Is there a fast product operation for PackedArrays?

Question

In Mathematica a vector (or rectangular array) containing all machine size integers or floats may be stored in a packed array. These objects take less memory, and some operatio

Answer 1

First, to avoid confusion, take a look at an example whose results are all representable as hardware machine precision numbers, which must all be less than

In[1]:= $MaxMachineNumber

Out[1]= 1.79769*10^308

Your Total example already had this nice (and fast) property. Here is a variant on your Times example using machine numbers:

In[2]:= lst = RandomReal[{0.99, 1.01}, 5000000];
Times @@ lst // Timing

Out[3]= {1.435, 1.38851*10^-38}

Now we can use Compile to make a compiled function to perform this operation efficiently:

In[4]:= listproduct = 
 Compile[{{l, _Real, 1}}, 
  Module[{tot = 1.}, Do[tot *= x, {x, l}]; tot]]

Out[4]= CompiledFunction[{l},Module[{tot=1.},Do[tot*=x,{x,l}];tot],-CompiledCode-]

It's much faster:

In[5]:= listproduct[lst] // Timing

Out[5]= {0.141, 1.38851*10^-38}

Assuming you have a C compiler and Mathematica 8, you can also automatically compile all the way to C code. A temporary DLL is created and linked back into Mathematica at run-time.

In[6]:= compiledlistproduct = 
 Compile[{{l, _Real, 1}}, 
  Module[{tot = 1.}, Do[tot *= x, {x, l}]; tot], 
  CompilationTarget -> "C"]

Out[6]= CompiledFunction[{l},Module[{tot=1.},Do[tot*=x,{x,l}];tot],-CompiledCode-]

This gives performance not much different to that which a built-in Mathematica function would have:

In[7]:= compiledlistproduct[lst] // Timing

Out[7]= {0.015, 1.38851*10^-38}

Note that if your product really will go beyond $MaxMachineNumber (or $MinMachineNumber), then you are better off sticking with Apply[Times, list]. The same comment applies to Total, if your results can get that big:

In[11]:= lst = RandomReal[10^305, 5000000];
Plus @@ lst // Timing

Out[12]= {1.435, 2.499873364498981*10^311}

In[13]:= lst = RandomReal[10^305, 5000000];
Total[lst] // Timing

Out[14]= {1.576, 2.500061580905602*10^311}