diff --git a/Alpha_shapes_2/include/CGAL/internal/Lazy_alpha_nt_2.h b/Alpha_shapes_2/include/CGAL/internal/Lazy_alpha_nt_2.h
index fa811c49222..07504bb2a2e 100644
--- a/Alpha_shapes_2/include/CGAL/internal/Lazy_alpha_nt_2.h
+++ b/Alpha_shapes_2/include/CGAL/internal/Lazy_alpha_nt_2.h
@@ -75,7 +75,7 @@ class Lazy_alpha_nt_2{
 //NT & kernels
   typedef CGAL::Interval_nt<mode>                                                               NT_approx;
   //Gmpq or Quotient<MP_float>
-  typedef Exact_type_selector<double>::Type                                                     NT_exact; 
+  typedef Exact_field_selector<double>::Type                                                    NT_exact;
   typedef CGAL::Simple_cartesian<NT_approx>                                                     Kernel_approx;
   typedef CGAL::Simple_cartesian<NT_exact>                                                      Kernel_exact;
   typedef typename Kernel_traits<typename Input_traits::Point_2>::Kernel                        Kernel_input;
diff --git a/Alpha_shapes_3/include/CGAL/internal/Lazy_alpha_nt_3.h b/Alpha_shapes_3/include/CGAL/internal/Lazy_alpha_nt_3.h
index 634342de55c..7ded7ec6ef0 100644
--- a/Alpha_shapes_3/include/CGAL/internal/Lazy_alpha_nt_3.h
+++ b/Alpha_shapes_3/include/CGAL/internal/Lazy_alpha_nt_3.h
@@ -74,7 +74,7 @@ class Lazy_alpha_nt_3{
 //NT & kernels
   typedef CGAL::Interval_nt<mode>                                                               NT_approx;
   //Gmpq or Quotient<MP_float>
-  typedef Exact_type_selector<double>::Type                                                     NT_exact; 
+  typedef Exact_field_selector<double>::Type                                                    NT_exact;
   typedef CGAL::Simple_cartesian<NT_approx>                                                     Kernel_approx;
   typedef CGAL::Simple_cartesian<NT_exact>                                                      Kernel_exact;
 //Helper class for weighted and non-weighted case  
diff --git a/Convex_hull_3/include/CGAL/convex_hull_3.h b/Convex_hull_3/include/CGAL/convex_hull_3.h
index bd26b6d9b1b..3bd79501c0e 100644
--- a/Convex_hull_3/include/CGAL/convex_hull_3.h
+++ b/Convex_hull_3/include/CGAL/convex_hull_3.h
@@ -123,7 +123,7 @@ public:
 //and in case of failure, exact arithmetic is used.
 template <class Kernel>
 class Is_on_positive_side_of_plane_3<Convex_hull_traits_3<Kernel>,Tag_true>{
-  typedef Simple_cartesian<CGAL::internal::Exact_type_selector<double>::Type>         PK;
+  typedef Simple_cartesian<CGAL::internal::Exact_field_selector<double>::Type>         PK;
   typedef Simple_cartesian<Interval_nt_advanced >                               CK;  
   typedef Convex_hull_traits_3<Kernel>                                          Traits;
   typedef typename Traits::Point_3                                              Point_3;
diff --git a/Filtered_kernel/include/CGAL/Filtered_kernel.h b/Filtered_kernel/include/CGAL/Filtered_kernel.h
index 992ba9db30e..468c1411dcb 100644
--- a/Filtered_kernel/include/CGAL/Filtered_kernel.h
+++ b/Filtered_kernel/include/CGAL/Filtered_kernel.h
@@ -26,6 +26,8 @@
 #include <CGAL/Filtered_predicate.h>
 #include <CGAL/Cartesian_converter.h>
 #include <CGAL/Simple_cartesian.h>
+#include <CGAL/Homogeneous_converter.h>
+#include <CGAL/Simple_homogeneous.h>
 #include <CGAL/Kernel/Type_equality_wrapper.h>
 
 #include <CGAL/MP_Float.h>
@@ -33,6 +35,7 @@
 #include <CGAL/internal/Exact_type_selector.h>
 
 #include <CGAL/internal/Static_filters/Static_filters.h>
+#include <boost/type_traits.hpp>
 
 // This file contains the definition of a generic kernel filter.
 //
@@ -46,6 +49,28 @@
 //   traits-like mechanism ?
 
 namespace CGAL {
+namespace Filter {
+template <class CK, class Rep=typename CK::Rep_tag /* Cartesian_tag */>
+struct Select_exact_kernel
+{
+  typedef typename internal::Exact_field_selector<typename CK::RT>::Type  Exact_nt;
+  typedef typename internal::Exact_ring_selector <typename CK::RT>::Type  Exact_rt;
+  typedef Simple_cartesian<Exact_nt>			Exact_kernel;
+  typedef Simple_cartesian<Exact_rt>			Exact_kernel_rt;
+  typedef Cartesian_converter<CK, Exact_kernel>		C2E;
+  typedef Cartesian_converter<CK, Exact_kernel_rt>	C2E_rt;
+};
+
+template <class CK>
+struct Select_exact_kernel <CK, Homogeneous_tag>
+{
+  typedef typename internal::Exact_ring_selector<typename CK::RT>::Type  Exact_nt;
+  typedef Simple_homogeneous<Exact_nt>			Exact_kernel;
+  typedef Homogeneous_converter<CK, Exact_kernel>	C2E;
+  typedef Exact_kernel					Exact_kernel_rt;
+  typedef C2E						C2E_rt;
+};
+}
 
 // CK = eventually rebound construction kernel (gets Point_2 from).
 // Exact_kernel = exact kernel called when needed by the filter.
@@ -54,11 +79,15 @@ template < typename CK >
 struct Filtered_kernel_base
   : public CK
 {
-    typedef typename internal::Exact_type_selector<typename CK::RT>::Type  Exact_nt;
-    typedef Simple_cartesian<Exact_nt>                           Exact_kernel;
-    typedef Simple_cartesian<Interval_nt_advanced>               Approximate_kernel;
-    typedef Cartesian_converter<CK, Exact_kernel>                C2E;
-    typedef Cartesian_converter<CK, Approximate_kernel>          C2F;
+  // Use Select_exact_kernel as a base class?
+    typedef typename Filter::Select_exact_kernel<CK>::Exact_nt Exact_nt;
+    typedef typename Filter::Select_exact_kernel<CK>::Exact_kernel Exact_kernel;
+    typedef typename Filter::Select_exact_kernel<CK>::Exact_kernel_rt Exact_kernel_rt;
+    typedef typename Filter::Select_exact_kernel<CK>::C2E C2E;
+    typedef typename Filter::Select_exact_kernel<CK>::C2E_rt C2E_rt;
+
+    typedef Simple_cartesian<Interval_nt_advanced>        Approximate_kernel;
+    typedef Cartesian_converter<CK, Approximate_kernel>   C2F;
 
     enum { Has_filtered_predicates = true };
     typedef Boolean_tag<Has_filtered_predicates> Has_filtered_predicates_tag;
@@ -84,6 +113,10 @@ struct Filtered_kernel_base
     typedef Filtered_predicate<typename Exact_kernel::P, typename Approximate_kernel::P, C2E, C2F> P; \
     P Pf() const { return P(); }
 
+#define CGAL_Kernel_pred_RT(P, Pf) \
+    typedef Filtered_predicate<typename Exact_kernel_rt::P, typename Approximate_kernel::P, C2E_rt, C2F> P; \
+    P Pf() const { return P(); }
+
     // We don't touch the constructions.
 #define CGAL_Kernel_cons(Y,Z)
 
diff --git a/Kernel_23/include/CGAL/Exact_predicates_exact_constructions_kernel.h b/Kernel_23/include/CGAL/Exact_predicates_exact_constructions_kernel.h
index 87208dbe19b..b29b5505076 100644
--- a/Kernel_23/include/CGAL/Exact_predicates_exact_constructions_kernel.h
+++ b/Kernel_23/include/CGAL/Exact_predicates_exact_constructions_kernel.h
@@ -37,8 +37,8 @@
 
 namespace CGAL {
 
-// Epeck_ft is either Gmpq of Quotient<MP_float>
-typedef internal::Exact_type_selector<double>::Type Epeck_ft;
+// Epeck_ft is either Gmpq or Quotient<MP_float>
+typedef internal::Exact_field_selector<double>::Type Epeck_ft;
 
 // The following are redefined kernels instead of simple typedefs in order to shorten
 // template name length (for error messages, mangling...).
diff --git a/Kernel_23/include/CGAL/Kernel/interface_macros.h b/Kernel_23/include/CGAL/Kernel/interface_macros.h
index 98936757b6f..2a4587e24bb 100644
--- a/Kernel_23/include/CGAL/Kernel/interface_macros.h
+++ b/Kernel_23/include/CGAL/Kernel/interface_macros.h
@@ -34,6 +34,12 @@
 #  define CGAL_Kernel_pred(X, Y)
 #endif
 
+// Those predicates for which Simple_cartesian is guaranteed not to use
+// any division.
+#ifndef CGAL_Kernel_pred_RT
+#  define CGAL_Kernel_pred_RT(X, Y) CGAL_Kernel_pred(X, Y)
+#endif
+
 #ifndef CGAL_Kernel_cons
 #  define CGAL_Kernel_cons(X, Y)
 #endif
@@ -534,23 +540,24 @@ CGAL_Kernel_pred(Less_y_3,
 		 less_y_3_object)
 CGAL_Kernel_pred(Less_z_3,
 		 less_z_3_object)
-CGAL_Kernel_pred(Orientation_2,
-		 orientation_2_object)
-CGAL_Kernel_pred(Orientation_3,
-		 orientation_3_object)
+CGAL_Kernel_pred_RT(Orientation_2,
+		    orientation_2_object)
+CGAL_Kernel_pred_RT(Orientation_3,
+		    orientation_3_object)
 CGAL_Kernel_pred(Oriented_side_2,
 		 oriented_side_2_object)
 CGAL_Kernel_pred(Oriented_side_3,
 		 oriented_side_3_object)
-CGAL_Kernel_pred(Side_of_bounded_circle_2,
-		 side_of_bounded_circle_2_object)
-CGAL_Kernel_pred(Side_of_bounded_sphere_3,
-		 side_of_bounded_sphere_3_object)
-CGAL_Kernel_pred(Side_of_oriented_circle_2,
-		 side_of_oriented_circle_2_object)
-CGAL_Kernel_pred(Side_of_oriented_sphere_3,
-		 side_of_oriented_sphere_3_object)
+CGAL_Kernel_pred_RT(Side_of_bounded_circle_2,
+		    side_of_bounded_circle_2_object)
+CGAL_Kernel_pred_RT(Side_of_bounded_sphere_3,
+		    side_of_bounded_sphere_3_object)
+CGAL_Kernel_pred_RT(Side_of_oriented_circle_2,
+		    side_of_oriented_circle_2_object)
+CGAL_Kernel_pred_RT(Side_of_oriented_sphere_3,
+		    side_of_oriented_sphere_3_object)
 
+#undef CGAL_Kernel_pred_RT
 #undef CGAL_Kernel_pred
 #undef CGAL_Kernel_cons
 #undef CGAL_Kernel_obj
diff --git a/Number_types/include/CGAL/GMP/Gmpzf_type.h b/Number_types/include/CGAL/GMP/Gmpzf_type.h
index e2e6e0300d9..4fdadcc227f 100644
--- a/Number_types/include/CGAL/GMP/Gmpzf_type.h
+++ b/Number_types/include/CGAL/GMP/Gmpzf_type.h
@@ -163,7 +163,7 @@ public:
       e = 0;
       return;
     }
-    static int p = std::numeric_limits<double>::digits;
+    const int p = std::numeric_limits<double>::digits;
     CGAL_assertion(CGAL_NTS is_finite(d) & is_valid(d));
     int exp;
     double x = std::frexp(d, &exp); // x in [1/2, 1], x*2^exp = d
@@ -274,7 +274,8 @@ Gmpzf& Gmpzf::operator*=( const Gmpzf& b)
   mpz_mul(result.man(), man(), b.man());
   e += b.exp();
   swap (result);
-  canonicalize();
+  if(is_zero()) e=0; // product is 0 or odd, so we can skip canonicalize()
+  CGAL_postcondition(is_canonical());
   return *this;
 }
 
@@ -456,8 +457,10 @@ void Gmpzf::canonicalize()
   if (!is_zero()) {
     // chop off trailing zeros in m
     unsigned long zeros = mpz_scan1(man(), 0);
-    mpz_tdiv_q_2exp( man(), man(), zeros);  // bit-wise right-shift
-    e += zeros;
+    if (zeros != 0) {
+      mpz_tdiv_q_2exp( man(), man(), zeros);  // bit-wise right-shift
+      e += zeros;
+    }
   } else {
     e = 0;
   }
diff --git a/Number_types/include/CGAL/Lazy_exact_nt.h b/Number_types/include/CGAL/Lazy_exact_nt.h
index e012d36f190..e8a6b6cd45e 100644
--- a/Number_types/include/CGAL/Lazy_exact_nt.h
+++ b/Number_types/include/CGAL/Lazy_exact_nt.h
@@ -46,6 +46,7 @@
 #include <CGAL/Lazy.h>
 
 #include <CGAL/Sqrt_extension_fwd.h>
+#include <CGAL/Kernel/mpl.h>
 
 /*
  * This file contains the definition of the number type Lazy_exact_nt<ET>,
@@ -1323,6 +1324,35 @@ struct NT_converter < Lazy_exact_nt<ET>, ET >
   { return a.exact(); }
 };
 
+// Forward declaration to break inclusion cycle
+namespace internal {
+template<class>struct Exact_field_selector;
+template<class>struct Exact_ring_selector;
+}
+// Compiler can deduce ET from the first argument.
+template < typename ET >
+struct NT_converter < Lazy_exact_nt<ET>,
+  typename First_if_different<
+    typename internal::Exact_field_selector<ET>::Type,
+    ET>::Type>
+{
+  typename internal::Exact_field_selector<ET>::Type
+    operator()(const Lazy_exact_nt<ET> &a) const
+  { return NT_converter<ET,typename internal::Exact_field_selector<ET>::Type>()(a.exact()); }
+};
+
+template < typename ET >
+struct NT_converter < Lazy_exact_nt<ET>,
+  typename First_if_different<
+   typename First_if_different<
+    typename internal::Exact_ring_selector<ET>::Type,
+    ET>::Type,
+   typename internal::Exact_field_selector<ET>::Type>::Type>
+{
+  typename internal::Exact_ring_selector<ET>::Type operator()(const Lazy_exact_nt<ET> &a) const
+  { return NT_converter<ET,typename internal::Exact_ring_selector<ET>::Type>()(a.exact()); }
+};
+
 namespace internal {
 // Returns true if the value is representable by a double and to_double()
 // would return it.  False means "don't know" (the exact number type is not
diff --git a/Number_types/include/CGAL/internal/Exact_type_selector.h b/Number_types/include/CGAL/internal/Exact_type_selector.h
index d5c57e350ff..3f8241ad139 100644
--- a/Number_types/include/CGAL/internal/Exact_type_selector.h
+++ b/Number_types/include/CGAL/internal/Exact_type_selector.h
@@ -35,6 +35,7 @@
 #ifdef CGAL_USE_GMP
 #  include <CGAL/Gmpz.h>
 #  include <CGAL/Gmpq.h>
+#  include <CGAL/Gmpzf.h>
 #endif
 #ifdef CGAL_USE_GMPXX
 #  include <CGAL/gmpxx.h>
@@ -53,76 +54,116 @@ class Expr;
 
 namespace CGAL { namespace internal {
 
-// A class which tells the prefered "exact number type" corresponding to a type.
+// Two classes which tell the prefered "exact number types" corresponding to a type.
 
 // The default template chooses Gmpq or Quotient<MP_Float>.
 // It should support the built-in types.
 template < typename >
-struct Exact_type_selector
+struct Exact_field_selector
 #ifdef CGAL_USE_GMP
 { typedef Gmpq Type; };
 #else
 { typedef Quotient<MP_Float> Type; };
 #endif
 
+// By default, a field is a safe choice of ring.
+template < typename T >
+struct Exact_ring_selector : Exact_field_selector < T > { };
+
 template <>
-struct Exact_type_selector<MP_Float>
+struct Exact_ring_selector<double>
+#if defined(CGAL_HAS_THREADS) || !defined(CGAL_USE_GMP)
+{ typedef MP_Float Type; };
+#else
+{ typedef Gmpzf Type; };
+#endif
+
+template <>
+struct Exact_ring_selector<float> : Exact_ring_selector<double> { };
+
+template <>
+struct Exact_field_selector<MP_Float>
 { typedef Quotient<MP_Float> Type; };
 
 template <>
-struct Exact_type_selector<Quotient<MP_Float> >
+struct Exact_ring_selector<MP_Float>
+{ typedef MP_Float Type; };
+
+template <>
+struct Exact_field_selector<Quotient<MP_Float> >
 { typedef Quotient<MP_Float> Type; };
 
 // And we specialize for the following types :
 #ifdef CGAL_USE_GMP
 template <>
-struct Exact_type_selector<Gmpz>
+struct Exact_field_selector<Gmpz>
 { typedef Gmpq  Type; };
 
 template <>
-struct Exact_type_selector<Gmpq>
+struct Exact_ring_selector<Gmpz>
+{ typedef Gmpz  Type; };
+
+template <>
+struct Exact_ring_selector<Gmpzf>
+{ typedef Gmpzf Type; };
+
+template <>
+struct Exact_field_selector<Gmpq>
 { typedef Gmpq  Type; };
 #endif
 
 #ifdef CGAL_USE_GMPXX
 template <>
-struct Exact_type_selector< ::mpz_class>
+struct Exact_field_selector< ::mpz_class>
 { typedef ::mpq_class  Type; };
 
 template <>
-struct Exact_type_selector< ::mpq_class>
+struct Exact_ring_selector< ::mpz_class>
+{ typedef ::mpz_class  Type; };
+
+template <>
+struct Exact_field_selector< ::mpq_class>
 { typedef ::mpq_class  Type; };
 #endif
 
 #ifdef CGAL_USE_LEDA
 template <>
-struct Exact_type_selector<leda_integer>
+struct Exact_field_selector<leda_integer>
 { typedef leda_rational  Type; };
 
 template <>
-struct Exact_type_selector<leda_rational>
+struct Exact_ring_selector<leda_integer>
+{ typedef leda_integer   Type; };
+
+template <>
+struct Exact_field_selector<leda_rational>
 { typedef leda_rational  Type; };
 
 template <>
-struct Exact_type_selector<leda_real>
+struct Exact_field_selector<leda_real>
 { typedef leda_real  Type; };
 #endif
 
 #ifdef CGAL_USE_CORE
 template <>
-struct Exact_type_selector<CORE::Expr>
+struct Exact_field_selector<CORE::Expr>
 { typedef CORE::Expr  Type; };
 #endif
 
 template < typename ET >
-struct Exact_type_selector<Lazy_exact_nt<ET> >
+struct Exact_field_selector<Lazy_exact_nt<ET> >
+: Exact_field_selector<ET>
 {
   // We have a choice here :
   // - using ET gets rid of the DAG computation as well as redoing the interval
   // - using Lazy_exact_nt<ET> might use sharper intervals.
-  typedef ET  Type;
+  // typedef ET  Type;
   // typedef Lazy_exact_nt<ET>  Type;
 };
+template < typename ET >
+struct Exact_ring_selector<Lazy_exact_nt<ET> >
+: Exact_ring_selector<ET>
+{};
 
 } } // namespace CGAL::internal
 
diff --git a/Number_types/test/Number_types/Sqrt_extension.cpp b/Number_types/test/Number_types/Sqrt_extension.cpp
index 52696fc389c..c67f9235b75 100644
--- a/Number_types/test/Number_types/Sqrt_extension.cpp
+++ b/Number_types/test/Number_types/Sqrt_extension.cpp
@@ -732,7 +732,7 @@ void sqrt_extension_test(){
 #include <CGAL/internal/Exact_type_selector.h>
 void test_nt_converter()
 {
-  typedef CGAL::internal::Exact_type_selector<int>::Type NT;
+  typedef CGAL::internal::Exact_field_selector<int>::Type NT;
   typedef CGAL::Sqrt_extension<double,double> Source;
   typedef CGAL::Sqrt_extension<NT,NT> Target;
   
diff --git a/Periodic_3_triangulation_3/include/CGAL/Periodic_3_triangulation_3.h b/Periodic_3_triangulation_3/include/CGAL/Periodic_3_triangulation_3.h
index 16d3b57859c..331ee2b3a8b 100644
--- a/Periodic_3_triangulation_3/include/CGAL/Periodic_3_triangulation_3.h
+++ b/Periodic_3_triangulation_3/include/CGAL/Periodic_3_triangulation_3.h
@@ -216,7 +216,7 @@ public:
       const Geometric_traits & gt = Geometric_traits())
     : _gt(gt), _tds(), _domain(domain), too_long_edge_counter(0) {
     _gt.set_domain(_domain);
-    typedef typename internal::Exact_type_selector<FT>::Type EFT;
+    typedef typename internal::Exact_field_selector<FT>::Type EFT;
     typedef NT_converter<FT,EFT> NTC;
     CGAL_USE_TYPE(NTC);
     CGAL_triangulation_assertion_code( NTC ntc; )