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.

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).

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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.

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).

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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.

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

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

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.

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).

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.

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.

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.

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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.

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:

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.

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.

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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.

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.

Return fmod(x, y), as defined by the platform C library. Note that the Python expression x % y may not return the same result. The intent of the C standard is that fmod(x, y) be exactly (mathematically; to infinite precision) equal to x - n*y for some integer n such that the result has the same sign as x and magnitude less than abs(y). Python’s x % y returns a result with the sign of y instead, and may not be exactly computable for float arguments. For example, fmod(-1e-100, 1e100) is -1e-100, but the result of Python’s -1e-100 % 1e100 is 1e100-1e-100, which cannot be represented exactly as a float, and rounds to the surprising 1e100. For this reason, function fmod() is generally preferred when working with floats, while Python’s x % y is preferred when working with integers.

CPython implementation detail: The math module consists mostly of thin wrappers around the platform C math library functions. Behavior in exceptional cases follows Annex F of the C99 standard where appropriate. The current implementation will raise ValueError for invalid operations like sqrt(-1.0) or log(0.0) (where C99 Annex F recommends signaling invalid operation or divide-by-zero), and OverflowError for results that overflow (for example, exp(1000.0)). A NaN will not be returned from any of the functions above unless one or more of the input arguments was a NaN; in that case, most functions will return a NaN, but (again following C99 Annex F) there are some exceptions to this rule, for example pow(float('nan'), 0.0) or hypot(float('nan'), float('inf')).

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).

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

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

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:

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

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

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

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

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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.

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).

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.

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.

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!

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.

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.

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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.

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These functions cannot be used with complex numbers; use the functions of the same name from the cmath module if you require support for complex numbers. The distinction between functions which support complex numbers and those which don’t is made since most users do not want to learn quite as much mathematics as required to understand complex numbers. Receiving an exception instead of a complex result allows earlier detection of the unexpected complex number used as a parameter, so that the programmer can determine how and why it was generated in the first place.

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.

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These mirrors are designed for high reflectance over a broad wavelength range. This makes them ideal for use with CW tunable lasers, laser diodes, and broadband light sources. These broadband dielectric mirrors are intended for use at 45 degrees or normal incidence.Please note that these items are on clearance and therefore limited quantities are available.

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).

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

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 x and rel_tol less than 1.0. So add an appropriate positive abs_tol argument to the call.

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).

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.

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).

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 a = 1 + isqrt(n - 1).

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.

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.

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.

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

The following functions are provided by this module. Except when explicitly noted otherwise, all return values are floats.

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

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