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New implementation of mpf_manager::rem.

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Partially addresses #561
This commit is contained in:
Christoph M. Wintersteiger 2016-05-12 14:15:24 +01:00
parent ed1861d90d
commit dd83495d5d
2 changed files with 152 additions and 106 deletions
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255
src/util/mpf.cpp
View file

@ -1214,149 +1214,192 @@ void mpf_manager::to_ieee_bv_mpz(const mpf & x, scoped_mpz & o) {
}
}
void mpf_manager::prem(scoped_mpf & x, scoped_mpf const & y, mpf_exp_t const & exp_diff, bool partial) {
void mpf_manager::renormalize(unsigned ebits, unsigned sbits, mpf_exp_t & exp, mpz & sig) {
if (m_mpz_manager.is_zero(sig))
return;
mpf_exp_t max_e = mk_max_exp(ebits);
mpf_exp_t min_e = mk_min_exp(ebits);
const mpz & pg = m_powers2(sbits);
while (m_mpz_manager.ge(sig, pg)) {
m_mpz_manager.machine_div2k(sig, 1);
exp++;
}
const mpz & pl = m_powers2(sbits-1);
while (m_mpz_manager.lt(sig, pl)) {
m_mpz_manager.mul2k(sig, 1);
exp--;
}
}
void mpf_manager::partial_remainder(scoped_mpf & x, scoped_mpf const & y, mpf_exp_t const & exp_diff, bool partial) {
unsigned ebits = x.get().ebits;
unsigned sbits = x.get().sbits;
bool sign = x.get().sign;
signed int D = (unsigned)(partial ? 0 : exp_diff);
mpf_exp_t N = sbits/2;
SASSERT(-1 <= exp_diff && exp_diff < INT64_MAX);
SASSERT(exp_diff < ebits+sbits || partial);
signed int D = (signed int)(exp_diff);
mpf_exp_t N = (sbits+ebits)/2;
TRACE("mpf_dbg_rem", tout << "x=" << to_string(x) << std::endl;
tout << "y=" << to_string(y) << std::endl;
tout << "d=" << D << std::endl;
tout << "partial=" << partial << std::endl;);
// 1. Compute a/b
unsigned extra_bits = sbits + 2;
scoped_mpz x_sig_shifted(m_mpz_manager), x_div_y_sig_lrg(m_mpz_manager);
scoped_mpz x_div_y_sig(m_mpz_manager), x_rem_y_sig(m_mpz_manager);
m_mpz_manager.mul2k(x.significand(), sbits + extra_bits, x_sig_shifted);
m_mpz_manager.machine_div(x_sig_shifted, y.significand(), x_div_y_sig_lrg);
m_mpz_manager.machine_div_rem(x_div_y_sig_lrg, m_powers2(extra_bits-2), x_div_y_sig, x_rem_y_sig);
TRACE("mpf_dbg_rem", tout << "X/Y_sig=" << m_mpz_manager.to_string(x_div_y_sig) << std::endl;
tout << "X%Y_sig=" << m_mpz_manager.to_string(x_rem_y_sig) << std::endl;
tout << "X/Y=" << to_string_hexfloat(
to_packed_mpf(x.sign(), D, x_div_y_sig, ebits, sbits, 3)) << std::endl;);
SASSERT(m_mpz_manager.lt(x_div_y_sig, m_powers2(sbits + 3))); // 3 extra bits in x_div_y_sig.
SASSERT(m_mpz_manager.ge(x_div_y_sig, m_powers2(sbits - 1 + 3)));
// Bug: a_rem_b_sig is not used.
SASSERT(m_mpz_manager.lt(x.significand(), m_powers2(sbits)));
SASSERT(m_mpz_manager.ge(x.significand(), m_powers2(sbits - 1)));
SASSERT(m_mpz_manager.lt(y.significand(), m_powers2(sbits)));
SASSERT(m_mpz_manager.ge(y.significand(), m_powers2(sbits - 1)));
// 1. Compute a/b
bool x_div_y_sgn = x.sign() ^ y.sign();
mpf_exp_t x_div_y_exp = D;
scoped_mpz x_sig_shifted(m_mpz_manager), x_div_y_sig_lrg(m_mpz_manager), x_div_y_rem(m_mpz_manager);
scoped_mpz x_rem_y_sig(m_mpz_manager);
m_mpz_manager.mul2k(x.significand(), 2*sbits + 2, x_sig_shifted);
m_mpz_manager.machine_div_rem(x_sig_shifted, y.significand(), x_div_y_sig_lrg, x_div_y_rem); // rem useful?
// x/y is in x_diuv_y_sig_lrg and has sbits+3 extra bits.
TRACE("mpf_dbg_rem", tout << "X/Y_exp=" << x_div_y_exp << std::endl;
tout << "X/Y_sig_lrg=" << m_mpz_manager.to_string(x_div_y_sig_lrg) << std::endl;
tout << "X/Y_rem=" << m_mpz_manager.to_string(x_div_y_rem) << std::endl;
scoped_mpf & XdY = to_packed_mpf(x_div_y_sgn, x_div_y_exp, x_div_y_sig_lrg, ebits, sbits, sbits+3);
tout << "X/Y~=" << to_string_hexfloat(XdY) << std::endl;);
// 2. Round a/b to integer Q/QQ
mpf_exp_t Q_exp = partial ? N : D;
bool Q_sgn = x_div_y_sgn;
mpf_exp_t Q_exp = x_div_y_exp;
scoped_mpz Q_sig(m_mpz_manager), Q_rem(m_mpz_manager);
if (partial) {
unsigned Q_shft = (sbits-1) + (sbits+3) - (unsigned) (partial ? N :Q_exp);
if (partial) {
// Round according to MPF_ROUND_TOWARD_ZERO
SASSERT(N < D && D < INT_MAX);
unsigned D_N = (unsigned)(D-N);
unsigned Q_shft = (sbits - 1) - D_N;
m_mpz_manager.machine_div_rem(x_div_y_sig, m_powers2(Q_shft+3), Q_sig, Q_rem);
m_mpz_manager.mul2k(Q_sig, Q_shft);
SASSERT(0 < N && N < Q_exp && Q_exp < INT_MAX);
m_mpz_manager.machine_div2k(x_div_y_sig_lrg, Q_shft, Q_sig);
}
else {
// Round according to MPF_ROUND_NEAREST_TEVEN
unsigned Q_shft = (sbits - 1) - D;
m_mpz_manager.machine_div_rem(x_div_y_sig, m_powers2(Q_shft+3), Q_sig, Q_rem);
m_mpz_manager.machine_div_rem(x_div_y_sig_lrg, m_powers2(Q_shft), Q_sig, Q_rem);
const mpz & shiftm1_p = m_powers2(Q_shft-1);
bool tie = m_mpz_manager.eq(Q_rem, shiftm1_p);
bool more_than_tie = m_mpz_manager.gt(Q_rem, shiftm1_p);
TRACE("mpf_dbg_rem", tout << "tie= " << tie << "; >tie= " << more_than_tie << std::endl;);
if ((tie && m_mpz_manager.is_odd(Q_sig)) || more_than_tie)
m_mpz_manager.inc(Q_sig);
m_mpz_manager.mul2k(Q_sig, Q_shft);
SASSERT(m_mpz_manager.ge(Q_sig, m_powers2(sbits-1)));
}
TRACE("mpf_dbg_rem", tout << "Q_sig'=" << m_mpz_manager.to_string(Q_sig) << std::endl;
tout << "Q_rem=" << m_mpz_manager.to_string(Q_rem) << std::endl; );
// re-normalize Q
while (m_mpz_manager.ge(Q_sig, m_powers2(sbits))) {
m_mpz_manager.machine_div2k(Q_sig, 1);
Q_exp++;
}
m_mpz_manager.mul2k(Q_sig, Q_shft);
m_mpz_manager.machine_div2k(Q_sig, sbits+3);
renormalize(ebits, sbits, Q_exp, Q_sig);
TRACE("mpf_dbg_rem", tout << "Q_exp=" << Q_exp << std::endl;
tout << "Q_sig=" << m_mpz_manager.to_string(Q_sig) << std::endl;
scoped_mpf & Q = to_packed_mpf(x.sign(), Q_exp, Q_sig, ebits, sbits, 0);
scoped_mpf & Q = to_packed_mpf(Q_sgn, Q_exp, Q_sig, ebits, sbits, 0);
tout << "Q=" << to_string_hexfloat(Q) << std::endl;);
if ((D == -1 || partial) && m_mpz_manager.is_zero(Q_sig))
return; // This means x % y = x.
// no extra bits in Q_sig.
SASSERT(!m_mpz_manager.is_zero(Q_sig));
SASSERT(m_mpz_manager.lt(Q_sig, m_powers2(sbits)));
SASSERT(m_mpz_manager.ge(Q_sig, m_powers2(sbits - 1)));
// 3. Compute Y*Q / Y*QQ*2^{D-N}
bool YQ_sgn = y.sign() ^ Q_sgn;
scoped_mpz YQ_sig(m_mpz_manager);
mpf_exp_t YQ_exp = (Q_exp + y.exponent()) + (partial ? D-N : 0);
m_mpz_manager.mul(y.significand(), Q_sig, YQ_sig);
mpf_exp_t YQ_exp = Q_exp + y.exponent();
m_mpz_manager.mul(y.significand(), Q_sig, YQ_sig);
renormalize(ebits, 2*sbits-1, YQ_exp, YQ_sig); // YQ_sig has `sbits-1' extra bits.
if (sbits >= 7) {
scoped_mpz sticky_rem(m_mpz_manager);
m_mpz_manager.machine_div_rem(YQ_sig, m_powers2(sbits-7), YQ_sig, sticky_rem);
if (!m_mpz_manager.is_zero(sticky_rem) && m_mpz_manager.is_even(YQ_sig))
m_mpz_manager.inc(YQ_sig);
}
else
m_mpz_manager.mul2k(YQ_sig, 7-sbits, YQ_sig);
TRACE("mpf_dbg_rem", tout << "YQ_exp=" << YQ_exp << std::endl;
TRACE("mpf_dbg_rem", tout << "YQ_sgn=" << YQ_sgn << std::endl;
tout << "YQ_exp=" << YQ_exp << std::endl;
tout << "YQ_sig=" << m_mpz_manager.to_string(YQ_sig) << std::endl;
scoped_mpf & YQ = to_packed_mpf(x.sign(), YQ_exp, YQ_sig, ebits, sbits, 6);
scoped_mpf & YQ = to_packed_mpf(YQ_sgn, YQ_exp, YQ_sig, ebits, sbits, sbits-1);
tout << "YQ=" << to_string_hexfloat(YQ) << std::endl;);
// 4. Compute X-Y*Q
bool YQ_sgn = !x.sign();
mpf_exp_t X_YQ_exp = x.exponent() - 3;
// `sbits-1' extra bits in YQ_sig.
SASSERT(m_mpz_manager.lt(YQ_sig, m_powers2(2*sbits-1)));
SASSERT(m_mpz_manager.ge(YQ_sig, m_powers2(2*sbits-2)) || YQ_exp <= mk_bot_exp(ebits));
// 4. Compute X-Y*Q
mpf_exp_t X_YQ_exp = x.exponent();
scoped_mpz X_YQ_sig(m_mpz_manager);
mpf_exp_t exp_delta = exp(x) - YQ_exp;
TRACE("mpf_dbg_rem", tout << "exp_delta=" << exp_delta << std::endl;);
SASSERT(INT_MIN < exp_delta && exp_delta <= INT_MAX);
scoped_mpz minuend(m_mpz_manager), subtrahend(m_mpz_manager);
if (exp_delta >= 0) {
m_mpz_manager.set(minuend, x.significand());
m_mpz_manager.set(subtrahend, YQ_sig);
}
else if (exp_delta < 0) {
m_mpz_manager.set(minuend, YQ_sig);
m_mpz_manager.set(subtrahend, x.significand());
YQ_sgn = !YQ_sgn;
exp_delta = -exp_delta;
}
scoped_mpz x_sig_lrg(m_mpz_manager);
m_mpz_manager.mul2k(x.significand(), sbits-1, x_sig_lrg); // sbits-1 extra bits into x
m_mpz_manager.set(minuend, x_sig_lrg);
m_mpz_manager.set(subtrahend, YQ_sig);
SASSERT(m_mpz_manager.lt(minuend, m_powers2(2*sbits-1)));
SASSERT(m_mpz_manager.ge(minuend, m_powers2(2*sbits-2)));
SASSERT(m_mpz_manager.lt(subtrahend, m_powers2(2*sbits-1)));
SASSERT(m_mpz_manager.ge(subtrahend, m_powers2(2*sbits-2)));
if (exp_delta != 0) {
scoped_mpz sticky_rem(m_mpz_manager);
if (exp_delta > sbits+5) {
m_mpz_manager.set(sticky_rem, 0);
sticky_rem.swap(subtrahend);
m_mpz_manager.set(sticky_rem, 0);
if (exp_delta > sbits+5)
sticky_rem.swap(subtrahend);
else if (exp_delta > 0)
m_mpz_manager.machine_div_rem(subtrahend, m_powers2((unsigned)exp_delta), subtrahend, sticky_rem);
else {
SASSERT(exp_delta < 0);
exp_delta = -exp_delta;
m_mpz_manager.mul2k(subtrahend, (int)exp_delta);
}
else
m_mpz_manager.machine_div_rem(subtrahend, m_powers2((unsigned)exp_delta), subtrahend, sticky_rem);
if (!m_mpz_manager.is_zero(sticky_rem) && m_mpz_manager.is_even(subtrahend))
m_mpz_manager.inc(subtrahend);
TRACE("mpf_dbg_rem", tout << "aligned subtrahend=" << m_mpz_manager.to_string(subtrahend) << std::endl;);
}
m_mpz_manager.sub(minuend, subtrahend, X_YQ_sig);
TRACE("mpf_dbg_rem", tout << "X_YQ_sig'=" << m_mpz_manager.to_string(X_YQ_sig) << std::endl;);
scoped_mpz minuend_lrg(m_mpz_manager);
m_mpz_manager.mul2k(minuend, 6, minuend_lrg); // + 6 extra bits into X
m_mpz_manager.sub(minuend_lrg, subtrahend, X_YQ_sig);
TRACE("mpf_dbg_rem", tout << "minuend=" << m_mpz_manager.to_string(minuend) << std::endl;
tout << "subtrahend=" << m_mpz_manager.to_string(subtrahend) << std::endl;
tout << "X_YQ_exp=" << X_YQ_exp << std::endl;
tout << "X_YQ_sig=" << m_mpz_manager.to_string(X_YQ_sig) << std::endl;
scoped_mpf & X_YQ = to_packed_mpf(YQ_sgn, X_YQ_exp, X_YQ_sig, ebits, sbits, 3);
tout << "X-YQ=" << to_string_hexfloat(X_YQ) << std::endl;);
SASSERT(m_mpz_manager.lt(X_YQ_sig, m_powers2(sbits+5)));
bool neg = m_mpz_manager.is_neg(X_YQ_sig);
if (neg) m_mpz_manager.neg(X_YQ_sig);
bool X_YQ_sgn = ((!x.sign() && !YQ_sgn && neg) ||
(x.sign() && YQ_sgn && !neg) ||
(x.sign() && !YQ_sgn));
// 5. Rounding
if (m_mpz_manager.is_zero(X_YQ_sig)) {
TRACE("mpf_dbg_rem", tout << "sig zero" << std::endl;);
if (m_mpz_manager.is_zero(X_YQ_sig))
mk_zero(ebits, sbits, x.sign(), x);
}
else {
bool neg = m_mpz_manager.is_neg(X_YQ_sig);
if (neg) m_mpz_manager.neg(X_YQ_sig);
bool X_YQ_sgn = ((!x.sign() && YQ_sgn && neg) ||
(x.sign() && !YQ_sgn && !neg) ||
(x.sign() && YQ_sgn));
renormalize(ebits, 2*sbits-1, X_YQ_exp, X_YQ_sig);
TRACE("mpf_dbg_rem", tout << "minuend=" << m_mpz_manager.to_string(minuend) << std::endl;
tout << "subtrahend=" << m_mpz_manager.to_string(subtrahend) << std::endl;
tout << "X_YQ_sgn=" << X_YQ_sgn << std::endl;
tout << "X_YQ_exp=" << X_YQ_exp << std::endl;
tout << "X_YQ_sig=" << m_mpz_manager.to_string(X_YQ_sig) << std::endl;
scoped_mpf & X_YQ = to_packed_mpf(X_YQ_sgn, X_YQ_exp, X_YQ_sig, ebits, sbits, sbits-1);
tout << "X-YQ=" << to_string_hexfloat(X_YQ) << std::endl;);
// `sbits-1' extra bits in X_YQ_sig
SASSERT(m_mpz_manager.lt(X_YQ_sig, m_powers2(2*sbits-1)));
scoped_mpz rnd_bits(m_mpz_manager);
m_mpz_manager.machine_div_rem(X_YQ_sig, m_powers2(sbits-1), X_YQ_sig, rnd_bits);
TRACE("mpf_dbg_rem", tout << "final sticky=" << m_mpz_manager.to_string(rnd_bits) << std::endl; );
// Round to nearest, ties to even.
if (m_mpz_manager.eq(rnd_bits, mpz(32))) { // tie.
if (m_mpz_manager.is_odd(X_YQ_sig)) {
m_mpz_manager.inc(X_YQ_sig);
}
}
else if (m_mpz_manager.gt(rnd_bits, mpz(32)))
m_mpz_manager.inc(X_YQ_sig);
set(x, ebits, sbits, X_YQ_sgn, X_YQ_exp, X_YQ_sig);
round(MPF_ROUND_NEAREST_TEVEN, x);
}
TRACE("mpf_dbg_rem", tout << "partial remainder = " << to_string_hexfloat(x) << std::endl;);
@ -1365,8 +1408,8 @@ void mpf_manager::prem(scoped_mpf & x, scoped_mpf const & y, mpf_exp_t const & e
void mpf_manager::rem(mpf const & x, mpf const & y, mpf & o) {
SASSERT(x.sbits == y.sbits && x.ebits == y.ebits);
TRACE("mpf_dbg", tout << "X = " << to_string(x) << std::endl;);
TRACE("mpf_dbg", tout << "Y = " << to_string(y) << std::endl;);
TRACE("mpf_dbg_rem", tout << "X = " << to_string(x) << "=" << to_string_hexfloat(x) << std::endl;
tout << "Y = " << to_string(y) << "=" << to_string_hexfloat(y) << std::endl;);
if (is_nan(x) || is_nan(y))
mk_nan(x.ebits, x.sbits, o);
@ -1380,7 +1423,6 @@ void mpf_manager::rem(mpf const & x, mpf const & y, mpf & o) {
set(o, x);
else {
SASSERT(is_regular(x) && is_regular(y));
SASSERT(x.exponent >= y.exponent);
// This is a generalized version of the algorithm for FPREM1 in the `Intel
// 64 and IA-32 Architectures Software Developer's Manual',
@ -1391,24 +1433,24 @@ void mpf_manager::rem(mpf const & x, mpf const & y, mpf & o) {
unpack(ST0, true);
unpack(ST1, true);
const mpf_exp_t B = x.sbits; // max bits per iteration.
const mpf_exp_t B = x.sbits+x.ebits;
mpf_exp_t D;
do {
D = ST0.exponent() - ST1.exponent();
SASSERT(0 <= D && D < INT64_MAX);
TRACE("mpf_dbg_rem", tout << "ST0=" << to_string_raw(ST0) << "=" << to_string_hexfloat(ST0) << std::endl;
tout << "ST1=" << to_string_raw(ST1) << "=" << to_string_hexfloat(ST1) << std::endl;
tout << "D=" << D << std::endl;);
prem(ST0, ST1, D, (D >= B));
} while (D >= B);
if (ST0.exponent() < (ST1.exponent()) - 1) {
D = 0;
}
else {
D = ST0.exponent() - ST1.exponent();
partial_remainder(ST0, ST1, D, (D >= B));
}
} while (D >= B && !ST0.is_zero());
m_mpz_manager.mul2k(ST0.significand(), 3);
set(o, x.ebits, x.sbits, MPF_ROUND_TOWARD_ZERO, ST0);
if (is_zero(o))
o.sign = x.sign;
round(MPF_ROUND_NEAREST_TEVEN, o);
}
TRACE("mpf_dbg_rem", tout << "R = " << to_string(o) << "=" << to_string_hexfloat(o) << std::endl; );
TRACE("mpf_dbg", tout << "REMAINDER = " << to_string(o) << std::endl;);
}
@ -1538,7 +1580,10 @@ scoped_mpf mpf_manager::to_packed_mpf(bool sgn, mpf_exp_t exp, scoped_mpz const
scoped_mpz q_sig(m_mpz_manager);
m_mpz_manager.set(q_sig, sig);
if (rbits != 0) m_mpz_manager.div(q_sig, m_powers2(rbits), q_sig); // restore scale
m_mpz_manager.sub(q_sig, m_powers2(sbits-1), q_sig); // strip hidden bit
if (m_mpz_manager.ge(q_sig, m_powers2(sbits-1)))
m_mpz_manager.sub(q_sig, m_powers2(sbits-1), q_sig); // strip hidden bit
else if (exp == mk_min_exp(ebits))
exp = mk_bot_exp(ebits);
set(q, ebits, sbits, sgn, exp, q_sig);
return q;
}

3
src/util/mpf.h
View file

@ -223,7 +223,8 @@ protected:
void round(mpf_rounding_mode rm, mpf & o);
void round_sqrt(mpf_rounding_mode rm, mpf & o);
void prem(scoped_mpf & x, scoped_mpf const & y, mpf_exp_t const & exp_diff, bool partial);
void renormalize(unsigned ebits, unsigned sbits, mpf_exp_t & exp, mpz & sig);
void partial_remainder(scoped_mpf & x, scoped_mpf const & y, mpf_exp_t const & exp_diff, bool partial);
void mk_round_inf(mpf_rounding_mode rm, mpf & o);