“Arctan of x can be defined as the inverse of the tangent function of x where x is a real number [x∈ℝ].” It is an angle between −π/2 and +π/2 radians [or between −90° and +90°] whose tan is x.
arctan Special Values of Angles
Below is a table of some special values of angels of arctan or inverse of tan.
Some more theoretical and detailed explanation of values of arctan. can be found in the following table.
atan[] Function in Python
math.atan[] function is from Standard math Library of Python Programming Language. The purpose of this function is to calculate arc tan or the inverse of tan of any given number.
These functions cannot be used with complex numbers; use the functions of the same name from the 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 , which should return an 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
>>> 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-050 and evaluates to zero when
>>> 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-051.
Also called the binomial coefficient because it is equivalent to the coefficient of k-th term in polynomial expansion of
>>> 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-052.
Raises if either of the arguments are not integers. Raises if either of the arguments are negative.
New 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,
>>> 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-055 returns -1.0.math.fabs[x]
Return the absolute value of x.
math.factorial[n]Return n factorial as an integer. Raises if n is not integral or is negative.
Deprecated since version 3.9: Accepting floats with integral values [like
>>> 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-057] is deprecated.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 , which should return an value.
math.fmod[x, y]Return
sqrt[sum[[px - qx] ** 2.0 for px, qx in zip[p, q]]]0, as defined by the platform C library. Note that the Python expression
sqrt[sum[[px - qx] ** 2.0 for px, qx in zip[p, q]]]1 may not return the same result. The intent of the C standard is that
sqrt[sum[[px - qx] ** 2.0 for px, qx in zip[p, q]]]0 be exactly [mathematically; to infinite precision] equal to
sqrt[sum[[px - qx] ** 2.0 for px, qx in zip[p, q]]]3 for some integer n such that the result has the same sign as x and magnitude less than
sqrt[sum[[px - qx] ** 2.0 for px, qx in zip[p, q]]]4. Python’s
sqrt[sum[[px - qx] ** 2.0 for px, qx in zip[p, q]]]1 returns a result with the sign of y instead, and may not be exactly computable for float arguments. For example,
sqrt[sum[[px - qx] ** 2.0 for px, qx in zip[p, q]]]6 is
sqrt[sum[[px - qx] ** 2.0 for px, qx in zip[p, q]]]7, but the result of Python’s
sqrt[sum[[px - qx] ** 2.0 for px, qx in zip[p, q]]]8 is
sqrt[sum[[px - qx] ** 2.0 for px, qx in zip[p, q]]]9, which cannot be represented exactly as a float, and rounds to the surprising
def phi[x]: 'Cumulative distribution function for the standard normal distribution' return [1.0 + erf[x / sqrt[2.0]]] / 2.00. For this reason, function is generally preferred when working with floats, while Python’s
sqrt[sum[[px - qx] ** 2.0 for px, qx in zip[p, q]]]1 is preferred when working with integers.math.frexp[x]
Return the mantissa and exponent of x as the pair
def phi[x]: 'Cumulative distribution function for the standard normal distribution' return [1.0 + erf[x / sqrt[2.0]]] / 2.03. m is a float and e is an integer such that
def phi[x]: 'Cumulative distribution function for the standard normal distribution' return [1.0 + erf[x / sqrt[2.0]]] / 2.04 exactly. If x is zero, returns
def phi[x]: 'Cumulative distribution function for the standard normal distribution' return [1.0 + erf[x / sqrt[2.0]]] / 2.05, otherwise
def phi[x]: 'Cumulative distribution function for the standard normal distribution' return [1.0 + erf[x / sqrt[2.0]]] / 2.06. This is used to “pick apart” the internal representation of a float in a portable way.math.fsum[iterable]
Return an accurate floating point sum of values in the iterable. Avoids loss of precision by tracking multiple intermediate partial sums:
>>> sum[[.1, .1, .1, .1, .1, .1, .1, .1, .1, .1]] 0.9999999999999999 >>> fsum[[.1, .1, .1, .1, .1, .1, .1, .1, .1, .1]] 1.0
The algorithm’s accuracy depends on IEEE-754 arithmetic guarantees and the typical case where the rounding mode is half-even. On some non-Windows builds, the underlying C library uses extended precision addition and may occasionally double-round an intermediate sum causing it to be off in its least significant bit.
For further discussion and two alternative approaches, see the ASPN cookbook recipes for accurate floating point summation.
math.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
def phi[x]: 'Cumulative distribution function for the standard normal distribution' return [1.0 + erf[x / sqrt[2.0]]] / 2.07.
def phi[x]: 'Cumulative distribution function for the standard normal distribution' return [1.0 + erf[x / sqrt[2.0]]] / 2.08 without arguments returns
def phi[x]: 'Cumulative distribution function for the standard normal distribution' return [1.0 + erf[x / sqrt[2.0]]] / 2.07.
New 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
>>> import math >>> math.nan == math.nan False >>> float['nan'] == float['nan'] False >>> math.isnan[math.nan] True >>> math.isnan[float['nan']] True0 if the values a and b are close to each other and
>>> import math >>> math.nan == math.nan False >>> float['nan'] == float['nan'] False >>> math.isnan[math.nan] True >>> math.isnan[float['nan']] True1 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
>>> import math >>> math.nan == math.nan False >>> float['nan'] == float['nan'] False >>> math.isnan[math.nan] True >>> math.isnan[float['nan']] True2. The default tolerance is
>>> import math >>> math.nan == math.nan False >>> float['nan'] == float['nan'] False >>> math.isnan[math.nan] True >>> math.isnan[float['nan']] True3, 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:
>>> import math >>> math.nan == math.nan False >>> float['nan'] == float['nan'] False >>> math.isnan[math.nan] True >>> math.isnan[float['nan']] True4.
The IEEE 754 special values of
>>> import math >>> math.nan == math.nan False >>> float['nan'] == float['nan'] False >>> math.isnan[math.nan] True >>> math.isnan[float['nan']] True5,
>>> import math >>> math.nan == math.nan False >>> float['nan'] == float['nan'] False >>> math.isnan[math.nan] True >>> math.isnan[float['nan']] True6, and
>>> import math >>> math.nan == math.nan False >>> float['nan'] == float['nan'] False >>> math.isnan[math.nan] True >>> math.isnan[float['nan']] True7 will be handled according to IEEE rules. Specifically,
>>> import math >>> math.nan == math.nan False >>> float['nan'] == float['nan'] False >>> math.isnan[math.nan] True >>> math.isnan[float['nan']] True5 is not considered close to any other value, including
>>> import math >>> math.nan == math.nan False >>> float['nan'] == float['nan'] False >>> math.isnan[math.nan] True >>> math.isnan[float['nan']] True5.
>>> import math >>> math.nan == math.nan False >>> float['nan'] == float['nan'] False >>> math.isnan[math.nan] True >>> math.isnan[float['nan']] True6 and
>>> import math >>> math.nan == math.nan False >>> float['nan'] == float['nan'] False >>> math.isnan[math.nan] True >>> math.isnan[float['nan']] True7 are only considered close to themselves.
New in version 3.5.
See also
PEP 485 – A function for testing approximate equality
Return
>>> import math >>> math.nan == math.nan False >>> float['nan'] == float['nan'] False >>> math.isnan[math.nan] True >>> math.isnan[float['nan']] True0 if x is neither an infinity nor a NaN, and
>>> import math >>> math.nan == math.nan False >>> float['nan'] == float['nan'] False >>> math.isnan[math.nan] True >>> math.isnan[float['nan']] True1 otherwise. [Note that
math4 is considered finite.]New in version 3.2.
math.isinf[x]Return
>>> import math >>> math.nan == math.nan False >>> float['nan'] == float['nan'] False >>> math.isnan[math.nan] True >>> math.isnan[float['nan']] True0 if x is a positive or negative infinity, and
>>> import math >>> math.nan == math.nan False >>> float['nan'] == float['nan'] False >>> math.isnan[math.nan] True >>> math.isnan[float['nan']] True1 otherwise.math.isnan[x]
Return
>>> import math >>> math.nan == math.nan False >>> float['nan'] == float['nan'] False >>> math.isnan[math.nan] True >>> math.isnan[float['nan']] True0 if x is a NaN [not a number], and
>>> import math >>> math.nan == math.nan False >>> float['nan'] == float['nan'] False >>> math.isnan[math.nan] True >>> math.isnan[float['nan']] True1 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 a² ≤ n.
For some applications, it may be more convenient to have the least integer a such that n ≤ a², or in other words the ceiling of the exact square root of n. For positive n, this can be computed using math9.
New 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
def phi[x]: 'Cumulative distribution function for the standard normal distribution' return [1.0 + erf[x / sqrt[2.0]]] / 2.07.
cmath1 without arguments returns cmath2.New in version 3.9.
math.ldexp[x, i]Return cmath3. This is essentially the inverse of function .
Return the fractional and integer parts of x. Both results carry the sign of x and are floats.
math.nextafter[x, y]Return the next floating-point value after x towards y.
If x is equal to y, return y.
Examples:
cmath5 goes up: towards positive infinity.cmath6 goes down: towards minus infinity.cmath7 goes towards zero.cmath8 goes away from zero.
See also .
New in version 3.9.
math.perm[n, k=None]Return the number of ways to choose k items from n items without repetition and with order.
Evaluates to x.__ceil__0 when
>>> 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-050 and evaluates to zero when
>>> 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-051.
If k is not specified or is None, then k defaults to n and the function returns x.__ceil__3.
Raises if either of the arguments are not integers. Raises if either of the arguments are negative.
New 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 cmath2.
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.
New 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
sqrt[sum[[px - qx] ** 2.0 for px, qx in zip[p, q]]]3, where
x.__ceil__8 is the closest integer to the exact value of the quotient x.__ceil__9. If x.__ceil__9 is exactly halfway between two consecutive integers, the nearest even integer is used for x.__ceil__8. The remainder Integral2 thus always satisfies Integral3.Special cases follow IEEE 754: in particular, Integral4 is x for any finite x, and Integral5 and Integral6 raise 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.
New in version 3.7.
math.trunc[x]Return x with the fractional part removed, leaving the integer part. This rounds toward 0: Integral8 is equivalent to for positive x, and equivalent to for negative x. If x is not a float, delegates to , which should return an value.
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
n! / [k! * [n - k]!]3.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, ].
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
n! / [k! * [n - k]!]5.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
n! / [k! * [n - k]!]6.
ULP stands for “Unit in the Last Place”.
See also and .
New in version 3.9.
Note that and 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 , , and 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
>>> 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-0504 necessarily has no fractional bits.
Power and logarithmic functions
math.cbrt[x]Return the cube root of x.
New 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
>>> 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-0505 or
>>> 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-0506.math.exp2[x]
Return 2 raised to the power x.
New 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
>>> 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-0507 can result in a significant loss of precision; the 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
New 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
>>> 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-0509.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
>>> 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-0510.
New in version 3.3.
See also
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
>>> 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-0512.math.pow[x, y]
Return
>>> 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-0513 raised to the power
>>> 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-0514. Exceptional cases follow the IEEE 754 standard as far as possible. In particular,
>>> 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-0515 and
>>> 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-0516 always return
>>> 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-0517, even when
>>> 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-0513 is a zero or a NaN. If both
>>> 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-0513 and
>>> 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-0514 are finite,
>>> 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-0513 is negative, and
>>> 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-0514 is not an integer then
>>> 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-0523 is undefined, and raises .
Unlike the built-in
>>> 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-0525 operator, converts both its arguments to type . Use
>>> 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-0525 or the built-in function for computing exact integer powers.
Changed in version 3.11: The special cases
>>> 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-0530 and
>>> 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-0531 were changed to return
>>> import math >>> math.nan == math.nan False >>> float['nan'] == float['nan'] False >>> math.isnan[math.nan] True >>> math.isnan[float['nan']] True6 instead of raising , 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
def phi[x]: 'Cumulative distribution function for the standard normal distribution' return [1.0 + erf[x / sqrt[2.0]]] / 2.07 and
>>> 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-0535.math.asin[x]
Return the arc sine of x, in radians. The result is between
>>> 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-0536 and
>>> 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-0537.math.atan[x]
Return the arc tangent of x, in radians. The result is between
>>> 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-0536 and
>>> 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-0537.math.atan2[y, x]
Return
>>> 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-0540, in radians. The result is between
>>> 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-0541 and
>>> 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-0535. The vector in the plane from the origin to point
>>> 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-0543 makes this angle with the positive X axis. The point of is that the signs of both inputs are known to it, so it can compute the correct quadrant for the angle. For example,
>>> 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-0545 and
>>> 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-0546 are both
>>> 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-0547, but
>>> 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-0548 is
>>> 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-0549.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]]]
New in version 3.8.
math.hypot[*coordinates]Return the Euclidean norm,
>>> 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-0550. This is the length of the vector from the origin to the point given by the coordinates.
For a two dimensional point
>>> 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-0543, this is equivalent to computing the hypotenuse of a right triangle using the Pythagorean theorem,
>>> 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-0552.
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 function can be used to compute traditional statistical functions such as the :
def phi[x]: 'Cumulative distribution function for the standard normal distribution' return [1.0 + erf[x / sqrt[2.0]]] / 2.0
New in version 3.2.
math.erfc[x]Return the complementary error function at x. The complementary error function is defined as
>>> 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-0554. It is used for large values of x where a subtraction from one would cause a loss of significance.
New in version 3.2.
math.gamma[x]Return the Gamma function at x.
New in version 3.2.
math.lgamma[x]Return the natural logarithm of the absolute value of the Gamma function at x.
New in version 3.2.
Constants
math.piThe mathematical constant π = 3.141592…, to available precision.
math.eThe mathematical constant e = 2.718281…, to available precision.
math.tauThe mathematical constant τ = 6.283185…, to available precision. Tau is a circle constant equal to 2π, 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!
New in version 3.6.
math.infA floating-point positive infinity. [For negative infinity, use
>>> 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-0555.] Equivalent to the output of
>>> 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-0556.
New in version 3.5.
math.nanA floating-point “not a number” [NaN] value. Equivalent to the output of
>>> 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-0557. Due to the requirements of the IEEE-754 standard,
>>> 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-0558 and
>>> 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-0557 are not considered to equal to any other numeric value, including themselves. To check whether a number is a NaN, use the function to test for NaNs instead of
>>> 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-0561 or
>>> 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-0562. Example:
>>> import math >>> math.nan == math.nan False >>> float['nan'] == float['nan'] False >>> math.isnan[math.nan] True >>> math.isnan[float['nan']] True
Changed in version 3.11: It is now always available.
New in version 3.5.
CPython implementation detail: The 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 for invalid operations like
>>> 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-0565 or
>>> 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-0566 [where C99 Annex F recommends signaling invalid operation or divide-by-zero], and for results that overflow [for example,
>>> 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-0568]. 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
>>> 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-0569 or
>>> 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-0570.
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.