
``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*. ``copysign`` copies the sign bit
   of an IEEE 754 float, ``copysign(1, -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.isinf(x)

   Checks if the float *x* is positive or negative infinite.

math.isnan(x)

   Checks if the float *x* is a NaN (not a number). NaNs are part of
   the IEEE 754 standards. Operation like but not limited to ``inf *
   0``, ``inf / inf`` or any operation involving a NaN, e.g. ``nan *
   1``, return a NaN.

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.log(x[, base])

   Return the logarithm of *x* to the given *base*. If the *base* is
   not specified, return the natural logarithm of *x* (that is, the
   logarithm to base *e*).

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*.

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

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*.


Constants
=========

math.pi

   The mathematical constant *pi*.

math.e

   The mathematical constant *e*.

Note: The ``math`` module consists mostly of thin wrappers around the
  platform C math library functions.  Behavior in exceptional cases is
  loosely specified by the C standards, and Python inherits much of
  its math-function error-reporting behavior from the platform C
  implementation.  As a result, the specific exceptions raised in
  error cases (and even whether some arguments are considered to be
  exceptional at all) are not defined in any useful cross-platform or
  cross-release way.  For example, whether ``math.log(0)`` returns
  ``-Inf`` or raises ``ValueError`` or ``OverflowError`` isn't
  defined, and in cases where ``math.log(0)`` raises
  ``OverflowError``, ``math.log(0L)`` may raise ``ValueError``
  instead.All functions return a quiet *NaN* if at least one of the
  args is *NaN*. Signaling *NaN*s raise an exception. The exception
  type still depends on the platform and libm implementation. It's
  usually ``ValueError`` for *EDOM* and ``OverflowError`` for errno
  *ERANGE*.

See also:

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