diff --git a/Number_types/include/CGAL/Number_types/internal/Exact_type_selector.h b/Number_types/include/CGAL/Number_types/internal/Exact_type_selector.h
index cdc3aea33df..4fc3a4e95a1 100644
--- a/Number_types/include/CGAL/Number_types/internal/Exact_type_selector.h
+++ b/Number_types/include/CGAL/Number_types/internal/Exact_type_selector.h
@@ -25,6 +25,7 @@
 #include <CGAL/Lazy_exact_nt.h>
 
 #include <CGAL/boost_mp.h>
+#  include <CGAL/cpp_float.h>
 #ifdef CGAL_USE_GMP
 #  include <CGAL/Gmpz.h>
 #  include <CGAL/Gmpq.h>
diff --git a/Number_types/include/CGAL/cpp_float.h b/Number_types/include/CGAL/cpp_float.h
new file mode 100644
index 00000000000..b3885c0e922
--- /dev/null
+++ b/Number_types/include/CGAL/cpp_float.h
@@ -0,0 +1,399 @@
+// Copyright (c) 2023 GeometryFactory (France).
+// All rights reserved.
+//
+// This file is part of CGAL (www.cgal.org)
+//
+// $URL$
+// $Id$
+// SPDX-License-Identifier: LGPL-3.0-or-later OR LicenseRef-Commercial
+//
+// Author(s)        :  Andreas Fabri
+
+#ifndef CGAL_CPP_FLOAT_H
+#define CGAL_CPP_FLOAT_H
+
+
+#include <CGAL/boost_mp_type.h>
+#include <CGAL/assertions.h>
+#include <boost/multiprecision/cpp_int.hpp>
+#include <iostream>
+
+namespace CGAL {
+
+
+namespace internal {
+  // Only used with an argument known not to be 0.
+  inline int low_bit (boost::uint64_t x) {
+#if defined(_MSC_VER)
+    unsigned long ret;
+    _BitScanForward64(&ret, x);
+    return (int)ret;
+#elif defined(__xlC__)
+    return __cnttz8 (x);
+#else
+    // Assume long long is 64 bits
+    return __builtin_ctzll (x);
+#endif
+  }
+  inline int high_bit (boost::uint64_t x) {
+#if defined(_MSC_VER)
+    unsigned long ret;
+    _BitScanReverse64(&ret, x);
+    return (int)ret; // AF:  was 63 - (int)ret;  The others have to be changed too
+#elif defined(__xlC__)
+    // Macro supposedly not defined on z/OS.
+    return __cntlz8 (x);
+#else
+    return __builtin_clzll (x);
+#endif
+  }
+
+} // namespace internal
+
+#if 0 // needs C++20
+  template <class T>
+  void fmt(const T& t)
+  {
+    std::cout << std::format("{:b}", t) << std::endl;
+  }
+#endif
+
+class cpp_float {
+
+  boost::multiprecision::cpp_int man;
+  int exp; /* The number man (an integer) * 2 ^ exp  */
+
+public:
+  cpp_float()
+    : man(), exp()
+  {}
+
+  cpp_float(int i)
+    : man(i),exp(0)
+  {}
+
+  cpp_float(const boost::multiprecision::cpp_int& man,  int exp)
+    : man(man),exp(exp)
+  {}
+
+
+  cpp_float(double d)
+  {
+      std::cout << "\ndouble = " << d << std::endl;
+    using boost::uint64_t;
+    union {
+#ifdef CGAL_LITTLE_ENDIAN
+      struct { uint64_t man:52; uint64_t exp:11; uint64_t sig:1; } s;
+#else /* CGAL_BIG_ENDIAN */
+      //WARNING: untested!
+      struct { uint64_t sig:1; uint64_t exp:11; uint64_t man:52; } s;
+#endif
+      double d;
+    } u;
+    u.d = d;
+
+    uint64_t m;
+    uint64_t dexp = u.s.exp;
+    CGAL_assertion_msg(dexp != 2047, "Creating an cpp_float from infinity or NaN.");
+    if (dexp == 0) {
+      if (d == 0) { exp=0; return; }
+      else { // denormal number
+        m = u.s.man;
+        ++dexp;
+      }
+    } else {
+      m = (1LL << 52) |  u.s.man;
+    }
+
+
+    int idexp = (int)dexp;
+    idexp -= 1023;
+
+    std::cout << "m    = "  << m << std::endl;
+    std::cout << "idexp = " << idexp << std::endl;
+
+    int shifted = internal::low_bit(m);
+
+    m >>= shifted;
+
+    int nbits = internal::high_bit(m);
+    std::cout << "nbits = " << nbits << std::endl;
+
+    exp = idexp - nbits;
+    if(u.s.sig){
+      m = -m;
+    }
+    std::cout << "m = " << m << " * 2^" << exp  << std::endl;
+    // fmt(m);
+    man = boost::multiprecision::cpp_int(m);
+  }
+
+  friend std::ostream& operator<<(std::ostream& os, const cpp_float& m)
+  {
+    return os << m.man << " * 2 ^ " << m.exp << " ( " << to_double(m) << ") ";
+  }
+
+  friend cpp_float operator-(cpp_float const&x)
+  {
+    return cpp_float(-x.man,x.exp);
+  }
+
+  cpp_float& operator*=(const cpp_float& other)
+  {
+    man *= other.man;
+    exp += other.exp;
+    return *this;
+  }
+
+  cpp_float operator+=(const cpp_float& other)
+  {
+    int shift = exp - other.exp;
+    if(shift > 0){
+      man <<= shift;
+      man += other.man;
+      exp = other.exp;
+    }else if(shift < 0){
+      boost::multiprecision::cpp_int cpy(other.man);
+      cpy << shift;
+      man += cpy;
+    }else{
+      man += other.man;
+    }
+    return *this;
+  }
+
+  cpp_float operator-=(const cpp_float& other)
+  {
+    int shift = exp - other.exp;
+    if(shift > 0){
+      man <<= shift;
+      man -= other.man;
+      exp = other.exp;
+    }else if(shift < 0){
+      boost::multiprecision::cpp_int cpy(other.man);
+      cpy << shift;
+      man -= cpy;
+    }else{
+      man -= other.man;
+    }
+    return *this;
+  }
+
+  bool positive() const
+  {
+    return is_positive(man);
+  }
+
+
+  friend bool operator<(const cpp_float& a, const cpp_float& b)
+  {
+    cpp_float d(b);
+    d -= a;
+    return d.positive();
+  }
+
+  friend bool operator>(cpp_float const&a, cpp_float const&b){
+    return b<a;
+  }
+  friend bool operator>=(cpp_float const&a, cpp_float const&b){
+    return !(a<b);
+  }
+  friend bool operator<=(cpp_float const&a, cpp_float const&b){
+    return !(a>b);
+  }
+
+
+  friend bool operator==(cpp_float const&a, cpp_float const&b){
+   int shift = a.exp - b.exp;
+    if(shift > 0){
+      boost::multiprecision::cpp_int ac(a.man);
+      ac <<= shift;
+      return ac == b.man;
+    }else if(shift < 0){
+      boost::multiprecision::cpp_int bc(b.man);
+      bc <<= shift;
+      return a.man == bc;
+    }
+    return a.man==b.man;
+  }
+
+  Comparison_result compare(const cpp_float& other) const
+  {
+    if(*this < other) return SMALLER;
+    if(*this > other) return LARGER;
+    return EQUAL;
+  }
+
+  friend bool operator!=(cpp_float const&a, cpp_float const&b){
+    return !(a==b);
+  }
+
+  friend double to_double(const cpp_float const&a)
+  {
+    if(a.exp == 0){
+      return to_double(a.man);
+    }
+    if(a.exp > 0){
+      boost::multiprecision::cpp_int as(a.man);
+      as <<= a.exp;
+      return to_double(as);
+    }
+    boost::multiprecision::cpp_int pow(1);
+    pow <<= -a.exp;
+    boost::multiprecision::cpp_rational rat(a.man, pow);
+    return to_double(rat);
+  }
+
+  friend std::pair<double,double> to_interval()
+  {
+    assert(false);
+    double zero(0);
+    return std::make_pair(zero,zero);
+  }
+
+
+  bool is_zero () const {
+    assert(false);
+    return man==0 && exp == 0;
+  }
+
+
+  bool is_one () const {
+    assert(false);
+    return true;
+    //  return exp==0 && size==1 && data()[0]==1;
+  }
+
+
+  CGAL::Sign sign () const
+  {
+    return CGAL::sign(man);
+  }
+
+};
+
+  cpp_float operator+(const cpp_float& a, const cpp_float&b){
+    cpp_float ret(a);
+    return ret += b;
+  }
+
+  cpp_float operator-(const cpp_float& a, const cpp_float&b){
+    cpp_float ret(a);
+    return ret -= b;
+  }
+
+  cpp_float operator*(const cpp_float& a, const cpp_float&b){
+    cpp_float ret(a);
+    return ret *= b;
+  }
+
+
+  template <> struct Algebraic_structure_traits< cpp_float >
+    : public Algebraic_structure_traits_base< cpp_float, Integral_domain_without_division_tag >  {
+      typedef Tag_true            Is_exact;
+      typedef Tag_false            Is_numerical_sensitive;
+
+      struct Is_zero
+        : public CGAL::cpp98::unary_function< Type, bool > {
+          bool operator()( const Type& x ) const {
+            return x.is_zero();
+          }
+        };
+
+      struct Is_one
+        : public CGAL::cpp98::unary_function< Type, bool > {
+          bool operator()( const Type& x ) const {
+            assert(false);
+            return false; // x.is_one();
+          }
+        };
+
+      struct Gcd
+        : public CGAL::cpp98::binary_function< Type, Type, Type > {
+          Type operator()(
+              const Type& x,
+              const Type& y ) const {
+            assert(false);
+            return Type(); // cpp_float_gcd(x, y);
+          }
+        };
+
+      struct Square
+        : public CGAL::cpp98::unary_function< Type, Type > {
+          Type operator()( const Type& x ) const {
+            return x*x ; // cpp_float_square(x);
+          }
+        };
+
+      struct Integral_division
+        : public CGAL::cpp98::binary_function< Type, Type, Type > {
+          Type operator()(
+              const Type& x,
+              const Type& y ) const {
+            assert(false);
+            return Type(); // x / y;
+          }
+        };
+
+      struct Sqrt
+        : public CGAL::cpp98::unary_function< Type, Type > {
+          Type operator()( const Type& x) const {
+            assert(false);
+            return Type(); // cpp_float_sqrt(x);
+          }
+        };
+
+      struct Is_square
+        : public CGAL::cpp98::binary_function< Type, Type&, bool > {
+          bool operator()( const Type& x, Type& y ) const {
+            // TODO: avoid doing 2 calls.
+            assert(false);
+            return true;
+          }
+          bool operator()( const Type& x) const {
+            assert(false);
+            return true;
+          }
+        };
+
+    };
+  template <> struct Real_embeddable_traits< cpp_float >
+    : public INTERN_RET::Real_embeddable_traits_base< cpp_float , CGAL::Tag_true > {
+      struct Sgn
+        : public CGAL::cpp98::unary_function< Type, ::CGAL::Sign > {
+          ::CGAL::Sign operator()( const Type& x ) const {
+            return x.sign();
+          }
+        };
+
+      struct To_double
+        : public CGAL::cpp98::unary_function< Type, double > {
+            double operator()( const Type& x ) const {
+              return x.to_double();
+            }
+        };
+
+      struct Compare
+        : public CGAL::cpp98::binary_function< Type, Type, Comparison_result > {
+            Comparison_result operator()(
+                const Type& x,
+                const Type& y ) const {
+              return x.compare(y);
+            }
+        };
+
+      struct To_interval
+        : public CGAL::cpp98::unary_function< Type, std::pair< double, double > > {
+            std::pair<double, double> operator()( const Type& x ) const {
+              return x.to_interval();
+            }
+        };
+
+    };
+
+
+
+} // namespace CGAL
+
+
+#endif // CGAL_CPP_FLOAT_H