[issue1580] Use shorter float repr when possible

[Mark Dickinson]

Mark Dickinson <dickinsm at gmail.com> added the comment:

> I think ANY attempt to rely on eval(repr(x))==x is asking for trouble,
> and it should probably be removed from the docs.

I disagree.  I've read the paper you refer to; nevertheless, it's still 
perfectly possible to guarantee eval(repr(x)) == x in spite of the 
various floating-point details one has to deal with.   In the worst 
case, assuming that C doubles are IEEE 754 binary64 format, for double -
> string conversion one can simply cast the double to an array of 8 
bytes, extract the sign, exponent and mantissa from those bytes, and 
then compute the appropriate digit string using nothing but integer 
arithmetic;  similarly for the reverse operation: take a digit string, 
do some integer arithmetic to figure out what the exponent, mantissa and 
sign should be, and pack those values back into the double.  Hey presto:  
correctly rounded string -> double and double -> string conversions!  
Add a whole bag of tricks to speed things up in common cases and you end 
up with something like David Gay's code.

It *is* true that the correctness of Gay's code depends on the FPU being 
set to use 53-bit precision, something that isn't true by default on x87 
FPUs.  But this isn't hard to deal with, either:  first, you make sure 
that you're not using the x87 FPU unless you really have to (all Intel 
machines that are newer than Pentium 3 have the SSE2 instruction set, 
which doesn't have the problems that x87 does);  second, if you *really* 
have to use x87 then you figure out how to set its control word to get 
53-bit precision and round-half-to-even rounding;  third, if you can't 
figure out how to set the control word then you use a fallback something 
like that described above.

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