Quick Test

What if we want a quick and dirty test that determines both the number of digits a calculator uses and whether or not it gives correct results to that number of digits?

Of course there is an unlimited number of possibilities of combining functions and values and I've simply picked one. It uses functions that most scientific calculators offer (using radians):
Operation 8 digit result 9 digit result 10 digit result 11 digit result 12 digit result 13 digit result 14 digit result
100 / 18  5.5555556E+0  5.55555556E+0  5.555555556E+0  5.5555555556E+0  5.55555555556E+0  5.555555555556E+0  5.5555555555556E+0
sin -6.6510148E-1 -6.65101512E-1 -6.651014600E-1 -6.6510151495E-1 -6.65101514976E-1 -6.651015149785E-1 -6.6510151497879E-1
e^x  5.1422134E-1  5.14221325E-1  5.142213410E-1  5.1422132390E-1  5.14221323892E-1  5.142213238903E-1  5.1422132389015E-1
asin  5.4009951E-1  5.40099493E-1  5.400994918E-1  5.4009949158E-1  5.40099491570E-1  5.400994915684E-1  5.4009949156822E-1
tan  5.9956490E-1  5.99564874E-1  5.995648720E-1  5.9956487171E-1  5.99564871699E-1  5.995648716971E-1  5.9956487169684E-1
* sqrt(2)  8.4791284E-1  8.47912775E-1  8.479127733E-1  8.4791277311E-1  8.47912773077E-1  8.479127730765E-1  8.4791277307616E-1
ln -1.6497743E-1 -1.64977508E-1 -1.649775102E-1 -1.6497751038E-1 -1.64977510418E-1 -1.649775104190E-1 -1.6497751041938E-1
* -5  8.2488715E-1  8.24887540E-1  8.248875510E-1  8.2488755190E-1  8.24887552090E-1  8.248875520950E-1  8.2488755209690E-1
atan  6.8973289E-1  6.89733125E-1  6.897331320E-1  6.8973313254E-1  6.89733132652E-1  6.897331326549E-1  6.8973313265606E-1
Previous precision  6.8973289 E-1  6.89733125 E-1  6.897331320 E-1  6.8973313254 E-1  6.89733132652 E-1  6.897331326549 E-1
Relative error -3.5E-07 -1.1E-08 -9.5E-10 -1.7E-10 -6.1E-12 -1.9E-12 -1.7E-13

The sequence of operations is carefully chosen to avoid unnecessary amplification of errors. But it also assures that errors of one stage are not cancelled out in the next one: Looking at the row labelled "Previous precision" it can be observed that at least the last digits of the final result is usually not correct - regardless of how many digits are used! This is also expressed by the relative error of the result given in the last line.
Sequence of operations
100  18  sin  e^x  asin tan  sqrt ln  -5  atan
100  18  sin  e^x  asin tan  sqrt ln  -5  atan