Floating point arithmetic¶

math.ceil(x)¶

Return the ceiling of x, the smallest integer greater than or equal to x. If x is not a float, delegates to x.__ceil__, which should return an Integral value.

math.fabs(x)¶

Return the absolute value of x.

math.floor(x)¶

Return the floor of x, the largest integer less than or equal to x. If x is not a float, delegates to x.__floor__, which should return an Integral value.

math.fma(x, y, z)¶

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

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

Added in version 3.13.

math.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, 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).

Added in version 3.15.

math.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, 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).

Added in version 3.15.

math.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 fmod() is generally preferred when working with floats, while Python’s x % y is preferred when working with integers.

math.modf(x)¶

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

Note that 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).

math.remainder(x, y)¶

Return the IEEE 754-style remainder of x with respect to y. For finite x and finite nonzero y, this is the difference x - n*y, where n is the closest integer to the exact value of the quotient x / y. If x / y is exactly halfway between two consecutive integers, the nearest even integer is used for n. The remainder r = remainder(x, y) thus always satisfies abs(r) <= 0.5 * abs(y).

Special cases follow IEEE 754: in particular, remainder(x, math.inf) is x for any finite x, and remainder(x, 0) and remainder(math.inf, x) raise ValueError for any non-NaN x. If the result of the remainder operation is zero, that zero will have the same sign as x.

On platforms using IEEE 754 binary floating point, the result of this operation is always exactly representable: no rounding error is introduced.

Added in version 3.7.

math.trunc(x)¶

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

For the ceil(), floor(), and modf() functions, note that all floating-point numbers of sufficiently large magnitude are exact integers. Python floats typically carry no more than 53 bits of precision (the same as the platform C double type), in which case any float x with abs(x) >= 2**52 necessarily has no fractional bits.

Floating point manipulation functions¶

math.copysign(x, y)¶

Return a float with the magnitude (absolute value) of x but the sign of y. On platforms that support signed zeros, copysign(1.0, -0.0) returns -1.0.

math.frexp(x)¶

Return the mantissa and exponent of x as the pair (m, e). If x is a finite nonzero number, then m is a float with 0.5 <= abs(m) < 1.0 and an integer e is such that x == m * 2**e exactly. Else, return (x, 0). This is used to “pick apart” the internal representation of a float in a portable way.

Note that 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).

math.isclose(a, b, *, rel_tol=1e-09, abs_tol=0.0)¶

Return True if the values a and b are close to each other and False otherwise.

Whether or not two values are considered close is determined according to given absolute and relative tolerances. 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.

Added in version 3.5.

See also

PEP 485 – A function for testing approximate equality

math.isfinite(x)¶

Return True if x is neither an infinity nor a NaN, and False otherwise. (Note that 0.0 is considered finite.)

Added in version 3.2.

math.isnormal(x)¶

Return True if x is a normal number, that is a finite nonzero number that is not a subnormal (see issubnormal()). Return False otherwise.

Added in version 3.15.

math.issubnormal(x)¶

Return True if x is a subnormal number, that is a finite nonzero number with a magnitude smaller than sys.float_info.min. Return False otherwise.

Added in version 3.15.

math.isinf(x)¶

Return True if x is a positive or negative infinity, and False otherwise.

math.isnan(x)¶

Return True if x is a NaN (not a number), and False otherwise.

math.ldexp(x, i)¶

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

math.nextafter(x, y, steps=1)¶

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

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

Examples:

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

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

• math.nextafter(x, 0.0) goes towards zero.

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

See also math.ulp().

Added in version 3.9.

Changed in version 3.12: Added the steps argument.

math.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.

Added in version 3.15.

math.ulp(x)¶

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

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

• If x is negative, return ulp(-x).

• If x is a positive infinity, return x.

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

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

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

ULP stands for “Unit in the Last Place”.

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

Added in version 3.9.

Power, exponential and logarithmic functions¶

math.cbrt(x)¶

Return the cube root of x.

Added in version 3.11.

math.exp(x)¶

Return e raised to the power x, where e = 2.718281… is the base of natural logarithms. This is usually more accurate than math.e ** x or pow(math.e, x).

math.exp2(x)¶

Return 2 raised to the power x.

Added in version 3.11.

math.expm1(x)¶

Return e raised to the power x, minus 1. Here e is the base of natural logarithms. For small floats x, the subtraction in exp(x) - 1 can result in a significant loss of precision; the expm1() function provides a way to compute this quantity to full precision:

>>> from math import exp, expm1
>>> exp(1e-5) - 1  # gives result accurate to 11 places
1.0000050000069649e-05
>>> expm1(1e-5)    # result accurate to full precision
1.0000050000166668e-05

Added in version 3.2.

math.log(x[, base])¶

With one argument, return the natural logarithm of x (to base e).

With two arguments, return the logarithm of x to the given base, calculated as log(x)/log(base).

math.log1p(x)¶

Return the natural logarithm of 1+x (base e). The result is calculated in a way which is accurate for x near zero.

math.log2(x)¶

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

Added in version 3.3.

See also

int.bit_length() returns the number of bits necessary to represent an integer in binary, excluding the sign and leading zeros.

math.log10(x)¶

Return the base-10 logarithm of x. This is usually more accurate than log(x, 10).

math.pow(x, y)¶

Return x raised to the power y. Exceptional cases follow the IEEE 754 standard as far as possible. In particular, pow(1.0, x) and pow(x, 0.0) always return 1.0, even when x is a zero or a NaN. If both x and y are finite, x is negative, and y is not an integer then pow(x, y) is undefined, and raises ValueError.

Unlike the built-in ** operator, math.pow() converts both its arguments to type float. Use ** or the built-in pow() function for computing exact integer powers.

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

math.sqrt(x)¶

Return the square root of x.

Summation and product functions¶

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, strict=True)))

Added in version 3.8.

math.fsum(iterable)¶

Return an accurate floating-point sum of values in the iterable. Avoids loss of precision by tracking multiple intermediate partial sums.

The algorithm’s accuracy depends on IEEE-754 arithmetic guarantees and the typical case where the rounding mode is half-even. On some non-Windows builds, the underlying C library uses extended precision addition and may occasionally double-round an intermediate sum causing it to be off in its least significant bit.

For further discussion and two alternative approaches, see the ASPN cookbook recipes for accurate floating-point summation.

math.hypot(*coordinates)¶

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

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

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

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

math.prod(iterable, *, start=1)¶

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

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

Added in version 3.8.

math.sumprod(p, q)¶

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

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

Roughly equivalent to:

sum(map(operator.mul, p, q, strict=True))

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

Added in version 3.12.

Angular conversion¶

math.degrees(x)¶

Convert angle x from radians to degrees.

math.radians(x)¶

Convert angle x from degrees to radians.

Trigonometric functions¶

math.acos(x)¶

Return the arc cosine of x, in radians. The result is between 0 and pi.

math.acospi(x)¶

Return the arc cosine of x, in half-turns. The result is between 0 and 1.

Added in version 3.16.

math.asin(x)¶

Return the arc sine of x, in radians. The result is between -pi/2 and pi/2.

math.asinpi(x)¶

Return the arc sine of x, in half-turns. The result is between -0.5 and 0.5.

Added in version 3.16.

math.atan(x)¶

Return the arc tangent of x, in radians. The result is between -pi/2 and pi/2.

math.atanpi(x)¶

Return the arc tangent of x, in half-turns. The result is between -0.5 and 0.5.

Added in version 3.16.

math.atan2(y, x)¶

Return atan(y / x), in radians. The result is between -pi and pi. The vector in the plane from the origin to point (x, y) makes this angle with the positive X axis. The point of atan2() is that the signs of both inputs are known to it, so it can compute the correct quadrant for the angle. For example, atan(1) and atan2(1, 1) are both pi/4, but atan2(-1, -1) is -3*pi/4.

math.atan2pi(y, x)¶

Return atanpi(y / x), in half-turns. The result is between -1 and 1. The vector in the plane from the origin to point (x, y) makes this angle with the positive X axis. The point of atan2pi() is that the signs of both inputs are known to it, so it can compute the correct quadrant for the angle. For example, atanpi(1) and atan2pi(1, 1) are both 0.25, but atan2pi(-1, -1) is -0.75.

Added in version 3.16.

math.cos(x)¶

Return the cosine of x radians.

math.cospi(x)¶

Return the cosine of x half-turns (x⋅π radians).

Added in version 3.16.

math.sin(x)¶

Return the sine of x radians.

math.sinpi(x)¶

Return the sine of x half-turns (x⋅π radians).

Added in version 3.16.

math.tan(x)¶

Return the tangent of x radians.

math.tanpi(x)¶

Return the tangent of x half-turns (x⋅π radians).

Added in version 3.16.

Hyperbolic functions¶

Hyperbolic functions are analogs of trigonometric functions that are based on hyperbolas instead of circles.

math.acosh(x)¶

Return the inverse hyperbolic cosine of x.

math.asinh(x)¶

Return the inverse hyperbolic sine of x.

math.atanh(x)¶

Return the inverse hyperbolic tangent of x.

math.cosh(x)¶

Return the hyperbolic cosine of x.

math.sinh(x)¶

Return the hyperbolic sine of x.

math.tanh(x)¶

Return the hyperbolic tangent of x.

Special functions¶

math.erf(x)¶

Return the error function at x.

The erf() function can be used to compute traditional statistical functions such as the cumulative standard normal distribution:

def phi(x):
    'Cumulative distribution function for the standard normal distribution'
    return (1.0 + erf(x / sqrt(2.0))) / 2.0

Added in version 3.2.

math.erfc(x)¶

Return the complementary error function at x. The complementary error function is defined as 1.0 - erf(x). It is used for large values of x where a subtraction from one would cause a loss of significance.

Added in version 3.2.

math.gamma(x)¶

Return the Gamma function at x.

Added in version 3.2.

math.lgamma(x)¶

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

Added in version 3.2.

Number-theoretic functions¶

For backward compatibility, the math module provides also aliases of the following functions from the math.integer module:

Constants¶

math.pi¶

The mathematical constant π = 3.141592…, to available precision.

math.e¶

The mathematical constant e = 2.718281…, to available precision.

math.tau¶

The mathematical constant τ = 6.283185…, to available precision. Tau is a circle constant equal to 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!

Added in version 3.6.

math.inf¶

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

Added in version 3.5.

math.nan¶

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

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

Added in version 3.5.

Changed in version 3.11: It is now always available.