Mathematical functions in GoogleSQL

GoogleSQL for Spanner 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   SIN   SINH   TAN   TANH  
Exponential and
logarithmic
EXP   LN   LOG   LOG10  
Rounding and
truncation
CEIL   CEILING   FLOOR   ROUND   TRUNC  
Power and
root
POW   POWER   SQRT  
Sign ABS   SIGN  
Distance COSINE_DISTANCE   EUCLIDEAN_DISTANCE  
Comparison GREATEST   LEAST  
Arithmetic and error handling DIV   IEEE_DIVIDE   IS_INF   IS_NAN   MOD   SAFE_ADD   SAFE_DIVIDE   SAFE_MULTIPLY   SAFE_NEGATE   SAFE_SUBTRACT  

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.
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.
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.
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.
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

INPUTINT64NUMERICFLOAT64
OUTPUTINT64NUMERICFLOAT64

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].

If X is NUMERIC then, the output is FLOAT64.

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.

If X is NUMERIC then, the output is FLOAT64.

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].

If X is NUMERIC then, the output is FLOAT64.

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.

If X is NUMERIC then, the output is FLOAT64.

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.

If X is NUMERIC then, the output is FLOAT64.

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].

If Y is NUMERIC then, the output is FLOAT64.

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).

If X is NUMERIC then, the output is FLOAT64.

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

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

INPUTINT64NUMERICFLOAT64
OUTPUTFLOAT64NUMERICFLOAT64

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.

If X is NUMERIC then, the output is FLOAT64.

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.
  • vector2: A vector that is represented by an ARRAY<T> 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]
    
  • Both 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,vectors are used to compute the cosine distance:

SELECT COSINE_DISTANCE([1.0, 2.0], [3.0, 4.0]) 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;
 /*----------*
  | 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;

Both 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;

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. If both inputs are NUMERIC and the result is overflow, then it returns a numeric overflow error.

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.

INPUTINT64NUMERIC
INT64INT64NUMERIC
NUMERICNUMERICNUMERIC

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

INPUTINT64NUMERICFLOAT64
OUTPUTFLOAT64NUMERICFLOAT64

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.
  • vector2: A vector that is represented by an ARRAY<T> 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]
    
  • Both 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, vectors are used to compute the Euclidean distance:

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

Both 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;

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

INPUTINT64NUMERICFLOAT64
OUTPUTFLOAT64NUMERICFLOAT64

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

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.

Returns FALSE for NUMERIC inputs since NUMERIC cannot be INF.

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.

Returns FALSE for NUMERIC inputs since NUMERIC cannot be NaN.

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

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

INPUTINT64NUMERICFLOAT64
OUTPUTFLOAT64NUMERICFLOAT64

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

INPUTINT64NUMERICFLOAT64
INT64FLOAT64NUMERICFLOAT64
NUMERICNUMERICNUMERICFLOAT64
FLOAT64FLOAT64FLOAT64FLOAT64

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

INPUTINT64NUMERICFLOAT64
OUTPUTFLOAT64NUMERICFLOAT64

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.

INPUTINT64NUMERIC
INT64INT64NUMERIC
NUMERICNUMERICNUMERIC

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.

INPUTINT64NUMERICFLOAT64
INT64FLOAT64NUMERICFLOAT64
NUMERICNUMERICNUMERICFLOAT64
FLOAT64FLOAT64FLOAT64FLOAT64

POWER

POWER(X, Y)

Description

Synonym of POW(X, Y).

ROUND

ROUND(X [, N])

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.

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

Return Data Type

INPUTINT64NUMERICFLOAT64
OUTPUTFLOAT64NUMERICFLOAT64

SAFE_ADD

SAFE_ADD(X, Y)

Description

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

X Y SAFE_ADD(X, Y)
5 4 9

Return Data Type

INPUTINT64NUMERICFLOAT64
INT64INT64NUMERICFLOAT64
NUMERICNUMERICNUMERICFLOAT64
FLOAT64FLOAT64FLOAT64FLOAT64

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

INPUTINT64NUMERICFLOAT64
INT64FLOAT64NUMERICFLOAT64
NUMERICNUMERICNUMERICFLOAT64
FLOAT64FLOAT64FLOAT64FLOAT64

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

INPUTINT64NUMERICFLOAT64
INT64INT64NUMERICFLOAT64
NUMERICNUMERICNUMERICFLOAT64
FLOAT64FLOAT64FLOAT64FLOAT64

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

INPUTINT64NUMERICFLOAT64
OUTPUTINT64NUMERICFLOAT64

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

INPUTINT64NUMERICFLOAT64
INT64INT64NUMERICFLOAT64
NUMERICNUMERICNUMERICFLOAT64
FLOAT64FLOAT64FLOAT64FLOAT64

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

INPUTINT64NUMERICFLOAT64
OUTPUTINT64NUMERICFLOAT64

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.

If X is NUMERIC then, the output is FLOAT64.

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

INPUTINT64NUMERICFLOAT64
OUTPUTFLOAT64NUMERICFLOAT64

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.

If X is NUMERIC then, the output is FLOAT64.

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

INPUTINT64NUMERICFLOAT64
OUTPUTFLOAT64NUMERICFLOAT64