float equality

[spir]

On 02/21/2011 05:32 AM, Walter Bright wrote:
> Kevin Bealer wrote:
>> == Quote from Walter Bright (newshound2 at digitalmars.com)'s article
>>> Kevin Bealer wrote:
>>>> You could switch to this:
>>>>
>>>> struct {
>>>> BigInt numerator;
>>>> BigInt denominator;
>>>> };
>>>>
>>>> Bingo -- no compromise.
>>> It cannot represent irrational numbers accurately.
>>
>> True but I did mention this a few lines later.
>
> I guess I'm not seeing the point of representing numbers as ratios. That works
> only if you stick to arithmetic. As soon as you do logs, trig functions, roots,
> pi, etc., you're back to square one.

"Naturally" non-representable numbers (irrationals), or results of "naturally" 
approximate operations (like trig), are not an issue because they are expected 
to yield inaccuracy.

This is different from numbers which are well definite in base ten, as well as 
results of operations which should yield well definite results. We think in 
base ten, thus for us 1.1 is exact. Two operations yielding 1.1 (including 
plain conversion to binary of the literal '1.1') may in fact yield unequal 
numbers at the binary level; this is a trap, and just wrong:
     assert (1.1 != 3.3 - 2.2);	// pass
Subsequent issue is (as I just discovered) that there is no remedy for that, 
since there is no way to know, in the general case, how many common magnitude 
bits are supposed to be shared by the numbers to be compared, when the results 
are supposed to be correct. (Is this worse if floating point format since the 
scale factor is variable?)

This lets me think that, for common use, fixed point with a single /decimal/ 
scale factor (for instance 10^-6) may be a better solution. While the 
underlying integer type may well be binary --this is orthogonal-- we don't need 
using eg BCD. Using longs, this example would allow representing numbers up to 
+/- (2^63)/(10^-9), equivalent to having about 43 bits for the integral part 
(*), together with a precision of 6 decimal digits for the fractional part. 
Seems correct for common use cases, I guess.


Denis

(*) 10^6 ~ 2^20
(2^63)/(10^-6) > 9_000_000_000_000
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