Some trivial identities and simplifications missed by Sage (pynac ?)

[charpent]

On 8.1.rc0. Consider :

sage: import sympy
sage: [arcsin(t) for t in [0, 1/2, sqrt(2)/2, sqrt(3)/2, 1]]
[0, 1/6*pi, arcsin(1/2*sqrt(2)), arcsin(1/2*sqrt(3)), 1/2*pi]
sage: [maxima.asin(t).sage() for t in [0, 1/2, sqrt(2)/2, sqrt(3)/2, 1]]
[0, 1/6*pi, 1/4*pi, 1/3*pi, 1/2*pi]
sage: [sympy.asin(t)._sage_() for t in [0, 1/2, sqrt(2)/2, sqrt(3)/2, 1]]
[0, 1/6*pi, 1/4*pi, 1/3*pi, 1/2*pi]
sage: [arctan(t) for t in [0, 1/sqrt(3), 1, sqrt(3)]]
[0, arctan(1/3*sqrt(3)), 1/4*pi, arctan(sqrt(3))]
sage: [maxima.atan(t).sage() for t in [0, 1/sqrt(3), 1, sqrt(3)]]
[0, 1/6*pi, 1/4*pi, 1/3*pi]
sage: [sympy.atan(t)._sage_() for t in [0, 1/sqrt(3), 1, sqrt(3)]]
[0, 1/6*pi, 1/4*pi, 1/3*pi]

[ Edit on 2017-11-23 ] In the same vein :

sage: [arccos(t) for t in [0, 1/2, sqrt(2)/2, sqrt(3)/2, 1]]
[1/2*pi, 1/3*pi, arccos(1/2*sqrt(2)), arccos(1/2*sqrt(3)), 0]
sage: [maxima.arccos(t).sage() for t in [0, 1/2, sqrt(2)/2, sqrt(3)/2, 1]]
[1/2*pi, 1/3*pi, arccos(1/2*sqrt(2)), arccos(1/2*sqrt(3)), 0]
sage: [sympy.acos(t)._sage_() for t in [0, 1/2, sqrt(2)/2, sqrt(3)/2, 1]]
[1/2*pi, 1/3*pi, 1/4*pi, 1/6*pi, 0]
sage: [arctan(t) for t in [0, 1/sqrt(3), 1, sqrt(3)]]
[0, arctan(1/3*sqrt(3)), 1/4*pi, arctan(sqrt(3))]
sage: [maxima.arctan(t).sage() for t in [0, 1/sqrt(3), 1, sqrt(3)]]
[0, arctan(1/3*sqrt(3)), 1/4*pi, arctan(sqrt(3))]
sage: [sympy.atan(t)._sage_() for t in [0, 1/sqrt(3), 1, sqrt(3)]]
[0, 1/6*pi, 1/4*pi, 1/3*pi]

This is not a bug stricto sensu (the answers are not false), but is certainly a lack of functionality.

The absence of this problem when using sympy or maxima fingerpoints to pynac.

Furthermore, arctan suffers from some lack of simplification:

sage: assume(-pi/2<x, x<pi/2)  
sage: arctan(tan(x)).simplify()  # OK
x
sage: arctan(sin(x)/cos(x)).simplify()   # should return x as above
arctan(sin(x)/cos(x))
sage: arctan(sin(x)/cos(x)).simplify_full()
arctan(sin(x)/cos(x))

arctan2 has the same issue:

sage: arctan2(sin(x), cos(x)).simplify_full()
arctan2(sin(x), cos(x))

In those cases, the problem is *not* solved by recourse to pynac...

[rws]

Is sin(x)/cos(x) simplifiable for all x to tan(x)?

[egourgoulhon]

Replying to rws:

Is sin(x)/cos(x) simplifiable for all x to tan(x)?

I would say yes, see e.g.

• ​http://planetmath.org/complextangentandcotangent
• ​https://fr.wikipedia.org/wiki/Trigonom%C3%A9trie_complexe

[egourgoulhon]

Replying to egourgoulhon:

Replying to rws:

Is sin(x)/cos(x) simplifiable for all x to tan(x)?

I would say yes, see e.g.

• ​http://planetmath.org/complextangentandcotangent
• ​https://fr.wikipedia.org/wiki/Trigonom%C3%A9trie_complexe

Moreover, Sage assumes it is the case, without any assumption on x:

sage: bool(tan(x) == sin(x)/cos(x))
True

[rws]

Replying to egourgoulhon:

Moreover, Sage assumes it is the case, without any assumption on x:

sage: bool(tan(x) == sin(x)/cos(x))
True

That comes from Maxima which simplifies tan(x) to sin(x)/cos(x)---but not vice versa, as the atan cases show. It may be difficult to find sin(x)/cos(x) in all expressions but I haven't looked further into it, and the simple cases should be implemented regardless.

[rws]

These doctest now work with Pynac master:

sage: [arcsin(t) for t in [0, 1/2, sqrt(2)/2, sqrt(3)/2, 1]]
[0, 1/6*pi, 1/4*pi, 1/3*pi, 1/2*pi]
sage: [arcsin(-t) for t in [0, 1/2, sqrt(2)/2, sqrt(3)/2, 1]]
[0, -1/6*pi, -1/4*pi, -1/3*pi, -1/2*pi]
sage: [arctan(t) for t in [0, 1/sqrt(3), 1, sqrt(3)]]
[0, 1/6*pi, 1/4*pi, 1/3*pi]
sage: [arctan(-t) for t in [0, 1/sqrt(3), 1, sqrt(3)]]
[0, -1/6*pi, -1/4*pi, -1/3*pi]

[charpent]

#24262 partially fixes some of these deficiencies. See its initial ​review.

[charpent]

Edit : added similar examples with arccos/arctan.