reciprocal_divide: update/correction of the algorithm

Jakub Zawadzki noticed that some divisions by reciprocal_divide()
were not correct [1][2], which he could also show with BPF code
after divisions are transformed into reciprocal_value() for runtime
invariance which can be passed to reciprocal_divide() later on;
reverse in BPF dump ended up with a different, off-by-one K in
some situations.

This has been fixed by Eric Dumazet in commit aee636c4809fa5
("bpf: do not use reciprocal divide"). This follow-up patch
improves reciprocal_value() and reciprocal_divide() to work in
all cases by using Granlund and Montgomery method, so that also
future use is safe and without any non-obvious side-effects.
Known problems with the old implementation were that division by 1
always returned 0 and some off-by-ones when the dividend and divisor
where very large. This seemed to not be problematic with its
current users, as far as we can tell. Eric Dumazet checked for
the slab usage, we cannot surely say so in the case of flex_array.
Still, in order to fix that, we propose an extension from the
original implementation from commit 6a2d7a955d8d resp. [3][4],
by using the algorithm proposed in "Division by Invariant Integers
Using Multiplication" [5], Torbjörn Granlund and Peter L.
Montgomery, that is, pseudocode for q = n/d where q, n, d is in
u32 universe:

1) Initialization:

  int l = ceil(log_2 d)
  uword m' = floor((1<<32)*((1<<l)-d)/d)+1
  int sh_1 = min(l,1)
  int sh_2 = max(l-1,0)

2) For q = n/d, all uword:

  uword t = (n*m')>>32
  q = (t+((n-t)>>sh_1))>>sh_2

The assembler implementation from Agner Fog [6] also helped a lot
while implementing. We have tested the implementation on x86_64,
ppc64, i686, s390x; on x86_64/haswell we're still half the latency
compared to normal divide.

Joint work with Daniel Borkmann.

  [1] http://www.wireshark.org/~darkjames/reciprocal-buggy.c
  [2] http://www.wireshark.org/~darkjames/set-and-dump-filter-k-bug.c
  [3] https://gmplib.org/~tege/division-paper.pdf
  [4] http://homepage.cs.uiowa.edu/~jones/bcd/divide.html
  [5] http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.1.2556
  [6] http://www.agner.org/optimize/asmlib.zip

Reported-by: Jakub Zawadzki <darkjames-ws@darkjames.pl>
Cc: Eric Dumazet <eric.dumazet@gmail.com>
Cc: Austin S Hemmelgarn <ahferroin7@gmail.com>
Cc: linux-kernel@vger.kernel.org
Cc: Jesse Gross <jesse@nicira.com>
Cc: Jamal Hadi Salim <jhs@mojatatu.com>
Cc: Stephen Hemminger <stephen@networkplumber.org>
Cc: Matt Mackall <mpm@selenic.com>
Cc: Pekka Enberg <penberg@kernel.org>
Cc: Christoph Lameter <cl@linux-foundation.org>
Cc: Andy Gospodarek <andy@greyhouse.net>
Cc: Veaceslav Falico <vfalico@redhat.com>
Cc: Jay Vosburgh <fubar@us.ibm.com>
Cc: Jakub Zawadzki <darkjames-ws@darkjames.pl>
Signed-off-by: Daniel Borkmann <dborkman@redhat.com>
Signed-off-by: Hannes Frederic Sowa <hannes@stressinduktion.org>
Signed-off-by: David S. Miller <davem@davemloft.net>

1 files changed, 21 insertions, 18 deletions

diff --git a/include/linux/reciprocal_div.h b/include/linux/reciprocal_div.h
index f9c90b33285b..8c5a3fb6c6c5 100644
--- a/include/linux/reciprocal_div.h
+++ b/include/linux/reciprocal_div.h
@@ -4,29 +4,32 @@
 #include <linux/types.h>
 
 /*
- * This file describes reciprocical division.
+ * This algorithm is based on the paper "Division by Invariant
+ * Integers Using Multiplication" by Torbjörn Granlund and Peter
+ * L. Montgomery.
  *
- * This optimizes the (A/B) problem, when A and B are two u32
- * and B is a known value (but not known at compile time)
+ * The assembler implementation from Agner Fog, which this code is
+ * based on, can be found here:
+ * http://www.agner.org/optimize/asmlib.zip
  *
- * The math principle used is :
- *   Let RECIPROCAL_VALUE(B) be (((1LL << 32) + (B - 1))/ B)
- *   Then A / B = (u32)(((u64)(A) * (R)) >> 32)
- *
- * This replaces a divide by a multiply (and a shift), and
- * is generally less expensive in CPU cycles.
+ * This optimization for A/B is helpful if the divisor B is mostly
+ * runtime invariant. The reciprocal of B is calculated in the
+ * slow-path with reciprocal_value(). The fast-path can then just use
+ * a much faster multiplication operation with a variable dividend A
+ * to calculate the division A/B.
  */
 
-/*
- * Computes the reciprocal value (R) for the value B of the divisor.
- * Should not be called before each reciprocal_divide(),
- * or else the performance is slower than a normal divide.
- */
-extern u32 reciprocal_value(u32 B);
+struct reciprocal_value {
+	u32 m;
+	u8 sh1, sh2;
+};
 
+struct reciprocal_value reciprocal_value(u32 d);
 
-static inline u32 reciprocal_divide(u32 A, u32 R)
+static inline u32 reciprocal_divide(u32 a, struct reciprocal_value R)
 {
-	return (u32)(((u64)A * R) >> 32);
+	u32 t = (u32)(((u64)a * R.m) >> 32);
+	return (t + ((a - t) >> R.sh1)) >> R.sh2;
 }
-#endif
+
+#endif /* _LINUX_RECIPROCAL_DIV_H */