Re: Ok, from the HHC email list, what is the proper answer to 8 ENTER 2 divide 2 divide on 10 digit calc Message #11 Posted by Dieter on 2 Oct 2011, 1:07 p.m., in response to message #10 by Jake Schwartz
Jake,
Quote:
Just as an added "celebration" of the 34S, for what it is worth, I wanted to report that...
(...)
...only the final two computations differing from the others by one count in the 16th place. I am impressed
I am afraid I have to pour some water into the wine. ;)
While I really appreciate the accuracy of the results returned by the WP34s, the reason for its good results here (and the differences in the 10digit results) mainly is the input value itself: Both sqrt(2), Pi and by the way also e happen to round very well to 16 digits, while on the other hand the error is quite large in 10 digit precision. Simply take a look at the correct values (blanks inserted after 10th, 12th and 16th significant digit):
sqrt(2) = 1,414213562 37 3095 048...
Pi = 3,141592653 58 9793 238...
e = 2,718281828 45 9045 235...
You see, rounding to 10 digits always causes an error of roughly 0,4 ULP, that's almost the worst case possible. On the other hand the rounded 16 digit value is only half that (roughly 0,2 ULP) or even merely 0,05 ULP for sqrt(2). Rounding to 12 digits also gives good results, especially for Pi and e (less than 0,1 ULP).
It even gets better: Set the 34s rounding mode to RM 1 (i.e. round 0.5 up, like most classic HPs), and this happens:
[Pi] 4 [÷] = 1/4 of 3,1415926535897932
= 0,78539 81633 97448 25 exactly
= 0,78539 81633 97448 2 in RM 0 (newer HPs)
= 0,78539 81633 97448 3 in RM 1 (classic HPs)
which agrees to the exact value...
Pi/4 = 0,78539 81633 97448 309...
...with an error of less than 0,1 ULP, i.e. less than 1 unit in the 17th (!) digit.
In other words, setting RM 1 (like the classic HPs) will even remove that last "inconsistency" in the (radians) trig functions:
[rad]
[Pi] 4 [÷] [sin] => 0,70710 67811 86547 5
[Pi] 4 [÷] [cos] => 0,70710 67811 86547 5
By the way  as shown above, Pi/4 rounds very well to 10 digits (error less than 0,03 ULP). So you can do this in a classic 10digit machine:
[rad]
45 [D>R] [sin] => 0,70710 67812
45 [D>R] [cos] => 0,70710 67812
q.e.d. ;)
Dieter
Edited: 2 Oct 2011, 1:18 p.m.
