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
[akpm@linux-foundation.org: avoid #defining a CONFIG_ variable]
Signed-off-by: Zhaoxiu Zeng <zhaoxiu.zeng@gmail.com>
Signed-off-by: George Spelvin <linux@horizon.com>
Signed-off-by: Andrew Morton <akpm@linux-foundation.org>
Signed-off-by: Linus Torvalds <torvalds@linux-foundation.org>
diff --git a/arch/Kconfig b/arch/Kconfig
index 8f84fd2..b16e74e 100644
--- a/arch/Kconfig
+++ b/arch/Kconfig
@@ -647,4 +647,7 @@
config ARCH_NO_COHERENT_DMA_MMAP
bool
+config CPU_NO_EFFICIENT_FFS
+ def_bool n
+
source "kernel/gcov/Kconfig"
diff --git a/arch/alpha/Kconfig b/arch/alpha/Kconfig
index fe99f89..7f312d8 100644
--- a/arch/alpha/Kconfig
+++ b/arch/alpha/Kconfig
@@ -26,6 +26,7 @@
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,
diff --git a/arch/arc/Kconfig b/arch/arc/Kconfig
index 8894f7e..0dcbacf 100644
--- a/arch/arc/Kconfig
+++ b/arch/arc/Kconfig
@@ -107,6 +107,7 @@
config ISA_ARCOMPACT
bool "ARCompact ISA"
+ select CPU_NO_EFFICIENT_FFS
help
The original ARC ISA of ARC600/700 cores
diff --git a/arch/arm/mm/Kconfig b/arch/arm/mm/Kconfig
index 5534766..cb569b6 100644
--- a/arch/arm/mm/Kconfig
+++ b/arch/arm/mm/Kconfig
@@ -421,18 +421,21 @@
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
diff --git a/arch/h8300/Kconfig b/arch/h8300/Kconfig
index 986ea84..aa232de 100644
--- a/arch/h8300/Kconfig
+++ b/arch/h8300/Kconfig
@@ -20,6 +20,7 @@
select HAVE_KERNEL_GZIP
select HAVE_KERNEL_LZO
select HAVE_ARCH_KGDB
+ select CPU_NO_EFFICIENT_FFS
config RWSEM_GENERIC_SPINLOCK
def_bool y
diff --git a/arch/m32r/Kconfig b/arch/m32r/Kconfig
index c82b292..3cc8498 100644
--- a/arch/m32r/Kconfig
+++ b/arch/m32r/Kconfig
@@ -17,6 +17,7 @@
select ARCH_USES_GETTIMEOFFSET
select MODULES_USE_ELF_RELA
select HAVE_DEBUG_STACKOVERFLOW
+ select CPU_NO_EFFICIENT_FFS
config SBUS
bool
diff --git a/arch/m68k/Kconfig.cpu b/arch/m68k/Kconfig.cpu
index c1beb5a..8ace920 100644
--- a/arch/m68k/Kconfig.cpu
+++ b/arch/m68k/Kconfig.cpu
@@ -40,6 +40,7 @@
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
@@ -51,6 +52,7 @@
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
@@ -130,6 +132,7 @@
depends on !MMU
select COLDFIRE_SW_A7
select HAVE_MBAR
+ select CPU_NO_EFFICIENT_FFS
help
Motorola ColdFire 5206 processor support.
@@ -138,6 +141,7 @@
depends on !MMU
select COLDFIRE_SW_A7
select HAVE_MBAR
+ select CPU_NO_EFFICIENT_FFS
help
Motorola ColdFire 5206e processor support.
@@ -163,6 +167,7 @@
depends on !MMU
select COLDFIRE_SW_A7
select HAVE_MBAR
+ select CPU_NO_EFFICIENT_FFS
help
Motorola ColdFire 5249 processor support.
@@ -171,6 +176,7 @@
depends on !MMU
select COLDFIRE_SW_A7
select HAVE_MBAR
+ select CPU_NO_EFFICIENT_FFS
help
Freescale (Motorola) Coldfire 5251/5253 processor support.
@@ -189,6 +195,7 @@
depends on !MMU
select COLDFIRE_SW_A7
select HAVE_MBAR
+ select CPU_NO_EFFICIENT_FFS
help
Motorola ColdFire 5272 processor support.
@@ -217,6 +224,7 @@
select COLDFIRE_SW_A7
select HAVE_CACHE_CB
select HAVE_MBAR
+ select CPU_NO_EFFICIENT_FFS
help
Motorola ColdFire 5307 processor support.
@@ -242,6 +250,7 @@
select COLDFIRE_SW_A7
select HAVE_CACHE_CB
select HAVE_MBAR
+ select CPU_NO_EFFICIENT_FFS
help
Motorola ColdFire 5407 processor support.
@@ -251,6 +260,7 @@
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.
@@ -260,6 +270,7 @@
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.
diff --git a/arch/metag/Kconfig b/arch/metag/Kconfig
index e47a08d..5b7a45d 100644
--- a/arch/metag/Kconfig
+++ b/arch/metag/Kconfig
@@ -30,6 +30,7 @@
select OF
select OF_EARLY_FLATTREE
select SPARSE_IRQ
+ select CPU_NO_EFFICIENT_FFS
config STACKTRACE_SUPPORT
def_bool y
diff --git a/arch/microblaze/Kconfig b/arch/microblaze/Kconfig
index 3d793b5..f17c3a4 100644
--- a/arch/microblaze/Kconfig
+++ b/arch/microblaze/Kconfig
@@ -32,6 +32,7 @@
select OF_EARLY_FLATTREE
select TRACING_SUPPORT
select VIRT_TO_BUS
+ select CPU_NO_EFFICIENT_FFS
config SWAP
def_bool n
diff --git a/arch/mips/include/asm/cpu-features.h b/arch/mips/include/asm/cpu-features.h
index e6f19fc..e961c8a 100644
--- a/arch/mips/include/asm/cpu-features.h
+++ b/arch/mips/include/asm/cpu-features.h
@@ -204,6 +204,16 @@
#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
diff --git a/arch/nios2/Kconfig b/arch/nios2/Kconfig
index 87ca653..51a56c8 100644
--- a/arch/nios2/Kconfig
+++ b/arch/nios2/Kconfig
@@ -15,6 +15,7 @@
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
diff --git a/arch/openrisc/Kconfig b/arch/openrisc/Kconfig
index e118c02..142cb05 100644
--- a/arch/openrisc/Kconfig
+++ b/arch/openrisc/Kconfig
@@ -25,6 +25,7 @@
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
diff --git a/arch/parisc/Kconfig b/arch/parisc/Kconfig
index 88cfaa8..3d498a6 100644
--- a/arch/parisc/Kconfig
+++ b/arch/parisc/Kconfig
@@ -32,6 +32,7 @@
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
diff --git a/arch/s390/Kconfig b/arch/s390/Kconfig
index 1c3c43d..a8c2590 100644
--- a/arch/s390/Kconfig
+++ b/arch/s390/Kconfig
@@ -123,6 +123,7 @@
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
diff --git a/arch/score/Kconfig b/arch/score/Kconfig
index 366e1b5..507d631 100644
--- a/arch/score/Kconfig
+++ b/arch/score/Kconfig
@@ -14,6 +14,7 @@
select VIRT_TO_BUS
select MODULES_USE_ELF_REL
select CLONE_BACKWARDS
+ select CPU_NO_EFFICIENT_FFS
choice
prompt "System type"
diff --git a/arch/sh/Kconfig b/arch/sh/Kconfig
index f625434..e803a83 100644
--- a/arch/sh/Kconfig
+++ b/arch/sh/Kconfig
@@ -20,6 +20,7 @@
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
diff --git a/arch/sparc/Kconfig b/arch/sparc/Kconfig
index 1012f7f..546293d 100644
--- a/arch/sparc/Kconfig
+++ b/arch/sparc/Kconfig
@@ -42,6 +42,7 @@
select ODD_RT_SIGACTION
select OLD_SIGSUSPEND
select ARCH_HAS_SG_CHAIN
+ select CPU_NO_EFFICIENT_FFS
config SPARC32
def_bool !64BIT
diff --git a/lib/gcd.c b/lib/gcd.c
index 3657f12..135ee64 100644
--- a/lib/gcd.c
+++ b/lib/gcd.c
@@ -2,20 +2,77 @@
#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);