avutil/mathematics: speed up av_gcd by using Stein's binary GCD algorithm

This uses Stein's binary GCD algorithm:
https://en.wikipedia.org/wiki/Binary_GCD_algorithm
to get a roughly 4x speedup over Euclidean GCD on standard architectures
with a compiler intrinsic for ctzll, and a roughly 2x speedup otherwise.
At the moment, the compiler intrinsic is used on GCC and Clang due to
its easy availability.

Quick note regarding overflow: yes, subtractions on int64_t can, but the
llabs takes care of that. The llabs is also guaranteed to be safe, with
no annoying INT64_MIN business since INT64_MIN being a power of 2, is
shifted down before being sent to llabs.

The binary GCD needs ff_ctzll, an extension of ff_ctz for long long (int64_t). On
GCC, this is provided by a built-in. On Microsoft, there is a
BitScanForward64 analog of BitScanForward that should work; but I can't confirm.
Apparently it is not available on 32 bit builds; so this may or may not
work correctly. On Intel, per the documentation there is only an
intrinsic for _bit_scan_forward and people have posted on forums
regarding _bit_scan_forward64, but often their documentation is
woeful. Again, I don't have it, so I can't test.

As such, to be safe, for now only the GCC/Clang intrinsic is added, the rest
use a compiled version based on the De-Bruijn method of Leiserson et al:
http://supertech.csail.mit.edu/papers/debruijn.pdf.

Tested with FATE, sample benchmark (x86-64, GCC 5.2.0, Haswell)
with a START_TIMER and STOP_TIMER in libavutil/rationsl.c, followed by a
make fate.

aac-am00_88.err:
builtin:
714 decicycles in av_gcd,    4095 runs,      1 skips

de-bruijn:
1440 decicycles in av_gcd,    4096 runs,      0 skips

previous:
2889 decicycles in av_gcd,    4096 runs,      0 skips

Signed-off-by: Ganesh Ajjanagadde <gajjanagadde@gmail.com>
Signed-off-by: Michael Niedermayer <michael@niedermayer.cc>

libavutil/intmath.h

@@ -114,6 +114,9 @@ static av_always_inline av_const int ff_log2_16bit_c(unsigned int v)
 #ifndef ff_ctz
 #define ff_ctz(v) __builtin_ctz(v)
 #endif
+#ifndef ff_ctzll
+#define ff_ctzll(v) __builtin_ctzll(v)
+#endif
 #endif
 #endif
 
@@ -158,6 +161,22 @@ static av_always_inline av_const int ff_ctz_c( int v )
 #endif
 #endif
 
+#ifndef ff_ctzll
+#define ff_ctzll ff_ctzll_c
+/* We use the De-Bruijn method outlined in:
+ * http://supertech.csail.mit.edu/papers/debruijn.pdf. */
+static av_always_inline av_const int ff_ctzll_c(long long v)
+{
+    static const int debruijn_ctz64[64] = {
+        0, 1, 2, 53, 3, 7, 54, 27, 4, 38, 41, 8, 34, 55, 48, 28,
+        62, 5, 39, 46, 44, 42, 22, 9, 24, 35, 59, 56, 49, 18, 29, 11,
+        63, 52, 6, 26, 37, 40, 33, 47, 61, 45, 43, 21, 23, 58, 17, 10,
+        51, 25, 36, 32, 60, 20, 57, 16, 50, 31, 19, 15, 30, 14, 13, 12
+    };
+    return debruijn_ctz64[(uint64_t)((v & -v) * 0x022FDD63CC95386D) >> 58];
+}
+#endif
+
 /**
  * Trailing zero bit count.
  *

libavutil/mathematics.c

@@ -27,16 +27,32 @@
 #include <limits.h>
 
 #include "mathematics.h"
+#include "libavutil/intmath.h"
 #include "libavutil/common.h"
 #include "avassert.h"
 #include "version.h"
 
-int64_t av_gcd(int64_t a, int64_t b)
-{
-    if (b)
-        return av_gcd(b, a % b);
-    else
+/* Stein's binary GCD algorithm:
+ * https://en.wikipedia.org/wiki/Binary_GCD_algorithm */
+int64_t av_gcd(int64_t a, int64_t b) {
+    int za, zb, k;
+    int64_t u, v;
+    if (a == 0)
+        return b;
+    if (b == 0)
         return a;
+    za = ff_ctzll(a);
+    zb = ff_ctzll(b);
+    k  = FFMIN(za, zb);
+    u = llabs(a >> za);
+    v = llabs(b >> zb);
+    while (u != v) {
+        if (u > v)
+            FFSWAP(int64_t, v, u);
+        v -= u;
+        v >>= ff_ctzll(v);
+    }
+    return u << k;
 }
 
 int64_t av_rescale_rnd(int64_t a, int64_t b, int64_t c, enum AVRounding rnd)