First, here's a common example of abbreviated operator precedence without Prefix. Do we need parens around the exponential coefficients?

Plot[E^-r t/.{r->.5},{t,-2,10}]

The graph shows something is wrong and Precedence analysis tells us yes, we need parens.

Precedence/@{Power,Minus,Times,ReplaceAll}

{590.,480.,400.,110.}

Plot[E^(-r t)/.{r->.5},{t,-2,10}]

Here's a typical example using Prefix. Will this function work as intended? (It's from S11: What's New in

*Mathematica*8 by Paul Wellin.)

f1@x_ := 1/x Cos[Log@x/x]; Plot[f1@x, {x, 0, .01}]

What would speed up Precedence analysis would be seeing an operator's Precedence popup on Mouseover, like a Tooltip. And a more minor recommendation to WRI is to make Precedence Listable to avoid needing Map. But that only saves one keystroke and I'm after much bigger game by recommending extensive usage of Prefix.

A tip is to list the operators in the order they appear in your function. Then it's easier to see which are 'stickier' than if they are out of order. Anyway, using Precedence, we see that the function will work as intended.

Unprotect@Precedence; SetAttributes[Precedence,Listable]; Protect@Precedence; Precedence@{Divide,Times,Prefix}

{470.,400.,640.}

f1@x_ := 1/x Cos[Log@x/x]; Plot[f1@x, {x, 0, .01}]

Now that we know the Precedence of Prefix and Divide, we can write the following interesting usage of Range with impunity; we don't need Table or Map. It works because Divide is Listable.

Range@5/6

{1/6,1/3,1/2,2/3,5/6}

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