Matrix Division on the 28C/S (and later)

[Albert Chan]

(03-04-2020 12:29 PM)Thomas Okken Wrote:  I'll write the new version of DOT next week, I think. I'll have to add FMA to the "phloat" functions, using bid128_fma() in the decimal version, and (I think) std::fma() in the binary version, and then implement the functions from your citations.
It looks pretty straightforward.

DotFMA probably will not gain much accuracy.
Example, if we fuse for every step, with tiny a, such that a+1=1

DotFMA([a 1 -1], [1 1 1]) = (a+1) - 1 = 1-1 = 0

However, fusing result at the end help (essentially "double" precision until final sum)
see CompDot, CompDotFMA, http://rnc7.loria.fr/louvet_poster.pdf

(03-04-2020 01:41 PM)J-F Garnier Wrote:  Slightly changing the test to DOT( [ a -1 1 ] , [ 1 1 1] ), I noticed this difference between the 71B and the 28S/42S
(and later RPL machines I guess):
Using a=1E-20 (or any values <5E-16):
71B: result is 1E-15 !
28S/42S: result is correctly rounded to 0.

It seems 71B internal 15-digits precision is using Round-to-zero (chopping excess digits)
Say, a = 1.00000000001e-15

DOT([a 1 -1], [1 1 1]) = (1.0000 00000 00000 10000 00000 01) - 1 ≈ (1.0000 00000 00000) - 1 = 0

DOT([a -1 1], [1 1 1]) = (-0.99999 99999 99998 99999 999999) + 1 ≈ (-0.99999 99999 99998) + 1 = 2E-15