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<!DOCTYPE HTML>

<head>

<title>Python 整数对象 · Python 3 源码分析</title>

<style>

@media only screen and (max-width: 640px) {

.book-header .hidden-mobile {

display: none;

}

</style>

window["gitbook-plugin-github-buttons"] = {"repo":"flaggo/python3-source-code-analysis","types":["star"],"size":"small"};

</script>

</head>

<body>

</div>

<li>

</li>

1.1.

前言

</a>

</li>

1.2.

Python 源代码的组织

</a>

</li>

1.3.

Windows 环境下编译 Python

</a>

</li>

1.4.

UNIX/Linux 环境下编译 Python

</a>

</li>

1.5.

修改 Python 源码

</a>

</li>

<li class="header">第 2 部分：Python 内建对象</li>

2.1.

Python 对象初探

</a>

</li>

2.2.

Python 整数对象

</a>

</li>

2.3.

Python 字符串对象

</a>

</li>

2.4.

Python List 对象

</a>

</li>

2.5.

Python Dict 对象

</a>

</li>

2.6.

Python Set 对象

</a>

</li>

2.7.

实现简版 Python

</a>

</li>

<li class="header">第 3 部分：Python 虚拟机</li>

<li>

本书使用 GitBook 发布

</a>

</li>

</ul>

</nav>

</div>

<h1>

<a href=".." >Python 整数对象</a>

</h1>

</div>

<h1 id="python-整数对象">Python 整数对象</h1>

CPython2 的整数对象 有 <code>PyIntObject</code> 和 <code>PyLongObject</code> 这两种类型，

CPython3 只保留了 <code>PyLongObject</code>

在 <code>源文件：</code><a href="https://github.com/python/cpython/blob/v3.7.0/Objects/longobject.c#L3" target="_blank">Objects/longobject.c</a>

的第三行有这么一句话 <code>XXX The functional organization of this file is terrible</code>

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

<h2 id="pylongobject">PyLongObject</h2>

<code>源文件：</code><a href="https://github.com/python/cpython/blob/v3.7.0/Include/longobject.h#L10" target="_blank">Include/longobject.h</a>

<pre class="language-"><code class="lang-c">// longobject.h

typedef struct _longobject PyLongObject; /* Revealed in longintrepr.h */

</code></pre>

<code>源文件：</code><a href="https://github.com/python/cpython/blob/v3.7.0/Include/longintrepr.h#L85" target="_blank">Include/longintrepr.h</a>

<pre class="language-"><code class="lang-c">// 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];

};

</code></pre>

从源码可以看出 PyLongObject 是变长对象

<h2 id="类型对象-pylongtype">类型对象 PyLong_Type</h2>

<code>源文件：</code><a href="https://github.com/python/cpython/blob/v3.7.0/Objects/longobject.c#L5379" target="_blank">Objects/longobject.c</a>

<pre class="language-"><code class="lang-c">// 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 */

};

</code></pre>

<h2 id="创建整数对象">创建整数对象</h2>

从 PyLong_Type 可以看出，创建一个整数对象的入口函数为 long_new

<code>源文件：</code><a href="https://github.com/python/cpython/blob/v3.7.0/Objects/clinic/longobject.c.h#L0" target="_blank">Objects/clinic/longobject.c.h</a>

<pre class="language-"><code class="lang-c">// 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;

}

</code></pre>

具体实现在 long_new_impl <code>源文件：</code><a href="https://github.com/python/cpython/blob/v3.7.0/Objects/longobject.c#L4785" target="_blank">Objects/longobject.c</a>

<pre class="language-"><code class="lang-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 *

/*[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;

}

</code></pre>

从 long_new_impl 函数可以看出有如下几种情况

<ul>

<li>x == NULL 且 obase != NULL 调用 PyLong_FromLong</li>

<li>obase 为NULL 调用 PyNumber_Long</li>

<li>x 和 obase 都不为 NULL<ul>

<li>PyUnicode 调用PyLong_FromUnicodeObject，最终调用PyLong_FromString</li>

<li>PyByteArray/PyBytes 调用_PyLong_FromBytes，最终调用PyLong_FromString</li>

</ul>

</li>

</ul>

<h2 id="小整数对象">小整数对象</h2>

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) <code>源文件：</code><a href="https://github.com/python/cpython/blob/v3.7.0/Objects/longobject.c#L17" target="_blank">Objects/longobject.c</a>

<pre class="language-"><code class="lang-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)

</code></pre>

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

<code>源文件：</code><a href="https://github.com/python/cpython/blob/v3.7.0/Objects/longobject.c#L239" target="_blank">Objects/longobject.c</a>

<pre class="language-"><code class="lang-c">// 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);

...

}

</code></pre>

<h3 id="小整数初始化">小整数初始化</h3>

<code>源文件：</code><a href="https://github.com/python/cpython/blob/v3.7.0/Objects/longobject.c#L5462" target="_blank">Objects/longobject.c</a>

<pre class="language-"><code class="lang-c">// 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;

}

</code></pre>

<h2 id="整数的存储结构">整数的存储结构</h2>

<code>源文件：</code><a href="https://github.com/python/cpython/blob/v3.7.0/Objects/longobject.c#L1581" target="_blank">Objects/longobject.c</a>

在 long_to_decimal_string_internal中添加如下代码并重新编译安装

<pre class="language-"><code class="lang-c">// 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]);

}

...

}

</code></pre>

编译安装后进入python解释器输入如下代码

<pre class="language-"><code class="lang-python">num = 9223372043297226753

print(num)

# output

>>> ob_size = 3

>>> ob_digit[0] = 1

>>> ob_digit[1] = 6

>>> ob_digit[2] = 8

>>> 9223372043297226753

</code></pre>

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

<h2 id="整数对象的数值操作">整数对象的数值操作</h2>

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

<code>源文件：</code><a href="https://github.com/python/cpython/blob/v3.7.0/Objects/longobject.c#L5341" target="_blank">Objects/longobject.c</a>

<pre class="language-"><code class="lang-c">// 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 */

};

</code></pre>

<code>源文件：</code><a href="https://github.com/python/cpython/blob/v3.7.0/Objects/longobject.c#L3081" target="_blank">Objects/longobject.c</a>

<pre class="language-"><code class="lang-c">// 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

}

return (PyObject *)z;

}

</code></pre>

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

<code>源文件：</code><a href="https://github.com/python/cpython/blob/v3.7.0/Objects/longobject.c#L2991" target="_blank">Objects/longobject.c</a>

<pre class="language-"><code class="lang-c">// 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 >>= PyLong_SHIFT;

}

return long_normalize(z);

}

</code></pre>

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

<code>源文件：</code><a href="https://github.com/python/cpython/blob/v3.7.0/Objects/longobject.c#L3025" target="_blank">Objects/longobject.c</a>

<pre class="language-"><code class="lang-c">// Objects/longobject.c

/* Subtract the absolute values of two integers. */

static PyLongObject *

x_sub(PyLongObject *a, PyLongObject *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

sign = -1;

{ 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;

}

size_a = size_b = i+1;

}

z = _PyLong_New(size_a);

if (z == NULL)

return NULL;

/* 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 */

}

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);

}

</code></pre>

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

<code>源文件：</code><a href="https://github.com/python/cpython/blob/v3.7.0/Objects/longobject.c#L3547" target="_blank">Objects/longobject.c</a>

<pre class="language-"><code class="lang-c">// Objects/longobject.c

static PyObject *

long_mul(PyLongObject *a, PyLongObject *b)

{

PyLongObject *z;

/* fast path for single-digit multiplication */

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;

}

</code></pre>

k_mul函数是一种快速乘法 <a href="https://github.com/python/cpython/blob/v3.7.0/Objects/longobject.c#L3268" target="_blank">源文件</a>

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。

</blockquote>

具体可以看wiki <a href="https://www.wikiwand.com/zh-hans/Karatsuba算法" target="_blank">Karatsuba算法</a>的实现

2019-06-08 20:34:04

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