Trac #15888: CC -> PARI: return t_REAL for real numbers

When converting Sage complex numbers to PARI, we should return a PARI
`t_REAL` if the original number was actually real. This fixes a problem
with `ellwp()` in #15767. In PARI's floating point model, the number
`1e-19 + 0.0*I` has less precision than the real `1e-19`.

It is also consistent with the PARI philosophy of returning simplified
types (examples: `(1.1*I)^2` and `ellj(1.1*I)` in PARI gives real
numbers).

URL: http://trac.sagemath.org/15888
Reported by: jdemeyer
Ticket author(s): Jeroen Demeyer
Reviewer(s): Peter Bruin

src/sage/libs/pari/gen.pyx

@@ -4862,7 +4862,7 @@ cdef class gen(sage.structure.element.RingElement):
             -2.18503986326152
             sage: C.<i> = ComplexField()
             sage: pari(i).tan()
-            0.E-19 + 0.761594155955765*I
+            0.761594155955765*I
         """
         pari_catch_sig_on()
         return P.new_gen(gtan(x.g, prec_bits_to_words(precision)))
@@ -4882,7 +4882,7 @@ cdef class gen(sage.structure.element.RingElement):
             0.761594155955765
             sage: C.<i> = ComplexField()
             sage: z = pari(i); z
-            0.E-19 + 1.00000000000000*I
+            1.00000000000000*I
             sage: result = z.tanh()
             sage: result.real() <= 1e-18
             True
@@ -4967,7 +4967,7 @@ cdef class gen(sage.structure.element.RingElement):
             1.18920711500272 + 0.E-19*I                 # 32-bit
             1.18920711500272 + 2.71050543121376 E-20*I  # 64-bit
             sage: pari(i).weber(1)
-            1.09050773266526 + 0.E-19*I
+            1.09050773266526
             sage: pari(i).weber(2)
             1.09050773266526
         """

src/sage/rings/complex_double.pyx

@@ -1054,21 +1054,33 @@ cdef class ComplexDoubleElement(FieldElement):
 
     cdef GEN _gen(self):
         cdef GEN y
-        y = cgetg(3, t_COMPLEX)    # allocate space for a complex number
-        set_gel(y, 1, pari.double_to_GEN(self._complex.dat[0]))
-        set_gel(y, 2, pari.double_to_GEN(self._complex.dat[1]))
+        if self._complex.dat[1] == 0:
+            # Return t_REAL
+            y = pari.double_to_GEN(self._complex.dat[0])
+        else:
+            # Return t_COMPLEX
+            y = cgetg(3, t_COMPLEX)
+            if self._complex.dat[0] == 0:
+                set_gel(y, 1, gen_0)
+            else:
+                set_gel(y, 1, pari.double_to_GEN(self._complex.dat[0]))
+            set_gel(y, 2, pari.double_to_GEN(self._complex.dat[1]))
         return y
 
     def _pari_(self):
         """
-        Return PARI version of ``self``.
+        Return PARI version of ``self``, as ``t_COMPLEX`` or ``t_REAL``.
 
         EXAMPLES::
 
             sage: CDF(1,2)._pari_()
             1.00000000000000 + 2.00000000000000*I
             sage: pari(CDF(1,2))
             1.00000000000000 + 2.00000000000000*I
+            sage: pari(CDF(2.0))
+            2.00000000000000
+            sage: pari(CDF(I))
+            1.00000000000000*I
         """
         pari_catch_sig_on()
         return pari.new_gen(self._gen())

src/sage/rings/complex_number.pyx

@@ -544,7 +544,8 @@ cdef class ComplexNumber(sage.structure.element.FieldElement):
 
     def _pari_(self):
         r"""
-        Coerces ``self`` into a Pari ``complex`` object.
+        Coerces ``self`` into a PARI ``t_COMPLEX`` object,
+        or a ``t_REAL`` if ``self`` is real.
 
         EXAMPLES:
 
@@ -553,14 +554,26 @@ cdef class ComplexNumber(sage.structure.element.FieldElement):
             sage: a = ComplexNumber(2,1)
             sage: pari(a)
             2.00000000000000 + 1.00000000000000*I
+            sage: pari(a).type()
+            't_COMPLEX'
             sage: type(pari(a))
             <type 'sage.libs.pari.gen.gen'>
             sage: a._pari_()
             2.00000000000000 + 1.00000000000000*I
             sage: type(a._pari_())
             <type 'sage.libs.pari.gen.gen'>
+            sage: a = CC(pi)
+            sage: pari(a)
+            3.14159265358979
+            sage: pari(a).type()
+            't_REAL'
+            sage: a = CC(-2).sqrt()
+            sage: pari(a)
+            1.41421356237310*I
         """
-        return sage.libs.pari.all.pari.complex(self.real()._pari_(), self.imag()._pari_())
+        if self.is_real():
+            return self.real()._pari_()
+        return sage.libs.pari.all.pari.complex(self.real() or 0, self.imag())
 
     def _mpmath_(self, prec=None, rounding=None):
         """

src/sage/schemes/elliptic_curves/heegner.py

@@ -3152,7 +3152,7 @@ def numerical_approx(self, prec=53, algorithm=None):
             sage: E = EllipticCurve('37a'); P = E.heegner_point(-40); P
             Heegner point of discriminant -40 on elliptic curve of conductor 37
             sage: P.numerical_approx()
-            (-6.68...e-16 + 1.41421356237310*I : 1.00000000000000 - 1.41421356237309*I : 1.00000000000000)
+            (-6.6...e-16 + 1.41421356237310*I : 1.00000000000000 - 1.41421356237309*I : 1.00000000000000)
 
         A rank 2 curve, where all Heegner points of conductor 1 are 0::