RFR: JDK-8277175 : Add a parallel multiply method to BigInteger [v2]

[kabutz]

> BigInteger currently uses three different algorithms for multiply. The simple quadratic algorithm, then the slightly better Karatsuba if we exceed a bit count and then Toom Cook 3 once we go into the several thousands of bits. Since Toom Cook 3 is a recursive algorithm, it is trivial to parallelize it. I have demonstrated this several times in conference talks. In order to be consistent with other classes such as Arrays and Collection, I have added a parallelMultiply() method. Internally we have added a parameter to the private multiply method to indicate whether the calculation should be done in parallel.
> 
> The performance improvements are as should be expected. Fibonacci of 100 million (using a single-threaded Dijkstra's sum of squares version) completes in 9.2 seconds with the parallelMultiply() vs 25.3 seconds with the sequential multiply() method. This is on my 1-8-2 laptop. The final multiplications are with very large numbers, which then benefit from the parallelization of Toom-Cook 3. Fibonacci 100 million is a 347084 bit number.
> 
> We have also parallelized the private square() method. Internally, the square() method defaults to be sequential.
> 
> Some benchmark results, run on my 1-6-2 server:
> 
> 
> Benchmark                                          (n)  Mode  Cnt      Score      Error  Units
> BigIntegerParallelMultiply.multiply            1000000    ss    4     51.707 ±   11.194  ms/op
> BigIntegerParallelMultiply.multiply           10000000    ss    4    988.302 ±  235.977  ms/op
> BigIntegerParallelMultiply.multiply          100000000    ss    4  24662.063 ± 1123.329  ms/op
> BigIntegerParallelMultiply.parallelMultiply    1000000    ss    4     49.337 ±   26.611  ms/op
> BigIntegerParallelMultiply.parallelMultiply   10000000    ss    4    527.560 ±  268.903  ms/op
> BigIntegerParallelMultiply.parallelMultiply  100000000    ss    4   9076.551 ± 1899.444  ms/op
> 
> 
> We can see that for larger calculations (fib 100m), the execution is 2.7x faster in parallel. For medium size (fib 10m) it is 1.873x faster. And for small (fib 1m) it is roughly the same. Considering that the fibonacci algorithm that we used was in itself sequential, and that the last 3 calculations would dominate, 2.7x faster should probably be considered quite good on a 1-6-2 machine.

kabutz has updated the pull request incrementally with one additional commit since the last revision:

  Reordered import statements to be consistent with canonical order

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Changes:
  - all: https://git.openjdk.java.net/jdk/pull/6409/files
  - new: https://git.openjdk.java.net/jdk/pull/6409/files/10faf938..66cec9bd

Webrevs:
 - full: https://webrevs.openjdk.java.net/?repo=jdk&pr=6409&range=01
 - incr: https://webrevs.openjdk.java.net/?repo=jdk&pr=6409&range=00-01

  Stats: 12 lines in 1 file changed: 6 ins; 6 del; 0 mod
  Patch: https://git.openjdk.java.net/jdk/pull/6409.diff
  Fetch: git fetch https://git.openjdk.java.net/jdk pull/6409/head:pull/6409

PR: https://git.openjdk.java.net/jdk/pull/6409