diff --git a/Alpha_shapes_3/doc/Alpha_shapes_3/CGAL/Alpha_shape_3.h b/Alpha_shapes_3/doc/Alpha_shapes_3/CGAL/Alpha_shape_3.h
index e1e14111ed0..9046a58a2eb 100644
--- a/Alpha_shapes_3/doc/Alpha_shapes_3/CGAL/Alpha_shape_3.h
+++ b/Alpha_shapes_3/doc/Alpha_shapes_3/CGAL/Alpha_shape_3.h
@@ -104,8 +104,11 @@ resorting to exact arithmetic). Access to the interval containing the exact valu
 `FT::Approximate_nt approx() const` where `FT::Approximate_nt` is `Interval_nt<Protected>` 
 with `Protected=true`. Access to the exact value is provided through the function 
 `FT::Exact_nt exact() const` where `FT::Exact_nt` depends on the configuration of %CGAL 
-(it is `Gmpq` if `gmp` is available and `Quotient<CGAL::MP_Float>` otherwise).
-An overload for the function `double to_double(FT)` is also available.
+(it may be `mpq_class`, `Gmpq`, `Quotient<CGAL::MP_Float>`, etc).
+An overload for the function `double to_double(FT)` is also available. Its
+precision is controlled through `FT::set_relative_precision_of_to_double()` in
+exactly the same way as with `Lazy_exact_nt<NT>`, so a call to `to_double` may
+trigger an exact evaluation.
 It must be noted that an object of type `FT` is valid as long as the alpha shapes class that creates 
 it is valid and has not been modified. 
 For convenience, classical comparison operators are provided for the type `FT`. 
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 97005825a79..3a0ffedc88e 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
@@ -177,8 +177,25 @@ class Lazy_alpha_nt_3{
   const Data_vector& data() const{ return input_points;}
   Data_vector& data(){ return input_points;}
 
+  static double & relative_precision_of_to_double_internal()
+  {
+    CGAL_STATIC_THREAD_LOCAL_VARIABLE(double, relative_precision_of_to_double, 0.00001);
+      return relative_precision_of_to_double;
+  }
+
 public:
 
+  static const double & get_relative_precision_of_to_double()
+  {
+    return relative_precision_of_to_double_internal();
+  }
+
+  static void set_relative_precision_of_to_double(double d)
+  {
+      CGAL_assertion((0 < d) & (d < 1));
+      relative_precision_of_to_double_internal() = d;
+  }
+
   typedef NT_exact               Exact_nt;
   typedef NT_approx              Approximate_nt;
 
@@ -419,7 +436,17 @@ struct Alpha_nt_selector_3
 template<class Input_traits, bool mode, class Weighted_tag>
 double to_double(const internal::Lazy_alpha_nt_3<Input_traits, mode, Weighted_tag>& a)
 {
-  return to_double(a.approx());
+  double r;
+  if (fit_in_double(a.approx(), r))
+    return r;
+
+  // If it isn't precise enough,
+  // we trigger the exact computation first,
+  // which will refine the approximation.
+  if (!has_smaller_relative_precision(a.approx(), a.get_relative_precision_of_to_double()))
+    a.exact();
+
+  return CGAL_NTS to_double(a.approx());
 }
 
 } //namespace CGAL
diff --git a/Number_types/include/CGAL/Lazy_exact_nt.h b/Number_types/include/CGAL/Lazy_exact_nt.h
index aa741a97207..d1ffc105ccf 100644
--- a/Number_types/include/CGAL/Lazy_exact_nt.h
+++ b/Number_types/include/CGAL/Lazy_exact_nt.h
@@ -478,7 +478,7 @@ public:
     return relative_precision_of_to_double_internal();
   }
 
-  static void set_relative_precision_of_to_double(const double & d)
+  static void set_relative_precision_of_to_double(double d)
   {
       CGAL_assertion((0 < d) & (d < 1));
       relative_precision_of_to_double_internal() = d;