Doc/library/math.rst
!math --- Mathematical functions.. module:: math :synopsis: Mathematical functions (sin() etc.).
.. testsetup::
from math import fsum
This module provides access to common mathematical functions and constants, including those 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.
:func:ceil(x) <ceil> Ceiling of x, the smallest integer greater than or equal to x
:func:fabs(x) <fabs> Absolute value of x
:func:floor(x) <floor> Floor of x, the largest integer less than or equal to x
:func:fma(x, y, z) <fma> Fused multiply-add operation: (x * y) + z
:func:fmax(x, y) <fmax> Maximum of two floating-point values
:func:fmin(x, y) <fmin> Minimum of two floating-point values
:func:fmod(x, y) <fmod> Remainder of division x / y
:func:modf(x) <modf> Fractional and integer parts of x
:func:remainder(x, y) <remainder> Remainder of x with respect to y
:func:trunc(x) <trunc> Integer part of x
:func:copysign(x, y) <copysign> Magnitude (absolute value) of x with the sign of y
:func:frexp(x) <frexp> Mantissa and exponent of x
:func:isclose(a, b, rel_tol, abs_tol) <isclose> Check if the values a and b are close to each other
:func:isfinite(x) <isfinite> Check if x is neither an infinity nor a NaN
:func:isnormal(x) <isnormal> Check if x is a normal number
:func:issubnormal(x) <issubnormal> Check if x is a subnormal number
:func:isinf(x) <isinf> Check if x is a positive or negative infinity
:func:isnan(x) <isnan> Check if x is a NaN (not a number)
:func:ldexp(x, i) <ldexp> x * (2**i), inverse of function :func:frexp
:func:nextafter(x, y, steps) <nextafter> Floating-point value steps steps after x towards y
:func:signbit(x) <signbit> Check if x is a negative number
:func:ulp(x) <ulp> Value of the least significant bit of x
:func:cbrt(x) <cbrt> Cube root of x
:func:exp(x) <exp> e raised to the power x
:func:exp2(x) <exp2> 2 raised to the power x
:func:expm1(x) <expm1> e raised to the power x, minus 1
:func:log(x, base) <log> Logarithm of x to the given base (e by default)
:func:log1p(x) <log1p> Natural logarithm of 1+x (base e)
:func:log2(x) <log2> Base-2 logarithm of x
:func:log10(x) <log10> Base-10 logarithm of x
:func:pow(x, y) <math.pow> x raised to the power y
:func:sqrt(x) <sqrt> Square root of x
:func:dist(p, q) <dist> Euclidean distance between two points p and q given as an iterable of coordinates
:func:fsum(iterable) <fsum> Sum of values in the input iterable
:func:hypot(*coordinates) <hypot> Euclidean norm of an iterable of coordinates
:func:prod(iterable, start) <prod> Product of elements in the input iterable with a start value
:func:sumprod(p, q) <sumprod> Sum of products from two iterables p and q
:func:degrees(x) <degrees> Convert angle x from radians to degrees
:func:radians(x) <radians> Convert angle x from degrees to radians
:func:acos(x) <acos> Arc cosine of x
:func:asin(x) <asin> Arc sine of x
:func:atan(x) <atan> Arc tangent of x
:func:atan2(y, x) <atan2> atan(y / x)
:func:cos(x) <cos> Cosine of x
:func:sin(x) <sin> Sine of x
:func:tan(x) <tan> Tangent of x
:func:acosh(x) <acosh> Inverse hyperbolic cosine of x
:func:asinh(x) <asinh> Inverse hyperbolic sine of x
:func:atanh(x) <atanh> Inverse hyperbolic tangent of x
:func:cosh(x) <cosh> Hyperbolic cosine of x
:func:sinh(x) <sinh> Hyperbolic sine of x
:func:tanh(x) <tanh> Hyperbolic tangent of x
:func:erf(x) <erf> Error function <https://en.wikipedia.org/wiki/Error_function>_ at x
:func:erfc(x) <erfc> Complementary error function <https://en.wikipedia.org/wiki/Error_function>_ at x
:func:gamma(x) <gamma> Gamma function <https://en.wikipedia.org/wiki/Gamma_function>_ at x
:func:lgamma(x) <lgamma> Natural logarithm of the absolute value of the Gamma function <https://en.wikipedia.org/wiki/Gamma_function>_ at x
:data:pi π = 3.141592...
:data:e e = 2.718281...
:data:tau τ = 2\ π = 6.283185...
:data:inf Positive infinity
:data:nan "Not a number" (NaN)
==================================================== ============================================
.. function:: ceil(x)
Return the ceiling of x, the smallest integer greater than or equal to x.
If x is not a float, delegates to :meth:x.__ceil__ <object.__ceil__>,
which should return an :class:~numbers.Integral value.
.. function:: fabs(x)
Return the absolute value of x.
.. function:: floor(x)
Return the floor of x, the largest integer less than or equal to x. If
x is not a float, delegates to :meth:x.__floor__ <object.__floor__>, which
should return an :class:~numbers.Integral value.
.. function:: 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.
.. versionadded:: 3.13
.. function:: fmax(x, y)
Get the larger of two floating-point values, treating NaNs as missing data.
When both operands are (signed) NaNs or zeroes, return nan and 0
respectively and the sign of the result is implementation-defined, that
is, :func:!fmax is not required to be sensitive to the sign of such
operands (see Annex F of the C11 standard, §F.10.0.3 and §F.10.9.2).
.. versionadded:: 3.15
.. function:: fmin(x, y)
Get the smaller of two floating-point values, treating NaNs as missing data.
When both operands are (signed) NaNs or zeroes, return nan and 0
respectively and the sign of the result is implementation-defined, that
is, :func:!fmin is not required to be sensitive to the sign of such
operands (see Annex F of the C11 standard, §F.10.0.3 and §F.10.9.3).
.. versionadded:: 3.15
.. function:: fmod(x, y)
Return the floating-point remainder of x / y,
as defined by the platform C library function fmod(x, y). 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:: modf(x)
Return the fractional and integer parts of x. Both results carry the sign of x and are floats.
Note that :func:modf has a different call/return pattern
than its C equivalents: it takes a single argument and return a pair of
values, rather than returning its second return value through an 'output
parameter' (there is no such thing in Python).
.. function:: 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 :exc: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.
.. versionadded:: 3.7
.. function:: trunc(x)
Return x with the fractional part
removed, leaving the integer part. This rounds toward 0: trunc() is
equivalent to :func:floor for positive x, and equivalent to :func:ceil
for negative x. If x is not a float, delegates to :meth:x.__trunc__ <object.__trunc__>, which should return an :class:~numbers.Integral value.
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.
.. function:: 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.
.. 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.
Note that :func:frexp has a different call/return pattern
than its C equivalents: it takes a single argument and return a pair of
values, rather than returning its second return value through an 'output
parameter' (there is no such thing in Python).
.. function:: 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. If no errors occur, the result will
be: abs(a-b) <= max(rel_tol * max(abs(a), abs(b)), abs_tol).
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 nonnegative and less
than 1.0.
abs_tol is the absolute tolerance; it defaults to 0.0 and it must be
nonnegative. When comparing x to 0.0, isclose(x, 0) is computed
as abs(x) <= rel_tol * abs(x), which is False for any nonzero x and
rel_tol less than 1.0. So add an appropriate positive abs_tol argument
to the call.
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.
.. versionadded:: 3.5
.. seealso::
:pep:`485` -- A function for testing approximate equality
.. function:: isfinite(x)
Return True if x is neither an infinity nor a NaN, and
False otherwise. (Note that 0.0 is considered finite.)
.. versionadded:: 3.2
.. function:: isnormal(x)
Return True if x is a normal number, that is a finite
nonzero number that is not a subnormal (see :func:issubnormal).
Return False otherwise.
.. versionadded:: 3.15
.. function:: issubnormal(x)
Return True if x is a subnormal number, that is a finite
nonzero number with a magnitude smaller than :data:sys.float_info.min.
Return False otherwise.
.. versionadded:: 3.15
.. function:: isinf(x)
Return True if x is a positive or negative infinity, and
False otherwise.
.. function:: isnan(x)
Return True if x is a NaN (not a number), and False otherwise.
.. function:: ldexp(x, i)
Return x * (2**i). This is essentially the inverse of function
:func:frexp.
.. function:: 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 :func:math.ulp.
.. versionadded:: 3.9
.. versionchanged:: 3.12 Added the steps argument.
.. function:: signbit(x)
Return True if the sign of x is negative and False otherwise.
This is useful to detect the sign bit of zeroes, infinities and NaNs.
.. versionadded:: 3.15
.. function:: ulp(x)
Return the value of the least significant bit of the float x:
ulp(-x).sys.float_info.min <sys.float_info>).x - ulp(x).x + ulp(x).ULP stands for "Unit in the Last Place".
See also :func:math.nextafter and :data:sys.float_info.epsilon <sys.float_info>.
.. versionadded:: 3.9
.. function:: cbrt(x)
Return the cube root of x.
.. versionadded:: 3.11
.. function:: 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).
.. function:: exp2(x)
Return 2 raised to the power x.
.. versionadded:: 3.11
.. function:: 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 <https://en.wikipedia.org/wiki/Loss_of_significance>_; the :func: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
.. versionadded:: 3.2
.. function:: 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).
.. 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.
.. function:: log2(x)
Return the base-2 logarithm of x. This is usually more accurate than
log(x, 2).
.. versionadded:: 3.3
.. seealso::
:meth:`int.bit_length` returns the number of bits necessary to represent
an integer in binary, excluding the sign and leading zeros.
.. function:: log10(x)
Return the base-10 logarithm of x. This is usually more accurate
than log(x, 10).
.. function:: 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 :exc:ValueError.
Unlike the built-in ** operator, :func:math.pow converts both
its arguments to type :class:float. Use ** or the built-in
:func:pow function for computing exact integer powers.
.. versionchanged:: 3.11
The special cases pow(0.0, -inf) and pow(-0.0, -inf) were
changed to return inf instead of raising :exc:ValueError,
for consistency with IEEE 754.
.. function:: sqrt(x)
Return the square root of x.
.. function:: 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, strict=True)))
.. versionadded:: 3.8
.. 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. 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 <https://code.activestate.com/recipes/393090-binary-floating-point-summation-accurate-to-full-p/>_.
.. function:: 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).
.. versionchanged:: 3.8 Added support for n-dimensional points. Formerly, only the two dimensional case was supported.
.. versionchanged:: 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.
.. function:: 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.
.. versionadded:: 3.8
.. function:: sumprod(p, q)
Return the sum of products of values from two iterables p and q.
Raises :exc:ValueError if the inputs do not have the same length.
Roughly equivalent to::
sum(map(operator.mul, p, q, strict=True))
For float and mixed int/float inputs, the intermediate products and sums are computed with extended precision.
.. versionadded:: 3.12
.. function:: degrees(x)
Convert angle x from radians to degrees.
.. function:: radians(x)
Convert angle x from degrees to radians.
.. function:: acos(x)
Return the arc cosine of x, in radians. The result is between 0 and
pi.
.. function:: asin(x)
Return the arc sine of x, in radians. The result is between -pi/2 and
pi/2.
.. function:: atan(x)
Return the arc tangent of x, in radians. The result is between -pi/2 and
pi/2.
.. 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:: sin(x)
Return the sine of x radians.
.. function:: tan(x)
Return the tangent of x radians.
Hyperbolic functions <https://en.wikipedia.org/wiki/Hyperbolic_functions>_
are analogs of trigonometric functions that are based on hyperbolas
instead of circles.
.. function:: acosh(x)
Return the inverse hyperbolic cosine of x.
.. function:: asinh(x)
Return the inverse hyperbolic sine of x.
.. function:: atanh(x)
Return the inverse hyperbolic tangent of x.
.. 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.
.. function:: erf(x)
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/Cumulative_distribution_function>_::
def phi(x):
'Cumulative distribution function for the standard normal distribution'
return (1.0 + erf(x / sqrt(2.0))) / 2.0
.. 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
For backward compatibility, the :mod:!math module provides also aliases of
the following functions from the :mod:math.integer module:
.. list-table::
.. function:: comb(n, k) :no-typesetting:
:func:comb(n, k) <math.integer.comb>
Number of ways to choose k items from n items without repetition and without order
.. function:: factorial(n) :no-typesetting:
:func:factorial(n) <math.integer.factorial>
n factorial
.. function:: gcd(*integers) :no-typesetting:
:func:gcd(*integers) <math.integer.gcd>
Greatest common divisor of the integer arguments
.. function:: isqrt(n) :no-typesetting:
:func:isqrt(n) <math.integer.isqrt>
Integer square root of a nonnegative integer n
.. function:: lcm(*integers) :no-typesetting:
:func:lcm(*integers) <math.integer.lcm>
Least common multiple of the integer arguments
.. function:: perm(n, k) :no-typesetting:
:func:perm(n, k) <math.integer.perm>
Number of ways to choose k items from n items without repetition and with order
.. versionadded:: 3.5
The :func:gcd function.
.. versionadded:: 3.8
The :func:comb, :func:perm and :func:isqrt functions.
.. versionadded:: 3.9
The :func:lcm function.
.. versionchanged:: 3.9
Added support for an arbitrary number of arguments in the :func:gcd
function.
Formerly, only two arguments were supported.
.. versionchanged:: 3.10
Floats with integral values (like 5.0) are no longer accepted in the
:func:factorial function.
.. deprecated:: 3.15
These aliases are :term:soft deprecated in favor of the
:mod:math.integer functions.
.. 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://vimeo.com/147792667>, 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'). Due to the requirements of the IEEE-754 standard <https://en.wikipedia.org/wiki/IEEE_754>_, 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 :func: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
.. versionadded:: 3.5
.. versionchanged:: 3.11 It is now always available.
.. 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.
Module :mod:math.integer
Integer-specific mathematics functions.