Online calculator simulators?

[Albert Chan]

(07-09-2022 02:15 PM)jonmoore Wrote:  In conclusion, the older HP hardware is still pretty accurate considering their lower precision, but the fact remains that they begin to deviate from 9 at the 4th decimal place, the post Voyager HP's begin to deviate at the 6th decimal place, and the TI's come out second from best by only deviating from 9 at the 8th decimal place. But it's the Casio's that win in this particular accuracy test, by only beginning to deviate from 9 at the 9th decimal place. Both the Casios calculate all functions to 15 significant digits under the hood, even though they only display 10 significant digits.

This has less to do with calculators, and more to do with math.

sin(9°) = sin(pi/20) ≈ pi/20
cos(sin(9°)°) ≈ cos(pi/20 * pi/180) ≈ 1 - (pi^2/3600)^2 / 2! ≈ 0.999996

We had lost 5+ digits precision, due to catastrophic cancellation.
That's why all calculator forensic test will lose 5 to 6 digits precisions.
This is true even if all calculations produce correctly rounded results!

Example, HP Prime CAS side work with 48 bits (truncation mode)
48 bits ≈ 14 decimal digits, we expected 8 to 9 decimals accuracy.

CAS> AAngle := 2; // Degree mode
CAS> asin(acos(atan(tan(cos(sin(9.))))))-9

5.91438720221e−9

If we replace cos(x) with versin(x), the issue goes away.

CAS> versin(x) := 2*sin(x/2)^2; // = 1 - cos(x)
CAS> aversin(y) := 2*asin(sqrt(y/2)); // = acos(1 - y)

CAS> asin(aversin(atan(tan(versin(sin(9.))))))-9

−1.70530256582e−13