
``math`` --- Mathematical functions
***********************************

This module is always available.  It provides access to the
mathematical functions defined by the C standard.

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.copysign(x, y)

   Return *x* with the sign of *y*.  On a platform that supports
   signed zeros, ``copysign(1.0, -0.0)`` returns *-1.0*.

math.fabs(x)

   Return the absolute value of *x*.

math.factorial(x)

   Return *x* factorial.  Raises ``ValueError`` if *x* is not integral
   or is negative.

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.isfinite(x)

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

   New in version 3.2.

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.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.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**x``.

math.expm1(x)

   Return ``e**x - 1``.  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

   New in version 3.2.

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.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``.

math.sqrt(x)

   Return the square root of *x*.


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

math.acos(x)

   Return the arc cosine of *x*, in radians.

math.asin(x)

   Return the arc sine of *x*, in radians.

math.atan(x)

   Return the arc tangent of *x*, in radians.

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.hypot(x, y)

   Return the Euclidean norm, ``sqrt(x*x + y*y)``. This is the length
   of the vector from the origin to point ``(x, y)``.

math.sin(x)

   Return the sine of *x* radians.

math.tan(x)

   Return the tangent of *x* radians.


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

math.degrees(x)

   Converts angle *x* from radians to degrees.

math.radians(x)

   Converts 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

   New in version 3.2.

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.

   New in version 3.2.

math.gamma(x)

   Return the Gamma function at *x*.

   New in version 3.2.

math.lgamma(x)

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

   New in version 3.2.


Constants
=========

math.pi

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

math.e

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

**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.

See also:

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