Mathematical Functions in Standard SQL

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.

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. Returns +inf for a +/-inf argument.

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. Returns NaN for a NaN argument.

IS_INF

IS_INF(X)

Description

Returns TRUE if the value is positive or negative infinity. Returns NULL for NULL inputs.

IS_NAN

IS_NAN(X)

Description

Returns TRUE if the value is a NaN value. Returns NULL for NULL inputs.

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.

Special cases:

If the result overflows, returns +/-inf.
If Y=0 and X=0, returns NaN.
If Y=0 and X!=0, returns +/-inf.
If X = +/-inf and Y = +/-inf, returns NaN.

The behavior of IEEE_DIVIDE is further illustrated in the table below.

Special cases for `IEEE_DIVIDE`

The following table lists special cases for IEEE_DIVIDE.

Numerator Data Type (X)	Denominator Data Type (Y)	Result Value
Anything except 0	0	`+/-inf`
0	0	`NaN`
0	`NaN`	`NaN`
`NaN`	0	`NaN`
`+/-inf`	`+/-inf`	`NaN`

RAND

RAND()

Description

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

SQRT

SQRT(X)

Description

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

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. Returns an error if one of the following is true:

X is a finite value less than 0 and Y is a noninteger
X is 0 and Y is a finite value less than 0

The behavior of POW() is further illustrated in the table below.

POWER

POWER(X, Y)

Description

Synonym of POW().

Special cases for `POW(X, Y)` and `POWER(X, Y)`

The following are special cases for POW(X, Y) and POWER(X, Y).

X	Y	POW(X, Y) or POWER(X, Y)
1.0	Any value including `NaN`	1.0
any including `NaN`	0	1.0
-1.0	`+/-inf`	1.0
ABS(X) < 1	`-inf`	`+inf`
ABS(X) > 1	`-inf`	0
ABS(X) < 1	`+inf`	0
ABS(X) > 1	`+inf`	`+inf`
`-inf`	Y < 0	0
`-inf`	Y > 0	`-inf` if Y is an odd integer, `+inf` otherwise
`+inf`	Y < 0	0
`+inf`	Y > 0	`+inf`

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. If X is +/-inf, then this function returns +inf or 0.

LN

LN(X)

Description

Computes the natural logarithm of X. Generates an error if X is less than or equal to zero. If X is +inf, then this function returns +inf.

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. Generates an error in these cases:

X is less than or equal to zero
Y is 1.0
Y is less than or equal to zero.

The behavior of LOG(X, Y) is further illustrated in the table below.

Special cases for `LOG(X, Y)`

X	Y	LOG(X, Y)
`-inf`	Any value	`NaN`
Any value	`+inf`	`NaN`
`+inf`	0.0 Y < 1.0	`-inf`
`+inf`	Y > 1.0	`+inf`

LOG10

LOG10(X)

Description

Similar to LOG, but computes logarithm to base 10.

GREATEST

GREATEST(X1,...,XN)

Description

Returns NULL if any of the inputs is NULL. Otherwise, returns NaN if any of the inputs is NaN. Otherwise, returns the largest value among X1,...,XN according to the < comparison.

LEAST

LEAST(X1,...,XN)

Description

Returns NULL if any of the inputs is NULL. Returns NaN if any of the inputs is NaN. Otherwise, returns the smallest value among X1,...,XN according to the > comparison.

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. See the table below for possible result types.

SAFE_DIVIDE

SAFE_DIVIDE(X, Y)

Description

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

SAFE_MULTIPLY

SAFE_MULTIPLY(X, Y)

Description

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

SAFE_NEGATE

SAFE_NEGATE(X)

Description

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

SAFE_ADD

SAFE_ADD(X, Y)

Description

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

SAFE_SUBTRACT

SAFE_SUBTRACT(X, Y)

Description

Equivalent to the subtraction operator (-), but returns NULL if overflow occurs.

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. See the table below for possible result types.

ROUND

ROUND(X [, N])

Description

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

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.

CEIL

CEIL(X)

Description

Returns the smallest integral value (with FLOAT64 type) that is not less than X.

CEILING

CEILING(X)

Description

Synonym of CEIL(X)

FLOOR

FLOOR(X)

Description

Returns the largest integral value (with FLOAT64 type) that is not greater than X.

Example rounding function behavior

Example behavior of BigQuery rounding functions:

Input "X"	ROUND(X)	TRUNC(X)	CEIL(X)	FLOOR(X)
2.0	2.0	2.0	2.0	2.0
2.3	2.0	2.0	3.0	2.0
2.8	3.0	2.0	3.0	2.0
2.5	3.0	2.0	3.0	2.0
-2.3	-2.0	-2.0	-2.0	-3.0
-2.8	-3.0	-2.0	-2.0	-3.0
-2.5	-3.0	-2.0	-2.0	-3.0
0	0	0	0	0
`+/-inf`	`+/-inf`	`+/-inf`	`+/-inf`	`+/-inf`
`NaN`	`NaN`	`NaN`	`NaN`	`NaN`

COS

COS(X)

Description

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

COSH

COSH(X)

Description

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

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].

ACOSH

ACOSH(X)

Description

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

SIN

SIN(X)

Description

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

SINH

SINH(X)

Description

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

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].

ASINH

ASINH(X)

Description

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

TAN

TAN(X)

Description

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

TANH

TANH(X)

Description

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

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.

ATANH

ATANH(X)

Description

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

ATAN2

ATAN2(Y, X)

Description

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

Special cases for `ATAN2()`

Y	X	ATAN2(Y, X)
`NaN`	Any value	`NaN`
Any value	`NaN`	`NaN`
0	0	0, π or -π depending on the sign of X and Y
Finite value	`-inf`	π or -π depending on the sign of Y
Finite value	`+inf`	0
`+/-inf`	Finite value	π/2 or π/2 depending on the sign of Y
`+/-inf`	`-inf`	¾π or -¾π depending on the sign of Y
`+/-inf`	`+inf`	π/4 or -π/4 depending on the sign of Y

Special cases for trigonometric and hyperbolic rounding functions

X	COS(X)	COSH(X)	ACOS(X)	ACOSH(X)	SIN(X)	SINH(X)	ASIN(X)	ASINH(X)	TAN(X)	TANH(X)	ATAN(X)	ATANH(X)
`+/-inf`	`NaN`	`=+inf`	`NaN`	`=+inf`	`NaN`	`=+inf`	`NaN`	`=+inf`	`NaN`	=+1.0	π/2	`NaN`
`-inf`	`NaN`	`=+inf`	`NaN`	`NaN`	`NaN`	`-inf`	`NaN`	`-inf`	`NaN`	-1.0	-π/2	`NaN`
`NaN`	`NaN`	`NaN`	`NaN`	`NaN`	`NaN`	`NaN`	`NaN`	`NaN`	`NaN`	`NaN`	`NaN`	`NaN`

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     |
+--------------+-------+

ABS

SIGN

IS_INF

IS_NAN

IEEE_DIVIDE

Special cases for IEEE_DIVIDE

RAND

SQRT

POW

POWER

Special cases for POW(X, Y) and POWER(X, Y)

EXP

LN

LOG

Special cases for LOG(X, Y)

LOG10

GREATEST

LEAST

DIV

SAFE_DIVIDE

SAFE_MULTIPLY

SAFE_NEGATE

SAFE_ADD

SAFE_SUBTRACT

MOD

ROUND

TRUNC

CEIL

CEILING

FLOOR

Example rounding function behavior

COS

COSH

ACOS

ACOSH

SIN

SINH

ASIN

ASINH

TAN

TANH

ATAN

ATANH

ATAN2

Special cases for ATAN2()

Special cases for trigonometric and hyperbolic rounding functions

RANGE_BUCKET

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Special cases for `IEEE_DIVIDE`

Special cases for `POW(X, Y)` and `POWER(X, Y)`

Special cases for `LOG(X, Y)`

Special cases for `ATAN2()`