"math" --- Mathematical functions
*********************************

======================================================================

该模块提供了对C标准定义的数学函数的访问。

These functions cannot be used with complex numbers; use the functions
of the same name from the "cmath" module if you require support for
complex numbers.  The distinction between functions which support
complex numbers and those which don't is made since most users do not
want to learn quite as much mathematics as required to understand
complex numbers.  Receiving an exception instead of a complex result
allows earlier detection of the unexpected complex number used as a
parameter, so that the programmer can determine how and why it was
generated in the first place.

The following functions are provided by this module.  Except when
explicitly noted otherwise, all return values are floats.


Number-theoretic and representation functions
=============================================

math.ceil(x)

   Return the ceiling of *x*, the smallest integer greater than or
   equal to *x*. If *x* is not a float, delegates to "x.__ceil__()",
   which should return an "Integral" value.

math.comb(n, k)

   返回不重复且无顺序地从 *n* 项中选择 *k* 项的方式总数。

   当 "k <= n" 时取值为 "n! / (k! * (n - k)!)"；当 "k > n" 时取值为零
   。

   也称为二项式系数，因为它等价于表达式 "(1 + x) ** n" 的多项式展开中
   第 k 项的系数。

   如果任一参数不为整数则会引发 "TypeError"。 如果任一参数为负数则会引
   发 "ValueError"。

   3.8 版新加入.

math.copysign(x, y)

   Return a float with the magnitude (absolute value) of *x* but the
   sign of *y*.  On platforms that support signed zeros,
   "copysign(1.0, -0.0)" returns *-1.0*.

math.fabs(x)

   Return the absolute value of *x*.

math.factorial(x)

   以一个整数返回 *x* 的阶乘。 如果 *x* 不是整数或为负数时则将引发
   "ValueError"。

   3.9 版後已棄用: Accepting floats with integral values (like "5.0")
   is deprecated.

math.floor(x)

   Return the floor of *x*, the largest integer less than or equal to
   *x*. If *x* is not a float, delegates to "x.__floor__()", which
   should return an "Integral" value.

math.fmod(x, y)

   Return "fmod(x, y)", as defined by the platform C library. Note
   that the Python expression "x % y" may not return the same result.
   The intent of the C standard is that "fmod(x, y)" be exactly
   (mathematically; to infinite precision) equal to "x - n*y" for some
   integer *n* such that the result has the same sign as *x* and
   magnitude less than "abs(y)".  Python's "x % y" returns a result
   with the sign of *y* instead, and may not be exactly computable for
   float arguments. For example, "fmod(-1e-100, 1e100)" is "-1e-100",
   but the result of Python's "-1e-100 % 1e100" is "1e100-1e-100",
   which cannot be represented exactly as a float, and rounds to the
   surprising "1e100".  For this reason, function "fmod()" is
   generally preferred when working with floats, while Python's "x %
   y" is preferred when working with integers.

math.frexp(x)

   Return the mantissa and exponent of *x* as the pair "(m, e)".  *m*
   is a float and *e* is an integer such that "x == m * 2**e" exactly.
   If *x* is zero, returns "(0.0, 0)", otherwise "0.5 <= abs(m) < 1".
   This is used to "pick apart" the internal representation of a float
   in a portable way.

math.fsum(iterable)

   Return an accurate floating point sum of values in the iterable.
   Avoids loss of precision by tracking multiple intermediate partial
   sums:

      >>> sum([.1, .1, .1, .1, .1, .1, .1, .1, .1, .1])
      0.9999999999999999
      >>> fsum([.1, .1, .1, .1, .1, .1, .1, .1, .1, .1])
      1.0

   The algorithm's accuracy depends on IEEE-754 arithmetic guarantees
   and the typical case where the rounding mode is half-even.  On some
   non-Windows builds, the underlying C library uses extended
   precision addition and may occasionally double-round an
   intermediate sum causing it to be off in its least significant bit.

   For further discussion and two alternative approaches, see the ASPN
   cookbook recipes for accurate floating point summation.

math.gcd(a, b)

   Return the greatest common divisor of the integers *a* and *b*.  If
   either *a* or *b* is nonzero, then the value of "gcd(a, b)" is the
   largest positive integer that divides both *a* and *b*.  "gcd(0,
   0)" returns "0".

   3.5 版新加入.

math.isclose(a, b, *, rel_tol=1e-09, abs_tol=0.0)

   Return "True" if the values *a* and *b* are close to each other and
   "False" otherwise.

   Whether or not two values are considered close is determined
   according to given absolute and relative tolerances.

   *rel_tol* is the relative tolerance -- it is the maximum allowed
   difference between *a* and *b*, relative to the larger absolute
   value of *a* or *b*. For example, to set a tolerance of 5%, pass
   "rel_tol=0.05".  The default tolerance is "1e-09", which assures
   that the two values are the same within about 9 decimal digits.
   *rel_tol* must be greater than zero.

   *abs_tol* is the minimum absolute tolerance -- useful for
   comparisons near zero. *abs_tol* must be at least zero.

   If no errors occur, the result will be: "abs(a-b) <= max(rel_tol *
   max(abs(a), abs(b)), abs_tol)".

   The IEEE 754 special values of "NaN", "inf", and "-inf" will be
   handled according to IEEE rules.  Specifically, "NaN" is not
   considered close to any other value, including "NaN".  "inf" and
   "-inf" are only considered close to themselves.

   3.5 版新加入.

   也參考: **PEP 485** -- A function for testing approximate
     equality

math.isfinite(x)

   Return "True" if *x* is neither an infinity nor a NaN, and "False"
   otherwise.  (Note that "0.0" *is* considered finite.)

   3.2 版新加入.

math.isinf(x)

   Return "True" if *x* is a positive or negative infinity, and
   "False" otherwise.

math.isnan(x)

   Return "True" if *x* is a NaN (not a number), and "False"
   otherwise.

math.isqrt(n)

   返回非负整数 *n* 的整数平方根。 这就是对 *n* 的实际平方根向下取整，
   或者相当于使得 *a*² ≤ *n* 的最大整数 *a*。

   对于某些应用来说，可以更适合取值为使得 *n* ≤ *a*² 的最小整数 *a* ，
   或者换句话说就是 *n* 的实际平方根向上取整。 对于正数 *n*，这可以使
   用 "a = 1 + isqrt(n - 1)" 来计算。

   3.8 版新加入.

math.ldexp(x, i)

   Return "x * (2**i)".  This is essentially the inverse of function
   "frexp()".

math.modf(x)

   Return the fractional and integer parts of *x*.  Both results carry
   the sign of *x* and are floats.

math.perm(n, k=None)

   返回不重复且无顺序地从 *n* 项中选择 *k* 项的方式总数。

   当 "k <= n" 时取值为 "n! / (n - k)!"；当 "k > n" 时取值为零。

   如果 *k* 未指定或为 None，则 *k* 默认值为 *n* 并且函数将返回 "n!"。

   如果任一参数不为整数则会引发 "TypeError"。 如果任一参数为负数则会引
   发 "ValueError"。

   3.8 版新加入.

math.prod(iterable, *, start=1)

   计算输入的 *iterable* 中所有元素的积。 积的默认 *start* 值为 "1"。

   当可迭代对象为空时，返回起始值。 此函数特别针对数字值使用，并会拒绝
   非数字类型。

   3.8 版新加入.

math.remainder(x, y)

   Return the IEEE 754-style remainder of *x* with respect to *y*.
   For finite *x* and finite nonzero *y*, this is the difference "x -
   n*y", where "n" is the closest integer to the exact value of the
   quotient "x / y".  If "x / y" is exactly halfway between two
   consecutive integers, the nearest *even* integer is used for "n".
   The remainder "r = remainder(x, y)" thus always satisfies "abs(r)
   <= 0.5 * abs(y)".

   Special cases follow IEEE 754: in particular, "remainder(x,
   math.inf)" is *x* for any finite *x*, and "remainder(x, 0)" and
   "remainder(math.inf, x)" raise "ValueError" for any non-NaN *x*. If
   the result of the remainder operation is zero, that zero will have
   the same sign as *x*.

   On platforms using IEEE 754 binary floating-point, the result of
   this operation is always exactly representable: no rounding error
   is introduced.

   3.7 版新加入.

math.trunc(x)

   Return the "Real" value *x* truncated to an "Integral" (usually an
   integer). Delegates to "x.__trunc__()".

Note that "frexp()" and "modf()" have a different call/return pattern
than their C equivalents: they take a single argument and return a
pair of values, rather than returning their second return value
through an 'output parameter' (there is no such thing in Python).

For the "ceil()", "floor()", and "modf()" functions, note that *all*
floating-point numbers of sufficiently large magnitude are exact
integers. Python floats typically carry no more than 53 bits of
precision (the same as the platform C double type), in which case any
float *x* with "abs(x) >= 2**52" necessarily has no fractional bits.


Power and logarithmic functions
===============================

math.exp(x)

   Return *e* raised to the power *x*, where *e* = 2.718281... is the
   base of natural logarithms.  This is usually more accurate than
   "math.e ** x" or "pow(math.e, x)".

math.expm1(x)

   Return *e* raised to the power *x*, minus 1.  Here *e* is the base
   of natural logarithms.  For small floats *x*, the subtraction in
   "exp(x) - 1" can result in a significant loss of precision; the
   "expm1()" function provides a way to compute this quantity to full
   precision:

      >>> from math import exp, expm1
      >>> exp(1e-5) - 1  # gives result accurate to 11 places
      1.0000050000069649e-05
      >>> expm1(1e-5)    # result accurate to full precision
      1.0000050000166668e-05

   3.2 版新加入.

math.log(x[, base])

   With one argument, return the natural logarithm of *x* (to base
   *e*).

   With two arguments, return the logarithm of *x* to the given
   *base*, calculated as "log(x)/log(base)".

math.log1p(x)

   Return the natural logarithm of *1+x* (base *e*). The result is
   calculated in a way which is accurate for *x* near zero.

math.log2(x)

   Return the base-2 logarithm of *x*. This is usually more accurate
   than "log(x, 2)".

   3.3 版新加入.

   也參考: "int.bit_length()" returns the number of bits necessary
     to represent an integer in binary, excluding the sign and leading
     zeros.

math.log10(x)

   Return the base-10 logarithm of *x*.  This is usually more accurate
   than "log(x, 10)".

math.pow(x, y)

   Return "x" raised to the power "y".  Exceptional cases follow Annex
   'F' of the C99 standard as far as possible.  In particular,
   "pow(1.0, x)" and "pow(x, 0.0)" always return "1.0", even when "x"
   is a zero or a NaN.  If both "x" and "y" are finite, "x" is
   negative, and "y" is not an integer then "pow(x, y)" is undefined,
   and raises "ValueError".

   Unlike the built-in "**" operator, "math.pow()" converts both its
   arguments to type "float".  Use "**" or the built-in "pow()"
   function for computing exact integer powers.

math.sqrt(x)

   Return the square root of *x*.


Trigonometric functions
=======================

math.acos(x)

   Return the arc cosine of *x*, in radians. The result is between "0"
   and "pi".

math.asin(x)

   Return the arc sine of *x*, in radians. The result is between
   "-pi/2" and "pi/2".

math.atan(x)

   Return the arc tangent of *x*, in radians. The result is between
   "-pi/2" and "pi/2".

math.atan2(y, x)

   Return "atan(y / x)", in radians. The result is between "-pi" and
   "pi". The vector in the plane from the origin to point "(x, y)"
   makes this angle with the positive X axis. The point of "atan2()"
   is that the signs of both inputs are known to it, so it can compute
   the correct quadrant for the angle. For example, "atan(1)" and
   "atan2(1, 1)" are both "pi/4", but "atan2(-1, -1)" is "-3*pi/4".

math.cos(x)

   Return the cosine of *x* radians.

math.dist(p, q)

   返回 *p* 与 *q* 两点之间的欧几里得距离，以一个坐标序列（或可迭代对
   象）的形式给出。 两个点必须具有相同的维度。

   大致相当于：

      sqrt(sum((px - qx) ** 2.0 for px, qx in zip(p, q)))

   3.8 版新加入.

math.hypot(*coordinates)

   返回欧几里得范数，"sqrt(sum(x**2 for x in coordinates))"。 这是从原
   点到坐标给定点的向量长度。

   对于一个二维点 "(x, y)"，这等价于使用毕达哥拉斯定义 "sqrt(x*x +
   y*y)" 计算一个直角三角形的斜边。

   3.8 版更變: 添加了对 n 维点的支持。 之前的版本只支持二维点。

math.sin(x)

   Return the sine of *x* radians.

math.tan(x)

   Return the tangent of *x* radians.


Angular conversion
==================

math.degrees(x)

   Convert angle *x* from radians to degrees.

math.radians(x)

   Convert angle *x* from degrees to radians.


Hyperbolic functions
====================

Hyperbolic functions are analogs of trigonometric functions that are
based on hyperbolas instead of circles.

math.acosh(x)

   Return the inverse hyperbolic cosine of *x*.

math.asinh(x)

   Return the inverse hyperbolic sine of *x*.

math.atanh(x)

   Return the inverse hyperbolic tangent of *x*.

math.cosh(x)

   Return the hyperbolic cosine of *x*.

math.sinh(x)

   Return the hyperbolic sine of *x*.

math.tanh(x)

   Return the hyperbolic tangent of *x*.


Special functions
=================

math.erf(x)

   Return the error function at *x*.

   The "erf()" function can be used to compute traditional statistical
   functions such as the cumulative standard normal distribution:

      def phi(x):
          'Cumulative distribution function for the standard normal distribution'
          return (1.0 + erf(x / sqrt(2.0))) / 2.0

   3.2 版新加入.

math.erfc(x)

   Return the complementary error function at *x*.  The complementary
   error function is defined as "1.0 - erf(x)".  It is used for large
   values of *x* where a subtraction from one would cause a loss of
   significance.

   3.2 版新加入.

math.gamma(x)

   Return the Gamma function at *x*.

   3.2 版新加入.

math.lgamma(x)

   Return the natural logarithm of the absolute value of the Gamma
   function at *x*.

   3.2 版新加入.


Constants
=========

math.pi

   The mathematical constant *π* = 3.141592..., to available
   precision.

math.e

   The mathematical constant *e* = 2.718281..., to available
   precision.

math.tau

   The mathematical constant *τ* = 6.283185..., to available
   precision. Tau is a circle constant equal to 2*π*, the ratio of a
   circle's circumference to its radius. To learn more about Tau,
   check out Vi Hart's video Pi is (still) Wrong, and start
   celebrating Tau day by eating twice as much pie!

   3.6 版新加入.

math.inf

   A floating-point positive infinity.  (For negative infinity, use
   "-math.inf".)  Equivalent to the output of "float('inf')".

   3.5 版新加入.

math.nan

   A floating-point "not a number" (NaN) value.  Equivalent to the
   output of "float('nan')".

   3.5 版新加入.

**CPython implementation detail:** The "math" module consists mostly
of thin wrappers around the platform C math library functions.
Behavior in exceptional cases follows Annex F of the C99 standard
where appropriate.  The current implementation will raise "ValueError"
for invalid operations like "sqrt(-1.0)" or "log(0.0)" (where C99
Annex F recommends signaling invalid operation or divide-by-zero), and
"OverflowError" for results that overflow (for example,
"exp(1000.0)").  A NaN will not be returned from any of the functions
above unless one or more of the input arguments was a NaN; in that
case, most functions will return a NaN, but (again following C99 Annex
F) there are some exceptions to this rule, for example
"pow(float('nan'), 0.0)" or "hypot(float('nan'), float('inf'))".

Note that Python makes no effort to distinguish signaling NaNs from
quiet NaNs, and behavior for signaling NaNs remains unspecified.
Typical behavior is to treat all NaNs as though they were quiet.

也參考:

  Module "cmath"
     Complex number versions of many of these functions.
