Different trig algorithms in CAS and Home?

[Dieter]

(01-05-2018 01:33 AM)chromos Wrote:  My question was rather a reaction to the posted example, where the accuracy of the calculations were compared and which differed in a fifth or so significant digit. In any case, my question was not a complaint, but rather a curiosity.

This is simply caused by the properties of the tangent function for arguments close to pi/2. Here the tangent and its derivative approach infinity. If you want the tangent of 355/226 to agree with the true result in 12 significant digits, you have to specify 355/226 with at least 20 (!) digits.

This is an example of a function where a change in the last 12th digit of the input may cause changes in the 4th digit of the result. More precisely: at this point the output varies by 5,62 E+13 times the input change. Since the input has an inherent roundoff error of ±5 E–12 the calculated tangent cannot be more precise than ±281 (!). This means that 4 significant digits are the best you can get.

BTW, if you really want higher precision you can of course have it: Free42 and the DM42 as well as the WP34s work with 34 digits, the latter may internally even use much more than these to ensure a correct result. And there are extended/arbitrary precision libraries available for some calculators, e.g. the longfloat library for the 50G. Maybe sometime something similar will be available for the Prime as well.

Dieter