Mathematical functions

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Google Standard SQL 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.

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

INPUTINT64NUMERICBIGNUMERICFLOAT64
OUTPUTINT64NUMERICBIGNUMERICFLOAT64

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

INPUTINT64NUMERICBIGNUMERICFLOAT64
OUTPUTINT64NUMERICBIGNUMERICFLOAT64

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

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

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.

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

Return Data Type

INPUTINT64NUMERICBIGNUMERICFLOAT64
OUTPUTFLOAT64NUMERICBIGNUMERICFLOAT64

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.

INPUTINT64NUMERICBIGNUMERICFLOAT64
INT64FLOAT64NUMERICBIGNUMERICFLOAT64
NUMERICNUMERICNUMERICBIGNUMERICFLOAT64
BIGNUMERICBIGNUMERICBIGNUMERICBIGNUMERICFLOAT64
FLOAT64FLOAT64FLOAT64FLOAT64FLOAT64

POWER

POWER(X, Y)

Description

Synonym of POW(X, Y).

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

INPUTINT64NUMERICBIGNUMERICFLOAT64
OUTPUTFLOAT64NUMERICBIGNUMERICFLOAT64

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

INPUTINT64NUMERICBIGNUMERICFLOAT64
OUTPUTFLOAT64NUMERICBIGNUMERICFLOAT64

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

INPUTINT64NUMERICBIGNUMERICFLOAT64
INT64FLOAT64NUMERICBIGNUMERICFLOAT64
NUMERICNUMERICNUMERICBIGNUMERICFLOAT64
BIGNUMERICBIGNUMERICBIGNUMERICBIGNUMERICFLOAT64
FLOAT64FLOAT64FLOAT64FLOAT64FLOAT64

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

INPUTINT64NUMERICBIGNUMERICFLOAT64
OUTPUTFLOAT64NUMERICBIGNUMERICFLOAT64

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.

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.

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.

INPUTINT64NUMERICBIGNUMERIC
INT64INT64NUMERICBIGNUMERIC
NUMERICNUMERICNUMERICBIGNUMERIC
BIGNUMERICBIGNUMERICBIGNUMERICBIGNUMERIC

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.

XYSAFE_DIVIDE(X, Y)
2045
0200
200NULL

Return Data Type

INPUTINT64NUMERICBIGNUMERICFLOAT64
INT64FLOAT64NUMERICBIGNUMERICFLOAT64
NUMERICNUMERICNUMERICBIGNUMERICFLOAT64
BIGNUMERICBIGNUMERICBIGNUMERICBIGNUMERICFLOAT64
FLOAT64FLOAT64FLOAT64FLOAT64FLOAT64

SAFE_MULTIPLY

SAFE_MULTIPLY(X, Y)

Description

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

XYSAFE_MULTIPLY(X, Y)
20480

Return Data Type

INPUTINT64NUMERICBIGNUMERICFLOAT64
INT64INT64NUMERICBIGNUMERICFLOAT64
NUMERICNUMERICNUMERICBIGNUMERICFLOAT64
BIGNUMERICBIGNUMERICBIGNUMERICBIGNUMERICFLOAT64
FLOAT64FLOAT64FLOAT64FLOAT64FLOAT64

SAFE_NEGATE

SAFE_NEGATE(X)

Description

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

XSAFE_NEGATE(X)
+1-1
-1+1
00

Return Data Type

INPUTINT64NUMERICBIGNUMERICFLOAT64
OUTPUTINT64NUMERICBIGNUMERICFLOAT64

SAFE_ADD

SAFE_ADD(X, Y)

Description

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

XYSAFE_ADD(X, Y)
549

Return Data Type

INPUTINT64NUMERICBIGNUMERICFLOAT64
INT64INT64NUMERICBIGNUMERICFLOAT64
NUMERICNUMERICNUMERICBIGNUMERICFLOAT64
BIGNUMERICBIGNUMERICBIGNUMERICBIGNUMERICFLOAT64
FLOAT64FLOAT64FLOAT64FLOAT64FLOAT64

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.

XYSAFE_SUBTRACT(X, Y)
541

Return Data Type

INPUTINT64NUMERICBIGNUMERICFLOAT64
INT64INT64NUMERICBIGNUMERICFLOAT64
NUMERICNUMERICNUMERICBIGNUMERICFLOAT64
BIGNUMERICBIGNUMERICBIGNUMERICBIGNUMERICFLOAT64
FLOAT64FLOAT64FLOAT64FLOAT64FLOAT64

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.

INPUTINT64NUMERICBIGNUMERIC
INT64INT64NUMERICBIGNUMERIC
NUMERICNUMERICNUMERICBIGNUMERIC
BIGNUMERICBIGNUMERICBIGNUMERICBIGNUMERIC

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 (Preview) 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

INPUTINT64NUMERICBIGNUMERICFLOAT64
OUTPUTFLOAT64NUMERICBIGNUMERICFLOAT64

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

INPUTINT64NUMERICBIGNUMERICFLOAT64
OUTPUTFLOAT64NUMERICBIGNUMERICFLOAT64

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

INPUTINT64NUMERICBIGNUMERICFLOAT64
OUTPUTFLOAT64NUMERICBIGNUMERICFLOAT64

CEILING

CEILING(X)

Description

Synonym of CEIL(X)

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

INPUTINT64NUMERICBIGNUMERICFLOAT64
OUTPUTFLOAT64NUMERICBIGNUMERICFLOAT64

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

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

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

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

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

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

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

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

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

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

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

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