# Crate exmex

Expand description

Exmex is an extendable mathematical expression parser and evaluator. Ease of use, flexibility, and efficient evaluations are its main design goals. Exmex can parse mathematical expressions possibly containing variables and operators. On the one hand, it comes with a list of default operators for floating point values. For differentiable default operators, Exmex can compute partial derivatives. On the other hand, users can define their own operators and work with different data types such as float, integer, bool, or other types that implement Clone, FromStr, Debug, and Default.

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 predefined operators containing ^, *, /, +, -, sin, cos, tan, exp, log10, ln, and log2. Further, the constants π, τ, and Euler’s number are refered to via π/PI, τ/TAU, and E, respectively. The full list is defined in FloatOpsFactory. 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 * ln(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 = "ln({👍+👎})";  // {👍+👎} 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, Express, and Calculate are the items made accessible by the wildcard import from prelude if the feature partial is not used.

### §Features

Exmex comes with three features that can be activated in the Cargo.toml via

[dependencies]
exmex = { ..., features = ["partial", "serde", "value"] }

partial allows the computation of partal derivatives, serde enables serialization and deserialization, and value makes a more general value type accessible.

#### §Partial Derivatives

Expressions with floating point data types can be transformed into their partial derivatives again represented by expressions after activating the feature partial. See the readme for examples.

#### §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 FlatEx.

#### §A more General Value Type

To use different data types within an expression, one can activate the feature value and use the more general type Val. The additional flexibility comes with higher parsing and evaluation run times, see the benchmarks.

### §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 can be accessed via the method 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.

All binary operators can be used either like a op b or like op(a, b). Thereby, the latter will be interpreted as ((a) op (b)). For instance atan2(y * 2, 1 / x) * 2 and ((y * 2) atan2 (1 / x)) * 2 are equivalent. We do not support n-ary operators like f(a, b, c) for n = 3.

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>::parse(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::{FloatOpsFactory, MakeOperators, Operator};
#[derive(Clone, Debug)]
struct ExtendedOpsFactory;
impl MakeOperators<f32> for ExtendedOpsFactory {
fn make<'a>() -> Vec<Operator<'a, f32>> {
let mut ops = FloatOpsFactory::<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>::parse(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’s literals in the string does not match the number regex r"^(\.?[0-9]+(\.[0-9]+)?)", you have to create a suitable matcher type that implements MatchLiteral. Given a suitable regex pattern, you can utilize the macro literal_matcher_from_pattern. Here is an example for bool.

use exmex::prelude::*;
use exmex::{
BinOp, MakeOperators, MatchLiteral, Operator,
literal_matcher_from_pattern, 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)
);
literal_matcher_from_pattern!(BooleanMatcher, "^(true|false)");
let to_be_parsed = "!(true && false) || (!false || (true && false))";
type FlatExBool = FlatEx::<bool, BooleanOpsFactory, BooleanMatcher>;
let expr = FlatExBool::parse(to_be_parsed)?;
assert_eq!(expr.eval(&[])?, true);

Two examples of exmex with non-trivial data types are:

• Numbers can be operators and operators can operate on operators, see, e.g., also a blog post on ninety.de.
• The value type implemented as part of the feature value allows expressions containing integers, floats, and bools. Therewith, Pythonesque expressions of the form "x if a > b else y" are possible.

### §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

Expressions can be displayed as string. This unparsed string coincides with the original string.

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) + 2^7 + E");

### §Calculating with Expression

Like partial derivatives, calculations need the nested expression type DeepEx that is slower to evaluate than the flattened expression type FlatEx. It is possible to calculate with flat expressions of type FlatEx. However, transformations to the nested expression DeepEx happen in the background.

use exmex::prelude::*;
let expr_1 = FlatEx::<f64>::parse("x")?;
let expr_2px = FlatEx::<f64>::parse("2 + x")?;
let expr_2p2x = expr_1.operate_binary(expr_2px, "+")?;
assert!(expr_2p2x.eval(&[-1.5])? < 1e-12);

To save transformations, we can start by parsing a deep expression to do multiple calculations and flatten eventually.

use exmex::{DeepEx, prelude::*};
let deep_cos_x = DeepEx::<f64>::parse("cos(x)")?;
let deep_identity = deep_cos_x.operate_unary("acos")?;
let one = DeepEx::one();
let deep_identity = deep_identity.operate_binary(one, "*")?;
let flat_identity = FlatEx::from_deepex(deep_identity)?;
assert!((flat_identity.eval(&[3.0])? - 3.0).abs() < 1e-12);

Alternatively, it is possible to transform a flat expression to a nested expression with FlatEx::to_deepex. Moreover, we have implemented the default operators as wrappers around Calculate::operate_unary and Calculate::operate_binary, see the following re-write of the snippet above.

use exmex::{DeepEx, prelude::*};
let deep_cos_x = DeepEx::<f64>::parse("cos(x)")?;
let deep_identity = deep_cos_x.acos()?;
let one = DeepEx::one();
let deep_identity = (deep_identity * one)?;
let flat_identity = FlatEx::from_deepex(deep_identity)?;
assert!((flat_identity.eval(&[3.0])? - 3.0).abs() < 1e-12);

## Modules§

• Exmex’ prelude can be imported via use exmex::prelude::*;.

## Macros§

• Creates an ExError with a formatted message.
• Helper to implement a struct called \$matcher_name that implements MatchLiteral and matches the regex pattern \$regex_pattern.
• 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§

• A binary operator that consists of a function pointer, a priority, and a commutativity-flag.
• A deep expression evaluates co-recursively since its nodes can contain other deep expressions. Compared to FlatEx, this is slower to evaluate but easier to calculate with.
• 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.
• Flattened expressions 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.
• Factory of default operators for floating point values.
• Default factory to match numeric literals.
• Operators can be unary such as sin, binary such as *, unary and binary such as -, or constants such as π. To use custom operators, see the short-cut-macro ops_factory implement the trait MakeOperators directly.
• Literal matcher type that was created with the macro literal_matcher_from_pattern.
• feature = "value" - Factory of default operators for the data type Val.

## Enums§

• Feature partial - What should happen in case for an operator the derivative is missing
• feature = "value" - The value type Val can contain an integer, float, bool, a vector of floats, none, or error. To use the value type, there are the is a parse function parse_val. In the following example, the ternary Python-style a if condition else b is used. This is equivalent to if condition {a} else {b} in Rust or condition ? a : b in C.

## Traits§

• Calculation with expression such as application of operators or substitution
• Gathers Clone, FromStr, Debug, and Default in one trait. Every type that is used as value needs to implement at least this.
• DataTypes of expressions that are differentiable need to implement additionally From<f32>, PartialEq, and NeutralElts. Vice-versa, if you have an expression with a datatype that implements DiffDataType, partial derivatives of that expression can be computed.
• feature = "partial" - Trait for partial differentiation. This is implemented for expressions with datatypes that implement DiffDataType.
• Expressions implementing this trait can be parsed from stings, evaluated for specific variable values, and unparsed, i.e., transformed into a string representation.
• 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.
• Implement this trait to create a matcher for custom literals of operands.
• The neutral elements of addition and multiplication are zero and one, respectively. An implementation is provided for all types that implement From<u8> + PartialEq.

## Functions§

• Parses a string, evaluates the expression, and returns the resulting number.
• Parses a string and returns the expression with default operators that can be evaluated.
• feature = "value" - Parses a string into an expression of type FlatExVal with datatype Val.