/* mpfr_sub1 -- internal function to perform a "real" subtraction 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. */ #include "mpfr-impl.h" /* compute sign(b) * (|b| - |c|), with |b| > |c|, diff_exp = EXP(b) - EXP(c) Returns 0 iff result is exact, a negative value when the result is less than the exact value, a positive value otherwise. */ int mpfr_sub1 (mpfr_ptr a, mpfr_srcptr b, mpfr_srcptr c, mpfr_rnd_t rnd_mode) { int sign; mpfr_exp_t diff_exp, exp_a, exp_b; mpfr_prec_t cancel, cancel1; mp_size_t cancel2, an, bn, cn, cn0; mp_limb_t *ap, *bp, *cp; mp_limb_t carry, bb, cc; mpfr_prec_t aq, bq; int inexact, shift_b, shift_c, add_exp = 0; int cmp_low = 0; /* used for rounding to nearest: 0 if low(b) = low(c), negative if low(b) < low(c), positive if low(b) > low(c) */ int sh, k; MPFR_TMP_DECL(marker); MPFR_TMP_MARK(marker); ap = MPFR_MANT(a); an = MPFR_LIMB_SIZE(a); (void) MPFR_GET_PREC (a); (void) MPFR_GET_PREC (b); (void) MPFR_GET_PREC (c); sign = mpfr_cmp2 (b, c, &cancel); if (MPFR_UNLIKELY(sign == 0)) { MPFR_LOG_MSG (("sign=0\n", 0)); if (rnd_mode == MPFR_RNDD) MPFR_SET_NEG (a); else MPFR_SET_POS (a); MPFR_SET_ZERO (a); MPFR_RET (0); } /* sign != 0, so that cancel has a valid value. */ MPFR_LOG_MSG (("sign=%d cancel=%Pd\n", sign, cancel)); MPFR_ASSERTD (cancel >= 0 && cancel <= MPFR_PREC_MAX); /* * If subtraction: sign(a) = sign * sign(b) * If addition: sign(a) = sign of the larger argument in absolute value. * * Both cases can be simplified in: * if (sign>0) * if addition: sign(a) = sign * sign(b) = sign(b) * if subtraction, b is greater, so sign(a) = sign(b) * else * if subtraction, sign(a) = - sign(b) * if addition, sign(a) = sign(c) (since c is greater) * But if it is an addition, sign(b) and sign(c) are opposed! * So sign(a) = - sign(b) */ if (sign < 0) /* swap b and c so that |b| > |c| */ { mpfr_srcptr t; MPFR_SET_OPPOSITE_SIGN (a,b); t = b; b = c; c = t; } else MPFR_SET_SAME_SIGN (a,b); if (MPFR_UNLIKELY (MPFR_IS_UBF (b) || MPFR_IS_UBF (c))) { exp_b = MPFR_UBF_GET_EXP (b); /* Early underflow detection. Rare, but a test is needed anyway since in the "MAX (aq, bq) + 2 <= diff_exp" branch, the exponent may decrease and MPFR_EXP_MIN would yield an integer overflow. */ if (MPFR_UNLIKELY (exp_b < __gmpfr_emin - 1)) { if (rnd_mode == MPFR_RNDN) rnd_mode = MPFR_RNDZ; return mpfr_underflow (a, rnd_mode, MPFR_SIGN(a)); } diff_exp = mpfr_ubf_diff_exp (b, c); MPFR_LOG_MSG (("UBF: exp_b=%" MPFR_EXP_FSPEC "d%s " "diff_exp=%" MPFR_EXP_FSPEC "d%s\n", (mpfr_eexp_t) exp_b, exp_b == MPFR_EXP_MAX ? "=MPFR_EXP_MAX" : "", (mpfr_eexp_t) diff_exp, diff_exp == MPFR_EXP_MAX ? "=MPFR_EXP_MAX" : "")); /* If diff_exp == MPFR_EXP_MAX, the actual value can be larger, but anyway, since mpfr_exp_t >= mp_size_t, this will be the case c small below, and the exact value does not matter. */ /* mpfr_set4 below used with MPFR_RNDF does not support UBF. */ if (rnd_mode == MPFR_RNDF) rnd_mode = MPFR_RNDN; } else { exp_b = MPFR_GET_EXP (b); diff_exp = exp_b - MPFR_GET_EXP (c); } MPFR_ASSERTD (diff_exp >= 0); aq = MPFR_GET_PREC (a); bq = MPFR_GET_PREC (b); /* Check if c is too small. A more precise test is to replace 2 by (rnd == MPFR_RNDN) + mpfr_power2_raw (b) but it is more expensive and not very useful */ if (MPFR_UNLIKELY (MAX (aq, bq) + 2 <= diff_exp)) { MPFR_LOG_MSG (("case c small\n", 0)); /* Remember, we can't have an exact result! */ /* A.AAAAAAAAAAAAAAAAA = B.BBBBBBBBBBBBBBB - C.CCCCCCCCCCCCC */ /* A = S*ABS(B) +/- ulp(a) */ /* since we can't have an exact result, for RNDF we can truncate b */ if (rnd_mode == MPFR_RNDF) return mpfr_set4 (a, b, MPFR_RNDZ, MPFR_SIGN (a)); exp_a = exp_b; /* may be any out-of-range value due to UBF */ MPFR_RNDRAW_EVEN (inexact, a, MPFR_MANT (b), bq, rnd_mode, MPFR_SIGN (a), if (exp_a != MPFR_EXP_MAX) exp_a ++); MPFR_LOG_MSG (("inexact=%d\n", inexact)); if (inexact == 0 && /* a = b, but the exact value of b - c is a bit below. Then, except for directed rounding similar to toward zero and before overflow checking: a is the correctly rounded value and since |b| - |c| < |a|, the ternary value value is given by the sign of a. */ ! MPFR_IS_LIKE_RNDZ (rnd_mode, MPFR_IS_NEG (a))) { MPFR_LOG_MSG (("c small, case 1\n", 0)); inexact = MPFR_INT_SIGN (a); } else if (inexact != 0 && /* A.AAAAAAAAAAAAAA = B.BBBBBBBBBBBBBBB - C.CCCCCCCCCCCCC */ /* It isn't exact, so PREC(b) > PREC(a) and the last PREC(b)-PREC(a) bits of b are not all zeros. Subtracting c from b will not have an effect on the rounding except in case of a midpoint in the round-to-nearest mode, when the even rounding was done away from zero instead of toward zero. In case of even rounding: 1.BBBBBBBBBBBBBx10 - 1.CCCCCCCCCCCC = 1.BBBBBBBBBBBBBx01 Rounded to PREC(b) = 1.BBBBBBBBBBBBBx Nearest / Rounded to PREC(a) Set gives: 1.BBBBBBBBBBBBB0 if inexact == EVEN_INEX (x == 0) 1.BBBBBBBBBBBBB1+1 if inexact == -EVEN_INEX (x == 1) which means we get a wrong rounded result if x == 1, i.e. inexact == MPFR_EVEN_INEX (for positive numbers). */ MPFR_LIKELY (inexact != MPFR_EVEN_INEX * MPFR_INT_SIGN (a))) { MPFR_LOG_MSG (("c small, case 2\n", 0)); /* nothing to do */ } else { /* We need to take the value preceding |a|. We can't use mpfr_nexttozero due to a possible out-of-range exponent. But this will allow us to have more specific code. */ MPFR_LOG_MSG (("c small, case 3: correcting the value of a\n", 0)); sh = (mpfr_prec_t) an * GMP_NUMB_BITS - aq; mpn_sub_1 (ap, ap, an, MPFR_LIMB_ONE << sh); if (MPFR_UNLIKELY (MPFR_LIMB_MSB (ap[an-1]) == 0)) { exp_a --; /* The following is valid whether an = 1 or an > 1. */ ap[an-1] |= MPFR_LIMB_HIGHBIT; } inexact = - MPFR_INT_SIGN (a); } /* The underflow case is possible only with UBF. The overflow case is also possible with normal FP due to rounding. */ if (MPFR_UNLIKELY (exp_a > __gmpfr_emax)) return mpfr_overflow (a, rnd_mode, MPFR_SIGN (a)); if (MPFR_UNLIKELY (exp_a < __gmpfr_emin)) { if (rnd_mode == MPFR_RNDN && (exp_a < __gmpfr_emin - 1 || (inexact * MPFR_INT_SIGN (a) >= 0 && mpfr_powerof2_raw (a)))) rnd_mode = MPFR_RNDZ; return mpfr_underflow (a, rnd_mode, MPFR_SIGN(a)); } MPFR_SET_EXP (a, exp_a); MPFR_RET (inexact); } /* reserve a space to store b aligned with the result, i.e. shifted by (-cancel) % GMP_NUMB_BITS to the right */ bn = MPFR_LIMB_SIZE (b); MPFR_UNSIGNED_MINUS_MODULO (shift_b, cancel); cancel1 = (cancel + shift_b) / GMP_NUMB_BITS; /* the high cancel1 limbs from b should not be taken into account */ if (MPFR_UNLIKELY (shift_b == 0)) { bp = MPFR_MANT(b); /* no need of an extra space */ /* Ensure ap != bp */ if (MPFR_UNLIKELY (ap == bp)) { bp = MPFR_TMP_LIMBS_ALLOC (bn); MPN_COPY (bp, ap, bn); } } else { bp = MPFR_TMP_LIMBS_ALLOC (bn + 1); bp[0] = mpn_rshift (bp + 1, MPFR_MANT(b), bn++, shift_b); } /* reserve a space to store c aligned with the result, i.e. shifted by (diff_exp-cancel) % GMP_NUMB_BITS to the right */ cn = MPFR_LIMB_SIZE (c); if (IS_POW2 (GMP_NUMB_BITS)) shift_c = ((mpfr_uexp_t) diff_exp - cancel) % GMP_NUMB_BITS; else { /* The above operation does not work if diff_exp - cancel < 0. */ shift_c = diff_exp - (cancel % GMP_NUMB_BITS); shift_c = (shift_c + GMP_NUMB_BITS) % GMP_NUMB_BITS; } MPFR_ASSERTD (shift_c >= 0 && shift_c < GMP_NUMB_BITS); if (MPFR_UNLIKELY(shift_c == 0)) { cp = MPFR_MANT(c); /* Ensure ap != cp */ if (ap == cp) { cp = MPFR_TMP_LIMBS_ALLOC (cn); MPN_COPY(cp, ap, cn); } } else { cp = MPFR_TMP_LIMBS_ALLOC (cn + 1); cp[0] = mpn_rshift (cp + 1, MPFR_MANT(c), cn++, shift_c); } #if 0 MPFR_LOG_MSG (("rnd=%s shift_b=%d shift_c=%d diffexp=%" MPFR_EXP_FSPEC "d\n", mpfr_print_rnd_mode (rnd_mode), shift_b, shift_c, (mpfr_eexp_t) diff_exp)); #endif MPFR_ASSERTD (ap != cp); MPFR_ASSERTD (bp != cp); /* here we have shift_c = (diff_exp - cancel) % GMP_NUMB_BITS, 0 <= shift_c < GMP_NUMB_BITS thus we want cancel2 = ceil((cancel - diff_exp) / GMP_NUMB_BITS) */ /* Possible optimization with a C99 compiler (i.e. well-defined integer division): if MPFR_PREC_MAX is reduced to ((mpfr_prec_t)((mpfr_uprec_t)(~(mpfr_uprec_t)0)>>1) - GMP_NUMB_BITS + 1) and diff_exp is of type mpfr_exp_t (no need for mpfr_uexp_t, since the sum or difference of 2 exponents must be representable, as used by the multiplication code), then the computation of cancel2 could be simplified to cancel2 = (cancel - (diff_exp - shift_c)) / GMP_NUMB_BITS; because cancel, diff_exp and shift_c are all nonnegative and these variables are signed. */ MPFR_ASSERTD (cancel >= 0); if (cancel >= diff_exp) /* Note that cancel is signed and will be converted to mpfr_uexp_t (type of diff_exp) in the expression below, so that this will work even if cancel is very large and diff_exp = 0. */ cancel2 = (cancel - diff_exp + (GMP_NUMB_BITS - 1)) / GMP_NUMB_BITS; else cancel2 = - (mp_size_t) ((diff_exp - cancel) / GMP_NUMB_BITS); /* the high cancel2 limbs from b should not be taken into account */ #if 0 MPFR_LOG_MSG (("cancel=%Pd cancel1=%Pd cancel2=%Pd\n", cancel, cancel1, cancel2)); #endif /* ap[an-1] ap[0] <----------------+-----------|----> <----------PREC(a)----------><-sh-> cancel1 limbs bp[bn-cancel1-1] <--...-----><----------------+-----------+-----------> cancel2 limbs cp[cn-cancel2-1] cancel2 >= 0 <--...--><----------------+----------------+----------------> (-cancel2) cancel2 < 0 limbs <----------------+----------------> */ /* first part: put in ap[0..an-1] the value of high(b) - high(c), where high(b) consists of the high an+cancel1 limbs of b, and high(c) consists of the high an+cancel2 limbs of c. */ /* copy high(b) into a */ if (MPFR_LIKELY(an + (mp_size_t) cancel1 <= bn)) /* a: <----------------+-----------|----> b: <-----------------------------------------> */ MPN_COPY (ap, bp + bn - (an + cancel1), an); else /* a: <----------------+-----------|----> b: <-------------------------> */ if ((mp_size_t) cancel1 < bn) /* otherwise b does not overlap with a */ { MPN_ZERO (ap, an + cancel1 - bn); MPN_COPY (ap + (an + cancel1 - bn), bp, bn - cancel1); } else MPN_ZERO (ap, an); /* subtract high(c) */ if (MPFR_LIKELY(an + cancel2 > 0)) /* otherwise c does not overlap with a */ { mp_limb_t *ap2; if (cancel2 >= 0) { if (an + cancel2 <= cn) /* a: <-----------------------------> c: <-----------------------------------------> */ mpn_sub_n (ap, ap, cp + cn - (an + cancel2), an); else /* a: <----------------------------> c: <-------------------------> */ { ap2 = ap + an + (cancel2 - cn); if (cn > cancel2) mpn_sub_n (ap2, ap2, cp, cn - cancel2); } } else /* cancel2 < 0 */ { mp_limb_t borrow; if (an + cancel2 <= cn) /* a: <-----------------------------> c: <-----------------------------> */ borrow = mpn_sub_n (ap, ap, cp + cn - (an + cancel2), an + cancel2); else /* a: <----------------------------> c: <----------------> */ { ap2 = ap + an + cancel2 - cn; borrow = mpn_sub_n (ap2, ap2, cp, cn); } ap2 = ap + an + cancel2; mpn_sub_1 (ap2, ap2, -cancel2, borrow); } } /* now perform rounding */ sh = (mpfr_prec_t) an * GMP_NUMB_BITS - aq; /* last unused bits from a */ carry = ap[0] & MPFR_LIMB_MASK (sh); ap[0] -= carry; if (rnd_mode == MPFR_RNDF) { inexact = 0; /* truncating is always correct since -1 ulp < low(b) - low(c) < 1 ulp */ goto truncate; } else if (rnd_mode == MPFR_RNDN) { if (MPFR_LIKELY(sh)) { /* can decide except when carry = 2^(sh-1) [middle] or carry = 0 [truncate, but cannot decide inexact flag] */ if (carry > (MPFR_LIMB_ONE << (sh - 1))) goto add_one_ulp; else if ((0 < carry) && (carry < (MPFR_LIMB_ONE << (sh - 1)))) { inexact = -1; /* result if smaller than exact value */ goto truncate; } /* now carry = 2^(sh-1), in which case cmp_low=2, or carry = 0, in which case cmp_low=0 */ cmp_low = (carry == 0) ? 0 : 2; } } else /* directed rounding: set rnd_mode to RNDZ iff toward zero */ { if (MPFR_IS_RNDUTEST_OR_RNDDNOTTEST(rnd_mode, MPFR_IS_NEG(a))) rnd_mode = MPFR_RNDZ; if (carry) { if (rnd_mode == MPFR_RNDZ) { inexact = -1; goto truncate; } else /* round away */ goto add_one_ulp; } } /* we have to consider the low (bn - (an+cancel1)) limbs from b, and the (cn - (an+cancel2)) limbs from c. */ bn -= an + cancel1; cn0 = cn; cn -= an + cancel2; #if 0 MPFR_LOG_MSG (("last sh=%d bits from a are %Mu, bn=%Pd, cn=%Pd\n", sh, carry, (mpfr_prec_t) bn, (mpfr_prec_t) cn)); #endif /* for rounding to nearest, we couldn't conclude up to here in the following cases: 1. sh = 0, then cmp_low=0: we can either truncate, subtract one ulp or add one ulp: -1 ulp < low(b)-low(c) < 1 ulp 2. sh > 0 but the low sh bits from high(b)-high(c) equal 2^(sh-1): -0.5 ulp <= -1/2^sh < low(b)-low(c)-0.5 < 1/2^sh <= 0.5 ulp we can't decide the rounding, in that case cmp_low=2: either we truncate and flag=-1, or we add one ulp and flag=1 3. the low sh>0 bits from high(b)-high(c) equal 0: we know we have to truncate but we can't decide the ternary value, here cmp_low=0: -0.5 ulp <= -1/2^sh < low(b)-low(c) < 1/2^sh <= 0.5 ulp we always truncate and inexact can be any of -1,0,1 */ /* note: here cn might exceed cn0, in which case we consider a zero limb */ for (k = 0; (bn > 0) || (cn > 0); k = 1) { /* if cmp_low < 0, we know low(b) - low(c) < 0 if cmp_low > 0, we know low(b) - low(c) > 0 (more precisely if cmp_low = 2, low(b) - low(c) = 0.5 ulp so far) if cmp_low = 0, so far low(b) - low(c) = 0 */ /* get next limbs */ bb = (bn > 0) ? bp[--bn] : 0; if ((cn > 0) && (cn-- <= cn0)) cc = cp[cn]; else cc = 0; /* cmp_low compares low(b) and low(c) */ if (cmp_low == 0) /* case 1 or 3 */ cmp_low = (bb < cc) ? -2+k : (bb > cc) ? 1 : 0; /* Case 1 for k=0 splits into 7 subcases: 1a: bb > cc + half 1b: bb = cc + half 1c: 0 < bb - cc < half 1d: bb = cc 1e: -half < bb - cc < 0 1f: bb - cc = -half 1g: bb - cc < -half Case 2 splits into 3 subcases: 2a: bb > cc 2b: bb = cc 2c: bb < cc Case 3 splits into 3 subcases: 3a: bb > cc 3b: bb = cc 3c: bb < cc */ /* the case rounding to nearest with sh=0 is special since one couldn't subtract above 1/2 ulp in the trailing limb of the result */ if (rnd_mode == MPFR_RNDN && sh == 0 && k == 0) /* case 1 for k=0 */ { mp_limb_t half = MPFR_LIMB_HIGHBIT; /* add one ulp if bb > cc + half truncate if cc - half < bb < cc + half sub one ulp if bb < cc - half */ if (cmp_low < 0) /* bb < cc: -1 ulp < low(b) - low(c) < 0, cases 1e, 1f and 1g */ { if (cc >= half) cc -= half; else /* since bb < cc < half, bb+half < 2*half */ bb += half; /* now we have bb < cc + half: we have to subtract one ulp if bb < cc, and truncate if bb > cc */ } else if (cmp_low >= 0) /* bb >= cc, cases 1a to 1d */ { if (cc < half) cc += half; else /* since bb >= cc >= half, bb - half >= 0 */ bb -= half; /* now we have bb > cc - half: we have to add one ulp if bb > cc, and truncate if bb < cc */ if (cmp_low > 0) cmp_low = 2; } } #if 0 MPFR_LOG_MSG (("k=%d bb=%Mu cc=%Mu cmp_low=%d\n", k, bb, cc, cmp_low)); #endif if (cmp_low < 0) /* low(b) - low(c) < 0: either truncate or subtract one ulp */ { if (rnd_mode == MPFR_RNDZ) goto sub_one_ulp; /* set inexact=-1 */ else if (rnd_mode != MPFR_RNDN) /* round away */ { inexact = 1; goto truncate; } else /* round to nearest */ { /* If cmp_low < 0 and bb > cc, then -0.5 ulp < low(b)-low(c) < 0, whatever the value of sh. If sh>0, then cmp_low < 0 implies that the initial neglected sh bits were 0 (otherwise cmp_low=2 initially), thus the weight of the new bits is less than 0.5 ulp too. If k > 0 (and sh=0) this means that either the first neglected limbs bb and cc were equal (thus cmp_low was 0 for k=0), or we had bb - cc = -0.5 ulp or 0.5 ulp. The last case is not possible here since we would have cmp_low > 0 which is sticky. In the first case (where we have cmp_low = -1), we truncate, whereas in the 2nd case we have cmp_low = -2 and we subtract one ulp. */ if (bb > cc || sh > 0 || cmp_low == -1) { /* -0.5 ulp < low(b)-low(c) < 0, bb > cc corresponds to cases 1e and 1f1 sh > 0 corresponds to cases 3c and 3b3 cmp_low = -1 corresponds to case 1d3 (also 3b3) */ inexact = 1; goto truncate; } else if (bb < cc) /* here sh = 0 and low(b)-low(c) < -0.5 ulp, this corresponds to cases 1g and 1f3 */ goto sub_one_ulp; /* the only case where we can't conclude is sh=0 and bb=cc, i.e., we have low(b) - low(c) = -0.5 ulp (up to now), thus we don't know if we must truncate or subtract one ulp. Note: for sh=0 we can't have low(b) - low(c) = -0.5 ulp up to now, since low(b) - low(c) > 1/2^sh */ } } else if (cmp_low > 0) /* 0 < low(b) - low(c): either truncate or add one ulp */ { if (rnd_mode == MPFR_RNDZ) { inexact = -1; goto truncate; } else if (rnd_mode != MPFR_RNDN) /* round away */ goto add_one_ulp; else /* round to nearest */ { if (bb > cc) { /* if sh=0, then bb>cc means that low(b)-low(c) > 0.5 ulp, and similarly when cmp_low=2 */ if (cmp_low == 2) /* cases 1a, 1b1, 2a and 2b1 */ goto add_one_ulp; /* sh > 0 and cmp_low > 0: this implies that the sh initial neglected bits were 0, and the remaining low(b)-low(c)>0, but its weight is less than 0.5 ulp */ else /* 0 < low(b) - low(c) < 0.5 ulp, this corresponds to cases 3a, 1d1 and 3b1 */ { inexact = -1; goto truncate; } } else if (bb < cc) /* 0 < low(b) - low(c) < 0.5 ulp, cases 1c, 1b3, 2b3 and 2c */ { inexact = -1; goto truncate; } /* the only case where we can't conclude is bb=cc, i.e., low(b) - low(c) = 0.5 ulp (up to now), thus we don't know if we must truncate or add one ulp. */ } } /* after k=0, we cannot conclude in the following cases, we split them according to the values of bb and cc for k=1: 1b. sh=0 and cmp_low = 1 and bb-cc = half [around 0.5 ulp] 1b1. bb > cc: add one ulp, inex = 1 1b2: bb = cc: cannot conclude 1b3: bb < cc: truncate, inex = -1 1d. sh=0 and cmp_low = 0 and bb-cc = 0 [around 0] 1d1: bb > cc: truncate, inex = -1 1d2: bb = cc: cannot conclude 1d3: bb < cc: truncate, inex = +1 1f. sh=0 and cmp_low = -1 and bb-cc = -half [around -0.5 ulp] 1f1: bb > cc: truncate, inex = +1 1f2: bb = cc: cannot conclude 1f3: bb < cc: sub one ulp, inex = -1 2b. sh > 0 and cmp_low = 2 and bb=cc [around 0.5 ulp] 2b1. bb > cc: add one ulp, inex = 1 2b2: bb = cc: cannot conclude 2b3: bb < cc: truncate, inex = -1 3b. sh > 0 and cmp_low = 0 [around 0] 3b1. bb > cc: truncate, inex = -1 3b2: bb = cc: cannot conclude 3b3: bb < cc: truncate, inex = +1 */ } if ((rnd_mode == MPFR_RNDN) && cmp_low != 0) { /* even rounding rule */ if ((ap[0] >> sh) & 1) { if (cmp_low < 0) goto sub_one_ulp; else goto add_one_ulp; } else inexact = (cmp_low > 0) ? -1 : 1; } else inexact = 0; goto truncate; sub_one_ulp: /* sub one unit in last place to a */ mpn_sub_1 (ap, ap, an, MPFR_LIMB_ONE << sh); inexact = -1; goto end_of_sub; add_one_ulp: /* add one unit in last place to a */ if (MPFR_UNLIKELY(mpn_add_1 (ap, ap, an, MPFR_LIMB_ONE << sh))) /* result is a power of 2: 11111111111111 + 1 = 1000000000000000 */ { ap[an-1] = MPFR_LIMB_HIGHBIT; add_exp = 1; } inexact = 1; /* result larger than exact value */ truncate: if (MPFR_UNLIKELY((ap[an-1] >> (GMP_NUMB_BITS - 1)) == 0)) /* case 1 - epsilon */ { ap[an-1] = MPFR_LIMB_HIGHBIT; add_exp = 1; } end_of_sub: /* we have to set MPFR_EXP(a) to MPFR_EXP(b) - cancel + add_exp, taking care of underflows/overflows in that computation, and of the allowed exponent range */ MPFR_TMP_FREE (marker); if (MPFR_LIKELY(cancel)) { cancel -= add_exp; /* OK: add_exp is an int equal to 0 or 1 */ MPFR_ASSERTD (cancel >= 0); /* Detect an underflow case to avoid a possible integer overflow with UBF in the computation of exp_a. */ if (MPFR_UNLIKELY (exp_b < __gmpfr_emin - 1)) { if (rnd_mode == MPFR_RNDN) rnd_mode = MPFR_RNDZ; return mpfr_underflow (a, rnd_mode, MPFR_SIGN(a)); } exp_a = exp_b - cancel; /* The following assertion corresponds to a limitation of the MPFR implementation. It may fail with a 32-bit ABI and huge precisions, but this is practically impossible with a 64-bit ABI. This kind of issue is not specific to this function. */ MPFR_ASSERTN (exp_b != MPFR_EXP_MAX || exp_a > __gmpfr_emax); if (MPFR_UNLIKELY (exp_a < __gmpfr_emin)) { underflow: if (rnd_mode == MPFR_RNDN && (exp_a < __gmpfr_emin - 1 || (inexact >= 0 && mpfr_powerof2_raw (a)))) rnd_mode = MPFR_RNDZ; return mpfr_underflow (a, rnd_mode, MPFR_SIGN(a)); } /* We cannot have an overflow here, except for UBFs. Indeed: exp_a = exp_b - cancel + add_exp <= emax - 1 + 1 <= emax. For UBFs, we can have exp_b > emax. */ if (exp_a > __gmpfr_emax) { MPFR_ASSERTD (exp_b > __gmpfr_emax); /* since exp_b >= exp_a */ return mpfr_overflow (a, rnd_mode, MPFR_SIGN (a)); } } else /* cancel = 0: MPFR_EXP(a) <- MPFR_EXP(b) + add_exp */ { /* in case cancel = 0, add_exp can still be 1, in case b is just below a power of two, c is very small, prec(a) < prec(b), and rnd=away or nearest */ MPFR_ASSERTD (add_exp == 0 || add_exp == 1); /* Overflow iff exp_b + add_exp > __gmpfr_emax in Z, but we do a subtraction below to avoid a potential integer overflow in the case exp_b == MPFR_EXP_MAX. */ if (MPFR_UNLIKELY (exp_b > __gmpfr_emax - add_exp)) return mpfr_overflow (a, rnd_mode, MPFR_SIGN (a)); exp_a = exp_b + add_exp; /* Warning: an underflow can happen for UBFs, for example when mpfr_add is called from mpfr_fmma or mpfr_fmms. */ if (MPFR_UNLIKELY (exp_a < __gmpfr_emin)) goto underflow; MPFR_ASSERTD (exp_a >= __gmpfr_emin); } MPFR_SET_EXP (a, exp_a); /* check that result is msb-normalized */ MPFR_ASSERTD(ap[an-1] > ~ap[an-1]); MPFR_RET (inexact * MPFR_INT_SIGN (a)); }