/* mpfr_hypot -- Euclidean distance 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 #include "mpfr-impl.h" /* The computation of hypot of x and y is done by * * hypot(x,y)= sqrt(x^2+y^2) = z */ int mpfr_hypot (mpfr_ptr z, mpfr_srcptr x, mpfr_srcptr y, mpfr_rnd_t rnd_mode) { int inexact; unsigned int exact; /* Warning: 0 will mean "exact" */ mpfr_t t, te, ti; /* auxiliary variables */ mpfr_prec_t N, Nz; /* size variables */ mpfr_prec_t Nt; /* precision of the intermediary variable */ mpfr_prec_t threshold; mpfr_exp_t Ex, sh; mpfr_uexp_t diff_exp; MPFR_SAVE_EXPO_DECL (expo); MPFR_ZIV_DECL (loop); MPFR_BLOCK_DECL (flags); MPFR_LOG_FUNC (("x[%Pu]=%.*Rg y[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, mpfr_get_prec (y), mpfr_log_prec, y, rnd_mode), ("z[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (z), mpfr_log_prec, z, inexact)); /* particular cases */ if (MPFR_ARE_SINGULAR (x, y)) { if (MPFR_IS_INF (x) || MPFR_IS_INF (y)) { /* Return +inf, even when the other number is NaN. */ MPFR_SET_INF (z); MPFR_SET_POS (z); MPFR_RET (0); } else if (MPFR_IS_NAN (x) || MPFR_IS_NAN (y)) { MPFR_SET_NAN (z); MPFR_RET_NAN; } else if (MPFR_IS_ZERO (x)) return mpfr_abs (z, y, rnd_mode); else /* y is necessarily 0 */ return mpfr_abs (z, x, rnd_mode); } /* TODO: It may be sufficient to just compare the exponents. The error analysis would need to be updated. */ if (mpfr_cmpabs (x, y) < 0) { mpfr_srcptr u; u = x; x = y; y = u; } /* now |x| >= |y| */ Ex = MPFR_GET_EXP (x); diff_exp = (mpfr_uexp_t) Ex - MPFR_GET_EXP (y); N = MPFR_PREC (x); /* Precision of input variable */ Nz = MPFR_PREC (z); /* Precision of output variable */ threshold = (MAX (N, Nz) + (rnd_mode == MPFR_RNDN ? 1 : 0)) << 1; if (rnd_mode == MPFR_RNDA) rnd_mode = MPFR_RNDU; /* since the result is positive, RNDA = RNDU */ /* Is |x| a suitable approximation to the precision Nz ? (see algorithms.tex for explanations) */ if (diff_exp > threshold) /* result is |x| or |x|+ulp(|x|,Nz) */ { if (MPFR_UNLIKELY (rnd_mode == MPFR_RNDU)) { /* If z > abs(x), then it was already rounded up; otherwise z = abs(x), and we need to add one ulp due to y. */ if (mpfr_abs (z, x, rnd_mode) == 0) { mpfr_nexttoinf (z); /* since mpfr_nexttoinf does not set the overflow flag, we have to check manually for overflow */ if (MPFR_UNLIKELY (MPFR_IS_INF (z))) MPFR_SET_OVERFLOW (); } MPFR_RET (1); } else /* MPFR_RNDZ, MPFR_RNDD, MPFR_RNDN */ { if (MPFR_LIKELY (Nz >= N)) { mpfr_abs (z, x, rnd_mode); /* exact */ MPFR_RET (-1); } else { MPFR_SET_EXP (z, Ex); MPFR_SET_SIGN (z, MPFR_SIGN_POS); MPFR_RNDRAW_GEN (inexact, z, MPFR_MANT (x), N, rnd_mode, 1, goto addoneulp, if (MPFR_UNLIKELY (++ MPFR_EXP (z) > __gmpfr_emax)) return mpfr_overflow (z, rnd_mode, 1); ); if (MPFR_UNLIKELY (inexact == 0)) inexact = -1; MPFR_RET (inexact); } } } /* General case */ N = MAX (MPFR_PREC (x), MPFR_PREC (y)); /* working precision */ Nt = Nz + MPFR_INT_CEIL_LOG2 (Nz) + 4; mpfr_init2 (t, Nt); mpfr_init2 (te, Nt); mpfr_init2 (ti, Nt); MPFR_SAVE_EXPO_MARK (expo); /* Scale x and y to avoid any overflow and underflow in x^2 (as |x| >= |y|), and to limit underflow cases for y or y^2. We have: x = Mx * 2^Ex with 1/2 <= |Mx| < 1, and we choose: sh = floor((Emax - 1) / 2) - Ex. Thus (x * 2^sh)^2 = Mx^2 * 2^(2*floor((Emax-1)/2)), which has an exponent of at most Emax - 1. Therefore (x * 2^sh)^2 + (y * 2^sh)^2 will have an exponent of at most Emax, even after rounding as we round toward zero. Using a larger sh wouldn't guarantee an absence of overflow. Note that the scaling of y can underflow only when the target precision is huge, otherwise the case would already have been handled by the diff_exp > threshold code; but this case is avoided thanks to a FMA (this problem is transferred to the FMA code). */ sh = (mpfr_get_emax () - 1) / 2 - Ex; /* FIXME: ti is subject to underflow. Solution: x and y could be aliased with MPFR_ALIAS, and if need be, the aliases be pre-scaled exactly as UBF, so that x^2 + y^2 is in range. Then call mpfr_fmma and the square root, and scale the result. The error analysis would be simpler. Note: mpfr_fmma is currently not optimized. */ MPFR_ZIV_INIT (loop, Nt); for (;;) { mpfr_prec_t err; exact = mpfr_mul_2si (te, x, sh, MPFR_RNDZ); exact |= mpfr_mul_2si (ti, y, sh, MPFR_RNDZ); exact |= mpfr_sqr (te, te, MPFR_RNDZ); /* Use fma in order to avoid underflow when diff_exp<=MPFR_EMAX_MAX-2 */ exact |= mpfr_fma (t, ti, ti, te, MPFR_RNDZ); exact |= mpfr_sqrt (t, t, MPFR_RNDZ); err = Nt < N ? 4 : 2; if (MPFR_LIKELY (exact == 0 || MPFR_CAN_ROUND (t, Nt-err, Nz, rnd_mode))) break; MPFR_ZIV_NEXT (loop, Nt); mpfr_set_prec (t, Nt); mpfr_set_prec (te, Nt); mpfr_set_prec (ti, Nt); } MPFR_ZIV_FREE (loop); MPFR_BLOCK (flags, inexact = mpfr_div_2si (z, t, sh, rnd_mode)); MPFR_ASSERTD (exact == 0 || inexact != 0 || rnd_mode == MPFR_RNDF); mpfr_clear (t); mpfr_clear (ti); mpfr_clear (te); /* exact inexact 0 0 result is exact, ternary flag is 0 0 non zero t is exact, ternary flag given by inexact 1 0 impossible (see above) 1 non zero ternary flag given by inexact */ MPFR_SAVE_EXPO_FREE (expo); if (MPFR_OVERFLOW (flags)) MPFR_SET_OVERFLOW (); /* hypot(x,y) >= |x|, thus underflow is not possible. */ return mpfr_check_range (z, inexact, rnd_mode); }