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+/* 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);
+}