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662 lines
20 KiB
ReStructuredText
:mod:`math` --- Mathematical functions
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======================================
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.. module:: math
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:synopsis: Mathematical functions (sin() etc.).
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.. testsetup::
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from math import fsum
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--------------
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This module provides access to the mathematical functions defined by the C
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standard.
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These functions cannot be used with complex numbers; use the functions of the
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same name from the :mod:`cmath` module if you require support for complex
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numbers. The distinction between functions which support complex numbers and
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those which don't is made since most users do not want to learn quite as much
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mathematics as required to understand complex numbers. Receiving an exception
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instead of a complex result allows earlier detection of the unexpected complex
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number used as a parameter, so that the programmer can determine how and why it
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was generated in the first place.
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The following functions are provided by this module. Except when explicitly
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noted otherwise, all return values are floats.
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Number-theoretic and representation functions
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---------------------------------------------
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.. function:: ceil(x)
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Return the ceiling of *x*, the smallest integer greater than or equal to *x*.
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If *x* is not a float, delegates to ``x.__ceil__()``, which should return an
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:class:`~numbers.Integral` value.
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.. function:: comb(n, k)
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Return the number of ways to choose *k* items from *n* items without repetition
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and without order.
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Evaluates to ``n! / (k! * (n - k)!)`` when ``k <= n`` and evaluates
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to zero when ``k > n``.
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Also called the binomial coefficient because it is equivalent
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to the coefficient of k-th term in polynomial expansion of the
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expression ``(1 + x) ** n``.
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Raises :exc:`TypeError` if either of the arguments are not integers.
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Raises :exc:`ValueError` if either of the arguments are negative.
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.. versionadded:: 3.8
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.. function:: copysign(x, y)
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Return a float with the magnitude (absolute value) of *x* but the sign of
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*y*. On platforms that support signed zeros, ``copysign(1.0, -0.0)``
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returns *-1.0*.
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.. function:: fabs(x)
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Return the absolute value of *x*.
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.. function:: factorial(x)
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Return *x* factorial as an integer. Raises :exc:`ValueError` if *x* is not integral or
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is negative.
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.. deprecated:: 3.9
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Accepting floats with integral values (like ``5.0``) is deprecated.
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.. function:: floor(x)
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Return the floor of *x*, the largest integer less than or equal to *x*.
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If *x* is not a float, delegates to ``x.__floor__()``, which should return an
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:class:`~numbers.Integral` value.
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.. function:: fmod(x, y)
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Return ``fmod(x, y)``, as defined by the platform C library. Note that the
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Python expression ``x % y`` may not return the same result. The intent of the C
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standard is that ``fmod(x, y)`` be exactly (mathematically; to infinite
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precision) equal to ``x - n*y`` for some integer *n* such that the result has
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the same sign as *x* and magnitude less than ``abs(y)``. Python's ``x % y``
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returns a result with the sign of *y* instead, and may not be exactly computable
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for float arguments. For example, ``fmod(-1e-100, 1e100)`` is ``-1e-100``, but
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the result of Python's ``-1e-100 % 1e100`` is ``1e100-1e-100``, which cannot be
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represented exactly as a float, and rounds to the surprising ``1e100``. For
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this reason, function :func:`fmod` is generally preferred when working with
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floats, while Python's ``x % y`` is preferred when working with integers.
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.. function:: frexp(x)
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Return the mantissa and exponent of *x* as the pair ``(m, e)``. *m* is a float
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and *e* is an integer such that ``x == m * 2**e`` exactly. If *x* is zero,
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returns ``(0.0, 0)``, otherwise ``0.5 <= abs(m) < 1``. This is used to "pick
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apart" the internal representation of a float in a portable way.
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.. function:: fsum(iterable)
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Return an accurate floating point sum of values in the iterable. Avoids
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loss of precision by tracking multiple intermediate partial sums::
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>>> sum([.1, .1, .1, .1, .1, .1, .1, .1, .1, .1])
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0.9999999999999999
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>>> fsum([.1, .1, .1, .1, .1, .1, .1, .1, .1, .1])
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1.0
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The algorithm's accuracy depends on IEEE-754 arithmetic guarantees and the
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typical case where the rounding mode is half-even. On some non-Windows
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builds, the underlying C library uses extended precision addition and may
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occasionally double-round an intermediate sum causing it to be off in its
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least significant bit.
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For further discussion and two alternative approaches, see the `ASPN cookbook
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recipes for accurate floating point summation
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<https://code.activestate.com/recipes/393090/>`_\.
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.. function:: gcd(*integers)
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Return the greatest common divisor of the specified integer arguments.
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If any of the arguments is nonzero, then the returned value is the largest
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positive integer that is a divisor of all arguments. If all arguments
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are zero, then the returned value is ``0``. ``gcd()`` without arguments
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returns ``0``.
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.. versionadded:: 3.5
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.. versionchanged:: 3.9
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Added support for an arbitrary number of arguments. Formerly, only two
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arguments were supported.
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.. function:: isclose(a, b, *, rel_tol=1e-09, abs_tol=0.0)
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Return ``True`` if the values *a* and *b* are close to each other and
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``False`` otherwise.
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Whether or not two values are considered close is determined according to
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given absolute and relative tolerances.
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*rel_tol* is the relative tolerance -- it is the maximum allowed difference
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between *a* and *b*, relative to the larger absolute value of *a* or *b*.
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For example, to set a tolerance of 5%, pass ``rel_tol=0.05``. The default
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tolerance is ``1e-09``, which assures that the two values are the same
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within about 9 decimal digits. *rel_tol* must be greater than zero.
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*abs_tol* is the minimum absolute tolerance -- useful for comparisons near
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zero. *abs_tol* must be at least zero.
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If no errors occur, the result will be:
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``abs(a-b) <= max(rel_tol * max(abs(a), abs(b)), abs_tol)``.
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The IEEE 754 special values of ``NaN``, ``inf``, and ``-inf`` will be
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handled according to IEEE rules. Specifically, ``NaN`` is not considered
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close to any other value, including ``NaN``. ``inf`` and ``-inf`` are only
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considered close to themselves.
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.. versionadded:: 3.5
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.. seealso::
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:pep:`485` -- A function for testing approximate equality
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.. function:: isfinite(x)
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Return ``True`` if *x* is neither an infinity nor a NaN, and
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``False`` otherwise. (Note that ``0.0`` *is* considered finite.)
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.. versionadded:: 3.2
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.. function:: isinf(x)
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Return ``True`` if *x* is a positive or negative infinity, and
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``False`` otherwise.
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.. function:: isnan(x)
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Return ``True`` if *x* is a NaN (not a number), and ``False`` otherwise.
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.. function:: isqrt(n)
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Return the integer square root of the nonnegative integer *n*. This is the
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floor of the exact square root of *n*, or equivalently the greatest integer
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*a* such that *a*\ ² |nbsp| ≤ |nbsp| *n*.
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For some applications, it may be more convenient to have the least integer
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*a* such that *n* |nbsp| ≤ |nbsp| *a*\ ², or in other words the ceiling of
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the exact square root of *n*. For positive *n*, this can be computed using
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``a = 1 + isqrt(n - 1)``.
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.. versionadded:: 3.8
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.. function:: lcm(*integers)
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Return the least common multiple of the specified integer arguments.
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If all arguments are nonzero, then the returned value is the smallest
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positive integer that is a multiple of all arguments. If any of the arguments
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is zero, then the returned value is ``0``. ``lcm()`` without arguments
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returns ``1``.
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.. versionadded:: 3.9
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.. function:: ldexp(x, i)
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Return ``x * (2**i)``. This is essentially the inverse of function
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:func:`frexp`.
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.. function:: modf(x)
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Return the fractional and integer parts of *x*. Both results carry the sign
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of *x* and are floats.
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.. function:: nextafter(x, y)
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Return the next floating-point value after *x* towards *y*.
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If *x* is equal to *y*, return *y*.
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Examples:
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* ``math.nextafter(x, math.inf)`` goes up: towards positive infinity.
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* ``math.nextafter(x, -math.inf)`` goes down: towards minus infinity.
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* ``math.nextafter(x, 0.0)`` goes towards zero.
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* ``math.nextafter(x, math.copysign(math.inf, x))`` goes away from zero.
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See also :func:`math.ulp`.
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.. versionadded:: 3.9
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.. function:: perm(n, k=None)
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Return the number of ways to choose *k* items from *n* items
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without repetition and with order.
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Evaluates to ``n! / (n - k)!`` when ``k <= n`` and evaluates
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to zero when ``k > n``.
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If *k* is not specified or is None, then *k* defaults to *n*
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and the function returns ``n!``.
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Raises :exc:`TypeError` if either of the arguments are not integers.
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Raises :exc:`ValueError` if either of the arguments are negative.
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.. versionadded:: 3.8
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.. function:: prod(iterable, *, start=1)
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Calculate the product of all the elements in the input *iterable*.
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The default *start* value for the product is ``1``.
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When the iterable is empty, return the start value. This function is
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intended specifically for use with numeric values and may reject
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non-numeric types.
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.. versionadded:: 3.8
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.. function:: remainder(x, y)
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Return the IEEE 754-style remainder of *x* with respect to *y*. For
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finite *x* and finite nonzero *y*, this is the difference ``x - n*y``,
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where ``n`` is the closest integer to the exact value of the quotient ``x /
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y``. If ``x / y`` is exactly halfway between two consecutive integers, the
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nearest *even* integer is used for ``n``. The remainder ``r = remainder(x,
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y)`` thus always satisfies ``abs(r) <= 0.5 * abs(y)``.
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Special cases follow IEEE 754: in particular, ``remainder(x, math.inf)`` is
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*x* for any finite *x*, and ``remainder(x, 0)`` and
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``remainder(math.inf, x)`` raise :exc:`ValueError` for any non-NaN *x*.
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If the result of the remainder operation is zero, that zero will have
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the same sign as *x*.
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On platforms using IEEE 754 binary floating-point, the result of this
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operation is always exactly representable: no rounding error is introduced.
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.. versionadded:: 3.7
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.. function:: trunc(x)
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Return the :class:`~numbers.Real` value *x* truncated to an
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:class:`~numbers.Integral` (usually an integer). Delegates to
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:meth:`x.__trunc__() <object.__trunc__>`.
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.. function:: ulp(x)
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Return the value of the least significant bit of the float *x*:
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* If *x* is a NaN (not a number), return *x*.
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* If *x* is negative, return ``ulp(-x)``.
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* If *x* is a positive infinity, return *x*.
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* If *x* is equal to zero, return the smallest positive
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*denormalized* representable float (smaller than the minimum positive
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*normalized* float, :data:`sys.float_info.min <sys.float_info>`).
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* If *x* is equal to the largest positive representable float,
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return the value of the least significant bit of *x*, such that the first
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float smaller than *x* is ``x - ulp(x)``.
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* Otherwise (*x* is a positive finite number), return the value of the least
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significant bit of *x*, such that the first float bigger than *x*
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is ``x + ulp(x)``.
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ULP stands for "Unit in the Last Place".
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See also :func:`math.nextafter` and :data:`sys.float_info.epsilon
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<sys.float_info>`.
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.. versionadded:: 3.9
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Note that :func:`frexp` and :func:`modf` have a different call/return pattern
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than their C equivalents: they take a single argument and return a pair of
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values, rather than returning their second return value through an 'output
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parameter' (there is no such thing in Python).
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For the :func:`ceil`, :func:`floor`, and :func:`modf` functions, note that *all*
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floating-point numbers of sufficiently large magnitude are exact integers.
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Python floats typically carry no more than 53 bits of precision (the same as the
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platform C double type), in which case any float *x* with ``abs(x) >= 2**52``
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necessarily has no fractional bits.
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Power and logarithmic functions
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-------------------------------
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.. function:: exp(x)
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Return *e* raised to the power *x*, where *e* = 2.718281... is the base
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of natural logarithms. This is usually more accurate than ``math.e ** x``
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or ``pow(math.e, x)``.
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.. function:: expm1(x)
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Return *e* raised to the power *x*, minus 1. Here *e* is the base of natural
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logarithms. For small floats *x*, the subtraction in ``exp(x) - 1``
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can result in a `significant loss of precision
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<https://en.wikipedia.org/wiki/Loss_of_significance>`_\; the :func:`expm1`
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function provides a way to compute this quantity to full precision::
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>>> from math import exp, expm1
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>>> exp(1e-5) - 1 # gives result accurate to 11 places
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1.0000050000069649e-05
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>>> expm1(1e-5) # result accurate to full precision
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1.0000050000166668e-05
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.. versionadded:: 3.2
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.. function:: log(x[, base])
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With one argument, return the natural logarithm of *x* (to base *e*).
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With two arguments, return the logarithm of *x* to the given *base*,
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calculated as ``log(x)/log(base)``.
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.. function:: log1p(x)
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Return the natural logarithm of *1+x* (base *e*). The
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result is calculated in a way which is accurate for *x* near zero.
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.. function:: log2(x)
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Return the base-2 logarithm of *x*. This is usually more accurate than
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``log(x, 2)``.
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.. versionadded:: 3.3
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.. seealso::
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:meth:`int.bit_length` returns the number of bits necessary to represent
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an integer in binary, excluding the sign and leading zeros.
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.. function:: log10(x)
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Return the base-10 logarithm of *x*. This is usually more accurate
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than ``log(x, 10)``.
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.. function:: pow(x, y)
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Return ``x`` raised to the power ``y``. Exceptional cases follow
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Annex 'F' of the C99 standard as far as possible. In particular,
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``pow(1.0, x)`` and ``pow(x, 0.0)`` always return ``1.0``, even
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when ``x`` is a zero or a NaN. If both ``x`` and ``y`` are finite,
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``x`` is negative, and ``y`` is not an integer then ``pow(x, y)``
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is undefined, and raises :exc:`ValueError`.
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Unlike the built-in ``**`` operator, :func:`math.pow` converts both
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its arguments to type :class:`float`. Use ``**`` or the built-in
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:func:`pow` function for computing exact integer powers.
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.. function:: sqrt(x)
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Return the square root of *x*.
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Trigonometric functions
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-----------------------
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.. function:: acos(x)
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Return the arc cosine of *x*, in radians. The result is between ``0`` and
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``pi``.
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.. function:: asin(x)
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Return the arc sine of *x*, in radians. The result is between ``-pi/2`` and
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``pi/2``.
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.. function:: atan(x)
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Return the arc tangent of *x*, in radians. The result is between ``-pi/2`` and
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``pi/2``.
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.. function:: atan2(y, x)
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Return ``atan(y / x)``, in radians. The result is between ``-pi`` and ``pi``.
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The vector in the plane from the origin to point ``(x, y)`` makes this angle
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with the positive X axis. The point of :func:`atan2` is that the signs of both
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inputs are known to it, so it can compute the correct quadrant for the angle.
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For example, ``atan(1)`` and ``atan2(1, 1)`` are both ``pi/4``, but ``atan2(-1,
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-1)`` is ``-3*pi/4``.
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.. function:: cos(x)
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Return the cosine of *x* radians.
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.. function:: dist(p, q)
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Return the Euclidean distance between two points *p* and *q*, each
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given as a sequence (or iterable) of coordinates. The two points
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must have the same dimension.
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Roughly equivalent to::
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sqrt(sum((px - qx) ** 2.0 for px, qx in zip(p, q)))
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.. versionadded:: 3.8
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.. function:: hypot(*coordinates)
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Return the Euclidean norm, ``sqrt(sum(x**2 for x in coordinates))``.
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This is the length of the vector from the origin to the point
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given by the coordinates.
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For a two dimensional point ``(x, y)``, this is equivalent to computing
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the hypotenuse of a right triangle using the Pythagorean theorem,
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``sqrt(x*x + y*y)``.
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.. versionchanged:: 3.8
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Added support for n-dimensional points. Formerly, only the two
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dimensional case was supported.
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.. versionchanged:: 3.10
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Improved the algorithm's accuracy so that the maximum error is
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under 1 ulp (unit in the last place). More typically, the result
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is almost always correctly rounded to within 1/2 ulp.
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.. function:: sin(x)
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Return the sine of *x* radians.
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.. function:: tan(x)
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Return the tangent of *x* radians.
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Angular conversion
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------------------
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.. function:: degrees(x)
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Convert angle *x* from radians to degrees.
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.. function:: radians(x)
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Convert angle *x* from degrees to radians.
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Hyperbolic functions
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--------------------
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`Hyperbolic functions <https://en.wikipedia.org/wiki/Hyperbolic_function>`_
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are analogs of trigonometric functions that are based on hyperbolas
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instead of circles.
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.. function:: acosh(x)
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Return the inverse hyperbolic cosine of *x*.
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.. function:: asinh(x)
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|
Return the inverse hyperbolic sine of *x*.
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.. function:: atanh(x)
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|
Return the inverse hyperbolic tangent of *x*.
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.. function:: cosh(x)
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|
Return the hyperbolic cosine of *x*.
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.. function:: sinh(x)
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Return the hyperbolic sine of *x*.
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.. function:: tanh(x)
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|
Return the hyperbolic tangent of *x*.
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Special functions
|
|
-----------------
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|
|
|
.. function:: erf(x)
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|
|
|
Return the `error function <https://en.wikipedia.org/wiki/Error_function>`_ at
|
|
*x*.
|
|
|
|
The :func:`erf` function can be used to compute traditional statistical
|
|
functions such as the `cumulative standard normal distribution
|
|
<https://en.wikipedia.org/wiki/Normal_distribution#Cumulative_distribution_function>`_::
|
|
|
|
def phi(x):
|
|
'Cumulative distribution function for the standard normal distribution'
|
|
return (1.0 + erf(x / sqrt(2.0))) / 2.0
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|
|
|
.. versionadded:: 3.2
|
|
|
|
|
|
.. function:: erfc(x)
|
|
|
|
Return the complementary error function at *x*. The `complementary error
|
|
function <https://en.wikipedia.org/wiki/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
|
|
<https://en.wikipedia.org/wiki/Loss_of_significance>`_\.
|
|
|
|
.. versionadded:: 3.2
|
|
|
|
|
|
.. function:: gamma(x)
|
|
|
|
Return the `Gamma function <https://en.wikipedia.org/wiki/Gamma_function>`_ at
|
|
*x*.
|
|
|
|
.. versionadded:: 3.2
|
|
|
|
|
|
.. function:: lgamma(x)
|
|
|
|
Return the natural logarithm of the absolute value of the Gamma
|
|
function at *x*.
|
|
|
|
.. versionadded:: 3.2
|
|
|
|
|
|
Constants
|
|
---------
|
|
|
|
.. data:: pi
|
|
|
|
The mathematical constant *π* = 3.141592..., to available precision.
|
|
|
|
|
|
.. data:: e
|
|
|
|
The mathematical constant *e* = 2.718281..., to available precision.
|
|
|
|
|
|
.. data:: 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 <https://www.youtube.com/watch?v=jG7vhMMXagQ>`_, and start celebrating
|
|
`Tau day <https://tauday.com/>`_ by eating twice as much pie!
|
|
|
|
.. versionadded:: 3.6
|
|
|
|
|
|
.. data:: inf
|
|
|
|
A floating-point positive infinity. (For negative infinity, use
|
|
``-math.inf``.) Equivalent to the output of ``float('inf')``.
|
|
|
|
.. versionadded:: 3.5
|
|
|
|
|
|
.. data:: nan
|
|
|
|
A floating-point "not a number" (NaN) value. Equivalent to the output of
|
|
``float('nan')``.
|
|
|
|
.. versionadded:: 3.5
|
|
|
|
|
|
.. impl-detail::
|
|
|
|
The :mod:`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
|
|
:exc:`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 :exc:`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.
|
|
|
|
|
|
.. seealso::
|
|
|
|
Module :mod:`cmath`
|
|
Complex number versions of many of these functions.
|
|
|
|
.. |nbsp| unicode:: 0xA0
|
|
:trim:
|