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ClickHouse/docs/en/sql-reference/functions/math-functions.md
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---
slug: /en/sql-reference/functions/math-functions
sidebar_position: 125
sidebar_label: Mathematical
---
# Mathematical Functions
## e
Returns e ([Euler's constant](https://en.wikipedia.org/wiki/Euler%27s_constant)).
**Syntax**
```sql
e()
```
**Returned value**
Type: [Float64](../data-types/float.md).
## pi
Returns π ([Pi](https://en.wikipedia.org/wiki/Pi)).
**Syntax**
```sql
pi()
```
**Returned value**
Type: [Float64](../data-types/float.md).
## exp
Returns e to the power of the given argument.
**Syntax**
```sql
exp(x)
```
**Arguments**
- `x` - [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
Type: [Float*](../data-types/float.md).
## log
Returns the natural logarithm of the argument.
**Syntax**
```sql
log(x)
```
Alias: `ln(x)`
**Arguments**
- `x` - [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
Type: [Float*](../data-types/float.md).
## exp2
Returns 2 to the power of the given argument
**Syntax**
```sql
exp2(x)
```
**Arguments**
- `x` - [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
Type: [Float*](../data-types/float.md).
## intExp2
Like `exp` but returns a UInt64.
**Syntax**
```sql
intExp2(x)
```
## log2
Returns the binary logarithm of the argument.
**Syntax**
```sql
log2(x)
```
**Arguments**
- `x` - [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
Type: [Float*](../data-types/float.md).
## exp10
Returns 10 to the power of the given argument.
**Syntax**
```sql
exp10(x)
```
**Arguments**
- `x` - [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
Type: [Float*](../data-types/float.md).
## intExp10
Like `exp10` but returns a UInt64.
**Syntax**
```sql
intExp10(x)
```
## log10
Returns the decimal logarithm of the argument.
**Syntax**
```sql
log10(x)
```
**Arguments**
- `x` - [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
Type: [Float*](../data-types/float.md).
## sqrt
Returns the square root of the argument.
```sql
sqrt(x)
```
**Arguments**
- `x` - [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
Type: [Float*](../data-types/float.md).
## cbrt
Returns the cubic root of the argument.
```sql
cbrt(x)
```
**Arguments**
- `x` - [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
Type: [Float*](../data-types/float.md).
## 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**
```sql
erf(x)
```
**Arguments**
- `x` - [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
Type: [Float*](../data-types/float.md).
**Example**
(three sigma rule)
``` sql
SELECT erf(3 / sqrt(2));
```
```result
┌─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**
```sql
erfc(x)
```
**Arguments**
- `x` - [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
Type: [Float*](../data-types/float.md).
## lgamma
Returns the logarithm of the gamma function.
**Syntax**
```sql
lgamma(x)
```
**Arguments**
- `x` - [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
Type: [Float*](../data-types/float.md).
## tgamma
Returns the gamma function.
**Syntax**
```sql
gamma(x)
```
**Arguments**
- `x` - [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
Type: [Float*](../data-types/float.md).
## sin
Returns the sine of the argument
**Syntax**
```sql
sin(x)
```
**Arguments**
- `x` - [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
Type: [Float*](../data-types/float.md).
**Example**
Query:
```sql
SELECT sin(1.23);
```
```response
0.9424888019316975
```
## cos
Returns the cosine of the argument.
**Syntax**
```sql
cos(x)
```
**Arguments**
- `x` - [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
Type: [Float*](../data-types/float.md).
## tan
Returns the tangent of the argument.
**Syntax**
```sql
tan(x)
```
**Arguments**
- `x` - [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
Type: [Float*](../data-types/float.md).
## asin
Returns the arc sine of the argument.
**Syntax**
```sql
asin(x)
```
**Arguments**
- `x` - [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
Type: [Float*](../data-types/float.md).
## acos
Returns the arc cosine of the argument.
**Syntax**
```sql
acos(x)
```
**Arguments**
- `x` - [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
Type: [Float*](../data-types/float.md).
## atan
Returns the arc tangent of the argument.
**Syntax**
```sql
atan(x)
```
**Arguments**
- `x` - [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
Type: [Float*](../data-types/float.md).
## pow
Returns `x` to the power of `y`.
**Syntax**
```sql
pow(x, y)
```
Alias: `power(x, y)`
**Arguments**
- `x` - [(U)Int8/16/32/64](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md)
- `y` - [(U)Int8/16/32/64](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md)
**Returned value**
Type: [Float64](../data-types/float.md).
## cosh
Returns the [hyperbolic cosine](https://in.mathworks.com/help/matlab/ref/cosh.html) of the argument.
**Syntax**
``` sql
cosh(x)
```
**Arguments**
- `x` — The angle, in radians. Values from the interval: `-∞ < x < +∞`. [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
- Values from the interval: `1 <= cosh(x) < +∞`.
Type: [Float64](../data-types/float.md#float32-float64).
**Example**
``` sql
SELECT cosh(0);
```
Result:
```result
┌─cosh(0)──┐
│ 1 │
└──────────┘
```
## acosh
Returns the [inverse hyperbolic cosine](https://www.mathworks.com/help/matlab/ref/acosh.html).
**Syntax**
``` sql
acosh(x)
```
**Arguments**
- `x` — Hyperbolic cosine of angle. Values from the interval: `1 <= x < +∞`. [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
- The angle, in radians. Values from the interval: `0 <= acosh(x) < +∞`.
Type: [Float64](../data-types/float.md#float32-float64).
**Example**
``` sql
SELECT acosh(1);
```
Result:
```result
┌─acosh(1)─┐
│ 0 │
└──────────┘
```
## sinh
Returns the [hyperbolic sine](https://www.mathworks.com/help/matlab/ref/sinh.html).
**Syntax**
``` sql
sinh(x)
```
**Arguments**
- `x` — The angle, in radians. Values from the interval: `-∞ < x < +∞`. [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
- Values from the interval: `-∞ < sinh(x) < +∞`.
Type: [Float64](../data-types/float.md#float32-float64).
**Example**
``` sql
SELECT sinh(0);
```
Result:
```result
┌─sinh(0)──┐
│ 0 │
└──────────┘
```
## asinh
Returns the [inverse hyperbolic sine](https://www.mathworks.com/help/matlab/ref/asinh.html).
**Syntax**
``` sql
asinh(x)
```
**Arguments**
- `x` — Hyperbolic sine of angle. Values from the interval: `-∞ < x < +∞`. [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
- The angle, in radians. Values from the interval: `-∞ < asinh(x) < +∞`.
Type: [Float64](../data-types/float.md#float32-float64).
**Example**
``` sql
SELECT asinh(0);
```
Result:
```result
┌─asinh(0)─┐
│ 0 │
└──────────┘
```
## tanh
Returns the [hyperbolic tangent](https://www.mathworks.com/help/matlab/ref/tanh.html).
**Syntax**
``` sql
tanh(x)
```
**Arguments**
- `x` — The angle, in radians. Values from the interval: `-∞ < x < +∞`. [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
- Values from the interval: `-1 < tanh(x) < 1`.
Type: [Float*](../data-types/float.md#float32-float64).
**Example**
``` sql
SELECT tanh(0);
```
Result:
```result
0
```
## atanh
Returns the [inverse hyperbolic tangent](https://www.mathworks.com/help/matlab/ref/atanh.html).
**Syntax**
``` sql
atanh(x)
```
**Arguments**
- `x` — Hyperbolic tangent of angle. Values from the interval: `–1 < x < 1`. [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
- The angle, in radians. Values from the interval: `-∞ < atanh(x) < +∞`.
Type: [Float64](../data-types/float.md#float32-float64).
**Example**
``` sql
SELECT atanh(0);
```
Result:
```result
┌─atanh(0)─┐
│ 0 │
└──────────┘
```
## atan2
Returns the [atan2](https://en.wikipedia.org/wiki/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**
``` sql
atan2(y, x)
```
**Arguments**
- `y` — y-coordinate of the point through which the ray passes. [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
- `x` — x-coordinate of the point through which the ray passes. [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
- The angle `θ` such that `−π < θ ≤ π`, in radians.
Type: [Float64](../data-types/float.md#float32-float64).
**Example**
``` sql
SELECT atan2(1, 1);
```
Result:
```result
┌────────atan2(1, 1)─┐
│ 0.7853981633974483 │
└────────────────────┘
```
## hypot
Returns the length of the hypotenuse of a right-angle triangle. [Hypot](https://en.wikipedia.org/wiki/Hypot) avoids problems that occur when squaring very large or very small numbers.
**Syntax**
``` sql
hypot(x, y)
```
**Arguments**
- `x` — The first cathetus of a right-angle triangle. [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
- `y` — The second cathetus of a right-angle triangle. [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
- The length of the hypotenuse of a right-angle triangle.
Type: [Float64](../data-types/float.md#float32-float64).
**Example**
``` sql
SELECT hypot(1, 1);
```
Result:
```result
┌────────hypot(1, 1)─┐
│ 1.4142135623730951 │
└────────────────────┘
```
## log1p
Calculates `log(1+x)`. The [calculation](https://en.wikipedia.org/wiki/Natural_logarithm#lnp1) `log1p(x)` is more accurate than `log(1+x)` for small values of x.
**Syntax**
``` sql
log1p(x)
```
**Arguments**
- `x` — Values from the interval: `-1 < x < +∞`. [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
- Values from the interval: `-∞ < log1p(x) < +∞`.
Type: [Float64](../data-types/float.md#float32-float64).
**Example**
``` sql
SELECT log1p(0);
```
Result:
```result
┌─log1p(0)─┐
│ 0 │
└──────────┘
```
## sign
Returns the sign of a real number.
**Syntax**
``` sql
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`
Type: [Int8](../data-types/int-uint.md).
**Examples**
Sign for the zero value:
``` sql
SELECT sign(0);
```
Result:
```result
┌─sign(0)─┐
│ 0 │
└─────────┘
```
Sign for the positive value:
``` sql
SELECT sign(1);
```
Result:
```result
┌─sign(1)─┐
│ 1 │
└─────────┘
```
Sign for the negative value:
``` sql
SELECT sign(-1);
```
Result:
```result
┌─sign(-1)─┐
│ -1 │
└──────────┘
```
## sigmoid
Returns the [sigmoid function](https://en.wikipedia.org/wiki/Sigmoid_function).
**Syntax**
```sql
sigmoid(x)
```
**Parameters**
- `x` — input value. Values from the interval: `-∞ < x < +∞`. [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
- Corresponding value along the sigmoid curve between 0 and 1. [Float64](../data-types/float.md).
**Example**
Query:
``` sql
SELECT round(sigmoid(x), 5) FROM (SELECT arrayJoin([-1, 0, 1]) AS x);
```
Result:
```result
0.26894
0.5
0.73106
```
## degrees
Converts radians to degrees.
**Syntax**
``` sql
degrees(x)
```
**Arguments**
- `x` — Input in radians. [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
- `x` — Input in radians. [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
- Value in degrees. [Float64](../data-types/float.md#float32-float64).
**Example**
``` sql
SELECT degrees(3.141592653589793);
```
Result:
```result
┌─degrees(3.141592653589793)─┐
│ 180 │
└────────────────────────────┘
```
## radians
Converts degrees to radians.
**Syntax**
``` sql
radians(x)
```
**Arguments**
- `x` — Input in degrees. [(U)Int*](../data-types/int-uint.md), [Float*](../data-types/float.md) or [Decimal*](../data-types/decimal.md).
**Returned value**
- Value in radians.
Type: [Float64](../data-types/float.md#float32-float64).
**Example**
``` sql
SELECT radians(180);
```
Result:
```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**
``` sql
factorial(n)
```
**Example**
``` sql
SELECT factorial(10);
```
Result:
```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**
```sql
widthBucket(operand, low, high, count)
```
Alias: `WIDTH_BUCKET`
**Example**
``` sql
SELECT widthBucket(10.15, -8.6, 23, 18);
```
Result:
```result
┌─widthBucket(10.15, -8.6, 23, 18)─┐
│ 11 │
└──────────────────────────────────┘
```
## proportionsZTest
Returns test statistics for the two proportion Z-test - a statistical test for comparing the proportions from two populations `x` and `y`.
**Syntax**
```sql
proportionsZTest(successes_x, successes_y, trials_x, trials_y, conf_level, pool_type)
```
**Arguments**
- `successes_x`: Number of successes in population `x`. [UInt64](../data-types/int-uint.md).
- `successes_y`: Number of successes in population `y`. [UInt64](../data-types/int-uint.md).
- `trials_x`: Number of trials in population `x`. [UInt64](../data-types/int-uint.md).
- `trials_y`: Number of trials in population `y`. [UInt64](../data-types/int-uint.md).
- `conf_level`: Confidence level for the test. [Float64](../data-types/float.md).
- `pool_type`: Selection of pooling (way in which the standard error is estimated). Can be either `unpooled` or `pooled`. [String](../data-types/string.md).
:::note
For argument `pool_type`: In the pooled version, the two proportions are averaged, and only one proportion is used to estimate the standard error. In the unpooled version, the two proportions are used separately.
:::
**Returned value**
- `z_stat`: Z statistic. [Float64](../data-types/float.md).
- `p_val`: P value. [Float64](../data-types/float.md).
- `ci_low`: The lower confidence interval. [Float64](../data-types/float.md).
- `ci_high`: The upper confidence interval. [Float64](../data-types/float.md).
**Example**
Query:
```sql
SELECT proportionsZTest(10, 11, 100, 101, 0.95, 'unpooled');
```
Result:
```response
┌─proportionsZTest(10, 11, 100, 101, 0.95, 'unpooled')───────────────────────────────┐
│ (-0.20656724435948853,0.8363478437079654,-0.09345975390115283,0.07563797172293502) │
└────────────────────────────────────────────────────────────────────────────────────┘
```
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