Inaccuracy of TAN near 75° in rad mode
Post: #21
 BarryMead Senior Member Posts: 416 Joined: Feb 2014
RE: [WP 34s] Innacuracy of TAN near 75° in rad mode
Another way to say what Dieter is saying is that ANY function that has a STEEP slope, causes input errors to be AMPLIFIED on the output. The steeper the slope the greater the affect. The concept of accuracy is not absolute (to a certain number of digits) in situations like this. Even very tiny errors in the input to the function can generate very significant changes (NOT REALLY ERRORS) on the output.

Hope this helps to clear up the issue. Sincerely, Barry
07-01-2015, 01:41 AM (This post was last modified: 07-01-2015 01:43 AM by Marcio.)
Post: #22
 Marcio Senior Member Posts: 438 Joined: Feb 2015
RE: Innacuracy of TAN near 75° in rad mode
Thanks everyone.

I compiled this to show what happens(ed) more closely:

$\begin{matrix} & WP-34s & HP-35s & HP-50g & HP-Prime & Hosoda(35s) \\ 60 & 1.73205080757 & 1.73205080758 & 1.73205080754 & 1.73205080754 & 1.73205080758 \\ 73 & 3.27085261848 & 3.27085261853 & 3.27085261853 & 3.27085261853 & 3.27085261853 \\ 86 & 14.3006662567 & 14.3006662577 & 14.3006662577 & 14.3006662577 & 14.3006662577 \\ 89 & 57.2899616308 & 57.2899616145 & 57.2899616473 & 57.2899616473 & 57.2899616145 \\ 89.999 & 57,295.7795083 & 57,295.7717270 & 57,295.7717051 & 57,295.7717051 & 57,295.7717051 \end{matrix}$

Marcio
07-01-2015, 06:36 PM (This post was last modified: 07-02-2015 11:58 AM by Dieter.)
Post: #23
 Dieter Senior Member Posts: 2,397 Joined: Dec 2013
RE: Inaccuracy of TAN near 75° in rad mode
(07-01-2015 01:41 AM)Marcio Wrote:  I compiled this to show what happens(ed) more closely:

The table suggests that there is a characteristic result for each calculator. But it isn't. The result depends on the calculated argument of the tangent function, i.e. the angle in radians. This value may be rounded up or down, and whether you get the one or other result depends on random influences like the order of the multiplication by Pi and the division by 180. Take a look at the 89° case.

89° = 1,5533 43034 27495 ... rad. This value is quite exactly in the middle between the two possible adjacent 12-digit values 1,5533 43034 27 and ...28. If this happens to get rounded down to ...27, the tangent is 57,2899616145. If it happens to get rounded up to ...28, the tangent is 57,2899616473. Now even the way you do the multiplication by pi/180 influences the result you get. Here's an example for the 35s, but other 12-digit HPs should behave similarly:

Code:
89 [pi] [x] 180 [/]   =>  1,55334303428 [TAN] => 57,2899616473 89 [ENTER] 180 [/] [pi] [x]   =>  1,55334303427 [TAN] => 57,2899616145 89 [->RAD]   =>  1,55334303427 [TAN] => 57,2899616145

What's "the 35s result" now? ..6473 or ...6145?

Another example:

Code:
73 [pi] [x] 180 [/]   =>  1,27409035396 [TAN] => 3,27085261853 [pi] 180 [/] 73 [x]   =>  1,27409035395 [TAN] => 3,27085261842

What's "the 35s result" now? ..1853 or ...1842?

And it gets even worse – try 55°:

Code:
55 [pi] [x] 180 [/]   =>  0,959931088594 [TAN] => 1,42814800673 55 [ENTER] 180 [/] [pi] [x]  =>  0,959931088598 [TAN] => 1,42814800675 55 [->RAD]  =>  0,959931088597 [TAN] => 1,42814800674

Three different ways of determining the angle in radians yield three different results for this angle that differ by no less than 4 ULP.
What's "the 35s result" for the tangent now? ..673 or ...675 or ...674?

And finally:

Code:
[MODE] DEG 89 [TAN]  => 57,2899616308

This result is exact – because the angle is given exactly. Unlike its representation in radians which is always slightly rounded up or down.

So I do not think there is a characteristic result for a certain calculator. The result depends on how the angle in radians is rounded. Which in turn is influenced by various parameters.

The observed differences are not due to errors or inaccuracies in the calculators' tangent functions. They are caused by variations during the calculation of the angle in radians which cannot be given exactly. The tangent results for the rounded angles are correct.

The 34s is not affected here because the result is calculated with at least 16-digit precision. In the 73° case it does not evaluate tan 1,27409035395 or ...96 but tan 1,274090353955861 or ...60. So the displayed first 12 digits are exact. But exactly as with the 12-digit calculators, the last one or two digits are off as well.

Dieter
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