bpf: Improve the general precision of tnum_mul

Drop the value-mask decomposition technique and adopt straightforward
long-multiplication with a twist: when LSB(a) is uncertain, find the
two partial products (for LSB(a) = known 0 and LSB(a) = known 1) and
take a union.

Experiment shows that applying this technique in long multiplication
improves the precision in a significant number of cases (at the cost
of losing precision in a relatively lower number of cases).

Signed-off-by: Nandakumar Edamana <[e-mail]>
Signed-off-by: Andrii Nakryiko <[e-mail]>
Tested-by: Harishankar Vishwanathan <[e-mail]>
Reviewed-by: Harishankar Vishwanathan <[e-mail]>
Acked-by: Eduard Zingerman <[e-mail]>
Link: https://lore.kernel.org/bpf/[e-mail]

include/linux/tnum.h

···
 /* Return a tnum representing numbers satisfying both @a and @b */
 struct tnum tnum_intersect(struct tnum a, struct tnum b);
 
+/* Returns a tnum representing numbers satisfying either @a or @b */
+struct tnum tnum_union(struct tnum t1, struct tnum t2);
+
 /* Return @a with all but the lowest @size bytes cleared */
 struct tnum tnum_cast(struct tnum a, u8 size);
 

kernel/bpf/tnum.c

···
 	return TNUM(v & ~mu, mu);
 }
 
-/* Generate partial products by multiplying each bit in the multiplier (tnum a)
- * with the multiplicand (tnum b), and add the partial products after
- * appropriately bit-shifting them. Instead of directly performing tnum addition
- * on the generated partial products, equivalenty, decompose each partial
- * product into two tnums, consisting of the value-sum (acc_v) and the
- * mask-sum (acc_m) and then perform tnum addition on them. The following paper
- * explains the algorithm in more detail: https://arxiv.org/abs/2105.05398.
+/* Perform long multiplication, iterating through the bits in a using rshift:
+ * - if LSB(a) is a known 0, keep current accumulator
+ * - if LSB(a) is a known 1, add b to current accumulator
+ * - if LSB(a) is unknown, take a union of the above cases.
+ *
+ * For example:
+ *
+ *               acc_0:        acc_1:
+ *
+ *     11 *  ->      11 *  ->      11 *  -> union(0011, 1001) == x0x1
+ *     x1            01            11
+ * ------        ------        ------
+ *     11            11            11
+ *    xx            00            11
+ * ------        ------        ------
+ *   ????          0011          1001
  */
 struct tnum tnum_mul(struct tnum a, struct tnum b)
 {
-	u64 acc_v = a.value * b.value;
-	struct tnum acc_m = TNUM(0, 0);
+	struct tnum acc = TNUM(0, 0);
 
 	while (a.value || a.mask) {
 		/* LSB of tnum a is a certain 1 */
 		if (a.value & 1)
-			acc_m = tnum_add(acc_m, TNUM(0, b.mask));
+			acc = tnum_add(acc, b);
 		/* LSB of tnum a is uncertain */
-		else if (a.mask & 1)
-			acc_m = tnum_add(acc_m, TNUM(0, b.value | b.mask));
+		else if (a.mask & 1) {
+			/* acc = tnum_union(acc_0, acc_1), where acc_0 and
+			 * acc_1 are partial accumulators for cases
+			 * LSB(a) = certain 0 and LSB(a) = certain 1.
+			 * acc_0 = acc + 0 * b = acc.
+			 * acc_1 = acc + 1 * b = tnum_add(acc, b).
+			 */
+
+			acc = tnum_union(acc, tnum_add(acc, b));
+		}
 		/* Note: no case for LSB is certain 0 */
 		a = tnum_rshift(a, 1);
 		b = tnum_lshift(b, 1);
 	}
-	return tnum_add(TNUM(acc_v, 0), acc_m);
+	return acc;
 }
 
 bool tnum_overlap(struct tnum a, struct tnum b)
···
 
 	v = a.value | b.value;
 	mu = a.mask & b.mask;
+	return TNUM(v & ~mu, mu);
+}
+
+/* Returns a tnum with the uncertainty from both a and b, and in addition, new
+ * uncertainty at any position that a and b disagree. This represents a
+ * superset of the union of the concrete sets of both a and b. Despite the
+ * overapproximation, it is optimal.
+ */
+struct tnum tnum_union(struct tnum a, struct tnum b)
+{
+	u64 v = a.value & b.value;
+	u64 mu = (a.value ^ b.value) | a.mask | b.mask;
+
 	return TNUM(v & ~mu, mu);
 }