Talk:Total circles area

Monte Carlo

A semi-important point and semi-nitpicking for doing Monte Carlo: you need to estimate the error in a credible way given the number of samples, and should not output more significant digits than the confidence allows. The current C code sample size of 100M can't possibly allow 8 decimal point precision. --Ledrug 04:29, 16 September 2012 (UTC)

Right, in the C/Python outputs only the first few digits are correct. Feel free to add a bit of error estimation code.

This is basically what I used on reddit, but I don't feel like changing the example on the task page because MC really isn't the right tool for this task.

<lang c>#include <stdio.h>

include <stdlib.h>
include <tgmath.h>
include <time.h>

typedef float flt; flt data[][3] = { { 1.6417233788, 1.6121789534, 0.0848270516}, ... };

define N sizeof(data)/sizeof(data[0])

int main(void) { flt x0 = 1/0., x1 = -1/0., y0 = 1/0., y1 = -1/0.;

inline flt min(flt x, flt y) { return x < y ? x : y; } inline flt max(flt x, flt y) { return x > y ? x : y; } inline flt sq(flt x) { return x * x; } inline flt rndf(flt x, flt d) { return x + d * rand() / RAND_MAX; }

inline int covered(flt x, flt y) { for (int i = 0; i < N; i++) if (sq(x - data[i][0]) + sq(y - data[i][1]) < data[i][2]) return 1; return 0; }

for (int i = 0; i < N; i++) { flt *p = data[i]; x0 = min(x0, p[0] - p[2]); x1 = max(x1, p[0] + p[2]); y0 = min(y0, p[1] - p[2]); y1 = max(y1, p[1] + p[2]);

p[2] *= p[2]; }

x1 -= x0, y1 -= y0;

flt total = x1 * y1; unsigned long to_try = 0x10000, tries = 0, hit = 0; srand(time(0));

while (1) { hit += covered(rndf(x0, x1), rndf(y0, y1));

if (++tries == to_try) { flt area = total * hit / tries; flt r = (flt) hit / tries; flt s = area * sqrt(r * (1 - r) / tries); printf("%.4f +/- %.4f (%lu samples)\n", area, s, tries); // stop at 3 sigmas if (s * 3 <= 1e-3) break; to_try *= 2; } } return 0; }</lang> --Ledrug 22:09, 16 September 2012 (UTC)

A bad solution algorithm is quite useful for didactic purposes (but I don't know how much Rosettacode cares for didactic).

Haskell analytical solution

I like the analytical solution in Haskell, but I think a tuples soup harms readability. So do you mind if I replace it with this code that introduces the Vec, Circle and Arc types? http://ideone.com/JjD4x bearophile 12:11, 21 September 2012 (UTC)

Go ahead. I'm a complete newbie in Haskell, and have no problem with people improving it. --Ledrug 15:04, 21 September 2012 (UTC)

Actually there is one thing that's not quite right: in an arc ((x,y,r), (a0,a1)) the (a0,a1) is not a vector, they are a pair of angles of the start and end points on the original circle. It doesn't hurt the computation, but as readability goes, nameing them "Vec" is misleading. --Ledrug 18:26, 21 September 2012 (UTC)

Right, I am sorry. In a (Haskell) program types contain lot of information (like telling apart two tuples that contain two doubles). For a second programmer that tries to add more of such information it's easy to miss part of the semantics and introduce wrong type information. I will try to introduce a Angle2 type and fix the code. I am an Haskell newbie myself, but we are learning. --bearophile 19:36, 21 September 2012 (UTC)

Done. Please take a look at the code, to make sure I have not used an Angle where instead goes a normal Double, and I have not used an Angle2 where a Vet instead goes.

I think it's fine. I changed the polyline area function, maybe it's clearer this way. Also added some comments explaining some odd stuff I did in the code. One thing sort of amuses me is, is there really a need for an Angle type? I mean, it does no harm, but angles are really just plain numbers and doesn't do anything to distinguish itself from a Double. --Ledrug 06:11, 22 September 2012 (UTC)

Your question is almost as interesting as the circles area Task itself.

Unless you go to extremes, in such situations there is no wrong/correct solutions, just better/worse ones. It's often a judgement call. And judgement sometimes requires having experience in the (Haskell) language.

I am learning Haskell so I have tried to see how well it manages subtypes of simple values. The answer is: mixed.

It's easy to remove the Angle newtype from this program, if not desired.

The program is a little more complex than before, as you see I have had to unpack the angle argument of arcPoint, I have defined a Pi angle, I have had to use a language extension (GeneralizedNewtypeDeriving) to do it simply, and so on.

Functional languages like Haskell recognize and show the value of giving different types to different entities. In F# you even add units of measures (http://msdn.microsoft.com/en-us/library/dd233243.aspx ) to variables (many other languages have libraries for it, but I think they are rarely used in small programs like this). The idea of encoding units of measure or the meaning of values like Angle in the type system is to help avoid mistakes like mixing things that must not mix.

In this program Angle has not spot bugs, but I think it helps self-document a bit better this sparsely comment code. One advantage of using types over comments is that the (Haskell) compiler enforces their correct usage. As example, the type of the aNorm function is "Angle -> Angle". Now it's easy to understand that "a" in its name refers to angle, and "Norm" of the function name means putting the angle value back in a simple range of values, no comments on this function are required now.

In this program there are other unspecified Doubles (beside the input data itself), the return types of arcArea, pathArea, polylineArea and circlesArea. All of them are the same type, of areas of surface. This means that if we use units of measures system like in F# code, such Doubles need to be products of two distances. Currently this is not encoded in the types of this program.

If you want and you have some time, I suggest you to add an image (like the ones in the Stack Overflow thread) to the Haskell code to help show how your program works, using only on four or five of the input circles for visual simplicity. --bearophile 11:57, 22 September 2012 (UTC)

Link

Stack Overflow. Nice diagram and comments.

Python Van der Corput

Is this solution converging faster or slower than the simpler grid or totally random sampling strategies?

Lets see,

The Python grid uses 500*500 = 250000 points to get a value of 21.561559772.
After 200000 points, the Van der Corput has a value of 21.565708203389384
The given analytical solution is 21.565036603856395

It looks as if the VdC does more with its 200K points than the grid with 250K, but I like the fact that the VdC generates a finer and finer grid as you take more points unlike the fixed box grid. (Plus I had stowed away the fact that VdC was used for Monte Carlo sims and wanted to try it out). --Paddy3118 15:07, 22 September 2012 (UTC)

In contrast, the C coded Monte Carlo method takes 100 million points to compute 21.56262288. --Paddy3118 15:15, 22 September 2012 (UTC)

With 100 million points, the Python VdC area estimate is 21.565056432232886. --Paddy3118 15:24, 22 September 2012 (UTC)

Maybe you want to write part of such information inside the page, instead of just here.

C89 vs C99

I have a suggestion: generally, don't convert C89 to C99 code. Compatibility aside, GCC sometimes does weird things in C99 mode. On my machine, for the second C entry, the current C99 code runtime is about twice as long as that of the old C89 (GCC 4.7, both with -O -msse -lm), even though they really shouldn't differ. ICC (intel) doesn't have this problem, so it's definintely GCC, but still, GCC is the more available compiler out there. --Ledrug 20:36, 24 September 2012 (UTC)

I prefer to use C99 where possible because it offers VLAs, better struct initializers, some nice standard libraries, and localized variable definitions that allow for more readable, safer and nicer looking C code. I try to show that just because it's C code it doesn't need to look ugly, unreadable and messy. And C11 is coming.

Sometimes I too noticed small performance reductions with GCC using C99 instead of C89, but I have never seen a 2X runtime increase, and even in this case I can't reproduce the magnitude of your speed decrease. On my system using GCC to compile this nearly-GNU-C89 version is only about 2% faster than the GNU-C99 you see on the page (gcc 4.7.1, -Ofast -s -flto): http://ideone.com/9xihK A further very small performance increase comes from using malloc+free instead of the variable length array (that are not present in C89), maybe because the heap memory has a bigger alignment. Maybe the performance decrease you are seeing doesn't come from C99 alone, but from other very small changes I have put in the code.

Generally I think the quality of the code is more important than performance, especially if the performance is already enough and the performance difference is small. The small increase of language guarantees given by C99 (like not leaving to the implementation the definition of %) should not decrease performance. On the other hand C99 allows to reduce the scope of variables because it allows to define loop variables in place and to define variables where they are initialized and used; this should help increase code performance (and this is what happens if I am using the DMD D compiler).

If the 2X speed decrease you are seeing is real (and I can't reproduce it), I suggest you to create a bug report for GCC devs, to help them fix the situation in the right place, inside GCC, instead of inside my code. --bearophile 12:13, 25 September 2012 (UTC)

Excessive digits

Maybe it's worth explaining after the Task description how the many digits of the reference answer were computed. Have you used this http://hackage.haskell.org/package/hmpfr with the Haskell analytical solution? --bearophile 11:22, 1 October 2012 (UTC)

No, it was calculated using the same method as Haskell, but in C with MPFR, at 2000 bits precision. I only did it as an exercise to know MPFR, and the code was too messy to post here -- I got too bored halfway and stopped keeping track of memory allocation, so there are hundreds of lost memory chunks and valgrind was extremely grumpy about it, but I do believe all the digits I posted are accurate. --Ledrug 20:06, 1 October 2012 (UTC)