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?

I traced the operations in my Emu75 and Emu71/DOS emulators.
First thing I immediately realized is that the HP-75C is using 16-digit extended accuracy for internal calculations,
when the Saturn is using only 15 digits, and this explains a lot !

Here are the intermediate results in extended precision on both machines for 3^729, computed as exp(729*ln(3)):

                        75c (16 digits)                71b (15 digits)
ln(3) =               1.098612288668109          1.09861228866810       exact value = 1.098612288668109(69)
729*ln(3) =        800.8883584390514          800.888358439044
exp(729*ln(3)) = 6.628186054237396e347    6.62818605418905e347
rounded to:        6.62818605424e347           6.62818605419e347

Each 71B step is correct, the loss of accuracy comes from the limited guard digits.
This confirms my opinion that 3 guard digits are just too short for the exponentiation on large numbers. It was fine for the 10-digit machines with numbers limited to 9.99..E99, but no more for the Saturn.

So the question is: why did the Saturn team (well, the initial 71B team) use only 15 digits for internal calculations instead of 16, since both the Capricorn and the Saturn CPUs have 64-bit, i.e. 16-nibble registers?
I don't have a clear answer, but managing only 15 digits simplifies the code (adding two 15-digit numbers naturally fits in a 16-digit register) and makes it faster, an important aspect for the quite slow HP-71B. Furthermore, this is how the 41C was doing: internal 13-digit numbers in 14-digit registers.
So it was probably a deliberate choice, a trade-off between efficiency and accuracy.

J-F