@Thomas Klemm -> CORDIC Article

[Claudio L.]

(06-04-2014 08:34 PM)pito Wrote:  I did following experiment when entering the 9degree formula in the wolfram - I added a small epsilon to the 9:

Code:

180/pi*(asin(acos(atan(tan(cos(sin((9.0+epsilon)*pi/180)))))))
epsilon = -1e-36
8.999999999999999999999999999999999999
epsilon = -1e-35
8.99999999999999999999999999999999999
epsilon = -1e-34
8.9999999999999999999999999999999999
epsilon = 0
9.0
epsilon = 1e-34
9.0000000000000000000000000000000001
epsilon = 1e-35
9.00000000000000000000000000000000001
epsilon = 1e-36
9.000000000000000000000000000000000001

Based on this result I think wolfram optimizes out the formula entered.

Wolfram has a very complex engine, either it symbolically cancels out the functions, or (more likely) uses adaptive precision. Basically, they internally keep recalculating the expression, adding as many precision digits as needed until the last digits they want to display don't change anymore.
If they want to display a result with 34 digits, they start maybe with a 34-digit calculation, then they redo the math with let's say 39 digits, and round only the final result to 34 digits. If it is different, they add 5 more digits of precision and redo the calculation with 44, until the rounded final result doesn't change at all.
So we see 34 digits, but all intermediate rounding was done at 39, 44 or higher, as needed to get a consistent final result.

And in general, we should do this on the calculator as well: if WP34S has 34 digits for calculations, set your display to 30 digits or less, and in your mind, think that you are using only 30 digits.
Then your 9 degree trig test will return exactly 9.0 with 30 digits, and the positive psychological effect of that "perfection" will put a smile on your face.

Claudio