/* mpfr_pow_si -- power function x^y with y a signed int Copyright 2001-2020 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 https://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 /* for MPFR_INT_CEIL_LOG2 */ #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, mpfr_rnd_t rnd) { MPFR_LOG_FUNC (("x[%Pu]=%.*Rg n=%ld rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, n, rnd), ("y[%Pu]=%.*Rg", mpfr_get_prec (y), mpfr_log_prec, 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 { int positive = MPFR_IS_POS (x) || ((unsigned long) n & 1) == 0; if (MPFR_IS_INF (x)) MPFR_SET_ZERO (y); else /* x is zero */ { MPFR_ASSERTD (MPFR_IS_ZERO (x)); MPFR_SET_INF (y); MPFR_SET_DIVBY0 (); } if (positive) MPFR_SET_POS (y); else MPFR_SET_NEG (y); MPFR_RET (0); } } /* 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) { mpfr_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. * http://www.freebsd.org/cgi/query-pr.cgi?pr=72024 * * 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); known affected versions: * cc: Sun C 5.8 2005/10/13 * cc: Sun C 5.8 Patch 121016-02 2006/03/31 * cc: Sun C 5.8 Patch 121016-04 2006/10/18 */ 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 */ mpfr_prec_t Ny; /* target precision */ mpfr_prec_t Nt; /* working precision */ mpfr_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; size_n = mpfr_nbits_ulong (abs_n); /* initial working precision */ Ny = MPFR_PREC (y); Nt = Ny + size_n + 3 + MPFR_INT_CEIL_LOG2 (Ny); MPFR_SAVE_EXPO_MARK (expo); /* initialize 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 ? MPFR_RNDZ : (MPFR_IS_POS (x) ? MPFR_RNDU : MPFR_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 == MPFR_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, MPFR_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, MPFR_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; /* actualization 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); } } }