[HP35S] Disassembling ROM

[Dieter]

(02-22-2015 10:26 AM)Tugdual Wrote:  Edit: ok found that these are constants used for various polynomial approximations. For example .918938533204673 is used during gamma log approximation.

Ah, that's interesting. I always thought the 35s used some kind of approximation for Gamma resp. factorials. You may try the x! key with some non-integers so that the 35s has to evaluate the Gamma function. The result appears almost immediately. Try some integers and you'll notice an increasing lag as the values increase. 10! is returned instantly, but 150! requires roughly a second. Try 160! and the result appears immediately again. In fact the 35s seems to calculate factorials up to 158! by iterative multiplication, while 159! and up is evaluated by means of the Gamma function.

In this case not Γ, but ln Γ seems to be calculated. This would also explain slight errors for large arguments. Try 225,5! and the result is exact (1,89218764486 E434 with the last digit correctly rounded up from ...856). Try 225,6! and the last digit is 3 ULP low: 3,25375380207 E434 instead of ...210). This is the point where ln Γ exceeds 1000. Even if ln Γ is evaluated accurately to all internal 15 digits, there still is a potential absolute error of 1 E–11 in ln Γ which means a relative error of this order in Γ itself so that not more than 11 correct digits can be expected.

Dieter