Imaginary type and IEC 60559-compatible complex arithmetic

[skirpichev]

"Generally, mixed-mode arithmetic combining real and complex variables should be performed directly, not by first coercing the real to complex, lest the sign of zero be rendered uninformative; the same goes for combinations of pure imaginary quantities with complex variables." (c) Kahan, W: Branch cuts for complex elementary functions.

That's why C standards since C99 introduce imaginary types. This patch implements similar extension to the Python language.

Lets consider (actually interrelated) problems, which will be solved on this way.

1. Now complex arithmetic could be used for implementation of mathematical functions without special "corner cases", with textbooks formulae. Take the inverse tangent as an example:

   >>> z = complex(-0.0, 2)
   >>> cmath.atan(z)
   (-1.5707963267948966+0.5493061443340549j)
   >>> atan = lambda z: 1j*(cmath.log(1 - 1j*z) - cmath.log(1 + 1j*z))/2
   >>> atan(z)  # real part had wrong sign before
   (-1.5707963267948966+0.5493061443340549j)
   

2. Previously, we have unsigned imaginary literals with the following semantics:

   a±bj = complex(a ± 0.0, ±b)
   complex(a, ±b)*cj = complex(a*0.0 ∓ b*c, a*c ± b*0.0)
   

   While this behaviour was well documented, most users would expect instead here:

   a±bj = complex(a, ±b)
   complex(a, ±b)*cj = complex(∓b*c, a*c)
   

   i.e. that it follows to the rectangular notation for complex numbers.

   For example:

   >>> -0.0+1j  # was 1j
   (-0.0+1j)
   >>> float('inf')*1j  # was (nan+infj)
   infj
   >>> -0.0+1j  # was 1j
   (-0.0+1j)
   >>> complex(-0.0, 1)  # was (-0+1j), note funny signed integer zero
   (-0.0+1j)
   

3. The eval(repr(x)) == x invariant now holds for the complex type.

What's changed:

• Added a new subtype (imaginary) of the complex type with few overloaded methods (conjugate() and __getnewargs__()).
• Complex and imaginary types implement IEC 60559-compatible complex arithmetic (as specified by C11 Annex G).
• Imaginary literals now produce instances of imaginary type.
• cmath.infj/nanj were changed to be of imaginary type.
• Modules ast, code, copy, marshal got support for imaginary type.
• Few tests adapted to use complex, instead of imaginary literals
• Print dot for signed zeros in the real part of repr(complex)
• repr(complex) now prints real part even if it's zero.