Trac #20064: Bug in sqrt in QQbar

See #18836.  This bug is holding up that (and also #20028).  The
following code creates an elemnt d of QQbar and tries to do d.sqrt().
It fails unless you call d.imag().is_zero() first.
{{{
K.<i> = QuadraticField(-1)

# define a low-precision embedding from K to CC:

emb = K.embeddings(CC)[1]

# extend this to the closest embedding into QQbar:

old_gen = emb(K.gen())
rr = K.defining_polynomial().roots(QQbar, multiplicities=False)
diffs = [(CC(r)-old_gen).abs() for r in rr]
new_gen = rr[diffs.index(min(diffs))]
emb0 = K.hom([new_gen], check=False)

# Take a polynomial with 3 roots in K:

e1 = -4+i
e2 = 1+i
e3 = 3-2*i
print("Original ei: %s with parent %s" % ([e1,e2,e3],parent(e1)))
x = polygen(K)
pol = (x-e1)*(x-e2)*(x-e3)

# Find the roots again in QQbar:

pol0 = PolynomialRing(QQbar,'x')([emb0(c) for c in list(pol)])
e1, e2, e3 = pol0.roots(QQbar,multiplicities=False)
print("Roots ei: %s with parent %s" % ([e1,e2,e3],parent(e1)))

# Attempt to compute sqrt(e1-e2) from these:

d = e1-e2
print("d=%s with parent %s" % (d,d.parent()))
# If the next 2 lines are commented out, an error is raised in the sqrt!
s = d.imag().is_zero()
print("d.imag().is_zero()=%s" % s)
print("d=%s with parent %s" % (d,d.parent()))
d = d.sqrt()
print("d=%s" % d)
}}}

URL: http://trac.sagemath.org/20064
Reported by: cremona
Ticket author(s): Nils Bruin
Reviewer(s): John Cremona

src/sage/rings/real_mpfi.pyx

@@ -3191,6 +3191,45 @@ cdef class RealIntervalFieldElement(RingElement):
         else:
             raise ValueError("interval does not have a unique sign")
 
+    def argument(self):
+        r"""
+        The argument of this interval, if it is well-defined, in the
+        complex sense. Otherwise raises a ``ValueError``.
+        
+        OUTPUT:
+        
+        - an element of the parent of this interval (0 or pi)
+        
+        EXAMPLES::
+        
+            sage: RIF(1).argument()
+            0
+            sage: RIF(-1).argument()
+            3.141592653589794?
+            sage: RIF(0,1).argument()
+            0
+            sage: RIF(-1,0).argument()
+            3.141592653589794?
+            sage: RIF(0).argument()
+            Traceback (most recent call last):
+            ...
+            ValueError: Can't take the argument of an exact zero
+            sage: RIF(-1,1).argument()
+            Traceback (most recent call last):
+            ...
+            ValueError: Can't take the argument of interval strictly containing zero
+
+        """
+        k=self.parent()
+        if mpfi_is_zero(self.value):
+            raise ValueError("Can't take the argument of an exact zero")
+        if mpfi_is_nonneg(self.value):
+            return k.zero()
+        elif mpfi_is_nonpos(self.value):
+            return k.pi()
+        else:
+            raise ValueError("Can't take the argument of interval strictly containing zero")
+
     def unique_floor(self):
         """
         Returns the unique floor of this interval, if it is well defined,