# Mathematical functions

GoogleSQL for BigQuery supports mathematical functions. All mathematical functions have the following behaviors:

• They return `NULL` if any of the input parameters is `NULL`.
• They return `NaN` if any of the arguments is `NaN`.

### Categories

Category Functions
Trigonometric `ACOS`   `ACOSH`   `ASIN`   `ASINH`   `ATAN`   `ATAN2`   `ATANH`   `COS`   `COSH`   `COT`   `COTH`   `CSC`   `CSCH`   `SEC`   `SECH`   `SIN`   `SINH`   `TAN`   `TANH`
Exponential and
logarithmic
`EXP`   `LN`   `LOG`   `LOG10`
Rounding and
truncation
`CEIL`   `CEILING`   `FLOOR`   `ROUND`   `TRUNC`
Power and
root
`CBRT`   `POW`   `POWER`   `SQRT`
Sign `ABS`   `SIGN`
Distance `COSINE_DISTANCE`   `EUCLIDEAN_DISTANCE`
Comparison `GREATEST`   `LEAST`
Random number generator `RAND`
Arithmetic and error handling `DIV`   `IEEE_DIVIDE`   `IS_INF`   `IS_NAN`   `MOD`   `SAFE_ADD`   `SAFE_DIVIDE`   `SAFE_MULTIPLY`   `SAFE_NEGATE`   `SAFE_SUBTRACT`
Bucket `RANGE_BUCKET`

### Function list

Name Summary
`ABS` Computes the absolute value of `X`.
`ACOS` Computes the inverse cosine of `X`.
`ACOSH` Computes the inverse hyperbolic cosine of `X`.
`ASIN` Computes the inverse sine of `X`.
`ASINH` Computes the inverse hyperbolic sine of `X`.
`ATAN` Computes the inverse tangent of `X`.
`ATAN2` Computes the inverse tangent of `X/Y`, using the signs of `X` and `Y` to determine the quadrant.
`ATANH` Computes the inverse hyperbolic tangent of `X`.
`CBRT` Computes the cube root of `X`.
`CEIL` Gets the smallest integral value that is not less than `X`.
`CEILING` Synonym of `CEIL`.
`COS` Computes the cosine of `X`.
`COSH` Computes the hyperbolic cosine of `X`.
`COSINE_DISTANCE` Computes the cosine distance between two vectors.
`COT` Computes the cotangent of `X`.
`COTH` Computes the hyperbolic cotangent of `X`.
`CSC` Computes the cosecant of `X`.
`CSCH` Computes the hyperbolic cosecant of `X`.
`DIV` Divides integer `X` by integer `Y`.
`EXP` Computes `e` to the power of `X`.
`EUCLIDEAN_DISTANCE` Computes the Euclidean distance between two vectors.
`FLOOR` Gets the largest integral value that is not greater than `X`.
`GREATEST` Gets the greatest value among `X1,...,XN`.
`IEEE_DIVIDE` Divides `X` by `Y`, but does not generate errors for division by zero or overflow.
`IS_INF` Checks if `X` is positive or negative infinity.
`IS_NAN` Checks if `X` is a `NaN` value.
`LEAST` Gets the least value among `X1,...,XN`.
`LN` Computes the natural logarithm of `X`.
`LOG` Computes the natural logarithm of `X` or the logarithm of `X` to base `Y`.
`LOG10` Computes the natural logarithm of `X` to base 10.
`MOD` Gets the remainder of the division of `X` by `Y`.
`POW` Produces the value of `X` raised to the power of `Y`.
`POWER` Synonym of `POW`.
`RAND` Generates a pseudo-random value of type `FLOAT64` in the range of `[0, 1)`.
`RANGE_BUCKET` Scans through a sorted array and returns the 0-based position of a point's upper bound.
`ROUND` Rounds `X` to the nearest integer or rounds `X` to `N` decimal places after the decimal point.
`SAFE_ADD` Equivalent to the addition operator (`X + Y`), but returns `NULL` if overflow occurs.
`SAFE_DIVIDE` Equivalent to the division operator (`X / Y`), but returns `NULL` if an error occurs.
`SAFE_MULTIPLY` Equivalent to the multiplication operator (`X * Y`), but returns `NULL` if overflow occurs.
`SAFE_NEGATE` Equivalent to the unary minus operator (`-X`), but returns `NULL` if overflow occurs.
`SAFE_SUBTRACT` Equivalent to the subtraction operator (`X - Y`), but returns `NULL` if overflow occurs.
`SEC` Computes the secant of `X`.
`SECH` Computes the hyperbolic secant of `X`.
`SIGN` Produces -1 , 0, or +1 for negative, zero, and positive arguments respectively.
`SIN` Computes the sine of `X`.
`SINH` Computes the hyperbolic sine of `X`.
`SQRT` Computes the square root of `X`.
`TAN` Computes the tangent of `X`.
`TANH` Computes the hyperbolic tangent of `X`.
`TRUNC` Rounds a number like `ROUND(X)` or `ROUND(X, N)`, but always rounds towards zero and never overflows.

### `ABS`

``````ABS(X)
``````

Description

Computes absolute value. Returns an error if the argument is an integer and the output value cannot be represented as the same type; this happens only for the largest negative input value, which has no positive representation.

X ABS(X)
25 25
-25 25
`+inf` `+inf`
`-inf` `+inf`

Return Data Type

INPUT`INT64``NUMERIC``BIGNUMERIC``FLOAT64`
OUTPUT`INT64``NUMERIC``BIGNUMERIC``FLOAT64`

### `ACOS`

``````ACOS(X)
``````

Description

Computes the principal value of the inverse cosine of X. The return value is in the range [0,π]. Generates an error if X is a value outside of the range [-1, 1].

X ACOS(X)
`+inf` `NaN`
`-inf` `NaN`
`NaN` `NaN`
X < -1 Error
X > 1 Error

### `ACOSH`

``````ACOSH(X)
``````

Description

Computes the inverse hyperbolic cosine of X. Generates an error if X is a value less than 1.

X ACOSH(X)
`+inf` `+inf`
`-inf` `NaN`
`NaN` `NaN`
X < 1 Error

### `ASIN`

``````ASIN(X)
``````

Description

Computes the principal value of the inverse sine of X. The return value is in the range [-π/2,π/2]. Generates an error if X is outside of the range [-1, 1].

X ASIN(X)
`+inf` `NaN`
`-inf` `NaN`
`NaN` `NaN`
X < -1 Error
X > 1 Error

### `ASINH`

``````ASINH(X)
``````

Description

Computes the inverse hyperbolic sine of X. Does not fail.

X ASINH(X)
`+inf` `+inf`
`-inf` `-inf`
`NaN` `NaN`

### `ATAN`

``````ATAN(X)
``````

Description

Computes the principal value of the inverse tangent of X. The return value is in the range [-π/2,π/2]. Does not fail.

X ATAN(X)
`+inf` π/2
`-inf` -π/2
`NaN` `NaN`

### `ATAN2`

``````ATAN2(X, Y)
``````

Description

Calculates the principal value of the inverse tangent of X/Y using the signs of the two arguments to determine the quadrant. The return value is in the range [-π,π].

X Y ATAN2(X, Y)
`NaN` Any value `NaN`
Any value `NaN` `NaN`
0.0 0.0 0.0
Positive Finite value `-inf` π
Negative Finite value `-inf`
Finite value `+inf` 0.0
`+inf` Finite value π/2
`-inf` Finite value -π/2
`+inf` `-inf` ¾π
`-inf` `-inf` -¾π
`+inf` `+inf` π/4
`-inf` `+inf` -π/4

### `ATANH`

``````ATANH(X)
``````

Description

Computes the inverse hyperbolic tangent of X. Generates an error if X is outside of the range (-1, 1).

X ATANH(X)
`+inf` `NaN`
`-inf` `NaN`
`NaN` `NaN`
X < -1 Error
X > 1 Error

### `CBRT`

``````CBRT(X)
``````

Description

Computes the cube root of `X`. `X` can be any data type that coerces to `FLOAT64`. Supports the `SAFE.` prefix.

X CBRT(X)
`+inf` `inf`
`-inf` `-inf`
`NaN` `NaN`
`0` `0`
`NULL` `NULL`

Return Data Type

`FLOAT64`

Example

``````SELECT CBRT(27) AS cube_root;

/*--------------------*
| cube_root          |
+--------------------+
| 3.0000000000000004 |
*--------------------*/
``````

### `CEIL`

``````CEIL(X)
``````

Description

Returns the smallest integral value that is not less than X.

X CEIL(X)
2.0 2.0
2.3 3.0
2.8 3.0
2.5 3.0
-2.3 -2.0
-2.8 -2.0
-2.5 -2.0
0 0
`+inf` `+inf`
`-inf` `-inf`
`NaN` `NaN`

Return Data Type

INPUT`INT64``NUMERIC``BIGNUMERIC``FLOAT64`
OUTPUT`FLOAT64``NUMERIC``BIGNUMERIC``FLOAT64`

### `CEILING`

``````CEILING(X)
``````

Description

Synonym of CEIL(X)

### `COS`

``````COS(X)
``````

Description

Computes the cosine of X where X is specified in radians. Never fails.

X COS(X)
`+inf` `NaN`
`-inf` `NaN`
`NaN` `NaN`

### `COSH`

``````COSH(X)
``````

Description

Computes the hyperbolic cosine of X where X is specified in radians. Generates an error if overflow occurs.

X COSH(X)
`+inf` `+inf`
`-inf` `+inf`
`NaN` `NaN`

### `COSINE_DISTANCE`

``````COSINE_DISTANCE(vector1, vector2)
``````

Description

Computes the cosine distance between two vectors.

Definitions

• `vector1`: A vector that is represented by an `ARRAY<T>` value or a sparse vector that is represented by an `ARRAY<STRUCT<dimension,magnitude>>` value.
• `vector2`: A vector that is represented by an `ARRAY<T>` value or a sparse vector that is represented by an `ARRAY<STRUCT<dimension,magnitude>>` value.

Details

• `ARRAY<T>` can be used to represent a vector. Each zero-based index in this array represents a dimension. The value for each element in this array represents a magnitude.

`T` can represent the following and must be the same for both vectors:

• `FLOAT64`

In the following example vector, there are four dimensions. The magnitude is `10.0` for dimension `0`, `55.0` for dimension `1`, `40.0` for dimension `2`, and `34.0` for dimension `3`:

``````[10.0, 55.0, 40.0, 34.0]
``````
• `ARRAY<STRUCT<dimension,magnitude>>` can be used to represent a sparse vector. With a sparse vector, you only need to include dimension-magnitude pairs for non-zero magnitudes. If a magnitude isn't present in the sparse vector, the magnitude is implicitly understood to be zero.

For example, if you have a vector with 10,000 dimensions, but only 10 dimensions have non-zero magnitudes, then the vector is a sparse vector. As a result, it's more efficient to describe a sparse vector by only mentioning its non-zero magnitudes.

In `ARRAY<STRUCT<dimension,magnitude>>`, `STRUCT<dimension,magnitude>` represents a dimension-magnitude pair for each non-zero magnitude in a sparse vector. These parts need to be included for each dimension-magnitude pair:

• `dimension`: A `STRING` or `INT64` value that represents a dimension in a vector.

• `magnitude`: A `FLOAT64` value that represents a non-zero magnitude for a specific dimension in a vector.

You don't need to include empty dimension-magnitude pairs in a sparse vector. For example, the following sparse vector and non-sparse vector are equivalent:

``````-- sparse vector ARRAY<STRUCT<INT64, FLOAT64>>
[(1, 10.0), (2: 30.0), (5, 40.0)]
``````
``````-- vector ARRAY<FLOAT64>
[0.0, 10.0, 30.0, 0.0, 0.0, 40.0]
``````

In a sparse vector, dimension-magnitude pairs don't need to be in any particular order. The following sparse vectors are equivalent:

``````[('a', 10.0), ('b': 30.0), ('d': 40.0)]
``````
``````[('d': 40.0), ('a', 10.0), ('b': 30.0)]
``````
• Both non-sparse vectors in this function must share the same dimensions, and if they don't, an error is produced.

• A vector can't be a zero vector. A vector is a zero vector if it has no dimensions or all dimensions have a magnitude of `0`, such as `[]` or `[0.0, 0.0]`. If a zero vector is encountered, an error is produced.

• An error is produced if a magnitude in a vector is `NULL`.

• If a vector is `NULL`, `NULL` is returned.

Return type

`FLOAT64`

Examples

In the following example, non-sparsevectors are used to compute the cosine distance:

``````SELECT COSINE_DISTANCE([1.0, 2.0], [3.0, 4.0]) AS results;

/*----------*
| results  |
+----------+
| 0.016130 |
*----------*/
``````

In the following example, sparse vectors are used to compute the cosine distance:

``````SELECT COSINE_DISTANCE(
[(1, 1.0), (2, 2.0)],
[(2, 4.0), (1, 3.0)]) AS results;

/*----------*
| results  |
+----------+
| 0.016130 |
*----------*/
``````

The ordering of numeric values in a vector doesn't impact the results produced by this function. For example these queries produce the same results even though the numeric values in each vector is in a different order:

``````SELECT COSINE_DISTANCE([1.0, 2.0], [3.0, 4.0]) AS results;
``````
``````SELECT COSINE_DISTANCE([2.0, 1.0], [4.0, 3.0]) AS results;
``````
``````SELECT COSINE_DISTANCE([(1, 1.0), (2, 2.0)], [(1, 3.0), (2, 4.0)]) AS results;
``````
`````` /*----------*
| results  |
+----------+
| 0.016130 |
*----------*/
``````

In the following example, the function can't compute cosine distance against the first vector, which is a zero vector:

``````-- ERROR
SELECT COSINE_DISTANCE([0.0, 0.0], [3.0, 4.0]) AS results;
``````
``````-- ERROR
SELECT COSINE_DISTANCE([(1, 0.0), (2, 0.0)], [(1, 3.0), (2, 4.0)]) AS results;
``````

Both non-sparse vectors must have the same dimensions. If not, an error is produced. In the following example, the first vector has two dimensions and the second vector has three:

``````-- ERROR
SELECT COSINE_DISTANCE([9.0, 7.0], [8.0, 4.0, 5.0]) AS results;
``````

If you use sparse vectors and you repeat a dimension, an error is produced:

``````-- ERROR
SELECT COSINE_DISTANCE(
[(1, 9.0), (2, 7.0), (2, 8.0)], [(1, 8.0), (2, 4.0), (3, 5.0)]) AS results;
``````

### `COT`

``````COT(X)
``````

Description

Computes the cotangent for the angle of `X`, where `X` is specified in radians. `X` can be any data type that coerces to `FLOAT64`. Supports the `SAFE.` prefix.

X COT(X)
`+inf` `NaN`
`-inf` `NaN`
`NaN` `NaN`
`0` `Error`
`NULL` `NULL`

Return Data Type

`FLOAT64`

Example

``````SELECT COT(1) AS a, SAFE.COT(0) AS b;

/*---------------------+------*
| a                   | b    |
+---------------------+------+
| 0.64209261593433065 | NULL |
*---------------------+------*/
``````

### `COTH`

``````COTH(X)
``````

Description

Computes the hyperbolic cotangent for the angle of `X`, where `X` is specified in radians. `X` can be any data type that coerces to `FLOAT64`. Supports the `SAFE.` prefix.

X COTH(X)
`+inf` `1`
`-inf` `-1`
`NaN` `NaN`
`0` `Error`
`NULL` `NULL`

Return Data Type

`FLOAT64`

Example

``````SELECT COTH(1) AS a, SAFE.COTH(0) AS b;

/*----------------+------*
| a              | b    |
+----------------+------+
| 1.313035285499 | NULL |
*----------------+------*/
``````

### `CSC`

``````CSC(X)
``````

Description

Computes the cosecant of the input angle, which is in radians. `X` can be any data type that coerces to `FLOAT64`. Supports the `SAFE.` prefix.

X CSC(X)
`+inf` `NaN`
`-inf` `NaN`
`NaN` `NaN`
`0` `Error`
`NULL` `NULL`

Return Data Type

`FLOAT64`

Example

``````SELECT CSC(100) AS a, CSC(-1) AS b, SAFE.CSC(0) AS c;

/*----------------+-----------------+------*
| a              | b               | c    |
+----------------+-----------------+------+
| -1.97485753142 | -1.188395105778 | NULL |
*----------------+-----------------+------*/
``````

### `CSCH`

``````CSCH(X)
``````

Description

Computes the hyperbolic cosecant of the input angle, which is in radians. `X` can be any data type that coerces to `FLOAT64`. Supports the `SAFE.` prefix.

X CSCH(X)
`+inf` `0`
`-inf` `0`
`NaN` `NaN`
`0` `Error`
`NULL` `NULL`

Return Data Type

`FLOAT64`

Example

``````SELECT CSCH(0.5) AS a, CSCH(-2) AS b, SAFE.CSCH(0) AS c;

/*----------------+----------------+------*
| a              | b              | c    |
+----------------+----------------+------+
| 1.919034751334 | -0.27572056477 | NULL |
*----------------+----------------+------*/
``````

### `DIV`

``````DIV(X, Y)
``````

Description

Returns the result of integer division of X by Y. Division by zero returns an error. Division by -1 may overflow.

X Y DIV(X, Y)
20 4 5
12 -7 -1
20 3 6
0 20 0
20 0 Error

Return Data Type

The return data type is determined by the argument types with the following table.

INPUT`INT64``NUMERIC``BIGNUMERIC`
`INT64``INT64``NUMERIC``BIGNUMERIC`
`NUMERIC``NUMERIC``NUMERIC``BIGNUMERIC`
`BIGNUMERIC``BIGNUMERIC``BIGNUMERIC``BIGNUMERIC`

### `EXP`

``````EXP(X)
``````

Description

Computes e to the power of X, also called the natural exponential function. If the result underflows, this function returns a zero. Generates an error if the result overflows.

X EXP(X)
0.0 1.0
`+inf` `+inf`
`-inf` 0.0

Return Data Type

INPUT`INT64``NUMERIC``BIGNUMERIC``FLOAT64`
OUTPUT`FLOAT64``NUMERIC``BIGNUMERIC``FLOAT64`

### `EUCLIDEAN_DISTANCE`

``````EUCLIDEAN_DISTANCE(vector1, vector2)
``````

Description

Computes the Euclidean distance between two vectors.

Definitions

• `vector1`: A vector that is represented by an `ARRAY<T>` value or a sparse vector that is represented by an `ARRAY<STRUCT<dimension,magnitude>>` value.
• `vector2`: A vector that is represented by an `ARRAY<T>` value or a sparse vector that is represented by an `ARRAY<STRUCT<dimension,magnitude>>` value.

Details

• `ARRAY<T>` can be used to represent a vector. Each zero-based index in this array represents a dimension. The value for each element in this array represents a magnitude.

`T` can represent the following and must be the same for both vectors:

• `FLOAT64`

In the following example vector, there are four dimensions. The magnitude is `10.0` for dimension `0`, `55.0` for dimension `1`, `40.0` for dimension `2`, and `34.0` for dimension `3`:

``````[10.0, 55.0, 40.0, 34.0]
``````
• `ARRAY<STRUCT<dimension,magnitude>>` can be used to represent a sparse vector. With a sparse vector, you only need to include dimension-magnitude pairs for non-zero magnitudes. If a magnitude isn't present in the sparse vector, the magnitude is implicitly understood to be zero.

For example, if you have a vector with 10,000 dimensions, but only 10 dimensions have non-zero magnitudes, then the vector is a sparse vector. As a result, it's more efficient to describe a sparse vector by only mentioning its non-zero magnitudes.

In `ARRAY<STRUCT<dimension,magnitude>>`, `STRUCT<dimension,magnitude>` represents a dimension-magnitude pair for each non-zero magnitude in a sparse vector. These parts need to be included for each dimension-magnitude pair:

• `dimension`: A `STRING` or `INT64` value that represents a dimension in a vector.

• `magnitude`: A `FLOAT64` value that represents a non-zero magnitude for a specific dimension in a vector.

You don't need to include empty dimension-magnitude pairs in a sparse vector. For example, the following sparse vector and non-sparse vector are equivalent:

``````-- sparse vector ARRAY<STRUCT<INT64, FLOAT64>>
[(1, 10.0), (2: 30.0), (5, 40.0)]
``````
``````-- vector ARRAY<FLOAT64>
[0.0, 10.0, 30.0, 0.0, 0.0, 40.0]
``````

In a sparse vector, dimension-magnitude pairs don't need to be in any particular order. The following sparse vectors are equivalent:

``````[('a', 10.0), ('b': 30.0), ('d': 40.0)]
``````
``````[('d': 40.0), ('a', 10.0), ('b': 30.0)]
``````
• Both non-sparse vectors in this function must share the same dimensions, and if they don't, an error is produced.

• A vector can be a zero vector. A vector is a zero vector if it has no dimensions or all dimensions have a magnitude of `0`, such as `[]` or `[0.0, 0.0]`.

• An error is produced if a magnitude in a vector is `NULL`.

• If a vector is `NULL`, `NULL` is returned.

Return type

`FLOAT64`

Examples

In the following example, non-sparse vectors are used to compute the Euclidean distance:

``````SELECT EUCLIDEAN_DISTANCE([1.0, 2.0], [3.0, 4.0]) AS results;

/*----------*
| results  |
+----------+
| 2.828    |
*----------*/
``````

In the following example, sparse vectors are used to compute the Euclidean distance:

``````SELECT EUCLIDEAN_DISTANCE(
[(1, 1.0), (2, 2.0)],
[(2, 4.0), (1, 3.0)]) AS results;

/*----------*
| results  |
+----------+
| 2.828    |
*----------*/
``````

The ordering of magnitudes in a vector doesn't impact the results produced by this function. For example these queries produce the same results even though the magnitudes in each vector is in a different order:

``````SELECT EUCLIDEAN_DISTANCE([1.0, 2.0], [3.0, 4.0]);
``````
``````SELECT EUCLIDEAN_DISTANCE([2.0, 1.0], [4.0, 3.0]);
``````
``````SELECT EUCLIDEAN_DISTANCE([(1, 1.0), (2, 2.0)], [(1, 3.0), (2, 4.0)]) AS results;
``````
`````` /*----------*
| results  |
+----------+
| 2.828    |
*----------*/
``````

Both non-sparse vectors must have the same dimensions. If not, an error is produced. In the following example, the first vector has two dimensions and the second vector has three:

``````-- ERROR
SELECT EUCLIDEAN_DISTANCE([9.0, 7.0], [8.0, 4.0, 5.0]) AS results;
``````

If you use sparse vectors and you repeat a dimension, an error is produced:

``````-- ERROR
SELECT EUCLIDEAN_DISTANCE(
[(1, 9.0), (2, 7.0), (2, 8.0)], [(1, 8.0), (2, 4.0), (3, 5.0)]) AS results;
``````

### `FLOOR`

``````FLOOR(X)
``````

Description

Returns the largest integral value that is not greater than X.

X FLOOR(X)
2.0 2.0
2.3 2.0
2.8 2.0
2.5 2.0
-2.3 -3.0
-2.8 -3.0
-2.5 -3.0
0 0
`+inf` `+inf`
`-inf` `-inf`
`NaN` `NaN`

Return Data Type

INPUT`INT64``NUMERIC``BIGNUMERIC``FLOAT64`
OUTPUT`FLOAT64``NUMERIC``BIGNUMERIC``FLOAT64`

### `GREATEST`

``````GREATEST(X1,...,XN)
``````

Description

Returns the greatest value among `X1,...,XN`. If any argument is `NULL`, returns `NULL`. Otherwise, in the case of floating-point arguments, if any argument is `NaN`, returns `NaN`. In all other cases, returns the value among `X1,...,XN` that has the greatest value according to the ordering used by the `ORDER BY` clause. The arguments `X1, ..., XN` must be coercible to a common supertype, and the supertype must support ordering.

X1,...,XN GREATEST(X1,...,XN)
3,5,1 5

This function supports specifying collation.

Return Data Types

Data type of the input values.

### `IEEE_DIVIDE`

``````IEEE_DIVIDE(X, Y)
``````

Description

Divides X by Y; this function never fails. Returns `FLOAT64`. Unlike the division operator (/), this function does not generate errors for division by zero or overflow.

X Y IEEE_DIVIDE(X, Y)
20.0 4.0 5.0
0.0 25.0 0.0
25.0 0.0 `+inf`
-25.0 0.0 `-inf`
0.0 0.0 `NaN`
0.0 `NaN` `NaN`
`NaN` 0.0 `NaN`
`+inf` `+inf` `NaN`
`-inf` `-inf` `NaN`

### `IS_INF`

``````IS_INF(X)
``````

Description

Returns `TRUE` if the value is positive or negative infinity.

X IS_INF(X)
`+inf` `TRUE`
`-inf` `TRUE`
25 `FALSE`

### `IS_NAN`

``````IS_NAN(X)
``````

Description

Returns `TRUE` if the value is a `NaN` value.

X IS_NAN(X)
`NaN` `TRUE`
25 `FALSE`

### `LEAST`

``````LEAST(X1,...,XN)
``````

Description

Returns the least value among `X1,...,XN`. If any argument is `NULL`, returns `NULL`. Otherwise, in the case of floating-point arguments, if any argument is `NaN`, returns `NaN`. In all other cases, returns the value among `X1,...,XN` that has the least value according to the ordering used by the `ORDER BY` clause. The arguments `X1, ..., XN` must be coercible to a common supertype, and the supertype must support ordering.

X1,...,XN LEAST(X1,...,XN)
3,5,1 1

This function supports specifying collation.

Return Data Types

Data type of the input values.

### `LN`

``````LN(X)
``````

Description

Computes the natural logarithm of X. Generates an error if X is less than or equal to zero.

X LN(X)
1.0 0.0
`+inf` `+inf`
`X < 0` Error

Return Data Type

INPUT`INT64``NUMERIC``BIGNUMERIC``FLOAT64`
OUTPUT`FLOAT64``NUMERIC``BIGNUMERIC``FLOAT64`

### `LOG`

``````LOG(X [, Y])
``````

Description

If only X is present, `LOG` is a synonym of `LN`. If Y is also present, `LOG` computes the logarithm of X to base Y.

X Y LOG(X, Y)
100.0 10.0 2.0
`-inf` Any value `NaN`
Any value `+inf` `NaN`
`+inf` 0.0 < Y < 1.0 `-inf`
`+inf` Y > 1.0 `+inf`
X <= 0 Any value Error
Any value Y <= 0 Error
Any value 1.0 Error

Return Data Type

INPUT`INT64``NUMERIC``BIGNUMERIC``FLOAT64`
`INT64``FLOAT64``NUMERIC``BIGNUMERIC``FLOAT64`
`NUMERIC``NUMERIC``NUMERIC``BIGNUMERIC``FLOAT64`
`BIGNUMERIC``BIGNUMERIC``BIGNUMERIC``BIGNUMERIC``FLOAT64`
`FLOAT64``FLOAT64``FLOAT64``FLOAT64``FLOAT64`

### `LOG10`

``````LOG10(X)
``````

Description

Similar to `LOG`, but computes logarithm to base 10.

X LOG10(X)
100.0 2.0
`-inf` `NaN`
`+inf` `+inf`
X <= 0 Error

Return Data Type

INPUT`INT64``NUMERIC``BIGNUMERIC``FLOAT64`
OUTPUT`FLOAT64``NUMERIC``BIGNUMERIC``FLOAT64`

### `MOD`

``````MOD(X, Y)
``````

Description

Modulo function: returns the remainder of the division of X by Y. Returned value has the same sign as X. An error is generated if Y is 0.

X Y MOD(X, Y)
25 12 1
25 0 Error

Return Data Type

The return data type is determined by the argument types with the following table.

INPUT`INT64``NUMERIC``BIGNUMERIC`
`INT64``INT64``NUMERIC``BIGNUMERIC`
`NUMERIC``NUMERIC``NUMERIC``BIGNUMERIC`
`BIGNUMERIC``BIGNUMERIC``BIGNUMERIC``BIGNUMERIC`

### `POW`

``````POW(X, Y)
``````

Description

Returns the value of X raised to the power of Y. If the result underflows and is not representable, then the function returns a value of zero.

X Y POW(X, Y)
2.0 3.0 8.0
1.0 Any value including `NaN` 1.0
Any value including `NaN` 0 1.0
-1.0 `+inf` 1.0
-1.0 `-inf` 1.0
ABS(X) < 1 `-inf` `+inf`
ABS(X) > 1 `-inf` 0.0
ABS(X) < 1 `+inf` 0.0
ABS(X) > 1 `+inf` `+inf`
`-inf` Y < 0 0.0
`-inf` Y > 0 `-inf` if Y is an odd integer, `+inf` otherwise
`+inf` Y < 0 0
`+inf` Y > 0 `+inf`
Finite value < 0 Non-integer Error
0 Finite value < 0 Error

Return Data Type

The return data type is determined by the argument types with the following table.

INPUT`INT64``NUMERIC``BIGNUMERIC``FLOAT64`
`INT64``FLOAT64``NUMERIC``BIGNUMERIC``FLOAT64`
`NUMERIC``NUMERIC``NUMERIC``BIGNUMERIC``FLOAT64`
`BIGNUMERIC``BIGNUMERIC``BIGNUMERIC``BIGNUMERIC``FLOAT64`
`FLOAT64``FLOAT64``FLOAT64``FLOAT64``FLOAT64`

### `POWER`

``````POWER(X, Y)
``````

Description

Synonym of `POW(X, Y)`.

### `RAND`

``````RAND()
``````

Description

Generates a pseudo-random value of type `FLOAT64` in the range of [0, 1), inclusive of 0 and exclusive of 1.

### `RANGE_BUCKET`

``````RANGE_BUCKET(point, boundaries_array)
``````

Description

`RANGE_BUCKET` scans through a sorted array and returns the 0-based position of the point's upper bound. This can be useful if you need to group your data to build partitions, histograms, business-defined rules, and more.

`RANGE_BUCKET` follows these rules:

• If the point exists in the array, returns the index of the next larger value.

``````RANGE_BUCKET(20, [0, 10, 20, 30, 40]) -- 3 is return value
RANGE_BUCKET(20, [0, 10, 20, 20, 40, 40]) -- 4 is return value
``````
• If the point does not exist in the array, but it falls between two values, returns the index of the larger value.

``````RANGE_BUCKET(25, [0, 10, 20, 30, 40]) -- 3 is return value
``````
• If the point is smaller than the first value in the array, returns 0.

``````RANGE_BUCKET(-10, [5, 10, 20, 30, 40]) -- 0 is return value
``````
• If the point is greater than or equal to the last value in the array, returns the length of the array.

``````RANGE_BUCKET(80, [0, 10, 20, 30, 40]) -- 5 is return value
``````
• If the array is empty, returns 0.

``````RANGE_BUCKET(80, []) -- 0 is return value
``````
• If the point is `NULL` or `NaN`, returns `NULL`.

``````RANGE_BUCKET(NULL, [0, 10, 20, 30, 40]) -- NULL is return value
``````
• The data type for the point and array must be compatible.

``````RANGE_BUCKET('a', ['a', 'b', 'c', 'd']) -- 1 is return value
RANGE_BUCKET(1.2, [1, 1.2, 1.4, 1.6]) -- 2 is return value
RANGE_BUCKET(1.2, [1, 2, 4, 6]) -- execution failure
``````

Execution failure occurs when:

• The array has a `NaN` or `NULL` value in it.

``````RANGE_BUCKET(80, [NULL, 10, 20, 30, 40]) -- execution failure
``````
• The array is not sorted in ascending order.

``````RANGE_BUCKET(30, [10, 30, 20, 40, 50]) -- execution failure
``````

Parameters

• `point`: A generic value.
• `boundaries_array`: A generic array of values.

Return Value

`INT64`

Examples

In a table called `students`, check to see how many records would exist in each `age_group` bucket, based on a student's age:

• age_group 0 (age < 10)
• age_group 1 (age >= 10, age < 20)
• age_group 2 (age >= 20, age < 30)
• age_group 3 (age >= 30)
``````WITH students AS
(
SELECT 9 AS age UNION ALL
SELECT 20 AS age UNION ALL
SELECT 25 AS age UNION ALL
SELECT 31 AS age UNION ALL
SELECT 32 AS age UNION ALL
SELECT 33 AS age
)
SELECT RANGE_BUCKET(age, [10, 20, 30]) AS age_group, COUNT(*) AS count
FROM students
GROUP BY 1

/*--------------+-------*
| age_group    | count |
+--------------+-------+
| 0            | 1     |
| 2            | 2     |
| 3            | 3     |
*--------------+-------*/
``````

### `ROUND`

``````ROUND(X [, N [, rounding_mode]])
``````

Description

If only X is present, rounds X to the nearest integer. If N is present, rounds X to N decimal places after the decimal point. If N is negative, rounds off digits to the left of the decimal point. Rounds halfway cases away from zero. Generates an error if overflow occurs.

If X is a `NUMERIC` or `BIGNUMERIC` type, then you can explicitly set `rounding_mode` to one of the following:

If you set the `rounding_mode` and X is not a `NUMERIC` or `BIGNUMERIC` type, then the function generates an error.

Expression Return Value
`ROUND(2.0)` 2.0
`ROUND(2.3)` 2.0
`ROUND(2.8)` 3.0
`ROUND(2.5)` 3.0
`ROUND(-2.3)` -2.0
`ROUND(-2.8)` -3.0
`ROUND(-2.5)` -3.0
`ROUND(0)` 0
`ROUND(+inf)` `+inf`
`ROUND(-inf)` `-inf`
`ROUND(NaN)` `NaN`
`ROUND(123.7, -1)` 120.0
`ROUND(1.235, 2)` 1.24
`ROUND(NUMERIC "2.25", 1, "ROUND_HALF_EVEN")` 2.2
`ROUND(NUMERIC "2.35", 1, "ROUND_HALF_EVEN")` 2.4
`ROUND(NUMERIC "2.251", 1, "ROUND_HALF_EVEN")` 2.3
`ROUND(NUMERIC "-2.5", 0, "ROUND_HALF_EVEN")` -2
`ROUND(NUMERIC "2.5", 0, "ROUND_HALF_AWAY_FROM_ZERO")` 3
`ROUND(NUMERIC "-2.5", 0, "ROUND_HALF_AWAY_FROM_ZERO")` -3

Return Data Type

INPUT`INT64``NUMERIC``BIGNUMERIC``FLOAT64`
OUTPUT`FLOAT64``NUMERIC``BIGNUMERIC``FLOAT64`

### `SAFE_ADD`

``````SAFE_ADD(X, Y)
``````

Description

Equivalent to the addition operator (`+`), but returns `NULL` if overflow occurs.

5 4 9

Return Data Type

INPUT`INT64``NUMERIC``BIGNUMERIC``FLOAT64`
`INT64``INT64``NUMERIC``BIGNUMERIC``FLOAT64`
`NUMERIC``NUMERIC``NUMERIC``BIGNUMERIC``FLOAT64`
`BIGNUMERIC``BIGNUMERIC``BIGNUMERIC``BIGNUMERIC``FLOAT64`
`FLOAT64``FLOAT64``FLOAT64``FLOAT64``FLOAT64`

### `SAFE_DIVIDE`

``````SAFE_DIVIDE(X, Y)
``````

Description

Equivalent to the division operator (`X / Y`), but returns `NULL` if an error occurs, such as a division by zero error.

X Y SAFE_DIVIDE(X, Y)
20 4 5
0 20 `0`
20 0 `NULL`

Return Data Type

INPUT`INT64``NUMERIC``BIGNUMERIC``FLOAT64`
`INT64``FLOAT64``NUMERIC``BIGNUMERIC``FLOAT64`
`NUMERIC``NUMERIC``NUMERIC``BIGNUMERIC``FLOAT64`
`BIGNUMERIC``BIGNUMERIC``BIGNUMERIC``BIGNUMERIC``FLOAT64`
`FLOAT64``FLOAT64``FLOAT64``FLOAT64``FLOAT64`

### `SAFE_MULTIPLY`

``````SAFE_MULTIPLY(X, Y)
``````

Description

Equivalent to the multiplication operator (`*`), but returns `NULL` if overflow occurs.

X Y SAFE_MULTIPLY(X, Y)
20 4 80

Return Data Type

INPUT`INT64``NUMERIC``BIGNUMERIC``FLOAT64`
`INT64``INT64``NUMERIC``BIGNUMERIC``FLOAT64`
`NUMERIC``NUMERIC``NUMERIC``BIGNUMERIC``FLOAT64`
`BIGNUMERIC``BIGNUMERIC``BIGNUMERIC``BIGNUMERIC``FLOAT64`
`FLOAT64``FLOAT64``FLOAT64``FLOAT64``FLOAT64`

### `SAFE_NEGATE`

``````SAFE_NEGATE(X)
``````

Description

Equivalent to the unary minus operator (`-`), but returns `NULL` if overflow occurs.

X SAFE_NEGATE(X)
+1 -1
-1 +1
0 0

Return Data Type

INPUT`INT64``NUMERIC``BIGNUMERIC``FLOAT64`
OUTPUT`INT64``NUMERIC``BIGNUMERIC``FLOAT64`

### `SAFE_SUBTRACT`

``````SAFE_SUBTRACT(X, Y)
``````

Description

Returns the result of Y subtracted from X. Equivalent to the subtraction operator (`-`), but returns `NULL` if overflow occurs.

X Y SAFE_SUBTRACT(X, Y)
5 4 1

Return Data Type

INPUT`INT64``NUMERIC``BIGNUMERIC``FLOAT64`
`INT64``INT64``NUMERIC``BIGNUMERIC``FLOAT64`
`NUMERIC``NUMERIC``NUMERIC``BIGNUMERIC``FLOAT64`
`BIGNUMERIC``BIGNUMERIC``BIGNUMERIC``BIGNUMERIC``FLOAT64`
`FLOAT64``FLOAT64``FLOAT64``FLOAT64``FLOAT64`

### `SEC`

``````SEC(X)
``````

Description

Computes the secant for the angle of `X`, where `X` is specified in radians. `X` can be any data type that coerces to `FLOAT64`.

X SEC(X)
`+inf` `NaN`
`-inf` `NaN`
`NaN` `NaN`
`NULL` `NULL`

Return Data Type

`FLOAT64`

Example

``````SELECT SEC(100) AS a, SEC(-1) AS b;

/*----------------+---------------*
| a              | b             |
+----------------+---------------+
| 1.159663822905 | 1.85081571768 |
*----------------+---------------*/
``````

### `SECH`

``````SECH(X)
``````

Description

Computes the hyperbolic secant for the angle of `X`, where `X` is specified in radians. `X` can be any data type that coerces to `FLOAT64`. Never produces an error.

X SECH(X)
`+inf` `0`
`-inf` `0`
`NaN` `NaN`
`NULL` `NULL`

Return Data Type

`FLOAT64`

Example

``````SELECT SECH(0.5) AS a, SECH(-2) AS b, SECH(100) AS c;

/*----------------+----------------+---------------------*
| a              | b              | c                   |
+----------------+----------------+---------------------+
| 0.88681888397  | 0.265802228834 | 7.4401519520417E-44 |
*----------------+----------------+---------------------*/
``````

### `SIGN`

``````SIGN(X)
``````

Description

Returns `-1`, `0`, or `+1` for negative, zero and positive arguments respectively. For floating point arguments, this function does not distinguish between positive and negative zero.

X SIGN(X)
25 +1
0 0
-25 -1
NaN NaN

Return Data Type

INPUT`INT64``NUMERIC``BIGNUMERIC``FLOAT64`
OUTPUT`INT64``NUMERIC``BIGNUMERIC``FLOAT64`

### `SIN`

``````SIN(X)
``````

Description

Computes the sine of X where X is specified in radians. Never fails.

X SIN(X)
`+inf` `NaN`
`-inf` `NaN`
`NaN` `NaN`

### `SINH`

``````SINH(X)
``````

Description

Computes the hyperbolic sine of X where X is specified in radians. Generates an error if overflow occurs.

X SINH(X)
`+inf` `+inf`
`-inf` `-inf`
`NaN` `NaN`

### `SQRT`

``````SQRT(X)
``````

Description

Computes the square root of X. Generates an error if X is less than 0.

X SQRT(X)
`25.0` `5.0`
`+inf` `+inf`
`X < 0` Error

Return Data Type

INPUT`INT64``NUMERIC``BIGNUMERIC``FLOAT64`
OUTPUT`FLOAT64``NUMERIC``BIGNUMERIC``FLOAT64`

### `TAN`

``````TAN(X)
``````

Description

Computes the tangent of X where X is specified in radians. Generates an error if overflow occurs.

X TAN(X)
`+inf` `NaN`
`-inf` `NaN`
`NaN` `NaN`

### `TANH`

``````TANH(X)
``````

Description

Computes the hyperbolic tangent of X where X is specified in radians. Does not fail.

X TANH(X)
`+inf` 1.0
`-inf` -1.0
`NaN` `NaN`

### `TRUNC`

``````TRUNC(X [, N])
``````

Description

If only X is present, `TRUNC` rounds X to the nearest integer whose absolute value is not greater than the absolute value of X. If N is also present, `TRUNC` behaves like `ROUND(X, N)`, but always rounds towards zero and never overflows.

X TRUNC(X)
2.0 2.0
2.3 2.0
2.8 2.0
2.5 2.0
-2.3 -2.0
-2.8 -2.0
-2.5 -2.0
0 0
`+inf` `+inf`
`-inf` `-inf`
`NaN` `NaN`

Return Data Type

INPUT`INT64``NUMERIC``BIGNUMERIC``FLOAT64`
OUTPUT`FLOAT64``NUMERIC``BIGNUMERIC``FLOAT64`
[{ "type": "thumb-down", "id": "hardToUnderstand", "label":"Hard to understand" },{ "type": "thumb-down", "id": "incorrectInformationOrSampleCode", "label":"Incorrect information or sample code" },{ "type": "thumb-down", "id": "missingTheInformationSamplesINeed", "label":"Missing the information/samples I need" },{ "type": "thumb-down", "id": "otherDown", "label":"Other" }]
[{ "type": "thumb-up", "id": "easyToUnderstand", "label":"Easy to understand" },{ "type": "thumb-up", "id": "solvedMyProblem", "label":"Solved my problem" },{ "type": "thumb-up", "id": "otherUp", "label":"Other" }]