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>

1 files changed, 95 insertions, 75 deletions

diff --git a/coreutils/factor.c b/coreutils/factor.c
index 1f24784fd..30498c73b 100644
--- a/coreutils/factor.c
+++ b/coreutils/factor.c
@@ -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;