| /* mpfr_expm1 -- Compute exp(x)-1 |
| |
| Copyright 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 |
| #include "mpfr-impl.h" |
| |
| /* The computation of expm1 is done by |
| expm1(x)=exp(x)-1 |
| */ |
| |
| int |
| mpfr_expm1 (mpfr_ptr y, mpfr_srcptr x , mp_rnd_t rnd_mode) |
| { |
| int inexact; |
| mp_exp_t ex; |
| MPFR_SAVE_EXPO_DECL (expo); |
| |
| if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x))) |
| { |
| if (MPFR_IS_NAN (x)) |
| { |
| MPFR_SET_NAN (y); |
| MPFR_RET_NAN; |
| } |
| /* check for inf or -inf (expm1(-inf)=-1) */ |
| else if (MPFR_IS_INF (x)) |
| { |
| if (MPFR_IS_POS (x)) |
| { |
| MPFR_SET_INF (y); |
| MPFR_SET_POS (y); |
| MPFR_RET (0); |
| } |
| else |
| return mpfr_set_si (y, -1, rnd_mode); |
| } |
| else |
| { |
| MPFR_ASSERTD (MPFR_IS_ZERO (x)); |
| MPFR_SET_ZERO (y); /* expm1(+/- 0) = +/- 0 */ |
| MPFR_SET_SAME_SIGN (y, x); |
| MPFR_RET (0); |
| } |
| } |
| |
| ex = MPFR_GET_EXP (x); |
| if (ex < 0) |
| { |
| /* For -1 < x < 0, abs(expm1(x)-x) < x^2/2. |
| For 0 < x < 1, abs(expm1(x)-x) < x^2. */ |
| if (MPFR_IS_POS (x)) |
| MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, - ex, 0, 1, rnd_mode, {}); |
| else |
| MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, - ex, 1, 0, rnd_mode, {}); |
| } |
| |
| MPFR_SAVE_EXPO_MARK (expo); |
| |
| if (MPFR_IS_NEG (x) && ex > 5) /* x <= -32 */ |
| { |
| mpfr_t minus_one, t; |
| mp_exp_t err; |
| |
| mpfr_init2 (minus_one, 2); |
| mpfr_init2 (t, 64); |
| mpfr_set_si (minus_one, -1, GMP_RNDN); |
| mpfr_const_log2 (t, GMP_RNDU); /* round upward since x is negative */ |
| mpfr_div (t, x, t, GMP_RNDU); /* > x / ln(2) */ |
| err = mpfr_cmp_si (t, MPFR_EMIN_MIN >= -LONG_MAX ? |
| MPFR_EMIN_MIN : -LONG_MAX) <= 0 ? |
| - (MPFR_EMIN_MIN >= -LONG_MAX ? MPFR_EMIN_MIN : -LONG_MAX) : |
| - mpfr_get_si (t, GMP_RNDU); |
| /* exp(x) = 2^(x/ln(2)) |
| <= 2^max(MPFR_EMIN_MIN,-LONG_MAX,ceil(x/ln(2)+epsilon)) |
| with epsilon > 0 */ |
| mpfr_clear (t); |
| MPFR_SMALL_INPUT_AFTER_SAVE_EXPO (y, minus_one, err, 0, 0, rnd_mode, |
| expo, { mpfr_clear (minus_one); }); |
| mpfr_clear (minus_one); |
| } |
| |
| /* General case */ |
| { |
| /* Declaration of the intermediary variable */ |
| mpfr_t t; |
| /* Declaration of the size variable */ |
| mp_prec_t Ny = MPFR_PREC(y); /* target precision */ |
| mp_prec_t Nt; /* working precision */ |
| mp_exp_t err, exp_te; /* 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 the |
| subtraction exp(x) - 1 */ |
| if (ex < 0) |
| Nt += - ex; |
| |
| /* initialize auxiliary variable */ |
| mpfr_init2 (t, Nt); |
| |
| /* First computation of expm1 */ |
| MPFR_ZIV_INIT (loop, Nt); |
| for (;;) |
| { |
| MPFR_BLOCK_DECL (flags); |
| |
| /* exp(x) may overflow and underflow */ |
| MPFR_BLOCK (flags, mpfr_exp (t, x, GMP_RNDN)); |
| if (MPFR_OVERFLOW (flags)) |
| { |
| inexact = mpfr_overflow (y, rnd_mode, MPFR_SIGN_POS); |
| MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_OVERFLOW); |
| break; |
| } |
| else if (MPFR_UNDERFLOW (flags)) |
| { |
| inexact = mpfr_set_si (y, -1, rnd_mode); |
| MPFR_ASSERTD (inexact == 0); |
| inexact = -1; |
| if (MPFR_IS_LIKE_RNDZ (rnd_mode, 1)) |
| { |
| inexact = 1; |
| mpfr_nexttozero (y); |
| } |
| break; |
| } |
| |
| exp_te = MPFR_GET_EXP (t); /* FIXME: exp(x) may overflow! */ |
| mpfr_sub_ui (t, t, 1, GMP_RNDN); /* exp(x)-1 */ |
| |
| /* error estimate */ |
| /*err=Nt-(__gmpfr_ceil_log2(1+pow(2,MPFR_EXP(te)-MPFR_EXP(t))));*/ |
| err = Nt - (MAX (exp_te - MPFR_GET_EXP (t), 0) + 1); |
| |
| if (MPFR_LIKELY (MPFR_CAN_ROUND (t, err, Ny, rnd_mode))) |
| { |
| inexact = mpfr_set (y, t, rnd_mode); |
| break; |
| } |
| |
| /* increase the precision */ |
| MPFR_ZIV_NEXT (loop, Nt); |
| mpfr_set_prec (t, Nt); |
| } |
| MPFR_ZIV_FREE (loop); |
| |
| mpfr_clear (t); |
| } |
| |
| MPFR_SAVE_EXPO_FREE (expo); |
| return mpfr_check_range (y, inexact, rnd_mode); |
| } |