Re: GUB and mpfr/mpc

[Masamichi HOSODA]

>>>>>>> I changed mingw.org's mingw-runtime to mingw-64 (32-bit).
>>>>>>> Then an error didn't occur and correct test.pdf was generated.
>>>>>>> 
>>>>>>> https://github.com/trueroad/gub/tree/binutils-2.24
>>>>>>> 
>>>>>>> I may pull request if you request it.
>>>>>>> 
>>>>>>> Further, even if the runtime is mingw-w64,
>>>>>>> bad_alloc occurred when optimization was turned on.
> 
> There is this at the end of skyline.cc:
> 
>       // Should add during ver2.19 (to avoid an endless loop
>       // when  merging identical skylines with a vertical segment)
>       // if (end >= s2->front().end_) s2->pop_front();
> 
> Also, the mention of optimization reminds me of one of the horrors of 
> floating point types: a value in a register can quietly be changed when it is 
> written to memory.  Take a look at some of the “myths” in section 1 of 
> https://hal.archives-ouvertes.fr/hal-00128124/en/ .
> 
> Do the skylines need to deal with the wide range of values that “double" 
> allows?  If not, replacing the Real class with a fixed-point math class would 
> make the code a lot less scaring.

I think that floating point can be used as long as it's used correct.

operator== shouldn't be used for comparison of floating point values.
operator!= shouldn't be used too.
operator< and operator> may be used.
It's better not to use operator<= and operator>=.

I tried the following patch.
But bad_alloc occurred.

Then, I think the floating point problem may be unrelated to this case.

```
--- a/lily/skyline.cc   2014-12-01 21:31:21.353083100 +0900
+++ b/lily/skyline.cc   2014-12-04 22:56:17.740379900 +0900
@@ -184,7 +184,7 @@

       // conceals and intersection_x involve multiplication and
       // division. Avoid that, if we can.
-      if (c.y_intercept_ == -infinity_f)
+      if ((isinf(c.y_intercept_) && signbit(c.y_intercept_)))
         {
           if (c.end_ > b.end_)
             return b.end_;
@@ -197,7 +197,7 @@
         return start_x;

       Real i = b.intersection_x (c);
-      if (i > start_x && i <= b.end_ && i <= c.end_)
+      if (i > start_x && (i - b.end_) < DBL_EPSILON && (i - c.end_) < 
DBL_EPSILON)
         return i;

       start_x = c.end_;
```