| /* mpfr_sub1 -- internal function to perform a "real" subtraction |
| |
| Copyright 2001, 2002, 2003, 2004, 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. |
| |
| 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" |
| |
| /* 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, mp_rnd_t rnd_mode) |
| { |
| int sign; |
| mp_exp_unsigned_t diff_exp; |
| mp_prec_t cancel, cancel1; |
| mp_size_t cancel2, an, bn, cn, cn0; |
| mp_limb_t *ap, *bp, *cp; |
| mp_limb_t carry, bb, cc, borrow = 0; |
| int inexact, shift_b, shift_c, is_exact = 1, down = 0, add_exp = 0; |
| int sh, k; |
| MPFR_TMP_DECL(marker); |
| |
| MPFR_TMP_MARK(marker); |
| ap = MPFR_MANT(a); |
| an = MPFR_LIMB_SIZE(a); |
| |
| sign = mpfr_cmp2 (b, c, &cancel); |
| if (MPFR_UNLIKELY(sign == 0)) |
| { |
| if (rnd_mode == GMP_RNDD) |
| MPFR_SET_NEG (a); |
| else |
| MPFR_SET_POS (a); |
| MPFR_SET_ZERO (a); |
| MPFR_RET (0); |
| } |
| |
| /* |
| * If subtraction: sign(a) = sign * sign(b) |
| * If addition: sign(a) = sign of the larger argument in absolute value. |
| * |
| * Both cases can be simplidied 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); |
| |
| /* Check if c is too small. |
| A more precise test is to replace 2 by |
| (rnd == GMP_RNDN) + mpfr_power2_raw (b) |
| but it is more expensive and not very useful */ |
| if (MPFR_UNLIKELY (MPFR_GET_EXP (c) <= MPFR_GET_EXP (b) |
| - (mp_exp_t) MAX (MPFR_PREC (a), MPFR_PREC (b)) - 2)) |
| { |
| /* Remember, we can't have an exact result! */ |
| /* A.AAAAAAAAAAAAAAAAA |
| = B.BBBBBBBBBBBBBBB |
| - C.CCCCCCCCCCCCC */ |
| /* A = S*ABS(B) +/- ulp(a) */ |
| MPFR_SET_EXP (a, MPFR_GET_EXP (b)); |
| MPFR_RNDRAW_EVEN (inexact, a, MPFR_MANT (b), MPFR_PREC (b), |
| rnd_mode, MPFR_SIGN (a), |
| if (MPFR_UNLIKELY ( ++MPFR_EXP (a) > __gmpfr_emax)) |
| inexact = mpfr_overflow (a, rnd_mode, MPFR_SIGN (a))); |
| /* inexact = mpfr_set4 (a, b, rnd_mode, MPFR_SIGN (a)); */ |
| if (inexact == 0) |
| { |
| /* a = b (Exact) |
| But we know it isn't (Since we have to remove `c') |
| So if we round to Zero, we have to remove one ulp. |
| Otherwise the result is correctly rounded. */ |
| if (MPFR_IS_LIKE_RNDZ (rnd_mode, MPFR_IS_NEG (a))) |
| { |
| mpfr_nexttozero (a); |
| MPFR_RET (- MPFR_INT_SIGN (a)); |
| } |
| MPFR_RET (MPFR_INT_SIGN (a)); |
| } |
| else |
| { |
| /* 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 zeros. |
| Which means that removing c from b can't generate a carry |
| execpt in case of even rounding. |
| In all other case the result and the inexact flag should be |
| correct (We can't have an exact result). |
| 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 */ |
| if (MPFR_UNLIKELY (inexact == MPFR_EVEN_INEX*MPFR_INT_SIGN (a))) |
| { |
| mpfr_nexttozero (a); |
| inexact = -MPFR_INT_SIGN (a); |
| } |
| MPFR_RET (inexact); |
| } |
| } |
| |
| diff_exp = (mp_exp_unsigned_t) MPFR_GET_EXP (b) - MPFR_GET_EXP (c); |
| |
| /* reserve a space to store b aligned with the result, i.e. shifted by |
| (-cancel) % BITS_PER_MP_LIMB to the right */ |
| bn = MPFR_LIMB_SIZE (b); |
| MPFR_UNSIGNED_MINUS_MODULO (shift_b, cancel); |
| cancel1 = (cancel + shift_b) / BITS_PER_MP_LIMB; |
| |
| /* 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 = (mp_ptr) MPFR_TMP_ALLOC(bn * BYTES_PER_MP_LIMB); |
| MPN_COPY (bp, ap, bn); |
| } |
| } |
| else |
| { |
| bp = (mp_ptr) MPFR_TMP_ALLOC ((bn + 1) * BYTES_PER_MP_LIMB); |
| 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) % BITS_PER_MP_LIMB to the right */ |
| cn = MPFR_LIMB_SIZE(c); |
| if ((UINT_MAX % BITS_PER_MP_LIMB) == (BITS_PER_MP_LIMB-1) |
| && ((-(unsigned) 1)%BITS_PER_MP_LIMB > 0)) |
| shift_c = (diff_exp - cancel) % BITS_PER_MP_LIMB; |
| else |
| { |
| shift_c = diff_exp - (cancel % BITS_PER_MP_LIMB); |
| shift_c = (shift_c + BITS_PER_MP_LIMB) % BITS_PER_MP_LIMB; |
| } |
| MPFR_ASSERTD( shift_c >= 0 && shift_c < BITS_PER_MP_LIMB); |
| |
| if (MPFR_UNLIKELY(shift_c == 0)) |
| { |
| cp = MPFR_MANT(c); |
| /* Ensure ap != cp */ |
| if (ap == cp) |
| { |
| cp = (mp_ptr) MPFR_TMP_ALLOC (cn * BYTES_PER_MP_LIMB); |
| MPN_COPY(cp, ap, cn); |
| } |
| } |
| else |
| { |
| cp = (mp_ptr) MPFR_TMP_ALLOC ((cn + 1) * BYTES_PER_MP_LIMB); |
| cp[0] = mpn_rshift (cp + 1, MPFR_MANT(c), cn++, shift_c); |
| } |
| |
| #ifdef DEBUG |
| printf ("shift_b=%d shift_c=%d diffexp=%lu\n", shift_b, shift_c, |
| (unsigned long) diff_exp); |
| #endif |
| |
| MPFR_ASSERTD (ap != cp); |
| MPFR_ASSERTD (bp != cp); |
| |
| /* here we have shift_c = (diff_exp - cancel) % BITS_PER_MP_LIMB, |
| 0 <= shift_c < BITS_PER_MP_LIMB |
| thus we want cancel2 = ceil((cancel - diff_exp) / BITS_PER_MP_LIMB) */ |
| |
| cancel2 = (long int) (cancel - (diff_exp - shift_c)) / BITS_PER_MP_LIMB; |
| /* the high cancel2 limbs from b should not be taken into account */ |
| #ifdef DEBUG |
| printf ("cancel=%lu cancel1=%lu cancel2=%ld\n", |
| (unsigned long) cancel, (unsigned long) cancel1, (long) 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); |
| |
| #ifdef DEBUG |
| printf("after copying high(b), a="); mpfr_print_binary(a); putchar('\n'); |
| #endif |
| |
| /* 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 */ |
| { |
| 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); |
| } |
| } |
| |
| #ifdef DEBUG |
| printf("after subtracting high(c), a="); |
| mpfr_print_binary(a); |
| putchar('\n'); |
| #endif |
| |
| /* now perform rounding */ |
| sh = (mp_prec_t) an * BITS_PER_MP_LIMB - MPFR_PREC(a); |
| /* last unused bits from a */ |
| carry = ap[0] & MPFR_LIMB_MASK (sh); |
| ap[0] -= carry; |
| |
| if (MPFR_LIKELY(rnd_mode == GMP_RNDN)) |
| { |
| if (MPFR_LIKELY(sh)) |
| { |
| is_exact = (carry == 0); |
| /* can decide except when carry = 2^(sh-1) [middle] |
| or carry = 0 [truncate, but cannot decide inexact flag] */ |
| down = (carry < (MPFR_LIMB_ONE << (sh - 1))); |
| if (carry > (MPFR_LIMB_ONE << (sh - 1))) |
| goto add_one_ulp; |
| else if ((0 < carry) && down) |
| { |
| inexact = -1; /* result if smaller than exact value */ |
| goto truncate; |
| } |
| } |
| } |
| else /* directed rounding: set rnd_mode to RNDZ iff towards zero */ |
| { |
| if (MPFR_IS_RNDUTEST_OR_RNDDNOTTEST(rnd_mode, MPFR_IS_NEG(a))) |
| rnd_mode = GMP_RNDZ; |
| |
| if (carry) |
| { |
| if (rnd_mode == GMP_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 -= (long int) an + cancel2; |
| |
| #ifdef DEBUG |
| printf ("last %d bits from a are %lu, bn=%ld, cn=%ld\n", |
| sh, (unsigned long) carry, (long) bn, (long) cn); |
| #endif |
| |
| for (k = 0; (bn > 0) || (cn > 0); k = 1) |
| { |
| /* get next limbs */ |
| bb = (bn > 0) ? bp[--bn] : 0; |
| if ((cn > 0) && (cn-- <= cn0)) |
| cc = cp[cn]; |
| else |
| cc = 0; |
| |
| /* down is set when low(b) < low(c) */ |
| if (down == 0) |
| down = (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 == GMP_RNDN) && sh == 0 && k == 0) |
| { |
| mp_limb_t half = MPFR_LIMB_HIGHBIT; |
| |
| is_exact = (bb == cc); |
| |
| /* add one ulp if bb > cc + half |
| truncate if cc - half < bb < cc + half |
| sub one ulp if bb < cc - half |
| */ |
| |
| if (down) |
| { |
| if (cc >= half) |
| cc -= half; |
| else |
| bb += half; |
| } |
| else /* bb >= cc */ |
| { |
| if (cc < half) |
| cc += half; |
| else |
| bb -= half; |
| } |
| } |
| |
| #ifdef DEBUG |
| printf (" bb=%lu cc=%lu down=%d is_exact=%d\n", |
| (unsigned long) bb, (unsigned long) cc, down, is_exact); |
| #endif |
| if (bb < cc) |
| { |
| if (rnd_mode == GMP_RNDZ) |
| goto sub_one_ulp; |
| else if (rnd_mode != GMP_RNDN) /* round away */ |
| { |
| inexact = 1; |
| goto truncate; |
| } |
| else /* round to nearest: special case here since for sh=k=0 |
| bb = bb0 - MPFR_LIMB_HIGHBIT */ |
| { |
| if (is_exact && sh == 0) |
| { |
| /* For k=0 we can't decide exactness since it may depend |
| from low order bits. |
| For k=1, the first low limbs matched: low(b)-low(c)<0. */ |
| if (k) |
| { |
| inexact = 1; |
| goto truncate; |
| } |
| } |
| else if (down && sh == 0) |
| goto sub_one_ulp; |
| else |
| { |
| inexact = (is_exact) ? 1 : -1; |
| goto truncate; |
| } |
| } |
| } |
| else if (bb > cc) |
| { |
| if (rnd_mode == GMP_RNDZ) |
| { |
| inexact = -1; |
| goto truncate; |
| } |
| else if (rnd_mode != GMP_RNDN) /* round away */ |
| goto add_one_ulp; |
| else /* round to nearest */ |
| { |
| if (is_exact) |
| { |
| inexact = -1; |
| goto truncate; |
| } |
| else if (down) |
| { |
| inexact = 1; |
| goto truncate; |
| } |
| else |
| goto add_one_ulp; |
| } |
| } |
| } |
| |
| if ((rnd_mode == GMP_RNDN) && !is_exact) |
| { |
| /* even rounding rule */ |
| if ((ap[0] >> sh) & 1) |
| { |
| if (down) |
| goto sub_one_ulp; |
| else |
| goto add_one_ulp; |
| } |
| else |
| inexact = (down) ? 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] >> (BITS_PER_MP_LIMB - 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 */ |
| if (MPFR_LIKELY(cancel)) |
| { |
| mp_exp_t exp_a; |
| |
| cancel -= add_exp; /* still valid as unsigned long */ |
| exp_a = MPFR_GET_EXP (b) - cancel; |
| if (MPFR_UNLIKELY(exp_a < __gmpfr_emin)) |
| { |
| MPFR_TMP_FREE(marker); |
| if (rnd_mode == GMP_RNDN && |
| (exp_a < __gmpfr_emin - 1 || |
| (inexact >= 0 && mpfr_powerof2_raw (a)))) |
| rnd_mode = GMP_RNDZ; |
| return mpfr_underflow (a, rnd_mode, MPFR_SIGN(a)); |
| } |
| MPFR_SET_EXP (a, exp_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 */ |
| mp_exp_t exp_b; |
| |
| exp_b = MPFR_GET_EXP (b); |
| if (MPFR_UNLIKELY(add_exp && exp_b == __gmpfr_emax)) |
| { |
| MPFR_TMP_FREE(marker); |
| return mpfr_overflow (a, rnd_mode, MPFR_SIGN(a)); |
| } |
| MPFR_SET_EXP (a, exp_b + add_exp); |
| } |
| MPFR_TMP_FREE(marker); |
| #ifdef DEBUG |
| printf ("result is a="); mpfr_print_binary(a); putchar('\n'); |
| #endif |
| /* check that result is msb-normalized */ |
| MPFR_ASSERTD(ap[an-1] > ~ap[an-1]); |
| MPFR_RET (inexact * MPFR_INT_SIGN (a)); |
| } |
