%YAML 1.2
---
scalar_functions:
  -
    name: "add"
    description: "Add two values."
    impls:
      - args:
          - name: x
            value: i8
          - name: y
            value: i8
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        return: i8
      - args:
          - name: x
            value: i16
          - name: y
            value: i16
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        return: i16
      - args:
          - name: x
            value: i32
          - name: y
            value: i32
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        return: i32
      - args:
          - value: i64
          - value: i64
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        return: i64
      - args:
          - name: x
            value: fp32
          - name: y
            value: fp32
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        return: fp32
      - args:
          - name: x
            value: fp64
          - name: y
            value: fp64
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        return: fp64
  -
    name: "subtract"
    description: "Subtract one value from another."
    impls:
      - args:
          - name: x
            value: i8
          - name: y
            value: i8
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        return: i8
      - args:
          - name: x
            value: i16
          - name: y
            value: i16
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        return: i16
      - args:
          - name: x
            value: i32
          - name: y
            value: i32
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        return: i32
      - args:
          - name: x
            value: i64
          - name: y
            value: i64
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        return: i64
      - args:
          - name: x
            value: fp32
          - name: y
            value: fp32
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        return: fp32
      - args:
          - name: x
            value: fp64
          - name: y
            value: fp64
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        return: fp64
  -
    name: "multiply"
    description: "Multiply two values."
    impls:
      - args:
          - name: x
            value: i8
          - name: y
            value: i8
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        return: i8
      - args:
          - name: x
            value: i16
          - name: y
            value: i16
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        return: i16
      - args:
          - name: x
            value: i32
          - name: y
            value: i32
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        return: i32
      - args:
          - name: x
            value: i64
          - name: y
            value: i64
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        return: i64
      - args:
          - name: x
            value: fp32
          - name: y
            value: fp32
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        return: fp32
      - args:
          - name: x
            value: fp64
          - name: y
            value: fp64
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        return: fp64
  -
    name: "divide"
    description: >
      Divide x by y. In the case of integer division, partial values are truncated (i.e. rounded towards 0).
      The `on_division_by_zero` option governs behavior in cases where y is 0 and x is not 0.
      `LIMIT` means positive or negative infinity (depending on the sign of x and y).
      If x and y are both 0 or both +/-infinity, behavior will be governed by `on_domain_error`.
    impls:
      - args:
          - name: x
            value: i8
          - name: y
            value: i8
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        return: i8
      - args:
          - name: x
            value: i16
          - name: y
            value: i16
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        return: i16
      - args:
          - name: x
            value: i32
          - name: y
            value: i32
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        return: i32
      - args:
          - name: x
            value: i64
          - name: y
            value: i64
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        return: i64
      - args:
          - name: x
            value: fp32
          - name: y
            value: fp32
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
          on_domain_error:
            values: [ NAN, ERROR ]
          on_division_by_zero:
            values: [ LIMIT, NAN, ERROR ]
        return: fp32
      - args:
          - name: x
            value: fp64
          - name: y
            value: fp64
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
          on_domain_error:
            values: [ NAN, ERROR ]
          on_division_by_zero:
            values: [ LIMIT, NAN, ERROR ]
        return: fp64
  -
    name: "negate"
    description: "Negation of the value"
    impls:
      - args:
          - name: x
            value: i8
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        return: i8
      - args:
          - name: x
            value: i16
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        return: i16
      - args:
          - name: x
            value: i32
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        return: i32
      - args:
          - name: x
            value: i64
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        return: i64
      - args:
          - name: x
            value: fp32
        return: fp32
      - args:
          - name: x
            value: fp64
        return: fp64
  -
    name: "modulus"
    description: >
      Calculate the remainder (r) when dividing dividend (x) by divisor (y).

      In mathematics, many conventions for the modulus (mod) operation exists. The result of a mod operation
      depends on the software implementation and underlying hardware. Substrait is a format for describing compute
      operations on structured data and designed for interoperability. Therefore the user is responsible for determining
      a definition of division as defined by the quotient (q).

      The following basic conditions of division are satisfied:
      (1) q ∈ ℤ (the quotient is an integer)
      (2) x = y * q + r (division rule)
      (3) abs(r) < abs(y)
      where q is the quotient.

      The `division_type` option determines the mathematical definition of quotient to use in the above definition of
      division.

      When `division_type`=TRUNCATE, q = trunc(x/y).
      When `division_type`=FLOOR, q = floor(x/y).

      In the cases of TRUNCATE and FLOOR division: remainder r = x - round_func(x/y)

      The `on_domain_error` option governs behavior in cases where y is 0, y is +/-inf, or x is +/-inf. In these cases
      the mod is undefined.
      The `overflow` option governs behavior when integer overflow occurs.
      If x and y are both 0 or both +/-infinity, behavior will be governed by `on_domain_error`.
    impls:
      - args:
          - name: x
            value: i8
          - name: y
            value: i8
        options:
          division_type:
            values: [ TRUNCATE, FLOOR ]
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
          on_domain_error:
            values: [ "NULL", ERROR ]
        return: i8
      - args:
          - name: x
            value: i16
          - name: y
            value: i16
        options:
          division_type:
            values: [ TRUNCATE, FLOOR ]
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
          on_domain_error:
            values: [ "NULL", ERROR ]
        return: i16
      - args:
          - name: x
            value: i32
          - name: y
            value: i32
        options:
          division_type:
            values: [ TRUNCATE, FLOOR ]
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
          on_domain_error:
            values: [ "NULL", ERROR ]
        return: i32
      - args:
          - name: x
            value: i64
          - name: y
            value: i64
        options:
          division_type:
            values: [ TRUNCATE, FLOOR ]
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
          on_domain_error:
            values: [ "NULL", ERROR ]
        return: i64
  -
    name: "power"
    description: "Take the power with x as the base and y as exponent."
    impls:
      - args:
          - name: x
            value: i64
          - name: y
            value: i64
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        return: i64
      - args:
          - name: x
            value: fp32
          - name: y
            value: fp32
        return: fp32
      - args:
          - name: x
            value: fp64
          - name: y
            value: fp64
        return: fp64
  -
    name: "sqrt"
    description: "Square root of the value"
    impls:
      - args:
          - name: x
            value: i64
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
          on_domain_error:
            values: [ NAN, ERROR ]
        return: fp64
      - args:
          - name: x
            value: fp32
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
          on_domain_error:
            values: [ NAN, ERROR ]
        return: fp32
      - args:
          - name: x
            value: fp64
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
          on_domain_error:
            values: [ NAN, ERROR ]
        return: fp64
  -
    name: "exp"
    description: "The mathematical constant e, raised to the power of the value."
    impls:
      - args:
          - name: x
            value: fp32
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        return: fp32
      - args:
          - name: x
            value: fp64
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        return: fp64
  -
    name: "cos"
    description: "Get the cosine of a value in radians."
    impls:
      - args:
          - name: x
            value: fp32
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        return: fp64
      - args:
          - name: x
            value: fp64
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        return: fp64
  -
    name: "sin"
    description: "Get the sine of a value in radians."
    impls:
      - args:
          - name: x
            value: fp32
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        return: fp64
      - args:
          - name: x
            value: fp64
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        return: fp64
  -
    name: "tan"
    description: "Get the tangent of a value in radians."
    impls:
      - args:
          - name: x
            value: fp32
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        return: fp64
      - args:
          - name: x
            value: fp64
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        return: fp64
  -
    name: "cosh"
    description: "Get the hyperbolic cosine of a value in radians."
    impls:
      - args:
          - name: x
            value: fp32
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        return: fp32
      - args:
          - name: x
            value: fp64
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        return: fp64
  -
    name: "sinh"
    description: "Get the hyperbolic sine of a value in radians."
    impls:
      - args:
          - name: x
            value: fp32
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        return: fp32
      - args:
          - name: x
            value: fp64
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        return: fp64
  -
    name: "tanh"
    description: "Get the hyperbolic tangent of a value in radians."
    impls:
      - args:
          - name: x
            value: fp32
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        return: fp32
      - args:
          - name: x
            value: fp64
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        return: fp64
  -
    name: "acos"
    description: "Get the arccosine of a value in radians."
    impls:
      - args:
          - name: x
            value: fp32
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
          on_domain_error:
            values: [ NAN, ERROR ]
        return: fp64
      - args:
          - name: x
            value: fp64
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
          on_domain_error:
            values: [ NAN, ERROR ]
        return: fp64
  -
    name: "asin"
    description: "Get the arcsine of a value in radians."
    impls:
      - args:
          - name: x
            value: fp32
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
          on_domain_error:
            values: [ NAN, ERROR ]
        return: fp64
      - args:
          - name: x
            value: fp64
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
          on_domain_error:
            values: [ NAN, ERROR ]
        return: fp64
  -
    name: "atan"
    description: "Get the arctangent of a value in radians."
    impls:
      - args:
          - name: x
            value: fp32
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        return: fp64
      - args:
          - name: x
            value: fp64
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        return: fp64
  -
    name: "acosh"
    description: "Get the hyperbolic arccosine of a value in radians."
    impls:
      - args:
          - name: x
            value: fp32
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
          on_domain_error:
            values: [ NAN, ERROR ]
        return: fp32
      - args:
          - name: x
            value: fp64
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
          on_domain_error:
            values: [ NAN, ERROR ]
        return: fp64
  -
    name: "asinh"
    description: "Get the hyperbolic arcsine of a value in radians."
    impls:
      - args:
          - name: x
            value: fp32
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        return: fp32
      - args:
          - name: x
            value: fp64
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        return: fp64
  -
    name: "atanh"
    description: "Get the hyperbolic arctangent of a value in radians."
    impls:
      - args:
          - name: x
            value: fp32
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
          on_domain_error:
            values: [ NAN, ERROR ]
        return: fp32
      - args:
          - name: x
            value: fp64
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
          on_domain_error:
            values: [ NAN, ERROR ]
        return: fp64
  -
    name: "atan2"
    description: "Get the arctangent of values given as x/y pairs."
    impls:
      - args:
          - name: x
            value: fp32
          - name: y
            value: fp32
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
          on_domain_error:
            values: [ NAN, ERROR ]
        return: fp64
      - args:
          - name: x
            value: fp64
          - name: y
            value: fp64
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
          on_domain_error:
            values: [ NAN, ERROR ]
        return: fp64
  -
    name: "radians"
    description: >
      Converts angle `x` in degrees to radians.

    impls:
      - args:
          - name: x
            value: fp32
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        return: fp32
      - args:
          - name: x
            value: fp64
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        return: fp64
  -
    name: "degrees"
    description: >
      Converts angle `x` in radians to degrees.

    impls:
      - args:
          - name: x
            value: fp32
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        return: fp32
      - args:
          - name: x
            value: fp64
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        return: fp64
  -
    name: "abs"
    description: >
      Calculate the absolute value of the argument.

      Integer values allow the specification of overflow behavior to handle the
      unevenness of the twos complement, e.g. Int8 range [-128 : 127].
    impls:
      - args:
          - name: x
            value: i8
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        return: i8
      - args:
          - name: x
            value: i16
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        return: i16
      - args:
          - name: x
            value: i32
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        return: i32
      - args:
          - name: x
            value: i64
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        return: i64
      - args:
          - name: x
            value: fp32
        return: fp32
      - args:
          - name: x
            value: fp64
        return: fp64
  -
    name: "sign"
    description: >
      Return the signedness of the argument.

      Integer values return signedness with the same type as the input.
      Possible return values are [-1, 0, 1]

      Floating point values return signedness with the same type as the input.
      Possible return values are [-1.0, -0.0, 0.0, 1.0, NaN]
    impls:
      - args:
          - name: x
            value: i8
        return: i8
      - args:
          - name: x
            value: i16
        return: i16
      - args:
          - name: x
            value: i32
        return: i32
      - args:
          - name: x
            value: i64
        return: i64
      - args:
          - name: x
            value: fp32
        return: fp32
      - args:
          - name: x
            value: fp64
        return: fp64
  -
    name: "factorial"
    description: >
      Return the factorial of a given integer input.

      The factorial of 0! is 1 by convention.

      Negative inputs will raise an error.
    impls:
      - args:
          - value: i32
            name: "n"
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        return: i32
      - args:
          - value: i64
            name: "n"
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        return: i64
  -
    name: "bitwise_not"
    description: >
      Return the bitwise NOT result for one integer input.

    impls:
      - args:
          - name: x
            value: i8
        return: i8
      - args:
          - name: x
            value: i16
        return: i16
      - args:
          - name: x
            value: i32
        return: i32
      - args:
          - name: x
            value: i64
        return: i64
  -
    name: "bitwise_and"
    description: >
      Return the bitwise AND result for two integer inputs.

    impls:
      - args:
          - name: x
            value: i8
          - name: y
            value: i8
        return: i8
      - args:
          - name: x
            value: i16
          - name: y
            value: i16
        return: i16
      - args:
          - name: x
            value: i32
          - name: y
            value: i32
        return: i32
      - args:
          - name: x
            value: i64
          - name: y
            value: i64
        return: i64
  -
    name: "bitwise_or"
    description: >
      Return the bitwise OR result for two given integer inputs.

    impls:
      - args:
          - name: x
            value: i8
          - name: y
            value: i8
        return: i8
      - args:
          - name: x
            value: i16
          - name: y
            value: i16
        return: i16
      - args:
          - name: x
            value: i32
          - name: y
            value: i32
        return: i32
      - args:
          - name: x
            value: i64
          - name: y
            value: i64
        return: i64
  -
    name: "bitwise_xor"
    description: >
      Return the bitwise XOR result for two integer inputs.

    impls:
      - args:
          - name: x
            value: i8
          - name: y
            value: i8
        return: i8
      - args:
          - name: x
            value: i16
          - name: y
            value: i16
        return: i16
      - args:
          - name: x
            value: i32
          - name: y
            value: i32
        return: i32
      - args:
          - name: x
            value: i64
          - name: y
            value: i64
        return: i64

aggregate_functions:
  - name: "sum"
    description: Sum a set of values. The sum of zero elements yields null.
    impls:
      - args:
          - name: x
            value: i8
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: i64?
        return: i64?
      - args:
          - name: x
            value: i16
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: i64?
        return: i64?
      - args:
          - name: x
            value: i32
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: i64?
        return: i64?
      - args:
          - name: x
            value: i64
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: i64?
        return: i64?
      - args:
          - name: x
            value: fp32
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: fp64?
        return: fp64?
      - args:
          - name: x
            value: fp64
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: fp64?
        return: fp64?
  - name: "sum0"
    description: >
      Sum a set of values. The sum of zero elements yields zero.

      Null values are ignored.
    impls:
      - args:
          - name: x
            value: i8
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: i64
        return: i64
      - args:
          - name: x
            value: i16
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: i64
        return: i64
      - args:
          - name: x
            value: i32
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: i64
        return: i64
      - args:
          - name: x
            value: i64
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: i64
        return: i64
      - args:
          - name: x
            value: fp32
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: fp64
        return: fp64
      - args:
          - name: x
            value: fp64
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: fp64
        return: fp64
  - name: "avg"
    description: Average a set of values. For integral types, this truncates partial values.
    impls:
      - args:
          - name: x
            value: i8
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: "STRUCT<i64,i64>"
        return: i8?
      - args:
          - name: x
            value: i16
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: "STRUCT<i64,i64>"
        return: i16?
      - args:
          - name: x
            value: i32
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: "STRUCT<i64,i64>"
        return: i32?
      - args:
          - name: x
            value: i64
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: "STRUCT<i64,i64>"
        return: i64?
      - args:
          - name: x
            value: fp32
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: "STRUCT<fp64,i64>"
        return: fp32?
      - args:
          - name: x
            value: fp64
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: "STRUCT<fp64,i64>"
        return: fp64?
  - name: "min"
    description: Min a set of values.
    impls:
      - args:
          - name: x
            value: i8
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: i8?
        return: i8?
      - args:
          - name: x
            value: i16
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: i16?
        return: i16?
      - args:
          - name: x
            value: i32
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: i32?
        return: i32?
      - args:
          - name: x
            value: i64
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: i64?
        return: i64?
      - args:
          - name: x
            value: fp32
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: fp32?
        return: fp32?
      - args:
          - name: x
            value: fp64
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: fp64?
        return: fp64?
      - args:
          - name: x
            value: timestamp
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: timestamp?
        return: timestamp?
      - args:
          - name: x
            value: timestamp_tz
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: timestamp_tz?
        return: timestamp_tz?
  - name: "max"
    description: Max a set of values.
    impls:
      - args:
          - name: x
            value: i8
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: i8?
        return: i8?
      - args:
          - name: x
            value: i16
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: i16?
        return: i16?
      - args:
          - name: x
            value: i32
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: i32?
        return: i32?
      - args:
          - name: x
            value: i64
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: i64?
        return: i64?
      - args:
          - name: x
            value: fp32
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: fp32?
        return: fp32?
      - args:
          - name: x
            value: fp64
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: fp64?
        return: fp64?
      - args:
          - name: x
            value: timestamp
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: timestamp?
        return: timestamp?
      - args:
          - name: x
            value: timestamp_tz
        nullability: DECLARED_OUTPUT
        decomposable: MANY
        intermediate: timestamp_tz?
        return: timestamp_tz?
  - name: "product"
    description: Product of a set of values. Returns 1 for empty input.
    impls:
      - args:
          - name: x
            value: i8
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        nullability: MIRROR
        decomposable: MANY
        intermediate: i64
        return: i8
      - args:
          - name: x
            value: i16
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        nullability: MIRROR
        decomposable: MANY
        intermediate: i64
        return: i16
      - args:
          - name: x
            value: i32
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        nullability: MIRROR
        decomposable: MANY
        intermediate: i64
        return: i32
      - args:
          - name: x
            value: i64
        options:
          overflow:
            values: [ SILENT, SATURATE, ERROR ]
        nullability: MIRROR
        decomposable: MANY
        intermediate: i64
        return: i64
      - args:
          - name: x
            value: fp32
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        nullability: MIRROR
        decomposable: MANY
        intermediate: fp64
        return: fp32
      - args:
          - name: x
            value: fp64
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        nullability: MIRROR
        decomposable: MANY
        intermediate: fp64
        return: fp64
  - name: "std_dev"
    description: Calculates standard-deviation for a set of values.
    impls:
      - args:
          - name: x
            value: fp32
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
          distribution:
            values: [ SAMPLE, POPULATION]
        nullability: DECLARED_OUTPUT
        return: fp32?
      - args:
          - name: x
            value: fp64
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
          distribution:
            values: [ SAMPLE, POPULATION]
        nullability: DECLARED_OUTPUT
        return: fp64?
  - name: "variance"
    description: Calculates variance for a set of values.
    impls:
      - args:
          - name: x
            value: fp32
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
          distribution:
            values: [ SAMPLE, POPULATION]
        nullability: DECLARED_OUTPUT
        return: fp32?
      - args:
          - name: x
            value: fp64
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
          distribution:
            values: [ SAMPLE, POPULATION]
        nullability: DECLARED_OUTPUT
        return: fp64?
  - name: "corr"
    description: >
      Calculates the value of Pearson's correlation coefficient between `x` and `y`.
      If there is no input, null is returned.
    impls:
      - args:
          - name: x
            value: fp32
          - name: y
            value: fp32
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        nullability: DECLARED_OUTPUT
        return: fp32?
      - args:
          - name: x
            value: fp64
          - name: y
            value: fp64
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        nullability: DECLARED_OUTPUT
        return: fp64?
  - name: "mode"
    description: >
      Calculates mode for a set of values.
      If there is no input, null is returned.
    impls:
      - args:
          - name: x
            value: i8
        nullability: DECLARED_OUTPUT
        return: i8?
      - args:
          - name: x
            value: i16
        nullability: DECLARED_OUTPUT
        return: i16?
      - args:
          - name: x
            value: i32
        nullability: DECLARED_OUTPUT
        return: i32?
      - args:
          - name: x
            value: i64
        nullability: DECLARED_OUTPUT
        return: i64?
      - args:
          - name: x
            value: fp32
        nullability: DECLARED_OUTPUT
        return: fp32?
      - args:
          - name: x
            value: fp64
        nullability: DECLARED_OUTPUT
        return: fp64?
  - name: "median"
    description: >
      Calculate the median for a set of values.

      Returns null if applied to zero records. For the integer implementations,
      the rounding option determines how the median should be rounded if it ends
      up midway between two values. For the floating point implementations,
      they specify the usual floating point rounding mode.
    impls:
      - args:
          - name: precision
            description: >
              Based on required operator performance and configured optimizations
              on saving memory bandwidth, the precision of the end result can be
              the highest possible accuracy or an approximation.

                - EXACT: provides the exact result, rounded if needed according
                  to the rounding option.
                - APPROXIMATE: provides only an estimate; the result must lie
                  between the minimum and maximum values in the input
                  (inclusive), but otherwise the accuracy is left up to the
                  consumer.
            options: [ EXACT, APPROXIMATE ]
          - name: x
            value: i8
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        nullability: DECLARED_OUTPUT
        return: i8?
      - args:
          - name: precision
            description: >
              Based on required operator performance and configured optimizations
              on saving memory bandwidth, the precision of the end result can be
              the highest possible accuracy or an approximation.

                - EXACT: provides the exact result, rounded if needed according
                  to the rounding option.
                - APPROXIMATE: provides only an estimate; the result must lie
                  between the minimum and maximum values in the input
                  (inclusive), but otherwise the accuracy is left up to the
                  consumer.
            options: [ EXACT, APPROXIMATE ]
          - name: x
            value: i16
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        nullability: DECLARED_OUTPUT
        return: i16?
      - args:
          - name: precision
            description: >
              Based on required operator performance and configured optimizations
              on saving memory bandwidth, the precision of the end result can be
              the highest possible accuracy or an approximation.

                - EXACT: provides the exact result, rounded if needed according
                  to the rounding option.
                - APPROXIMATE: provides only an estimate; the result must lie
                  between the minimum and maximum values in the input
                  (inclusive), but otherwise the accuracy is left up to the
                  consumer.
            options: [ EXACT, APPROXIMATE ]
          - name: x
            value: i32
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        nullability: DECLARED_OUTPUT
        return: i32?
      - args:
          - name: precision
            description: >
              Based on required operator performance and configured optimizations
              on saving memory bandwidth, the precision of the end result can be
              the highest possible accuracy or an approximation.

                - EXACT: provides the exact result, rounded if needed according
                  to the rounding option.
                - APPROXIMATE: provides only an estimate; the result must lie
                  between the minimum and maximum values in the input
                  (inclusive), but otherwise the accuracy is left up to the
                  consumer.
            options: [ EXACT, APPROXIMATE ]
          - name: x
            value: i64
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        nullability: DECLARED_OUTPUT
        return: i64?
      - args:
          - name: precision
            description: >
              Based on required operator performance and configured optimizations
              on saving memory bandwidth, the precision of the end result can be
              the highest possible accuracy or an approximation.

                - EXACT: provides the exact result, rounded if needed according
                  to the rounding option.
                - APPROXIMATE: provides only an estimate; the result must lie
                  between the minimum and maximum values in the input
                  (inclusive), but otherwise the accuracy is left up to the
                  consumer.
            options: [ EXACT, APPROXIMATE ]
          - name: x
            value: fp32
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        nullability: DECLARED_OUTPUT
        return: fp32?
      - args:
          - name: precision
            description: >
              Based on required operator performance and configured optimizations
              on saving memory bandwidth, the precision of the end result can be
              the highest possible accuracy or an approximation.

                - EXACT: provides the exact result, rounded if needed according
                  to the rounding option.
                - APPROXIMATE: provides only an estimate; the result must lie
                  between the minimum and maximum values in the input
                  (inclusive), but otherwise the accuracy is left up to the
                  consumer.
            options: [ EXACT, APPROXIMATE ]
          - name: x
            value: fp64
        options:
          rounding:
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        nullability: DECLARED_OUTPUT
        return: fp64?
  - name: "quantile"
    description: >
      Calculates quantiles for a set of values.

      This function will divide the aggregated values (passed via the
      distribution argument) over N equally-sized bins, where N is passed
      via a constant argument. It will then return the values at the
      boundaries of these bins in list form. If the input is appropriately
      sorted, this computes the quantiles of the distribution.

      The function can optionally return the first and/or last element of
      the input, as specified by the `boundaries` argument. If the input is
      appropriately sorted, this will thus be the minimum and/or maximum
      values of the distribution.

      When the boundaries do not lie exactly on elements of the incoming
      distribution, the function will interpolate between the two nearby
      elements. If the interpolated value cannot be represented exactly,
      the `rounding` option controls how the value should be selected or
      computed.

      The function fails and returns null in the following cases:
        - `n` is null or less than one;
        - any value in `distribution` is null.

      The function returns an empty list if `n` equals 1 and `boundaries` is
      set to `NEITHER`.

    impls:
      - args:
          - name: boundaries
            description: >
              Which boundaries to include. For NEITHER, the output will have
              n-1 elements, for MINIMUM and MAXIMUM it will have n elements,
              and for BOTH it will have n+1 elements.
            options: [ NEITHER, MINIMUM, MAXIMUM, BOTH ]
          - name: precision
            description: >
              Based on required operator performance and configured optimizations
              on saving memory bandwidth, the precision of the end result can be
              the highest possible accuracy or an approximation.

                - EXACT: provides the exact result, rounded if needed according
                  to the rounding option.
                - APPROXIMATE: provides only an estimate; the result must lie
                  between the minimum and maximum values in the input
                  (inclusive), but otherwise the accuracy is left up to the
                  consumer.
            options: [ EXACT, APPROXIMATE ]
          - value: i64
            constant: true
            name: n
            description: >
              A positive integer which defines the number of quantile
              partitions.
          - value: any
            name: distribution
            description: >
              The data for which the quantiles should be computed.
        options:
          rounding:
            description: >
              When a boundary is computed to lie somewhere between two values,
              and this value cannot be exactly represented, this specifies how
              to round it. For floating point numbers, it specifies the IEEE
              754 rounding mode (as it does for all other floating point
              operations). For integer types:

                - TIE_TO_EVEN: round to nearest value; if exactly halfway, tie
                  to the even option.
                - TIE_AWAY_FROM_ZERO: round to nearest value; if exactly
                  halfway, tie away from zero.
                - TRUNCATE: always round toward zero.
                - CEILING: always round toward positive infinity.
                - FLOOR: always round toward negative infinity.

              For non-numeric types, the behavior is the same as for integer
              types, but applied to the index of the value in distribution.
            values: [ TIE_TO_EVEN, TIE_AWAY_FROM_ZERO, TRUNCATE, CEILING, FLOOR ]
        nullability: DECLARED_OUTPUT
        ordered: true
        return: LIST?<any>

window_functions:
  - name: "row_number"
    description: "the number of the current row within its partition."
    impls:
      - args: []
        nullability: DECLARED_OUTPUT
        decomposable: NONE
        return: i64?
        window_type: PARTITION
  - name: "rank"
    description: "the rank of the current row, with gaps."
    impls:
      - args: []
        nullability: DECLARED_OUTPUT
        decomposable: NONE
        return: i64?
        window_type: PARTITION
  - name: "dense_rank"
    description: "the rank of the current row, without gaps."
    impls:
      - args: []
        nullability: DECLARED_OUTPUT
        decomposable: NONE
        return: i64?
        window_type: PARTITION
  - name: "percent_rank"
    description: "the relative rank of the current row."
    impls:
      - args: []
        nullability: DECLARED_OUTPUT
        decomposable: NONE
        return: fp64?
        window_type: PARTITION
  - name: "cume_dist"
    description: "the cumulative distribution."
    impls:
      - args: []
        nullability: DECLARED_OUTPUT
        decomposable: NONE
        return: fp64?
        window_type: PARTITION
  - name: "ntile"
    description: "Return an integer ranging from 1 to the argument value,dividing the partition as equally as possible."
    impls:
      - args:
          - name: x
            value: i32
        nullability: DECLARED_OUTPUT
        decomposable: NONE
        return: i32?
        window_type: PARTITION
      - args:
          - name: x
            value: i64
        nullability: DECLARED_OUTPUT
        decomposable: NONE
        return: i64?
        window_type: PARTITION
  - name: "first_value"
    description: >
      Returns the first value in the window.
    impls:
      - args:
          - value: any1
            name: expression
        nullability: DECLARED_OUTPUT
        decomposable: NONE
        return: any1
        window_type: PARTITION
  - name: "last_value"
    description: >
      Returns the last value in the window.
    impls:
      - args:
          - value: any1
            name: expression
        nullability: DECLARED_OUTPUT
        decomposable: NONE
        return: any1
        window_type: PARTITION
  - name: "nth_value"
    description: >
      Returns a value from the nth row based on the `window_offset`. `window_offset` should
      be a positive integer. If the value of the `window_offset` is outside the range
      of the window, `null` is returned.

      The `on_domain_error` option governs behavior in cases where `window_offset` is not
      a positive integer or `null`.
    impls:
      - args:
          - value: any1
            name: expression
          - value: i32
            name: window_offset
        options:
          on_domain_error:
            values: [ NAN, ERROR ]
        nullability: DECLARED_OUTPUT
        decomposable: NONE
        return: any1?
        window_type: PARTITION
  - name: "lead"
    description: >
      Return a value from a following row based on a specified physical offset.
      This allows you to compare a value in the current row against a following row.

      The `expression` is evaluated against a row that comes after the current row based
      on the `row_offset`.  The `row_offset` should be a positive integer and is set to
      1 if not specified explicitly. If the `row_offset` is negative, the expression
      will be evaluated against a row coming before the current row, similar to the `lag`
      function. A `row_offset` of `null` will return `null`. The function returns the
      `default` input value if `row_offset` goes beyond the scope of the window.
      If a `default` value is not specified, it is set to `null`.

      Example comparing the sales of the current year to the following year.
      `row_offset` of 1.
      | year | sales  | next_year_sales |
      | 2019 | 20.50  | 30.00           |
      | 2020 | 30.00  | 45.99           |
      | 2021 | 45.99  | null            |
    impls:
      - args:
          - value: any1
            name: expression
        nullability: DECLARED_OUTPUT
        decomposable: NONE
        return: any1?
        window_type: PARTITION
      - args:
          - value: any1
            name: expression
          - value: i32
            name: row_offset
        nullability: DECLARED_OUTPUT
        decomposable: NONE
        return: any1?
        window_type: PARTITION
      - args:
          - value: any1
            name: expression
          - value: i32
            name: row_offset
          - value: any1
            name: default
        nullability: DECLARED_OUTPUT
        decomposable: NONE
        return: any1?
        window_type: PARTITION
  - name: "lag"
    description: >
      Return a column value from a previous row based on a specified physical offset.
      This allows you to compare a value in the current row against a previous row.

      The `expression` is evaluated against a row that comes before the current row based
      on the `row_offset`.  The `expression` can be a column, expression or subquery that
      evaluates to a single value. The `row_offset` should be a positive integer and is set to
      1 if not specified explicitly. If the `row_offset` is negative, the expression will
      be evaluated against a row coming after the current row, similar to the `lead` function.
      A `row_offset` of `null` will return `null`. The function returns the `default`
      input value if `row_offset` goes beyond the scope of the partition. If a `default`
      value is not specified, it is set to `null`.

      Example comparing the sales of the current year to the previous year.
      `row_offset` of 1.
      | year | sales  | previous_year_sales |
      | 2019 | 20.50  | null                |
      | 2020 | 30.00  | 20.50               |
      | 2021 | 45.99  | 30.00               |
    impls:
      - args:
          - value: any1
            name: expression
        nullability: DECLARED_OUTPUT
        decomposable: NONE
        return: any1?
        window_type: PARTITION
      - args:
          - value: any1
            name: expression
          - value: i32
            name: row_offset
        nullability: DECLARED_OUTPUT
        decomposable: NONE
        return: any1?
        window_type: PARTITION
      - args:
          - value: any1
            name: expression
          - value: i32
            name: row_offset
          - value: any1
            name: default
        nullability: DECLARED_OUTPUT
        decomposable: NONE
        return: any1?
        window_type: PARTITION