| /* 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); |
| } |
| } |
| } |
