Semantics of ^^

[Lars T. Kyllingstad]

Don wrote:
> Based on everyone's comments, this is what I have come up with:
> 
> --------------------
> x ^^ y is right associative, and has a precedence intermediate between 
> multiplication and unary operators.
> 
> * The type of x ^^ y is the same as the type of x * y.
> * If y == 0,  x ^^ y is 1.
> * If both x and y are integers, and y > 0,  x^^y is equivalent to
>    { auto u = x; foreach(i; 1..y) { u *= x; } return u; }
> * If both x and y are integers, and y < 0, an integer divide error 
> occurs, regardless of the value of x. This error is detected at compile 
> time, if possible.
> * If either x or y are floating-point, the result is pow(x, y).
> --------------------
> Rationale:
> (1) Although the following special cases could be defined...
>  * If x == 1,  x ^^ y is 1
>  * If x == -1 and y is even, x^^y == 1
>  * If x == -1 and y is odd, x^^y == -1
> ... they are not sufficiently useful to justify the major increase in 
> complexity which they introduce. In all other cases, a negative exponent 
> indicates an error; it should be rewritten as (cast(real)x) ^^ y. Making 
> these cases errors makes everything much simpler, and allows the 
> compiler to use range propagation on the value of y to detect most 
> exponentiation errors at compile time. (If those cases are legal, the 
> compiler can't generate an error on x^^-2, because of the possibility 
> that x might be 1 or -1).
> Also note that making it an error leaves open the possibility of 
> changing it to a non-error later, without breaking code; but going from 
> non-error to error would be more difficult.

I agree.


> (2) USE OF THE INTEGER DIVIDE ERROR
> Note that on x86 at least, a hardware "integer divide error", although 
> commonly referred to as "division by zero", also occurs when the DIV 
> instruction, which performs uint = ulong/uint, results in a value 
> greater than uint.max. Raising a number to a negative power does involve 
> a division, so it seems to me not unreasonable to use it for this case 
> as well.
> Note that 0 ^^ -1 is a division by zero.
> This means that, just as you should check that y!=0 before performing 
> x/y, you should check that y>=0 before performing x^^y.

This is reasonable.


> (3) OVERFLOW
> int ^^ int returns an int, not a long. Although a long would allow 
> representation of larger numbers, even doubling the number of bits 
> doesn't help much in avoiding overflow, because x^^y is exponential. 
> Even a floating-point representation can easily overflow:
> 5000^5000 easily overflows an 80-bit real.
> So, it's preferable to retain the simplicity that typeof(x^^y) is 
> typeof(x*y).

Absolutely. I think people who use exponentiation will be well aware of 
how easily it overflows, and will use long or BigInt (which should of 
course overload ^^) if they are worried about this. In any case, the 
most common uses of int^^int will be x^^2, x^^3, and x^^4.

It looks like you've given this quite some thought. Thanks!

-Lars