# Ranking Expressions

This is a complete reference to the ranking expressions used to configure application specific ranking functions. For examples and an overview of how to use ranking expressions, see the ranking overview.

Ranking expressions are written in a simple language similar to ordinary functional notation. The atoms in ranking expressions are rank features and constants. These atoms can be combined by arithmetic operations and built-in functions.

Rank Features A rank feature is a named value calculated or looked up by vespa for each query/document combination. See the rank feature reference for a list of all the rank features available to ranking expressions. A constant is either a floating point number, or a quoted string. Strings are only useful for equality comparison in the if function.

## Arithmetic operations

The four basic operations, addition, subtraction, multiplication and division are expressed in in-fix notation:

a + b - c * d / (e + f)

Multiplication and division has higher precedence than addition and subtraction. Terms can be ordered using parentheses when it is necessary to override default precedence.

## Mathematical Functions

 cosh(x) Hyperbolic cosine of x sinh(x) Hyperbolic sine of x tanh(x) Hyperbolic tangent of x cos(x) Cosine of x sin(x) Sine of x tan(x) Tangent of x acos(x) Inverse cosine of x asin(x) Inverse sine of x atan2(y, x) Inverse tangent of y / x, using signs of both arguments to determine correct quadrant. atan(x) Inverse tangent of x exp(x) Base-e exponential function. ldexp(x, exp) Multiply x by 2 to the power of exp log10(x) Base-10 logarithm of x log(x) Base-e logarithm of x pow(x, y) Return x raised to the power of y sqrt(x) Square root of x ceil(x) Lowest integral value not less than x fabs(x) Absolute value of (floating-point) number x floor(x) Largest integral value not greater than x isNan(x) Returns 1.0 if x is NaN, 0.0 otherwise fmod(x, y) Remainder of x / y min(x, y) Smaller of x and y max(x, y) Larger of x and y

x and y may be any ranking expression.

## The if Function

The if function chooses between two sub-expressions based on the truth value of a condition.

if (expression1 operator expression2, trueExpression, falseExpression)

If the condition given in the first argument is true, the expression in argument 2 is used, otherwise argument 3. The four expressions may be any ranking expression. Conditional operators in ranking expression if functions:

 <= Less than or equal < Less than == Equal ~= Approximately equal >= Greater than or equal > Greater than

The in membership operator uses a slightly modified if-syntax:

if (expression1 in [expression2, expression3, ..., expressionN], trueExpression, falseExpression)

If expression1 is equal to either of expression1 through expressionN, then trueExpression is used, otherwise falseExpresion.

## The foreach Function

The foreach function is not really part of the expression language but implemented as a rank feature.

## Tensor Functions

The following set of tensors functions are available to use in ranking expressions. The functions are grouped in primitive functions and convenience functions that can be implemented in terms of the primitive ones.

### Primitive functions

map(tensor, f(x)(expr)) Returns a new tensor with the lambda function defined in f(x)(expr) applied to each cell. Arguments: tensor: a tensor expression. f(x)(expr): a lambda function with one argument. Returns a new tensor where the expression in the lambda function is evaluated in each cell in tensor. Examples: map(t, f(x)(abs(x))) map(t, f(i)(if(i < 0, 0, i)))  Returns a new tensor with the aggregator applied across dimensions dim1, dim2, etc. If no dimensions are specified, reduce over all dimensions. Arguments: tensor: a tensor expression. aggregator: the aggregator to use. See below. dim1, dim2, ...: the dimensions to reduce over. Optional. Returns a new tensor with the aggregator applied across dimensions dim1, dim2, etc. If no dimensions are specified, reduce over all dimensions. Available aggregators are: avg: arithmetic mean count: number of elements max: maximum value median: median value min: minimum value prod: product of all values sum: sum of all values Examples: reduce(t, sum) # Sum all values in tensor reduce(t, count, x) # Count number of cells along dimension x  Returns a new tensor constructed from the natural join between tensor1 and tensor2, with the resulting cells having the value as calculated from f(x,y)(expr), where x is the cell value from tensor1 and y from tensor2. Arguments: tensor1: a tensor expression. tensor2: a tensor expression. f(x,y)(expr): a lambda function with two arguments. Returns a new tensor constructed from the natural join between tensor1 and tensor2, with the resulting cells having the value as calculated from f(x,y)(expr), where x is the cell value from tensor1 and y from tensor2. Formally, the result of the join is a new tensor with dimensions the union of dimension between tensor1 and tensor2. The cells are the set of all combinations of cells that have equal values on their common dimensions. Examples: t1 = {{x:0}: 1.0, {x:1}: 2.0} t2 = {{x:0,y:0}: 3.0, {x:0,y:1}: 4.0, {x:1,y:0}: 5.0, {x:1,y:1}: 6.0} join(t1, t2, f(x,y)(x * y)) = {{x:0,y:0}: 3.0, {x:0,y:1}: 4.0, {x:1,y:0}: 10.0, {x:1,y:1}: 12.0} reduce(join(t1, t2, f(x,y)(x * y)), sum) = 29.0  Returns a new tensor consisting of all cells from both the arguments, where the lambda function is used to produce a single value in the cases where both arguments provide a value for a cell. Arguments: tensor1: a tensor expression. tensor2: a tensor expression. f(x,y)(expr): a lambda function with two arguments. Returns a new tensor having all the cells of both arguments, where the lambda is invoked to produce a single value only when both arguments have a value for the same cell. The argument tensors must have the same type, and that will be the type of the resulting tensor. Example: t1 = tensor(key{},x):{a:[1,2],b:[3,4]} t2 = tensor(key{},x):{b:[5,6],c:[7,8]} merge(t1, t2, f(left,right)(right)) = tensor(key{},x):{a:[1,2],b:[5,6],c:[7,8]}  Generates new tensors according to type specification and expression expr. Arguments: tensor-type-spec: an indexed tensor type specification. (expression): a lambda function expressing how to generate the tensor. Generates new tensors according to the type specification and expression expr. The tensor type must be an indexed tensor (e.g. tensor(x)). The expression in expr will be evaluated for each cell. The arguments in the expression is implicitly the names of the dimensions defined in the type spec. Useful for creating transformation tensors. Examples: t1 = tensor(x):[1.0, 2.0, 3.0, 4.0] tensor(x)(x) = [0.0, 1.0, 2.0] tensor(x,y)(x == y) = {{x:0,y:0}: 1.0, {x:0,y:1}: 0.0, {x:1,y:0}: 0.0, {x:1,y:1}: 1.0} tensor(f)(t1{x: f + f()})) - Create a size 2 1d tensor by accessing t1 at two indexes given by adding the index of it (f - which is 0 and 1) by a function also named f.  Renames one or more dimensions in the tensor. Arguments: tensor: a tensor expression. dim-to-rename: a dimension, or list of dimensions, to rename. new-names: new names for the dimensions listed above. Returns a new tensor with one or more dimension renamed. Examples: t1 = {{x:0,y:0}: 1.0, {x:0,y:1}: 0.0, {x:1,y:0}: 0.0, {x:1,y:1}: 1.0} rename(t1,x,z) = {{z:0,y:0}: 1.0, {z:0,y:1}: 0.0, {z:1,y:0}: 0.0, {z:1,y:1}: 1.0} rename(t1,(x,y),(i,j)) = {{i:0,j:0}: 1.0, {i:0,j:1}: 0.0, {i:1,j:0}: 0.0, {i:1,j:1}: 1.0}  Concatenates two tensors along dimension dim. Arguments: tensor1: a tensor expression. tensor2: a tensor expression. dim: the dimension to concatenate along. Returns a new tensor with the two tensors tensor1 and tensor2 concatenated along dimension dim. Examples: t1 = {{x:0}: 0.0, {x:1}: 1.0} t2 = {{x:0}: 2.0, {x:1}: 3.0} concat(t1,t2,x) = {{x:0}: 0.0, {x:1}: 1.0}, {x:2}: 2.0, {x:3}: 3.0}}  Slice - returns a new tensor containing the cells matching the partial address. Arguments: tensor: a tensor expression. partial-address: Can be given in the form of a tensor address {dimension:label,..}, or for tensors having a single mapped or indexed type respectively as just a label in curly brackets {label} or just an index in square brackets, [index]. Index labels may be specified by a lambda expression enclosed in parentheses. Returns a new tensor containing the cells matching the partial address. A common special case is producing a single value by specifying a full address. The type of the resulting tensor is the dimensions of the argument tensor not specified by the partial address. Examples: t1 = tensor(x):[1.0, 2.0]] t2 = tensor(key{}):{ key1:1.0, key2:2.0 } t3 = tensor(key{},x):{ key1:[1.0, 2.0], key2:[3.0, 4.0] } t1 = {2.0} t2{key1} = {1.0} t3{key1} = tensor(x):[1.0, 2.0] t3 = tensor(key{}):{key1:2.0, key2:4.0} t3{key:key1,x:1} = {2.0} t3{key:key1,x:(3-2)} = {2.0}  Returns a new tensor having the type and cell values given explicitly. Each cell value may be supplied by a lambda which can access other features. Returns a new tensor from the literal form, where the type must be specified explicitly. Each value may be supplied by a lambda, which - in contrast to all other lambdas - may refer to features and expressions from the context. Examples: t1 = tensor(x):[1.0, 2.0]] tensor(x{}):{x1:3, x2:4} = tensor(x{}):{x1:3.0, x2:4.0} tensor(x{}):{x1:t1, x2:t1} = tensor(x{}):{x1:2.0, x2:1.0} - Used in conjunction with slice, to convert an indexed tensor to mapped form. 

### Lambda functions in primitive functions

Some of the primitive functions accept lambda functions that are evaluated and applied to a set of tensor cells. The functions contain a single expression that have the same format and built-in functions as general ranking expressions. However, the atoms are the arguments defined in the argument list of the lambda.

The expression cannot access variables or data structures outside of the lambda, i.e. they are not closures.

Examples:

f(x)(abs(x))
f(x,y)(if(x < y, 0, 1))


### Non-primitive functions

Non-primitive functions can be implemented by primitive functions, but are not necessarily so for performance reasons.

 abs(t) map(t, f(x)(abs(x))) Absolute value of all elements. acos(t) map(t, f(x)(acos(x))) Arc cosine of all elements. t1 + t2 (add) join(t1, t2, f(x,y)(x + y)) Join and sum tensors t1 and t2. argmax(t, dim) join(t, max(t, dim), f(x,y)(if (x == y, 1, 0))) Returns a tensor with cell(s) of the highest value(s) in the tensor set to 1. The dimension argument follows the same format as reduce as multiple dimensions can be given and is optional. argmin(t, dim) join(t, min(t), f(x,y)(if (x >= y, 0, 1))) Returns a tensor with cell(s) of the lowest value(s) in the tensor set to 1. The dimension argument follows the same format as reduce as multiple dimensions can be given and is optional. asin(t) map(t, f(x)(asin(x))) Arc sine of all elements. atan(t) map(t, f(x)(atan(x))) Arc tangent of all elements. atan2(t1,t2) join(t1, t2, f(x,y)(atan2(x,y))) Arctangent of t1 and t2/ avg(t, dim) reduce(t, avg, dim) Reduce the tensor with the average aggregator along dimension dim. If the dimension argument is omitted, this reduces over all dimensions. ceil(t) map(t, f(x)(ceil(x))) Ceiling of all elements. count(t, dim) reduce(t, count, dim) Reduce the tensor with the count aggregator along dimension dim. If the dimension argument is omitted, this reduces over all dimensions. cos(t) map(t, f(x)(cos(x))) Cosine of all elements. cosh(t) map(t, f(x)(cosh(x))) Hyperbolic cosine of all elements. diag(n1,n2) tensor(i[n1],j[n2])(if (i==j, 1.0, 0.0))) Returns a tensor with the diagonal set to 1.0. t1 / t2 (div) join(t1, t2, f(x,y)(x / y)) Join and divide tensors t1 and t2. elu(t) map(t, f(x)(if(x < 0, exp(x)-1, x))) Exponential linear unit. t1 == t2 (equal) join(t1, t2, f(x,y)(x == y)) Join and determine if each element in t1 and t2 are equal. exp(t) map(t, f(x)(exp(x))) Exponential function (e^x) of each element. floor(t) map(t, f(x)(floor(x))) Floor of each element. t1 > t2 (greater) join(t1, t2, f(x,y)(x > y)) Join and determine if each element in t1 is greater than t2. t1 >= t2 (greater or equals) join(t1, t2, f(x,y)(x >= y)) Join and determine if each element in t1 is greater than or equals t2. t1 < t2 (less) join(t1, t2, f(x,y)(x < y)) Join and determine if each element in t1 is less than t2. t1 <= t2 (less equals) join(t1, t2, f(x,y)(x <= y)) Join and determine if each element in t1 is less than or equals t2. l1_normalize(t, dim) join(t, reduce(t, sum, dim), f(x,y) (x / y)) L1 normalization: t / sum(t, dim). l2_normalize(t, dim) join(t, map(reduce(map(t, f(x)(x * x)), sum, dim), f(x)(sqrt(x))), f(x,y)(x / y)) L2 normalization: t / sqrt(sum(t^2, dim). log(t) map(t, f(x)(log(x))) Natural logarithm of each element. log10(t) map(t, f(x)(log10(x))) Logarithm with base 10 of each element. matmul(t1, t2, dim) reduce(join(t1, t2, f(x,y)(x * y)), sum, dim) Matrix multiplication of two tensors. This is the product of the two tensors summed along a shared dimension. max(t, dim) reduce(t, max, dim) Reduce the tensor with the max aggregator along dimension dim. If the dimension argument is omitted, this reduces over all dimensions. max(t1,t2) join(t1, t2, f(x,y)(max(x,y))) Join and return the max of t1 or t2. Arguments can be scalars. median(t, dim) reduce(t, median, dim) Reduce the tensor with the median aggregator along dimension dim. If the dimension argument is omitted, this reduces over all dimensions. min(t, dim) reduce(t, min, dim) Reduce the tensor with the min aggregator along dimension dim. If the dimension argument is omitted, this reduces over all dimensions. min(t1,t2) join(t1, t2, f(x,y)(min(x,y))) Join and return the minimum of t1 or t2. Arguments can be scalars. mod(t,constant) map(t, f(x)(mod(x,constant))) Modulus of constant with each element. t1 * t2 (mul) join(t1, t2, f(x,y)(x * y)) Join and multiply tensors t1 and t2. t1 != t2 (not equal) join(t1, t2, f(x,y)(x != y)) Join and determine if each element in t1 and t2 are not equal. pow(t,constant) map(t, f(x)(pow(x,constant))) Raise each element to the power of constant. prod(t, dim) reduce(t, prod, dim) Reduce the tensor with the product aggregator along dimension dim. If the dimension argument is omitted, this reduces over all dimensions. random(n1, n2, ...) tensor(i1[n1],i2[n2],...)(random(1.0)) Returns a tensor with random values between 0.0 and 1.0, uniform distribution. range(n) tensor(i[n])(i) Returns a tensor with increasing values. relu(t) map(t, f(x)(max(0,x))) Rectified linear unit. round(t) map(t, f(x)(round(x))) Round each element. sigmoid(t) map(t, f(x)(1.0 / (1.0 + exp(0.0-x)))) Returns the sigmoid of each element. sin(t) map(t, f(x)(sin(x))) Sinus of each element. sinh(t) map(t, f(x)(sinh(x))) Hyperbolic sinus of each element. sign(t) map(t, f(x)(if(x < 0, -1.0, 1.0))) The sign of each element. softmax(t, dim) join(map(t, f(x)(exp(x))), reduce(map(t, f(x)(exp(x))), sum, dim), f(x,y)(x / y)) The softmax of the tensor, e.g. e^x / sum(e^x). sqrt(t) map(t, f(x)(sqrt(x))) The square root of each element. square(t) map(t, f(x)(square(x))) The square of each element. t1 - t2 (subtract) join(t1, t2, f(x,y)(x - y)) Join and subtract tensors t1 and t2. sum(t, dim) reduce(t, sum, dim) Reduce the tensor with the summation aggregator along dimension dim. If the dimension argument is omitted, this reduces over all dimensions. tan(t) map(t, f(x)(tan(x))) The tangent of each element. tanh(t) map(t, f(x)(tanh(x))) The hyperbolic tangent of each element. xw_plus_b(x,w,b,dim) join(reduce(join(x, w, f(x,y)(x * y)), sum, dim), b, f(x,y)(x+y)) Matrix multiplication of x (usually a vector) and w (weights), with b added (bias). A typical operation for activations in a neural network layer, e.g. sigmoid(xw_plus_b(x,w,b))).