What is required printf precision for a __float128 to not lose information?

Question

I'm trying to printf a __float128 using libquadmath, eg:

quadmath_snprintf(s, sizeof(s), "%.30Qg", f);

With the following three constaints:

1. The output must match the following production:

    number = [ minus ] int [ frac ] [ exp ]
   
    decimal-point = %x2E       ; .
   
    digit1-9 = %x31-39         ; 1-9
   
    e = %x65 / %x45            ; e E
   
    exp = e [ minus / plus ] 1*DIGIT
   
    frac = decimal-point 1*DIGIT
   
    int = zero / ( digit1-9 *DIGIT )
   
    minus = %x2D               ; -
   
    plus = %x2B                ; +
   
    zero = %x30                ; 0
   

2. Given any input __float128 "i" that has been printfed to a string matching the above production "s" and and then "s" is scanfed back into a __float128 "j" - "i" must be bitwise identical to "j" - ie no information should be lost. For at least some values this is not possible (NaN, Infinity), what is the complete list of those values?

3. There should be no other string satisfying the above two criteria, that is shorter than the candidate.

Is there a quadmath_snprintf format string that satisfies the above (1, 3 and 2 when possible)? If so what is it?

What are the values of __float128 that cannot be represented accurately enough to satisfy point 2 by the above production? (eg Nan, +/-Infinity, etc) How do I detect if a __float128 is holding one of these values?

Answer 1

If you're on x86, then the GCC __float128 type is a software implementation of the IEEE 754-2008 binary128 format. The IEEE 754 standard requires that a binary -> char -> binary roundtrip recovers the original value if the character representation contains 36 significant (decimal) digits. Thus the format string %.36Qg ought to do it.

It is not required that a NaN roundtrip recover the original bitwise value.

As for your requirement #3, libquadmath does not contain code for this kind of "shortest representation" formatting, e.g. in the spirit of the Steele & White paper or the code by David Gay.