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}

Answer 2

I've also wondered if there was a multiplicative equivalent to Total.

A really not so bad solution is

In[1]:= lst=RandomReal[2,5000000];
        Times@@lst//Timing
        Exp[Total[Log[lst]]]//Timing
Out[2]= {2.54,4.370467929041*10^-666614}
Out[3]= {0.47,4.370467940*10^-666614}

As long as the numbers are positive and aren't too big or small, then the rounding errors aren't too bad. A guess as to what might be happening during evaluation is that: (1) Provided the numbers are positive floats, the Log operation can be quickly applied to the packed array. (2) The numbers can then be quickly added using Total's packed array method. (3) Then it's only the final step where a non-machine sized float need arise.

See this SO answer for a solution that works for both positive and negative floats.

Let's quickly check that this solution works with floats that yield a non-machine sized answer. Compare with Andrew's (much faster) compiledListProduct:

In[10]:= compiledListProduct = 
          Compile[{{l, _Real, 1}}, 
           Module[{tot = 1.}, Do[tot *= x, {x, l}]; tot], 
           CompilationTarget -> "C"]

In[11]:= lst=RandomReal[{0.05,.10},15000000];
         Times@@lst//Timing
         Exp[Total[Log[lst]]]//Timing
         compiledListProduct[lst]//Timing
Out[12]= {7.49,2.49105025389*10^-16998863}
Out[13]= {0.5,2.4910349*10^-16998863}
Out[14]= {0.07,0.}

If you choose larger (>1) reals, then compiledListProduct will yield the warning CompiledFunction::cfne: Numerical error encountered; proceeding with uncompiled evaluation. and will take some time to give a result...

---

One curio is that both Sum and Product can take arbitrary lists. Sum works fine

In[4]:= lst=RandomReal[2,5000000];
         Sum[i,{i,lst}]//Timing
         Total[lst]//Timing
Out[5]= {0.58,5.00039*10^6}
Out[6]= {0.02,5.00039*10^6}

but for long PackedArrays, such as the test examples here, Product fails since the automatically compiled code (in version 8.0) does not catch underflows/overflows properly:

In[7]:= lst=RandomReal[2,5000000];
         Product[i,{i,lst}]//Timing
         Times@@lst//Timing
Out[8]= {0.,Compile`AutoVar12!}
Out[9]= {2.52,1.781498881673*10^-666005}

The work around supplied by the helpful WRI tech support is to turn off the product compilation using SetSystemOptions["CompileOptions" -> {"ProductCompileLength" -> Infinity}]. Another option is to use lst=Developer`FromPackedArray[lst].

Answer 3

Simon's method is fast, but it fails on negative values. Combining it with his answer to my other question, here is a fast solution that handles negatives. Thanks, Simon.

Function

f = (-1)^(-1 /. Rule @@@ Tally@Sign@# /. -1 -> 0) * Exp@Total@Log@Abs@# &;

---

Testing

lst = RandomReal[{-50, 50}, 5000000];

Times @@ lst // Timing
f@lst // Timing

lst = Developer`FromPackedArray[lst];

Times @@ lst // Timing
f@lst // Timing

---

{0.844, -4.42943661963*10^6323240}

{0.062, -4.4294366*10^6323240}

{0.64, -4.42943661963*10^6323240}

{0.203, -4.4294366*10^6323240}