lib/GCD.c: use binary GCD algorithm instead of Euclidean

The binary GCD algorithm is based on the following facts:
1. If a and b are all evens, then gcd(a,b) = 2 * gcd(a/2, b/2)
2. If a is even and b is odd, then gcd(a,b) = gcd(a/2, b)
3. If a and b are all odds, then gcd(a,b) = gcd((a-b)/2, b) = gcd((a+b)/2, b)

Even on x86 machines with reasonable division hardware, the binary
algorithm runs about 25% faster (80% the execution time) than the
division-based Euclidian algorithm.

On platforms like Alpha and ARMv6 where division is a function call to
emulation code, it's even more significant.

There are two variants of the code here, depending on whether a fast
__ffs (find least significant set bit) instruction is available. This
allows the unpredictable branches in the bit-at-a-time shifting loop to
be eliminated.

If fast __ffs is not available, the "even/odd" GCD variant is used.

I use the following code to benchmark:

#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <string.h>
#include <time.h>
#include <unistd.h>

#define swap(a, b) \
do { \
a ^= b; \
b ^= a; \
a ^= b; \
} while (0)

unsigned long gcd0(unsigned long a, unsigned long b)
{
unsigned long r;

if (a < b) {
swap(a, b);
}

if (b == 0)
return a;

while ((r = a % b) != 0) {
a = b;
b = r;
}

return b;
}

unsigned long gcd1(unsigned long a, unsigned long b)
{
unsigned long r = a | b;

if (!a || !b)
return r;

b >>= __builtin_ctzl(b);

for (;;) {
a >>= __builtin_ctzl(a);
if (a == b)
return a << __builtin_ctzl(r);

if (a < b)
swap(a, b);
a -= b;
}
}

unsigned long gcd2(unsigned long a, unsigned long b)
{
unsigned long r = a | b;

if (!a || !b)
return r;

r &= -r;

while (!(b & r))
b >>= 1;

for (;;) {
while (!(a & r))
a >>= 1;
if (a == b)
return a;

if (a < b)
swap(a, b);
a -= b;
a >>= 1;
if (a & r)
a += b;
a >>= 1;
}
}

unsigned long gcd3(unsigned long a, unsigned long b)
{
unsigned long r = a | b;

if (!a || !b)
return r;

b >>= __builtin_ctzl(b);
if (b == 1)
return r & -r;

for (;;) {
a >>= __builtin_ctzl(a);
if (a == 1)
return r & -r;
if (a == b)
return a << __builtin_ctzl(r);

if (a < b)
swap(a, b);
a -= b;
}
}

unsigned long gcd4(unsigned long a, unsigned long b)
{
unsigned long r = a | b;

if (!a || !b)
return r;

r &= -r;

while (!(b & r))
b >>= 1;
if (b == r)
return r;

for (;;) {
while (!(a & r))
a >>= 1;
if (a == r)
return r;
if (a == b)
return a;

if (a < b)
swap(a, b);
a -= b;
a >>= 1;
if (a & r)
a += b;
a >>= 1;
}
}

static unsigned long (*gcd_func[])(unsigned long a, unsigned long b) = {
gcd0, gcd1, gcd2, gcd3, gcd4,
};

#define TEST_ENTRIES (sizeof(gcd_func) / sizeof(gcd_func[0]))

#if defined(__x86_64__)

#define rdtscll(val) do { \
unsigned long __a,__d; \
__asm__ __volatile__("rdtsc" : "=a" (__a), "=d" (__d)); \
(val) = ((unsigned long long)__a) | (((unsigned long long)__d)<<32); \
} while(0)

static unsigned long long benchmark_gcd_func(unsigned long (*gcd)(unsigned long, unsigned long),
unsigned long a, unsigned long b, unsigned long *res)
{
unsigned long long start, end;
unsigned long long ret;
unsigned long gcd_res;

rdtscll(start);
gcd_res = gcd(a, b);
rdtscll(end);

if (end >= start)
ret = end - start;
else
ret = ~0ULL - start + 1 + end;

*res = gcd_res;
return ret;
}

#else

static inline struct timespec read_time(void)
{
struct timespec time;
clock_gettime(CLOCK_PROCESS_CPUTIME_ID, &time);
return time;
}

static inline unsigned long long diff_time(struct timespec start, struct timespec end)
{
struct timespec temp;

if ((end.tv_nsec - start.tv_nsec) < 0) {
temp.tv_sec = end.tv_sec - start.tv_sec - 1;
temp.tv_nsec = 1000000000ULL + end.tv_nsec - start.tv_nsec;
} else {
temp.tv_sec = end.tv_sec - start.tv_sec;
temp.tv_nsec = end.tv_nsec - start.tv_nsec;
}

return temp.tv_sec * 1000000000ULL + temp.tv_nsec;
}

static unsigned long long benchmark_gcd_func(unsigned long (*gcd)(unsigned long, unsigned long),
unsigned long a, unsigned long b, unsigned long *res)
{
struct timespec start, end;
unsigned long gcd_res;

start = read_time();
gcd_res = gcd(a, b);
end = read_time();

*res = gcd_res;
return diff_time(start, end);
}

#endif

static inline unsigned long get_rand()
{
if (sizeof(long) == 8)
return (unsigned long)rand() << 32 | rand();
else
return rand();
}

int main(int argc, char **argv)
{
unsigned int seed = time(0);
int loops = 100;
int repeats = 1000;
unsigned long (*res)[TEST_ENTRIES];
unsigned long long elapsed[TEST_ENTRIES];
int i, j, k;

for (;;) {
int opt = getopt(argc, argv, "n:r:s:");
/* End condition always first */
if (opt == -1)
break;

switch (opt) {
case 'n':
loops = atoi(optarg);
break;
case 'r':
repeats = atoi(optarg);
break;
case 's':
seed = strtoul(optarg, NULL, 10);
break;
default:
/* You won't actually get here. */
break;
}
}

res = malloc(sizeof(unsigned long) * TEST_ENTRIES * loops);
memset(elapsed, 0, sizeof(elapsed));

srand(seed);
for (j = 0; j < loops; j++) {
unsigned long a = get_rand();
/* Do we have args? */
unsigned long b = argc > optind ? strtoul(argv[optind], NULL, 10) : get_rand();
unsigned long long min_elapsed[TEST_ENTRIES];
for (k = 0; k < repeats; k++) {
for (i = 0; i < TEST_ENTRIES; i++) {
unsigned long long tmp = benchmark_gcd_func(gcd_func[i], a, b, &res[j][i]);
if (k == 0 || min_elapsed[i] > tmp)
min_elapsed[i] = tmp;
}
}
for (i = 0; i < TEST_ENTRIES; i++)
elapsed[i] += min_elapsed[i];
}

for (i = 0; i < TEST_ENTRIES; i++)
printf("gcd%d: elapsed %llu\n", i, elapsed[i]);

k = 0;
srand(seed);
for (j = 0; j < loops; j++) {
unsigned long a = get_rand();
unsigned long b = argc > optind ? strtoul(argv[optind], NULL, 10) : get_rand();
for (i = 1; i < TEST_ENTRIES; i++) {
if (res[j][i] != res[j][0])
break;
}
if (i < TEST_ENTRIES) {
if (k == 0) {
k = 1;
fprintf(stderr, "Error:\n");
}
fprintf(stderr, "gcd(%lu, %lu): ", a, b);
for (i = 0; i < TEST_ENTRIES; i++)
fprintf(stderr, "%ld%s", res[j][i], i < TEST_ENTRIES - 1 ? ", " : "\n");
}
}

if (k == 0)
fprintf(stderr, "PASS\n");

free(res);

return 0;
}

Compiled with "-O2", on "VirtualBox 4.4.0-22-generic #38-Ubuntu x86_64" got:

zhaoxiuzeng@zhaoxiuzeng-VirtualBox:~/develop$ ./gcd -r 500000 -n 10
gcd0: elapsed 10174
gcd1: elapsed 2120
gcd2: elapsed 2902
gcd3: elapsed 2039
gcd4: elapsed 2812
PASS
zhaoxiuzeng@zhaoxiuzeng-VirtualBox:~/develop$ ./gcd -r 500000 -n 10
gcd0: elapsed 9309
gcd1: elapsed 2280
gcd2: elapsed 2822
gcd3: elapsed 2217
gcd4: elapsed 2710
PASS
zhaoxiuzeng@zhaoxiuzeng-VirtualBox:~/develop$ ./gcd -r 500000 -n 10
gcd0: elapsed 9589
gcd1: elapsed 2098
gcd2: elapsed 2815
gcd3: elapsed 2030
gcd4: elapsed 2718
PASS
zhaoxiuzeng@zhaoxiuzeng-VirtualBox:~/develop$ ./gcd -r 500000 -n 10
gcd0: elapsed 9914
gcd1: elapsed 2309
gcd2: elapsed 2779
gcd3: elapsed 2228
gcd4: elapsed 2709
PASS

[[e-mail]: avoid #defining a CONFIG_ variable]
Signed-off-by: Zhaoxiu Zeng <[e-mail]>
Signed-off-by: George Spelvin <[e-mail]>
Signed-off-by: Andrew Morton <[e-mail]>
Signed-off-by: Linus Torvalds <[e-mail]>

arch/Kconfig

···
 config ARCH_NO_COHERENT_DMA_MMAP
 	bool
 
+config CPU_NO_EFFICIENT_FFS
+	def_bool n
+
 source "kernel/gcov/Kconfig"

arch/alpha/Kconfig

···
 	select MODULES_USE_ELF_RELA
 	select ODD_RT_SIGACTION
 	select OLD_SIGSUSPEND
+	select CPU_NO_EFFICIENT_FFS if !ALPHA_EV67
 	help
 	  The Alpha is a 64-bit general-purpose processor designed and
 	  marketed by the Digital Equipment Corporation of blessed memory,

arch/arc/Kconfig

···
 
 config ISA_ARCOMPACT
 	bool "ARCompact ISA"
+	select CPU_NO_EFFICIENT_FFS
 	help
 	  The original ARC ISA of ARC600/700 cores
 

arch/arm/mm/Kconfig

···
 	select CPU_USE_DOMAINS if MMU
 	select NEED_KUSER_HELPERS
 	select TLS_REG_EMUL if SMP || !MMU
+	select CPU_NO_EFFICIENT_FFS
 
 config CPU_32v4
 	bool
 	select CPU_USE_DOMAINS if MMU
 	select NEED_KUSER_HELPERS
 	select TLS_REG_EMUL if SMP || !MMU
+	select CPU_NO_EFFICIENT_FFS
 
 config CPU_32v4T
 	bool
 	select CPU_USE_DOMAINS if MMU
 	select NEED_KUSER_HELPERS
 	select TLS_REG_EMUL if SMP || !MMU
+	select CPU_NO_EFFICIENT_FFS
 
 config CPU_32v5
 	bool

arch/h8300/Kconfig

···
 	select HAVE_KERNEL_GZIP
 	select HAVE_KERNEL_LZO
 	select HAVE_ARCH_KGDB
+	select CPU_NO_EFFICIENT_FFS
 
 config RWSEM_GENERIC_SPINLOCK
 	def_bool y

arch/m32r/Kconfig

···
 	select ARCH_USES_GETTIMEOFFSET
 	select MODULES_USE_ELF_RELA
 	select HAVE_DEBUG_STACKOVERFLOW
+	select CPU_NO_EFFICIENT_FFS
 
 config SBUS
 	bool

arch/m68k/Kconfig.cpu

···
 	select CPU_HAS_NO_MULDIV64
 	select CPU_HAS_NO_UNALIGNED
 	select GENERIC_CSUM
+	select CPU_NO_EFFICIENT_FFS
 	help
 	  The Freescale (was Motorola) 68000 CPU is the first generation of
 	  the well known M68K family of processors. The CPU core as well as
···
 	bool
 	select CPU_HAS_NO_BITFIELDS
 	select CPU_HAS_NO_UNALIGNED
+	select CPU_NO_EFFICIENT_FFS
 	help
 	  The Freescale (was then Motorola) CPU32 is a CPU core that is
 	  based on the 68020 processor. For the most part it is used in
···
 	depends on !MMU
 	select COLDFIRE_SW_A7
 	select HAVE_MBAR
+	select CPU_NO_EFFICIENT_FFS
 	help
 	  Motorola ColdFire 5206 processor support.
 
···
 	depends on !MMU
 	select COLDFIRE_SW_A7
 	select HAVE_MBAR
+	select CPU_NO_EFFICIENT_FFS
 	help
 	  Motorola ColdFire 5206e processor support.
 
···
 	depends on !MMU
 	select COLDFIRE_SW_A7
 	select HAVE_MBAR
+	select CPU_NO_EFFICIENT_FFS
 	help
 	  Motorola ColdFire 5249 processor support.
 
···
 	depends on !MMU
 	select COLDFIRE_SW_A7
 	select HAVE_MBAR
+	select CPU_NO_EFFICIENT_FFS
 	help
 	  Freescale (Motorola) Coldfire 5251/5253 processor support.
 
···
 	depends on !MMU
 	select COLDFIRE_SW_A7
 	select HAVE_MBAR
+	select CPU_NO_EFFICIENT_FFS
 	help
 	  Motorola ColdFire 5272 processor support.
 
···
 	select COLDFIRE_SW_A7
 	select HAVE_CACHE_CB
 	select HAVE_MBAR
+	select CPU_NO_EFFICIENT_FFS
 	help
 	  Motorola ColdFire 5307 processor support.
 
···
 	select COLDFIRE_SW_A7
 	select HAVE_CACHE_CB
 	select HAVE_MBAR
+	select CPU_NO_EFFICIENT_FFS
 	help
 	  Motorola ColdFire 5407 processor support.
 
···
 	select MMU_COLDFIRE if MMU
 	select HAVE_CACHE_CB
 	select HAVE_MBAR
+	select CPU_NO_EFFICIENT_FFS
 	help
 	  Freescale ColdFire 5470/5471/5472/5473/5474/5475 processor support.
 
···
 	select M54xx
 	select HAVE_CACHE_CB
 	select HAVE_MBAR
+	select CPU_NO_EFFICIENT_FFS
 	help
 	  Freescale ColdFire 5480/5481/5482/5483/5484/5485 processor support.
 

arch/metag/Kconfig

···
 	select OF
 	select OF_EARLY_FLATTREE
 	select SPARSE_IRQ
+	select CPU_NO_EFFICIENT_FFS
 
 config STACKTRACE_SUPPORT
 	def_bool y

arch/microblaze/Kconfig

···
 	select OF_EARLY_FLATTREE
 	select TRACING_SUPPORT
 	select VIRT_TO_BUS
+	select CPU_NO_EFFICIENT_FFS
 
 config SWAP
 	def_bool n

arch/mips/include/asm/cpu-features.h

···
 #endif
 #endif
 
+/* __builtin_constant_p(cpu_has_mips_r) && cpu_has_mips_r */
+#if !((defined(cpu_has_mips32r1) && cpu_has_mips32r1) || \
+	  (defined(cpu_has_mips32r2) && cpu_has_mips32r2) || \
+	  (defined(cpu_has_mips32r6) && cpu_has_mips32r6) || \
+	  (defined(cpu_has_mips64r1) && cpu_has_mips64r1) || \
+	  (defined(cpu_has_mips64r2) && cpu_has_mips64r2) || \
+	  (defined(cpu_has_mips64r6) && cpu_has_mips64r6))
+#define CPU_NO_EFFICIENT_FFS 1
+#endif
+
 #ifndef cpu_has_mips_1
 # define cpu_has_mips_1		(!cpu_has_mips_r6)
 #endif

arch/nios2/Kconfig

···
 	select SOC_BUS
 	select SPARSE_IRQ
 	select USB_ARCH_HAS_HCD if USB_SUPPORT
+	select CPU_NO_EFFICIENT_FFS
 
 config GENERIC_CSUM
 	def_bool y

arch/openrisc/Kconfig

···
 	select MODULES_USE_ELF_RELA
 	select HAVE_DEBUG_STACKOVERFLOW
 	select OR1K_PIC
+	select CPU_NO_EFFICIENT_FFS if !OPENRISC_HAVE_INST_FF1
 
 config MMU
 	def_bool y

arch/parisc/Kconfig

···
 	select HAVE_ARCH_AUDITSYSCALL
 	select HAVE_ARCH_SECCOMP_FILTER
 	select ARCH_NO_COHERENT_DMA_MMAP
+	select CPU_NO_EFFICIENT_FFS
 
 	help
 	  The PA-RISC microprocessor is designed by Hewlett-Packard and used

arch/s390/Kconfig

···
 	select HAVE_ARCH_AUDITSYSCALL
 	select HAVE_ARCH_EARLY_PFN_TO_NID
 	select HAVE_ARCH_JUMP_LABEL
+	select CPU_NO_EFFICIENT_FFS if !HAVE_MARCH_Z9_109_FEATURES
 	select HAVE_ARCH_SECCOMP_FILTER
 	select HAVE_ARCH_SOFT_DIRTY
 	select HAVE_ARCH_TRACEHOOK

arch/score/Kconfig

···
 	select VIRT_TO_BUS
 	select MODULES_USE_ELF_REL
 	select CLONE_BACKWARDS
+	select CPU_NO_EFFICIENT_FFS
 
 choice
 	prompt "System type"

arch/sh/Kconfig

···
 	select PERF_USE_VMALLOC
 	select HAVE_DEBUG_KMEMLEAK
 	select HAVE_KERNEL_GZIP
+	select CPU_NO_EFFICIENT_FFS
 	select HAVE_KERNEL_BZIP2
 	select HAVE_KERNEL_LZMA
 	select HAVE_KERNEL_XZ

arch/sparc/Kconfig

···
 	select ODD_RT_SIGACTION
 	select OLD_SIGSUSPEND
 	select ARCH_HAS_SG_CHAIN
+	select CPU_NO_EFFICIENT_FFS
 
 config SPARC32
 	def_bool !64BIT

lib/gcd.c

···
 #include <linux/gcd.h>
 #include <linux/export.h>
 
-/* Greatest common divisor */
+/*
+ * This implements the binary GCD algorithm. (Often attributed to Stein,
+ * but as Knuth has noted, appears in a first-century Chinese math text.)
+ *
+ * This is faster than the division-based algorithm even on x86, which
+ * has decent hardware division.
+ */
+
+#if !defined(CONFIG_CPU_NO_EFFICIENT_FFS) && !defined(CPU_NO_EFFICIENT_FFS)
+
+/* If __ffs is available, the even/odd algorithm benchmarks slower. */
 unsigned long gcd(unsigned long a, unsigned long b)
 {
-	unsigned long r;
+	unsigned long r = a | b;
 
-	if (a < b)
-		swap(a, b);
+	if (!a || !b)
+		return r;
 
-	if (!b)
-		return a;
-	while ((r = a % b) != 0) {
-		a = b;
-		b = r;
+	b >>= __ffs(b);
+	if (b == 1)
+		return r & -r;
+
+	for (;;) {
+		a >>= __ffs(a);
+		if (a == 1)
+			return r & -r;
+		if (a == b)
+			return a << __ffs(r);
+
+		if (a < b)
+			swap(a, b);
+		a -= b;
 	}
-	return b;
 }
+
+#else
+
+/* If normalization is done by loops, the even/odd algorithm is a win. */
+unsigned long gcd(unsigned long a, unsigned long b)
+{
+	unsigned long r = a | b;
+
+	if (!a || !b)
+		return r;
+
+	/* Isolate lsbit of r */
+	r &= -r;
+
+	while (!(b & r))
+		b >>= 1;
+	if (b == r)
+		return r;
+
+	for (;;) {
+		while (!(a & r))
+			a >>= 1;
+		if (a == r)
+			return r;
+		if (a == b)
+			return a;
+
+		if (a < b)
+			swap(a, b);
+		a -= b;
+		a >>= 1;
+		if (a & r)
+			a += b;
+		a >>= 1;
+	}
+}
+
+#endif
+
 EXPORT_SYMBOL_GPL(gcd);