arc cosine surprising numeric results

[rwst]

Here under the hood, RR.acos gets called. I think this should return meaningful values even if the value is outside the domain.

sage: cos(1.*I)
1.54308063481524        <--- CC element
sage: acos(_)
1.00000000000000*I
sage: acos(1.54308063481524)
NaN
sage: acos(CC(1.54308063481524))
0.999999999999997*I
sage: acos(RR(1.54308063481524))
NaN

Neither SymPy nor Pari nor Maxima do this:

In [1]: acos(1.543)
Out[1]: 0.999931383282944⋅ⅈ
? acos(1.543)
%1 = 0.99993138328294395810833497263866793658*I
(%i1) acos(1.543);
(%o1)                        0.9999313832829438 %i

Component: symbolics

Issue created by migration from https://trac.sagemath.org/ticket/20089

[rwst]

Description changed:

--- 
+++ 
@@ -12,12 +12,14 @@
 sage: acos(RR(1.54308063481524))
 NaN
 ```
-Neither SymPy nor Pari do this: 
+Neither SymPy nor Pari nor Maxima do this: 
 
 ```
 In [1]: acos(1.543)
 Out[1]: 0.999931383282944⋅ⅈ
 ? acos(1.543)
 %1 = 0.99993138328294395810833497263866793658*I
+(%i1) acos(1.543);
+(%o1)                        0.9999313832829438 %i
 ```
 

[jdemeyer]

Replying to @rwst:

why are floats of type RR not CC?

Are you suggesting

sage: parent(1.0)
Complex Field with 53 bits of precision

[rwst]

comment:3

The alternative would be to return complex in RR.acos.

[rwst]

comment:4

It's also a symptom of inconsistency in symbolic/pynac.pyx where we call RR.function, i.e. mpfr but also mpmath directly. Do you have a preference? Maybe it's best to use mpmath throughout in pynac.pyx.

[rwst]

comment:5

I have come to a conclusion, and adapted the ticket description.

[rwst]

Description changed:

--- 
+++ 
@@ -1,8 +1,8 @@
-The default `RR` type of input FP numbers is surprising. If we have complex variables as default, why are floats of type `RR` not `CC`?
+Here under the hood, `RR.acos` gets called. I think this should return meaningful values even if the value is outside the domain.
 
 ```
 sage: cos(1.*I)
-1.54308063481524
+1.54308063481524        <--- CC element
 sage: acos(_)
 1.00000000000000*I
 sage: acos(1.54308063481524)

[rwst]

comment:6

Ok, since this is at least consistent with arb behaviour I'll file this as oddity not bug. It would be interesting to know what the big Ms return.