--- /dev/null
+/* mpfr_pow_si -- power function x^y with y a signed int
+
+Copyright 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 MPFR_NEED_LONGLONG_H
+#include "mpfr-impl.h"
+
+/* The computation of y = pow_si(x,n) is done by
+ * y = pow_ui(x,n) if n >= 0
+ * y = 1 / pow_ui(x,-n) if n < 0
+ */
+
+int
+mpfr_pow_si (mpfr_ptr y, mpfr_srcptr x, long int n, mp_rnd_t rnd)
+{
+ MPFR_LOG_FUNC (("x[%#R]=%R n=%ld rnd=%d", x, x, n, rnd),
+ ("y[%#R]=%R", y, y));
+
+ if (n >= 0)
+ return mpfr_pow_ui (y, x, n, rnd);
+ else
+ {
+ if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
+ {
+ if (MPFR_IS_NAN (x))
+ {
+ MPFR_SET_NAN (y);
+ MPFR_RET_NAN;
+ }
+ else if (MPFR_IS_INF (x))
+ {
+ MPFR_SET_ZERO (y);
+ if (MPFR_IS_POS (x) || ((unsigned) n & 1) == 0)
+ MPFR_SET_POS (y);
+ else
+ MPFR_SET_NEG (y);
+ MPFR_RET (0);
+ }
+ else /* x is zero */
+ {
+ MPFR_ASSERTD (MPFR_IS_ZERO (x));
+ MPFR_SET_INF(y);
+ if (MPFR_IS_POS (x) || ((unsigned) n & 1) == 0)
+ MPFR_SET_POS (y);
+ else
+ MPFR_SET_NEG (y);
+ MPFR_RET(0);
+ }
+ }
+ MPFR_CLEAR_FLAGS (y);
+
+ /* detect exact powers: x^(-n) is exact iff x is a power of 2 */
+ if (mpfr_cmp_si_2exp (x, MPFR_SIGN(x), MPFR_EXP(x) - 1) == 0)
+ {
+ mp_exp_t expx = MPFR_EXP (x) - 1, expy;
+ MPFR_ASSERTD (n < 0);
+ /* Warning: n * expx may overflow!
+ * Some systems (apparently alpha-freebsd) abort with
+ * LONG_MIN / 1, and LONG_MIN / -1 is undefined.
+ * Proof of the overflow checking. The expressions below are
+ * assumed to be on the rational numbers, but the word "overflow"
+ * still has its own meaning in the C context. / still denotes
+ * the integer (truncated) division, and // denotes the exact
+ * division.
+ * - First, (__gmpfr_emin - 1) / n and (__gmpfr_emax - 1) / n
+ * cannot overflow due to the constraints on the exponents of
+ * MPFR numbers.
+ * - If n = -1, then n * expx = - expx, which is representable
+ * because of the constraints on the exponents of MPFR numbers.
+ * - If expx = 0, then n * expx = 0, which is representable.
+ * - If n < -1 and expx > 0:
+ * + If expx > (__gmpfr_emin - 1) / n, then
+ * expx >= (__gmpfr_emin - 1) / n + 1
+ * > (__gmpfr_emin - 1) // n,
+ * and
+ * n * expx < __gmpfr_emin - 1,
+ * i.e.
+ * n * expx <= __gmpfr_emin - 2.
+ * This corresponds to an underflow, with a null result in
+ * the rounding-to-nearest mode.
+ * + If expx <= (__gmpfr_emin - 1) / n, then n * expx cannot
+ * overflow since 0 < expx <= (__gmpfr_emin - 1) / n and
+ * 0 > n * expx >= n * ((__gmpfr_emin - 1) / n)
+ * >= __gmpfr_emin - 1.
+ * - If n < -1 and expx < 0:
+ * + If expx < (__gmpfr_emax - 1) / n, then
+ * expx <= (__gmpfr_emax - 1) / n - 1
+ * < (__gmpfr_emax - 1) // n,
+ * and
+ * n * expx > __gmpfr_emax - 1,
+ * i.e.
+ * n * expx >= __gmpfr_emax.
+ * This corresponds to an overflow (2^(n * expx) has an
+ * exponent > __gmpfr_emax).
+ * + If expx >= (__gmpfr_emax - 1) / n, then n * expx cannot
+ * overflow since 0 > expx >= (__gmpfr_emax - 1) / n and
+ * 0 < n * expx <= n * ((__gmpfr_emax - 1) / n)
+ * <= __gmpfr_emax - 1.
+ * Note: one could use expx bounds based on MPFR_EXP_MIN and
+ * MPFR_EXP_MAX instead of __gmpfr_emin and __gmpfr_emax. The
+ * current bounds do not lead to noticeably slower code and
+ * allow us to avoid a bug in Sun's compiler for Solaris/x86
+ * (when optimizations are enabled).
+ */
+ expy =
+ n != -1 && expx > 0 && expx > (__gmpfr_emin - 1) / n ?
+ MPFR_EMIN_MIN - 2 /* Underflow */ :
+ n != -1 && expx < 0 && expx < (__gmpfr_emax - 1) / n ?
+ MPFR_EMAX_MAX /* Overflow */ : n * expx;
+ return mpfr_set_si_2exp (y, n % 2 ? MPFR_INT_SIGN (x) : 1,
+ expy, rnd);
+ }
+
+ /* General case */
+ {
+ /* Declaration of the intermediary variable */
+ mpfr_t t;
+ /* Declaration of the size variable */
+ mp_prec_t Ny; /* target precision */
+ mp_prec_t Nt; /* working precision */
+ mp_rnd_t rnd1;
+ int size_n;
+ int inexact;
+ unsigned long abs_n;
+ MPFR_SAVE_EXPO_DECL (expo);
+ MPFR_ZIV_DECL (loop);
+
+ abs_n = - (unsigned long) n;
+ count_leading_zeros (size_n, (mp_limb_t) abs_n);
+ size_n = BITS_PER_MP_LIMB - size_n;
+
+ /* initial working precision */
+ Ny = MPFR_PREC (y);
+ Nt = Ny + size_n + 3 + MPFR_INT_CEIL_LOG2 (Ny);
+
+ MPFR_SAVE_EXPO_MARK (expo);
+
+ /* initialise of intermediary variable */
+ mpfr_init2 (t, Nt);
+
+ /* We will compute rnd(rnd1(1/x) ^ |n|), where rnd1 is the rounding
+ toward sign(x), to avoid spurious overflow or underflow, as in
+ mpfr_pow_z. */
+ rnd1 = MPFR_EXP (x) < 1 ? GMP_RNDZ :
+ (MPFR_SIGN (x) > 0 ? GMP_RNDU : GMP_RNDD);
+
+ MPFR_ZIV_INIT (loop, Nt);
+ for (;;)
+ {
+ MPFR_BLOCK_DECL (flags);
+
+ /* compute (1/x)^|n| */
+ MPFR_BLOCK (flags, mpfr_ui_div (t, 1, x, rnd1));
+ MPFR_ASSERTD (! MPFR_UNDERFLOW (flags));
+ /* t = (1/x)*(1+theta) where |theta| <= 2^(-Nt) */
+ if (MPFR_UNLIKELY (MPFR_OVERFLOW (flags)))
+ goto overflow;
+ MPFR_BLOCK (flags, mpfr_pow_ui (t, t, abs_n, rnd));
+ /* t = (1/x)^|n|*(1+theta')^(|n|+1) where |theta'| <= 2^(-Nt).
+ If (|n|+1)*2^(-Nt) <= 1/2, which is satisfied as soon as
+ Nt >= bits(n)+2, then we can use Lemma \ref{lemma_graillat}
+ from algorithms.tex, which yields x^n*(1+theta) with
+ |theta| <= 2(|n|+1)*2^(-Nt), thus the error is bounded by
+ 2(|n|+1) ulps <= 2^(bits(n)+2) ulps. */
+ if (MPFR_UNLIKELY (MPFR_OVERFLOW (flags)))
+ {
+ overflow:
+ MPFR_ZIV_FREE (loop);
+ mpfr_clear (t);
+ MPFR_SAVE_EXPO_FREE (expo);
+ MPFR_LOG_MSG (("overflow\n", 0));
+ return mpfr_overflow (y, rnd, abs_n & 1 ?
+ MPFR_SIGN (x) : MPFR_SIGN_POS);
+ }
+ if (MPFR_UNLIKELY (MPFR_UNDERFLOW (flags)))
+ {
+ MPFR_ZIV_FREE (loop);
+ mpfr_clear (t);
+ MPFR_LOG_MSG (("underflow\n", 0));
+ if (rnd == GMP_RNDN)
+ {
+ mpfr_t y2, nn;
+
+ /* We cannot decide now whether the result should be
+ rounded toward zero or away from zero. So, like
+ in mpfr_pow_pos_z, let's use the general case of
+ mpfr_pow in precision 2. */
+ MPFR_ASSERTD (mpfr_cmp_si_2exp (x, MPFR_SIGN (x),
+ MPFR_EXP (x) - 1) != 0);
+ mpfr_init2 (y2, 2);
+ mpfr_init2 (nn, sizeof (long) * CHAR_BIT);
+ inexact = mpfr_set_si (nn, n, GMP_RNDN);
+ MPFR_ASSERTN (inexact == 0);
+ inexact = mpfr_pow_general (y2, x, nn, rnd, 1,
+ (mpfr_save_expo_t *) NULL);
+ mpfr_clear (nn);
+ mpfr_set (y, y2, GMP_RNDN);
+ mpfr_clear (y2);
+ MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_UNDERFLOW);
+ goto end;
+ }
+ else
+ {
+ MPFR_SAVE_EXPO_FREE (expo);
+ return mpfr_underflow (y, rnd, abs_n & 1 ?
+ MPFR_SIGN (x) : MPFR_SIGN_POS);
+ }
+ }
+ /* error estimate -- see pow function in algorithms.ps */
+ if (MPFR_LIKELY (MPFR_CAN_ROUND (t, Nt - size_n - 2, Ny, rnd)))
+ break;
+
+ /* actualisation of the precision */
+ MPFR_ZIV_NEXT (loop, Nt);
+ mpfr_set_prec (t, Nt);
+ }
+ MPFR_ZIV_FREE (loop);
+
+ inexact = mpfr_set (y, t, rnd);
+ mpfr_clear (t);
+
+ end:
+ MPFR_SAVE_EXPO_FREE (expo);
+ return mpfr_check_range (y, inexact, rnd);
+ }
+ }
+}
projects / msp430-gcc.git / blobdiff