--- /dev/null
+/* mpfr_exp_2 -- exponential of a floating-point number
+ using algorithms in O(n^(1/2)*M(n)) and O(n^(1/3)*M(n))
+
+Copyright 1999, 2000, 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 DEBUG */
+#define MPFR_NEED_LONGLONG_H /* for count_leading_zeros */
+#include "mpfr-impl.h"
+
+static unsigned long
+mpfr_exp2_aux (mpz_t, mpfr_srcptr, mp_prec_t, mp_exp_t *);
+static unsigned long
+mpfr_exp2_aux2 (mpz_t, mpfr_srcptr, mp_prec_t, mp_exp_t *);
+static mp_exp_t
+mpz_normalize (mpz_t, mpz_t, mp_exp_t);
+static mp_exp_t
+mpz_normalize2 (mpz_t, mpz_t, mp_exp_t, mp_exp_t);
+
+#define MY_INIT_MPZ(x, s) { \
+ (x)->_mp_alloc = (s); \
+ PTR(x) = (mp_ptr) MPFR_TMP_ALLOC((s)*BYTES_PER_MP_LIMB); \
+ (x)->_mp_size = 0; }
+
+/* if k = the number of bits of z > q, divides z by 2^(k-q) and returns k-q.
+ Otherwise do nothing and return 0.
+ */
+static mp_exp_t
+mpz_normalize (mpz_t rop, mpz_t z, mp_exp_t q)
+{
+ size_t k;
+
+ MPFR_MPZ_SIZEINBASE2 (k, z);
+ MPFR_ASSERTD (k == (mpfr_uexp_t) k);
+ if (q < 0 || (mpfr_uexp_t) k > (mpfr_uexp_t) q)
+ {
+ mpz_div_2exp(rop, z, (unsigned long) ((mpfr_uexp_t) k - q));
+ return (mp_exp_t) k - q;
+ }
+ if (MPFR_UNLIKELY(rop != z))
+ mpz_set(rop, z);
+ return 0;
+}
+
+/* if expz > target, shift z by (expz-target) bits to the left.
+ if expz < target, shift z by (target-expz) bits to the right.
+ Returns target.
+*/
+static mp_exp_t
+mpz_normalize2 (mpz_t rop, mpz_t z, mp_exp_t expz, mp_exp_t target)
+{
+ if (target > expz)
+ mpz_div_2exp(rop, z, target-expz);
+ else
+ mpz_mul_2exp(rop, z, expz-target);
+ return target;
+}
+
+/* use Brent's formula exp(x) = (1+r+r^2/2!+r^3/3!+...)^(2^K)*2^n
+ where x = n*log(2)+(2^K)*r
+ together with the Paterson-Stockmeyer O(t^(1/2)) algorithm for the
+ evaluation of power series. The resulting complexity is O(n^(1/3)*M(n)).
+ This function returns with the exact flags due to exp.
+*/
+int
+mpfr_exp_2 (mpfr_ptr y, mpfr_srcptr x, mp_rnd_t rnd_mode)
+{
+ long n;
+ unsigned long K, k, l, err; /* FIXME: Which type ? */
+ int error_r;
+ mp_exp_t exps;
+ mp_prec_t q, precy;
+ int inexact;
+ mpfr_t r, s;
+ mpz_t ss;
+ MPFR_ZIV_DECL (loop);
+ MPFR_TMP_DECL(marker);
+
+ MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", x, x, rnd_mode),
+ ("y[%#R]=%R inexact=%d", y, y, inexact));
+
+ precy = MPFR_PREC(y);
+
+ /* Warning: we cannot use the 'double' type here, since on 64-bit machines
+ x may be as large as 2^62*log(2) without overflow, and then x/log(2)
+ is about 2^62: not every integer of that size can be represented as a
+ 'double', thus the argument reduction would fail. */
+ if (MPFR_GET_EXP (x) <= -2)
+ /* |x| <= 0.25, thus n = round(x/log(2)) = 0 */
+ n = 0;
+ else
+ {
+ mpfr_init2 (r, sizeof (long) * CHAR_BIT);
+ mpfr_const_log2 (r, GMP_RNDZ);
+ mpfr_div (r, x, r, GMP_RNDN);
+ n = mpfr_get_si (r, GMP_RNDN);
+ mpfr_clear (r);
+ }
+ MPFR_LOG_MSG (("d(x)=%1.30e n=%ld\n", mpfr_get_d1(x), n));
+
+ /* error bounds the cancelled bits in x - n*log(2) */
+ if (MPFR_UNLIKELY (n == 0))
+ error_r = 0;
+ else
+ count_leading_zeros (error_r, (mp_limb_t) SAFE_ABS (unsigned long, n));
+ error_r = BITS_PER_MP_LIMB - error_r + 2;
+
+ /* for the O(n^(1/2)*M(n)) method, the Taylor series computation of
+ n/K terms costs about n/(2K) multiplications when computed in fixed
+ point */
+ K = (precy < MPFR_EXP_2_THRESHOLD) ? __gmpfr_isqrt ((precy + 1) / 2)
+ : __gmpfr_cuberoot (4*precy);
+ l = (precy - 1) / K + 1;
+ err = K + MPFR_INT_CEIL_LOG2 (2 * l + 18);
+ /* add K extra bits, i.e. failure probability <= 1/2^K = O(1/precy) */
+ q = precy + err + K + 5;
+
+ mpfr_init2 (r, q + error_r);
+ mpfr_init2 (s, q + error_r);
+
+ /* the algorithm consists in computing an upper bound of exp(x) using
+ a precision of q bits, and see if we can round to MPFR_PREC(y) taking
+ into account the maximal error. Otherwise we increase q. */
+ MPFR_ZIV_INIT (loop, q);
+ for (;;)
+ {
+ MPFR_LOG_MSG (("n=%ld K=%lu l=%lu q=%lu error_r=%d\n",
+ n, K, l, (unsigned long) q, error_r));
+
+ /* First reduce the argument to r = x - n * log(2),
+ so that r is small in absolute value. We want an upper
+ bound on r to get an upper bound on exp(x). */
+
+ /* if n<0, we have to get an upper bound of log(2)
+ in order to get an upper bound of r = x-n*log(2) */
+ mpfr_const_log2 (s, (n >= 0) ? GMP_RNDZ : GMP_RNDU);
+ /* s is within 1 ulp of log(2) */
+
+ mpfr_mul_ui (r, s, (n < 0) ? -n : n, (n >= 0) ? GMP_RNDZ : GMP_RNDU);
+ /* r is within 3 ulps of |n|*log(2) */
+ if (n < 0)
+ MPFR_CHANGE_SIGN (r);
+ /* r <= n*log(2), within 3 ulps */
+
+ MPFR_LOG_VAR (x);
+ MPFR_LOG_VAR (r);
+
+ mpfr_sub (r, x, r, GMP_RNDU);
+ /* possible cancellation here: if r is zero, increase the working
+ precision (Ziv's loop); otherwise, the error on r is at most
+ 3*2^(EXP(old_r)-EXP(new_r)) ulps */
+
+ if (MPFR_IS_PURE_FP (r))
+ {
+ mp_exp_t cancel;
+
+ /* number of cancelled bits */
+ cancel = MPFR_GET_EXP (x) - MPFR_GET_EXP (r);
+ if (cancel < 0) /* this might happen in the second loop if x is
+ tiny negative: the initial n is 0, then in the
+ first loop n becomes -1 and r = x + log(2) */
+ cancel = 0;
+ while (MPFR_IS_NEG (r))
+ { /* initial approximation n was too large */
+ n--;
+ mpfr_add (r, r, s, GMP_RNDU);
+ }
+ mpfr_prec_round (r, q, GMP_RNDU);
+ MPFR_LOG_VAR (r);
+ MPFR_ASSERTD (MPFR_IS_POS (r));
+ mpfr_div_2ui (r, r, K, GMP_RNDU); /* r = (x-n*log(2))/2^K, exact */
+
+ MPFR_TMP_MARK(marker);
+ MY_INIT_MPZ(ss, 3 + 2*((q-1)/BITS_PER_MP_LIMB));
+ exps = mpfr_get_z_exp (ss, s);
+ /* s <- 1 + r/1! + r^2/2! + ... + r^l/l! */
+ MPFR_ASSERTD (MPFR_IS_PURE_FP (r) && MPFR_EXP (r) < 0);
+ l = (precy < MPFR_EXP_2_THRESHOLD)
+ ? mpfr_exp2_aux (ss, r, q, &exps) /* naive method */
+ : mpfr_exp2_aux2 (ss, r, q, &exps); /* Paterson/Stockmeyer meth */
+
+ MPFR_LOG_MSG (("l=%lu q=%lu (K+l)*q^2=%1.3e\n",
+ l, (unsigned long) q, (K + l) * (double) q * q));
+
+ for (k = 0; k < K; k++)
+ {
+ mpz_mul (ss, ss, ss);
+ exps <<= 1;
+ exps += mpz_normalize (ss, ss, q);
+ }
+ mpfr_set_z (s, ss, GMP_RNDN);
+
+ MPFR_SET_EXP(s, MPFR_GET_EXP (s) + exps);
+ MPFR_TMP_FREE(marker); /* don't need ss anymore */
+
+ /* error is at most 2^K*l, plus cancel+2 to take into account of
+ the error of 3*2^(EXP(old_r)-EXP(new_r)) on r */
+ K += MPFR_INT_CEIL_LOG2 (l) + cancel + 2;
+
+ MPFR_LOG_MSG (("before mult. by 2^n:\n", 0));
+ MPFR_LOG_VAR (s);
+ MPFR_LOG_MSG (("err=%lu bits\n", K));
+
+ if (MPFR_LIKELY (MPFR_CAN_ROUND (s, q - K, precy, rnd_mode)))
+ {
+ mpfr_clear_flags ();
+ inexact = mpfr_mul_2si (y, s, n, rnd_mode);
+ break;
+ }
+ }
+
+ MPFR_ZIV_NEXT (loop, q);
+ mpfr_set_prec (r, q);
+ mpfr_set_prec (s, q);
+ }
+ MPFR_ZIV_FREE (loop);
+
+ mpfr_clear (r);
+ mpfr_clear (s);
+
+ return inexact;
+}
+
+/* s <- 1 + r/1! + r^2/2! + ... + r^l/l! while MPFR_EXP(r^l/l!)+MPFR_EXPR(r)>-q
+ using naive method with O(l) multiplications.
+ Return the number of iterations l.
+ The absolute error on s is less than 3*l*(l+1)*2^(-q).
+ Version using fixed-point arithmetic with mpz instead
+ of mpfr for internal computations.
+ s must have at least qn+1 limbs (qn should be enough, but currently fails
+ since mpz_mul_2exp(s, s, q-1) reallocates qn+1 limbs)
+*/
+static unsigned long
+mpfr_exp2_aux (mpz_t s, mpfr_srcptr r, mp_prec_t q, mp_exp_t *exps)
+{
+ unsigned long l;
+ mp_exp_t dif, expt, expr;
+ mp_size_t qn;
+ mpz_t t, rr;
+ mp_size_t sbit, tbit;
+ MPFR_TMP_DECL(marker);
+
+ MPFR_ASSERTN (MPFR_IS_PURE_FP (r));
+
+ MPFR_TMP_MARK(marker);
+ qn = 1 + (q-1)/BITS_PER_MP_LIMB;
+ expt = 0;
+ *exps = 1 - (mp_exp_t) q; /* s = 2^(q-1) */
+ MY_INIT_MPZ(t, 2*qn+1);
+ MY_INIT_MPZ(rr, qn+1);
+ mpz_set_ui(t, 1);
+ mpz_set_ui(s, 1);
+ mpz_mul_2exp(s, s, q-1);
+ expr = mpfr_get_z_exp(rr, r); /* no error here */
+
+ l = 0;
+ for (;;) {
+ l++;
+ mpz_mul(t, t, rr);
+ expt += expr;
+ MPFR_MPZ_SIZEINBASE2 (sbit, s);
+ MPFR_MPZ_SIZEINBASE2 (tbit, t);
+ dif = *exps + sbit - expt - tbit;
+ /* truncates the bits of t which are < ulp(s) = 2^(1-q) */
+ expt += mpz_normalize(t, t, (mp_exp_t) q-dif); /* error at most 2^(1-q) */
+ mpz_div_ui(t, t, l); /* error at most 2^(1-q) */
+ /* the error wrt t^l/l! is here at most 3*l*ulp(s) */
+ MPFR_ASSERTD (expt == *exps);
+ if (mpz_sgn (t) == 0)
+ break;
+ mpz_add(s, s, t); /* no error here: exact */
+ /* ensures rr has the same size as t: after several shifts, the error
+ on rr is still at most ulp(t)=ulp(s) */
+ MPFR_MPZ_SIZEINBASE2 (tbit, t);
+ expr += mpz_normalize(rr, rr, tbit);
+ }
+
+ MPFR_TMP_FREE(marker);
+ return 3*l*(l+1);
+}
+
+/* s <- 1 + r/1! + r^2/2! + ... + r^l/l! while MPFR_EXP(r^l/l!)+MPFR_EXPR(r)>-q
+ using Paterson-Stockmeyer algorithm with O(sqrt(l)) multiplications.
+ Return l.
+ Uses m multiplications of full size and 2l/m of decreasing size,
+ i.e. a total equivalent to about m+l/m full multiplications,
+ i.e. 2*sqrt(l) for m=sqrt(l).
+ Version using mpz. ss must have at least (sizer+1) limbs.
+ The error is bounded by (l^2+4*l) ulps where l is the return value.
+*/
+static unsigned long
+mpfr_exp2_aux2 (mpz_t s, mpfr_srcptr r, mp_prec_t q, mp_exp_t *exps)
+{
+ mp_exp_t expr, *expR, expt;
+ mp_size_t sizer;
+ mp_prec_t ql;
+ unsigned long l, m, i;
+ mpz_t t, *R, rr, tmp;
+ mp_size_t sbit, rrbit;
+ MPFR_TMP_DECL(marker);
+
+ /* estimate value of l */
+ MPFR_ASSERTD (MPFR_GET_EXP (r) < 0);
+ l = q / (- MPFR_GET_EXP (r));
+ m = __gmpfr_isqrt (l);
+ /* we access R[2], thus we need m >= 2 */
+ if (m < 2)
+ m = 2;
+
+ MPFR_TMP_MARK(marker);
+ R = (mpz_t*) MPFR_TMP_ALLOC((m+1)*sizeof(mpz_t)); /* R[i] is r^i */
+ expR = (mp_exp_t*) MPFR_TMP_ALLOC((m+1)*sizeof(mp_exp_t)); /* exponent for R[i] */
+ sizer = 1 + (MPFR_PREC(r)-1)/BITS_PER_MP_LIMB;
+ mpz_init(tmp);
+ MY_INIT_MPZ(rr, sizer+2);
+ MY_INIT_MPZ(t, 2*sizer); /* double size for products */
+ mpz_set_ui(s, 0);
+ *exps = 1-q; /* 1 ulp = 2^(1-q) */
+ for (i = 0 ; i <= m ; i++)
+ MY_INIT_MPZ(R[i], sizer+2);
+ expR[1] = mpfr_get_z_exp(R[1], r); /* exact operation: no error */
+ expR[1] = mpz_normalize2(R[1], R[1], expR[1], 1-q); /* error <= 1 ulp */
+ mpz_mul(t, R[1], R[1]); /* err(t) <= 2 ulps */
+ mpz_div_2exp(R[2], t, q-1); /* err(R[2]) <= 3 ulps */
+ expR[2] = 1-q;
+ for (i = 3 ; i <= m ; i++)
+ {
+ mpz_mul(t, R[i-1], R[1]); /* err(t) <= 2*i-2 */
+ mpz_div_2exp(R[i], t, q-1); /* err(R[i]) <= 2*i-1 ulps */
+ expR[i] = 1-q;
+ }
+ mpz_set_ui (R[0], 1);
+ mpz_mul_2exp (R[0], R[0], q-1);
+ expR[0] = 1-q; /* R[0]=1 */
+ mpz_set_ui (rr, 1);
+ expr = 0; /* rr contains r^l/l! */
+ /* by induction: err(rr) <= 2*l ulps */
+
+ l = 0;
+ ql = q; /* precision used for current giant step */
+ do
+ {
+ /* all R[i] must have exponent 1-ql */
+ if (l != 0)
+ for (i = 0 ; i < m ; i++)
+ expR[i] = mpz_normalize2 (R[i], R[i], expR[i], 1-ql);
+ /* the absolute error on R[i]*rr is still 2*i-1 ulps */
+ expt = mpz_normalize2 (t, R[m-1], expR[m-1], 1-ql);
+ /* err(t) <= 2*m-1 ulps */
+ /* computes t = 1 + r/(l+1) + ... + r^(m-1)*l!/(l+m-1)!
+ using Horner's scheme */
+ for (i = m-1 ; i-- != 0 ; )
+ {
+ mpz_div_ui (t, t, l+i+1); /* err(t) += 1 ulp */
+ mpz_add (t, t, R[i]);
+ }
+ /* now err(t) <= (3m-2) ulps */
+
+ /* now multiplies t by r^l/l! and adds to s */
+ mpz_mul (t, t, rr);
+ expt += expr;
+ expt = mpz_normalize2 (t, t, expt, *exps);
+ /* err(t) <= (3m-1) + err_rr(l) <= (3m-2) + 2*l */
+ MPFR_ASSERTD (expt == *exps);
+ mpz_add (s, s, t); /* no error here */
+
+ /* updates rr, the multiplication of the factors l+i could be done
+ using binary splitting too, but it is not sure it would save much */
+ mpz_mul (t, rr, R[m]); /* err(t) <= err(rr) + 2m-1 */
+ expr += expR[m];
+ mpz_set_ui (tmp, 1);
+ for (i = 1 ; i <= m ; i++)
+ mpz_mul_ui (tmp, tmp, l + i);
+ mpz_fdiv_q (t, t, tmp); /* err(t) <= err(rr) + 2m */
+ l += m;
+ if (MPFR_UNLIKELY (mpz_sgn (t) == 0))
+ break;
+ expr += mpz_normalize (rr, t, ql); /* err_rr(l+1) <= err_rr(l) + 2m+1 */
+ if (MPFR_UNLIKELY (mpz_sgn (rr) == 0))
+ rrbit = 1;
+ else
+ MPFR_MPZ_SIZEINBASE2 (rrbit, rr);
+ MPFR_MPZ_SIZEINBASE2 (sbit, s);
+ ql = q - *exps - sbit + expr + rrbit;
+ /* TODO: Wrong cast. I don't want what is right, but this is
+ certainly wrong */
+ }
+ while ((size_t) expr+rrbit > (size_t) (int) -q);
+
+ MPFR_TMP_FREE(marker);
+ mpz_clear(tmp);
+ return l*(l+4);
+}
projects / msp430-gcc.git / blobdiff