|Re: Rounding of 10^x|
Message #9 Posted by Dieter on 28 Aug 2013, 6:33 a.m.,
in response to message #8 by Paul Dale
Compare it to the inaccuracy that's still present on today's machines: let's consider the yx function, which is evaluated as e x ln y.
On classic HPs with a working range up to 9,999...E+99, the largest possible exponent in this equation is 230,25..., so that even a perfectly rounded 10-digit result still may have an absolute error of +/- 5E-8 in the exponent. Which quite exactly equals a relative error of 5E-8 in the final result, i.e. up to 5 units in the 8th significant digit. So the result has merely 7 digits that can be trusted. The loss of accuracy is caused by the three digits in the integer portion of the exponent (230). Maybe this was (one) reason why HP chose to add three guard digits in the mid-seventies, and not two or four. So far, so good.
Things get worse as soon as the mentioned exponent may get larger: in current 12-digit machines with working range up to 9,999...E+499 it may exceed 1000 (up to 1151,29...) an thus four digits are lost. Since there are still no more than three internal guard digits, roundoff errors similar to those discussed for early HPs may show up in the last digit. Try this e.g. with an HP35s:
10 [ENTER] 450,9 [yx] 7,943 2823 4729 E+450
450,9 [10x] 7,943 2823 4729 E+450
correct result 7,943 2823 4724 E+450
10 [ENTER] -450,1 [yx] 7,943 2823 4719 E-451
-450,1 [10x] 7,943 2823 4719 E-451
correct result 7,943 2823 4724 E-451
Edited: 28 Aug 2013, 6:41 a.m.