Small challenge

[Albert Chan]

(04-23-2023 11:10 AM)robve Wrote:  It is common to find exponentiation by squaring for integer powers, which tends to be more accurate than repeated multiplication and log*exp closed forms.

Unfortunately, this is true only if we don't cause intermediate rounding errors.

(03-22-2022 07:11 PM)Albert Chan Wrote:  When we keep squaring, the errors also blows up exponentially.

x = 1e8 = 0b101 111 101 011 110 000 100 000 000
b = 1+1/x

Free42 (internally only do squarings)
b^(2^08) = 1.000002560003264002763521747927282
b^(2^13) = 1.000081923355125194292436470243333
...
b^(2^26) = 1.956365933428064586618947538663749

All terms to multiply has errors on the *same* side; losing 7 digits precision is normal.

We had this discussion in Free42 possible accuracy flaw thread
Free42/Plus42 were updated to avoid above explosive error issues.

This is result of Plus42 1.0.14 (RHS now all exact)

b^(2^08) = 1.000002560003264002763521747927281
b^(2^13) = 1.000081923355125194292436470243294
b^(2^26) = 1.956365933428064586618947538036231

(03-24-2022 05:01 PM)Thomas Okken Wrote:  

(03-24-2022 04:53 PM)Albert Chan Wrote:  Can I assume new implementation is never worse than internal bid128_pow() ?

Yes: it only uses multiplications when the result of that calculation is exact, and bid128_pow() in all other cases.