# Mathematical Functions

All the functions return a Float64 number. The accuracy of the result is close to the maximum precision possible, but the result might not coincide with the machine representable number nearest to the corresponding real number.

## e()

Returns a Float64 number that is close to the number e.

## pi()

Returns a Float64 number that is close to the number π.

## exp(x)

Accepts a numeric argument and returns a Float64 number close to the exponent of the argument.

## log(x), ln(x)

Accepts a numeric argument and returns a Float64 number close to the natural logarithm of the argument.

## exp2(x)

Accepts a numeric argument and returns a Float64 number close to 2 to the power of x.

## log2(x)

Accepts a numeric argument and returns a Float64 number close to the binary logarithm of the argument.

## exp10(x)

Accepts a numeric argument and returns a Float64 number close to 10 to the power of x.

## log10(x)

Accepts a numeric argument and returns a Float64 number close to the decimal logarithm of the argument.

## sqrt(x)

Accepts a numeric argument and returns a Float64 number close to the square root of the argument.

## cbrt(x)

Accepts a numeric argument and returns a Float64 number close to the cubic root of the argument.

## erf(x)

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’.

Example (three sigma rule):

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

## erfc(x)

Accepts a numeric argument and returns a Float64 number close to 1 - erf(x), but without loss of precision for large ‘x’ values.

## lgamma(x)

The logarithm of the gamma function.

Gamma function.

The sine.

The cosine.

The tangent.

The arc sine.

The arc cosine.

The arc tangent.

## pow(x, y), power(x, y)

Takes two numeric arguments x and y. Returns a Float64 number close to x to the power of y.

## intExp2

Accepts a numeric argument and returns a UInt64 number close to 2 to the power of x.

## intExp10

Accepts a numeric argument and returns a UInt64 number close to 10 to the power of x.

## cosh(x)

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

Query:

``````SELECT cosh(0);
``````

Result:

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

## acosh(x)

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

Query:

``````SELECT acosh(1);
``````

Result:

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

## sinh(x)

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

Query:

``````SELECT sinh(0);
``````

Result:

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

## asinh(x)

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

Query:

``````SELECT asinh(0);
``````

Result:

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

## atanh(x)

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

Query:

``````SELECT atanh(0);
``````

Result:

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

## atan2(y, x)

The function calculates 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

Query:

``````SELECT atan2(1, 1);
``````

Result:

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

## hypot(x, y)

Calculates the length of the hypotenuse of a right-angle triangle. The function 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

Query:

``````SELECT hypot(1, 1);
``````

Result:

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

## log1p(x)

Calculates `log(1+x)`. The function `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

Query:

``````SELECT log1p(0);
``````

Result:

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

## sign(x)

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 │
└──────────┘
``````