What algorithm does Python employ in fractions.gcd()?
Question:
I’m using the fractions module in Python v3.1 to compute the greatest common divisor. I would like to know what algorithm is used. I’m guessing the Euclidean method, but would like to be sure. The docs (http://docs.python.org/py3k/library/fractions.html?highlight=fractions.gcd#fractions.gcd) don’t help. Can anybody clue me in?
Answers:
According to the 3.1.2 source code online, here’s gcd as defined in Python-3.1.2/Lib/fractions.py:
def gcd(a, b):
"""Calculate the Greatest Common Divisor of a and b.
Unless b==0, the result will have the same sign as b (so that when
b is divided by it, the result comes out positive).
"""
while b:
a, b = b, a%b
return a
So yes, it’s the Euclidean algorithm, written in pure Python.
From fractions python
"Deprecated since version 3.5: Use math.gcd() instead."
I was looking for the algorithm as well. I hope it helped.
Since Python 3.5, the GCD code has been moved to math.gcd. Since Python 3.9, math.gcd takes an arbitrary number of arguments.
The actual GCD code is now implemented in C (for CPython), making it significantly faster than the original pure Python implementation.
static PyObject *
math_gcd(PyObject *module, PyObject * const *args, Py_ssize_t nargs)
{
PyObject *res, *x;
Py_ssize_t i;
if (nargs == 0) {
return PyLong_FromLong(0);
}
res = PyNumber_Index(args[0]);
if (res == NULL) {
return NULL;
}
if (nargs == 1) {
Py_SETREF(res, PyNumber_Absolute(res));
return res;
}
PyObject *one = _PyLong_GetOne(); // borrowed ref
for (i = 1; i < nargs; i++) {
x = _PyNumber_Index(args[i]);
if (x == NULL) {
Py_DECREF(res);
return NULL;
}
if (res == one) {
/* Fast path: just check arguments.
It is okay to use identity comparison here. */
Py_DECREF(x);
continue;
}
Py_SETREF(res, _PyLong_GCD(res, x));
Py_DECREF(x);
if (res == NULL) {
return NULL;
}
}
return res;
}
The actual computation uses Lehmer’s GCD algorithm:
PyObject *
_PyLong_GCD(PyObject *aarg, PyObject *barg)
{
PyLongObject *a, *b, *c = NULL, *d = NULL, *r;
stwodigits x, y, q, s, t, c_carry, d_carry;
stwodigits A, B, C, D, T;
int nbits, k;
Py_ssize_t size_a, size_b, alloc_a, alloc_b;
digit *a_digit, *b_digit, *c_digit, *d_digit, *a_end, *b_end;
a = (PyLongObject *)aarg;
b = (PyLongObject *)barg;
size_a = Py_SIZE(a);
size_b = Py_SIZE(b);
if (-2 <= size_a && size_a <= 2 && -2 <= size_b && size_b <= 2) {
Py_INCREF(a);
Py_INCREF(b);
goto simple;
}
/* Initial reduction: make sure that 0 <= b <= a. */
a = (PyLongObject *)long_abs(a);
if (a == NULL)
return NULL;
b = (PyLongObject *)long_abs(b);
if (b == NULL) {
Py_DECREF(a);
return NULL;
}
if (long_compare(a, b) < 0) {
r = a;
a = b;
b = r;
}
/* We now own references to a and b */
alloc_a = Py_SIZE(a);
alloc_b = Py_SIZE(b);
/* reduce until a fits into 2 digits */
while ((size_a = Py_SIZE(a)) > 2) {
nbits = bit_length_digit(a->ob_digit[size_a-1]);
/* extract top 2*PyLong_SHIFT bits of a into x, along with
corresponding bits of b into y */
size_b = Py_SIZE(b);
assert(size_b <= size_a);
if (size_b == 0) {
if (size_a < alloc_a) {
r = (PyLongObject *)_PyLong_Copy(a);
Py_DECREF(a);
}
else
r = a;
Py_DECREF(b);
Py_XDECREF(c);
Py_XDECREF(d);
return (PyObject *)r;
}
x = (((twodigits)a->ob_digit[size_a-1] << (2*PyLong_SHIFT-nbits)) |
((twodigits)a->ob_digit[size_a-2] << (PyLong_SHIFT-nbits)) |
(a->ob_digit[size_a-3] >> nbits));
y = ((size_b >= size_a - 2 ? b->ob_digit[size_a-3] >> nbits : 0) |
(size_b >= size_a - 1 ? (twodigits)b->ob_digit[size_a-2] << (PyLong_SHIFT-nbits) : 0) |
(size_b >= size_a ? (twodigits)b->ob_digit[size_a-1] << (2*PyLong_SHIFT-nbits) : 0));
/* inner loop of Lehmer's algorithm; A, B, C, D never grow
larger than PyLong_MASK during the algorithm. */
A = 1; B = 0; C = 0; D = 1;
for (k=0;; k++) {
if (y-C == 0)
break;
q = (x+(A-1))/(y-C);
s = B+q*D;
t = x-q*y;
if (s > t)
break;
x = y; y = t;
t = A+q*C; A = D; B = C; C = s; D = t;
}
if (k == 0) {
/* no progress; do a Euclidean step */
if (l_mod(a, b, &r) < 0)
goto error;
Py_SETREF(a, b);
b = r;
alloc_a = alloc_b;
alloc_b = Py_SIZE(b);
continue;
}
/*
a, b = A*b-B*a, D*a-C*b if k is odd
a, b = A*a-B*b, D*b-C*a if k is even
*/
if (k&1) {
T = -A; A = -B; B = T;
T = -C; C = -D; D = T;
}
if (c != NULL) {
Py_SET_SIZE(c, size_a);
}
else if (Py_REFCNT(a) == 1) {
c = (PyLongObject*)Py_NewRef(a);
}
else {
alloc_a = size_a;
c = _PyLong_New(size_a);
if (c == NULL)
goto error;
}
if (d != NULL) {
Py_SET_SIZE(d, size_a);
}
else if (Py_REFCNT(b) == 1 && size_a <= alloc_b) {
d = (PyLongObject*)Py_NewRef(b);
Py_SET_SIZE(d, size_a);
}
else {
alloc_b = size_a;
d = _PyLong_New(size_a);
if (d == NULL)
goto error;
}
a_end = a->ob_digit + size_a;
b_end = b->ob_digit + size_b;
/* compute new a and new b in parallel */
a_digit = a->ob_digit;
b_digit = b->ob_digit;
c_digit = c->ob_digit;
d_digit = d->ob_digit;
c_carry = 0;
d_carry = 0;
while (b_digit < b_end) {
c_carry += (A * *a_digit) - (B * *b_digit);
d_carry += (D * *b_digit++) - (C * *a_digit++);
*c_digit++ = (digit)(c_carry & PyLong_MASK);
*d_digit++ = (digit)(d_carry & PyLong_MASK);
c_carry >>= PyLong_SHIFT;
d_carry >>= PyLong_SHIFT;
}
while (a_digit < a_end) {
c_carry += A * *a_digit;
d_carry -= C * *a_digit++;
*c_digit++ = (digit)(c_carry & PyLong_MASK);
*d_digit++ = (digit)(d_carry & PyLong_MASK);
c_carry >>= PyLong_SHIFT;
d_carry >>= PyLong_SHIFT;
}
assert(c_carry == 0);
assert(d_carry == 0);
Py_INCREF(c);
Py_INCREF(d);
Py_DECREF(a);
Py_DECREF(b);
a = long_normalize(c);
b = long_normalize(d);
}
Py_XDECREF(c);
Py_XDECREF(d);
simple:
assert(Py_REFCNT(a) > 0);
assert(Py_REFCNT(b) > 0);
/* Issue #24999: use two shifts instead of ">> 2*PyLong_SHIFT" to avoid
undefined behaviour when LONG_MAX type is smaller than 60 bits */
#if LONG_MAX >> PyLong_SHIFT >> PyLong_SHIFT
/* a fits into a long, so b must too */
x = PyLong_AsLong((PyObject *)a);
y = PyLong_AsLong((PyObject *)b);
#elif LLONG_MAX >> PyLong_SHIFT >> PyLong_SHIFT
x = PyLong_AsLongLong((PyObject *)a);
y = PyLong_AsLongLong((PyObject *)b);
#else
# error "_PyLong_GCD"
#endif
x = Py_ABS(x);
y = Py_ABS(y);
Py_DECREF(a);
Py_DECREF(b);
/* usual Euclidean algorithm for longs */
while (y != 0) {
t = y;
y = x % y;
x = t;
}
#if LONG_MAX >> PyLong_SHIFT >> PyLong_SHIFT
return PyLong_FromLong(x);
#elif LLONG_MAX >> PyLong_SHIFT >> PyLong_SHIFT
return PyLong_FromLongLong(x);
#else
# error "_PyLong_GCD"
#endif
error:
Py_DECREF(a);
Py_DECREF(b);
Py_XDECREF(c);
Py_XDECREF(d);
return NULL;
}
I’m using the fractions module in Python v3.1 to compute the greatest common divisor. I would like to know what algorithm is used. I’m guessing the Euclidean method, but would like to be sure. The docs (http://docs.python.org/py3k/library/fractions.html?highlight=fractions.gcd#fractions.gcd) don’t help. Can anybody clue me in?
According to the 3.1.2 source code online, here’s gcd as defined in Python-3.1.2/Lib/fractions.py:
def gcd(a, b):
"""Calculate the Greatest Common Divisor of a and b.
Unless b==0, the result will have the same sign as b (so that when
b is divided by it, the result comes out positive).
"""
while b:
a, b = b, a%b
return a
So yes, it’s the Euclidean algorithm, written in pure Python.
From fractions python
"Deprecated since version 3.5: Use math.gcd() instead."
I was looking for the algorithm as well. I hope it helped.
Since Python 3.5, the GCD code has been moved to math.gcd. Since Python 3.9, math.gcd takes an arbitrary number of arguments.
The actual GCD code is now implemented in C (for CPython), making it significantly faster than the original pure Python implementation.
static PyObject *
math_gcd(PyObject *module, PyObject * const *args, Py_ssize_t nargs)
{
PyObject *res, *x;
Py_ssize_t i;
if (nargs == 0) {
return PyLong_FromLong(0);
}
res = PyNumber_Index(args[0]);
if (res == NULL) {
return NULL;
}
if (nargs == 1) {
Py_SETREF(res, PyNumber_Absolute(res));
return res;
}
PyObject *one = _PyLong_GetOne(); // borrowed ref
for (i = 1; i < nargs; i++) {
x = _PyNumber_Index(args[i]);
if (x == NULL) {
Py_DECREF(res);
return NULL;
}
if (res == one) {
/* Fast path: just check arguments.
It is okay to use identity comparison here. */
Py_DECREF(x);
continue;
}
Py_SETREF(res, _PyLong_GCD(res, x));
Py_DECREF(x);
if (res == NULL) {
return NULL;
}
}
return res;
}
The actual computation uses Lehmer’s GCD algorithm:
PyObject *
_PyLong_GCD(PyObject *aarg, PyObject *barg)
{
PyLongObject *a, *b, *c = NULL, *d = NULL, *r;
stwodigits x, y, q, s, t, c_carry, d_carry;
stwodigits A, B, C, D, T;
int nbits, k;
Py_ssize_t size_a, size_b, alloc_a, alloc_b;
digit *a_digit, *b_digit, *c_digit, *d_digit, *a_end, *b_end;
a = (PyLongObject *)aarg;
b = (PyLongObject *)barg;
size_a = Py_SIZE(a);
size_b = Py_SIZE(b);
if (-2 <= size_a && size_a <= 2 && -2 <= size_b && size_b <= 2) {
Py_INCREF(a);
Py_INCREF(b);
goto simple;
}
/* Initial reduction: make sure that 0 <= b <= a. */
a = (PyLongObject *)long_abs(a);
if (a == NULL)
return NULL;
b = (PyLongObject *)long_abs(b);
if (b == NULL) {
Py_DECREF(a);
return NULL;
}
if (long_compare(a, b) < 0) {
r = a;
a = b;
b = r;
}
/* We now own references to a and b */
alloc_a = Py_SIZE(a);
alloc_b = Py_SIZE(b);
/* reduce until a fits into 2 digits */
while ((size_a = Py_SIZE(a)) > 2) {
nbits = bit_length_digit(a->ob_digit[size_a-1]);
/* extract top 2*PyLong_SHIFT bits of a into x, along with
corresponding bits of b into y */
size_b = Py_SIZE(b);
assert(size_b <= size_a);
if (size_b == 0) {
if (size_a < alloc_a) {
r = (PyLongObject *)_PyLong_Copy(a);
Py_DECREF(a);
}
else
r = a;
Py_DECREF(b);
Py_XDECREF(c);
Py_XDECREF(d);
return (PyObject *)r;
}
x = (((twodigits)a->ob_digit[size_a-1] << (2*PyLong_SHIFT-nbits)) |
((twodigits)a->ob_digit[size_a-2] << (PyLong_SHIFT-nbits)) |
(a->ob_digit[size_a-3] >> nbits));
y = ((size_b >= size_a - 2 ? b->ob_digit[size_a-3] >> nbits : 0) |
(size_b >= size_a - 1 ? (twodigits)b->ob_digit[size_a-2] << (PyLong_SHIFT-nbits) : 0) |
(size_b >= size_a ? (twodigits)b->ob_digit[size_a-1] << (2*PyLong_SHIFT-nbits) : 0));
/* inner loop of Lehmer's algorithm; A, B, C, D never grow
larger than PyLong_MASK during the algorithm. */
A = 1; B = 0; C = 0; D = 1;
for (k=0;; k++) {
if (y-C == 0)
break;
q = (x+(A-1))/(y-C);
s = B+q*D;
t = x-q*y;
if (s > t)
break;
x = y; y = t;
t = A+q*C; A = D; B = C; C = s; D = t;
}
if (k == 0) {
/* no progress; do a Euclidean step */
if (l_mod(a, b, &r) < 0)
goto error;
Py_SETREF(a, b);
b = r;
alloc_a = alloc_b;
alloc_b = Py_SIZE(b);
continue;
}
/*
a, b = A*b-B*a, D*a-C*b if k is odd
a, b = A*a-B*b, D*b-C*a if k is even
*/
if (k&1) {
T = -A; A = -B; B = T;
T = -C; C = -D; D = T;
}
if (c != NULL) {
Py_SET_SIZE(c, size_a);
}
else if (Py_REFCNT(a) == 1) {
c = (PyLongObject*)Py_NewRef(a);
}
else {
alloc_a = size_a;
c = _PyLong_New(size_a);
if (c == NULL)
goto error;
}
if (d != NULL) {
Py_SET_SIZE(d, size_a);
}
else if (Py_REFCNT(b) == 1 && size_a <= alloc_b) {
d = (PyLongObject*)Py_NewRef(b);
Py_SET_SIZE(d, size_a);
}
else {
alloc_b = size_a;
d = _PyLong_New(size_a);
if (d == NULL)
goto error;
}
a_end = a->ob_digit + size_a;
b_end = b->ob_digit + size_b;
/* compute new a and new b in parallel */
a_digit = a->ob_digit;
b_digit = b->ob_digit;
c_digit = c->ob_digit;
d_digit = d->ob_digit;
c_carry = 0;
d_carry = 0;
while (b_digit < b_end) {
c_carry += (A * *a_digit) - (B * *b_digit);
d_carry += (D * *b_digit++) - (C * *a_digit++);
*c_digit++ = (digit)(c_carry & PyLong_MASK);
*d_digit++ = (digit)(d_carry & PyLong_MASK);
c_carry >>= PyLong_SHIFT;
d_carry >>= PyLong_SHIFT;
}
while (a_digit < a_end) {
c_carry += A * *a_digit;
d_carry -= C * *a_digit++;
*c_digit++ = (digit)(c_carry & PyLong_MASK);
*d_digit++ = (digit)(d_carry & PyLong_MASK);
c_carry >>= PyLong_SHIFT;
d_carry >>= PyLong_SHIFT;
}
assert(c_carry == 0);
assert(d_carry == 0);
Py_INCREF(c);
Py_INCREF(d);
Py_DECREF(a);
Py_DECREF(b);
a = long_normalize(c);
b = long_normalize(d);
}
Py_XDECREF(c);
Py_XDECREF(d);
simple:
assert(Py_REFCNT(a) > 0);
assert(Py_REFCNT(b) > 0);
/* Issue #24999: use two shifts instead of ">> 2*PyLong_SHIFT" to avoid
undefined behaviour when LONG_MAX type is smaller than 60 bits */
#if LONG_MAX >> PyLong_SHIFT >> PyLong_SHIFT
/* a fits into a long, so b must too */
x = PyLong_AsLong((PyObject *)a);
y = PyLong_AsLong((PyObject *)b);
#elif LLONG_MAX >> PyLong_SHIFT >> PyLong_SHIFT
x = PyLong_AsLongLong((PyObject *)a);
y = PyLong_AsLongLong((PyObject *)b);
#else
# error "_PyLong_GCD"
#endif
x = Py_ABS(x);
y = Py_ABS(y);
Py_DECREF(a);
Py_DECREF(b);
/* usual Euclidean algorithm for longs */
while (y != 0) {
t = y;
y = x % y;
x = t;
}
#if LONG_MAX >> PyLong_SHIFT >> PyLong_SHIFT
return PyLong_FromLong(x);
#elif LLONG_MAX >> PyLong_SHIFT >> PyLong_SHIFT
return PyLong_FromLongLong(x);
#else
# error "_PyLong_GCD"
#endif
error:
Py_DECREF(a);
Py_DECREF(b);
Py_XDECREF(c);
Py_XDECREF(d);
return NULL;
}