Rethink mathematical simplifications

[hkiel]

I propose to have in this ticket a list of (more complex) mathematical simplifications to be implemented in OMC (ExpressionSimplify).

To quote sjoelund.se from github discussion:

acos(cos(x)) is not equal to x...
For example acos(cos(13)) = 0.4336293856408271

And cos(acos(13)) is an error, not 13...

acos(cos(x)) could be mod(x, pi)?

The same argumentation holds for typical math simplification
e^(ln(x)) = x which is an error for x<0

always correct simplifications

• ln(e^x) = x
• sin²(x)+cos²(x) = 1
• cosh²(x) - sinh²(x) = 1
• sin(acos(x)) = sqrt(1-x²)
• cos(asin(x)) = sqrt(1-x²)

problematic simplifications

• asin(sin(x)) = ?
• tan(atan(x) = ?
• atan(tan(x)) = ?
• sin(asin(x)) = ?
• acos(cos(x)) = mod(x, pi) -> is wrong, it is a triangle function
• asin(sin(x)) is shifted triangle function

incorrect simplifications

• cos(acos(x)) = x (only true for -1<=x<=1)
• sin(asin(x)) = x (only true for -1<=x<=1)
• e^ln(x)= x (only true for x>0)