--- /dev/null
+/* mpfr_exp -- exponential of a floating-point number
+
+Copyright 1999, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc.
+Contributed by the Arenaire and Cacao 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 2.1 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.LIB. If not, 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, mp_prec_t *mult)
+{
+ unsigned long n, i, j;
+ mpz_t *S, *ptoj;
+ mp_prec_t *log2_nb_terms;
+ mp_exp_t diff, expo;
+ mp_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 = (mp_exp_t) prec_i_have - 2 * (mp_exp_t) precy;
+ expo = diff;
+ if (diff >= 0)
+ mpz_div_2exp (S[0], S[0], diff);
+ else
+ mpz_mul_2exp (S[0], S[0], -diff);
+
+ MPFR_MPZ_SIZEINBASE2 (prec_i_have, Q[0]);
+ diff = (mp_exp_t) prec_i_have - (mp_prec_t) precy;
+ expo -= diff;
+ if (diff > 0)
+ mpz_div_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], GMP_RNDD);
+ MPFR_SET_EXP (y, MPFR_GET_EXP (y) + expo - r * (i - 1) );
+}
+
+#define shift (BITS_PER_MP_LIMB/2)
+
+int
+mpfr_exp_3 (mpfr_ptr y, mpfr_srcptr x, mp_rnd_t rnd_mode)
+{
+ mpfr_t t, x_copy, tmp;
+ mpz_t uk;
+ mp_exp_t ttt, shift_x;
+ unsigned long twopoweri;
+ mpz_t *P;
+ mp_prec_t *mult;
+ int i, k, loop;
+ int prec_x;
+ mp_prec_t realprec, Prec;
+ int iter;
+ int inexact = 0;
+ MPFR_SAVE_EXPO_DECL (expo);
+ MPFR_ZIV_DECL (ziv_loop);
+
+ MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", x, x, rnd_mode),
+ ("y[%#R]=%R inexact=%d", y, 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_BITS_PER_MP_LIMB;
+ if (prec_x < 0)
+ prec_x = 0;
+
+ ttt = MPFR_GET_EXP (x);
+ mpfr_init2 (x_copy, MPFR_PREC(x));
+ mpfr_set (x_copy, x, GMP_RNDD);
+
+ /* we shift to get a number less than 1 */
+ if (ttt > 0)
+ {
+ shift_x = ttt;
+ mpfr_div_2ui (x_copy, x, ttt, GMP_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_BITS_PER_MP_LIMB;
+
+ /* now we have to extract */
+ twopoweri = BITS_PER_MP_LIMB;
+
+ /* 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 = (mp_prec_t*) (*__gmp_allocate_func) (2*(k+2)*sizeof(mp_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, GMP_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, GMP_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(mp_prec_t));
+
+ if (shift_x > 0)
+ {
+ MPFR_BLOCK (flags, {
+ for (loop = 0; loop < shift_x - 1; loop++)
+ mpfr_sqr (tmp, tmp, GMP_RNDD);
+ mpfr_sqr (t, tmp, GMP_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, GMP_RNDD); /* no overflow, exact */
+ mpfr_sqr (t, tmp, GMP_RNDD);
+ if (MPFR_IS_ZERO (t))
+ {
+ /* approximate result < 2^(emin - 3), thus
+ exact result < 2^(emin - 2). */
+ inexact = mpfr_underflow (y, (rnd_mode == GMP_RNDN) ?
+ GMP_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, GMP_RNDD, GMP_RNDZ,
+ MPFR_PREC(y) + (rnd_mode == GMP_RNDN)))
+ {
+ inexact = mpfr_set (y, shift_x > 0 ? t : tmp, rnd_mode);
+ if (MPFR_UNLIKELY (scaled && MPFR_IS_PURE_FP (y)))
+ {
+ int inex2;
+ mp_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 == GMP_RNDN && inexact < 0 &&
+ MPFR_IS_ZERO (y) && ey == __gmpfr_emin + 1)
+ {
+ /* Double rounding case: in GMP_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, GMP_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;
+}
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