Further BigInteger performance improvements

Alan Eliasen eliasen at mindspring.com
Thu Jun 4 12:02:50 UTC 2009

   16 months ago, I posted the first of several patches to vastly
improve the performance of the BigInteger class.  These included
implementation of Karatsuba and 3-way Toom-Cook multiplication and
Karatsuba and Toom-Cook squaring.  The particular algorithm used for
multiplication or squaring is chosen according to the size of the
numbers, and are these algorithms are asymptotically faster than the
O(n^2) algorithms in previous versions of Java. (Karatsuba is
O(n^1.585), 3-way Toom Cook is O(n^1.465)).

   Unfortunately, none of these patches has yet been applied or reviewed
 in any way by Sun.  (Although they have been reviewed by some of the
top researchers in the field of high-performance multiplication.)  Nor
have I received answers to my questions about how long Sun wants
regression tests to run, size of output, etc., nor have I been provided
with promised existing regression tests for BigInteger to see if they
need improvement.  (If I can find these somewhere, please let me know.
They're not apparently in my checkout.)

   I'm still hoping to get these changes into Java 1.7, because the
performance increase is so significant, and because this is necessary to
make Java a feasible platform for number theory and numerics.

   Recently, Xiaobin Lu submitted patches to improve the performance of
BigDecimal, and also improved BigInteger's performance in the process.
Great job!

   Luckily, the overlap between our patches was negligible.  Xiaobin may
now be the person most suited to look over my patches, as he has worked
intimately in the class recently.  My fixes are further improved by his
fixes to give some truly spectacular improvements in performance for big
numbers, which should also help large BigDecimals.  I was disappointed,
though, that Sun knew about my larger performance improvements for a
long time and didn't use this opportunity to evaluate and apply them,
nor communicate with me, despite my repeated attempts and my frankly
hard work on this.

  Below, interspersed among my previous statements, are
some updated performance numbers with my patches and Xiaobin's.

   For multiplying the numbers 3^1000000 * 3^1000001,
(with my fixes to do the exponentiation hundreds of thousands of times
faster factored out; without these, JDK 1.6 would be thousands of times
slower,) the times for 20 iterations are:

   JDK 1.6                    292987 ms
   OpenJDK1.7 + my Karatsuba   28650 ms
   OpenJDK1.7 + my Toom/Cook   18054 ms
   Plus Xiaobin's improvements 12880 ms

   *** Performance improvement over Java 1.6:  22.7x

   For multiplying numbers 3^14000000 * 3^14000001, the time for 1
iteration is:

   JDK 1.6	             3499115 ms
   OpenJDK1.7 + my Karatsuba   89505 ms
   OpenJDK1.7 + my Toom/Cook   43636 ms
   Plus Xiaobin's improvements 29813 ms

   *** Performance improvement over Java 1.6:  117.3x

   You can see that operations that weren't even feasible to consider in
JDK 1.6 are now possible in Java.  Operations that used to take almost
an hour can be done in less than 30 seconds.

   This encompasses a lot of different bug reports:

4228681:  Some BigInteger operations are slow with very large numbers

   (This was closed but never fixed.)

4837946: Implement Karatsuba multiplication algorithm in BigInteger

    This is done, along with Toom-Cook multiplication.  My
implementation is intended to be easy to read, understand, and check.
It significantly improves multiplication performance for large numbers.

4646474: BigInteger.pow() algorithm slow in 1.4.0

   This is improved in many ways:

   * Rewrote pow() to use Karatsuba and Toom-Cook multiplication
   * Implemented Karatsuba squaring
   * Immplemented 3-way Toom-Cook squaring
   * Found good threshholds for Karatsuba and Toom-Cook squaring
   * Added an optimization to use left-shifting for multiples of 2 in
the base.  This improved speed by thousands of times for things like
Mersenne numbers, and may be one of the most significant improvements
for practical prgrams.

4641897: BigInteger.toString() algorithm slow for large numbers

   This algorithm uses a very inefficient algorithm for large numbers.
 I plan to replace it with a recursive divide-and-conquer algorithm
devised by Schoenhage and Strassen.  I have developed and tested this in
my own software.  This operates hundreds or thousands of times faster
than the current version for large numbers.  What takes a day in Java
1.6, I can do in 5 minutes.  It will also benefit from faster
multiplication and exponentiation.  This is likely to be the most
controversial algorithm because it has potential threading and
synchronization and caching and memory-vs-speed tradeoffs, so I'm going
to submit it separately, if I can get Sun to review the multiply and pow
patches first.

   My patches are designed to be as readable and simple as possible.
They all build on existing functions, and eschew lots of low-level
bit-fiddling, as those types of changes are harder to understand and
debug.  I think it's best to get working algorithms with better
asymptotic efficiency, as those will vastly improve performance
for large numbers, and tune them by doing more low-level bit fiddling
later if necessary.  Even without being tuned to the nth degree, the new
algorithms are vastly faster for large numbers, and identical for small

   I've been asked by Sun to submit my patches in small chunks, so I
plan to submit just the multiplication and squaring patch, and leave the
patches for pow() for later unless I hear otherwise.  I'd rather they
had it *all* and got it tested and integrated as soon as possible, after
all this waiting, and the work it takes to separate out working code to
make a smaller patch.

   I will be submitting a patch in the next day or two.  I'm running my
*huge* regression tests which produce about 250 GB of output and take at
least a day to run.  (And you have to run it twice against a known good
platform to have something to compare to... luckily I've done that.)

   For those who'd just like to replace their file with my improved
version (includes Xiaobin's patches), it's available at:

   From the queries I get, this is important to a lot of people.  The
performance of BigInteger can be improved by tens or hundreds or
thousands of times (or even more in the case of certain arguments of
pow()), and should be done to make Java a more viable platform for numerics.

   This work has been in Sun's hands for a long time, and really needs
to get into 1.7.

   Alan Eliasen
   eliasen at mindspring.com

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