Mathematical functions | ClickHouse Docs

acos

Introduced in: v1.1.0

Returns the arc cosine of the argument.

Syntax

acos(x)

Arguments

x — The value for which to find arc cosine of. (U)Int* or Float* or Decimal*

Returned value

Returns the arc cosine of x Float*

Examples

Usage example

SELECT acos(0.5);

1.0471975511965974

acosh

Introduced in: v20.12.0

Returns the inverse hyperbolic cosine.

Syntax

acosh(x)

Arguments

x — Hyperbolic cosine of angle. Values from the interval: 1 ≤ x < +∞. (U)Int* or Float* or Decimal*

Returned value

Returns the angle, in radians. Values from the interval: 0 ≤ acosh(x) < +∞. Float64

Examples

Usage example

SELECT acosh(1)

asin

Introduced in: v1.1.0

Calculates the arcsine of the provided argument. For arguments in the range [-1, 1] it returns the value in the range of [-pi() / 2, pi() / 2].

Syntax

asin(x)

Arguments

x — Argument for which to calculate arcsine of. (U)Int* or Float* or Decimal

Returned value

Returns the arcsine value of the provided argument x Float64

Examples

inverse

SELECT asin(1.0) = pi() / 2, sin(asin(1)), asin(sin(1))

1	1	0.9999999999999999

float32

SELECT toTypeName(asin(1.0::Float32))

Float64

nan

SELECT asin(1.1), asin(-2), asin(inf), asin(nan)

nan	nan	nan	nan

asinh

Introduced in: v20.12.0

Returns the inverse hyperbolic sine.

Syntax

asinh(x)

Arguments

x — Hyperbolic sine of angle. Values from the interval: -∞ < x < +∞. (U)Int* or Float* or Decimal*

Returned value

Returns the angle, in radians. Values from the interval: -∞ < asinh(x) < +∞. Float64

Examples

Basic usage

SELECT asinh(0)

atan

Introduced in: v1.1.0

Returns the arc tangent of the argument.

Syntax

atan(x)

Arguments

x — The value for which to find arc tangent of. (U)Int* or Float* or Decimal*

Returned value

Returns the arc tangent of x. Float*

Examples

Usage example

SELECT atan(1);

0.7853981633974483

atan2

Introduced in: v20.12.0

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. (U)Int* or Float* or BFloat16 or Decimal*
x — x-coordinate of the point through which the ray passes. (U)Int* or Float* or BFloat16 or Decimal*

Returned value

Returns the angle θ such that -π < θ ≤ π, in radians Float64

Examples

Usage example

SELECT atan2(1, 1)

0.7853981633974483

atanh

Introduced in: v20.12.0

Returns the inverse hyperbolic tangent.

Syntax

atanh(x)

Arguments

x — Hyperbolic tangent of angle. Values from the interval: -1 < x < 1. (U)Int*, Float* or Decimal*. (U)Int* or Float* or Decimal*

Returned value

Returns the angle, in radians. Values from the interval: -∞ < atanh(x) < +∞ Float64

Examples

Usage example

SELECT atanh(0)

cbrt

Introduced in: v1.1.0

Returns the cubic root of the argument.

Syntax

cbrt(x)

Arguments

x — The value for which to find the cubic root of. (U)Int* or Float* or Decimal*

Returned value

Returns the cubic root of x. Float*

Examples

Usage example

SELECT cbrt(8);

cos

Introduced in: v1.1.0

Returns the cosine of the argument.

Syntax

cos(x)

Arguments

x — The angle in radians. (U)Int* or Float* or Decimal*

Returned value

Returns the cosine of x. Float*

Examples

Usage example

SELECT cos(0);

cosh

Introduced in: v20.12.0

Returns the hyperbolic cosine of the argument.

Syntax

cosh(x)

Arguments

x — The angle, in radians. Values from the interval: -∞ < x < +∞. (U)Int* or Float* or Decimal*

Returned value

Returns values from the interval: 1 ≤ cosh(x) < +∞ Float64

Examples

Basic usage

SELECT cosh(0)

degrees

Introduced in: v22.2.0

Converts radians to degrees.

Syntax

degrees(x)

Arguments

x — Input in radians. (U)Int* or Float* or Decimal*

Returned value

Returns the value of x in degrees. Float64

Examples

Basic usage

SELECT degrees(3.141592653589793)

e

Introduced in: v1.1.0

Returns Euler’s constant (e).

Syntax

e()

Arguments

None.

Returned value

Returns Euler’s constant Float64

Examples

Usage example

SELECT e();

2.718281828459045

erf

Introduced in: v1.1.0

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)

Arguments

x — The value for which to compute the error function value. (U)Int* or Float* or Decimal*

Returned value

Returns the error function value Float*

Examples

Three sigma rule

SELECT erf(3 / sqrt(2))

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

erfc

Introduced in: v1.1.0

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

Syntax

erfc(x)

Arguments

x — The value for which to find the error function value. (U)Int* or Float* or Decimal*

Returned value

Returns the complementary error function value Float*

Examples

Usage example

SELECT erfc(0);

exp

Introduced in: v1.1.0

Returns e raised to the power of x, where x is the given argument to the function.

Syntax

exp(x)

Arguments

x — The exponent. (U)Int* or Float* or Decimal*

Returned value

Returns e^x Float*

Examples

Basic usage

SELECT round(exp(-1), 4)

┌─round(exp(-1), 4)─┐
│            0.3679 │
└───────────────────┘

exp10

Introduced in: v1.1.0

Returns 10 to the power of the given argument.

Syntax

exp10(x)

Arguments

x — The exponent. (U)Int* or Float* or Decimal*

Returned value

Returns 10^x Float*

Examples

Usage example

SELECT exp10(2);

exp2

Introduced in: v1.1.0

Returns 2 to the power of the given argument.

Syntax

exp2(x)

Arguments

x — The exponent. (U)Int* or Float* or Decimal*

Returned value

Returns 2^x Float*

Examples

Usage example

SELECT exp2(3);

factorial

Introduced in: v22.11.0

Computes the factorial of an integer value. 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 an exception.

Syntax

factorial(n)

Arguments

n — Integer value for which to calculate the factorial. Maximum value is 20. (U)Int8/16/32/64

Returned value

Returns the factorial of the input as UInt64. Returns 1 for input 0 or any negative value. UInt64

Examples

Usage example

SELECT factorial(10)

hypot

Introduced in: v20.12.0

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. (U)Int* or Float* or BFloat16 or Decimal*
y — The second cathetus of a right-angle triangle. (U)Int* or Float* or BFloat16 or Decimal*

Returned value

Returns the length of the hypotenuse of a right-angle triangle. Float64

Examples

Basic usage

SELECT hypot(1, 1)

1.4142135623730951

intExp10

Introduced in: v1.1.0

Like exp10 but returns a UInt64 number.

Syntax

intExp10(x)

Arguments

x — The exponent. Int* or UInt* or Float*

Returned value

Returns 10^x. UInt64

Examples

Usage example

SELECT intExp10(2);

intExp2

Introduced in: v1.1.0

Like exp2 but returns a UInt64 number.

Syntax

intExp2(x)

Arguments

x — The exponent. Int* or UInt* or Float*

Returned value

Returns 2^x. UInt64

Examples

Usage example

SELECT intExp2(3);

isPrime

Introduced in: v26.5.0

Returns 1 if the argument is a prime number, otherwise 0.

Uses an exact lookup bitmap for small values and a deterministic Miller-Rabin test for larger values. The result is exact for every supported input type.

For wider unsigned integer types (UInt128, UInt256), use isProbablePrime instead.

Syntax

isPrime(n)

Arguments

n — Unsigned integer to test for primality. UInt8 or UInt16 or UInt32 or UInt64

Returned value

Returns 1 if n is prime, 0 otherwise. UInt8

Examples

Prime number

SELECT isPrime(17)

Composite number

SELECT isPrime(18)

Large UInt64 prime

SELECT isPrime(18446744073709551557)

Maximum UInt64 value

SELECT isPrime(18446744073709551615)

isProbablePrime

Introduced in: v26.5.0

Returns 1 if the argument is probably prime, 0 if it is definitely composite.

For UInt8, UInt16, UInt32, and UInt64, the result is exact and matches isPrime. The rounds argument is ignored.

For UInt128 and UInt256, a return value of 1 is probabilistic. The optional rounds argument controls how many Miller-Rabin rounds are used: more rounds reduce the chance of a false positive and increase the running time. With uniformly random witnesses, the false-positive rate for a fixed composite is bounded by 4^(-rounds); the default of 25 keeps this bound below 10^-15, and the maximum of 256 keeps it below 10^-154.

The function is deterministic: witnesses are seeded from n, so the same (n, rounds) pair always produces the same result. The 4^(-rounds) bound is the per-input probability under uniformly random witnesses; with our deterministic seeding it instead describes a fraction over inputs — a composite that fools its witness sequence will reproducibly return 1.

Syntax

isProbablePrime(n[, rounds])

Arguments

n — Unsigned integer to test for primality. UInt8 or UInt16 or UInt32 or UInt64 or UInt128 or UInt256
rounds — Optional positive integer constant in [1, 256]. Number of Miller-Rabin rounds for UInt128/UInt256 (ignored for narrower types). Default 25. UInt8 or UInt16 or UInt32 or UInt64

Returned value

Returns 1 if n is probably prime, 0 if it is definitely composite. UInt8

Examples

Small prime

SELECT isProbablePrime(17)

Small composite

SELECT isProbablePrime(18)

Largest UInt64 prime (exact result)

SELECT isProbablePrime(18446744073709551557)

Mersenne prime M_127 (UInt128)

SELECT isProbablePrime(toUInt128('170141183460469231731687303715884105727'))

Curve25519 base field prime 2^255 - 19 (UInt256)

SELECT isProbablePrime(toUInt256('57896044618658097711785492504343953926634992332820282019728792003956564819949'))

Faster, lower-confidence check: 5 rounds

SELECT isProbablePrime(toUInt256('57896044618658097711785492504343953926634992332820282019728792003956564819949'), 5)

lgamma

Introduced in: v1.1.0

Returns the logarithm of the gamma function.

Syntax

lgamma(x)

Arguments

x — The number for which to compute the logarithm of the gamma function. (U)Int* or Float* or Decimal*

Returned value

Returns the logarithm of the gamma function of x. Float*

Examples

Usage example

SELECT lgamma(5);

3.1780538303479458

log

Introduced in: v1.1.0

Returns the natural logarithm of the argument.

Syntax

log(x)

Aliases: ln

Arguments

x — The number for which to compute the natural logarithm of. (U)Int* or Float* or Decimal*

Returned value

Returns the natural logarithm of x. Float*

Examples

Usage example

SELECT log(10);

2.3025850938475476

log10

Introduced in: v1.1.0

Returns the decimal logarithm of the argument.

Syntax

log10(x)

Arguments

x — Number for which to compute the decimal logarithm of. (U)Int* or Float* or Decimal*

Returned value

Returns the decimal logarithm of x. Float*

Examples

Usage example

SELECT log10(100);

log1p

Introduced in: v20.12.0

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 < +∞. (U)Int* or Float* or Decimal*

Returned value

Returns values from the interval: -∞ < log1p(x) < +∞ Float64

Examples

Usage example

SELECT log1p(0)

log2

Introduced in: v1.1.0

Returns the binary logarithm of the argument.

Syntax

log2(x)

Arguments

x — The number for which to compute the binary logarithm of. (U)Int* or Float* or Decimal*

Returned value

Returns the binary logarithm of x. Float*

Examples

Usage example

SELECT log2(8);

pi

Introduced in: v1.1.0

Returns pi (π).

Syntax

pi()

Arguments

None.

Returned value

Returns pi Float64

Examples

Usage example

SELECT pi();

3.141592653589793

pow

Introduced in: v1.1.0

Returns x raised to the power of y.

Syntax

pow(x, y)

Aliases: power

Arguments

x — The base. (U)Int8/16/32/64 or Float* or BFloat16 or Decimal*
y — The exponent. (U)Int8/16/32/64 or Float* or BFloat16 or Decimal*

Returned value

Returns x^y Float64

Examples

Usage example

SELECT pow(2, 3);

proportionsZTest

Introduced in: v22.3.0

Returns test statistics for the two proportion Z-test - a statistical test for comparing the proportions from two populations x and y. The function supports both pooled and unpooled estimation methods for the standard error. 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.

Syntax

proportionsZTest(successes_x, successes_y, trials_x, trials_y, conf_level, pool_type)

Arguments

successes_x — Number of successes in population x. UInt64
successes_y — Number of successes in population y. UInt64
trials_x — Number of trials in population x. UInt64
trials_y — Number of trials in population y. UInt64
conf_level — Confidence level for the test. Float64
pool_type — Selection of pooling method for standard error estimation. Can be either ‘unpooled’ or ‘pooled’. String

Returned value

Returns a tuple containing: z_stat (Z statistic), p_val (P value), ci_low (lower confidence interval), ci_high (upper confidence interval). Tuple(Float64, Float64, Float64, Float64)

Examples

Usage example

SELECT proportionsZTest(10, 11, 100, 101, 0.95, 'unpooled');

┌─proportionsZTest(10, 11, 100, 101, 0.95, 'unpooled')───────────────────────────────┐
│ (-0.20656724435948853,0.8363478437079654,-0.09345975390115283,0.07563797172293502) │
└────────────────────────────────────────────────────────────────────────────────────┘

radians

Introduced in: v22.2.0

Converts degrees to radians.

Syntax

radians(x)

Arguments

x — Input in degrees. (U)Int* or Float* or Decimal*

Returned value

Returns value in radians Float64

Examples

Usage example

SELECT radians(180)

3.141592653589793

sigmoid

Introduced in: v20.1.0

Calculates the sigmoid function: 1 / (1 + exp(-x)). The sigmoid function maps any real number to the range (0, 1) and is commonly used in machine learning.

Syntax

sigmoid(x)

Arguments

x — The input value. (U)Int* or Float*

Returned value

Returns the sigmoid of the input value, in the range (0, 1). Float64

Examples

Basic usage

SELECT sigmoid(0)

0.5

sign

Introduced in: v21.2.0

Returns the sign of a real number.

Syntax

sign(x)

Arguments

x — Values from -∞ to +∞. (U)Int* or Decimal* or Float*

Returned value

Returns -1 for x < 0, 0 for x = 0, 1 for x > 0. Int8

Examples

Sign for zero

SELECT sign(0)

Sign for positive

SELECT sign(1)

Sign for negative

SELECT sign(-1)

-1

sin

Introduced in: v1.1.0

Returns the sine of the argument.

Syntax

sin(x)

Arguments

x — The number whose sine will be returned. (U)Int* or Float* or Decimal*

Returned value

Returns the sine of x.

Examples

simple

SELECT sin(1.23)

0.9424888019316975

sinh

Introduced in: v20.12.0

Returns the hyperbolic sine.

Syntax

sinh(x)

Arguments

x — The angle, in radians. Values from the interval: -∞ < x < +∞. (U)Int* or Float* or Decimal*

Returned value

Returns values from the interval: -∞ < sinh(x) < +∞ Float64

Examples

Usage example

SELECT sinh(0)

sqrt

Introduced in: v1.1.0

Returns the square root of the argument.

Syntax

sqrt(x)

Arguments

x — The number for which to find the square root of. (U)Int* or Float* or Decimal*

Returned value

Returns the square root of x Float*

Examples

Usage example

SELECT sqrt(16);

tan

Introduced in: v1.1.0

Returns the tangent of the argument.

Syntax

tan(x)

Arguments

x — The angle in radians. (U)Int* or Float* or Decimal*

Returned value

Returns the tangent of x. Float*

Examples

Usage example

SELECT tan(0);

tanh

Introduced in: v20.1.0

Returns the hyperbolic tangent.

Syntax

tanh(x)

Arguments

x — The angle in radians. Values from the interval: -∞ < x < +∞. (U)Int* or Float* or Decimal*

Returned value

Returns values from the interval: -1 < tanh(x) < 1 Float*

Examples

Usage example

SELECT tanh(0)

tgamma

Introduced in: v1.1.0

Returns the gamma function.

Syntax

tgamma(x)

Arguments

x — The number for which to compute the gamma function of. (U)Int* or Float* or Decimal*

Returned value

Returns the gamma function value Float*

Examples

Usage example

SELECT tgamma(5);

widthBucket

Introduced in: v23.3.0

Returns the number of the bucket in which parameter operand falls in a histogram having count equal-width buckets spanning the range low to high. Returns 0 if operand is less than low, and returns count+1 if operand is greater than or equal to high. There is also a case insensitive alias called WIDTH_BUCKET to provide compatibility with other databases.

Syntax

widthBucket(operand, low, high, count)

Aliases: width_bucket

Arguments

operand — The value for which to determine the bucket. (U)Int8/16/32/64
low — The lower bound of the histogram range. (U)Int8/16/32/64
high — The upper bound of the histogram range. (U)Int8/16/32/64
count — The number of equal-width buckets. Cannot be zero. UInt8/16/32/64

Returned value

Returns the bucket number as an integer. Returns 0 if operand < low, returns count+1 if operand >= high. UInt8/16/32/64

Examples

Usage example

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