Accuracy of HP Financial Calculators - Canadian mortgage

[Dieter]

(04-07-2014 11:16 PM)Thomas Klemm Wrote:  But it's still good to know when you better avoid it.

OK, you could check whether (1+x) – 1 equals x and then evaluate ln(1+x) literally (1 + LN). But is it worth the effort? Without a dedicated function the last digit cannot be trusted anyway.

(04-07-2014 11:16 PM)Thomas Klemm Wrote:  OTOH: how does your method compare to HP's trick to calculate \(log(1+x)\) for small values?

The artanh-method was not intended to replace HP's method. It's an alternative way that requires only two stack levels. But anyway I did a quick test. I used the 34s emulator with a short program that rounds every intermediate result to a given number of significant digits and compares the approximate result with that of the the exact ln1+x function. This was done for 100.000 random numbers between 1 and 1E-15, using x = 10^(–15 · ran#). The seed for the RNG was always the same.

Here are the results:

10 digit working precision:
HP method: error within –6 ... +9 ULP
artanh method: error within –6 ... +6 ULP

12 digit working precision:
HP method: error within –6 ... +9 ULP
artanh method: error within –7 ... +8 ULP

16 digit working precision:
HP method: error within –5 ... +8 ULP
artanh method: error within –4 ... +4 ULP

So the artanh method seems to perform on par or even slightly better. On the other hand the HP method also works for large x where the artanh method fails (argument rounds to 1).

Dieter