Basic math expression parser built with Point•Free's swift-parsing package (v0.12.0). See the API documentation 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 doc for additional details). If you need to use an older version, use the tagged 3.0.1 release instead.

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.

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 of a number (!) ¹
• Constants: pi (π) and e
• 1-argument functions: sin, asin, cos, acos, tan, atan, log10, ln (loge), log2, exp, ceil, floor, round, sqrt (√), cbrt (cube root), abs, sgn, and ! for factorial (see ¹)
• 2-argument functions: atan, hypot, pow ²
• 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.

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.

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

The usual math operations follow the traditional precedence hierarchy: multiplication and division is above 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)).

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 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):

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.

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.