|Vintage logarithm round up|
Message #1 Posted by McAllan on 15 Mar 2012, 4:59 p.m.
I've played around a little with the logarithms or more accurately the powers function of a small selection of vintage calculators. Here I've done some very interesting discoveries.
While 2^8 gives more or less correct answers on all (like 259.9999999 or just above 256 on the hidden digits) especially interesting seems the two cases 2^26 and 2^30.
Before I start I should add that only the SR-50 seems to use hidden digits in an "honest way". I suspect the SR-51-II uses hidden digits too but if that's the case they're very hard to get for logarithms (easy for Pi). Most likely it's using hidden digits but the result of the isolated calculation is rounded to 10 (as many as can be displayed). HP and Canon for logarithm only seems to use as many digits as can be displayed - that is I can't seem to extract any more digits of the result that is displayed. Perhaps someone has something to add?
The results for 2^26
Correct result: 67.108.864 - please note I'm using European format ;)
TI SR-50: 67.108.864 + 8ee-5
TI SR-51-II: 67.108.864 (+ 0)
Canon F-71: 67.108.863,9 (note only 9 digits while it's a 10 digit machine)
Canon F-7: 67.108.857
The results for 2^30
Correct result: 1.073.741.824
TI SR-50: 1.073.741.824 + 1ee-3
TI SR-51-II: 1.073.741.824 (+ 0)
Canon F-71: 1.073.741.820
Canon F-7: 1,0737417ee9
Seems a bit odd. Especially HP and Canon seems to have some quirks. While the F-7 seems quite bad at 2^26 the HP is even worse if you take into account that is has more digits available. F-7 is off by 7 at the least significant digit while HP is off by 31(!). To add to oddity the F-7 seems substantially more precise if doing powers manually:
Keystrokes: 26 [x] 2 [ln] [)] [e^x]
Which results in: 67.108.863. Not 100% correct but very acceptable considering it's available digits.Using that method 2^8 is spot on.
On closer look it actually seems like F-71 rounds to 9 significant digits on powers despite is has 10 on the display for mantissa. Playing a little using the manually method on F-71 gets 2^26 = 67.108.864,03 and 2^30 = 1.073.741.825. While not 100% correct then a litte more correct than using dedicated power function - and with the 10th digit available... Odd... :S
On HP "manual" always seems to get same result as the dedicated button.
Seems like TI is really trumps the others. On the other hand the SR-50 seems to use logarithms to calculate square root. It's slower and sqrt(256) = 16 - 7ee-11. All others get square root 100% correct and are significantly faster here despite more or less same speed on other functions. The SR-51 (not the -II) seems equal to the SR-50 except for square root where it is also spot on. Perhaps the SR-50 implements it using logarithms because of ROM space (the SR-51 and SR-51-II has two ROMs where SR-50 only has one - so perhaps it's fully stuffed).
Regarding trigonometric I haven't found any real deal breakers although I should say I haven't played that much around with those functions. The largest reason I've found the SR-50 to be superior is because of it's larger internal precision. The SR-50 seems to use all 13 internal digits for triginometrics while the others only seems to use as many digits as is displayed. So after all the F-7 isn't as bad as writtin on some site. It may not beat neither HP nor SR-50 but it still beats the TI-30. May do trigonometric round up when I've played a little bit more with those functions.
Despite their flaws I somehow still like the Canons. They're well built and has nice reliable buttons. They got something about the casing and buttons correct. Second best imho SR-50 followed closely by it's replacement SR-50A then the HPs and last SR-51-II with it's low cost version of the "Klixon". Oh and who doesn't don't like the conversions on the F-7? While likely not much used they're well implemented and easy reachable in contrast to SR-51 or very limited like on SR-51-II.