These functions determine a "floating point" approximation to an integer or rational. The base of the representation is either 2 or 10.
See also: ToString for functions producing readable numbers.
MantissaAndExponent2(x,prec) determine the MantExp2 structure for x with precision prec
FloatApprox(x,prec) apply MantissaAndExponent2 then convert the result into BigRat.
The value of prec is the number of bits in the mantissa; if unspecified, it defaults to 53.
A MantExp2 structure contains 4 public data fields:
mySign an int having value -1 or 1
myExponent a long
myMantissa a BigInt (between 2^(prec-1) and 2^prec-1)
myNumDigits a long (just the value of prec)
As an exception if x=0 then all fields are set to 0.
The structure represents the value mySign * (myMantissa/2^(myNumDigits-1)) * 2^myExponent.
MantissaAndExponent10(x,prec) determine the MantExp10 structure for x with precision prec
The value of prec is the number of (decimal) digits in the mantissa;
if unspecified, it defaults to 5.
A MantExp10 structure contains 4 public data fields:
mySign an int having value -1 or 1
myExponent a long
myMantissa a BigInt (between 10^(prec-1) and 10^prec-1)
myNumDigits a long (just the value of prec)
As an exception if x=0 then all fields are set to 0.
The structure represents the value mySign * (myMantissa/10^(myNumDigits-1)) * 10^myExponent.
The implementation is simple rather than efficient. The current design ensures that 0.5ulp is rounded consistently (currently towards zero).
The only tricky parts were deciding how to round in the case of a tie,
and correct behaviour when the mantissa "overflows". I finally
decided to delegate rounding to RoundDiv: it is easy to implement,
and I wanted a solution which was symmetric about zero, so that the two
MantissaAndExponent fns applied to N and to -N would always
give the same result except for sign.
Mantissa overflow requires special handling, but it's quite easy.
Printing of a MantExp2 or MantExp10 structure is simple rather
than elegant.
Using mpfr would surely be better.
The fields of a MantExp2 and MantExp10 are publicly accessible;
I'm undecided whether it is really better to supply the obvious accessor fns.
The conversion in MantissaAndExponent10 is rather slow when the input
number is large.
In principle the call to FloorLog2 could fail because of overflow;
but in that case FloorLog2 itself should report the problem.
In principle a mantissa overflow could trigger an exponent overflow (i.e. if the exponent was already the largest possible long).
2014