Inaccuracy of TAN near 75° in rad mode
06-30-2015, 07:41 PM (This post was last modified: 06-30-2015 07:56 PM by Dieter.)
Post: #15
 Dieter Senior Member Posts: 2,397 Joined: Dec 2013
RE: [WP 34s] Innacuracy of TAN near 75° in rad mode
(06-30-2015 03:18 PM)Marcio Wrote:  It seems the 34s is having troubles getting the 10th dec number correct when it evaluates the tangent of angles near 75 degrees in radians mode. I know this looks kinda silly but I thought it would be worth reporting.

1. The 34s is fine.
A bit of simple calculus explains it all.

2. tan(75°) = 3,7320 50807 56887 72935 27446 34150 58723 ...
And that's exactly what the 34s returns.

3. 75° = 1,3089 96938 99574 71826 92768 07636 64595 ... rad
And that's exactly what the 34s returns.

However, on a simple 12-digit calculator, 75° is rounded to 1,3089 96939 00 rad. Due to roundoff in the last digit, the value in radians may be off by 5 units in the 13th digit. Since the derivative of tan x at this point is approx. 15, this translates to a potential error of 7 units in the last (12th) digit:

The exact argument (75*pi/180) is about halfway between 1,3089 96938 99 and ...900.
If it happens to get rounded down, you get tan(1,3089 96938 99) = 3,73205080748
If it happens to get (correctly) rounded up, you get tan(1,3089 96939 00) = 3,73205080763

You see that a change in 1 ULP of the argument makes the tanget change by 15 ULP (!), just as expected.
Since the argument always has a potential roundoff error of 0,5 ULP, it may and will be up to 7 ULP off.

Now try this with 89,9° and the last four (!) digits will be off.
Try 90° and the result becomes completely useless. ;-)

So the problem is the limited accuracy of the manually calculated argument. 75 degress is not exactly 1,30899693900 radians, so the tangent is inaccurate. On the other hand the 34s with its internal 39-digit precision gets it right. And so does the 35s or the 50G if you calculate tan(75°) in degrees mode: the internal calculations are done with three additional digits so the error does not show up. However, manually converting 75° to radians introduces a very slight error that is large enough to throw the tangent off in the last one or two digits. And that's exactly what you see here. The 34s returns a correct 12-digit result because even in SP mode it uses 16 digit precision and the error only affects the (not displayed) 15th digit.

Dieter
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