Mathematica 7 compares to other languages

Jon Harrop jon at ffconsultancy.com
Tue Dec 2 01:06:22 CET 2008


Xah Lee wrote:
> And on this page, there are sections where Mathematica is compared to
> programing langs, such as C, C++, Java, and research langs Lisp,
> ML, ..., and scripting langs Python, Perl, Ruby...

Have they implemented any of the following features in the latest version:

1. Redistributable standalone executables.

2. Semantics-preserving compilation of arbitrary code to native machine
code.

3. A concurrent run-time to make efficient parallelism easy.

4. Static type checking.

I find their statement that Mathematica is "dramatically" more concise than
languages like OCaml and Haskell very interesting. I ported my ray tracer
language comparison to Mathematica:

  http://www.ffconsultancy.com/languages/ray_tracer/

My Mathematica code weighs in at 50 LOC compared to 43 LOC for OCaml and 44
LOC for Haskell. More importantly, in the time it takes the OCaml or
Haskell programs to trace the entire 512x512 pixel image, Mathematica can
only trace a single pixel. Overall, Mathematica is a whopping 700,000 times
slower!

Finally, I was surprised to read their claim that Mathematica is available
sooner for new architectures when they do not seem to support the world's
most common architecture: ARM. Also, 64-bit Mathematica came 12 years after
the first 64-bit ML...

Here's my Mathematica code for the ray tracer benchmark:

delta = Sqrt[$MachineEpsilon];

RaySphere[o_, d_, c_, r_] :=
  Block[{v, b, disc, t1, t2},
    v = c - o;
    b = v.d;
    disc = Sqrt[b^2 - v.v + r^2];
    t2 = b + disc;
    If[Im[disc] != 0 || t2 <= 0, \[Infinity],
      t1 = b - disc;
      If[t1 > 0, t1, t2]]
    ]

Intersect[o_, d_][{lambda_, n_}, Sphere[c_, r_]] :=
 Block[{lambda2 = RaySphere[o, d, c, r]},
  If[lambda2 >= lambda, {lambda, n}, {lambda2, 
    Normalize[o + lambda2 d - c]}]
  ]
Intersect[o_, d_][{lambda_, n_}, Bound[c_, r_, s_]] :=
 Block[{lambda2 = RaySphere[o, d, c, r]},
  If[lambda2 >= lambda, {lambda, n}, 
   Fold[Intersect[o, d], {lambda, n}, s]]
  ]

neglight = N at Normalize[{1, 3, -2}];

nohit = {\[Infinity], {0, 0, 0}};

RayTrace[o_, d_, scene_] :=
 Block[{lambda, n, g, p},
  {lambda, n} = Intersect[o, d][nohit, scene];
  If[lambda == \[Infinity], 0,
   g = n.neglight;
   If[g <= 0, 0,
    {lambda, n} = 
     Intersect[o + lambda d + delta n, neglight][nohit, scene];
    If[lambda < \[Infinity], 0, g]]]
  ]

Create[level_, c_, r_] :=
 Block[{obj = Sphere[c, r]},
  If[level == 1, obj,
   Block[{a = 3*r/Sqrt[12], Aux},
    Aux[x1_, z1_] := Create[level - 1, c + {x1, a, z1}, 0.5 r];
    Bound[c, 
     3 r, {obj, Aux[-a, -a], Aux[a, -a], Aux[-a, a], Aux[a, a]}]]]]

scene = Create[1, {0, -1, 4}, 1];

Main[level_, n_, ss_] :=
 Block[{scene = Create[level, {0, -1, 4}, 1]},
  Table[
   Sum[
     RayTrace[{0, 0, 0}, 
      N at Normalize[{(x + s/ss/ss)/n - 1/2, (y + Mod[s, ss]/ss)/n - 1/2,
          1}], scene], {s, 0, ss^2 - 1}]/ss^2, {y, 0, n - 1},
   {x, 0, n - 1}]]

AbsoluteTiming[Export["image.pgm", Graphics at Raster@Main[9, 512, 4]]]

-- 
Dr Jon D Harrop, Flying Frog Consultancy Ltd.
http://www.ffconsultancy.com/?u



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