/* mpfr_sub1sp -- internal function to perform a "real" substraction All the op must have the same precision Copyright 2003-2015 Free Software Foundation, Inc. Contributed by the AriC and Caramel 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 http://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" /* Check if we have to check the result of mpfr_sub1sp with mpfr_sub1 */ #ifdef MPFR_WANT_ASSERT # if MPFR_WANT_ASSERT >= 2 int mpfr_sub1sp2 (mpfr_ptr a, mpfr_srcptr b, mpfr_srcptr c, mpfr_rnd_t rnd_mode); int mpfr_sub1sp (mpfr_ptr a, mpfr_srcptr b, mpfr_srcptr c, mpfr_rnd_t rnd_mode) { mpfr_t tmpa, tmpb, tmpc; int inexb, inexc, inexact, inexact2; mpfr_init2 (tmpa, MPFR_PREC (a)); mpfr_init2 (tmpb, MPFR_PREC (b)); mpfr_init2 (tmpc, MPFR_PREC (c)); inexb = mpfr_set (tmpb, b, MPFR_RNDN); MPFR_ASSERTN (inexb == 0); inexc = mpfr_set (tmpc, c, MPFR_RNDN); MPFR_ASSERTN (inexc == 0); inexact2 = mpfr_sub1 (tmpa, tmpb, tmpc, rnd_mode); inexact = mpfr_sub1sp2(a, b, c, rnd_mode); if (mpfr_cmp (tmpa, a) || inexact != inexact2) { fprintf (stderr, "sub1 & sub1sp return different values for %s\n" "Prec_a = %lu, Prec_b = %lu, Prec_c = %lu\nB = ", mpfr_print_rnd_mode (rnd_mode), (unsigned long) MPFR_PREC (a), (unsigned long) MPFR_PREC (b), (unsigned long) MPFR_PREC (c)); mpfr_fprint_binary (stderr, tmpb); fprintf (stderr, "\nC = "); mpfr_fprint_binary (stderr, tmpc); fprintf (stderr, "\nSub1 : "); mpfr_fprint_binary (stderr, tmpa); fprintf (stderr, "\nSub1sp: "); mpfr_fprint_binary (stderr, a); fprintf (stderr, "\nInexact sp = %d | Inexact = %d\n", inexact, inexact2); MPFR_ASSERTN (0); } mpfr_clears (tmpa, tmpb, tmpc, (mpfr_ptr) 0); return inexact; } # define mpfr_sub1sp mpfr_sub1sp2 # endif #endif /* Debugging support */ #ifdef DEBUG # undef DEBUG # define DEBUG(x) (x) #else # define DEBUG(x) /**/ #endif /* Rounding Sub */ /* compute sgn(b)*(|b| - |c|) if |b|>|c| else -sgn(b)*(|c| -|b|) Returns 0 iff result is exact, a negative value when the result is less than the exact value, a positive value otherwise. */ /* A0...Ap-1 * Cp Cp+1 .... * <- C'p+1 -> * Cp = -1 if calculated from c mantissa * Cp = 0 if 0 from a or c * Cp = 1 if calculated from a. * C'p+1 = First bit not null or 0 if there isn't one * * Can't have Cp=-1 and C'p+1=1*/ /* RND = MPFR_RNDZ: * + if Cp=0 and C'p+1=0,1, Truncate. * + if Cp=0 and C'p+1=-1, SubOneUlp * + if Cp=-1, SubOneUlp * + if Cp=1, AddOneUlp * RND = MPFR_RNDA (Away) * + if Cp=0 and C'p+1=0,-1, Truncate * + if Cp=0 and C'p+1=1, AddOneUlp * + if Cp=1, AddOneUlp * + if Cp=-1, Truncate * RND = MPFR_RNDN * + if Cp=0, Truncate * + if Cp=1 and C'p+1=1, AddOneUlp * + if Cp=1 and C'p+1=-1, Truncate * + if Cp=1 and C'p+1=0, Truncate if Ap-1=0, AddOneUlp else * + if Cp=-1 and C'p+1=-1, SubOneUlp * + if Cp=-1 and C'p+1=0, Truncate if Ap-1=0, SubOneUlp else * * If AddOneUlp: * If carry, then it is 11111111111 + 1 = 10000000000000 * ap[n-1]=MPFR_HIGHT_BIT * If SubOneUlp: * If we lose one bit, it is 1000000000 - 1 = 0111111111111 * Then shift, and put as last bit x which is calculated * according Cp, Cp-1 and rnd_mode. * If Truncate, * If it is a power of 2, * we may have to suboneulp in some special cases. * * To simplify, we don't use Cp = 1. * */ int mpfr_sub1sp (mpfr_ptr a, mpfr_srcptr b, mpfr_srcptr c, mpfr_rnd_t rnd_mode) { mpfr_exp_t bx,cx; mpfr_uexp_t d; mpfr_prec_t p, sh, cnt; mp_size_t n; mp_limb_t *ap, *bp, *cp; mp_limb_t limb; int inexact; mp_limb_t bcp,bcp1; /* Cp and C'p+1 */ mp_limb_t bbcp = (mp_limb_t) -1, bbcp1 = (mp_limb_t) -1; /* Cp+1 and C'p+2, gcc claims that they might be used uninitialized. We fill them with invalid values, which should produce a failure if so. See README.dev file. */ MPFR_TMP_DECL(marker); MPFR_TMP_MARK(marker); MPFR_ASSERTD(MPFR_PREC(a) == MPFR_PREC(b) && MPFR_PREC(b) == MPFR_PREC(c)); MPFR_ASSERTD(MPFR_IS_PURE_FP(b)); MPFR_ASSERTD(MPFR_IS_PURE_FP(c)); /* Read prec and num of limbs */ p = MPFR_PREC (b); n = MPFR_PREC2LIMBS (p); /* Fast cmp of |b| and |c|*/ bx = MPFR_GET_EXP (b); cx = MPFR_GET_EXP (c); if (MPFR_UNLIKELY(bx == cx)) { mp_size_t k = n - 1; /* Check mantissa since exponent are equals */ bp = MPFR_MANT(b); cp = MPFR_MANT(c); while (k>=0 && MPFR_UNLIKELY(bp[k] == cp[k])) k--; if (MPFR_UNLIKELY(k < 0)) /* b == c ! */ { /* Return exact number 0 */ if (rnd_mode == MPFR_RNDD) MPFR_SET_NEG(a); else MPFR_SET_POS(a); MPFR_SET_ZERO(a); MPFR_RET(0); } else if (bp[k] > cp[k]) goto BGreater; else { MPFR_ASSERTD(bp[k] c */ BGreater: MPFR_SET_SAME_SIGN(a,b); } /* Now b > c */ MPFR_ASSERTD(bx >= cx); d = (mpfr_uexp_t) bx - cx; DEBUG (printf ("New with diff=%lu\n", (unsigned long) d)); if (MPFR_UNLIKELY(d <= 1)) { if (MPFR_LIKELY(d < 1)) { /* <-- b --> <-- c --> : exact sub */ ap = MPFR_MANT(a); mpn_sub_n (ap, MPFR_MANT(b), MPFR_MANT(c), n); /* Normalize */ ExactNormalize: limb = ap[n-1]; if (MPFR_LIKELY(limb)) { /* First limb is not zero. */ count_leading_zeros(cnt, limb); /* cnt could be == 0 <= SubD1Lose */ if (MPFR_LIKELY(cnt)) { mpn_lshift(ap, ap, n, cnt); /* Normalize number */ bx -= cnt; /* Update final expo */ } /* Last limb should be ok */ MPFR_ASSERTD(!(ap[0] & MPFR_LIMB_MASK((unsigned int) (-p) % GMP_NUMB_BITS))); } else { /* First limb is zero */ mp_size_t k = n-1, len; /* Find the first limb not equal to zero. FIXME:It is assume it exists (since |b| > |c| and same prec)*/ do { MPFR_ASSERTD( k > 0 ); limb = ap[--k]; } while (limb == 0); MPFR_ASSERTD(limb != 0); count_leading_zeros(cnt, limb); k++; len = n - k; /* Number of last limb */ MPFR_ASSERTD(k >= 0); if (MPFR_LIKELY(cnt)) mpn_lshift(ap+len, ap, k, cnt); /* Normalize the High Limb*/ else { /* Must use DECR since src and dest may overlap & dest>=src*/ MPN_COPY_DECR(ap+len, ap, k); } MPN_ZERO(ap, len); /* Zeroing the last limbs */ bx -= cnt + len*GMP_NUMB_BITS; /* Update Expo */ /* Last limb should be ok */ MPFR_ASSERTD(!(ap[len]&MPFR_LIMB_MASK((unsigned int) (-p) % GMP_NUMB_BITS))); } /* Check expo underflow */ if (MPFR_UNLIKELY(bx < __gmpfr_emin)) { MPFR_TMP_FREE(marker); /* inexact=0 */ DEBUG( printf("(D==0 Underflow)\n") ); if (rnd_mode == MPFR_RNDN && (bx < __gmpfr_emin - 1 || (/*inexact >= 0 &&*/ mpfr_powerof2_raw (a)))) rnd_mode = MPFR_RNDZ; return mpfr_underflow (a, rnd_mode, MPFR_SIGN(a)); } MPFR_SET_EXP (a, bx); /* No rounding is necessary since the result is exact */ MPFR_ASSERTD(ap[n-1] > ~ap[n-1]); MPFR_TMP_FREE(marker); return 0; } else /* if (d == 1) */ { /* | <-- b --> | <-- c --> */ mp_limb_t c0, mask; mp_size_t k; MPFR_UNSIGNED_MINUS_MODULO(sh, p); /* If we lose at least one bit, compute 2*b-c (Exact) * else compute b-c/2 */ bp = MPFR_MANT(b); cp = MPFR_MANT(c); k = n-1; limb = bp[k] - cp[k]/2; if (limb > MPFR_LIMB_HIGHBIT) { /* We can't lose precision: compute b-c/2 */ /* Shift c in the allocated temporary block */ SubD1NoLose: c0 = cp[0] & (MPFR_LIMB_ONE< ~ap[n-1]); /* No rounding is necessary since the result is exact */ MPFR_TMP_FREE(marker); return 0; } ap = MPFR_MANT(a); mask = ~MPFR_LIMB_MASK(sh); cp[0] &= mask; /* Delete last bit of c */ mpn_sub_n (ap, bp, cp, n); MPFR_SET_EXP(a, bx); /* No expo overflow! */ MPFR_ASSERTD( !(ap[0] & ~mask) ); /* Check last bits */ /* No normalize is needed */ MPFR_ASSERTD(ap[n-1] > ~ap[n-1]); /* Rounding is necessary since c0 = 1*/ /* Cp =-1 and C'p+1=0 */ bcp = 1; bcp1 = 0; if (MPFR_LIKELY(rnd_mode == MPFR_RNDN)) { /* Even Rule apply: Check Ap-1 */ if (MPFR_LIKELY( (ap[0] & (MPFR_LIMB_ONE< We lose at least one bit*/ /* If b[k]>c'[k] => We don't lose any bit */ /* If k==-1 => We don't lose any bit AND the result is 100000000000 0000000000 00000000000 */ mp_limb_t carry; do { carry = cp[k]&MPFR_LIMB_ONE; k--; } while (k>=0 && bp[k]==(carry=cp[k]/2+(carry<<(GMP_NUMB_BITS-1)))); if (MPFR_UNLIKELY(k<0)) { /*If carry then (sh==0 and Virtual c'[-1] > Virtual b[-1]) */ if (MPFR_UNLIKELY(carry)) /* carry = cp[0]&MPFR_LIMB_ONE */ { /* FIXME: Can be faster? */ MPFR_ASSERTD(sh == 0); goto SubD1Lose; } /* Result is a power of 2 */ ap = MPFR_MANT (a); MPN_ZERO (ap, n); ap[n-1] = MPFR_LIMB_HIGHBIT; MPFR_SET_EXP (a, bx); /* No expo overflow! */ /* No Normalize is needed*/ /* No Rounding is needed */ MPFR_TMP_FREE (marker); return 0; } /* carry = cp[k]/2+(cp[k-1]&1)<<(GMP_NUMB_BITS-1) = c'[k]*/ else if (bp[k] > carry) goto SubD1NoLose; else { MPFR_ASSERTD(bp[k]= p)) { ap = MPFR_MANT(a); MPFR_UNSIGNED_MINUS_MODULO(sh, p); /* We can't set A before since we use cp for rounding... */ /* Perform rounding: check if a=b or a=b-ulp(b) */ if (MPFR_UNLIKELY(d == p)) { /* cp == -1 and c'p+1 = ? */ bcp = 1; /* We need Cp+1 later for a very improbable case. */ bbcp = (MPFR_MANT(c)[n-1] & (MPFR_LIMB_ONE<<(GMP_NUMB_BITS-2))); /* We need also C'p+1 for an even more unprobable case... */ if (MPFR_LIKELY( bbcp )) bcp1 = 1; else { cp = MPFR_MANT(c); if (MPFR_UNLIKELY(cp[n-1] == MPFR_LIMB_HIGHBIT)) { mp_size_t k = n-1; do { k--; } while (k>=0 && cp[k]==0); bcp1 = (k>=0); } else bcp1 = 1; } DEBUG( printf("(D=P) Cp=-1 Cp+1=%d C'p+1=%d \n", bbcp!=0, bcp1!=0) ); bp = MPFR_MANT (b); /* Even if src and dest overlap, it is ok using MPN_COPY */ if (MPFR_LIKELY(rnd_mode == MPFR_RNDN)) { if (MPFR_UNLIKELY( bcp && bcp1==0 )) /* Cp=-1 and C'p+1=0: Even rule Apply! */ /* Check Ap-1 = Bp-1 */ if ((bp[0] & (MPFR_LIMB_ONE<P) Cp=%d Cp+1=%d C'p+1=%d\n", bcp!=0,bbcp!=0,bcp1!=0) ); /* Need to compute C'p+2 if d==p+1 and if rnd_mode=NEAREST (Because of a very improbable case) */ if (MPFR_UNLIKELY(d==p+1 && rnd_mode==MPFR_RNDN)) { cp = MPFR_MANT(c); if (MPFR_UNLIKELY(cp[n-1] == MPFR_LIMB_HIGHBIT)) { mp_size_t k = n-1; do { k--; } while (k>=0 && cp[k]==0); bbcp1 = (k>=0); } else bbcp1 = 1; DEBUG( printf("(D>P) C'p+2=%d\n", bbcp1!=0) ); } /* Copy mantissa B in A */ MPN_COPY(ap, MPFR_MANT(b), n); /* Round */ if (MPFR_LIKELY(rnd_mode == MPFR_RNDN)) goto truncate; MPFR_UPDATE_RND_MODE(rnd_mode, MPFR_IS_NEG(a)); if (rnd_mode == MPFR_RNDZ) goto sub_one_ulp; else /* rnd_mode = AWAY */ goto truncate; } } else { mpfr_uexp_t dm; mp_size_t m; mp_limb_t mask; /* General case: 2 <= d < p */ MPFR_UNSIGNED_MINUS_MODULO(sh, p); cp = MPFR_TMP_LIMBS_ALLOC (n); /* Shift c in temporary allocated place */ dm = d % GMP_NUMB_BITS; m = d / GMP_NUMB_BITS; if (MPFR_UNLIKELY(dm == 0)) { /* dm = 0 and m > 0: Just copy */ MPFR_ASSERTD(m!=0); MPN_COPY(cp, MPFR_MANT(c)+m, n-m); MPN_ZERO(cp+n-m, m); } else if (MPFR_LIKELY(m == 0)) { /* dm >=2 and m == 0: just shift */ MPFR_ASSERTD(dm >= 2); mpn_rshift(cp, MPFR_MANT(c), n, dm); } else { /* dm > 0 and m > 0: shift and zero */ mpn_rshift(cp, MPFR_MANT(c)+m, n-m, dm); MPN_ZERO(cp+n-m, m); } DEBUG( mpfr_print_mant_binary("Before", MPFR_MANT(c), p) ); DEBUG( mpfr_print_mant_binary("B= ", MPFR_MANT(b), p) ); DEBUG( mpfr_print_mant_binary("After ", cp, p) ); /* Compute bcp=Cp and bcp1=C'p+1 */ if (MPFR_LIKELY(sh)) { /* Try to compute them from C' rather than C (FIXME: Faster?) */ bcp = (cp[0] & (MPFR_LIMB_ONE<<(sh-1))) ; if (MPFR_LIKELY( cp[0] & MPFR_LIMB_MASK(sh-1) )) bcp1 = 1; else { /* We can't compute C'p+1 from C'. Compute it from C */ /* Start from bit x=p-d+sh in mantissa C (+sh since we have already looked sh bits in C'!) */ mpfr_prec_t x = p-d+sh-1; if (MPFR_LIKELY(x>p)) /* We are already looked at all the bits of c, so C'p+1 = 0*/ bcp1 = 0; else { mp_limb_t *tp = MPFR_MANT(c); mp_size_t kx = n-1 - (x / GMP_NUMB_BITS); mpfr_prec_t sx = GMP_NUMB_BITS-1-(x%GMP_NUMB_BITS); DEBUG (printf ("(First) x=%lu Kx=%ld Sx=%lu\n", (unsigned long) x, (long) kx, (unsigned long) sx)); /* Looks at the last bits of limb kx (if sx=0 does nothing)*/ if (tp[kx] & MPFR_LIMB_MASK(sx)) bcp1 = 1; else { /*kx += (sx==0);*/ /*If sx==0, tp[kx] hasn't been checked*/ do { kx--; } while (kx>=0 && tp[kx]==0); bcp1 = (kx >= 0); } } } } else { /* Compute Cp and C'p+1 from C with sh=0 */ mp_limb_t *tp = MPFR_MANT(c); /* Start from bit x=p-d in mantissa C */ mpfr_prec_t x = p-d; mp_size_t kx = n-1 - (x / GMP_NUMB_BITS); mpfr_prec_t sx = GMP_NUMB_BITS-1-(x%GMP_NUMB_BITS); MPFR_ASSERTD(p >= d); bcp = (tp[kx] & (MPFR_LIMB_ONE<=0 && tp[kx]==0); bcp1 = (kx>=0); } } DEBUG( printf("sh=%lu Cp=%d C'p+1=%d\n", sh, bcp!=0, bcp1!=0) ); /* Check if we can lose a bit, and if so compute Cp+1 and C'p+2 */ bp = MPFR_MANT(b); if (MPFR_UNLIKELY((bp[n-1]-cp[n-1]) <= MPFR_LIMB_HIGHBIT)) { /* We can lose a bit so we precompute Cp+1 and C'p+2 */ /* Test for trivial case: since C'p+1=0, Cp+1=0 and C'p+2 =0 */ if (MPFR_LIKELY(bcp1 == 0)) { bbcp = 0; bbcp1 = 0; } else /* bcp1 != 0 */ { /* We can lose a bit: compute Cp+1 and C'p+2 from mantissa C */ mp_limb_t *tp = MPFR_MANT(c); /* Start from bit x=(p+1)-d in mantissa C */ mpfr_prec_t x = p+1-d; mp_size_t kx = n-1 - (x/GMP_NUMB_BITS); mpfr_prec_t sx = GMP_NUMB_BITS-1-(x%GMP_NUMB_BITS); MPFR_ASSERTD(p > d); DEBUG (printf ("(pre) x=%lu Kx=%ld Sx=%lu\n", (unsigned long) x, (long) kx, (unsigned long) sx)); bbcp = (tp[kx] & (MPFR_LIMB_ONE<=0 && tp[kx]==0); bbcp1 = (kx>=0); DEBUG (printf ("(Pre) Scan done for %ld\n", (long) kx)); } } /*End of Bcp1 != 0*/ DEBUG( printf("(Pre) Cp+1=%d C'p+2=%d\n", bbcp!=0, bbcp1!=0) ); } /* End of "can lose a bit" */ /* Clean shifted C' */ mask = ~MPFR_LIMB_MASK (sh); cp[0] &= mask; /* Subtract the mantissa c from b in a */ ap = MPFR_MANT(a); mpn_sub_n (ap, bp, cp, n); DEBUG( mpfr_print_mant_binary("Sub= ", ap, p) ); /* Normalize: we lose at max one bit*/ if (MPFR_UNLIKELY(MPFR_LIMB_MSB(ap[n-1]) == 0)) { /* High bit is not set and we have to fix it! */ /* Ap >= 010000xxx001 */ mpn_lshift(ap, ap, n, 1); /* Ap >= 100000xxx010 */ if (MPFR_UNLIKELY(bcp!=0)) /* Check if Cp = -1 */ /* Since Cp == -1, we have to substract one more */ { mpn_sub_1(ap, ap, n, MPFR_LIMB_ONE<= 10000xxx001 */ /* Final exponent -1 since we have shifted the mantissa */ bx--; /* Update bcp and bcp1 */ MPFR_ASSERTN(bbcp != (mp_limb_t) -1); MPFR_ASSERTN(bbcp1 != (mp_limb_t) -1); bcp = bbcp; bcp1 = bbcp1; /* We dont't have anymore a valid Cp+1! But since Ap >= 100000xxx001, the final sub can't unnormalize!*/ } MPFR_ASSERTD( !(ap[0] & ~mask) ); /* Rounding */ if (MPFR_LIKELY(rnd_mode == MPFR_RNDN)) { if (MPFR_LIKELY(bcp==0)) goto truncate; else if ((bcp1) || ((ap[0] & (MPFR_LIMB_ONE<=0 && ap[k]==0); if (MPFR_UNLIKELY(k<0)) { /* It is a power of 2! */ /* Compute Cp+1 if it isn't already compute (ie d==1) */ /* FIXME: Is this case possible? */ if (d == 1) bbcp=0; DEBUG( printf("(Truncate) Cp=%d, Cp+1=%d C'p+1=%d C'p+2=%d\n", \ bcp!=0, bbcp!=0, bcp1!=0, bbcp1!=0) ); MPFR_ASSERTN(bbcp != (mp_limb_t) -1); MPFR_ASSERTN((rnd_mode != MPFR_RNDN) || (bcp != 0) || (bbcp == 0) || (bbcp1 != (mp_limb_t) -1)); if (((rnd_mode != MPFR_RNDZ) && bcp) || ((rnd_mode == MPFR_RNDN) && (bcp == 0) && (bbcp) && (bbcp1))) { DEBUG( printf("(Truncate) Do sub\n") ); mpn_sub_1 (ap, ap, n, MPFR_LIMB_ONE << sh); mpn_lshift(ap, ap, n, 1); ap[0] |= MPFR_LIMB_ONE< MPFR_EXP(c) > emin if d == 1 : bx=MPFR_EXP(b). If we could lose any bits, the exact normalisation is called. if d >= p : bx=MPFR_EXP(b) >= MPFR_EXP(c) + p > emin After SubOneUlp, we could have one bit less. if 1 < d < p : bx >= MPFR_EXP(b)-2 >= MPFR_EXP(c) > emin if d == 1 : bx >= MPFR_EXP(b)-1 = MPFR_EXP(c) > emin. if d >= p : bx >= MPFR_EXP(b)-1 > emin since p>=2. */ MPFR_ASSERTD( bx >= __gmpfr_emin); /* if (MPFR_UNLIKELY(bx < __gmpfr_emin)) { DEBUG( printf("(Final Underflow)\n") ); if (rnd_mode == MPFR_RNDN && (bx < __gmpfr_emin - 1 || (inexact >= 0 && mpfr_powerof2_raw (a)))) rnd_mode = MPFR_RNDZ; MPFR_TMP_FREE(marker); return mpfr_underflow (a, rnd_mode, MPFR_SIGN(a)); } */ MPFR_SET_EXP (a, bx); MPFR_TMP_FREE(marker); MPFR_RET (inexact * MPFR_INT_SIGN (a)); }