Crate exmex

Exmex is a fast, simple, and extendable mathematical expression evaluator with the ability to compute partial derivatives of expressions.

The following snippet shows how to evaluate a string.

use exmex;
let eval_result = exmex::eval_str::<f64>("1.5 * ((cos(2*π) + 23.0) / 2.0)")?;
assert!((eval_result - 18.0).abs() < 1e-12);

For floats, we have a list of predifined operators containing ^, *, /, +, -, sin, cos, tan, exp, log, and log2. Further, the constants π and Euler’s number can be used via π/PI and E, respectively. The full list is defined in DefaultOpsFactory. Library users can also create their own operators and constants as shown below in the section about extendability.

Variables

To define variables we can use strings that are not in the list of operators as shown in the following expression. Additionally, variables should consist only of letters, greek letters, numbers, and underscores. More precisely, they need to fit the regular expression r"[a-zA-Zα-ωΑ-Ω_]+[a-zA-Zα-ωΑ-Ω_0-9]*", if they are not between curly brackets.

Variables’ values are passed as slices to eval.

use exmex::prelude::*;
let to_be_parsed = "α * log(z) + 2* (-z^2 + sin(4*y))";
let expr = exmex::parse::<f64>(to_be_parsed)?;
assert!((expr.eval(&[3.7, 2.5, 1.0])? - 14.992794866624788 as f64).abs() < 1e-12);

The n-th number in the slice corresponds to the n-th variable. Thereby, the alphabetical order of the variables is relevant. More precisely, the order is defined by the way how Rust sorts strings. In the example above we have y=3.7, z=2.5, and α=1. Note that α is the Greek letter Alpha. If variables are between curly brackets, they can have arbitrary names, e.g., {456/549*(}, {x}, and also {👍+👎} are valid variable names as shown in the following.

use exmex::prelude::*;
let x = 2.1f64;
let y = 0.1f64;
let to_be_parsed = "log({👍+👎})";  // {👍+👎} is the name of one variable 😕.
let expr = exmex::parse::<f64>(to_be_parsed)?;
assert!((expr.eval(&[x+y])? - 2.2f64.ln()).abs() < 1e-12);

The value returned by parse is an instance of the struct FlatEx that implements the Express trait. Moreover, FlatEx and Express are the only items made accessible by the wildcard import from prelude.

Partial Derivatives

For default operators, expressions can be transformed into their partial derivatives again represented by expressions. To this end, there exists the method partial.

use exmex::prelude::*;
let expr = exmex::parse::<f64>("x^2 + y^2")?;
let dexpr_dx = expr.clone().partial(0)?;
let dexpr_dy = expr.partial(1)?;
assert!((dexpr_dx.eval(&[3.0, 2.0])? - 6.0).abs() < 1e-12);
assert!((dexpr_dy.eval(&[3.0, 2.0])? - 4.0).abs() < 1e-12);

Owned Expression

You cannot return all expression types from a function without a lifetime parameter. For instance, expressions that are instances of FlatEx keep &strs instead of Strings of variable or operator names to make faster parsing possible.

use exmex::prelude::*;
use exmex::ExResult;
fn create_expr<'a>() -> ExResult<FlatEx::<'a, f64>> {
//              |                          |
//              lifetime parameter necessary

    let to_be_parsed = "log(z) + 2* (-z^2 + sin(4*y))";
    exmex::parse::<f64>(to_be_parsed)
}
let expr = create_expr()?;
assert!((expr.eval(&[3.7, 2.5])? - 14.992794866624788 as f64).abs() < 1e-12);

If you are willing to pay the price of roughly doubled parsing times, you can obtain an expression that is an instance of OwnedFlatEx and owns its strings. Evaluation times should be comparable. However, a lifetime parameter is not needed anymore as shown in the following.

use exmex::{ExResult, Express, OwnedFlatEx};
fn create_expr() -> ExResult<OwnedFlatEx::<f64>> {
    let to_be_parsed = "log(z) + 2* (-z^2 + sin(4*y))";
    OwnedFlatEx::<f64>::from_str(to_be_parsed)
}
let expr_owned = create_expr()?;
assert!((expr_owned.eval(&[3.7, 2.5])? - 14.992794866624788 as f64).abs() < 1e-12);

Extendability

How to use custom operators as well as custom data types of the operands even with non-numeric literals is described in the following sub-sections.

Custom Operators and Constants

Operators are instances of the struct Operator. Constants are defined in terms of constant operators. More precisely, operators can be

• binary such as *,
• unary such as sin,
• binary as well as unary such as -, or
• constant such as PI.

An operator’s representation is defined in the field repr. A token of the string-to-be-parsed is identified as operator if it matches the operator’s representation exactly. For instance, PI will be parsed as the constant π while PI5 will be parsed as a variable with name PI5. When an operator’s representation is used in a string-to-be-parsed, the following applies:

• Binary operators are positioned between their operands, e.g., 4 ^ 5.
• Unary operators are positioned in front of their operands, e.g., -1 or sin(4). Note that sin4 is parsed as variable name, but sin 4 is equivalent to sin(4).
• Constant operators are handled as if they were numbers and are replaced by their numeric values during parsing. They can be used as in sin(PI) or 4 + E. Note that the calling notation of constant operators such as PI() is invalid.

Binary, unary, and constant operators can be created with the functions make_bin, make_unary, and make_constant, respectively. Operators need to be created by factories to make serialization via serde possible as shown in the following.

use exmex::prelude::*;
use exmex::{BinOp, MakeOperators, Operator, ops_factory};
ops_factory!(
    IntegerOpsFactory,  // name of the factory type
    i32,                // data type of the operands
    Operator::make_bin(
        "%",
        BinOp{
            apply: |a, b| a % b,
            prio: 1,
            is_commutative: false,
        }
    ),
    Operator::make_bin(
        "/",
        BinOp{
            apply: |a, b| a / b,
            prio: 1,
            is_commutative: false,
        }
    ),
    Operator::make_constant("TWO", 2)
);
let to_be_parsed = "19 % 5 / TWO / a";
let expr = FlatEx::<_, IntegerOpsFactory>::from_str(to_be_parsed)?;
assert_eq!(expr.eval(&[1])?, 2);

To extend an existing list of operators, the macro ops_factory is not sufficient. In this case one has to create a factory struct and implement the MakeOperators trait with a little boilerplate code.

use exmex::prelude::*;
use exmex::{DefaultOpsFactory, MakeOperators, Operator};
#[derive(Clone)]
struct ExtendedOpsFactory;
impl MakeOperators<f32> for ExtendedOpsFactory {
    fn make<'a>() -> Vec<Operator<'a, f32>> {
        let mut ops = DefaultOpsFactory::<f32>::make();
        ops.push(
            Operator::make_unary("invert", |a| 1.0 / a)
        );
        ops
    }
}
let to_be_parsed = "1 / a + invert(a)";
let expr = FlatEx::<_, ExtendedOpsFactory>::from_str(to_be_parsed)?;
assert!((expr.eval(&[3.0])? - 2.0/3.0).abs() < 1e-12);

Custom Data Types of Numbers

You can use any type that implements Clone, FromStr, and Debug. In case the representation of your data type in the string does not match the number regex r"\.?[0-9]+(\.[0-9]+)?", you have to pass a suitable regex and use the function from_pattern instead of parse or from_str. Here is an example for bool.

use exmex::prelude::*;
use exmex::{BinOp, MakeOperators, Operator, ops_factory};
ops_factory!(
    BooleanOpsFactory,
    bool,
    Operator::make_bin(
        "&&",
        BinOp{
            apply: |a, b| a && b,
            prio: 1,
            is_commutative: true,
        }
    ),
    Operator::make_bin(
        "||",
        BinOp{
            apply: |a, b| a || b,
            prio: 1,
            is_commutative: true,
        }
    ),
    Operator::make_unary("!", |a| !a)
);
let to_be_parsed = "!(true && false) || (!false || (true && false))";
let expr = FlatEx::<_, BooleanOpsFactory>::from_pattern(to_be_parsed, "true|false")?;
assert_eq!(expr.eval(&[])?, true);

Priorities and Parentheses

In Exmex-land, unary operators always have higher priority than binary operators, e.g., -2^2=4 instead of -2^2=-4. Moreover, we are not too strict regarding parentheses. For instance

use exmex;
assert_eq!(exmex::eval_str::<f64>("---1")?, -1.0);

If you want to be on the safe side, we suggest using parentheses.

Display

Instances of FlatEx and OwnedFlatEx can be displayed as string. Note that this unparsed string does not necessarily coincide with the original string, since, e.g., curly brackets are added, expressions are compiled, and constants are replaced by their numeric values during parsing.

use exmex::prelude::*;
let expr = exmex::parse::<f64>("-sin(z)/cos(mother_of_names) + 2^7 + E")?;
assert_eq!(format!("{}", expr), "-(sin({z}))/cos({mother_of_names})+130.71828182845906");

Serialization and Deserialization

To use serde you can activate the feature serde. The implementation un-parses and re-parses the whole expression. Deserialize and Serialize are implemented for both, FlatEx and OwnedFlatEx.

Modules

prelude

To use the expression trait Express and its implementation FlatEx one can use exmex::prelude::*;.

Macros

ops_factory

This macro creates an operator factory struct that implements the trait MakeOperators. You have to pass the name of the struct as first, the type of the operands as second, and the Operators as third to n-th argument.

Structs

BinOp

A binary operator that consists of a function pointer, a priority, and a commutativity-flag.

DefaultOpsFactory

Factory of default operators for floating point values.

ExError

This will be thrown at you if the somehting within Exmex went wrong. Ok, obviously it is not an exception, so thrown needs to be understood figuratively.

FlatEx

This is the core data type representing a flattened expression and the result of parsing a string. We use flattened expressions to make efficient evaluation possible. Simplified, a flat expression consists of a SmallVec of nodes and a SmallVec of operators that are applied to the nodes in an order following operator priorities.

Operator

Operators can be custom-defined by the library-user in terms of this struct.

OwnedFlatEx

This is another representation of a flattened expression besides FlatEx. The difference is that OwnedFlatEx can be used without a lifetime parameter. All the data that FlatEx borrowed is kept in a buffer by OwnedFlatEx. The drawback is that parsing takes longer, since additional allocations are necessary. Evaluation time should be about the same for FlatEx and OwnedFlatEx.

Traits

Express

Expressions implementing this trait can be evaluated for specific variable values, differentiated partially, and unparsed, i.e., transformed into a string representation.

MakeOperators

To use custom operators one needs to create a factory that implements this trait. In this way, we make sure that we can deserialize expressions with serde with the correct operators based on the type.

Functions

eval_str

Parses a string, evaluates the expression, and returns the resulting number.

parse

Parses a string and returns the expression that can be evaluated.

Type Definitions

ExResult

Exmex’ result type with ExError as error type.