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Diffstat (limited to 'Build/source/libs/mpfr/mpfr-src/src/log_ui.c')
-rw-r--r-- | Build/source/libs/mpfr/mpfr-src/src/log_ui.c | 234 |
1 files changed, 234 insertions, 0 deletions
diff --git a/Build/source/libs/mpfr/mpfr-src/src/log_ui.c b/Build/source/libs/mpfr/mpfr-src/src/log_ui.c new file mode 100644 index 00000000000..3dc39c73415 --- /dev/null +++ b/Build/source/libs/mpfr/mpfr-src/src/log_ui.c @@ -0,0 +1,234 @@ +/* mpfr_log_ui -- compute natural logarithm of an unsigned long + +Copyright 2014-2017 Free Software Foundation, Inc. +Contributed by the AriC and Caramba 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 MPFR_NEED_LONGLONG_H +#include "mpfr-impl.h" + +/* FIXME: mpfr_log_ui is much slower than mpfr_log on some values of n, + e.g. about 4 times as slow for n around ULONG_MAX/3 on an + x86_64 Linux machine, for 10^6 bits of precision. The reason is that + for say n=6148914691236517205 and prec=10^6, the value of T computed + has more than 50M bits, which is much more than needed. Indeed the + binary splitting algorithm for series with a finite radius of convergence + gives rationals of size n*log(n) for a target precision n. One might + truncate the rationals inside the algorithm, but then the error analysis + should be redone. */ + +/* Cf http://www.ginac.de/CLN/binsplit.pdf: the Taylor series of log(1+x) + up to order N for x=p/2^k is T/(B*Q). + P[0] <- (-p)^(n2-n1) [with opposite sign when n1=1] + q <- k*(n2-n1) [corresponding to Q[0] = 2^q] + B[0] <- n1 * (n1+1) * ... * (n2-1) + T[0] <- B[0]*Q[0] * S(n1,n2) + where S(n1,n2) = -sum((-x)^(i-n1+1)/i, i=n1..n2-1) + Assumes p is odd or zero, and -1/3 <= x = p/2^k <= 1/3. +*/ +static void +S (mpz_t *P, unsigned long *q, mpz_t *B, mpz_t *T, unsigned long n1, + unsigned long n2, long p, unsigned long k, int need_P) +{ + MPFR_ASSERTD (n1 < n2); + MPFR_ASSERTD (p == 0 || ((unsigned long) p & 1) != 0); + if (n2 == n1 + 1) + { + mpz_set_si (P[0], (n1 == 1) ? p : -p); + *q = k; + mpz_set_ui (B[0], n1); + /* T = B*Q*S where S = P/(B*Q) thus T = P */ + mpz_set (T[0], P[0]); + /* since p is odd (or zero), there is no common factor 2 between + P and Q, or T and B */ + } + else + { + unsigned long m = (n1 / 2) + (n2 / 2) + (n1 & 1UL & n2), q1; + /* m = floor((n1+n2)/2) */ + + MPFR_ASSERTD (n1 < m && m < n2); + S (P, q, B, T, n1, m, p, k, 1); + S (P + 1, &q1, B + 1, T + 1, m, n2, p, k, need_P); + + /* T0 <- T0*B1*Q1 + P0*B0*T1 */ + mpz_mul (T[1], T[1], P[0]); + mpz_mul (T[1], T[1], B[0]); + mpz_mul (T[0], T[0], B[1]); + /* Q[1] = 2^q1 */ + mpz_mul_2exp (T[0], T[0], q1); /* mpz_mul (T[0], T[0], Q[1]) */ + mpz_add (T[0], T[0], T[1]); + if (need_P) + mpz_mul (P[0], P[0], P[1]); + *q += q1; /* mpz_mul (Q[0], Q[0], Q[1]) */ + mpz_mul (B[0], B[0], B[1]); + + /* there should be no common factors 2 between P, Q and T, + since P is odd (or zero) */ + } +} + +int +mpfr_log_ui (mpfr_ptr x, unsigned long n, mpfr_rnd_t rnd_mode) +{ + unsigned long k; + mpfr_prec_t w; /* working precision */ + mpz_t three_n, *P, *B, *T; + mpfr_t t, q; + int inexact; + unsigned long N, lgN, i, kk; + long p; + MPFR_GROUP_DECL(group); + MPFR_TMP_DECL(marker); + MPFR_ZIV_DECL(loop); + MPFR_SAVE_EXPO_DECL (expo); + + if (n <= 2) + { + if (n == 0) + { + MPFR_SET_INF (x); + MPFR_SET_NEG (x); + MPFR_SET_DIVBY0 (); + MPFR_RET (0); /* log(0) is an exact -infinity */ + } + else if (n == 1) + { + MPFR_SET_ZERO (x); + MPFR_SET_POS (x); + MPFR_RET (0); /* only "normal" case where the result is exact */ + } + /* now n=2 */ + return mpfr_const_log2 (x, rnd_mode); + } + + /* here n >= 3 */ + + /* Argument reduction: compute k such that 2/3 <= n/2^k < 4/3, + i.e., 2^(k+1) <= 3n < 2^(k+2). + + FIXME: we could do better by considering n/(2^k*3^i*5^j), + which reduces the maximal distance to 1 from 1/3 to 1/8, + thus needing about 1.89 less terms in the Taylor expansion of + the reduced argument. Then log(2^k*3^i*5^j) can be computed + using a combination of log(16/15), log(25/24) and log(81/80), + see Section 6.5 of "A Fortran Multiple-Precision Arithmetic Package", + Richard P. Brent, ACM Transactions on Mathematical Software, 1978. */ + + mpz_init_set_ui (three_n, n); + mpz_mul_ui (three_n, three_n, 3); + k = mpz_sizeinbase (three_n, 2) - 2; + MPFR_ASSERTD (k >= 2); + mpz_clear (three_n); + + /* The reduced argument is n/2^k - 1 = (n-2^k)/2^k. + Compute p = n-2^k. One has: |p| = |n-2^k| < 2^k/3 < n/2 <= LONG_MAX, + so that p and -p both fit in a long. */ + if (k < sizeof (unsigned long) * CHAR_BIT) + n -= 1UL << k; + /* n is now the value of p mod ULONG_MAX+1 */ + p = n > LONG_MAX ? - (long) - n : (long) n; + + MPFR_TMP_MARK(marker); + w = MPFR_PREC(x) + MPFR_INT_CEIL_LOG2 (MPFR_PREC(x)) + 10; + MPFR_GROUP_INIT_2(group, w, t, q); + MPFR_SAVE_EXPO_MARK (expo); + + kk = k; + if (p != 0) + while ((p % 2) == 0) /* replace p/2^kk by (p/2)/2^(kk-1) */ + { + p /= 2; + kk --; + } + + MPFR_ZIV_INIT (loop, w); + for (;;) + { + mpfr_t tmp; + unsigned int err; + unsigned long q0; + + /* we need at most w/log2(2^kk/|p|) terms for an accuracy of w bits */ + mpfr_init2 (tmp, 32); + mpfr_set_ui (tmp, (p > 0) ? p : -p, MPFR_RNDU); + mpfr_log2 (tmp, tmp, MPFR_RNDU); + mpfr_ui_sub (tmp, kk, tmp, MPFR_RNDD); + MPFR_ASSERTN (w <= ULONG_MAX); + mpfr_ui_div (tmp, w, tmp, MPFR_RNDU); + N = mpfr_get_ui (tmp, MPFR_RNDU); + if (N < 2) + N = 2; + lgN = MPFR_INT_CEIL_LOG2 (N) + 1; + mpfr_clear (tmp); + P = (mpz_t *) MPFR_TMP_ALLOC (3 * lgN * sizeof (mpz_t)); + B = P + lgN; + T = B + lgN; + for (i = 0; i < lgN; i++) + { + mpz_init (P[i]); + mpz_init (B[i]); + mpz_init (T[i]); + } + + S (P, &q0, B, T, 1, N, p, kk, 0); + /* mpz_mul (Q[0], B[0], Q[0]); */ + /* mpz_mul_2exp (B[0], B[0], q0); */ + + mpfr_set_z (t, T[0], MPFR_RNDN); /* t = P[0] * (1 + theta_1) */ + mpfr_set_z (q, B[0], MPFR_RNDN); /* q = B[0] * (1 + theta_2) */ + mpfr_mul_2exp (q, q, q0, MPFR_RNDN); /* B[0]*Q[0] */ + mpfr_div (t, t, q, MPFR_RNDN); /* t = T[0]/(B[0]*Q[0])*(1 + theta_3)^3 + = log(n/2^k) * (1 + theta_4)^4 + for |theta_i| < 2^(-w) */ + + /* argument reconstruction: add k*log(2) */ + mpfr_const_log2 (q, MPFR_RNDN); + mpfr_mul_ui (q, q, k, MPFR_RNDN); + mpfr_add (t, t, q, MPFR_RNDN); + for (i = 0; i < lgN; i++) + { + mpz_clear (P[i]); + mpz_clear (B[i]); + mpz_clear (T[i]); + } + /* The maximal error is 5 ulps for P/Q, since |(1+/-u)^4 - 1| < 5*u + for u < 2^(-12), k ulps for k*log(2), and 1 ulp for the addition, + thus at most k+6 ulps. + Note that there might be some cancellation in the addition: the worst + case is when log(1 + p/2^kk) = log(2/3) ~ -0.405, and with n=3 which + gives k=2, thus we add 2*log(2) = 1.386. Thus in the worst case we + have an exponent decrease of 1, which accounts for +1 in the error. */ + err = MPFR_INT_CEIL_LOG2 (k + 6) + 1; + if (MPFR_LIKELY (MPFR_CAN_ROUND (t, w - err, MPFR_PREC(x), rnd_mode))) + break; + + MPFR_ZIV_NEXT (loop, w); + MPFR_GROUP_REPREC_2(group, w, t, q); + } + MPFR_ZIV_FREE (loop); + + inexact = mpfr_set (x, t, rnd_mode); + + MPFR_GROUP_CLEAR(group); + MPFR_TMP_FREE(marker); + + MPFR_SAVE_EXPO_FREE (expo); + return mpfr_check_range (x, inexact, rnd_mode); +} |