Why are numeric division and subtraction not handled better in Mathematica?

Question

There is something that has been troubling me for a while. At least through version 10.0 the performance of a / b and a - b is not equivalent to, and significantly inferior to, Divide[a, b] and Subtract[a, b], despite the fact that these are treated as equivalent in the documentation. As a terse code fanatic it chafes me to have to write out the longer forms, but that is only the beginning of my concern, because the latter, faster forms are transparently converted into the former, slower forms.

Examples from the documentation for Divide:

x/y or Divide[x,y] is equivalent to x y^-1.

Divide[x,y] can be entered in StandardForm and InputForm as x ÷ y, x EscdivEsc y or x \[Divide] y.

Proof that these statements are false:

{a, b} = List @@ RandomReal[{-50, 50}, {2, 1*^7}];

Do[a/b, {50}]          // Timing // First
Do[a b^-1, {50}]       // Timing // First
Do[a\[Divide]b, {50}]  // Timing // First
Do[Divide[a, b], {50}] // Timing // First

2.262
2.231
2.184
0.437

Divide[a, b] is not evaluated in the same manner as the rest. Similarly for subtraction:

Do[a - b, {50}]            // Timing // First
Do[a + (-1 b), {50}]       // Timing // First
Do[Subtract[a, b], {50}]   // Timing // First

3.651
3.65
1.404

Although the examples above use packed arrays (see Why does list assignment with a packed array result in unpacked values? for an explanation of List @@) the same performance differential exists with unpacked lists.

One can see with Trace that the short forms induce additional operations:

a = Range[1, 2];
b = Range[3, 4];

a/b          // Trace
Divide[a, b] // Trace

{{a,{1,2}}, {{b,{3,4}}, 1/{3,4}, {1/3,1/4}}, {1,2} {1/3,1/4}, {1/3,1/4} {1,2}, 1/3,1/2}}
{{a,{1,2}}, {b,{3,4}}, {1,2}/{3,4}, {1/3,1/2}}

a - b          // Trace
Subtract[a, b] // Trace

{{a,{1,2}}, {{b,{3,4}}, -{3,4}, {-3,-4}}, {1,2} + {-3,-4}, {-3,-4} + {1,2}, {-2,-2}}
{{a,{1,2}}, {b,{3,4}}, {1,2} - {3,4}, {-2,-2}}

Note the false equivalence in this output; the fast Divide and Subtract operations are formatted as {1,2}/{3,4} and {1,2} - {3,4}, yet these slow forms are not part of their evaluation process.

More problematic is when this false equivalence changes evaluation. For example, if you try to work symbolically with Subtract and Divide in an evaluated expression and use the result these faster forms are replaced with the slower ones:

expr = Divide[x, y] + Subtract[x, y];
fn = Function[{x, y}, Evaluate @ expr]
fn // FullForm

Function[{x, y}, x + x/y - y]
Function[List[x,y], Plus[x, Times[x, Power[y,-1]], Times[-1,y]]]

Note also that Divide and Subtract are stripped when converting forms in the Front End (menu Cell > Convert To) so even without evaluation these operators may be lost.

Therefore I have these questions:

• Why are these operations universally treated as equivalent when they are clearly not programmatically equivalent?

• Why does Mathematica not include optimization for cases of numeric division and subtraction to eliminate the additional evaluation steps?

• Is there a practical way to add global optimizations to automatically convert these operations to the fast forms before numeric evaluation?

• Failing the above, what is the best way to work symbolically with actual division and subtraction operators for the sake of performance?

• Is internally representing division and subtraction as multiplication and addition really the only mathematically valid design option? Couldn't these instead be first-class operators that are recognized as equivalent to Times and Plus for the purpose of pattern matching but not converted into Times and Plus (which introduces additional Times and Power operations)?

Answer 1

An extended comment. I'm not sure if this has been realized, please correct me if it has. The result of the Divide[a,b] operation is not the same as the first 3 which are identical.

{a, b} = List @@ RandomReal[{-50, 50}, {2, 1*^7}];

x1 = a/b;
x2 = a b^-1;
x3 = a/b;
x4 = Divide[a, b];

Now...

Tally[x1 - x2]
Tally[x2 - x3]

Both give 10^7 zeros.

Tally[x3 - x4]

Gives ~ 7,500,000 zeros and 2,500,000 slight differences.