Euler Identity in Home

[Manolo Sobrino]

(04-14-2014 02:48 PM)ColinJDenman Wrote:  I'd note sin(n*pi) - sin(pi) for n=1...6000000 suggests there is no mapping of x to 0<= x < 2pi, which I do think is appropriately correctable for sin, as is the appropriate short circuiting of the internal algorithm for 'well known' values of x and sin.

There's no need for "short circuiting" and people really do get zeroes without it. This is a bounded (and periodical) function that maps [0, 2 Pi) to [-1, 1]. If you work with 12 significant digits, then:

A) 3.141592653585
B) 3.141592653586
C) 3.141592653587
D) 3.141592653588
E) 3.141592653589
F) 3.141592653590
G) 3.141592653591
H) 3.141592653592
I) 3.141592653593
J) 3.141592653594

Are all the same and (should) yield the same result than:

3.14159265359

There's no point in having as an answer:

-.000000000000206761537357

And swearing that every bit of it is true, because if those results were meaningful to E-13, then you could effectively tell F) from B), C)... J), and you can't. See how it works:

Let's generate A)-J):

(first line: Mathematica command, I hope the syntax is self-explanatory, results below)

Table[3.14159265359 + (i - 6)*10^-12, {i, 10}] // InputForm

/* i starts at i=1, // is the pipe to InputForm, so we can see all the digits from the output*/

{3.141592653585, 3.141592653586, 3.141592653587, 3.141592653588, 3.141592653589, 3.141592653590, 3.141592653591, 3.141592653592, 3.141592653593, 3.141592653594}


The true values of Sin for them are:

(first line: Mathematica command, results below, -I'm doing it the easy way-)

Table[Sin[3.14159265359 + (i - 6)*10^-12], {i, 9}]

{4.79318*10^-12, 3.79309*10^-12, 2.793*10^-12, 1.79335*10^-12,
7.93266*10^-13, -2.06823*10^-13, -1.20691*10^-12, -2.207*10^-12, -3.20665*10^-12}


Which value did the calc pick? The sixth? Why? But they are all good! in fact any value in (-3.20665*10^-12, 4.79318*10^-12) would be a good answer.


This is not a floating point issue. It's just continuous functions and significant digits' common sense. The best accurate and meaningful value your calc should give you is 0E-12, that is 2.06761537357E-13 rounded to 12, the rest is garbage. Yet people are not willing to get/believe it, go figure!


HP, or their fans, have taken historically a very strange viewpoint in which having obvious accuracy in the display is not that important, but raw results are, leaving to the user the task of making sense of them, with awful consequences on the average, I'm sad to say.


It's more sad that being burdened with mantras of 12 significant digits are enough, and you don't keep guard digits (even though apparently HP calculators adopted them later -I don't know for sure- and this, which makes some sense for an RPN machine, does not for a textbook entry), even when performing intermediate operations the user can't see, a powerful machine capable of a lot ends up doing things nobody else would. Anyway...