lib: reciprocal_div: implement the improved algorithm on the paper mentioned

The new added "reciprocal_value_adv" implements the advanced version of the
algorithm described in Figure 4.2 of the paper except when
"divisor > (1U << 31)" whose ceil(log2(d)) result will be 32 which then
requires u128 divide on host. The exception case could be easily handled
before calling "reciprocal_value_adv".

The advanced version requires more complex calculation to get the
reciprocal multiplier and other control variables, but then could reduce
the required emulation operations.

It makes no sense to use this advanced version for host divide emulation,
those extra complexities for calculating multiplier etc could completely
waive our saving on emulation operations.

However, it makes sense to use it for JIT divide code generation (for
example eBPF JIT backends) for which we are willing to trade performance of
JITed code with that of host. As shown by the following pseudo code, the
required emulation operations could go down from 6 (the basic version) to 3
or 4.

To use the result of "reciprocal_value_adv", suppose we want to calculate
n/d, the C-style pseudo code will be the following, it could be easily
changed to real code generation for other JIT targets.

  struct reciprocal_value_adv rvalue;
  u8 pre_shift, exp;

  // handle exception case.
  if (d >= (1U << 31)) {
    result = n >= d;
    return;
  }
  rvalue = reciprocal_value_adv(d, 32)
  exp = rvalue.exp;
  if (rvalue.is_wide_m && !(d & 1)) {
    // floor(log2(d & (2^32 -d)))
    pre_shift = fls(d & -d) - 1;
    rvalue = reciprocal_value_adv(d >> pre_shift, 32 - pre_shift);
  } else {
    pre_shift = 0;
  }

  // code generation starts.
  if (imm == 1U << exp) {
    result = n >> exp;
  } else if (rvalue.is_wide_m) {
    // pre_shift must be zero when reached here.
    t = (n * rvalue.m) >> 32;
    result = n - t;
    result >>= 1;
    result += t;
    result >>= rvalue.sh - 1;
  } else {
    if (pre_shift)
      result = n >> pre_shift;
    result = ((u64)result * rvalue.m) >> 32;
    result >>= rvalue.sh;
  }
Signed-off-by: Jiong Wang <jiong.wang@netronome.com>
Reviewed-by: Jakub Kicinski <jakub.kicinski@netronome.com>
Signed-off-by: Daniel Borkmann <daniel@iogearbox.net>

include/linux/reciprocal_div.h

@@ -25,6 +25,9 @@ struct reciprocal_value {
 	u8 sh1, sh2;
 };
 
+/* "reciprocal_value" and "reciprocal_divide" together implement the basic
+ * version of the algorithm described in Figure 4.1 of the paper.
+ */
 struct reciprocal_value reciprocal_value(u32 d);
 
 static inline u32 reciprocal_divide(u32 a, struct reciprocal_value R)
@@ -33,4 +36,69 @@ static inline u32 reciprocal_divide(u32 a, struct reciprocal_value R)
 	return (t + ((a - t) >> R.sh1)) >> R.sh2;
 }
 
+struct reciprocal_value_adv {
+	u32 m;
+	u8 sh, exp;
+	bool is_wide_m;
+};
+
+/* "reciprocal_value_adv" implements the advanced version of the algorithm
+ * described in Figure 4.2 of the paper except when "divisor > (1U << 31)" whose
+ * ceil(log2(d)) result will be 32 which then requires u128 divide on host. The
+ * exception case could be easily handled before calling "reciprocal_value_adv".
+ *
+ * The advanced version requires more complex calculation to get the reciprocal
+ * multiplier and other control variables, but then could reduce the required
+ * emulation operations.
+ *
+ * It makes no sense to use this advanced version for host divide emulation,
+ * those extra complexities for calculating multiplier etc could completely
+ * waive our saving on emulation operations.
+ *
+ * However, it makes sense to use it for JIT divide code generation for which
+ * we are willing to trade performance of JITed code with that of host. As shown
+ * by the following pseudo code, the required emulation operations could go down
+ * from 6 (the basic version) to 3 or 4.
+ *
+ * To use the result of "reciprocal_value_adv", suppose we want to calculate
+ * n/d, the pseudo C code will be:
+ *
+ *   struct reciprocal_value_adv rvalue;
+ *   u8 pre_shift, exp;
+ *
+ *   // handle exception case.
+ *   if (d >= (1U << 31)) {
+ *     result = n >= d;
+ *     return;
+ *   }
+ *
+ *   rvalue = reciprocal_value_adv(d, 32)
+ *   exp = rvalue.exp;
+ *   if (rvalue.is_wide_m && !(d & 1)) {
+ *     // floor(log2(d & (2^32 -d)))
+ *     pre_shift = fls(d & -d) - 1;
+ *     rvalue = reciprocal_value_adv(d >> pre_shift, 32 - pre_shift);
+ *   } else {
+ *     pre_shift = 0;
+ *   }
+ *
+ *   // code generation starts.
+ *   if (imm == 1U << exp) {
+ *     result = n >> exp;
+ *   } else if (rvalue.is_wide_m) {
+ *     // pre_shift must be zero when reached here.
+ *     t = (n * rvalue.m) >> 32;
+ *     result = n - t;
+ *     result >>= 1;
+ *     result += t;
+ *     result >>= rvalue.sh - 1;
+ *   } else {
+ *     if (pre_shift)
+ *       result = n >> pre_shift;
+ *     result = ((u64)result * rvalue.m) >> 32;
+ *     result >>= rvalue.sh;
+ *   }
+ */
+struct reciprocal_value_adv reciprocal_value_adv(u32 d, u8 prec);
+
 #endif /* _LINUX_RECIPROCAL_DIV_H */

lib/reciprocal_div.c

 // SPDX-License-Identifier: GPL-2.0
+#include <linux/bug.h>
 #include <linux/kernel.h>
 #include <asm/div64.h>
 #include <linux/reciprocal_div.h>
@@ -26,3 +27,43 @@ struct reciprocal_value reciprocal_value(u32 d)
 	return R;
 }
 EXPORT_SYMBOL(reciprocal_value);
+
+struct reciprocal_value_adv reciprocal_value_adv(u32 d, u8 prec)
+{
+	struct reciprocal_value_adv R;
+	u32 l, post_shift;
+	u64 mhigh, mlow;
+
+	/* ceil(log2(d)) */
+	l = fls(d - 1);
+	/* NOTE: mlow/mhigh could overflow u64 when l == 32. This case needs to
+	 * be handled before calling "reciprocal_value_adv", please see the
+	 * comment at include/linux/reciprocal_div.h.
+	 */
+	WARN(l == 32,
+	     "ceil(log2(0x%08x)) == 32, %s doesn't support such divisor",
+	     d, __func__);
+	post_shift = l;
+	mlow = 1ULL << (32 + l);
+	do_div(mlow, d);
+	mhigh = (1ULL << (32 + l)) + (1ULL << (32 + l - prec));
+	do_div(mhigh, d);
+
+	for (; post_shift > 0; post_shift--) {
+		u64 lo = mlow >> 1, hi = mhigh >> 1;
+
+		if (lo >= hi)
+			break;
+
+		mlow = lo;
+		mhigh = hi;
+	}
+
+	R.m = (u32)mhigh;
+	R.sh = post_shift;
+	R.exp = l;
+	R.is_wide_m = mhigh > U32_MAX;
+
+	return R;
+}
+EXPORT_SYMBOL(reciprocal_value_adv);