This file is indexed.

/usr/include/CGAL/MP_Float.h is in libcgal-dev 4.2-5ubuntu1.

This file is owned by root:root, with mode 0o644.

The actual contents of the file can be viewed below.

  1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
// Copyright (c) 2001-2007  INRIA Sophia-Antipolis (France).
// All rights reserved.
//
// This file is part of CGAL (www.cgal.org); you can redistribute it and/or
// modify it under the terms of the GNU Lesser General Public License as
// published by the Free Software Foundation; either version 3 of the License,
// or (at your option) any later version.
//
// Licensees holding a valid commercial license may use this file in
// accordance with the commercial license agreement provided with the software.
//
// This file is provided AS IS with NO WARRANTY OF ANY KIND, INCLUDING THE
// WARRANTY OF DESIGN, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
//
// $URL$
// $Id$
//
//
// Author(s)     : Sylvain Pion

#ifndef CGAL_MP_FLOAT_H
#define CGAL_MP_FLOAT_H

#include <CGAL/number_type_basic.h>
#include <CGAL/Algebraic_structure_traits.h>
#include <CGAL/Real_embeddable_traits.h>
#include <CGAL/Coercion_traits.h>
#include <CGAL/Quotient.h>
#include <CGAL/Sqrt_extension.h>

#include <CGAL/utils.h>

#include <CGAL/Interval_nt.h>
#include <iostream>
#include <vector>
#include <algorithm>

// MP_Float : multiprecision scaled integers.

// Some invariants on the internal representation :
// - zero is represented by an empty vector, and whatever exp.
// - no leading or trailing zero in the vector => unique

// The main algorithms are :
// - Addition/Subtraction
// - Multiplication
// - Integral division div(), gcd(), operator%().
// - Comparison
// - to_double() / to_interval()
// - Construction from a double.
// - IOs

// TODO :
// - The exponent really overflows sometimes -> make it multiprecision.
// - Write a generic wrapper that adds an exponent to be used by MP integers.
// - Karatsuba (or other) ?  Would be fun to implement at least.
// - Division, sqrt... : different options :
//   - nothing
//   - convert to double, take approximation, compute over double, reconstruct

namespace CGAL {

class MP_Float;

template < typename > class Quotient; // Needed for overloaded To_double

namespace INTERN_MP_FLOAT {

Comparison_result compare(const MP_Float&, const MP_Float&);

MP_Float square(const MP_Float&);

// to_double() returns, not the closest double, but a one bit error is allowed.
// We guarantee : to_double(MP_Float(double d)) == d.

double to_double(const MP_Float&);

double to_double(const Quotient<MP_Float>&);

std::pair<double,double> to_interval(const MP_Float &);

std::pair<double,double> to_interval(const Quotient<MP_Float>&);

MP_Float div(const MP_Float& n1, const MP_Float& n2);

MP_Float gcd(const MP_Float& a, const MP_Float& b);
  
} //namespace INTERN_MP_FLOAT

std::pair<double, int>
to_double_exp(const MP_Float &b);

// Returns (first * 2^second), an interval surrounding b.
std::pair<std::pair<double, double>, int>
to_interval_exp(const MP_Float &b);

std::ostream &
operator<< (std::ostream & os, const MP_Float &b);

// This one is for debug.
std::ostream &
print (std::ostream & os, const MP_Float &b);

std::istream &
operator>> (std::istream & is, MP_Float &b);

MP_Float operator+(const MP_Float &a, const MP_Float &b);

MP_Float operator-(const MP_Float &a, const MP_Float &b);

MP_Float operator*(const MP_Float &a, const MP_Float &b);

MP_Float operator%(const MP_Float &a, const MP_Float &b);


class MP_Float
{
public:
  typedef short      limb;
  typedef int        limb2;
  typedef double     exponent_type;

  typedef std::vector<limb>  V;
  typedef V::const_iterator  const_iterator;
  typedef V::iterator        iterator;

private:

  void remove_leading_zeros()
  {
    while (!v.empty() && v.back() == 0)
      v.pop_back();
  }

  void remove_trailing_zeros()
  {
    if (v.empty() || v.front() != 0)
      return;

    iterator i = v.begin();
    for (++i; *i == 0; ++i)
      ;
    exp += i-v.begin();
    v.erase(v.begin(), i);
  }

  // The constructors from float/double/long_double are factorized in the
  // following template :
  template < typename T >
  void construct_from_builtin_fp_type(T d);

public:
#ifdef CGAL_ROOT_OF_2_ENABLE_HISTOGRAM_OF_NUMBER_OF_DIGIT_ON_THE_COMPLEX_CONSTRUCTOR
    int tam() const { return v.size(); }
#endif

  // Splits a limb2 into 2 limbs (high and low).
  static
  void split(limb2 l, limb & high, limb & low)
  {
    const unsigned int sizeof_limb=8*sizeof(limb);
    const limb2 mask= ~( static_cast<limb2>(-1) << sizeof_limb ); //0000ffff
    //Note: For Integer type, if the destination type is signed, the value is unchanged 
    //if it can be represented in the destination type)
    low=static_cast<limb>(l & mask); //extract low bits from l 
    high = (l - low) >> sizeof_limb; //extract high bits from l
    
    CGAL_postcondition ( l == low + ( static_cast<limb2>(high) << sizeof_limb ) );
  }

  // Given a limb2, returns the higher limb.
  static
  limb higher_limb(limb2 l)
  {
      limb high, low;
      split(l, high, low);
      return high;
  }

  void canonicalize()
  {
    remove_leading_zeros();
    remove_trailing_zeros();
  }

  MP_Float()
      : exp(0)
  {
    CGAL_assertion(sizeof(limb2) == 2*sizeof(limb));
    CGAL_assertion(v.empty());
    // Creates zero.
  }

#if 0
  // Causes ambiguities
  MP_Float(limb i)
  : v(1,i), exp(0)
  {
    remove_leading_zeros();
  }
#endif

  MP_Float(limb2 i)
  : v(2), exp(0)
  {
    split(i, v[1], v[0]);
    canonicalize();
  }

  MP_Float(float d);

  MP_Float(double d);

  MP_Float(long double d);

  MP_Float operator+() const {
    return *this;
  }

  MP_Float operator-() const
  {
    return MP_Float() - *this;
  }

  MP_Float& operator+=(const MP_Float &a) { return *this = *this + a; }
  MP_Float& operator-=(const MP_Float &a) { return *this = *this - a; }
  MP_Float& operator*=(const MP_Float &a) { return *this = *this * a; }
  MP_Float& operator%=(const MP_Float &a) { return *this = *this % a; }

  exponent_type max_exp() const
  {
    return v.size() + exp;
  }

  exponent_type min_exp() const
  {
    return exp;
  }

  limb of_exp(exponent_type i) const
  {
    if (i < exp || i >= max_exp())
      return 0;
    return v[static_cast<int>(i-exp)];
  }

  bool is_zero() const
  {
    return v.empty();
  }

  Sign sign() const
  {
    if (v.empty())
      return ZERO;
    if (v.back()>0)
      return POSITIVE;
    CGAL_assertion(v.back()<0);
    return NEGATIVE;
  }

  void clear()
  {
    v.clear();
    exp = 0;
  }

  void swap(MP_Float &m)
  {
    std::swap(v, m.v);
    std::swap(exp, m.exp);
  }

  // Converts to a rational type (e.g. Gmpq).
  template < typename T >
  T to_rational() const
  {
    const unsigned log_limb = 8 * sizeof(MP_Float::limb);

    if (is_zero())
      return 0;

    MP_Float::const_iterator i;
    exponent_type exp2 = min_exp() * log_limb;
    T res = 0;

    for (i = v.begin(); i != v.end(); ++i)
    {
      res += T(std::ldexp(static_cast<double>(*i),static_cast<int>(exp2)));
      exp2 += log_limb;
    }

    return res;
  }

  std::size_t size() const
  {
    return v.size();
  }

  // Returns a scaling factor (in limbs) which would be good to extract to get
  // a value with an exponent close to 0.
  exponent_type find_scale() const
  {
    return exp + v.size();
  }

  // Rescale the value by some factor (in limbs).  (substract the exponent)
  void rescale(exponent_type scale)
  {
    if (v.size() != 0)
      exp -= scale;
  }

  // Accessory function that finds the least significant bit set (its position).
  static unsigned short 
  lsb(limb l)
  {
    unsigned short nb = 0;
    for (; (l&1)==0; ++nb, l>>=1)
      ;
    return nb;
  }

  // This one is needed for normalizing gcd so that the mantissa is odd
  // and non-negative, and the exponent is 0.
  void gcd_normalize()
  {
    const unsigned log_limb = 8 * sizeof(MP_Float::limb);
    if (is_zero())
      return;
    // First find how many least significant bits are 0 in the last digit.
    unsigned short nb = lsb(v[0]);
    if (nb != 0)
      *this = *this * (1<<(log_limb-nb));
    CGAL_assertion((v[0]&1) != 0);
    exp=0;
    if (sign() == NEGATIVE)
      *this = - *this;
  }

  MP_Float unit_part() const
  {
    if (is_zero())
      return 1;
    MP_Float r = (sign() > 0) ? *this : - *this;
    CGAL_assertion(r.v.begin() != r.v.end());
    unsigned short nb = lsb(r.v[0]);
    r.v.clear();
    r.v.push_back(1<<nb);
    return (sign() > 0) ? r : -r;
  }

  bool is_integer() const
  {
    return is_zero() || (exp >= 0);
  }

  V v;
  exponent_type exp;
};

namespace internal{
std::pair<MP_Float, MP_Float> // <quotient, remainder>
division(const MP_Float & n, const MP_Float & d);
} // namespace internal

inline
void swap(MP_Float &m, MP_Float &n)
{ m.swap(n); }

inline
bool operator<(const MP_Float &a, const MP_Float &b)
{ return INTERN_MP_FLOAT::compare(a, b) == SMALLER; }

inline
bool operator>(const MP_Float &a, const MP_Float &b)
{ return b < a; }

inline
bool operator>=(const MP_Float &a, const MP_Float &b)
{ return ! (a < b); }

inline
bool operator<=(const MP_Float &a, const MP_Float &b)
{ return ! (a > b); }

inline
bool operator==(const MP_Float &a, const MP_Float &b)
{ return (a.v == b.v) && (a.v.empty() || (a.exp == b.exp)); }

inline
bool operator!=(const MP_Float &a, const MP_Float &b)
{ return ! (a == b); }

MP_Float
approximate_sqrt(const MP_Float &d);

MP_Float
approximate_division(const MP_Float &n, const MP_Float &d);



// Algebraic structure traits specialization
template <> class Algebraic_structure_traits< MP_Float >
  : public Algebraic_structure_traits_base< MP_Float,
                                            Unique_factorization_domain_tag
	    // with some work on mod/div it could be Euclidean_ring_tag
                                          >  {
  public:

    typedef Tag_true            Is_exact;
    typedef Tag_true            Is_numerical_sensitive;

    struct Unit_part
      : public std::unary_function< Type , Type >
    {
      Type operator()(const Type &x) const {
        return x.unit_part();
      }
    };

    struct Integral_division
        : public std::binary_function< Type,
                                 Type,
                                 Type > {
    public:
        Type operator()(
                const Type& x,
                const Type& y ) const {
            std::pair<MP_Float, MP_Float> res = internal::division(x, y);
            CGAL_assertion_msg(res.second == 0,
                "exact_division() called with operands which do not divide");
            return res.first;
        }
    };


    class Square
      : public std::unary_function< Type, Type > {
      public:
        Type operator()( const Type& x ) const {
          return INTERN_MP_FLOAT::square(x);
        }
    };

    class Gcd
      : public std::binary_function< Type, Type,
                                Type > {
      public:
        Type operator()( const Type& x,
                                        const Type& y ) const {
          return INTERN_MP_FLOAT::gcd( x, y );
        }
    };

    class Div
      : public std::binary_function< Type, Type,
                                Type > {
      public:
        Type operator()( const Type& x,
                                        const Type& y ) const {
          return INTERN_MP_FLOAT::div( x, y );
        }
    };

  typedef INTERN_AST::Mod_per_operator< Type > Mod;
// Default implementation of Divides functor for unique factorization domains
  // x divides y if gcd(y,x) equals x up to inverses 
  class Divides 
    : public std::binary_function<Type,Type,bool>{ 
  public:
    bool operator()( const Type& x,  const Type& y) const {  
      return internal::division(y,x).second == 0 ;
    }
    // second operator computing q = x/y 
    bool operator()( const Type& x,  const Type& y, Type& q) const {    
      std::pair<Type,Type> qr = internal::division(y,x);
      q=qr.first;
      return qr.second == 0;
      
    }
    CGAL_IMPLICIT_INTEROPERABLE_BINARY_OPERATOR_WITH_RT( Type , bool)
  };
};

// Real embeddable traits
template <> class Real_embeddable_traits< MP_Float >
  : public INTERN_RET::Real_embeddable_traits_base< MP_Float , CGAL::Tag_true > {
  public:

    class Sgn
      : public std::unary_function< Type, ::CGAL::Sign > {
      public:
        ::CGAL::Sign operator()( const Type& x ) const {
          return x.sign();
        }
    };

    class Compare
      : public std::binary_function< Type, Type,
                                Comparison_result > {
      public:
        Comparison_result operator()( const Type& x,
                                            const Type& y ) const {
          return INTERN_MP_FLOAT::compare( x, y );
        }
    };

    class To_double
      : public std::unary_function< Type, double > {
      public:
        double operator()( const Type& x ) const {
          return INTERN_MP_FLOAT::to_double( x );
        }
    };

    class To_interval
      : public std::unary_function< Type, std::pair< double, double > > {
      public:
        std::pair<double, double> operator()( const Type& x ) const {
          return INTERN_MP_FLOAT::to_interval( x );
        }
    };
};



namespace INTERN_MP_FLOAT{

//Sqrt_extension internally uses Algebraic_structure_traits
template <class ACDE_TAG_, class FP_TAG>  
double
to_double(const Sqrt_extension<MP_Float,MP_Float,ACDE_TAG_,FP_TAG> &x)
{
  typedef MP_Float RT;
  typedef Quotient<RT> FT;
  typedef CGAL::Rational_traits< FT > Rational;
  Rational r;
  const RT r1 = r.numerator(x.a0());
  const RT d1 = r.denominator(x.a0());

  if(x.is_rational()) {
    std::pair<double, int> n = to_double_exp(r1);
    std::pair<double, int> d = to_double_exp(d1);
    double scale = std::ldexp(1.0, n.second - d.second);
    return (n.first / d.first) * scale;
  }

  const RT r2 = r.numerator(x.a1());
  const RT d2 = r.denominator(x.a1());
  const RT r3 = r.numerator(x.root());
  const RT d3 = r.denominator(x.root());

  std::pair<double, int> n1 = to_double_exp(r1);
  std::pair<double, int> v1 = to_double_exp(d1);
  double scale1 = std::ldexp(1.0, n1.second - v1.second);

  std::pair<double, int> n2 = to_double_exp(r2);
  std::pair<double, int> v2 = to_double_exp(d2);
  double scale2 = std::ldexp(1.0, n2.second - v2.second);

  std::pair<double, int> n3 = to_double_exp(r3);
  std::pair<double, int> v3 = to_double_exp(d3);
  double scale3 = std::ldexp(1.0, n3.second - v3.second);

  return ((n1.first / v1.first) * scale1) + 
         ((n2.first / v2.first) * scale2) *
         std::sqrt((n3.first / v3.first) * scale3);
}

} //namespace INTERN_MP_FLOAT


namespace internal {
// This compares the absolute values of the odd-mantissa.
// (take the mantissas, get rid of all powers of 2, compare
// the absolute values)
inline
Sign
compare_bitlength(const MP_Float &a, const MP_Float &b)
{
  if (a.is_zero())
    return b.is_zero() ? EQUAL : SMALLER;
  if (b.is_zero())
    return LARGER;

  //Real_embeddable_traits<MP_Float>::Abs abs;

  MP_Float aa = CGAL_NTS abs(a);
  MP_Float bb = CGAL_NTS abs(b);

  if (aa.size() > (bb.size() + 2)) return LARGER;
  if (bb.size() > (aa.size() + 2)) return SMALLER;

  // multiply by 2 till last bit is 1.
  while (((aa.v[0]) & 1) == 0) // last bit is zero
    aa = aa + aa;

  while (((bb.v[0]) & 1) == 0) // last bit is zero
    bb = bb + bb;

  // sizes might have changed
  if (aa.size() > bb.size()) return LARGER;
  if (aa.size() < bb.size()) return SMALLER;

  for (std::size_t i = aa.size(); i > 0; --i)
  {
    if (aa.v[i-1] > bb.v[i-1]) return LARGER;
    if (aa.v[i-1] < bb.v[i-1]) return SMALLER;
  }
  return EQUAL;
}

inline // Move it to libCGAL once it's stable.
std::pair<MP_Float, MP_Float> // <quotient, remainder>
division(const MP_Float & n, const MP_Float & d)
{
  typedef MP_Float::exponent_type  exponent_type;

  MP_Float remainder = n, divisor = d;

  CGAL_precondition(divisor != 0);

  // Rescale d to have a to_double() value with reasonnable exponent.
  exponent_type scale_d = divisor.find_scale();
  divisor.rescale(scale_d);
  const double dd = INTERN_MP_FLOAT::to_double(divisor);

  MP_Float res = 0;
  exponent_type scale_remainder = 0;

  bool first_time_smaller_than_divisor = true;

  // School division algorithm.

  while ( remainder != 0 )
  {
    // We have to rescale, since remainder can diminish towards 0.
    exponent_type tmp_scale = remainder.find_scale();
    remainder.rescale(tmp_scale);
    res.rescale(tmp_scale);
    scale_remainder += tmp_scale;

    // Compute a double approximation of the quotient
    // (imagine school division with base ~2^53).
    double approx = INTERN_MP_FLOAT::to_double(remainder) / dd;
    CGAL_assertion(approx != 0);
    res += approx;
    remainder -= approx * divisor;

    if (remainder == 0)
      break;

    // Then we need to fix it up by checking if neighboring double values
    // are closer to the exact result.
    // There should not be too many iterations, because approx is only a few ulps
    // away from the optimal.
    // If we don't do the fixup, then spurious bits can be introduced, which
    // will require an unbounded amount of additional iterations to be eliminated.

    // The direction towards which we need to try to move from "approx".
    double direction = (CGAL_NTS sign(remainder) == CGAL_NTS sign(dd))
                     ?  std::numeric_limits<double>::infinity()
                     : -std::numeric_limits<double>::infinity();

    while (true)
    {
      const double approx2 = nextafter(approx, direction);
      const double delta = approx2 - approx;
      MP_Float new_remainder = remainder - delta * divisor;
      if (CGAL_NTS abs(new_remainder) < CGAL_NTS abs(remainder)) {
        remainder = new_remainder;
        res += delta;
        approx = approx2;
      }
      else {
        break;
      }
    }

    if (remainder == 0)
      break;

    // Test condition for non-exact division (with remainder).
    if (compare_bitlength(remainder, divisor) == SMALLER)
    {
      if (! first_time_smaller_than_divisor)
      {
        // Scale back.
        res.rescale(scale_d - scale_remainder);
        remainder.rescale(- scale_remainder);
        CGAL_postcondition(res * d  + remainder == n);
        return std::make_pair(res, remainder);
      }
      first_time_smaller_than_divisor = false;
    }
  }

  // Scale back the result.
  res.rescale(scale_d - scale_remainder);
  CGAL_postcondition(res * d == n);
  return std::make_pair(res, MP_Float(0));
}

inline // Move it to libCGAL once it's stable.
bool
divides(const MP_Float & d, const MP_Float & n)
{
  return internal::division(n, d).second == 0;
}

} // namespace internal

inline
bool
is_integer(const MP_Float &m)
{
  return m.is_integer();
}



inline
MP_Float
operator%(const MP_Float& n1, const MP_Float& n2)
{
  return internal::division(n1, n2).second;
}


// The namespace INTERN_MP_FLOAT contains global functions like square or sqrt
// which collide with the global functor adapting functions provided by the new
// AST/RET concept.
//
// TODO: IMHO, a better solution would be to put the INTERN_MP_FLOAT-functions
//       into the MP_Float-class... But there is surely a reason why this is not
//       the case..?


namespace INTERN_MP_FLOAT {
  inline
  MP_Float
  div(const MP_Float& n1, const MP_Float& n2)
  {
    return internal::division(n1, n2).first;
  }

  inline
  MP_Float
  gcd( const MP_Float& a, const MP_Float& b)
  {
    if (a == 0) {
      if (b == 0)
        return 0;
      MP_Float tmp=b;
      tmp.gcd_normalize();
      return tmp;
    }
    if (b == 0) {
      MP_Float tmp=a;
      tmp.gcd_normalize();
      return tmp;
    }

    MP_Float x = a, y = b;
    while (true) {
      x = x % y;
      if (x == 0) {
        CGAL_postcondition(internal::divides(y, a) & internal::divides(y, b));
        y.gcd_normalize();
        return y;
      }
      swap(x, y);
    }
  }

} // INTERN_MP_FLOAT


inline
void
simplify_quotient(MP_Float & numerator, MP_Float & denominator)
{
  // Currently only simplifies the two exponents.
#if 0
  // This better version causes problems as the I/O is changed for
  // Quotient<MP_Float>, which then does not appear as rational 123/345,
  // 1.23/3.45, this causes problems in the T2 test-suite (to be investigated).
  numerator.exp -= denominator.exp
                    + (MP_Float::exponent_type) denominator.v.size();
  denominator.exp = - (MP_Float::exponent_type) denominator.v.size();
#elif 1
  numerator.exp -= denominator.exp;
  denominator.exp = 0;
#else
  if (numerator != 0 && denominator != 0) {
    numerator.exp -= denominator.exp;
    denominator.exp = 0;
    const MP_Float g = gcd(numerator, denominator);
    numerator = integral_division(numerator, g);
    denominator = integral_division(denominator, g);
  }
  numerator.exp -= denominator.exp;
  denominator.exp = 0;
#endif
}

inline void simplify_root_of_2(MP_Float &/*a*/, MP_Float &/*b*/, MP_Float&/*c*/) {
#if 0
  if(is_zero(a)) {
  	simplify_quotient(b,c); return;
  } else if(is_zero(b)) {
  	simplify_quotient(a,c); return;
  } else if(is_zero(c)) {
  	simplify_quotient(a,b); return;
  }
  MP_Float::exponent_type va = a.exp +
    (MP_Float::exponent_type) a.v.size();
  MP_Float::exponent_type vb = b.exp +
    (MP_Float::exponent_type) b.v.size();
  MP_Float::exponent_type vc = c.exp +
    (MP_Float::exponent_type) c.v.size();
  MP_Float::exponent_type min = (std::min)((std::min)(va,vb),vc);
  MP_Float::exponent_type max = (std::max)((std::max)(va,vb),vc);
  MP_Float::exponent_type med = (min+max)/2.0;
  a.exp -= med;
  b.exp -= med;
  c.exp -= med;
#endif
}

namespace internal {
  inline void simplify_3_exp(int &a, int &b, int &c) {
    int min = (std::min)((std::min)(a,b),c);
    int max = (std::max)((std::max)(a,b),c);
    int med = (min+max)/2;
    a -= med;
    b -= med;
    c -= med;
  }
}


// specialization of to double functor
template<>
class Real_embeddable_traits< Quotient<MP_Float> >
    : public INTERN_QUOTIENT::Real_embeddable_traits_quotient_base<
Quotient<MP_Float> >{
public:
    struct To_double: public std::unary_function<Quotient<MP_Float>, double>{
         inline
         double operator()(const Quotient<MP_Float>& q) const {
            return INTERN_MP_FLOAT::to_double(q);
        }
    };
    struct To_interval
        : public std::unary_function<Quotient<MP_Float>, std::pair<double,double> > {
        inline
        std::pair<double,double> operator()(const Quotient<MP_Float>& q) const {
            return INTERN_MP_FLOAT::to_interval(q);
        }
    };
};

inline MP_Float min BOOST_PREVENT_MACRO_SUBSTITUTION(const MP_Float& x,const MP_Float& y){
  return (x<=y)?x:y; 
}
inline MP_Float max BOOST_PREVENT_MACRO_SUBSTITUTION(const MP_Float& x,const MP_Float& y){
  return (x>=y)?x:y; 
}


// Coercion_traits
CGAL_DEFINE_COERCION_TRAITS_FOR_SELF(MP_Float)
CGAL_DEFINE_COERCION_TRAITS_FROM_TO(int, MP_Float)


} //namespace CGAL

namespace Eigen {
  template<class> struct NumTraits;
  template<> struct NumTraits<CGAL::MP_Float>
  {
    typedef CGAL::MP_Float Real;
    typedef CGAL::Quotient<CGAL::MP_Float> NonInteger;
    typedef CGAL::MP_Float Nested;

    static inline Real epsilon() { return 0; }

    enum {
      IsInteger = 1, // Is this lie right?
      IsSigned = 1,
      IsComplex = 0,
      RequireInitialization = 1,
      ReadCost = 6,
      AddCost = 40,
      MulCost = 40
    };
  };
}

#include <CGAL/MP_Float_impl.h>

//specialization for Get_arithmetic_kernel
#include <CGAL/MP_Float_arithmetic_kernel.h>

#endif // CGAL_MP_FLOAT_H