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

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

This module 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.comb(n, k)

   Return the number of ways to choose *k* items from *n* items
   without repetition and without order.

   Evaluates to "n! / (k! * (n - k)!)" when "k <= n" and evaluates to
   zero when "k > n".

   Also called the binomial coefficient because it is equivalent to
   the coefficient of k-th term in polynomial expansion of "(1 + x)ⁿ".

   Raises "TypeError" if either of the arguments are not integers.
   Raises "ValueError" if either of the arguments are negative.

   Added in version 3.8.

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(n)

   Return *n* factorial as an integer.  Raises "ValueError" if *n* is
   not integral or is negative.

   Changed in version 3.10: Floats with integral values (like "5.0")
   are no longer accepted.

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.fma(x, y, z)

   Fused multiply-add operation. Return "(x * y) + z", computed as
   though with infinite precision and range followed by a single round
   to the "float" format. This operation often provides better
   accuracy than the direct expression "(x * y) + z".

   This function follows the specification of the fusedMultiplyAdd
   operation described in the IEEE 754 standard. The standard leaves
   one case implementation-defined, namely the result of "fma(0, inf,
   nan)" and "fma(inf, 0, nan)". In these cases, "math.fma" returns a
   NaN, and does not raise any exception.

   Added in version 3.13.

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.

   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(*integers)

   Return the greatest common divisor of the specified integer
   arguments. If any of the arguments is nonzero, then the returned
   value is the largest positive integer that is a divisor of all
   arguments.  If all arguments are zero, then the returned value is
   "0".  "gcd()" without arguments returns "0".

   Added in version 3.5.

   Changed in version 3.9: Added support for an arbitrary number of
   arguments. Formerly, only two arguments were supported.

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.

   Added in version 3.5.

   See also:

     **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.)

   Added 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.isqrt(n)

   Return the integer square root of the nonnegative integer *n*. This
   is the floor of the exact square root of *n*, or equivalently the
   greatest integer *a* such that *a*² ≤ *n*.

   For some applications, it may be more convenient to have the least
   integer *a* such that *n* ≤ *a*², or in other words the ceiling of
   the exact square root of *n*. For positive *n*, this can be
   computed using "a = 1 + isqrt(n - 1)".

   Added in version 3.8.

math.lcm(*integers)

   Return the least common multiple of the specified integer
   arguments. If all arguments are nonzero, then the returned value is
   the smallest positive integer that is a multiple of all arguments.
   If any of the arguments is zero, then the returned value is "0".
   "lcm()" without arguments returns "1".

   Added in version 3.9.

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.nextafter(x, y, steps=1)

   Return the floating-point value *steps* steps after *x* towards
   *y*.

   If *x* is equal to *y*, return *y*, unless *steps* is zero.

   Examples:

   * "math.nextafter(x, math.inf)" goes up: towards positive infinity.

   * "math.nextafter(x, -math.inf)" goes down: towards minus infinity.

   * "math.nextafter(x, 0.0)" goes towards zero.

   * "math.nextafter(x, math.copysign(math.inf, x))" goes away from
     zero.

   See also "math.ulp()".

   Added in version 3.9.

   Changed in version 3.12: Added the *steps* argument.

math.perm(n, k=None)

   Return the number of ways to choose *k* items from *n* items
   without repetition and with order.

   Evaluates to "n! / (n - k)!" when "k <= n" and evaluates to zero
   when "k > n".

   If *k* is not specified or is None, then *k* defaults to *n* and
   the function returns "n!".

   Raises "TypeError" if either of the arguments are not integers.
   Raises "ValueError" if either of the arguments are negative.

   Added in version 3.8.

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

   Calculate the product of all the elements in the input *iterable*.
   The default *start* value for the product is "1".

   When the iterable is empty, return the start value.  This function
   is intended specifically for use with numeric values and may reject
   non-numeric types.

   Added in version 3.8.

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.

   Added in version 3.7.

math.sumprod(p, q)

   Return the sum of products of values from two iterables *p* and
   *q*.

   Raises "ValueError" if the inputs do not have the same length.

   Roughly equivalent to:

      sum(itertools.starmap(operator.mul, zip(p, q, strict=True)))

   For float and mixed int/float inputs, the intermediate products and
   sums are computed with extended precision.

   Added in version 3.12.

math.trunc(x)

   Return *x* with the fractional part removed, leaving the integer
   part.  This rounds toward 0: "trunc()" is equivalent to "floor()"
   for positive *x*, and equivalent to "ceil()" for negative *x*. If
   *x* is not a float, delegates to "x.__trunc__", which should return
   an "Integral" value.

math.ulp(x)

   Return the value of the least significant bit of the float *x*:

   * If *x* is a NaN (not a number), return *x*.

   * If *x* is negative, return "ulp(-x)".

   * If *x* is a positive infinity, return *x*.

   * If *x* is equal to zero, return the smallest positive
     *denormalized* representable float (smaller than the minimum
     positive *normalized* float, "sys.float_info.min").

   * If *x* is equal to the largest positive representable float,
     return the value of the least significant bit of *x*, such that
     the first float smaller than *x* is "x - ulp(x)".

   * Otherwise (*x* is a positive finite number), return the value of
     the least significant bit of *x*, such that the first float
     bigger than *x* is "x + ulp(x)".

   ULP stands for "Unit in the Last Place".

   See also "math.nextafter()" and "sys.float_info.epsilon".

   Added in version 3.9.

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.cbrt(x)

   Return the cube root of *x*.

   Added in version 3.11.

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.exp2(x)

   Return *2* raised to the power *x*.

   Added in version 3.11.

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

   Added 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.log2(x)

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

   Added in version 3.3.

   See also:

     "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 the
   IEEE 754 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.

   Changed in version 3.11: The special cases "pow(0.0, -inf)" and
   "pow(-0.0, -inf)" were changed to return "inf" instead of raising
   "ValueError", for consistency with IEEE 754.

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)

   Return the Euclidean distance between two points *p* and *q*, each
   given as a sequence (or iterable) of coordinates.  The two points
   must have the same dimension.

   Roughly equivalent to:

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

   Added in version 3.8.

math.hypot(*coordinates)

   Return the Euclidean norm, "sqrt(sum(x**2 for x in coordinates))".
   This is the length of the vector from the origin to the point given
   by the coordinates.

   For a two dimensional point "(x, y)", this is equivalent to
   computing the hypotenuse of a right triangle using the Pythagorean
   theorem, "sqrt(x*x + y*y)".

   Changed in version 3.8: Added support for n-dimensional points.
   Formerly, only the two dimensional case was supported.

   Changed in version 3.10: Improved the algorithm's accuracy so that
   the maximum error is under 1 ulp (unit in the last place).  More
   typically, the result is almost always correctly rounded to within
   1/2 ulp.

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

   Added 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.

   Added in version 3.2.

math.gamma(x)

   Return the Gamma function at *x*.

   Added in version 3.2.

math.lgamma(x)

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

   Added 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.

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!

   Added in version 3.6.

math.inf

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

   Added in version 3.5.

math.nan

   A floating-point "not a number" (NaN) value. Equivalent to the
   output of "float('nan')". Due to the requirements of the IEEE-754
   standard, "math.nan" and "float('nan')" are not considered to equal
   to any other numeric value, including themselves. To check whether
   a number is a NaN, use the "isnan()" function to test for NaNs
   instead of "is" or "==". Example:

   >>> import math
   >>> math.nan == math.nan
   False
   >>> float('nan') == float('nan')
   False
   >>> math.isnan(math.nan)
   True
   >>> math.isnan(float('nan'))
   True

   Added in version 3.5.

   Changed in version 3.11: It is now always available.

**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.
