ClickHouse/docs/en/sql-reference/functions/math-functions.md

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---
toc_priority: 44
toc_title: Mathematical
---
# Mathematical Functions {#mathematical-functions}
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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.
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## e() {#e}
Returns a Float64 number that is close to the number e.
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## pi() {#pi}
Returns a Float64 number that is close to the number π.
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## exp(x) {#expx}
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Accepts a numeric argument and returns a Float64 number close to the exponent of the argument.
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## log(x), ln(x) {#logx-lnx}
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Accepts a numeric argument and returns a Float64 number close to the natural logarithm of the argument.
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## exp2(x) {#exp2x}
Accepts a numeric argument and returns a Float64 number close to 2 to the power of x.
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## log2(x) {#log2x}
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Accepts a numeric argument and returns a Float64 number close to the binary logarithm of the argument.
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## exp10(x) {#exp10x}
Accepts a numeric argument and returns a Float64 number close to 10 to the power of x.
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## log10(x) {#log10x}
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Accepts a numeric argument and returns a Float64 number close to the decimal logarithm of the argument.
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## sqrt(x) {#sqrtx}
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Accepts a numeric argument and returns a Float64 number close to the square root of the argument.
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## cbrt(x) {#cbrtx}
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Accepts a numeric argument and returns a Float64 number close to the cubic root of the argument.
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## erf(x) {#erfx}
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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.
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Example (three sigma rule):
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``` sql
SELECT erf(3 / sqrt(2))
```
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``` text
┌─erf(divide(3, sqrt(2)))─┐
│ 0.9973002039367398 │
└─────────────────────────┘
```
CLICKHOUSE-2720: progress on website (#865) * update presentations * CLICKHOUSE-2936: redirect from clickhouse.yandex.ru and clickhouse.yandex.com * update submodule * lost files * CLICKHOUSE-2981: prefer sphinx docs over original reference * CLICKHOUSE-2981: docs styles more similar to main website + add flags to switch language links * update presentations * Less confusing directory structure (docs -> doc/reference/) * Minify sphinx docs too * Website release script: fail fast + pass docker hash on deploy * Do not underline links in docs * shorter * cleanup docker images * tune nginx config * CLICKHOUSE-3043: get rid of habrastorage links * Lost translation * CLICKHOUSE-2936: temporary client-side redirect * behaves weird in test * put redirect back * CLICKHOUSE-3047: copy docs txts to public too * move to proper file * remove old pages to avoid confusion * Remove reference redirect warning for now * Refresh README.md * Yellow buttons in docs * Use svg flags instead of unicode ones in docs * fix test website instance * Put flags to separate files * wrong flag * Copy Yandex.Metrica introduction from main page to docs * Yet another home page structure change, couple new blocks (CLICKHOUSE-3045) * Update Contacts section * CLICKHOUSE-2849: more detailed legal information * CLICKHOUSE-2978 preparation - split by files * More changes in Contacts block * Tune texts on index page * update presentations * One more benchmark * Add usage sections to index page, adapted from slides * Get the roadmap started, based on slides from last ClickHouse Meetup * CLICKHOUSE-2977: some rendering tuning * Get rid of excessive section in the end of getting started * Make headers linkable * CLICKHOUSE-2981: links to editing reference - https://github.com/yandex/ClickHouse/issues/849 * CLICKHOUSE-2981: fix mobile styles in docs * Ban crawling of duplicating docs * Open some external links in new tab * Ban old docs too * Lots of trivial fixes in english docs * Lots of trivial fixes in russian docs * Remove getting started copies in markdown * Add Yandex.Webmaster * Fix some sphinx warnings * More warnings fixed in english docs * More sphinx warnings fixed * Add code-block:: text * More code-block:: text * These headers look not that well * Better switch between documentation languages * merge use_case.rst into ya_metrika_task.rst * Edit the agg_functions.rst texts * Add lost empty lines
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## erfc(x) {#erfcx}
CLICKHOUSE-2720: progress on website (#865) * update presentations * CLICKHOUSE-2936: redirect from clickhouse.yandex.ru and clickhouse.yandex.com * update submodule * lost files * CLICKHOUSE-2981: prefer sphinx docs over original reference * CLICKHOUSE-2981: docs styles more similar to main website + add flags to switch language links * update presentations * Less confusing directory structure (docs -> doc/reference/) * Minify sphinx docs too * Website release script: fail fast + pass docker hash on deploy * Do not underline links in docs * shorter * cleanup docker images * tune nginx config * CLICKHOUSE-3043: get rid of habrastorage links * Lost translation * CLICKHOUSE-2936: temporary client-side redirect * behaves weird in test * put redirect back * CLICKHOUSE-3047: copy docs txts to public too * move to proper file * remove old pages to avoid confusion * Remove reference redirect warning for now * Refresh README.md * Yellow buttons in docs * Use svg flags instead of unicode ones in docs * fix test website instance * Put flags to separate files * wrong flag * Copy Yandex.Metrica introduction from main page to docs * Yet another home page structure change, couple new blocks (CLICKHOUSE-3045) * Update Contacts section * CLICKHOUSE-2849: more detailed legal information * CLICKHOUSE-2978 preparation - split by files * More changes in Contacts block * Tune texts on index page * update presentations * One more benchmark * Add usage sections to index page, adapted from slides * Get the roadmap started, based on slides from last ClickHouse Meetup * CLICKHOUSE-2977: some rendering tuning * Get rid of excessive section in the end of getting started * Make headers linkable * CLICKHOUSE-2981: links to editing reference - https://github.com/yandex/ClickHouse/issues/849 * CLICKHOUSE-2981: fix mobile styles in docs * Ban crawling of duplicating docs * Open some external links in new tab * Ban old docs too * Lots of trivial fixes in english docs * Lots of trivial fixes in russian docs * Remove getting started copies in markdown * Add Yandex.Webmaster * Fix some sphinx warnings * More warnings fixed in english docs * More sphinx warnings fixed * Add code-block:: text * More code-block:: text * These headers look not that well * Better switch between documentation languages * merge use_case.rst into ya_metrika_task.rst * Edit the agg_functions.rst texts * Add lost empty lines
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Accepts a numeric argument and returns a Float64 number close to 1 - erf(x), but without loss of precision for large x values.
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## lgamma(x) {#lgammax}
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The logarithm of the gamma function.
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## tgamma(x) {#tgammax}
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Gamma function.
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## sin(x) {#sinx}
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The sine.
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## cos(x) {#cosx}
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The cosine.
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## tan(x) {#tanx}
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The tangent.
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## asin(x) {#asinx}
The arc sine.
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## acos(x) {#acosx}
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The arc cosine.
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## atan(x) {#atanx}
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The arc tangent.
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## pow(x, y), power(x, y) {#powx-y-powerx-y}
Takes two numeric arguments x and y. Returns a Float64 number close to x to the power of y.
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## intExp2 {#intexp2}
Accepts a numeric argument and returns a UInt64 number close to 2 to the power of x.
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## intExp10 {#intexp10}
Accepts a numeric argument and returns a UInt64 number close to 10 to the power of x.
## cosh(x) {#coshx}
[Hyperbolic cosine](https://in.mathworks.com/help/matlab/ref/cosh.html).
**Syntax**
``` sql
cosh(x)
```
**Parameters**
- `x` — The angle, in radians. Values from the interval: `-∞ < x < +∞`. [Float64](../../sql-reference/data-types/float.md#float32-float64).
**Returned value**
- Values from the interval: `1 <= cosh(x) < +∞`.
Type: [Float64](../../sql-reference/data-types/float.md#float32-float64).
**Example**
Query:
``` sql
SELECT cosh(0);
```
Result:
``` text
┌─cosh(0)──┐
│ 1 │
└──────────┘
```
## acosh(x) {#acoshx}
[Inverse hyperbolic cosine](https://www.mathworks.com/help/matlab/ref/acosh.html).
**Syntax**
``` sql
acosh(x)
```
**Parameters**
- `x` — Hyperbolic cosine of angle. Values from the interval: `1 <= x < +∞`. [Float64](../../sql-reference/data-types/float.md#float32-float64).
**Returned value**
- The angle, in radians. Values from the interval: `0 <= acosh(x) < +∞`.
Type: [Float64](../../sql-reference/data-types/float.md#float32-float64).
**Example**
Query:
``` sql
SELECT acosh(1);
```
Result:
``` text
┌─acosh(1)─┐
│ 0 │
└──────────┘
```
**See Also**
- [cosh(x)](../../sql-reference/functions/math-functions.md#coshx)
## sinh(x) {#sinhx}
[Hyperbolic sine](https://www.mathworks.com/help/matlab/ref/sinh.html).
**Syntax**
``` sql
sinh(x)
```
**Parameters**
- `x` — The angle, in radians. Values from the interval: `-∞ < x < +∞`. [Float64](../../sql-reference/data-types/float.md#float32-float64).
**Returned value**
- Values from the interval: `-∞ < sinh(x) < +∞`.
Type: [Float64](../../sql-reference/data-types/float.md#float32-float64).
**Example**
Query:
``` sql
SELECT sinh(0);
```
Result:
``` text
┌─sinh(0)──┐
│ 0 │
└──────────┘
```
## asinh(x) {#asinhx}
[Inverse hyperbolic sine](https://www.mathworks.com/help/matlab/ref/asinh.html).
**Syntax**
``` sql
asinh(x)
```
**Parameters**
- `x` — Hyperbolic sine of angle. Values from the interval: `-∞ < x < +∞`. [Float64](../../sql-reference/data-types/float.md#float32-float64).
**Returned value**
- The angle, in radians. Values from the interval: `-∞ < asinh(x) < +∞`.
Type: [Float64](../../sql-reference/data-types/float.md#float32-float64).
**Example**
Query:
``` sql
SELECT asinh(0);
```
Result:
``` text
┌─asinh(0)─┐
│ 0 │
└──────────┘
```
**See Also**
- [sinh(x)](../../sql-reference/functions/math-functions.md#sinhx)
## atanh(x) {#atanhx}
[Inverse hyperbolic tangent](https://www.mathworks.com/help/matlab/ref/atanh.html).
**Syntax**
``` sql
atanh(x)
```
**Parameters**
- `x` — Hyperbolic tangent of angle. Values from the interval: `1 < x < 1`. [Float64](../../sql-reference/data-types/float.md#float32-float64).
**Returned value**
- The angle, in radians. Values from the interval: `-∞ < atanh(x) < +∞`.
Type: [Float64](../../sql-reference/data-types/float.md#float32-float64).
**Example**
Query:
``` sql
SELECT atanh(0);
```
Result:
``` text
┌─atanh(0)─┐
│ 0 │
└──────────┘
```
## atan2(y, x) {#atan2yx}
The [function](https://en.wikipedia.org/wiki/Atan2) 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**
``` sql
atan2(y, x)
```
**Parameters**
- `y` — y-coordinate of the point through which the ray passes. [Float64](../../sql-reference/data-types/float.md#float32-float64).
- `x` — x-coordinate of the point through which the ray passes. [Float64](../../sql-reference/data-types/float.md#float32-float64).
**Returned value**
- The angle `θ` such that `−π < θ ≤ π`, in radians.
Type: [Float64](../../sql-reference/data-types/float.md#float32-float64).
**Example**
Query:
``` sql
SELECT atan2(1, 1);
```
Result:
``` text
┌────────atan2(1, 1)─┐
│ 0.7853981633974483 │
└────────────────────┘
```
## hypot(x, y) {#hypotxy}
Calculates the length of the hypotenuse of a right-angle triangle. The [function](https://en.wikipedia.org/wiki/Hypot) avoids problems that occur when squaring very large or very small numbers.
**Syntax**
``` sql
hypot(x, y)
```
**Parameters**
- `x` — The first cathetus of a right-angle triangle. [Float64](../../sql-reference/data-types/float.md#float32-float64).
- `y` — The second cathetus of a right-angle triangle. [Float64](../../sql-reference/data-types/float.md#float32-float64).
**Returned value**
- The length of the hypotenuse of a right-angle triangle.
Type: [Float64](../../sql-reference/data-types/float.md#float32-float64).
**Example**
Query:
``` sql
SELECT hypot(1, 1);
```
Result:
``` text
┌────────hypot(1, 1)─┐
│ 1.4142135623730951 │
└────────────────────┘
```
## log1p(x) {#log1px}
Calculates `log(1+x)`. The [function](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)
```
**Parameters**
- `x` — Values from the interval: `-1 < x < +∞`. [Float64](../../sql-reference/data-types/float.md#float32-float64).
**Returned value**
- Values from the interval: `-∞ < log1p(x) < +∞`.
Type: [Float64](../../sql-reference/data-types/float.md#float32-float64).
**Example**
Query:
``` sql
SELECT log1p(0);
```
Result:
``` text
┌─log1p(0)─┐
│ 0 │
└──────────┘
```
**See Also**
- [log(x)](../../sql-reference/functions/math-functions.md#logx-lnx)
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## sign(x) {#signx}
The `sign` function can extract the sign of a real number.
**Syntax**
``` sql
sign(x)
```
**Parameters**
- `x` — Values from `-∞` to `+∞`. Support all numeric types in ClickHouse.
**Returned value**
- -1 for `x < 0`
- 0 for `x = 0`
- 1 for `x > 0`
**Example**
Query:
``` sql
SELECT sign(0);
```
Result:
``` text
┌─sign(0)─┐
│ 0 │
└─────────┘
```
Query:
``` sql
SELECT sign(1);
```
Result:
``` text
┌─sign(1)─┐
│ 1 │
└─────────┘
```
Query:
``` sql
SELECT sign(-1);
```
Result:
``` text
┌─sign(-1)─┐
│ -1 │
└──────────┘
```
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[Original article](https://clickhouse.tech/docs/en/query_language/functions/math_functions/) <!--hide-->