factor: 30% faster trial division (better sieve)

function                                             old     new   delta
packed_wheel                                           -     192    +192
factor_main                                          108     176     +68
factorize                                            284     218     -66
------------------------------------------------------------------------------
(add/remove: 1/0 grow/shrink: 1/1 up/down: 260/-66)           Total: 194 bytes

Signed-off-by: Denys Vlasenko <vda.linux@googlemail.com>
This commit is contained in:
Denys Vlasenko 2020-12-20 21:37:29 +01:00
parent bb4e32befa
commit f079f91371

View File

@ -19,6 +19,7 @@
//usage: "Print prime factors"
#include "libbb.h"
#include "common_bufsiz.h"
#if 0
# define dbg(...) bb_error_msg(__VA_ARGS__)
@ -42,6 +43,83 @@ typedef unsigned long half_t;
#error Cant find an integer type which is half as wide as ullong
#endif
/* The trial divisor increment wheel. Use it to skip over divisors that
* are composites of 2, 3, 5, 7, or 11.
* Larger wheels improve sieving only slightly, but quickly grow in size
* (adding just one prime, 13, results in 5766 element sieve).
*/
#define R(a,b,c,d,e,f,g,h,i,j,A,B,C,D,E,F,G,H,I,J) \
(((uint64_t)(a<<0) | (b<<3) | (c<<6) | (d<<9) | (e<<12) | (f<<15) | (g<<18) | (h<<21) | (i<<24) | (j<<27)) ) | \
(((uint64_t)(A<<0) | (B<<3) | (C<<6) | (D<<9) | (E<<12) | (F<<15) | (G<<18) | (H<<21) | (I<<24) | (J<<27)) << 30) | \
(((uint64_t)7 << 60))
#define P(a,b,c,d,e,f,g,h,i,j,A,B,C,D,E,F,G,H,I,J) \
R( (a/2-1),(b/2-1),(c/2-1),(d/2-1),(e/2-1),(f/2-1),(g/2-1),(h/2-1),(i/2-1),(j/2-1), \
(A/2-1),(B/2-1),(C/2-1),(D/2-1),(E/2-1),(F/2-1),(G/2-1),(H/2-1),(I/2-1),(J/2-1) )
static const uint64_t packed_wheel[] = {
/*1, 2, 2, 4, 2,*/
P( 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4), //01
P( 2, 4, 2, 4,14, 4, 6, 2,10, 2, 6, 6, 4, 2, 4, 6, 2,10, 2, 4), //02
P( 2,12,10, 2, 4, 2, 4, 6, 2, 6, 4, 6, 6, 6, 2, 6, 4, 2, 6, 4), //03
P( 6, 8, 4, 2, 4, 6, 8, 6,10, 2, 4, 6, 2, 6, 6, 4, 2, 4, 6, 2), //04
P( 6, 4, 2, 6,10, 2,10, 2, 4, 2, 4, 6, 8, 4, 2, 4,12, 2, 6, 4), //05
P( 2, 6, 4, 6,12, 2, 4, 2, 4, 8, 6, 4, 6, 2, 4, 6, 2, 6,10, 2), //06
P( 4, 6, 2, 6, 4, 2, 4, 2,10, 2,10, 2, 4, 6, 6, 2, 6, 6, 4, 6), //07
P( 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 6, 4, 8, 6, 4, 6, 2, 4, 6), //08
P( 8, 6, 4, 2,10, 2, 6, 4, 2, 4, 2,10, 2,10, 2, 4, 2, 4, 8, 6), //09
P( 4, 2, 4, 6, 6, 2, 6, 4, 8, 4, 6, 8, 4, 2, 4, 2, 4, 8, 6, 4), //10
P( 6, 6, 6, 2, 6, 6, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2,10, 2,10, 2), //11
P( 6, 4, 6, 2, 6, 4, 2, 4, 6, 6, 8, 4, 2, 6,10, 8, 4, 2, 4, 2), //12
P( 4, 8,10, 6, 2, 4, 8, 6, 6, 4, 2, 4, 6, 2, 6, 4, 6, 2,10, 2), //13
P(10, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 6, 6, 4, 6, 8), //14
P( 4, 2, 4, 2, 4, 8, 6, 4, 8, 4, 6, 2, 6, 6, 4, 2, 4, 6, 8, 4), //15
P( 2, 4, 2,10, 2,10, 2, 4, 2, 4, 6, 2,10, 2, 4, 6, 8, 6, 4, 2), //16
P( 6, 4, 6, 8, 4, 6, 2, 4, 8, 6, 4, 6, 2, 4, 6, 2, 6, 6, 4, 6), //17
P( 6, 2, 6, 6, 4, 2,10, 2,10, 2, 4, 2, 4, 6, 2, 6, 4, 2,10, 6), //18
P( 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2,12, 6, 4, 6, 2, 4, 6, 2), //19
P(12, 4, 2, 4, 8, 6, 4, 2, 4, 2,10, 2,10, 6, 2, 4, 6, 2, 6, 4), //20
P( 2, 4, 6, 6, 2, 6, 4, 2,10, 6, 8, 6, 4, 2, 4, 8, 6, 4, 6, 2), //21
P( 4, 6, 2, 6, 6, 6, 4, 6, 2, 6, 4, 2, 4, 2,10,12, 2, 4, 2,10), //22
P( 2, 6, 4, 2, 4, 6, 6, 2,10, 2, 6, 4,14, 4, 2, 4, 2, 4, 8, 6), //23
P( 4, 6, 2, 4, 6, 2, 6, 6, 4, 2, 4, 6, 2, 6, 4, 2, 4,12, 2,12), //24
};
#undef P
#undef r
#define WHEEL_START 5
#define WHEEL_SIZE (5 + 24 * 20)
#define wheel_tab ((uint8_t*)&bb_common_bufsiz1)
/*
* Why, you ask?
* plain byte array:
* function old new delta
* wheel_tab - 485 +485
* 3-bit-packed insanity:
* packed_wheel - 192 +192
* factor_main 108 176 +68
*/
static void unpack_wheel(void)
{
int i;
uint8_t *p;
setup_common_bufsiz();
wheel_tab[0] = 1;
wheel_tab[1] = 2;
wheel_tab[2] = 2;
wheel_tab[3] = 4;
wheel_tab[4] = 2;
p = &wheel_tab[5];
for (i = 0; i < ARRAY_SIZE(packed_wheel); i++) {
uint64_t v = packed_wheel[i];
do {
*p = ((unsigned)(v & 7) + 1) * 2;
//printf("%2u,", *p);
p++;
v >>= 3;
} while ((unsigned)v != 7);
//printf("\n");
}
}
static half_t isqrt_odd(wide_t N)
{
half_t s = isqrt(N);
@ -53,43 +131,20 @@ static half_t isqrt_odd(wide_t N)
static NOINLINE void factorize(wide_t N)
{
unsigned w;
half_t factor;
half_t max_factor;
// unsigned count3;
// unsigned count5;
// unsigned count7;
// ^^^^^^^^^^^^^^^ commented-out simple sieving code (easier to grasp).
// Faster sieving, using one word for potentially up to 6 counters:
// count upwards in each mask, counter "triggers" when it sets its mask to "100[0]..."
// 10987654321098765432109876543210 - bits 31-0 in 32-bit word
// 17777713333311111777775555333 - bit masks for counters for primes 3,5,7,11,13,17
// 100000100001000010001001 - value for adding 1 to each mask
// 10000010000010000100001000100 - value for checking that any mask reached msb
enum {
SHIFT_3 = 1 << 0,
SHIFT_5 = 1 << 3,
SHIFT_7 = 1 << 7,
INCREMENT_EACH = SHIFT_3 | SHIFT_5 | SHIFT_7,
MULTIPLE_OF_3 = 1 << 2,
MULTIPLE_OF_5 = 1 << 6,
MULTIPLE_OF_7 = 1 << 11,
MULTIPLE_DETECTED = MULTIPLE_OF_3 | MULTIPLE_OF_5 | MULTIPLE_OF_7,
};
unsigned sieve_word;
if (N < 4)
goto end;
while (!(N & 1)) {
printf(" 2");
N >>= 1;
}
/* The code needs to be optimized for the case where
* there are large prime factors. For example,
* this is not hard:
* 8262075252869367027 = 3 7 17 23 47 101 113 127 131 137 823
* (the largest factor to test is only ~sqrt(823) = 28)
* (the largest divisor to test for largest factor 823
* is only ~sqrt(823) = 28, the entire factorization needs
* only ~33 trial divisions)
* but this is:
* 18446744073709551601 = 53 348051774975651917
* the last factor requires testing up to
@ -98,20 +153,11 @@ static NOINLINE void factorize(wide_t N)
* factor 18446744073709551557 (0xffffffffffffffc5).
*/
max_factor = isqrt_odd(N);
// count3 = 3;
// count5 = 6;
// count7 = 9;
sieve_word = 0
/* initial count for SHIFT_n is (n-1)/2*3: */
+ (MULTIPLE_OF_3 - 3 * SHIFT_3)
+ (MULTIPLE_OF_5 - 6 * SHIFT_5)
+ (MULTIPLE_OF_7 - 9 * SHIFT_7)
//+ (MULTIPLE_OF_11 - 15 * SHIFT_11)
//+ (MULTIPLE_OF_13 - 18 * SHIFT_13)
//+ (MULTIPLE_OF_17 - 24 * SHIFT_17)
;
factor = 3;
factor = 2;
w = 0;
for (;;) {
half_t fw;
/* The division is the most costly part of the loop.
* On 64bit CPUs, takes at best 12 cycles, often ~20.
*/
@ -120,44 +166,16 @@ static NOINLINE void factorize(wide_t N)
printf(" %"HALF_FMT"u", factor);
max_factor = isqrt_odd(N);
}
next_factor:
if (factor >= max_factor)
break;
factor += 2;
/* Rudimentary wheel sieving: skip multiples of 3, 5 and 7:
* Every third odd number is divisible by three and thus isn't a prime:
* 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47...
* ^ ^ ^ ^ ^ ^ ^ _ ^ ^ _ ^ ^ ^ ^
* (^ = primes, _ = would-be-primes-if-not-divisible-by-5)
* The numbers with space under them are excluded by sieve 3.
*/
// count7--;
// count5--;
// count3--;
// if (count3 && count5 && count7)
// continue;
sieve_word += INCREMENT_EACH;
if (!(sieve_word & MULTIPLE_DETECTED))
fw = factor + wheel_tab[w];
if (fw < factor)
break; /* overflow */
factor = fw;
w++;
if (w < WHEEL_SIZE)
continue;
/*
* "factor" is multiple of 3 33% of the time (count3 reached 0),
* else, multiple of 5 13% of the time,
* else, multiple of 7 7.6% of the time.
* Cumulatively, with 3,5,7 sieving we are here 54.3% of the time.
*/
// if (count3 == 0)
// count3 = 3;
if (sieve_word & MULTIPLE_OF_3)
sieve_word -= SHIFT_3 * 3;
// if (count5 == 0)
// count5 = 5;
if (sieve_word & MULTIPLE_OF_5)
sieve_word -= SHIFT_5 * 5;
// if (count7 == 0)
// count7 = 7;
if (sieve_word & MULTIPLE_OF_7)
sieve_word -= SHIFT_7 * 7;
goto next_factor;
w = WHEEL_START;
}
end:
if (N > 1)
@ -182,6 +200,8 @@ static void factorize_numstr(const char *numstr)
int factor_main(int argc, char **argv) MAIN_EXTERNALLY_VISIBLE;
int factor_main(int argc UNUSED_PARAM, char **argv)
{
unpack_wheel();
//// coreutils has undocumented option ---debug (three dashes)
//getopt32(argv, "");
//argv += optind;