/* mpfr_root -- kth root. Copyright 2005-2021 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 #include "mpfr-impl.h" /* The computation of y = x^(1/k) is done as follows, except for large values of k, for which this would be inefficient or yield internal integer overflows: Let x = sign * m * 2^(k*e) where m is an integer with 2^(k*(n-1)) <= m < 2^(k*n) where n = PREC(y) and m = s^k + t where 0 <= t and m < (s+1)^k we want that s has n bits i.e. s >= 2^(n-1), or m >= 2^(k*(n-1)) i.e. m must have at least k*(n-1)+1 bits then, not taking into account the sign, the result will be x^(1/k) = s * 2^e or (s+1) * 2^e according to the rounding mode. */ static int mpfr_root_aux (mpfr_ptr y, mpfr_srcptr x, unsigned long k, mpfr_rnd_t rnd_mode); int mpfr_rootn_ui (mpfr_ptr y, mpfr_srcptr x, unsigned long k, mpfr_rnd_t rnd_mode) { mpz_t m; mpfr_exp_t e, r, sh, f; mpfr_prec_t n, size_m, tmp; int inexact, negative; MPFR_SAVE_EXPO_DECL (expo); MPFR_LOG_FUNC (("x[%Pu]=%.*Rg k=%lu rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, k, rnd_mode), ("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y, inexact)); if (MPFR_UNLIKELY (k <= 1)) { if (k == 0) { /* rootn(x,0) is NaN (IEEE 754-2008). */ MPFR_SET_NAN (y); MPFR_RET_NAN; } else /* y = x^(1/1) = x */ return mpfr_set (y, x, rnd_mode); } /* Singular values */ if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x))) { if (MPFR_IS_NAN (x)) { MPFR_SET_NAN (y); /* NaN^(1/k) = NaN */ MPFR_RET_NAN; } if (MPFR_IS_INF (x)) /* (+Inf)^(1/k) = +Inf (-Inf)^(1/k) = -Inf if k odd (-Inf)^(1/k) = NaN if k even */ { if (MPFR_IS_NEG (x) && (k & 1) == 0) { MPFR_SET_NAN (y); MPFR_RET_NAN; } MPFR_SET_INF (y); MPFR_SET_SAME_SIGN (y, x); } else /* x is necessarily 0: (+0)^(1/k) = +0 (-0)^(1/k) = -0 */ { MPFR_ASSERTD (MPFR_IS_ZERO (x)); MPFR_SET_ZERO (y); if (MPFR_IS_POS (x) || (k & 1) == 0) MPFR_SET_POS (y); else MPFR_SET_NEG (y); } MPFR_RET (0); } /* Returns NAN for x < 0 and k even */ if (MPFR_UNLIKELY (MPFR_IS_NEG (x) && (k & 1) == 0)) { MPFR_SET_NAN (y); MPFR_RET_NAN; } /* Special case |x| = 1. Note that if x = -1, then k is odd (NaN results have already been filtered), so that y = -1. */ if (mpfr_cmpabs (x, __gmpfr_one) == 0) return mpfr_set (y, x, rnd_mode); /* General case */ /* For large k, use exp(log(x)/k). The threshold of 100 seems to be quite good when the precision goes to infinity. */ if (k > 100) return mpfr_root_aux (y, x, k, rnd_mode); MPFR_SAVE_EXPO_MARK (expo); mpz_init (m); e = mpfr_get_z_2exp (m, x); /* x = m * 2^e */ if ((negative = MPFR_IS_NEG(x))) mpz_neg (m, m); r = e % (mpfr_exp_t) k; if (r < 0) r += k; /* now r = e (mod k) with 0 <= r < k */ MPFR_ASSERTD (0 <= r && r < k); /* x = (m*2^r) * 2^(e-r) where e-r is a multiple of k */ MPFR_MPZ_SIZEINBASE2 (size_m, m); /* for rounding to nearest, we want the round bit to be in the root */ n = MPFR_PREC (y) + (rnd_mode == MPFR_RNDN); /* we now multiply m by 2^sh so that root(m,k) will give exactly n bits: we want k*(n-1)+1 <= size_m + sh <= k*n i.e. sh = k*f + r with f = max(floor((k*n-size_m-r)/k),0) */ if ((mpfr_exp_t) size_m + r >= k * (mpfr_exp_t) n) f = 0; /* we already have too many bits */ else f = (k * (mpfr_exp_t) n - (mpfr_exp_t) size_m - r) / k; sh = k * f + r; mpz_mul_2exp (m, m, sh); e = e - sh; /* invariant: x = m*2^e, with e divisible by k */ /* we reuse the variable m to store the kth root, since it is not needed any more: we just need to know if the root is exact */ inexact = mpz_root (m, m, k) == 0; MPFR_MPZ_SIZEINBASE2 (tmp, m); sh = tmp - n; if (sh > 0) /* we have to flush to 0 the last sh bits from m */ { inexact = inexact || (mpz_scan1 (m, 0) < sh); mpz_fdiv_q_2exp (m, m, sh); e += k * sh; } if (inexact) { if (negative) rnd_mode = MPFR_INVERT_RND (rnd_mode); if (rnd_mode == MPFR_RNDU || rnd_mode == MPFR_RNDA || (rnd_mode == MPFR_RNDN && mpz_tstbit (m, 0))) inexact = 1, mpz_add_ui (m, m, 1); else inexact = -1; } /* either inexact is not zero, and the conversion is exact, i.e. inexact is not changed; or inexact=0, and inexact is set only when rnd_mode=MPFR_RNDN and bit (n+1) from m is 1 */ inexact += mpfr_set_z (y, m, MPFR_RNDN); MPFR_SET_EXP (y, MPFR_GET_EXP (y) + e / (mpfr_exp_t) k); if (negative) { MPFR_CHANGE_SIGN (y); inexact = -inexact; } mpz_clear (m); MPFR_SAVE_EXPO_FREE (expo); return mpfr_check_range (y, inexact, rnd_mode); } /* Compute y <- x^(1/k) using exp(log(x)/k). Assume all special cases have been eliminated before. In the extended exponent range, overflows/underflows are not possible. Assume x > 0, or x < 0 and k odd. Also assume |x| <> 1 because log(1) = 0, which does not have an exponent and would yield a failure in the error bound computation. A priori, this constraint is quite artificial because if |x| is close enough to 1, then the exponent of log|x| does not need to be used (in the code, err would be 1 in such a domain). So this constraint |x| <> 1 could be avoided in the code. However, this is an exact case easy to detect, so that such a change would be useless. Values very close to 1 are not an issue, since an underflow is not possible before the MPFR_GET_EXP. */ static int mpfr_root_aux (mpfr_ptr y, mpfr_srcptr x, unsigned long k, mpfr_rnd_t rnd_mode) { int inexact, exact_root = 0; mpfr_prec_t w; /* working precision */ mpfr_t absx, t; MPFR_GROUP_DECL(group); MPFR_TMP_DECL(marker); MPFR_ZIV_DECL(loop); MPFR_SAVE_EXPO_DECL (expo); MPFR_TMP_INIT_ABS (absx, x); MPFR_TMP_MARK(marker); w = MPFR_PREC(y) + 10; /* Take some guard bits to prepare for the 'expt' lost bits below. If |x| < 2^k, then log|x| < k, thus taking log2(k) bits should be fine. */ if (MPFR_GET_EXP(x) > 0) w += MPFR_INT_CEIL_LOG2 (MPFR_GET_EXP(x)); MPFR_GROUP_INIT_1(group, w, t); MPFR_SAVE_EXPO_MARK (expo); MPFR_ZIV_INIT (loop, w); for (;;) { mpfr_exp_t expt; unsigned int err; mpfr_log (t, absx, MPFR_RNDN); /* t = log|x| * (1 + theta) with |theta| <= 2^(-w) */ mpfr_div_ui (t, t, k, MPFR_RNDN); /* No possible underflow in mpfr_log and mpfr_div_ui. */ expt = MPFR_GET_EXP (t); /* assumes t <> 0 */ /* t = log|x|/k * (1 + theta) + eps with |theta| <= 2^(-w) and |eps| <= 1/2 ulp(t), thus the total error is bounded by 1.5 * 2^(expt - w) */ mpfr_exp (t, t, MPFR_RNDN); /* t = |x|^(1/k) * exp(tau) * (1 + theta1) with |tau| <= 1.5 * 2^(expt - w) and |theta1| <= 2^(-w). For |tau| <= 0.5 we have |exp(tau)-1| < 4/3*tau, thus for w >= expt + 2 we have: t = |x|^(1/k) * (1 + 2^(expt+2)*theta2) * (1 + theta1) with |theta1|, |theta2| <= 2^(-w). If expt+2 > 0, as long as w >= 1, we have: t = |x|^(1/k) * (1 + 2^(expt+3)*theta3) with |theta3| < 2^(-w). For expt+2 = 0, we have: t = |x|^(1/k) * (1 + 2^2*theta3) with |theta3| < 2^(-w). Finally for expt+2 < 0 we have: t = |x|^(1/k) * (1 + 2*theta3) with |theta3| < 2^(-w). */ err = (expt + 2 > 0) ? expt + 3 : (expt + 2 == 0) ? 2 : 1; /* now t = |x|^(1/k) * (1 + 2^(err-w)) thus the error is at most 2^(EXP(t) - w + err) */ if (MPFR_LIKELY (MPFR_CAN_ROUND(t, w - err, MPFR_PREC(y), rnd_mode))) break; /* If we fail to round correctly, check for an exact result or a midpoint result with MPFR_RNDN (regarded as hard-to-round in all precisions in order to determine the ternary value). */ { mpfr_t z, zk; mpfr_init2 (z, MPFR_PREC(y) + (rnd_mode == MPFR_RNDN)); mpfr_init2 (zk, MPFR_PREC(x)); mpfr_set (z, t, MPFR_RNDN); inexact = mpfr_pow_ui (zk, z, k, MPFR_RNDN); exact_root = !inexact && mpfr_equal_p (zk, absx); if (exact_root) /* z is the exact root, thus round z directly */ inexact = mpfr_set4 (y, z, rnd_mode, MPFR_SIGN (x)); mpfr_clear (zk); mpfr_clear (z); if (exact_root) break; } MPFR_ZIV_NEXT (loop, w); MPFR_GROUP_REPREC_1(group, w, t); } MPFR_ZIV_FREE (loop); if (!exact_root) inexact = mpfr_set4 (y, t, rnd_mode, MPFR_SIGN (x)); MPFR_GROUP_CLEAR(group); MPFR_TMP_FREE(marker); MPFR_SAVE_EXPO_FREE (expo); return mpfr_check_range (y, inexact, rnd_mode); } int mpfr_root (mpfr_ptr y, mpfr_srcptr x, unsigned long k, mpfr_rnd_t rnd_mode) { MPFR_LOG_FUNC (("x[%Pu]=%.*Rg k=%lu rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, k, rnd_mode), ("y[%Pu]=%.*Rg", mpfr_get_prec (y), mpfr_log_prec, y)); /* Like mpfr_rootn_ui... */ if (MPFR_UNLIKELY (k <= 1)) { if (k == 0) { /* rootn(x,0) is NaN (IEEE 754-2008). */ MPFR_SET_NAN (y); MPFR_RET_NAN; } else /* y = x^(1/1) = x */ return mpfr_set (y, x, rnd_mode); } if (MPFR_UNLIKELY (MPFR_IS_ZERO (x))) { /* The only case that may differ from mpfr_rootn_ui. */ MPFR_SET_ZERO (y); MPFR_SET_SAME_SIGN (y, x); MPFR_RET (0); } else return mpfr_rootn_ui (y, x, k, rnd_mode); }