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Mathematical Functions

All the functions return a Float64 number. Results are generally as close to the actual result as possible, but in some cases less precise than the machine-representable number.

e​

Returns e.

Syntax

e()

pi​

Returns π.

Syntax

pi()

exp​

Returns e to the power of the given argument.

Syntax

exp(x)

log​

Returns the natural logarithm of the argument.

Syntax

log(x)

Alias: ln(x)

exp2​

Returns 2 to the power of the given argument

Syntax

exp2(x)

intExp2​

Like exp but returns a UInt64.

Syntax

intExp2(x)

log2​

Returns the binary logarithm of the argument.

Syntax

log2(x)

exp10​

Returns 10 to the power of the given argument.

Syntax

exp10(x)

intExp10​

Like exp10 but returns a UInt64.

Syntax

intExp10(x)

log10​

Returns the decimal logarithm of the argument.

Syntax

log10(x)

sqrt​

Returns the square root of the argument.

sqrt(x)

cbrt​

Returns the cubic root of the argument.

cbrt(x)

erf​

If x is non-negative, then erf(x / σ√2) is the probability that a random variable having a normal distribution with standard deviation σ takes the value that is separated from the expected value by more than x.

Syntax

erf(x)

Example

(three sigma rule)

SELECT erf(3 / sqrt(2));
┌─erf(divide(3, sqrt(2)))─┐
│      0.9973002039367398 │
└─────────────────────────┘

erfc​

Returns a number close to 1 - erf(x) without loss of precision for large ‘x’ values.

Syntax

erfc(x)

lgamma​

Returns the logarithm of the gamma function.

Syntax

lgamma(x)

tgamma​

Returns the gamma function.

Syntax

gamma(x)

sin​

Returns the sine of the argument

Syntax

sin(x)

cos​

Returns the cosine of the argument.

Syntax

cos(x)

tan​

Returns the tangent of the argument.

Syntax

tan(x)

asin​

Returns the arc sine of the argument.

Syntax

asin(x)

acos​

Returns the arc cosine of the argument.

Syntax

acos(x)

atan​

Returns the arc tangent of the argument.

Syntax

atan(x)

pow​

Returns x to the power of y.

Syntax

pow(x, y)

Alias: power(x, y)

cosh​

Returns the hyperbolic cosine of the argument.

Syntax

cosh(x)

Arguments

  • x — The angle, in radians. Values from the interval: -∞ < x < +∞. Float64.

Returned value

  • Values from the interval: 1 <= cosh(x) < +∞.

Type: Float64.

Example

SELECT cosh(0);

Result:

┌─cosh(0)──┐
│        1 │
└──────────┘

acosh​

Returns the inverse hyperbolic cosine.

Syntax

acosh(x)

Arguments

  • x — Hyperbolic cosine of angle. Values from the interval: 1 <= x < +∞. Float64.

Returned value

  • The angle, in radians. Values from the interval: 0 <= acosh(x) < +∞.

Type: Float64.

Example

SELECT acosh(1);

Result:

┌─acosh(1)─┐
│        0 │
└──────────┘

sinh​

Returns the hyperbolic sine.

Syntax

sinh(x)

Arguments

  • x — The angle, in radians. Values from the interval: -∞ < x < +∞. Float64.

Returned value

  • Values from the interval: -∞ < sinh(x) < +∞.

Type: Float64.

Example

SELECT sinh(0);

Result:

┌─sinh(0)──┐
│        0 │
└──────────┘

asinh​

Returns the inverse hyperbolic sine.

Syntax

asinh(x)

Arguments

  • x — Hyperbolic sine of angle. Values from the interval: -∞ < x < +∞. Float64.

Returned value

  • The angle, in radians. Values from the interval: -∞ < asinh(x) < +∞.

Type: Float64.

Example

SELECT asinh(0);

Result:

┌─asinh(0)─┐
│        0 │
└──────────┘

atanh​

Returns the inverse hyperbolic tangent.

Syntax

atanh(x)

Arguments

  • x — Hyperbolic tangent of angle. Values from the interval: –1 < x < 1. Float64.

Returned value

  • The angle, in radians. Values from the interval: -∞ < atanh(x) < +∞.

Type: Float64.

Example

SELECT atanh(0);

Result:

┌─atanh(0)─┐
│        0 │
└──────────┘

atan2​

Returns the atan2 as the angle in the Euclidean plane, given in radians, between the positive x axis and the ray to the point (x, y) ≠ (0, 0).

Syntax

atan2(y, x)

Arguments

  • y — y-coordinate of the point through which the ray passes. Float64.
  • x — x-coordinate of the point through which the ray passes. Float64.

Returned value

  • The angle θ such that −π < θ ≤ π, in radians.

Type: Float64.

Example

SELECT atan2(1, 1);

Result:

┌────────atan2(1, 1)─┐
│ 0.7853981633974483 │
└────────────────────┘

hypot​

Returns the length of the hypotenuse of a right-angle triangle. Hypot avoids problems that occur when squaring very large or very small numbers.

Syntax

hypot(x, y)

Arguments

  • x — The first cathetus of a right-angle triangle. Float64.
  • y — The second cathetus of a right-angle triangle. Float64.

Returned value

  • The length of the hypotenuse of a right-angle triangle.

Type: Float64.

Example

SELECT hypot(1, 1);

Result:

┌────────hypot(1, 1)─┐
│ 1.4142135623730951 │
└────────────────────┘

log1p​

Calculates log(1+x). The calculation log1p(x) is more accurate than log(1+x) for small values of x.

Syntax

log1p(x)

Arguments

  • x — Values from the interval: -1 < x < +∞. Float64.

Returned value

  • Values from the interval: -∞ < log1p(x) < +∞.

Type: Float64.

Example

SELECT log1p(0);

Result:

┌─log1p(0)─┐
│        0 │
└──────────┘

sign​

Returns the sign of a real number.

Syntax

sign(x)

Arguments

  • x — Values from -∞ to +∞. Support all numeric types in ClickHouse.

Returned value

  • -1 for x < 0
  • 0 for x = 0
  • 1 for x > 0

Examples

Sign for the zero value:

SELECT sign(0);

Result:

┌─sign(0)─┐
│       0 │
└─────────┘

Sign for the positive value:

SELECT sign(1);

Result:

┌─sign(1)─┐
│       1 │
└─────────┘

Sign for the negative value:

SELECT sign(-1);

Result:

┌─sign(-1)─┐
│       -1 │
└──────────┘

degrees​

Converts radians to degrees.

Syntax

degrees(x)

Arguments

Returned value

  • Value in degrees.

Type: Float64.

Example

SELECT degrees(3.141592653589793);

Result:

┌─degrees(3.141592653589793)─┐
│                        180 │
└────────────────────────────┘

radians​

Converts degrees to radians.

Syntax

radians(x)

Arguments

Returned value

  • Value in radians.

Type: Float64.

Example

SELECT radians(180);

Result:

┌──────radians(180)─┐
│ 3.141592653589793 │
└───────────────────┘

factorial​

Computes the factorial of an integer value. Works with any native integer type including UInt(8|16|32|64) and Int(8|16|32|64). The return type is UInt64.

The factorial of 0 is 1. Likewise, the factorial() function returns 1 for any negative value. The maximum positive value for the input argument is 20, a value of 21 or greater will cause exception throw.

Syntax

factorial(n)

Example

SELECT factorial(10);

Result:

┌─factorial(10)─┐
│       3628800 │
└───────────────┘

width_bucket​

Returns the number of the bucket in which operand falls in a histogram having count equal-width buckets spanning the range low to high. Returns 0 if operand < low, and returns count+1 if operand >= high.

operand, low, high can be any native number type. count can only be unsigned native integer and its value cannot be zero.

Syntax

widthBucket(operand, low, high, count)

Alias: WIDTH_BUCKET

Example

SELECT widthBucket(10.15, -8.6, 23, 18);

Result:

┌─widthBucket(10.15, -8.6, 23, 18)─┐
│                               11 │
└──────────────────────────────────┘