Better FFT sizes for large inputs.

benchmark                old ns/op    new ns/op    delta
BenchmarkMulFFT_20Mb    1137847000   1116964000   -1.84%
BenchmarkMulFFT_50Mb    3056503000   2913291000   -4.69%
BenchmarkMulFFT_100Mb   8142517000   5962457000  -26.77%

README

@@ -4,25 +4,25 @@ Number size      old ns/op    new ns/op    delta
   1kb                 1599         1668   +4.32%
  10kb                61533        62427   +1.45%
 100kb              2567995      2683862   +4.51%
-  1Mb            105237800     33648160  -68.03%
-  5Mb           1272947000    219950000  -82.72%
- 10Mb           3834354000    578960600  -84.90%
- 20Mb          11514488000   1219312000  -89.41%
- 50Mb          49199945000   3203904000  -93.49%
-100Mb         147599836000   8546686000  -94.21%
+  1Mb            105237800     28586820  -72.84%
+  5Mb           1272947000    195314800  -84.66%
+ 10Mb           3834354000    404398400  -89.45%
+ 20Mb          11514488000   1116964000  -90.30%
+ 50Mb          49199945000   2913291000  -94.08%
+100Mb         147599836000   5962457000  -95.96%
 
 Benchmarking GMP vs bigfft
 
 Number size   GMP ns/op     Go ns/op    delta
   1kb               536         1668  +211.19%
  10kb             26669        62427  +134.08%
 100kb            686813      2683862  +290.77%
-  1Mb          12100000     33648160  +178.08%
-  5Mb         111731843    219950000  +96.86%
- 10Mb         212314000    578960600  +172.69%
- 20Mb         490196000   1219312000  +148.74%
- 50Mb        1280000000   3203904000  +150.31%
-100Mb        2673000000   8546686000  +219.74%
-
+  1Mb          12100000     28586820  +136.25%
+  5Mb         111731843    195314800  +74.81%
+ 10Mb         212314000    404398400  +90.47%
+ 20Mb         490196000   1116964000  +127.86%
+ 50Mb        1280000000   2913291000  +127.60%
+100Mb        2673000000   5962457000  +123.06%
+ 
 Benchmarks were run on a Core 2 Quad Q8200 (2.33GHz).
 FFT is enabled when input numbers are over 200kbits.

benchmarks/bench.fft

@@ -1,11 +1,11 @@
 # Benchmarks using the mulFFT function only.
-BenchmarkMul_1kb	   50000	     31247 ns/op
-BenchmarkMul_10kb	    5000	    311773 ns/op
-BenchmarkMul_100kb	    1000	   2745436 ns/op
-BenchmarkMul_1Mb	      50	  29235680 ns/op
-BenchmarkMul_5Mb	      10	 198136700 ns/op
-BenchmarkMul_10Mb	       5	 522288600 ns/op
-BenchmarkMul_20Mb	       1	1125621000 ns/op
-BenchmarkMul_50Mb	       1	2942162000 ns/op
-BenchmarkMul_100Mb	       1	8000678000 ns/op
+BenchmarkMul_1kb     100000             28712 ns/op
+BenchmarkMul_10kb     10000            294135 ns/op
+BenchmarkMul_100kb     1000           2659472 ns/op
+BenchmarkMul_1Mb         50          28586820 ns/op
+BenchmarkMul_5Mb         10         195314800 ns/op
+BenchmarkMul_10Mb         5         404398400 ns/op
+BenchmarkMul_20Mb         1        1116964000 ns/op
+BenchmarkMul_50Mb         1        2913291000 ns/op
+BenchmarkMul_100Mb        1        5962457000 ns/op
 

fft.go

@@ -64,8 +64,8 @@ func fftmul(x, y nat) nat {
 // fft size i.
 var fftSizeThreshold = [...]int64{0, 0, 0,
 	4 << 10, 8 << 10, 16 << 10, // 5 
-	32 << 10, 64 << 10, 1 << 18, 1 << 20, 1 << 22, // 10
-	1 << 24, 1 << 26, 1 << 28, 1 << 30,
+	32 << 10, 64 << 10, 1 << 18, 1 << 20, 3 << 20, // 10
+	16 << 20, 50 << 20, 160 << 20, 500 << 20, 2 << 30,
 }
 
 // returns the FFT length k, m the number of words per chunk

fft_test.go

@@ -60,6 +60,7 @@ func TestFourierSizes(t *testing.T) {
 		2e5, 3e5, 5e5, 7e5, 10e5, 14e5,
 		2e6, 3e6, 5e6, 7e6, 10e6, 14e6,
 		2e7, 3e7, 5e7, 7e7, 10e7, 14e7,
+		2e8, 3e8, 5e8, 7e8, 10e8, 14e8,
 	}
 	for _, s := range sizes {
 		k, m := fftSize(make(nat, s/_W), make(nat, s/_W))