The Museum of HP Calculators

HP Forum Archive 17

 Re: Significant digits -- well, yes and no...Message #1 Posted by Massimo Gnerucci (Italy) on 21 Feb 2007, 2:49 a.m. Quote: HP-42S: sin (3.14159265358 rad) = 9.79323846264 x 10-12 -- correct result to 12 significant digits HP-41: sin (3.141592653 rad) = 5.9 x 10-10 -- correct result to 2 significant digits Sorry Karl, shouldn't that be: HP-41: sin (3.141592654 rad) = -4.1 x 10-10 vs -4.10206761537 x 10-10 ? Greetings,Massimo

 Missing pi digits calculationMessage #2 Posted by Karl Schneider on 21 Feb 2007, 3:47 a.m.,in response to message #1 by Massimo Gnerucci (Italy) Hi, Massimo -- Quote: Sorry Karl, shouldn't that be: HP-41: sin (3.141592654 rad) = -4.1 x 10-10 vs -4.10206761537 x 10-10 ? That also is a correct calculation, but my point was to reveal the ensuing digits of pi by calculating a truncated (not rounded) value of pi in radians mode. I've gone through the exercise several times in the Forum, but didn't save a bookmark to those posts: ```sin(pi - x) = sin(pi)*cos(x) - cos(pi)*sin(x) = 0 * cos(x) - (-1)*sin(x) = sin(x) ``` x represents the truncated digits. For very small x, sin(x) ~= x, so the result produces a limited string of those digits. The excellent mathematical routines developed for the Saturn microprocessor (debuting with the HP-71B) were ported to the Pioneer-series calculators. No other calculator I own matches the quality of the Saturn mathematics, although the TI-89 might. It also seems likely that Valentin's vintage Sharp pocket computers could meet or exceed the accuracy. -- KS

 SHARP accuracyMessage #3 Posted by Valentin Albillo on 21 Feb 2007, 5:15 p.m.,in response to message #2 by Karl Schneider Hi, Karl: Karl posted: "It also seems likely that Valentin's vintage Sharp pocket computers could meet or exceed the accuracy." Likely. These are your results as computed by my SHARP PC-1475, rounded to 12 digits: ``` sin (3.14159265358 rad) = 9.79323846265E-12 sin (3.141592653 rad) = 5.89793238463E-10 cos (89.9999999 deg) = 1.74532925199E-09 ``` Matter of fact, this last result actually comes out in full precision as: ``` cos (89.9999999 deg) = 1.7453292519943295760D-09 ``` which has all its 20 significant digits absolutely correct. Best regards from V.

 Re: SHARP accuracyMessage #4 Posted by Karl Schneider on 23 Feb 2007, 1:39 a.m.,in response to message #3 by Valentin Albillo Hi, Valntin -- Quote: These are your results as computed by my SHARP PC-1475, rounded to 12 digits: sin (3.14159265358 rad) = 9.79323846265E-12 But, V! The last mantissa digit should be a "4", whether it's the actual digit or rounded. Who was responsible for the rounding? :-) Seriously though, 20 significant digits was quite impressive for the era. I safely considered the 12+ digit performance "likely" from your photos I had previously viewed, at least one of which depicted a Sharp "PC" with many digits showing in its display. Best regards, -- KS

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