:mod:`math` --- Mathematical functions

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

.. module:: math

   :synopsis: Mathematical functions (sin() etc.).

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 :mod:`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:

.. function:: ceil(x)

   Return the ceiling of *x* as a float, the smallest integer value greater than or

   equal to *x*.

.. function:: 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*.

   .. versionadded:: 2.6

.. function:: fabs(x)

   Return the absolute value of *x*.

.. function:: factorial(x)

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

   is negative.

   .. versionadded:: 2.6

.. function:: floor(x)

   Return the floor of *x* as a float, the largest integer value less than or equal

   to *x*.

   .. versionchanged:: 2.6

      Added :meth:`__floor__` delegation.

.. function:: 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 :func:`fmod` is generally preferred when working with

   floats, while Python's ``x % y`` is preferred when working with integers.

.. function:: 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.

.. function:: fsum(iterable)

   Return an accurate floating point sum of values in the iterable.  Avoids

   loss of precision by tracking multiple intermediate partial sums.  The

   algorithm's accuracy depends on IEEE-754 arithmetic guarantees and the

   typical case where the rounding mode is half-even.

   .. note::

      The accuracy of fsum() may be impaired on builds that use

      extended precision addition and then double-round the results.

   .. versionadded:: 2.6

.. function:: isinf(x)

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

   .. versionadded:: 2.6

.. function:: 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.

   .. versionadded:: 2.6

.. function:: ldexp(x, i)

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

   :func:`frexp`.

.. function:: modf(x)

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

   *x*, and both are floats.

.. function:: trunc(x)

   Return the :class:`Real` value *x* truncated to an :class:`Integral` (usually

   a long integer). Delegates to ``x.__trunc__()``.

   .. versionadded:: 2.6

Note that :func:`frexp` and :func:`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 :func:`ceil`, :func:`floor`, and :func:`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:

.. function:: exp(x)

   Return ``e**x``.

.. function:: 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*).

   .. versionchanged:: 2.3

      *base* argument added.

.. function:: 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.

   .. versionadded:: 2.6

.. function:: log10(x)

   Return the base-10 logarithm of *x*.

.. function:: 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 :exc:`ValueError`.

   .. versionchanged:: 2.6

      The outcome of ``1**nan`` and ``nan**0`` was undefined.

.. function:: sqrt(x)

   Return the square root of *x*.

Trigonometric functions:

.. function:: acos(x)

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

.. function:: asin(x)

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

.. function:: atan(x)

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

.. function:: 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 :func:`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``.

.. function:: cos(x)

   Return the cosine of *x* radians.

.. function:: 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)``.

.. function:: sin(x)

   Return the sine of *x* radians.

.. function:: tan(x)

   Return the tangent of *x* radians.

Angular conversion:

.. function:: degrees(x)

   Converts angle *x* from radians to degrees.

.. function:: radians(x)

   Converts angle *x* from degrees to radians.

Hyperbolic functions:

.. function:: acosh(x)

   Return the inverse hyperbolic cosine of *x*.

   .. versionadded:: 2.6

.. function:: asinh(x)

   Return the inverse hyperbolic sine of *x*.

   .. versionadded:: 2.6

.. function:: atanh(x)

   Return the inverse hyperbolic tangent of *x*.

   .. versionadded:: 2.6

.. function:: cosh(x)

   Return the hyperbolic cosine of *x*.

.. function:: sinh(x)

   Return the hyperbolic sine of *x*.

.. function:: tanh(x)

   Return the hyperbolic tangent of *x*.

The module also defines two mathematical constants:

.. data:: pi

   The mathematical constant *pi*.

.. data:: e

   The mathematical constant *e*.

.. note::

   The :mod:`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 :exc:`ValueError` or

   :exc:`OverflowError` isn't defined, and in cases where ``math.log(0)`` raises

   :exc:`OverflowError`, ``math.log(0L)`` may raise :exc:`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 :exc:`ValueError` for *EDOM*

   and :exc:`OverflowError` for errno *ERANGE*.

   .. versionchanged:: 2.6

      In earlier versions of Python the outcome of an operation with NaN as

      input depended on platform and libm implementation.

.. seealso::

   Module :mod:`cmath`

      Complex number versions of many of these functions.