After thinking about this more, I would only propose changing one thing. Implicit multiplication between numbers and symbols, or symbols and symbols, would take higher precedence than regular multiplication and division. Currently it has equal precedence.

Current operator precedence:

Operators	Description
...	...
``, `/`, `.`, `./`, `%`, `mod`	Multiply, divide, modulus, implicit multiply
...	...

Proposed operator precedence:

Operators	Description
...	...
`symbol symbol`, `number symbol`	Implicit multiplication
``, `/`, `.`, `./`, `%`, `mod`	Multiply, divide, modulus
...	...

Examples:

Expression	Parsed as
`1 / 2 * x`	`(1 / 2) * x`
`1 / 2 x`	`1 / (2 * x)`
`1 / 2 x y`	`1 / ((2 * x) * y)`
`1 / 2 x * y`	`(1 / (2 * x)) * y`
`2 1 / 2`	Syntax error: `number number` is not supported

If any users are bothered by the higher precedence of implicit multiplication, they would always be able to add back in the *'s to restore the original behavior. This actually gives users more control over how their expression is parsed.

I've also done a little more investigating into Google's behavior for fractions. As impressive as it is, we would not reproduce that behavior right now. We could add this later, though I doubt there would be much interest for it.

For reference, here is Google's operator precedence from what I've observed:

Operators	Description
`number number/number`	Compound fraction
...	...
`number / number`	Simple fraction
`symbol symbol`, `number symbol`	Implicit multiplication
``, `/`, `.`, `./`, `%`, `mod`	Multiply, divide, modulus
...	...

Google's different precedence for simple and compound fractions leads to some pretty intelligent behavior. 1 / 2 ^ 2 evaluates to 1 / (2 ^ 2) = 0.25, while 1 1 / 2 ^ 2 evaluates to (1 1 / 2) ^ 2 = 2.25. Another example: 1 1/2 ^ 4/5 parses as ((1 1/2)^4) / 5, while 1 1/2 ^ 3 4/5 parses as (1 1/2) ^ (3 4/5).

Uh oh!

Implicit multiplication and unit division #792

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