CPython2 的整数对象 有 PyIntObject 和 PyLongObject 这两种类型，
CPython3 只保留了 PyLongObject
在 源文件：Objects/longobject.c
的第三行有这么一句话 XXX The functional organization of this file is terrible
可见这个变化不是一蹴而就的，有比较艰辛的过程，大家有兴趣可以去挖掘一下
// longobject.h
typedef struct _longobject PyLongObject; /* Revealed in longintrepr.h */// longintrepr.h
/* Long integer representation.
The absolute value of a number is equal to
一个数的绝对值等价于下面的表达式
SUM(for i=0 through abs(ob_size)-1) ob_digit[i] * 2**(SHIFT*i)
Negative numbers are represented with ob_size < 0;
负数表示为 ob_size < 0
zero is represented by ob_size == 0.
整数0 用 ob_size == 0表示
In a normalized number, ob_digit[abs(ob_size)-1] (the most significant
digit) is never zero. Also, in all cases, for all valid i,
在一个规范的数字ob_digit[abs(ob_size)-1]（）永不为0。而且，所有有效的 i 都满足以下要求
0 <= ob_digit[i] <= MASK.
The allocation function takes care of allocating extra memory
so that ob_digit[0] ... ob_digit[abs(ob_size)-1] are actually available.
CAUTION: Generic code manipulating subtypes of PyVarObject has to
aware that ints abuse ob_size's sign bit.
警告: 通用代码操作 PyVarObject 的子类型必须注意 ob_size的符号滥用问题。
*/
struct _longobject {
PyObject_VAR_HEAD
digit ob_digit[1];
};从源码可以看出 PyLongObject 是变长对象
// Objects/longobject.c
PyTypeObject PyLong_Type = {
PyVarObject_HEAD_INIT(&PyType_Type, 0)
"int", /* tp_name */
offsetof(PyLongObject, ob_digit), /* tp_basicsize */
sizeof(digit), /* tp_itemsize */
long_dealloc, /* tp_dealloc */
0, /* tp_print */
0, /* tp_getattr */
0, /* tp_setattr */
0, /* tp_reserved */
long_to_decimal_string, /* tp_repr */
&long_as_number, /* tp_as_number */
0, /* tp_as_sequence */
0, /* tp_as_mapping */
(hashfunc)long_hash, /* tp_hash */
0, /* tp_call */
long_to_decimal_string, /* tp_str */
PyObject_GenericGetAttr, /* tp_getattro */
0, /* tp_setattro */
0, /* tp_as_buffer */
Py_TPFLAGS_DEFAULT | Py_TPFLAGS_BASETYPE |
Py_TPFLAGS_LONG_SUBCLASS, /* tp_flags */
long_doc, /* tp_doc */
0, /* tp_traverse */
0, /* tp_clear */
long_richcompare, /* tp_richcompare */
0, /* tp_weaklistoffset */
0, /* tp_iter */
0, /* tp_iternext */
long_methods, /* tp_methods */
0, /* tp_members */
long_getset, /* tp_getset */
0, /* tp_base */
0, /* tp_dict */
0, /* tp_descr_get */
0, /* tp_descr_set */
0, /* tp_dictoffset */
0, /* tp_init */
0, /* tp_alloc */
long_new, /* tp_new */
PyObject_Del, /* tp_free */
};从 PyLong_Type 可以看出，创建一个整数对象的入口函数为 long_new
源文件：Objects/clinic/longobject.c.h
// Objects/clinic/longobject.c.h
/*[clinic input]
preserve
[clinic start generated code]*/
static PyObject *
long_new_impl(PyTypeObject *type, PyObject *x, PyObject *obase);
static PyObject *
long_new(PyTypeObject *type, PyObject *args, PyObject *kwargs)
{
PyObject *return_value = NULL;
static const char * const _keywords[] = {"", "base", NULL};
static _PyArg_Parser _parser = {"|OO:int", _keywords, 0};
PyObject *x = NULL;
PyObject *obase = NULL;
if (!_PyArg_ParseTupleAndKeywordsFast(args, kwargs, &_parser,
&x, &obase)) {
goto exit;
}
return_value = long_new_impl(type, x, obase);
exit:
return return_value;
}具体实现在 long_new_impl 源文件：Objects/longobject.c
// Objects/longobject.c
/*[clinic input]
@classmethod
int.__new__ as long_new
x: object(c_default="NULL") = 0
/
base as obase: object(c_default="NULL") = 10
[clinic start generated code]*/
static PyObject *
long_new_impl(PyTypeObject *type, PyObject *x, PyObject *obase)
/*[clinic end generated code: output=e47cfe777ab0f24c input=81c98f418af9eb6f]*/
{
Py_ssize_t base;
if (type != &PyLong_Type)
return long_subtype_new(type, x, obase); /* Wimp out */
if (x == NULL) {
if (obase != NULL) {
PyErr_SetString(PyExc_TypeError,
"int() missing string argument");
return NULL;
}
return PyLong_FromLong(0L);
}
if (obase == NULL)
return PyNumber_Long(x);
base = PyNumber_AsSsize_t(obase, NULL);
if (base == -1 && PyErr_Occurred())
return NULL;
if ((base != 0 && base < 2) || base > 36) {
PyErr_SetString(PyExc_ValueError,
"int() base must be >= 2 and <= 36, or 0");
return NULL;
}
if (PyUnicode_Check(x))
return PyLong_FromUnicodeObject(x, (int)base);
else if (PyByteArray_Check(x) || PyBytes_Check(x)) {
char *string;
if (PyByteArray_Check(x))
string = PyByteArray_AS_STRING(x);
else
string = PyBytes_AS_STRING(x);
return _PyLong_FromBytes(string, Py_SIZE(x), (int)base);
}
else {
PyErr_SetString(PyExc_TypeError,
"int() can't convert non-string with explicit base");
return NULL;
}
}从 long_new_impl 函数可以看出有如下几种情况
- x == NULL 且 obase != NULL 调用 PyLong_FromLong
- obase 为 NULL 调用 PyNumber_Long
- x 和 obase 都不为 NULL
- PyUnicode 调用 PyLong_FromUnicodeObject，最终调用 PyLong_FromString
- PyByteArray/PyBytes 调用_PyLong_FromBytes，最终调用 PyLong_FromString
一些整数在一开始就会被初始化一直留存，当再次使用直接从小整数对象池中获取，不用频繁的申请内存。
默认的小整数范围是 [-5, 257) 源文件：Objects/longobject.c
// Objects/longobject.c
#ifndef NSMALLPOSINTS
#define NSMALLPOSINTS 257
#endif
#ifndef NSMALLNEGINTS
#define NSMALLNEGINTS 5
#endif
#if NSMALLNEGINTS + NSMALLPOSINTS > 0
/* Small integers are preallocated in this array so that they
can be shared.
The integers that are preallocated are those in the range
-NSMALLNEGINTS (inclusive) to NSMALLPOSINTS (not inclusive). */
static PyLongObject small_ints[NSMALLNEGINTS + NSMALLPOSINTS];
#ifdef COUNT_ALLOCS
Py_ssize_t quick_int_allocs, quick_neg_int_allocs;
#endif
static PyObject *
get_small_int(sdigit ival)
{
PyObject *v;
assert(-NSMALLNEGINTS <= ival && ival < NSMALLPOSINTS);
v = (PyObject *)&small_ints[ival + NSMALLNEGINTS];
Py_INCREF(v);
#ifdef COUNT_ALLOCS
if (ival >= 0)
quick_int_allocs++;
else
quick_neg_int_allocs++;
#endif
return v;
}
#define CHECK_SMALL_INT(ival) \
do if (-NSMALLNEGINTS <= ival && ival < NSMALLPOSINTS) { \
return get_small_int((sdigit)ival); \
} while(0)
宏 CHECK_SMALL_INT 会检查传入的数是否在小整数范围内，如果是直接返回。 可以在创建或复制整数对象等函数中找到 CHECK_SMALL_INT 的身影，以下只列出了 PyLong_FromLong，就不一一列举了
// Object/longobject.c
PyObject *
PyLong_FromLong(long ival)
{
PyLongObject *v;
unsigned long abs_ival;
unsigned long t; /* unsigned so >> doesn't propagate sign bit */
int ndigits = 0;
int sign;
CHECK_SMALL_INT(ival);
...
}// Objects/longobject.c
int
_PyLong_Init(void)
{
#if NSMALLNEGINTS + NSMALLPOSINTS > 0
int ival, size;
PyLongObject *v = small_ints;
for (ival = -NSMALLNEGINTS; ival < NSMALLPOSINTS; ival++, v++) {
size = (ival < 0) ? -1 : ((ival == 0) ? 0 : 1);
if (Py_TYPE(v) == &PyLong_Type) {
/* The element is already initialized, most likely
* the Python interpreter was initialized before.
*/
Py_ssize_t refcnt;
PyObject* op = (PyObject*)v;
refcnt = Py_REFCNT(op) < 0 ? 0 : Py_REFCNT(op);
_Py_NewReference(op);
/* _Py_NewReference sets the ref count to 1 but
* the ref count might be larger. Set the refcnt
* to the original refcnt + 1 */
Py_REFCNT(op) = refcnt + 1;
assert(Py_SIZE(op) == size);
assert(v->ob_digit[0] == (digit)abs(ival));
}
else {
(void)PyObject_INIT(v, &PyLong_Type);
}
Py_SIZE(v) = size;
v->ob_digit[0] = (digit)abs(ival);
}
#endif
_PyLong_Zero = PyLong_FromLong(0);
if (_PyLong_Zero == NULL)
return 0;
_PyLong_One = PyLong_FromLong(1);
if (_PyLong_One == NULL)
return 0;
/* initialize int_info */
if (Int_InfoType.tp_name == NULL) {
if (PyStructSequence_InitType2(&Int_InfoType, &int_info_desc) < 0)
return 0;
}
return 1;
}在 long_to_decimal_string_internal中添加如下代码并重新编译安装
// Objects/longobject.c
static int
long_to_decimal_string_internal(PyObject *aa,
PyObject **p_output,
_PyUnicodeWriter *writer,
_PyBytesWriter *bytes_writer,
char **bytes_str)
{
PyLongObject *scratch, *a;
PyObject *str = NULL;
Py_ssize_t size, strlen, size_a, i, j;
digit *pout, *pin, rem, tenpow;
int negative;
int d;
enum PyUnicode_Kind kind;
a = (PyLongObject *)aa;
// 添加打印代码
printf("ob_size = %d\n", Py_SIZE(a));
for (int index = 0; index < Py_SIZE(a); ++index) {
printf("ob_digit[%d] = %d\n", index, a->ob_digit[index]);
}
...
}编译安装后进入 python 解释器输入如下代码
num = 9223372043297226753
print(num)
# output
>>> ob_size = 3
>>> ob_digit[0] = 1
>>> ob_digit[1] = 6
>>> ob_digit[2] = 8
>>> 9223372043297226753如下图所示
注：这里的 30 是由 PyLong_SHIFT 决定的，64 位系统中，PyLong_SHIFT 为 30，否则 PyLong_SHIFT 为 15
可以看到整数对象的数值操作较多，由于篇幅限制无法一一分析，这里只分析整数的部分操作
// Objects/longobject.c
static PyNumberMethods long_as_number = {
(binaryfunc)long_add, /*nb_add 加法 */
(binaryfunc)long_sub, /*nb_subtract 减法 */
(binaryfunc)long_mul, /*nb_multiply 乘法 */
long_mod, /*nb_remainder 取余 */
long_divmod, /*nb_divmod */
long_pow, /*nb_power 求幂 */
(unaryfunc)long_neg, /*nb_negative */
(unaryfunc)long_long, /*tp_positive */
(unaryfunc)long_abs, /*tp_absolute 绝对值 */
(inquiry)long_bool, /*tp_bool 求bool值 */
(unaryfunc)long_invert, /*nb_invert 反转 */
long_lshift, /*nb_lshift 逻辑左移 */
(binaryfunc)long_rshift, /*nb_rshift 逻辑右移 */
long_and, /*nb_and 与操作 */
long_xor, /*nb_xor 异或 */
long_or, /*nb_or 或操作 */
long_long, /*nb_int*/
0, /*nb_reserved*/
long_float, /*nb_float*/
0, /* nb_inplace_add */
0, /* nb_inplace_subtract */
0, /* nb_inplace_multiply */
0, /* nb_inplace_remainder */
0, /* nb_inplace_power */
0, /* nb_inplace_lshift */
0, /* nb_inplace_rshift */
0, /* nb_inplace_and */
0, /* nb_inplace_xor */
0, /* nb_inplace_or */
long_div, /* nb_floor_divide */
long_true_divide, /* nb_true_divide */
0, /* nb_inplace_floor_divide */
0, /* nb_inplace_true_divide */
long_long, /* nb_index */
};// Objects/longobject.c
static PyObject *
long_add(PyLongObject *a, PyLongObject *b)
{
PyLongObject *z;
CHECK_BINOP(a, b);
if (Py_ABS(Py_SIZE(a)) <= 1 && Py_ABS(Py_SIZE(b)) <= 1) {
return PyLong_FromLong(MEDIUM_VALUE(a) + MEDIUM_VALUE(b));
}
if (Py_SIZE(a) < 0) {
if (Py_SIZE(b) < 0) {
z = x_add(a, b);
if (z != NULL) {
/* x_add received at least one multiple-digit int,
and thus z must be a multiple-digit int.
That also means z is not an element of
small_ints, so negating it in-place is safe. */
assert(Py_REFCNT(z) == 1);
Py_SIZE(z) = -(Py_SIZE(z));
}
}
else
z = x_sub(b, a);
}
else {
if (Py_SIZE(b) < 0)
z = x_sub(a, b);
else
z = x_add(a, b);
}
return (PyObject *)z;
}可以看到整数的加法运算函数 long_add 根据 a、b 的 ob_size 又细分为两个函数 (x_add 和 x_sub) 做处理
// Objects/longobject.c
/* Add the absolute values of two integers. */
static PyLongObject *
x_add(PyLongObject *a, PyLongObject *b)
{
Py_ssize_t size_a = Py_ABS(Py_SIZE(a)), size_b = Py_ABS(Py_SIZE(b));
PyLongObject *z;
Py_ssize_t i;
digit carry = 0;
/* Ensure a is the larger of the two: */
// 确保 a 大于 b
if (size_a < size_b) {
{ PyLongObject *temp = a; a = b; b = temp; }
{ Py_ssize_t size_temp = size_a;
size_a = size_b;
size_b = size_temp; }
}
z = _PyLong_New(size_a+1);
if (z == NULL)
return NULL;
for (i = 0; i < size_b; ++i) {
carry += a->ob_digit[i] + b->ob_digit[i];
z->ob_digit[i] = carry & PyLong_MASK;
carry >>= PyLong_SHIFT;
}
for (; i < size_a; ++i) {
carry += a->ob_digit[i];
z->ob_digit[i] = carry & PyLong_MASK;
carry >>= PyLong_SHIFT;
}
z->ob_digit[i] = carry;
return long_normalize(z);
}加法运算函数 x_add 从 ob_digit 数组的低位开始依次按位相加，carry 做进位处理，然后处理 a 对象的高位数字，最后使用 long_normalize 函数调整 ob_size，确保 ob_digit[abs(ob_size)-1]不为零，这与普通四则运算的加法运算相同，只不过进位单元不同而已
// Objects/longobject.c
/* Subtract the absolute values of two integers. */
static PyLongObject *
x_sub(PyLongObject *a, PyLongObject *b)
{
Py_ssize_t size_a = Py_ABS(Py_SIZE(a)), size_b = Py_ABS(Py_SIZE(b));
PyLongObject *z;
Py_ssize_t i;
int sign = 1;
digit borrow = 0;
/* Ensure a is the larger of the two: */
// 确保 a 大于 b
if (size_a < size_b) {
sign = -1;
{ PyLongObject *temp = a; a = b; b = temp; }
{ Py_ssize_t size_temp = size_a;
size_a = size_b;
size_b = size_temp; }
}
else if (size_a == size_b) {
/* Find highest digit where a and b differ: */
// 找到最高位 a 与 b的差异
i = size_a;
while (--i >= 0 && a->ob_digit[i] == b->ob_digit[i])
;
if (i < 0)
return (PyLongObject *)PyLong_FromLong(0);
if (a->ob_digit[i] < b->ob_digit[i]) {
sign = -1;
{ PyLongObject *temp = a; a = b; b = temp; }
}
size_a = size_b = i+1;
}
z = _PyLong_New(size_a);
if (z == NULL)
return NULL;
for (i = 0; i < size_b; ++i) {
/* The following assumes unsigned arithmetic
works module 2**N for some N>PyLong_SHIFT. */
borrow = a->ob_digit[i] - b->ob_digit[i] - borrow;
z->ob_digit[i] = borrow & PyLong_MASK;
borrow >>= PyLong_SHIFT;
borrow &= 1; /* Keep only one sign bit */
}
for (; i < size_a; ++i) {
borrow = a->ob_digit[i] - borrow;
z->ob_digit[i] = borrow & PyLong_MASK;
borrow >>= PyLong_SHIFT;
borrow &= 1; /* Keep only one sign bit */
}
assert(borrow == 0);
if (sign < 0) {
Py_SIZE(z) = -Py_SIZE(z);
}
return long_normalize(z);
}与普通四则运算减法相同，数不够大则向高一位借位， 减法运算函数 x_sub 的示例图如下，注：PyLong_SHIFT 为 30
// Objects/longobject.c
static PyObject *
long_mul(PyLongObject *a, PyLongObject *b)
{
PyLongObject *z;
CHECK_BINOP(a, b);
/* fast path for single-digit multiplication */
if (Py_ABS(Py_SIZE(a)) <= 1 && Py_ABS(Py_SIZE(b)) <= 1) {
stwodigits v = (stwodigits)(MEDIUM_VALUE(a)) * MEDIUM_VALUE(b);
return PyLong_FromLongLong((long long)v);
}
z = k_mul(a, b);
/* Negate if exactly one of the inputs is negative. */
if (((Py_SIZE(a) ^ Py_SIZE(b)) < 0) && z) {
_PyLong_Negate(&z);
if (z == NULL)
return NULL;
}
return (PyObject *)z;
}k_mul 函数是一种快速乘法 源文件
Karatsuba 的算法主要是用于两个大数的乘法，极大提高了运算效率，相较于普通乘法降低了复杂度，并在其中运用了递归的思想。 基本的原理和做法是将位数很多的两个大数 x 和 y 分成位数较少的数，每个数都是原来 x 和 y 位数的一半。 这样处理之后，简化为做三次乘法，并附带少量的加法操作和移位操作。
具体可以看 wiki Karatsuba 算法的实现