Re: Division using FMAC, reciprocal estimates and Newton-Raphson - eg ia64, rs6000, SSE, ARM MaverickCrunch?

[Andrew Haley <aph at redhat dot com>]

Paolo Bonzini wrote:
> 
>> I'd like to implement something similar for MaverickCrunch, using the
>> integer 32-bit MAC functions, but there is no reciprocal estimate
>> function on the MaverickCrunch.  I guess a lookup table could be
>> implemented, but how many entries will need to be generated, and how
>> accurate will it have to be IEEE754 compliant (in the swdiv routine)?
> 
> I think sh does something like that.  It is quite a mess, as it has half
> a dozen ways to implement division.
> 
> The idea is to use integer arithmetic to compute the right exponent, and
> the lookup table to estimate the mantissa.  I used something like this
> for square root:
> 
> 1) shift the entire FP number by 1 to the right (logical right shift)
> 2) sum 0x20000000 so that the exponent is still offset by 64
> 3) extract the 8 bits from 14 to 22 and look them up in a 256-entry,
> 32-bit table
> 4) sum the value (as a 32-bit integer!) with the content of the table
> 5) perform 2 Newton-Raphson iterations as necessary

To avoid the lookup table, calculate

   x = (a/2) + (8^(1/4) - 1)^2

which gives relative errors less than 0.036 over the range 1/2 <= a <= 2
at a cost of one shift and one addition.  The errors after 1,2,3, and 4
iterations of Heron's rule are 0.64E-3, 0.204E-6, 0.211E-13, and 0.222E-27.

So, this requires one more iteration but avoids the use of a table and the
corresponding memory hit.

Source: Computer Approximations, Hart et al.

Andrew.