Summation based benchmark for calculators

[Eddie W. Shore]

(12-22-2017 09:41 PM)pier4r Wrote:  I did not forget this post (nor the topic about common calculators).

So as said the summation based test will be very similar to the savage benchmark and I wonder how many similar tests were developed on various forum or groups regarding calculators.

Anyway, inspired by the test done in the thread of Eddie linked in the 1st post, I decided to pick randomly some scientific functions to assemble a summation that may give an idea of the performances of a calculator for common math functions.

The idea is to have a loop working on an incrementing 'x' value that is somewhat independent from the step before (while the savage test uses the value picked from the step before) . Furthermore while the idea of using a function and its inverse is pretty neat (see savage benchmark), some calculators may be very carefully coded figuring this out and therefore simplifying the expression.

I am not that interested in the accuracy, as it is well analyzed in many other posts using other examples (especially by Dieter, that for me will always be the ULPs man).

Surely speed without accuracy is not that neat and so one could go on and build a metric that weights accuracy and speed, as done by a recent HHC (about egyptian fractions), anyway I will collect only timings.

Timings will be collected here, and then moved on the wiki4hp if there are enough of them.

People are encouraged to post timings and results.

So to recap the idea of this test is there because:
- with a summation with increasing x (not dependent on the previous computed value) also some advanced scientific calculators can be tested. See casio fx991EX.
- it is easier to type in and execute. It does not take much time (well unless one is forced to write a program to optimize some parts).
- it may be used with a metric that combines speed and accuracy (a problem would be to get the right accuracy for the problem).
- it avoids to use a function and its inverse to skip very careful optimizations done by the parser. (especially systems with CAS may use this)


Here is the assembled formula (I guess there can be an infinity of formulas that capture the idea I am going to expose).

\[ tan^{-1}(x) \]
So x can be incremented without problems. Picked among the trigonometric functions.

\[ sin \left (tan^{-1}(x) \right ) \]
To let the value increase towards 1, although, after a while, very slowly. Still trig.

\[ e^{ sin \left (tan^{-1}(x) \right ) } \]
to use either a common exponential or a logarithmic function without producing large numbers

\[ \sqrt[3]{e^{ sin \left (tan^{-1}(x) \right ) }} \]
To make the final number even smaller. Using powers.

Final formula to use (adapt the max 'x' value for speedy calculators. For example using 10k or 100k and so on)
\[ \sum_{x=1}^{1000} \sqrt[3]{e^{ sin \left (tan^{-1}(x) \right ) }} \]

As said, it is an arbitrary formula picking functions among some of the most common types of functions: trig, exp/ln, powers. Trying to keep the final numbers small. I may have picked some other functions easier to check (to compare the accuracy). This one is pretty nasty. Anyway as I said I am interested in timings assuming that the calculators are as precise as they could with the digits that they have.

I am really interested in what scientific calculators can do. I hope Eddie W. Shore sees this posts and contributes with a couple of models. Or jebem.

Results

max = 10000
1. ~ 20 sec - ti nspire handheld (2006). Degrees, float 12, approx. 13955.8579044 . OS 3.9.0.463

max = 1000
~ 0.223s - HW-Prime , Home, Teval().
~ 12.582s - HW-Prime , CAS, approx mode.
~ 24.5 seconds - Hp 50g, 2.15, RPN mode, DEG (done with the beautiful eq matrix and then EVAL. TEVAL did not help) . 1395.3462877 (approx mode)
~ 33.8 sec - Hp 50g, 2.15, RPN mode, DEG (quick userRPL FOR loop, surprisingly slower than the summation) . 1395.3462877 (approx mode)
~ 48 secs - ti 89 titanium, 12 digits, approx mode, degrees. OS 3.10 (code sum(seq((e^(...))^(1/3), x, 1, 1000))) . 1395.34628774

max = 10
1. 1< seconds - Hp 50g, 2.15, RPN mode 13.7118350167 (approx mode)
2. ~ 47 seconds - sharp 506w (using X, formula memory F4 and M to sum) manually. 13.71183502 . I did not yet found out what is the equivalent of the summation if I abuse the integration function. (that should use the trapezoidal rule)

PS: I forgot how full of apps was the ti89 and how awful is the official command reference compared to the 50g AUR.

Casio fx-115ES Plus: 1395.346288, time 8 minutes and 45 seconds