Small challenge

[J-F Garnier]

(04-23-2023 06:41 PM)John Keith Wrote:  

(04-23-2023 02:13 PM)J-F Garnier Wrote:  The implementation of the exponentiation changed between the Capricorn platform (Series 80 / 75C) and the Saturn machines (starting with the 71B).

Does anyone know exactly how the code changed and, more importantly, is either implementation generally better than the other?

Here is a comparison between the 75C, the 71B (or any Saturn), and the 35S for a few other known test cases:
- the 729^33.5 vs 3^201 test cited by Pr. Kahan in Mathematics Written In Sand, for 10-digit calculators,
- the 3^729 vs (3^9)^81 test, cited here , source unclear, more suited for 12-digit machines.
- the (1e44)^10.5 worst case for the Saturn (9 ulp error), personal find, see here.

                       75C                71B (Saturn)                35S

729^33.5   7.96841966628e95   7.96841966628e95   7.96841966628e95   correct
3^201       7.96841966627e95   7.96841966626e95   7.96841966626e95   75c slightly better

3^729        6.62818605424e347  6.62818605419e347  6.62818605419e347   75c perfect !
(3^9)^81    6.62818605424e347  6.62818605425e347  6.62818605424e347

(1e44)^10.5  9.99999999998e461 9.99999999991e461 9.99999999991e461   75c clearly better

For the last test, the 35S closely mimics the Saturn, but the 75C is better !

You are right to relate the problem of the exponentiation accuracy to the guard digits, this has been discussed several times (the "2^3=8 New Accuracy" story), see here for instance.

J-F