mirror of
https://github.com/python/cpython.git
synced 2024-11-23 18:04:37 +08:00
Merged revisions 70542 via svnmerge from
svn+ssh://pythondev@svn.python.org/python/trunk ........ r70542 | mark.dickinson | 2009-03-23 18:25:13 +0000 (Mon, 23 Mar 2009) | 14 lines Issue #5512: speed up the long division algorithm for Python longs. The basic algorithm remains the same; the most significant speedups come from the following three changes: (1) normalize by shifting instead of multiplying and dividing (2) the old algorithm usually did an unnecessary extra iteration of the outer loop; remove this. As a special case, this means that long divisions with a single-digit result run twice as fast as before. (3) make inner loop much tighter. Various benchmarks show speedups of between 50% and 150% for long integer divisions and modulo operations. ........
This commit is contained in:
Mark Dickinson
parent
798ee1a4c6
commit
17e4fddb57
@ -12,6 +12,10 @@ What's New in Python 3.1 alpha 2?
|
||||
Core and Builtins
|
||||
-----------------
|
||||
|
||||
- Issue #5512: Rewrite PyLong long division algorithm (x_divrem) to
|
||||
improve its performance. Long divisions and remainder operations
|
||||
are now between 50% and 150% faster.
|
||||
|
||||
- Issue #4258: Make it possible to use base 2**30 instead of base
|
||||
2**15 for the internal representation of integers, for performance
|
||||
reasons. Base 2**30 is enabled by default on 64-bit machines. Add
|
||||
|
||||
@ -1363,6 +1363,26 @@ PyLong_AsUnsignedLongLongMask(register PyObject *op)
|
||||
return Py_NotImplemented; \
|
||||
}
|
||||
|
||||
/* bits_in_digit(d) returns the unique integer k such that 2**(k-1) <= d <
|
||||
2**k if d is nonzero, else 0. */
|
||||
|
||||
static const unsigned char BitLengthTable[32] = {
|
||||
0, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4,
|
||||
5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5
|
||||
};
|
||||
|
||||
static int
|
||||
bits_in_digit(digit d)
|
||||
{
|
||||
int d_bits = 0;
|
||||
while (d >= 32) {
|
||||
d_bits += 6;
|
||||
d >>= 6;
|
||||
}
|
||||
d_bits += (int)BitLengthTable[d];
|
||||
return d_bits;
|
||||
}
|
||||
|
||||
/* x[0:m] and y[0:n] are digit vectors, LSD first, m >= n required. x[0:n]
|
||||
* is modified in place, by adding y to it. Carries are propagated as far as
|
||||
* x[m-1], and the remaining carry (0 or 1) is returned.
|
||||
@ -1415,25 +1435,41 @@ v_isub(digit *x, Py_ssize_t m, digit *y, Py_ssize_t n)
|
||||
return borrow;
|
||||
}
|
||||
|
||||
/* Multiply by a single digit, ignoring the sign. */
|
||||
|
||||
static PyLongObject *
|
||||
mul1(PyLongObject *a, digit n)
|
||||
/* Shift digit vector a[0:m] d bits left, with 0 <= d < PyLong_SHIFT. Put
|
||||
* result in z[0:m], and return the d bits shifted out of the top.
|
||||
*/
|
||||
static digit
|
||||
v_lshift(digit *z, digit *a, Py_ssize_t m, int d)
|
||||
{
|
||||
Py_ssize_t size_a = ABS(Py_SIZE(a));
|
||||
PyLongObject *z = _PyLong_New(size_a+1);
|
||||
twodigits carry = 0;
|
||||
Py_ssize_t i;
|
||||
digit carry = 0;
|
||||
|
||||
if (z == NULL)
|
||||
return NULL;
|
||||
for (i = 0; i < size_a; ++i) {
|
||||
carry += (twodigits)a->ob_digit[i] * n;
|
||||
z->ob_digit[i] = (digit) (carry & PyLong_MASK);
|
||||
carry >>= PyLong_SHIFT;
|
||||
assert(0 <= d && d < PyLong_SHIFT);
|
||||
for (i=0; i < m; i++) {
|
||||
twodigits acc = (twodigits)a[i] << d | carry;
|
||||
z[i] = (digit)acc & PyLong_MASK;
|
||||
carry = (digit)(acc >> PyLong_SHIFT);
|
||||
}
|
||||
z->ob_digit[i] = (digit) carry;
|
||||
return long_normalize(z);
|
||||
return carry;
|
||||
}
|
||||
|
||||
/* Shift digit vector a[0:m] d bits right, with 0 <= d < PyLong_SHIFT. Put
|
||||
* result in z[0:m], and return the d bits shifted out of the bottom.
|
||||
*/
|
||||
static digit
|
||||
v_rshift(digit *z, digit *a, Py_ssize_t m, int d)
|
||||
{
|
||||
Py_ssize_t i;
|
||||
digit carry = 0;
|
||||
digit mask = ((digit)1 << d) - 1U;
|
||||
|
||||
assert(0 <= d && d < PyLong_SHIFT);
|
||||
for (i=m; i-- > 0;) {
|
||||
twodigits acc = (twodigits)carry << PyLong_SHIFT | a[i];
|
||||
carry = (digit)acc & mask;
|
||||
z[i] = (digit)(acc >> d);
|
||||
}
|
||||
return carry;
|
||||
}
|
||||
|
||||
/* Divide long pin, w/ size digits, by non-zero digit n, storing quotient
|
||||
@ -2089,104 +2125,131 @@ long_divrem(PyLongObject *a, PyLongObject *b,
|
||||
return 0;
|
||||
}
|
||||
|
||||
/* Unsigned long division with remainder -- the algorithm */
|
||||
/* Unsigned long division with remainder -- the algorithm. The arguments v1
|
||||
and w1 should satisfy 2 <= ABS(Py_SIZE(w1)) <= ABS(Py_SIZE(v1)). */
|
||||
|
||||
static PyLongObject *
|
||||
x_divrem(PyLongObject *v1, PyLongObject *w1, PyLongObject **prem)
|
||||
{
|
||||
Py_ssize_t size_v = ABS(Py_SIZE(v1)), size_w = ABS(Py_SIZE(w1));
|
||||
digit d = (digit) ((twodigits)PyLong_BASE / (w1->ob_digit[size_w-1] + 1));
|
||||
PyLongObject *v = mul1(v1, d);
|
||||
PyLongObject *w = mul1(w1, d);
|
||||
PyLongObject *a;
|
||||
Py_ssize_t j, k;
|
||||
PyLongObject *v, *w, *a;
|
||||
Py_ssize_t i, k, size_v, size_w;
|
||||
int d;
|
||||
digit wm1, wm2, carry, q, r, vtop, *v0, *vk, *w0, *ak;
|
||||
twodigits vv;
|
||||
sdigit zhi;
|
||||
stwodigits z;
|
||||
|
||||
if (v == NULL || w == NULL) {
|
||||
Py_XDECREF(v);
|
||||
Py_XDECREF(w);
|
||||
/* We follow Knuth [The Art of Computer Programming, Vol. 2 (3rd
|
||||
edn.), section 4.3.1, Algorithm D], except that we don't explicitly
|
||||
handle the special case when the initial estimate q for a quotient
|
||||
digit is >= PyLong_BASE: the max value for q is PyLong_BASE+1, and
|
||||
that won't overflow a digit. */
|
||||
|
||||
/* allocate space; w will also be used to hold the final remainder */
|
||||
size_v = ABS(Py_SIZE(v1));
|
||||
size_w = ABS(Py_SIZE(w1));
|
||||
assert(size_v >= size_w && size_w >= 2); /* Assert checks by div() */
|
||||
v = _PyLong_New(size_v+1);
|
||||
if (v == NULL) {
|
||||
*prem = NULL;
|
||||
return NULL;
|
||||
}
|
||||
w = _PyLong_New(size_w);
|
||||
if (w == NULL) {
|
||||
Py_DECREF(v);
|
||||
*prem = NULL;
|
||||
return NULL;
|
||||
}
|
||||
|
||||
assert(size_v >= size_w && size_w > 1); /* Assert checks by div() */
|
||||
assert(Py_REFCNT(v) == 1); /* Since v will be used as accumulator! */
|
||||
assert(size_w == ABS(Py_SIZE(w))); /* That's how d was calculated */
|
||||
/* normalize: shift w1 left so that its top digit is >= PyLong_BASE/2.
|
||||
shift v1 left by the same amount. Results go into w and v. */
|
||||
d = PyLong_SHIFT - bits_in_digit(w1->ob_digit[size_w-1]);
|
||||
carry = v_lshift(w->ob_digit, w1->ob_digit, size_w, d);
|
||||
assert(carry == 0);
|
||||
carry = v_lshift(v->ob_digit, v1->ob_digit, size_v, d);
|
||||
if (carry != 0 || v->ob_digit[size_v-1] >= w->ob_digit[size_w-1]) {
|
||||
v->ob_digit[size_v] = carry;
|
||||
size_v++;
|
||||
}
|
||||
|
||||
size_v = ABS(Py_SIZE(v));
|
||||
/* Now v->ob_digit[size_v-1] < w->ob_digit[size_w-1], so quotient has
|
||||
at most (and usually exactly) k = size_v - size_w digits. */
|
||||
k = size_v - size_w;
|
||||
a = _PyLong_New(k + 1);
|
||||
|
||||
for (j = size_v; a != NULL && k >= 0; --j, --k) {
|
||||
digit vj = (j >= size_v) ? 0 : v->ob_digit[j];
|
||||
twodigits q;
|
||||
stwodigits carry = 0;
|
||||
Py_ssize_t i;
|
||||
assert(k >= 0);
|
||||
a = _PyLong_New(k);
|
||||
if (a == NULL) {
|
||||
Py_DECREF(w);
|
||||
Py_DECREF(v);
|
||||
*prem = NULL;
|
||||
return NULL;
|
||||
}
|
||||
v0 = v->ob_digit;
|
||||
w0 = w->ob_digit;
|
||||
wm1 = w0[size_w-1];
|
||||
wm2 = w0[size_w-2];
|
||||
for (vk = v0+k, ak = a->ob_digit + k; vk-- > v0;) {
|
||||
/* inner loop: divide vk[0:size_w+1] by w0[0:size_w], giving
|
||||
single-digit quotient q, remainder in vk[0:size_w]. */
|
||||
|
||||
SIGCHECK({
|
||||
Py_DECREF(a);
|
||||
a = NULL;
|
||||
break;
|
||||
Py_DECREF(w);
|
||||
Py_DECREF(v);
|
||||
*prem = NULL;
|
||||
return NULL;
|
||||
})
|
||||
if (vj == w->ob_digit[size_w-1])
|
||||
q = PyLong_MASK;
|
||||
else
|
||||
q = (((twodigits)vj << PyLong_SHIFT) + v->ob_digit[j-1]) /
|
||||
w->ob_digit[size_w-1];
|
||||
|
||||
while (w->ob_digit[size_w-2]*q >
|
||||
((
|
||||
((twodigits)vj << PyLong_SHIFT)
|
||||
+ v->ob_digit[j-1]
|
||||
- q*w->ob_digit[size_w-1]
|
||||
) << PyLong_SHIFT)
|
||||
+ v->ob_digit[j-2])
|
||||
/* estimate quotient digit q; may overestimate by 1 (rare) */
|
||||
vtop = vk[size_w];
|
||||
assert(vtop <= wm1);
|
||||
vv = ((twodigits)vtop << PyLong_SHIFT) | vk[size_w-1];
|
||||
q = (digit)(vv / wm1);
|
||||
r = (digit)(vv - (twodigits)wm1 * q); /* r = vv % wm1 */
|
||||
while ((twodigits)wm2 * q > (((twodigits)r << PyLong_SHIFT)
|
||||
| vk[size_w-2])) {
|
||||
--q;
|
||||
r += wm1;
|
||||
if (r >= PyLong_BASE)
|
||||
break;
|
||||
}
|
||||
assert(q <= PyLong_BASE);
|
||||
|
||||
for (i = 0; i < size_w && i+k < size_v; ++i) {
|
||||
twodigits z = w->ob_digit[i] * q;
|
||||
digit zz = (digit) (z >> PyLong_SHIFT);
|
||||
carry += v->ob_digit[i+k] - z
|
||||
+ ((twodigits)zz << PyLong_SHIFT);
|
||||
v->ob_digit[i+k] = (digit)(carry & PyLong_MASK);
|
||||
carry = Py_ARITHMETIC_RIGHT_SHIFT(stwodigits,
|
||||
carry, PyLong_SHIFT);
|
||||
carry -= zz;
|
||||
/* subtract q*w0[0:size_w] from vk[0:size_w+1] */
|
||||
zhi = 0;
|
||||
for (i = 0; i < size_w; ++i) {
|
||||
/* invariants: -PyLong_BASE <= -q <= zhi <= 0;
|
||||
-PyLong_BASE * q <= z < PyLong_BASE */
|
||||
z = (sdigit)vk[i] + zhi -
|
||||
(stwodigits)q * (stwodigits)w0[i];
|
||||
vk[i] = (digit)z & PyLong_MASK;
|
||||
zhi = (sdigit)Py_ARITHMETIC_RIGHT_SHIFT(stwodigits,
|
||||
z, PyLong_SHIFT);
|
||||
}
|
||||
|
||||
if (i+k < size_v) {
|
||||
carry += v->ob_digit[i+k];
|
||||
v->ob_digit[i+k] = 0;
|
||||
}
|
||||
|
||||
if (carry == 0)
|
||||
a->ob_digit[k] = (digit) q;
|
||||
else {
|
||||
assert(carry == -1);
|
||||
a->ob_digit[k] = (digit) q-1;
|
||||
/* add w back if q was too large (this branch taken rarely) */
|
||||
assert((sdigit)vtop + zhi == -1 || (sdigit)vtop + zhi == 0);
|
||||
if ((sdigit)vtop + zhi < 0) {
|
||||
carry = 0;
|
||||
for (i = 0; i < size_w && i+k < size_v; ++i) {
|
||||
carry += v->ob_digit[i+k] + w->ob_digit[i];
|
||||
v->ob_digit[i+k] = (digit)(carry & PyLong_MASK);
|
||||
carry = Py_ARITHMETIC_RIGHT_SHIFT(
|
||||
stwodigits,
|
||||
carry, PyLong_SHIFT);
|
||||
for (i = 0; i < size_w; ++i) {
|
||||
carry += vk[i] + w0[i];
|
||||
vk[i] = carry & PyLong_MASK;
|
||||
carry >>= PyLong_SHIFT;
|
||||
}
|
||||
--q;
|
||||
}
|
||||
} /* for j, k */
|
||||
|
||||
if (a == NULL)
|
||||
*prem = NULL;
|
||||
else {
|
||||
a = long_normalize(a);
|
||||
*prem = divrem1(v, d, &d);
|
||||
/* d receives the (unused) remainder */
|
||||
if (*prem == NULL) {
|
||||
Py_DECREF(a);
|
||||
a = NULL;
|
||||
}
|
||||
/* store quotient digit */
|
||||
assert(q < PyLong_BASE);
|
||||
*--ak = q;
|
||||
}
|
||||
|
||||
/* unshift remainder; we reuse w to store the result */
|
||||
carry = v_rshift(w0, v0, size_w, d);
|
||||
assert(carry==0);
|
||||
Py_DECREF(v);
|
||||
Py_DECREF(w);
|
||||
return a;
|
||||
|
||||
*prem = long_normalize(w);
|
||||
return long_normalize(a);
|
||||
}
|
||||
|
||||
/* Methods */
|
||||
@ -3793,11 +3856,6 @@ long_sizeof(PyLongObject *v)
|
||||
return PyLong_FromSsize_t(res);
|
||||
}
|
||||
|
||||
static const unsigned char BitLengthTable[32] = {
|
||||
0, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4,
|
||||
5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5
|
||||
};
|
||||
|
||||
static PyObject *
|
||||
long_bit_length(PyLongObject *v)
|
||||
{
|
||||
|
||||
Loading…
Reference in New Issue
Block a user