1 /* mpfr_expm1 -- Compute exp(x)-1 2 3 Copyright 2001-2018 Free Software Foundation, Inc. 4 Contributed by the AriC and Caramba projects, INRIA. 5 6 This file is part of the GNU MPFR Library. 7 8 The GNU MPFR Library is free software; you can redistribute it and/or modify 9 it under the terms of the GNU Lesser General Public License as published by 10 the Free Software Foundation; either version 3 of the License, or (at your 11 option) any later version. 12 13 The GNU MPFR Library is distributed in the hope that it will be useful, but 14 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY 15 or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public 16 License for more details. 17 18 You should have received a copy of the GNU Lesser General Public License 19 along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see 20 http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc., 21 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */ 22 23 #define MPFR_NEED_LONGLONG_H 24 #include "mpfr-impl.h" 25 26 /* The computation of expm1 is done by 27 expm1(x)=exp(x)-1 28 */ 29 30 int 31 mpfr_expm1 (mpfr_ptr y, mpfr_srcptr x , mpfr_rnd_t rnd_mode) 32 { 33 int inexact; 34 mpfr_exp_t ex; 35 MPFR_SAVE_EXPO_DECL (expo); 36 37 MPFR_LOG_FUNC 38 (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode), 39 ("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y, 40 inexact)); 41 42 if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x))) 43 { 44 if (MPFR_IS_NAN (x)) 45 { 46 MPFR_SET_NAN (y); 47 MPFR_RET_NAN; 48 } 49 /* check for inf or -inf (expm1(-inf)=-1) */ 50 else if (MPFR_IS_INF (x)) 51 { 52 if (MPFR_IS_POS (x)) 53 { 54 MPFR_SET_INF (y); 55 MPFR_SET_POS (y); 56 MPFR_RET (0); 57 } 58 else 59 return mpfr_set_si (y, -1, rnd_mode); 60 } 61 else 62 { 63 MPFR_ASSERTD (MPFR_IS_ZERO (x)); 64 MPFR_SET_ZERO (y); /* expm1(+/- 0) = +/- 0 */ 65 MPFR_SET_SAME_SIGN (y, x); 66 MPFR_RET (0); 67 } 68 } 69 70 ex = MPFR_GET_EXP (x); 71 if (ex < 0) 72 { 73 /* For -1 < x < 0, abs(expm1(x)-x) < x^2/2. 74 For 0 < x < 1, abs(expm1(x)-x) < x^2. */ 75 if (MPFR_IS_POS (x)) 76 MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, - ex, 0, 1, rnd_mode, {}); 77 else 78 MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, - ex, 1, 0, rnd_mode, {}); 79 } 80 81 MPFR_SAVE_EXPO_MARK (expo); 82 83 if (MPFR_IS_NEG (x) && ex > 5) /* x <= -32 */ 84 { 85 mp_limb_t t_limb[(64 - 1) / GMP_NUMB_BITS + 1]; 86 mpfr_t t; 87 mpfr_exp_t err; 88 89 MPFR_TMP_INIT1(t_limb, t, 64); 90 mpfr_div (t, x, __gmpfr_const_log2_RNDU, MPFR_RNDU); /* > x / ln(2) */ 91 err = mpfr_cmp_si (t, MPFR_EMIN_MIN >= -LONG_MAX ? 92 MPFR_EMIN_MIN : -LONG_MAX) <= 0 ? 93 - (MPFR_EMIN_MIN >= -LONG_MAX ? MPFR_EMIN_MIN : -LONG_MAX) : 94 - mpfr_get_si (t, MPFR_RNDU); 95 /* exp(x) = 2^(x/ln(2)) 96 <= 2^max(MPFR_EMIN_MIN,-LONG_MAX,ceil(x/ln(2)+epsilon)) 97 with epsilon > 0 */ 98 MPFR_SMALL_INPUT_AFTER_SAVE_EXPO (y, __gmpfr_mone, err, 0, 0, 99 rnd_mode, expo, {}); 100 } 101 102 /* General case */ 103 { 104 /* Declaration of the intermediary variable */ 105 mpfr_t t; 106 /* Declaration of the size variable */ 107 mpfr_prec_t Ny = MPFR_PREC(y); /* target precision */ 108 mpfr_prec_t Nt; /* working precision */ 109 mpfr_exp_t err, exp_te; /* error */ 110 MPFR_ZIV_DECL (loop); 111 112 /* compute the precision of intermediary variable */ 113 /* the optimal number of bits : see algorithms.tex */ 114 Nt = Ny + MPFR_INT_CEIL_LOG2 (Ny) + 6; 115 116 /* if |x| is smaller than 2^(-e), we will loose about e bits in the 117 subtraction exp(x) - 1 */ 118 if (ex < 0) 119 Nt += - ex; 120 121 /* initialize auxiliary variable */ 122 mpfr_init2 (t, Nt); 123 124 /* First computation of expm1 */ 125 MPFR_ZIV_INIT (loop, Nt); 126 for (;;) 127 { 128 MPFR_BLOCK_DECL (flags); 129 130 /* exp(x) may overflow and underflow */ 131 MPFR_BLOCK (flags, mpfr_exp (t, x, MPFR_RNDN)); 132 if (MPFR_OVERFLOW (flags)) 133 { 134 inexact = mpfr_overflow (y, rnd_mode, MPFR_SIGN_POS); 135 MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_OVERFLOW); 136 break; 137 } 138 else if (MPFR_UNDERFLOW (flags)) 139 { 140 inexact = mpfr_set_si (y, -1, rnd_mode); 141 MPFR_ASSERTD (inexact == 0); 142 inexact = -1; 143 if (MPFR_IS_LIKE_RNDZ (rnd_mode, 1)) 144 { 145 inexact = 1; 146 mpfr_nexttozero (y); 147 } 148 break; 149 } 150 151 exp_te = MPFR_GET_EXP (t); /* FIXME: exp(x) may overflow! */ 152 mpfr_sub_ui (t, t, 1, MPFR_RNDN); /* exp(x)-1 */ 153 154 /* error estimate */ 155 /*err=Nt-(__gmpfr_ceil_log2(1+pow(2,MPFR_EXP(te)-MPFR_EXP(t))));*/ 156 err = Nt - (MAX (exp_te - MPFR_GET_EXP (t), 0) + 1); 157 158 if (MPFR_LIKELY (MPFR_CAN_ROUND (t, err, Ny, rnd_mode))) 159 { 160 inexact = mpfr_set (y, t, rnd_mode); 161 break; 162 } 163 164 /* increase the precision */ 165 MPFR_ZIV_NEXT (loop, Nt); 166 mpfr_set_prec (t, Nt); 167 } 168 MPFR_ZIV_FREE (loop); 169 170 mpfr_clear (t); 171 } 172 173 MPFR_SAVE_EXPO_FREE (expo); 174 return mpfr_check_range (y, inexact, rnd_mode); 175 } 176