Annotation of OpenXM_contrib/gmp/mpfr/exp.c, Revision 1.1.1.1
1.1 maekawa 1: /* mpfr_exp -- exponential of a floating-point number
2:
3: Copyright (C) 1999 PolKA project, Inria Lorraine and Loria
4:
5: This file is part of the MPFR Library.
6:
7: The MPFR Library is free software; you can redistribute it and/or modify
8: it under the terms of the GNU Library General Public License as published by
9: the Free Software Foundation; either version 2 of the License, or (at your
10: option) any later version.
11:
12: The MPFR Library is distributed in the hope that it will be useful, but
13: WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
14: or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Library General Public
15: License for more details.
16:
17: You should have received a copy of the GNU Library General Public License
18: along with the MPFR Library; see the file COPYING.LIB. If not, write to
19: the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
20: MA 02111-1307, USA. */
21:
22: #include <stdio.h>
23: #include <math.h>
24: #include "gmp.h"
25: #include "gmp-impl.h"
26: #include "mpfr.h"
27:
28: /* #define DEBUG */
29:
30: #define LOG2 0.69314718055994528622 /* log(2) rounded to zero on 53 bits */
31:
32: /* use Brent's formula exp(x) = (1+r+r^2/2!+r^3/3!+...)^(2^K)*2^n
33: where x = n*log(2)+(2^K)*r
34: number of operations = O(K+prec(r)/K)
35: */
36: int
37: #if __STDC__
38: mpfr_exp(mpfr_ptr y, mpfr_srcptr x, unsigned char rnd_mode)
39: #else
40: mpfr_exp(y, x, rnd_mode)
41: mpfr_ptr y;
42: mpfr_srcptr x;
43: unsigned char rnd_mode;
44: #endif
45: {
46: int n, expx, K, precy, q, k, l, expr, err;
47: mpfr_t r, s, t;
48:
49: if (FLAG_NAN(x)) { SET_NAN(y); return 1; }
50: if (!NOTZERO(x)) { mpfr_set_ui(y, 1, GMP_RNDN); return 0; }
51:
52: expx = EXP(x);
53: precy = PREC(y);
54: #ifdef DEBUG
55: printf("EXP(x)=%d\n",expx);
56: #endif
57:
58: /* if x > (2^31-1)*ln(2), then exp(x) > 2^(2^31-1) i.e. gives +infinity */
59: if (expx > 30) {
60: if (SIGN(x)>0) { printf("+infinity"); return 1; }
61: else { SET_ZERO(y); return 1; }
62: }
63:
64: /* if x < 2^(-precy), then exp(x) i.e. gives 1 +/- 1 ulp(1) */
65: if (expx < -precy) { int signx = SIGN(x);
66: mpfr_set_ui(y, 1, rnd_mode);
67: if (signx>0 && rnd_mode==GMP_RNDU) mpfr_add_one_ulp(y);
68: else if (signx<0 && (rnd_mode==GMP_RNDD || rnd_mode==GMP_RNDZ))
69: mpfr_sub_one_ulp(y);
70: return 1; }
71:
72: n = (int) floor(mpfr_get_d(x)/LOG2);
73:
74: K = (int) sqrt( (double) precy );
75: l = (precy-1)/K + 1;
76: err = K + (int) ceil(log(2.0*(double)l+18.0)/LOG2);
77: /* add K extra bits, i.e. failure probability <= 1/2^K = O(1/precy) */
78: q = precy + err + K + 3;
79: mpfr_init2(r, q); mpfr_init2(s, q); mpfr_init2(t, q);
80: /* the algorithm consists in computing an upper bound of exp(x) using
81: a precision of q bits, and see if we can round to PREC(y) taking
82: into account the maximal error. Otherwise we increase q. */
83: do {
84: #ifdef DEBUG
85: printf("n=%d K=%d l=%d q=%d\n",n,K,l,q);
86: #endif
87:
88: /* if n<0, we have to get an upper bound of log(2)
89: in order to get an upper bound of r = x-n*log(2) */
90: mpfr_log2(s, (n>=0) ? GMP_RNDZ : GMP_RNDU);
91: #ifdef DEBUG
92: printf("n=%d log(2)=",n); mpfr_print_raw(s); putchar('\n');
93: #endif
94: mpfr_mul_ui(r, s, (n<0) ? -n : n, (n>=0) ? GMP_RNDZ : GMP_RNDU);
95: if (n<0) mpfr_neg(r, r, GMP_RNDD);
96: /* r = floor(n*log(2)) */
97:
98: #ifdef DEBUG
99: printf("x=%1.20e\n",mpfr_get_d(x));
100: printf(" ="); mpfr_print_raw(x); putchar('\n');
101: printf("r=%1.20e\n",mpfr_get_d(r));
102: printf(" ="); mpfr_print_raw(r); putchar('\n');
103: #endif
104: mpfr_sub(r, x, r, GMP_RNDU);
105: if (SIGN(r)<0) { /* initial approximation n was too large */
106: n--;
107: mpfr_mul_ui(r, s, (n<0) ? -n : n, GMP_RNDZ);
108: if (n<0) mpfr_neg(r, r, GMP_RNDD);
109: mpfr_sub(r, x, r, GMP_RNDU);
110: }
111: #ifdef DEBUG
112: printf("x-r=%1.20e\n",mpfr_get_d(r));
113: printf(" ="); mpfr_print_raw(r); putchar('\n');
114: if (SIGN(r)<0) { fprintf(stderr,"Error in mpfr_exp: r<0\n"); exit(1); }
115: #endif
116: mpfr_div_2exp(r, r, K, GMP_RNDU); /* r = (x-n*log(2))/2^K */
117: mpfr_set_ui(s, 1, GMP_RNDU);
118: mpfr_set_ui(t, 1, GMP_RNDU);
119:
120: l = 1; expr = EXP(r);
121: do {
122: mpfr_mul(t, t, r, GMP_RNDU);
123: mpfr_div_ui(t, t, l, GMP_RNDU);
124: mpfr_add(s, s, t, GMP_RNDU);
125: #ifdef DEBUG
126: printf("l=%d t=%1.20e\n",l,mpfr_get_d(t));
127: printf("s=%1.20e\n",mpfr_get_d(s));
128: #endif
129: l++;
130: } while (EXP(t)+expr > -q);
131: #ifdef DEBUG
132: printf("l=%d q=%d (K+l)*q^2=%1.3e\n", l, q, (K+l)*(double)q*q);
133: #endif
134:
135: /* add 2 ulp to take into account rest of summation */
136: mpfr_add_one_ulp(s);
137: mpfr_add_one_ulp(s);
138:
139: for (k=0;k<K;k++) {
140: mpfr_mul(s, s, s, GMP_RNDU);
141: #ifdef DEBUG
142: printf("k=%d s=%1.20e\n",k,mpfr_get_d(s));
143: #endif
144: }
145:
146: if (n>0) mpfr_mul_2exp(s, s, n, GMP_RNDU);
147: else mpfr_div_2exp(s, s, -n, GMP_RNDU);
148:
149: /* error is at most 2^K*(2l+18) ulp */
150: l = 2*l+17; k=0; while (l) { k++; l >>= 1; }
151: /* now k = ceil(log(2l+18)/log(2)) */
152: K += k;
153: #ifdef DEBUG
154: printf("after mult. by 2^n:\n");
155: if (EXP(s)>-1024) printf("s=%1.20e\n",mpfr_get_d(s));
156: printf(" ="); mpfr_print_raw(s); putchar('\n');
157: printf("err=%d bits\n", K);
158: #endif
159:
160: l = mpfr_can_round(s, q-K, GMP_RNDU, rnd_mode, precy);
161: if (l==0) {
162: #ifdef DEBUG
163: printf("not enough precision, use %d\n", q+BITS_PER_MP_LIMB);
164: printf("q=%d q-K=%d precy=%d\n",q,q-K,precy);
165: #endif
166: q += BITS_PER_MP_LIMB;
167: mpfr_set_prec(r, q); mpfr_set_prec(s, q); mpfr_set_prec(t, q);
168: }
169: } while (l==0);
170:
171: mpfr_set(y, s, rnd_mode);
172:
173: mpfr_clear(r); mpfr_clear(s); mpfr_clear(t);
174: return 1;
175: }
176:
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