1 /* mpfr_exp_2 -- exponential of a floating-point number
2 using algorithms in O(n^(1/2)*M(n)) and O(n^(1/3)*M(n))
4 Copyright 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011 Free Software Foundation, Inc.
5 Contributed by the Arenaire and Caramel projects, INRIA.
7 This file is part of the GNU MPFR Library.
9 The GNU MPFR Library is free software; you can redistribute it and/or modify
10 it under the terms of the GNU Lesser General Public License as published by
11 the Free Software Foundation; either version 3 of the License, or (at your
12 option) any later version.
14 The GNU MPFR Library is distributed in the hope that it will be useful, but
15 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
16 or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
17 License for more details.
19 You should have received a copy of the GNU Lesser General Public License
20 along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see
21 http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
22 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
25 #define MPFR_NEED_LONGLONG_H /* for count_leading_zeros */
26 #include "mpfr-impl.h"
29 mpfr_exp2_aux (mpz_t, mpfr_srcptr, mpfr_prec_t, mpfr_exp_t *);
31 mpfr_exp2_aux2 (mpz_t, mpfr_srcptr, mpfr_prec_t, mpfr_exp_t *);
33 mpz_normalize (mpz_t, mpz_t, mpfr_exp_t);
35 mpz_normalize2 (mpz_t, mpz_t, mpfr_exp_t, mpfr_exp_t);
37 /* if k = the number of bits of z > q, divides z by 2^(k-q) and returns k-q.
38 Otherwise do nothing and return 0.
41 mpz_normalize (mpz_t rop, mpz_t z, mpfr_exp_t q)
45 MPFR_MPZ_SIZEINBASE2 (k, z);
46 MPFR_ASSERTD (k == (mpfr_uexp_t) k);
47 if (q < 0 || (mpfr_uexp_t) k > (mpfr_uexp_t) q)
49 mpz_fdiv_q_2exp (rop, z, (unsigned long) ((mpfr_uexp_t) k - q));
50 return (mpfr_exp_t) k - q;
52 if (MPFR_UNLIKELY(rop != z))
57 /* if expz > target, shift z by (expz-target) bits to the left.
58 if expz < target, shift z by (target-expz) bits to the right.
62 mpz_normalize2 (mpz_t rop, mpz_t z, mpfr_exp_t expz, mpfr_exp_t target)
65 mpz_fdiv_q_2exp (rop, z, target - expz);
67 mpz_mul_2exp (rop, z, expz - target);
71 /* use Brent's formula exp(x) = (1+r+r^2/2!+r^3/3!+...)^(2^K)*2^n
72 where x = n*log(2)+(2^K)*r
73 together with the Paterson-Stockmeyer O(t^(1/2)) algorithm for the
74 evaluation of power series. The resulting complexity is O(n^(1/3)*M(n)).
75 This function returns with the exact flags due to exp.
78 mpfr_exp_2 (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
81 unsigned long K, k, l, err; /* FIXME: Which type ? */
83 mpfr_exp_t exps, expx;
91 (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec(x), mpfr_log_prec, x, rnd_mode),
92 ("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec(y), mpfr_log_prec, y,
95 expx = MPFR_GET_EXP (x);
98 /* Warning: we cannot use the 'double' type here, since on 64-bit machines
99 x may be as large as 2^62*log(2) without overflow, and then x/log(2)
100 is about 2^62: not every integer of that size can be represented as a
101 'double', thus the argument reduction would fail. */
103 /* |x| <= 0.25, thus n = round(x/log(2)) = 0 */
107 mpfr_init2 (r, sizeof (long) * CHAR_BIT);
108 mpfr_const_log2 (r, MPFR_RNDZ);
109 mpfr_div (r, x, r, MPFR_RNDN);
110 n = mpfr_get_si (r, MPFR_RNDN);
113 /* we have |x| <= (|n|+1)*log(2) */
114 MPFR_LOG_MSG (("d(x)=%1.30e n=%ld\n", mpfr_get_d1(x), n));
116 /* error_r bounds the cancelled bits in x - n*log(2) */
117 if (MPFR_UNLIKELY (n == 0))
121 count_leading_zeros (error_r, (mp_limb_t) SAFE_ABS (unsigned long, n) + 1);
122 error_r = GMP_NUMB_BITS - error_r;
123 /* we have |x| <= 2^error_r * log(2) */
126 /* for the O(n^(1/2)*M(n)) method, the Taylor series computation of
127 n/K terms costs about n/(2K) multiplications when computed in fixed
129 K = (precy < MPFR_EXP_2_THRESHOLD) ? __gmpfr_isqrt ((precy + 1) / 2)
130 : __gmpfr_cuberoot (4*precy);
131 l = (precy - 1) / K + 1;
132 err = K + MPFR_INT_CEIL_LOG2 (2 * l + 18);
133 /* add K extra bits, i.e. failure probability <= 1/2^K = O(1/precy) */
134 q = precy + err + K + 8;
135 /* if |x| >> 1, take into account the cancelled bits */
139 /* Note: due to the mpfr_prec_round below, it is not possible to use
140 the MPFR_GROUP_* macros here. */
142 mpfr_init2 (r, q + error_r);
143 mpfr_init2 (s, q + error_r);
145 /* the algorithm consists in computing an upper bound of exp(x) using
146 a precision of q bits, and see if we can round to MPFR_PREC(y) taking
147 into account the maximal error. Otherwise we increase q. */
148 MPFR_ZIV_INIT (loop, q);
151 MPFR_LOG_MSG (("n=%ld K=%lu l=%lu q=%lu error_r=%d\n",
152 n, K, l, (unsigned long) q, error_r));
154 /* First reduce the argument to r = x - n * log(2),
155 so that r is small in absolute value. We want an upper
156 bound on r to get an upper bound on exp(x). */
158 /* if n<0, we have to get an upper bound of log(2)
159 in order to get an upper bound of r = x-n*log(2) */
160 mpfr_const_log2 (s, (n >= 0) ? MPFR_RNDZ : MPFR_RNDU);
161 /* s is within 1 ulp(s) of log(2) */
163 mpfr_mul_ui (r, s, (n < 0) ? -n : n, (n >= 0) ? MPFR_RNDZ : MPFR_RNDU);
164 /* r is within 3 ulps of |n|*log(2) */
166 MPFR_CHANGE_SIGN (r);
167 /* r <= n*log(2), within 3 ulps */
172 mpfr_sub (r, x, r, MPFR_RNDU);
174 if (MPFR_IS_PURE_FP (r))
176 while (MPFR_IS_NEG (r))
177 { /* initial approximation n was too large */
179 mpfr_add (r, r, s, MPFR_RNDU);
182 /* since there was a cancellation in x - n*log(2), the low error_r
183 bits from r are zero and thus non significant, thus we can reduce
184 the working precision */
186 mpfr_prec_round (r, q, MPFR_RNDU);
187 /* the error on r is at most 3 ulps (3 ulps if error_r = 0,
188 and 1 + 3/2 if error_r > 0) */
190 MPFR_ASSERTD (MPFR_IS_POS (r));
191 mpfr_div_2ui (r, r, K, MPFR_RNDU); /* r = (x-n*log(2))/2^K, exact */
194 exps = mpfr_get_z_2exp (ss, s);
195 /* s <- 1 + r/1! + r^2/2! + ... + r^l/l! */
196 MPFR_ASSERTD (MPFR_IS_PURE_FP (r) && MPFR_EXP (r) < 0);
197 l = (precy < MPFR_EXP_2_THRESHOLD)
198 ? mpfr_exp2_aux (ss, r, q, &exps) /* naive method */
199 : mpfr_exp2_aux2 (ss, r, q, &exps); /* Paterson/Stockmeyer meth */
201 MPFR_LOG_MSG (("l=%lu q=%lu (K+l)*q^2=%1.3e\n",
202 l, (unsigned long) q, (K + l) * (double) q * q));
204 for (k = 0; k < K; k++)
206 mpz_mul (ss, ss, ss);
208 exps += mpz_normalize (ss, ss, q);
210 mpfr_set_z (s, ss, MPFR_RNDN);
212 MPFR_SET_EXP(s, MPFR_GET_EXP (s) + exps);
215 /* error is at most 2^K*l, plus 2 to take into account of
216 the error of 3 ulps on r */
217 err = K + MPFR_INT_CEIL_LOG2 (l) + 2;
219 MPFR_LOG_MSG (("before mult. by 2^n:\n", 0));
221 MPFR_LOG_MSG (("err=%lu bits\n", K));
223 if (MPFR_LIKELY (MPFR_CAN_ROUND (s, q - err, precy, rnd_mode)))
226 inexact = mpfr_mul_2si (y, s, n, rnd_mode);
231 MPFR_ZIV_NEXT (loop, q);
232 mpfr_set_prec (r, q + error_r);
233 mpfr_set_prec (s, q + error_r);
235 MPFR_ZIV_FREE (loop);
243 /* s <- 1 + r/1! + r^2/2! + ... + r^l/l! while MPFR_EXP(r^l/l!)+MPFR_EXPR(r)>-q
244 using naive method with O(l) multiplications.
245 Return the number of iterations l.
246 The absolute error on s is less than 3*l*(l+1)*2^(-q).
247 Version using fixed-point arithmetic with mpz instead
248 of mpfr for internal computations.
249 NOTE[VL]: the following sentence seems to be obsolete since MY_INIT_MPZ
250 is no longer used (r6919); qn was the number of limbs of q.
251 s must have at least qn+1 limbs (qn should be enough, but currently fails
252 since mpz_mul_2exp(s, s, q-1) reallocates qn+1 limbs)
255 mpfr_exp2_aux (mpz_t s, mpfr_srcptr r, mpfr_prec_t q, mpfr_exp_t *exps)
258 mpfr_exp_t dif, expt, expr;
260 mp_size_t sbit, tbit;
262 MPFR_ASSERTN (MPFR_IS_PURE_FP (r));
265 *exps = 1 - (mpfr_exp_t) q; /* s = 2^(q-1) */
270 mpz_mul_2exp(s, s, q-1);
271 expr = mpfr_get_z_2exp(rr, r); /* no error here */
278 MPFR_MPZ_SIZEINBASE2 (sbit, s);
279 MPFR_MPZ_SIZEINBASE2 (tbit, t);
280 dif = *exps + sbit - expt - tbit;
281 /* truncates the bits of t which are < ulp(s) = 2^(1-q) */
282 expt += mpz_normalize(t, t, (mpfr_exp_t) q-dif); /* error at most 2^(1-q) */
283 mpz_fdiv_q_ui (t, t, l); /* error at most 2^(1-q) */
284 /* the error wrt t^l/l! is here at most 3*l*ulp(s) */
285 MPFR_ASSERTD (expt == *exps);
286 if (mpz_sgn (t) == 0)
288 mpz_add(s, s, t); /* no error here: exact */
289 /* ensures rr has the same size as t: after several shifts, the error
290 on rr is still at most ulp(t)=ulp(s) */
291 MPFR_MPZ_SIZEINBASE2 (tbit, t);
292 expr += mpz_normalize(rr, rr, tbit);
298 return 3 * l * (l + 1);
301 /* s <- 1 + r/1! + r^2/2! + ... + r^l/l! while MPFR_EXP(r^l/l!)+MPFR_EXPR(r)>-q
302 using Paterson-Stockmeyer algorithm with O(sqrt(l)) multiplications.
304 Uses m multiplications of full size and 2l/m of decreasing size,
305 i.e. a total equivalent to about m+l/m full multiplications,
306 i.e. 2*sqrt(l) for m=sqrt(l).
307 NOTE[VL]: The following sentence seems to be obsolete since MY_INIT_MPZ
308 is no longer used (r6919); sizer was the number of limbs of r.
309 Version using mpz. ss must have at least (sizer+1) limbs.
310 The error is bounded by (l^2+4*l) ulps where l is the return value.
313 mpfr_exp2_aux2 (mpz_t s, mpfr_srcptr r, mpfr_prec_t q, mpfr_exp_t *exps)
315 mpfr_exp_t expr, *expR, expt;
317 unsigned long l, m, i;
318 mpz_t t, *R, rr, tmp;
319 mp_size_t sbit, rrbit;
320 MPFR_TMP_DECL(marker);
322 /* estimate value of l */
323 MPFR_ASSERTD (MPFR_GET_EXP (r) < 0);
324 l = q / (- MPFR_GET_EXP (r));
325 m = __gmpfr_isqrt (l);
326 /* we access R[2], thus we need m >= 2 */
330 MPFR_TMP_MARK(marker);
331 R = (mpz_t*) MPFR_TMP_ALLOC ((m + 1) * sizeof (mpz_t)); /* R[i] is r^i */
332 expR = (mpfr_exp_t*) MPFR_TMP_ALLOC((m + 1) * sizeof (mpfr_exp_t));
333 /* expR[i] is the exponent for R[i] */
338 *exps = 1 - q; /* 1 ulp = 2^(1-q) */
339 for (i = 0 ; i <= m ; i++)
341 expR[1] = mpfr_get_z_2exp (R[1], r); /* exact operation: no error */
342 expR[1] = mpz_normalize2 (R[1], R[1], expR[1], 1 - q); /* error <= 1 ulp */
343 mpz_mul (t, R[1], R[1]); /* err(t) <= 2 ulps */
344 mpz_fdiv_q_2exp (R[2], t, q - 1); /* err(R[2]) <= 3 ulps */
346 for (i = 3 ; i <= m ; i++)
349 mpz_mul (t, R[i-1], R[1]); /* err(t) <= 2*i-2 */
351 mpz_mul (t, R[i/2], R[i/2]);
352 mpz_fdiv_q_2exp (R[i], t, q - 1); /* err(R[i]) <= 2*i-1 ulps */
355 mpz_set_ui (R[0], 1);
356 mpz_mul_2exp (R[0], R[0], q-1);
357 expR[0] = 1-q; /* R[0]=1 */
359 expr = 0; /* rr contains r^l/l! */
360 /* by induction: err(rr) <= 2*l ulps */
363 ql = q; /* precision used for current giant step */
366 /* all R[i] must have exponent 1-ql */
368 for (i = 0 ; i < m ; i++)
369 expR[i] = mpz_normalize2 (R[i], R[i], expR[i], 1 - ql);
370 /* the absolute error on R[i]*rr is still 2*i-1 ulps */
371 expt = mpz_normalize2 (t, R[m-1], expR[m-1], 1 - ql);
372 /* err(t) <= 2*m-1 ulps */
373 /* computes t = 1 + r/(l+1) + ... + r^(m-1)*l!/(l+m-1)!
374 using Horner's scheme */
375 for (i = m-1 ; i-- != 0 ; )
377 mpz_fdiv_q_ui (t, t, l+i+1); /* err(t) += 1 ulp */
378 mpz_add (t, t, R[i]);
380 /* now err(t) <= (3m-2) ulps */
382 /* now multiplies t by r^l/l! and adds to s */
385 expt = mpz_normalize2 (t, t, expt, *exps);
386 /* err(t) <= (3m-1) + err_rr(l) <= (3m-2) + 2*l */
387 MPFR_ASSERTD (expt == *exps);
388 mpz_add (s, s, t); /* no error here */
390 /* updates rr, the multiplication of the factors l+i could be done
391 using binary splitting too, but it is not sure it would save much */
392 mpz_mul (t, rr, R[m]); /* err(t) <= err(rr) + 2m-1 */
395 for (i = 1 ; i <= m ; i++)
396 mpz_mul_ui (tmp, tmp, l + i);
397 mpz_fdiv_q (t, t, tmp); /* err(t) <= err(rr) + 2m */
399 if (MPFR_UNLIKELY (mpz_sgn (t) == 0))
401 expr += mpz_normalize (rr, t, ql); /* err_rr(l+1) <= err_rr(l) + 2m+1 */
402 if (MPFR_UNLIKELY (mpz_sgn (rr) == 0))
405 MPFR_MPZ_SIZEINBASE2 (rrbit, rr);
406 MPFR_MPZ_SIZEINBASE2 (sbit, s);
407 ql = q - *exps - sbit + expr + rrbit;
408 /* TODO: Wrong cast. I don't want what is right, but this is
411 while ((size_t) expr + rrbit > (size_t) -q);
413 for (i = 0 ; i <= m ; i++)
415 MPFR_TMP_FREE(marker);