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big: Move division internals.
This commit is contained in:
Jeroen van Rijn
@@ -290,351 +290,6 @@ int_choose_digit :: proc(res: ^Int, n, k: int) -> (err: Error) {
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}
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choose :: proc { int_choose_digit, };
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/*
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Low level squaring, b = a*a, HAC pp.596-597, Algorithm 14.16
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*/
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_int_sqr :: proc(dest, src: ^Int) -> (err: Error) {
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pa := src.used;
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t := &Int{}; ix, iy: int;
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/*
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Grow `t` to maximum needed size, or `_DEFAULT_DIGIT_COUNT`, whichever is bigger.
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*/
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if err = grow(t, max((2 * pa) + 1, _DEFAULT_DIGIT_COUNT)); err != nil { return err; }
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t.used = (2 * pa) + 1;
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#no_bounds_check for ix = 0; ix < pa; ix += 1 {
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carry := DIGIT(0);
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/*
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First calculate the digit at 2*ix; calculate double precision result.
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*/
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r := _WORD(t.digit[ix+ix]) + (_WORD(src.digit[ix]) * _WORD(src.digit[ix]));
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/*
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Store lower part in result.
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*/
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t.digit[ix+ix] = DIGIT(r & _WORD(_MASK));
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/*
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Get the carry.
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*/
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carry = DIGIT(r >> _DIGIT_BITS);
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#no_bounds_check for iy = ix + 1; iy < pa; iy += 1 {
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/*
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First calculate the product.
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*/
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r = _WORD(src.digit[ix]) * _WORD(src.digit[iy]);
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/* Now calculate the double precision result. Nóte we use
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* addition instead of *2 since it's easier to optimize
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*/
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r = _WORD(t.digit[ix+iy]) + r + r + _WORD(carry);
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/*
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Store lower part.
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*/
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t.digit[ix+iy] = DIGIT(r & _WORD(_MASK));
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/*
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Get carry.
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*/
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carry = DIGIT(r >> _DIGIT_BITS);
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}
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/*
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Propagate upwards.
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*/
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#no_bounds_check for carry != 0 {
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r = _WORD(t.digit[ix+iy]) + _WORD(carry);
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t.digit[ix+iy] = DIGIT(r & _WORD(_MASK));
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carry = DIGIT(r >> _WORD(_DIGIT_BITS));
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iy += 1;
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}
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}
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err = clamp(t);
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swap(dest, t);
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destroy(t);
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return err;
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}
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/*
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Divide by three (based on routine from MPI and the GMP manual).
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*/
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_int_div_3 :: proc(quotient, numerator: ^Int) -> (remainder: DIGIT, err: Error) {
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/*
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b = 2**MP_DIGIT_BIT / 3
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*/
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b := _WORD(1) << _WORD(_DIGIT_BITS) / _WORD(3);
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q := &Int{};
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if err = grow(q, numerator.used); err != nil { return 0, err; }
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q.used = numerator.used;
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q.sign = numerator.sign;
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w, t: _WORD;
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for ix := numerator.used; ix >= 0; ix -= 1 {
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w = (w << _WORD(_DIGIT_BITS)) | _WORD(numerator.digit[ix]);
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if w >= 3 {
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/*
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Multiply w by [1/3].
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*/
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t = (w * b) >> _WORD(_DIGIT_BITS);
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/*
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Now subtract 3 * [w/3] from w, to get the remainder.
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*/
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w -= t+t+t;
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/*
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Fixup the remainder as required since the optimization is not exact.
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*/
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for w >= 3 {
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t += 1;
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w -= 3;
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}
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} else {
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t = 0;
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}
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q.digit[ix] = DIGIT(t);
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}
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remainder = DIGIT(w);
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/*
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[optional] store the quotient.
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*/
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if quotient != nil {
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err = clamp(q);
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swap(q, quotient);
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}
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destroy(q);
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return remainder, nil;
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}
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/*
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Signed Integer Division
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c*b + d == a [i.e. a/b, c=quotient, d=remainder], HAC pp.598 Algorithm 14.20
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Note that the description in HAC is horribly incomplete.
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For example, it doesn't consider the case where digits are removed from 'x' in
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the inner loop.
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It also doesn't consider the case that y has fewer than three digits, etc.
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The overall algorithm is as described as 14.20 from HAC but fixed to treat these cases.
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*/
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_int_div_school :: proc(quotient, remainder, numerator, denominator: ^Int) -> (err: Error) {
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if err = error_if_immutable(quotient, remainder); err != nil { return err; }
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if err = clear_if_uninitialized(quotient, numerator, denominator); err != nil { return err; }
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q, x, y, t1, t2 := &Int{}, &Int{}, &Int{}, &Int{}, &Int{};
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defer destroy(q, x, y, t1, t2);
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if err = grow(q, numerator.used + 2); err != nil { return err; }
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q.used = numerator.used + 2;
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if err = init_multi(t1, t2); err != nil { return err; }
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if err = copy(x, numerator); err != nil { return err; }
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if err = copy(y, denominator); err != nil { return err; }
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/*
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Fix the sign.
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*/
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neg := numerator.sign != denominator.sign;
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x.sign = .Zero_or_Positive;
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y.sign = .Zero_or_Positive;
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/*
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Normalize both x and y, ensure that y >= b/2, [b == 2**MP_DIGIT_BIT]
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*/
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norm, _ := count_bits(y);
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norm %= _DIGIT_BITS;
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if norm < _DIGIT_BITS - 1 {
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norm = (_DIGIT_BITS - 1) - norm;
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if err = shl(x, x, norm); err != nil { return err; }
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if err = shl(y, y, norm); err != nil { return err; }
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} else {
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norm = 0;
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}
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/*
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Note: HAC does 0 based, so if used==5 then it's 0,1,2,3,4, i.e. use 4
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*/
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n := x.used - 1;
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t := y.used - 1;
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/*
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while (x >= y*b**n-t) do { q[n-t] += 1; x -= y*b**{n-t} }
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y = y*b**{n-t}
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*/
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if err = shl_digit(y, n - t); err != nil { return err; }
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c, _ := cmp(x, y);
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for c != -1 {
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q.digit[n - t] += 1;
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if err = sub(x, x, y); err != nil { return err; }
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c, _ = cmp(x, y);
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}
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/*
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Reset y by shifting it back down.
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*/
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shr_digit(y, n - t);
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/*
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Step 3. for i from n down to (t + 1).
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*/
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for i := n; i >= (t + 1); i -= 1 {
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if (i > x.used) { continue; }
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/*
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step 3.1 if xi == yt then set q{i-t-1} to b-1, otherwise set q{i-t-1} to (xi*b + x{i-1})/yt
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*/
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if x.digit[i] == y.digit[t] {
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q.digit[(i - t) - 1] = 1 << (_DIGIT_BITS - 1);
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} else {
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tmp := _WORD(x.digit[i]) << _DIGIT_BITS;
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tmp |= _WORD(x.digit[i - 1]);
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tmp /= _WORD(y.digit[t]);
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if tmp > _WORD(_MASK) {
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tmp = _WORD(_MASK);
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}
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q.digit[(i - t) - 1] = DIGIT(tmp & _WORD(_MASK));
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}
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/* while (q{i-t-1} * (yt * b + y{t-1})) >
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xi * b**2 + xi-1 * b + xi-2
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do q{i-t-1} -= 1;
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*/
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iter := 0;
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q.digit[(i - t) - 1] = (q.digit[(i - t) - 1] + 1) & _MASK;
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for {
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q.digit[(i - t) - 1] = (q.digit[(i - t) - 1] - 1) & _MASK;
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/*
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Find left hand.
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*/
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zero(t1);
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t1.digit[0] = ((t - 1) < 0) ? 0 : y.digit[t - 1];
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t1.digit[1] = y.digit[t];
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t1.used = 2;
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if err = mul(t1, t1, q.digit[(i - t) - 1]); err != nil { return err; }
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/*
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Find right hand.
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*/
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t2.digit[0] = ((i - 2) < 0) ? 0 : x.digit[i - 2];
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t2.digit[1] = x.digit[i - 1]; /* i >= 1 always holds */
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t2.digit[2] = x.digit[i];
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t2.used = 3;
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if t1_t2, _ := cmp_mag(t1, t2); t1_t2 != 1 {
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break;
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}
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iter += 1; if iter > 100 { return .Max_Iterations_Reached; }
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}
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/*
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Step 3.3 x = x - q{i-t-1} * y * b**{i-t-1}
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*/
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if err = int_mul_digit(t1, y, q.digit[(i - t) - 1]); err != nil { return err; }
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if err = shl_digit(t1, (i - t) - 1); err != nil { return err; }
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if err = sub(x, x, t1); err != nil { return err; }
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/*
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if x < 0 then { x = x + y*b**{i-t-1}; q{i-t-1} -= 1; }
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*/
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if x.sign == .Negative {
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if err = copy(t1, y); err != nil { return err; }
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if err = shl_digit(t1, (i - t) - 1); err != nil { return err; }
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if err = add(x, x, t1); err != nil { return err; }
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q.digit[(i - t) - 1] = (q.digit[(i - t) - 1] - 1) & _MASK;
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}
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}
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/*
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Now q is the quotient and x is the remainder, [which we have to normalize]
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Get sign before writing to c.
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*/
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z, _ := is_zero(x);
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x.sign = .Zero_or_Positive if z else numerator.sign;
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if quotient != nil {
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clamp(q);
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swap(q, quotient);
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quotient.sign = .Negative if neg else .Zero_or_Positive;
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}
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if remainder != nil {
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if err = shr(x, x, norm); err != nil { return err; }
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swap(x, remainder);
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}
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return nil;
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}
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/*
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Slower bit-bang division... also smaller.
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*/
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@(deprecated="Use `_int_div_school`, it's 3.5x faster.")
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_int_div_small :: proc(quotient, remainder, numerator, denominator: ^Int) -> (err: Error) {
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ta, tb, tq, q := &Int{}, &Int{}, &Int{}, &Int{};
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c: int;
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goto_end: for {
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if err = one(tq); err != nil { break goto_end; }
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num_bits, _ := count_bits(numerator);
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den_bits, _ := count_bits(denominator);
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n := num_bits - den_bits;
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if err = abs(ta, numerator); err != nil { break goto_end; }
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if err = abs(tb, denominator); err != nil { break goto_end; }
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if err = shl(tb, tb, n); err != nil { break goto_end; }
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if err = shl(tq, tq, n); err != nil { break goto_end; }
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for n >= 0 {
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if c, _ = cmp_mag(ta, tb); c == 0 || c == 1 {
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// ta -= tb
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if err = sub(ta, ta, tb); err != nil { break goto_end; }
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// q += tq
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if err = add( q, q, tq); err != nil { break goto_end; }
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}
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if err = shr1(tb, tb); err != nil { break goto_end; }
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if err = shr1(tq, tq); err != nil { break goto_end; }
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n -= 1;
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}
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/*
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Now q == quotient and ta == remainder.
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*/
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neg := numerator.sign != denominator.sign;
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if quotient != nil {
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swap(quotient, q);
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z, _ := is_zero(quotient);
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quotient.sign = .Negative if neg && !z else .Zero_or_Positive;
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}
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if remainder != nil {
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swap(remainder, ta);
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z, _ := is_zero(numerator);
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remainder.sign = .Zero_or_Positive if z else numerator.sign;
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}
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break goto_end;
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}
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destroy(ta, tb, tq, q);
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return err;
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}
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/*
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Function computing both GCD and (if target isn't `nil`) also LCM.
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*/
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