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https://github.com/bradhowes/swift-math-parser

Math expression parser built with Point•Free's swift-parsing package
https://github.com/bradhowes/swift-math-parser

math parser parser-combinators pointfree swift wolfram-alpha

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Math expression parser built with Point•Free's swift-parsing package

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# swift-math-parser



Basic math expression parser built with [Point•Free's](https://www.pointfree.co/)

[swift-parsing](https://github.com/pointfreeco/swift-parsing) package (v0.12.0). See the [API documentation](https://bradhowes.github.io/swift-math-parser/documentation/mathparser/) for developer info.



NOTE: v3.1.0 uses swift-parsing v0.12 which requires Xcode 14 and ideally Swift 5.8 

(see their [What's Changed](https://github.com/pointfreeco/swift-parsing/releases/tag/0.12.0) doc for additional details).

If you need to use an older version, use the tagged 3.0.1 release instead.



# Usage Example



```swift

let parser = MathParser()

let evaluator = parser.parse("4 × sin(t × π) + 2 × sin(t × π)")

evaluator.eval("t", value: 0.0) // => 0.0

evaluator.eval("t", value: 0.25) // => 4.2426406871192848

evaluator.eval("t", value: 0.5) // => 6

evaluator.eval("t", value: 1.0) // => 0

```



The parser will return `nil` if it is unable to completely parse the expression. Alternatively, you can call the

`parseResult` to obtain a Swift `Result` enum that will have a `MathParserError` value when parsing fails. This

will contain a description of the parsing failure that comes from the swift-parsing library.



```swift

let evaluator = parser.parseResult("4 × sin(t × π")

print(evaluator)

failure(error: unexpected input

 --> input:1:8

1 | 4 × sin(t × π

  |        ^ expected end of input)



```



By default, the expression parser and evaluator handle the following symbols and functions:



* Standard math operations: addition (`+`), subtraction (`-`), multiplication (`*`), division (`/`), 

and exponentiation (`^`)

* The factorial operator (`!`) [^2]

* Constants: `pi` (`π`) and `e`

* 1-argument functions: 

  * trigonometric functions: `sin`, `asin`, `cos`, `acos`, `tan`, `atan`, `sec`, `csc`, `ctn`

  * hyperbolic functions: `sinh`, `asinh`, `cosh`, `acosh`, `tanh`, `atanh`

  * logarithmic and exponential functions: `log10`, `ln` (`loge`), `log2`, `exp`

  * others: `ceil`, `floor`, `round`, `sqrt` (`√`), `cbrt` (cube root), `abs`, and `sgn`

* 2-argument functions: `atan2`, `hypot`, `pow` [^1]

* alternative math operator symbols: `×` for multiplication and `÷` for division (see example above for use of `×`)



You can reference additional symbols or variables and functions by providing your own mapping functions. There are two

places where this can be done:



* `MathParser.init`

* `Evaluator.eval`



If a symbol or function does not exist during an `eval` call, the final result will be `NaN`. If a symbol is resolved

during parsing, it will be replaced with the symbol's value. Otherwise, it will be resolved during a future `eval` call.

Same for function calls -- if the function is known during parsing _and_ all arguments have a known value, then it will

be replaced with the function result. Otherwise, the function call will take place during an `eval` call.



You can get the unresolved symbol names from the `Evaluator.unresolved` attribute. It returns three collections for

unresolved variables, unary functions, and binary function names. You can also use the `evalResult` to attempt an

evaluation but also obtain a description of the failure when the evaluation fails.



# Custom Symbols



Below is an example that provides a custom unary function that returns the twice the value it receives. There is also a

custom variable called `foo` which holds the constant `123.4`.



```swift

let myVariables = ["foo": 123.4]

let myFuncs: [String:(Double)->Double] = ["twice": {$0 + $0}]

let parser = MathParser(variables: myVariables.producer, unaryFunctions: myFuncs.producer)

let evaluator = parser.parse("power(twice(foo))")



# Expression parsed and `twice(foo)` resolved to `246.8` but `power` is still unknown

evaluator?.value // => nan

evaluator?.unresolved.unaryFunctions // => ['power']'

# Give evaluator way to resolve `power(246.8)`

let myEvalFuncs: [String:(Double)->Double] = ["power": {$0 * $0}]

evaluator?.eval(unaryFunctions: myEvalFuncs.producer) // => 60910.240000000005

```



Instead of passing a closure to access the dictionary of symbols, you can pass the dictionary itself:



```

let parser = MathParser(variableDict: myVariables, unaryFunctionDict: myFuncs)

evaluator?.eval(unaryFunctionDict: myEvalFuncs) // => 60910.240000000005

```



# Precedence



The usual math operations follow the traditional precedence hierarchy: multiplication and division operations happen

before addition and subtraction, so `1 + 2 * 3 - 4 / 5 + 6` evaluates the same as `1 + (2 * 3) - (4 / 5) + 6`. 

There are three additional operators, one for exponentiations (^) which is higher than the previous ones, 

so `2 * 3 ^ 4 + 5` is the same as `2 * (3 ^ 4) + 5`. It is also right-associative, so `2 ^ 3 ^ 4` is evaluated as 

`2 ^ (3 ^ 4)` instead of `(2 ^ 3) ^ 4`.



There are two other operations that are even higher in precedence than exponentiation:



* negation (`-`) -- `-3.4`

* factorial (`!`) -- `12!`



Note that factorial of a negative number is undefined, so negation and factorial cannot be combined. In other words,

parsing `-3!` returns `nil`. Also, factorial is only done on the integral portion of a number, so `12.3!` will parse but

the resulting value will be the same as `12!`. In effect, factorial always operates as `floor(x)!` or `!(floor(x))`.



# Implied Multiplication



One of the original goals of this parser was to be able to accept a Wolfram Alpha math expression more or less as-is

-- for instance the definition https://www.wolframalpha.com/input/?i=Sawsbuck+Winter+Form%E2%80%90like+curve -- without

any editing. Here is the start of the textual representation from the above link:



```

x(t) = ((-2/9 sin(11/7 - 4 t) + 78/11 sin(t + 11/7) + 2/7 sin(2 t + 8/5) ...

```



Skipping over the assignment one can readily see that the representation includes implied multiplication between terms

when there are no explicit math operators present (eg `-2/9` __x__ `sin(11/7 - 4` __x__ `t)`). There is support for this

sort of operation in the parser that can be enabled by setting `enableImpliedMultiplication` when creating a new

`MathParser` instance (it defaults to `false`). Note that when enabled, an expression such as `2^3 2^4` would be

considered a valid expression, resolving to `2^3 * 2^4 = 128`, and `4sin(t(pi))` would become `4 * sin(t * pi)`.



You can see the entire Wolfram example in the [TestWolfram](Tests/MathParserTests/TestWolfram.swift) test case.



Here is the original example expression from the start of this README file with implied multiplication in use (all of 

the muliplication symbols have been removed):



```swift

let parser = MathParser(enableImpliedMultiplication: true)

let evaluator = parser.parse("4sin(t π) + 2sin(t π)")

evaluator.eval("t", value: 0.0) // => 0.0

evaluator.eval("t", value: 0.25) // => 4.2426406871192848

evaluator.eval("t", value: 0.5) // => 6

evaluator.eval("t", value: 1.0) // => 0

```



Be aware that with implied multiplication enabled, you could encounter strange parsing if you do not use spaces between

the "-" operator:



* `2-3` => -6

* `2 -3` -> -6

* `2 - 3` => -1



However, for "+" all is well:



* `2+3` => 5

* `2 +3` -> 5

* `2 + 3` => 5



Unfortunately, there is no way to handle this ambiguity between implied multiplication, subtraction and negation when 

spaces are not used to signify intent. 



## Symbol Splitting



When implied multiplication mode is active and the name of a variable or a 1-parameter (unary) function is not found in

their corresponding map, the token evaluation routine will attempt to resolve them by splitting the names into two or

more pieces that all resolve to known variables and/or functions. For example, using the default variable map and 

unary function map from `MathParser`:



* `pie` => `pi * e`

* `esin(2π)` => `e * sin(2 * pi)`

* `eeesgn(-1)` => `e * e * e * -1`



As you can see, this could lead to erroneous resolution of variable names and functions, but this behavior is only used

when the initial lookup of the name fails, and it is never performed when the symbol names are separated by a space.

However, if you make a mistake and forget to provide the definition of a custom variable or function, it could provide

a value instead of an error. For instance, consider evaluating `tabs(-3)` where `t` is a custom variable set to `1.2`

and `tabs` is a custom function but it is not provided for in the custom unary function map:



* `tabs(-3)` => `1.2 * abs(-3)` => `3.6`



If implied multiplication had not been active, the evaluator would have correctly reported an issue -- either returning

NaN or a `Result.failure` describing the missing function.



[^1]: Redundant since there is already the `^` operator.

[^2]: Exact up to 20! -- larger numbers are approximations