[metapost] Metafun : is "atan" the arctangent function?

[Franck Pastor]

Le 12 janv. 2014 à 16:32, John Kitzmiller a écrit :

> On Jan 12, 2014, at 6:28 AM, Franck Pastor <franck.pastor at me.com> wrote:
> 
>> Hello,
>> 
>> Reading the "Metafun manual" (2010 edition), I noticed that it mentions p. 324 the existence of a specific "atan" unary operator. I assumed it was the arctangent function and I was pleasantly surprised, since this important inverse trigonometric function was not referenced in chapter 9 ("Functions") of the same Metafun manual. So I gave this atan operator a try but it returned me surprising results. For example, this short line
>> 
>> show atan(1);
>> 
>> gives me "0.7282" as result ("0.7238095238095239" with numbersystem set to "double"), whereas the right result  should be something like "0.7854" in radians (approximation of the exact result, pi/4) and exactly "45" in degrees.
>> 
>> Is "atan" thus actually meant as the arctangent function, of something else? Or is it a bug?
> 
> 
> atan is vardef'd in mp-tool.mpiv as: x = x-(x**3)/3+(x**5)/5-(x**7)/7 , part of the infinite series form. Maybe better to use asin(a/sqrt(a**2+1))?
> 
> John
> 

I didn't know about this definition (haven't been smart enough to search into mp-tool.mpiv…). Well, I suppose that this definition becomes increasingly unaccurate as x gets away from zero, which would explain the bad result above.

asin(a/sqrt(a**2+1)) gives indeed the good result in degrees. But in the meantime I realised that the atan function in degrees could be more simply defined as "angle(1, —)". And in radians, using

pi/180*angle(1, 1)

gives 0.78552 in scaled number system, not very precise but quite a better result than the infinite series form.

(Note that the angle function with the "double" number representation is somewhat broken in MetaPost 1.803, as I reported a few days previously — Taco Hoekwater told me it was fixed in the latest development versions).

Franck