Minor enhancements

[Kalashnikovni]

The following minor enhancements are proposed:

• New implementation of rational without using boost. This won't affect its interface, only the underlying structure inside LIB::RATIONAL (no need to modify anything in the rest of the library).
• Parse intervals of step =1 without the need of writing the step. Currently the step is mandatory for all cases.
• Optimize error messages in the eval_expr.cpp file.
• As noted in pr-61 the member_class and member_imp macros need to be redefined.

[Kalashnikovni]

It was also noted by @LucasCavagna that this portion of UnordPWMap::operator- is incorrect:

// Increasing expression
else if (m > 0) {
   RATIONAL cross = -h / m;
   if (cross > 0 || cross == 0) {
     begin_pos = boost::rational_cast<NAT>(cross.value());
     if (begin_pos > 0)
       end_neg = begin_pos - 1;
     else {
        begin_neg = 1;
        begin_pos = 0;
      }
    }
 }

instead of begin_pos = 0; it should be end_neg = 0;, to return an empty interval for negative values.

[Kalashnikovni]

It was also discovered that the current implementation of minMap:

PWMapDelegPtr UnordPWMap::minMap(const PWMapDelegate &other) const
{
  if (isEmpty() || other.isEmpty())
    return std::make_unique<UnordPWMap>(fact_);

  PWMapDelegPtr aux1 = restrict(other.dom()), aux2 = other.restrict(dom());
  Set zero = fact_.createSet(SetPiece(arity(), Interval(0, 1, 0)));
  Set to_zero = (*aux1 - *aux2)->preImage(zero);
  Set not_zero = aux2->dom().difference(to_zero);
  aux1 = aux1->restrict(to_zero);
  aux2 = aux2->restrict(not_zero);

  return aux1->combine(*aux2);
}

only works for unidimensional maps. For example:

minMap(<<{[1:1:100] x [1:1:100]} ↦ x|N*x+100>>, <<{[1:1:100] x [1:1:100]} ↦ N*x+100|x>>)
  --> <<{[1:49]x[51:100], [1:49]x[50:50], [50:50]x[51:100], [50:50]x[50:50]} -> x|-1x+100, {[51:100]x[1:100], [50:50]x[1:49], [1:49]x[1:49]} -> -1x+100|x>>

where res(1, 1) = (99, 1) and it should be res(1, 1) = (1, 99).

[Kalashnikovni]

I'll also add automatic detection to use ordered dense sets. If the user selects a specific implementation it will override the default option.

[Kalashnikovni]

During the correction of minMap the following was obtained for UnordPWMap::operator-:

<<{[1:1:10] x [1:1:10]} ↦ x+1|N*x+11>>-<<{[1:1:10] x [1:1:10]} ↦ N*x+10|x>>
  --> <<{[1:3]x[6:10]} -> 0|0, {[1:3]x[1:5]} -> 0|-2x+11, {[4:10]x[6:10]} -> 2x-9|0, {[4:10]x[1:5]} -> 2x-9|-2x+11>>

which is clearly incorrect, i.e. maps to negative values for (4, 6:10).

[Kalashnikovni]

During the correction of minMap the following was obtained for UnordPWMap::operator-:

<<{[1:1:10] x [1:1:10]} ↦ x+1|N*x+11>>-<<{[1:1:10] x [1:1:10]} ↦ N*x+10|x>>
  --> <<{[1:3]x[6:10]} -> 0|0, {[1:3]x[1:5]} -> 0|-2x+11, {[4:10]x[6:10]} -> 2x-9|0, {[4:10]x[1:5]} -> 2x-9|-2x+11>>

which is clearly incorrect, i.e. maps to negative values for (4, 6:10).

This error appeared because of the following code:

// Increasing expression
else if (m > 0) {
  RATIONAL cross = -h / m;
  if (cross > 0 || cross == 0) {
    begin_pos = boost::rational_cast<NAT>(cross.value());
    if (begin_pos > 0)
      end_neg = begin_pos - 1;
    else {
      begin_neg = 1;
      end_neg = 0;
    }
  }
}

particularly, calculating begin_pos = boost::rational_cast<NAT>(cross.value()) returned the floor of the rational value, for which the map calculates a non-positive result. Replacing this line with begin_pos = boost::rational_cast<NAT>(cross.value()) + 1 solved the issue.

[Kalashnikovni]

I first decided to take out boost::rational and implement myself LIB::RATIONAL. After some consideration, I decided to use boost::multiprecision::cpp_rational since rl1.test with N=100000000 gives error:

terminate called after throwing an instance of 'boost::wrapexcept<boost::bad_rational>'
  what():  bad rational: zero denominator
Abortado (`core' generado)

I'll test if using this type doesn't affect efficiency.