arccosh works for much larger numbers than arccos

[endolith]

arccosh(x) = i arccos(x), so np.arccos() and np.arccosh() should produce the same output but with real and imaginary flipped:

In [39]: arccos(0.1+0j), arccosh(0.1+0j)
Out[39]: ((1.4706289056333368-0j), 1.4706289056333368j)

In [40]: arccos(1.1+0j), arccosh(1.1+0j)
Out[40]: (0.44356825438511532j, (0.44356825438511532+0j))

but at 10**7, arccos is inaccurate:

In [47]: arccos(10**7+0j), arccosh(10**7+0j)
Out[47]: (16.805431370234086j, (16.811242831518264+0j))

and then gives up at 10**8 and outputs infs:

In [48]: arccos(10**8+0j), arccosh(10**8+0j)
Out[48]: (inf*j, (19.113827924512311+0j))

but arccosh keeps going to almost the full float range:

In [86]: arccos(10**307+0j), arccosh(10**307+0j)
Out[86]: ((-inf-inf*j), (707.58677072973194+0j))

and is still accurate

so whatever algorithm is used for arccosh could presumably be used for arccos also, just with the imaginariness flipped.

[endolith]

arccosh appears to be far slower though :/

In [51]: timeit cos(x)
1000 loops, best of 3: 259 us per loop

In [52]: timeit cosh(x)
1000 loops, best of 3: 638 us per loop

In [53]: timeit arccos(x)
1000 loops, best of 3: 336 us per loop

In [54]: timeit arccosh(x)
100 loops, best of 3: 3.9 ms per loop

[mattip]

Closing. In the examples given arccos is now the complex conjugate of arccosh. Please reopen with failures if relevant.