/* mpfr_round_near_x -- Round a floating point number nears another one. Copyright 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, and was contributed by Mathieu Dutour. 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. */ #include "mpfr-impl.h" /* Use MPFR_FAST_COMPUTE_IF_SMALL_INPUT instead (a simple wrapper) */ /* int mpfr_round_near_x (mpfr_ptr y, mpfr_srcptr v, mpfr_uexp_t err, int dir, mp_rnd_t rnd) TODO: fix this description. Assuming y = o(f(x)) = o(x + g(x)) with |g(x)| < 2^(EXP(v)-error) If x is small enough, y ~= v. This function checks and does this. It assumes that f(x) is not representable exactly as a FP number. v must not be a singular value (NAN, INF or ZERO), usual values are v=1 or v=x. y is the destination (a mpfr_t), v the value to set (a mpfr_t), err the error term (a mpfr_uexp_t) such that |g(x)| < 2^(EXP(x)-err), dir (an int) is the direction of the error (if dir = 0, it rounds towards 0, if dir=1, it rounds away from 0), rnd the rounding mode. It returns 0 if it can't round. Otherwise it returns the ternary flag (It can't return an exact value). */ /* What "small enough" means? We work with the positive values. Assuming err > Prec (y)+1 i = [ y = o(x)] // i = inexact flag If i == 0 Setting x in y is exact. We have: y = [XXXXXXXXX[...]]0[...] + error where [..] are optional zeros if dirError = ToInf, x < f(x) < x + 2^(EXP(x)-err) since x=y, and ulp (y)/2 > 2^(EXP(x)-err), we have: y < f(x) < y+ulp(y) and |y-f(x)| < ulp(y)/2 if rnd = RNDN, nothing if rnd = RNDZ, nothing if rnd = RNDA, addoneulp elif dirError = ToZero x -2^(EXP(x)-err) < f(x) < x since x=y, and ulp (y)/2 > 2^(EXP(x)-err), we have: y-ulp(y) < f(x) < y and |y-f(x)| < ulp(y)/2 if rnd = RNDN, nothing if rnd = RNDZ, nexttozero if rnd = RNDA, nothing NOTE: err > prec (y)+1 is needed only for RNDN. elif i > 0 and i = EVEN_ROUNDING So rnd = RNDN and we have y = x + ulp(y)/2 if dirError = ToZero, we have x -2^(EXP(x)-err) < f(x) < x so y - ulp(y)/2 - 2^(EXP(x)-err) < f(x) < y-ulp(y)/2 so y -ulp(y) < f(x) < y-ulp(y)/2 => nexttozero(y) elif dirError = ToInf we have x < f(x) < x + 2^(EXP(x)-err) so y - ulp(y)/2 < f(x) < y+ulp(y)/2-ulp(y)/2 so y - ulp(y)/2 < f(x) < y => do nothing elif i < 0 and i = -EVEN_ROUNDING So rnd = RNDN and we have y = x - ulp(y)/2 if dirError = ToZero, y < f(x) < y + ulp(y)/2 => do nothing if dirError = ToInf y + ulp(y)/2 < f(x) < y + ulp(y) => AddOneUlp elif i > 0 we can't have rnd = RNDZ, and prec(x) > prec(y), so ulp(x) < ulp(y) we have y - ulp (y) < x < y or more exactly y - ulp(y) + ulp(x)/2 <= x <= y - ulp(x)/2 if rnd = RNDA, if dirError = ToInf, we have x < f(x) < x + 2^(EXP(x)-err) if err > prec (x), we have 2^(EXP(x)-err) < ulp(x), so 2^(EXP(x)-err) <= ulp(x)/2 so f(x) <= y - ulp(x)/2+ulp(x)/2 <= y and y - ulp(y) < x < f(x) so we have y - ulp(y) < f(x) < y so do nothing. elif we can round, ie y - ulp(y) < x + 2^(EXP(x)-err) < y we have y - ulp(y) < x < f(x) < x + 2^(EXP(x)-err) < y so do nothing otherwise Wrong. Example X=[0.11101]111111110000 + 1111111111111111111.... elif dirError = ToZero we have x - 2^(EXP(x)-err) < f(x) < x so f(x) < x < y if err > prec (x) x-2^(EXP(x)-err) >= x-ulp(x)/2 >= y - ulp(y) + ulp(x)/2-ulp(x)/2 so y - ulp(y) < f(x) < y so do nothing elif we can round, ie y - ulp(y) < x - 2^(EXP(x)-err) < y y - ulp(y) < x - 2^(EXP(x)-err) < f(x) < y so do nothing otherwise Wrong. Example: X=[1.111010]00000010 - 10000001000000000000100.... elif rnd = RNDN, y - ulp(y)/2 < x < y and we can't have x = y-ulp(y)/2: so we have: y - ulp(y)/2 + ulp(x)/2 <= x <= y - ulp(x)/2 if dirError = ToInf we have x < f(x) < x+2^(EXP(x)-err) and ulp(y) > 2^(EXP(x)-err) so y - ulp(y)/2 + ulp (x)/2 < f(x) < y + ulp (y)/2 - ulp (x)/2 we can round but we can't compute inexact flag. if err > prec (x) y - ulp(y)/2 + ulp (x)/2 < f(x) < y + ulp(x)/2 - ulp(x)/2 so y - ulp(y)/2 + ulp (x)/2 < f(x) < y we can round and compute inexact flag. do nothing elif we can round, ie y - ulp(y)/2 < x + 2^(EXP(x)-err) < y we have y - ulp(y)/2 + ulp (x)/2 < f(x) < y so do nothing otherwise Wrong elif dirError = ToZero we have x -2^(EXP(x)-err) < f(x) < x and ulp(y)/2 > 2^(EXP(x)-err) so y-ulp(y)+ulp(x)/2 < f(x) < y - ulp(x)/2 if err > prec (x) x- ulp(x)/2 < f(x) < x so y - ulp(y)/2+ulp(x)/2 - ulp(x)/2 < f(x) < x <= y - ulp(x)/2 < y do nothing elif we can round, ie y-ulp(y)/2 < x-2^(EXP(x)-err) < y we have y-ulp(y)/2 < x-2^(EXP(x)-err) < f(x) < x < y do nothing otherwise Wrong elif i < 0 same thing? */ int mpfr_round_near_x (mpfr_ptr y, mpfr_srcptr v, mpfr_uexp_t err, int dir, mp_rnd_t rnd) { int inexact, sign; unsigned int old_flags = __gmpfr_flags; MPFR_ASSERTD (!MPFR_IS_SINGULAR (v)); MPFR_ASSERTD (dir == 0 || dir == 1); /* First check if we can round. The test is more restrictive than necessary. Note that if err is not representable in an mp_exp_t, then err > MPFR_PREC (v) and the conversion to mp_exp_t will not occur. */ if (!(err > MPFR_PREC (y) + 1 && (err > MPFR_PREC (v) || mpfr_round_p (MPFR_MANT (v), MPFR_LIMB_SIZE (v), (mp_exp_t) err, MPFR_PREC (y) + (rnd == GMP_RNDN))))) /* If we assume we can not round, return 0, and y is not modified */ return 0; /* First round v in y */ sign = MPFR_SIGN (v); MPFR_SET_EXP (y, MPFR_GET_EXP (v)); MPFR_SET_SIGN (y, sign); MPFR_RNDRAW_GEN (inexact, y, MPFR_MANT (v), MPFR_PREC (v), rnd, sign, if (dir == 0) { inexact = -sign; goto trunc_doit; } else goto addoneulp; , if (MPFR_UNLIKELY (++MPFR_EXP (y) > __gmpfr_emax)) mpfr_overflow (y, rnd, sign) ); /* Fix it in some cases */ MPFR_ASSERTD (!MPFR_IS_NAN (y) && !MPFR_IS_ZERO (y)); /* If inexact == 0, setting y from v is exact but we haven't take into account yet the error term */ if (inexact == 0) { if (dir == 0) /* The error term is negative for v positive */ { inexact = sign; if (MPFR_IS_LIKE_RNDZ (rnd, MPFR_IS_NEG_SIGN (sign))) { /* case nexttozero */ /* The underflow flag should be set if the result is zero */ __gmpfr_flags = old_flags; inexact = -sign; mpfr_nexttozero (y); if (MPFR_UNLIKELY (MPFR_IS_ZERO (y))) mpfr_set_underflow (); } } else /* The error term is positive for v positive */ { inexact = -sign; /* Round Away */ if (rnd != GMP_RNDN && rnd != GMP_RNDZ && MPFR_IS_RNDUTEST_OR_RNDDNOTTEST (rnd, MPFR_IS_POS_SIGN(sign))) { /* case nexttoinf */ /* The overflow flag should be set if the result is infinity */ inexact = sign; mpfr_nexttoinf (y); if (MPFR_UNLIKELY (MPFR_IS_INF (y))) mpfr_set_overflow (); } } } /* the inexact flag cannot be 0, since this would mean an exact value, and in this case we cannot round correctly */ MPFR_ASSERTD(inexact != 0); MPFR_RET (inexact); }