|rational to floating point? firstname.lastname@example.org (Thant Tessman) (2003-10-13)|
|Re: rational to floating point? email@example.com (Miloslav Trmac) (2003-10-13)|
|Re: rational to floating point? firstname.lastname@example.org (2003-10-13)|
|Re: rational to floating point? email@example.com (2003-10-13)|
|Re: rational to floating point? firstname.lastname@example.org (Thant Tessman) (2003-10-14)|
|Re: rational to floating point? email@example.com (Fergus Henderson) (2003-10-18)|
|Re: rational to floating point? firstname.lastname@example.org (2003-10-18)|
|Re: rational to floating point? Peter-Lawrence.Montgomery@cwi.nl (2003-10-18)|
|Re: rational to floating point? email@example.com (Thant Tessman) (2003-10-27)|
|From:||Peter-Lawrence.Montgomery@cwi.nl (Peter L. Montgomery)|
|Date:||18 Oct 2003 15:35:40 -0400|
|Posted-Date:||18 Oct 2003 15:35:40 EDT|
[Followup to comp.arch.arithmetic]
Thant Tessman <firstname.lastname@example.org> writes:
>Can anyone recommend a source of information on converting
>arbitrary-precision rationals to floating-point numbers? I'm sure I
>could make something work, but doing it accurately, efficiently, and
>portably seems a little tricky. Google turns up lots of man pages of
>languages that support the capability, but a description of *how* they
>do it is harder to unearth.
Assume the input is x = n/d where n, d are positive integers (if you
can handle this case, you can handle negative and zero inputs too).
Assume you want a b-bit mantissa for your floating point output.
Find an exponent e such that 2^(e-1) <= n/d < 2^e
This e is approximately LOG2(n) - LOG2(d) where LOG2
denotes the truncated base-2 logarithm.
Let n1/d1 = (n/d)*2^(b - e).
The simple way to multiply by 2^(b-e) multiplies the numerator of n/d
by 2^(b-e) when b >= e, and multiplies the denominator of n/d
by 2^(e-b) when b < e. A more sophisticated algorithm
ensures the new n1 and d1 are not both even.
We observe that 2^(b-1) <= n1/d1 < 2^b.
Carry out the integer division n1 div d1,
getting quotient q and remainder r.
For round toward zero, or if the remainder r is zero,
answer is q*2^(e-b). You should check whether
the exponent is too large (overflow) for floating point arithmetic.
For round toward +infinity, answer is (q+1)*2^(e-b).
Be aware that q+1 may overflow to 2^b, requiring an exponent adjustment.
For round to nearest, with round to even in case of a tie,
compare r to d1/2. More precisely,
If q is even, then round down if r <= d1 - r, up if r > d1 - r.
If q is odd, then round down if r < d1 - r, up if r >= d1 - r.
Peter-Lawrence.Montgomery@cwi.nl Home: San Rafael, California
Microsoft Research and CWI
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