Trac #24371: Move real/complex interval fields to new coercion model

Also fix the following:
{{{
sage: NF.<a> = QuadraticField(-2)
sage: CIF(a)
...
ValueError: can not convert complex algebraic number to real interval
}}}

We also deal with Python 3 compatibility regarding `int`/`long` and
`bytes`/`unicode`.

URL: https://trac.sagemath.org/24371
Reported by: jdemeyer
Ticket author(s): Jeroen Demeyer
Reviewer(s): Travis Scrimshaw

src/doc/en/reference/rings_numerical/index.rst

@@ -36,7 +36,6 @@ ComplexBallField).
    :maxdepth: 2
 
    sage/rings/real_mpfi
-   sage/rings/real_interval_field
    sage/rings/real_interval_absolute
    sage/rings/complex_interval_field
    sage/rings/complex_interval

src/module_list.py

@@ -1201,6 +1201,8 @@ def uname_specific(name, value, alternative):
     Extension('sage.rings.ring',
               sources = ['sage/rings/ring.pyx']),
 
+    Extension('*', ['sage/rings/convert/*.pyx']),
+
     ################################
     ##
     ## sage.rings.finite_rings

src/sage/arith/constants.pxd

@@ -21,3 +21,5 @@ cdef extern from *:
     double M_1_LN2 "0x1.71547652b82fep+0"    # 1/log(2)
     double M_1_LN10 "0x6.f2dec549b9438p-4"   # 1/log(10)
     double M_1_LNPI "0xd.fa22fdd8cfd98p-4"   # 1/log(π)
+
+    double LOG_TEN_TWO_PLUS_EPSILON "0x3.5269e12f346e4p+0"  # log(10,2) rounded up

src/sage/categories/rings.py

@@ -946,9 +946,6 @@ def __getitem__(self, arg):
                   From: Number Field in i with defining polynomial x^2 + 1
                   To:   Complex Lazy Field
                   Defn: i -> 1*I
-                sage: QQi.<i> = QuadraticField(-1, embedding=None)
-                sage: QQ[i].coerce_embedding() is None
-                True
 
             TESTS:
 

src/sage/rings/complex_interval.pxd

@@ -2,23 +2,26 @@ from sage.libs.mpfr.types cimport mpfr_prec_t
 from sage.libs.mpfi.types cimport mpfi_t
 
 cimport sage.structure.element
-from .real_mpfi cimport RealIntervalFieldElement
+from .real_mpfi cimport RealIntervalFieldElement, RealIntervalField_class
 
 
 cdef class ComplexIntervalFieldElement(sage.structure.element.FieldElement):
     cdef mpfi_t __re
     cdef mpfi_t __im
     cdef mpfr_prec_t _prec
 
-    cpdef _add_(self, other)
-    cpdef _mul_(self, other)
-    cdef RealIntervalFieldElement abs_c(ComplexIntervalFieldElement self)
-    cdef RealIntervalFieldElement norm_c(ComplexIntervalFieldElement self)
-
     cdef inline ComplexIntervalFieldElement _new(self):
         """
         Quickly create a new complex interval with the same parent as
         ``self``.
         """
         cdef type t = type(self)
         return t.__new__(t, self._parent)
+
+    cdef inline RealIntervalFieldElement _new_real(self):
+        """
+        Quickly create a new real interval with the same precision as
+        ``self``.
+        """
+        P = <RealIntervalField_class>(self._parent.real_field())
+        return P._new()