Add and document functions that make MP_Float a EuclideanRingNumberType.

Number_types/doc_tex/NumberTypeSupport/Numbertype.tex

@@ -102,17 +102,24 @@ With the built-in integer types overflow might occur.
 \cgal\ provides several number types that are that can be used for 
 exact computation.  These include the \ccc{Quotient} class that can
 be used to create, for example, a number type that behaves like a rational
-number.  When used in conjunction with the number type \ccc{MP_Float} that
-is able to represent multi-precision floating point values, you achieve
-an exact rational number representation.
+number when parameterized with a number type which can represent integers.
+
+The number type \ccc{MP_Float} is able to represent multi-precision floating
+point values, a generalization of integers scaled by a (potentially negative)
+power of 2.  It allows to deal with ring operations over floating-point values
+with requiring rational numbers.  By plugging it in \ccc{Quotient}, one obtains
+rational numbers.  Note that \ccc{MP_Float} may not be as efficient as the
+integer types provided by \ccc{GMP} or \ccc{LEDA}, but it has the advantage
+to make more parts of \cgal\ independent on these external libraries for
+handling robustness issues.
 
 The templated number type \ccc{Lazy_exact_nt<NT>} is able to represent any
 number that \ccc{NT} is able to represent, but because it first tries to use an
-approximate value to perform computations it can be faster than the provided
-number type \ccc{NT}.  \cgal\ also provides a fixed-precision number type,
+approximate value to perform computations it can be faster than \ccc{NT}.
 
 A number type for doing interval arithmetic, \ccc{Interval_nt}, is provided.
-This number type helps in doing filtering of predicates.
+This number type helps in doing arithmetic filtering in many places such
+as \ccc{Filtered_predicate}.
 
 \ccc{CGAL::Root_of_2} is a number type that allows to represent algebraic
 numbers of degree up to~2 over a \ccc{RingNumberType}.  A generic function

Number_types/doc_tex/NumberTypeSupport_ref/EuclideanRingNumberType.tex

@@ -39,6 +39,7 @@ RingNumberType
 \ccc{int} \\
 \ccc{long} \\
 \ccc{long long} \\
+\ccc{CGAL::MP_Float} \\
 \ccc{CGAL::Gmpz} \\
 \ccc{leda_integer}
 

Number_types/doc_tex/NumberTypeSupport_ref/FieldNumberType.tex

@@ -33,7 +33,6 @@ RingNumberType
 \ccc{CGAL::Interval_nt} \\
 \ccc{CGAL::Interval_nt_advanced} \\
 \ccc{CGAL::Lazy_exact_nt<FieldNumberType>} \\
-\ccc{CGAL::MP_Float} \\
 \ccc{CGAL::Quotient<RingNumberType>} \\
 \ccc{leda_rational} \\
 \ccc{leda_bigfloat} \\

Number_types/doc_tex/NumberTypeSupport_ref/MP_Float.tex

@@ -8,18 +8,17 @@
 
 \ccDefinition
 An object of the class \ccc{MP_Float} is able to represent a floating point
-value with arbitrary precision.  This number type has the property that
+value with arbitrary precision, that is numbers of the form $m2^e$ where
+$m$ and $e$ are integers.  This number type has the property that
 additions, subtractions and multiplications are computed exactly, as well as
 the construction from \ccc{float}, \ccc{double} and \ccc{long double}.
+It also provides integral divisions related operations.
 
-Division and square root are not enabled by default since \cgal\ release 3.2,
-since they are computed approximately.  We suggest that you use
-rationals like \ccc{Quotient<MP_Float>} when you need exact (rational)
-divisions.
-To enable division and square root, you have to define the preprocessor
-macro \ccc{CGAL_MP_FLOAT_ALLOW_INEXACT}.
-
-An exact (integral) division is also provided.
+Square root and (field) division are not enabled by default since
+\cgal\ release 3.2, since they are computed approximately.  We suggest that you
+use rationals like \ccc{Quotient<MP_Float>} when you need exact (rational)
+divisions.  To enable division and square root, you have to define the
+preprocessor macro \ccc{CGAL_MP_FLOAT_ALLOW_INEXACT}.
 
 Note on the implementation : although the mantissa length is basically only
 limited by the available memory, the exponent is currently represented by a
@@ -29,8 +28,7 @@ plan to also have a multiprecision exponent to fix this issue.
 \ccInclude{CGAL/MP_Float.h}
 
 \ccIsModel
-\ccc{RingNumberType}
-or \ccc{SqrtFieldNumberType} if \ccc{CGAL_MP_FLOAT_ALLOW_INEXACT} is set.
+\ccc{EuclideanRingNumberType}
 
 \ccCreation
 \ccCreationVariable{m}
@@ -62,16 +60,33 @@ or \ccc{SqrtFieldNumberType} if \ccc{CGAL_MP_FLOAT_ALLOW_INEXACT} is set.
 {reads a \ccc{double} from \ccc{in}, then converts it to an \ccc{MP_Float}.}
 
 \ccFunction{bool divides(const MP_Float &a, const MP_Float &b);}
-{returns true iff \ccc{b} divides \ccc{b}.}
+{returns true iff \ccc{b} divides \ccc{a}.}
 
 \ccFunction{MP_Float exact_division(const MP_Float &a, const MP_Float &b);}
 {computes the result of the division of \ccc{a} by \ccc{b}.
 \ccPrecond{divides(a, b)}.}
 
+\ccFunction{MP_Float div(const MP_Float &a, const MP_Float &b);}
+{returns the result of the integral division of \ccc{a} by \ccc{b}.
+For \ccc{MP_Float}, this is not uniquely defined, but this function
+guarantees that the remainder has a smaller bitlength than \ccc{b}
+(considering the bitlength of the odd mantissa).}
+
+\ccFunction{MP_Float operator%(const MP_Float &a, const MP_Float &b);}
+{returns the remainder of the integral division of \ccc{a} by \ccc{b}.
+For \ccc{MP_Float}, this is not uniquely defined, but this function
+guarantees that the remainder has a smaller bitlength than \ccc{b}
+(considering the bitlength of the odd mantissa).}
+
+\ccFunction{MP_Float gcd(const MP_Float &a, const MP_Float &b);}
+{returns the greatest common divisor of \ccc{a} and \ccc{b}.
+For \ccc{MP_Float}, this is defined up to a power of 2.}
+\ccPrecond{\ccc{a} and \ccc{b} are non zero.}}
+
 \ccFunction{MP_Float approximate_division(const MP_Float &a, const MP_Float &b);}
-{computes an approximation of the division by converting the operands to
-\ccc{double}, performing the division on \ccc{double}, and converting back to
-\ccc{MP_Float}.}
+{computes an approximation of the (field) division by converting the operands
+to \ccc{double}, performing the division on \ccc{double}, and converting back
+to \ccc{MP_Float}.}
 
 \ccFunction{MP_Float approximate_sqrt(const MP_Float &a);}
 {computes an approximation of the square root by converting the operand to

Number_types/include/CGAL/MP_Float.h

@@ -37,6 +37,7 @@
 // The main algorithms are :
 // - Addition/Subtraction
 // - Multiplication
+// - Integral division div(), exact_division(), gcd(), operator%().
 // - Comparison
 // - to_double() / to_interval()
 // - Construction from a double.
@@ -49,8 +50,6 @@
 // - Division, sqrt... : different options :
 //   - nothing
 //   - convert to double, take approximation, compute over double, reconstruct
-//   - exact division as separate function (for Bareiss...)
-//   - returns the quotient of the division, forgetting the rest.
 
 CGAL_BEGIN_NAMESPACE
 
@@ -62,6 +61,7 @@ MP_Float operator*(const MP_Float &a, const MP_Float &b);
 #ifdef CGAL_MP_FLOAT_ALLOW_INEXACT
 MP_Float operator/(const MP_Float &a, const MP_Float &b);
 #endif
+MP_Float operator%(const MP_Float &a, const MP_Float &b);
 
 Comparison_result
 compare (const MP_Float & a, const MP_Float & b);
@@ -191,6 +191,7 @@ public:
 #ifdef CGAL_MP_FLOAT_ALLOW_INEXACT
   MP_Float& operator/=(const MP_Float &a) { return *this = *this / a; }
 #endif
+  MP_Float& operator%=(const MP_Float &a) { return *this = *this % a; }
 
   exponent_type max_exp() const
   {
@@ -597,6 +598,37 @@ divides(const MP_Float & n, const MP_Float & d)
   return CGALi::division(n, d).second == 0;
 }
 
+inline
+MP_Float
+operator%(const MP_Float& n1, const MP_Float& n2)
+{
+  return CGALi::division(n1, n2).second;
+}
+
+inline
+MP_Float
+div(const MP_Float& n1, const MP_Float& n2)
+{
+  return CGALi::division(n1, n2).first;
+}
+
+inline
+MP_Float
+gcd( const MP_Float& a, const MP_Float& b)
+{
+  CGAL_precondition( a != 0 );
+  CGAL_precondition( b != 0 );
+  MP_Float x = a, y = b;
+  while (true) {
+    x = x % y;
+    if (x == 0) {
+      CGAL_postcondition(divides(a, y) && divides(b, y));
+      return y;
+    }
+    swap(x, y);
+  }
+}
+
 CGAL_END_NAMESPACE
 
 #endif // CGAL_MP_FLOAT_H