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Diffstat (limited to 'src/exp_2.c')
| -rw-r--r-- | src/exp_2.c | 421 |
1 files changed, 421 insertions, 0 deletions
diff --git a/src/exp_2.c b/src/exp_2.c new file mode 100644 index 0000000..7f9ef6f --- /dev/null +++ b/src/exp_2.c @@ -0,0 +1,421 @@ +/* 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, 2010, 2011, 2012, 2013 Free Software Foundation, Inc. +Contributed by the AriC and Caramel 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 3 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.LESSER. If not, see +http://www.gnu.org/licenses/ or 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, mpfr_prec_t, mpfr_exp_t *); +static unsigned long +mpfr_exp2_aux2 (mpz_t, mpfr_srcptr, mpfr_prec_t, mpfr_exp_t *); +static mpfr_exp_t +mpz_normalize (mpz_t, mpz_t, mpfr_exp_t); +static mpfr_exp_t +mpz_normalize2 (mpz_t, mpz_t, mpfr_exp_t, mpfr_exp_t); + +/* 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 mpfr_exp_t +mpz_normalize (mpz_t rop, mpz_t z, mpfr_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_fdiv_q_2exp (rop, z, (unsigned long) ((mpfr_uexp_t) k - q)); + return (mpfr_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 mpfr_exp_t +mpz_normalize2 (mpz_t rop, mpz_t z, mpfr_exp_t expz, mpfr_exp_t target) +{ + if (target > expz) + mpz_fdiv_q_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, mpfr_rnd_t rnd_mode) +{ + long n; + unsigned long K, k, l, err; /* FIXME: Which type ? */ + int error_r; + mpfr_exp_t exps, expx; + mpfr_prec_t q, precy; + int inexact; + mpfr_t r, s; + mpz_t ss; + MPFR_ZIV_DECL (loop); + + MPFR_LOG_FUNC + (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec(x), mpfr_log_prec, x, rnd_mode), + ("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec(y), mpfr_log_prec, y, + inexact)); + + expx = MPFR_GET_EXP (x); + 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 (expx <= -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, MPFR_RNDZ); + mpfr_div (r, x, r, MPFR_RNDN); + n = mpfr_get_si (r, MPFR_RNDN); + mpfr_clear (r); + } + /* we have |x| <= (|n|+1)*log(2) */ + MPFR_LOG_MSG (("d(x)=%1.30e n=%ld\n", mpfr_get_d1(x), n)); + + /* error_r 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) + 1); + error_r = GMP_NUMB_BITS - error_r; + /* we have |x| <= 2^error_r * log(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 + 8; + /* if |x| >> 1, take into account the cancelled bits */ + if (expx > 0) + q += expx; + + /* Note: due to the mpfr_prec_round below, it is not possible to use + the MPFR_GROUP_* macros here. */ + + 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) ? MPFR_RNDZ : MPFR_RNDU); + /* s is within 1 ulp(s) of log(2) */ + + mpfr_mul_ui (r, s, (n < 0) ? -n : n, (n >= 0) ? MPFR_RNDZ : MPFR_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, MPFR_RNDU); + + if (MPFR_IS_PURE_FP (r)) + { + while (MPFR_IS_NEG (r)) + { /* initial approximation n was too large */ + n--; + mpfr_add (r, r, s, MPFR_RNDU); + } + + /* since there was a cancellation in x - n*log(2), the low error_r + bits from r are zero and thus non significant, thus we can reduce + the working precision */ + if (error_r > 0) + mpfr_prec_round (r, q, MPFR_RNDU); + /* the error on r is at most 3 ulps (3 ulps if error_r = 0, + and 1 + 3/2 if error_r > 0) */ + MPFR_LOG_VAR (r); + MPFR_ASSERTD (MPFR_IS_POS (r)); + mpfr_div_2ui (r, r, K, MPFR_RNDU); /* r = (x-n*log(2))/2^K, exact */ + + mpz_init (ss); + exps = mpfr_get_z_2exp (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, MPFR_RNDN); + + MPFR_SET_EXP(s, MPFR_GET_EXP (s) + exps); + mpz_clear (ss); + + /* error is at most 2^K*l, plus 2 to take into account of + the error of 3 ulps on r */ + err = K + MPFR_INT_CEIL_LOG2 (l) + 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 - err, 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 + error_r); + mpfr_set_prec (s, q + error_r); + } + 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. + NOTE[VL]: the following sentence seems to be obsolete since MY_INIT_MPZ + is no longer used (r6919); qn was the number of limbs of q. + 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, mpfr_prec_t q, mpfr_exp_t *exps) +{ + unsigned long l; + mpfr_exp_t dif, expt, expr; + mpz_t t, rr; + mp_size_t sbit, tbit; + + MPFR_ASSERTN (MPFR_IS_PURE_FP (r)); + + expt = 0; + *exps = 1 - (mpfr_exp_t) q; /* s = 2^(q-1) */ + mpz_init (t); + mpz_init (rr); + mpz_set_ui(t, 1); + mpz_set_ui(s, 1); + mpz_mul_2exp(s, s, q-1); + expr = mpfr_get_z_2exp(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, (mpfr_exp_t) q-dif); /* error at most 2^(1-q) */ + mpz_fdiv_q_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); + } + + mpz_clear (t); + mpz_clear (rr); + + 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). + NOTE[VL]: The following sentence seems to be obsolete since MY_INIT_MPZ + is no longer used (r6919); sizer was the number of limbs of r. + 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, mpfr_prec_t q, mpfr_exp_t *exps) +{ + mpfr_exp_t expr, *expR, expt; + mpfr_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 = (mpfr_exp_t*) MPFR_TMP_ALLOC((m + 1) * sizeof (mpfr_exp_t)); + /* expR[i] is the exponent for R[i] */ + mpz_init (tmp); + mpz_init (rr); + mpz_init (t); + mpz_set_ui (s, 0); + *exps = 1 - q; /* 1 ulp = 2^(1-q) */ + for (i = 0 ; i <= m ; i++) + mpz_init (R[i]); + expR[1] = mpfr_get_z_2exp (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_fdiv_q_2exp (R[2], t, q - 1); /* err(R[2]) <= 3 ulps */ + expR[2] = 1 - q; + for (i = 3 ; i <= m ; i++) + { + if ((i & 1) == 1) + mpz_mul (t, R[i-1], R[1]); /* err(t) <= 2*i-2 */ + else + mpz_mul (t, R[i/2], R[i/2]); + mpz_fdiv_q_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_fdiv_q_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) -q); + + for (i = 0 ; i <= m ; i++) + mpz_clear (R[i]); + MPFR_TMP_FREE(marker); + mpz_clear (rr); + mpz_clear (t); + mpz_clear (tmp); + + return l * (l + 4); +} |