k]/Q[k] <= 2^(-mult[k]) */
log2_nb_terms[0] = 0; /* log2(#terms) [exact in 1st loop where 2^k] */
prec_i_have = 0;
/* Main Loop */
n = 1UL << m;
for (i = 1; (prec_i_have < precy) && (i < n); i++)
{
/* invariant: Q[0]*Q[1]*...*Q[k] equals i! */
k++;
log2_nb_terms[k] = 0; /* 1 term */
mpz_set_ui (Q[k], i + 1);
mpz_set_ui (S[k], i + 1);
j = i + 1; /* we have computed j = i+1 terms so far */
l = 0;
while ((j & 1) == 0) /* combine and reduce */
{
/* invariant: S[k] corresponds to 2^l consecutive terms */
mpz_mul (S[k], S[k], ptoj[l]);
mpz_mul (S[k-1], S[k-1], Q[k]);
/* Q[k] corresponds to 2^l consecutive terms too.
Since it does not contains the factor 2^(r*2^l),
when going from l to l+1 we need to multiply
by 2^(r*2^(l+1))/2^(r*2^l) = 2^(r*2^l) */
mpz_mul_2exp (S[k-1], S[k-1], r << l);
mpz_add (S[k-1], S[k-1], S[k]);
mpz_mul (Q[k-1], Q[k-1], Q[k]);
log2_nb_terms[k-1] ++; /* number of terms in S[k-1]
is a power of 2 by construction */
MPFR_MPZ_SIZEINBASE2 (prec_i_have, Q[k]);
MPFR_MPZ_SIZEINBASE2 (prec_ptoj, ptoj[l]);
mult[k-1] += prec_i_have + (r << l) - prec_ptoj - 1;
prec_i_have = mult[k] = mult[k-1];
/* since mult[k] >= mult[k-1] + nbits(Q[k]),
we have Q[0]*...*Q[k] <= 2^mult[k] = 2^prec_i_have */
l ++;
j >>= 1;
k --;
}
}
/* accumulate all products in S[0] and Q[0]. Warning: contrary to above,
here we do not have log2_nb_terms[k-1] = log2_nb_terms[k]+1. */
l = 0; /* number of accumulated terms in the right part S[k]/Q[k] */
while (k > 0)
{
j = log2_nb_terms[k-1];
mpz_mul (S[k], S[k], ptoj[j]);
mpz_mul (S[k-1], S[k-1], Q[k]);
l += 1 << log2_nb_terms[k];
mpz_mul_2exp (S[k-1], S[k-1], r * l);
mpz_add (S[k-1], S[k-1], S[k]);
mpz_mul (Q[k-1], Q[k-1], Q[k]);
k--;
}
/* Q[0] now equals i! */
MPFR_MPZ_SIZEINBASE2 (prec_i_have, S[0]);
diff = (mpfr_exp_t) prec_i_have - 2 * (mpfr_exp_t) precy;
expo = diff;
if (diff >= 0)
mpz_fdiv_q_2exp (S[0], S[0], diff);
else
mpz_mul_2exp (S[0], S[0], -diff);
MPFR_MPZ_SIZEINBASE2 (prec_i_have, Q[0]);
diff = (mpfr_exp_t) prec_i_have - (mpfr_prec_t) precy;
expo -= diff;
if (diff > 0)
mpz_fdiv_q_2exp (Q[0], Q[0], diff);
else
mpz_mul_2exp (Q[0], Q[0], -diff);
mpz_tdiv_q (S[0], S[0], Q[0]);
mpfr_set_z (y, S[0], MPFR_RNDD);
MPFR_SET_EXP (y, MPFR_GET_EXP (y) + expo - r * (i - 1) );
}
#define shift (GMP_NUMB_BITS/2)
int
mpfr_exp_3 (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
mpfr_t t, x_copy, tmp;
mpz_t uk;
mpfr_exp_t ttt, shift_x;
unsigned long twopoweri;
mpz_t *P;
mpfr_prec_t *mult;
int i, k, loop;
int prec_x;
mpfr_prec_t realprec, Prec;
int iter;
int inexact = 0;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_ZIV_DECL (ziv_loop);
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));
MPFR_SAVE_EXPO_MARK (expo);
/* decompose x */
/* we first write x = 1.xxxxxxxxxxxxx
----- k bits -- */
prec_x = MPFR_INT_CEIL_LOG2 (MPFR_PREC (x)) - MPFR_LOG2_GMP_NUMB_BITS;
if (prec_x < 0)
prec_x = 0;
ttt = MPFR_GET_EXP (x);
mpfr_init2 (x_copy, MPFR_PREC(x));
mpfr_set (x_copy, x, MPFR_RNDD);
/* we shift to get a number less than 1 */
if (ttt > 0)
{
shift_x = ttt;
mpfr_div_2ui (x_copy, x, ttt, MPFR_RNDN);
ttt = MPFR_GET_EXP (x_copy);
}
else
shift_x = 0;
MPFR_ASSERTD (ttt <= 0);
/* Init prec and vars */
realprec = MPFR_PREC (y) + MPFR_INT_CEIL_LOG2 (prec_x + MPFR_PREC (y));
Prec = realprec + shift + 2 + shift_x;
mpfr_init2 (t, Prec);
mpfr_init2 (tmp, Prec);
mpz_init (uk);
/* Main loop */
MPFR_ZIV_INIT (ziv_loop, realprec);
for (;;)
{
int scaled = 0;
MPFR_BLOCK_DECL (flags);
k = MPFR_INT_CEIL_LOG2 (Prec) - MPFR_LOG2_GMP_NUMB_BITS;
/* now we have to extract */
twopoweri = GMP_NUMB_BITS;
/* Allocate tables */
P = (mpz_t*) (*__gmp_allocate_func) (3*(k+2)*sizeof(mpz_t));
for (i = 0; i < 3*(k+2); i++)
mpz_init (P[i]);
mult = (mpfr_prec_t*) (*__gmp_allocate_func) (2*(k+2)*sizeof(mpfr_prec_t));
/* Particular case for i==0 */
mpfr_extract (uk, x_copy, 0);
MPFR_ASSERTD (mpz_cmp_ui (uk, 0) != 0);
mpfr_exp_rational (tmp, uk, shift + twopoweri - ttt, k + 1, P, mult);
for (loop = 0; loop < shift; loop++)
mpfr_sqr (tmp, tmp, MPFR_RNDD);
twopoweri *= 2;
/* General case */
iter = (k <= prec_x) ? k : prec_x;
for (i = 1; i <= iter; i++)
{
mpfr_extract (uk, x_copy, i);
if (MPFR_LIKELY (mpz_cmp_ui (uk, 0) != 0))
{
mpfr_exp_rational (t, uk, twopoweri - ttt, k - i + 1, P, mult);
mpfr_mul (tmp, tmp, t, MPFR_RNDD);
}
MPFR_ASSERTN (twopoweri <= LONG_MAX/2);
twopoweri *=2;
}
/* Clear tables */
for (i = 0; i < 3*(k+2); i++)
mpz_clear (P[i]);
(*__gmp_free_func) (P, 3*(k+2)*sizeof(mpz_t));
(*__gmp_free_func) (mult, 2*(k+2)*sizeof(mpfr_prec_t));
if (shift_x > 0)
{
MPFR_BLOCK (flags, {
for (loop = 0; loop < shift_x - 1; loop++)
mpfr_sqr (tmp, tmp, MPFR_RNDD);
mpfr_sqr (t, tmp, MPFR_RNDD);
} );
if (MPFR_UNLIKELY (MPFR_OVERFLOW (flags)))
{
/* tmp <= exact result, so that it is a real overflow. */
inexact = mpfr_overflow (y, rnd_mode, 1);
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_OVERFLOW);
break;
}
if (MPFR_UNLIKELY (MPFR_UNDERFLOW (flags)))
{
/* This may be a spurious underflow. So, let's scale
the result. */
mpfr_mul_2ui (tmp, tmp, 1, MPFR_RNDD); /* no overflow, exact */
mpfr_sqr (t, tmp, MPFR_RNDD);
if (MPFR_IS_ZERO (t))
{
/* approximate result < 2^(emin - 3), thus
exact result < 2^(emin - 2). */
inexact = mpfr_underflow (y, (rnd_mode == MPFR_RNDN) ?
MPFR_RNDZ : rnd_mode, 1);
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_UNDERFLOW);
break;
}
scaled = 1;
}
}
if (mpfr_can_round (shift_x > 0 ? t : tmp, realprec, MPFR_RNDN, MPFR_RNDZ,
MPFR_PREC(y) + (rnd_mode == MPFR_RNDN)))
{
inexact = mpfr_set (y, shift_x > 0 ? t : tmp, rnd_mode);
if (MPFR_UNLIKELY (scaled && MPFR_IS_PURE_FP (y)))
{
int inex2;
mpfr_exp_t ey;
/* The result has been scaled and needs to be corrected. */
ey = MPFR_GET_EXP (y);
inex2 = mpfr_mul_2si (y, y, -2, rnd_mode);
if (inex2) /* underflow */
{
if (rnd_mode == MPFR_RNDN && inexact < 0 &&
MPFR_IS_ZERO (y) && ey == __gmpfr_emin + 1)
{
/* Double rounding case: in MPFR_RNDN, the scaled
result has been rounded downward to 2^emin.
As the exact result is > 2^(emin - 2), correct
rounding must be done upward. */
/* TODO: make sure in coverage tests that this line
is reached. */
inexact = mpfr_underflow (y, MPFR_RNDU, 1);
}
else
{
/* No double rounding. */
inexact = inex2;
}
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_UNDERFLOW);
}
}
break;
}
MPFR_ZIV_NEXT (ziv_loop, realprec);
Prec = realprec + shift + 2 + shift_x;
mpfr_set_prec (t, Prec);
mpfr_set_prec (tmp, Prec);
}
MPFR_ZIV_FREE (ziv_loop);
mpz_clear (uk);
mpfr_clear (tmp);
mpfr_clear (t);
mpfr_clear (x_copy);
MPFR_SAVE_EXPO_FREE (expo);
return inexact;
}