k_mul() comments: Explained why there's always enough room to subtract

tim-one · tim-one · commit 8e966ee49aed · 2002-08-14T16:36:23.000Z
ah*bh and al*bl.  This is much easier than explaining why that's true
for (ah+al)*(bh+bl), and follows directly from the simple part of the
(ah+al)*(bh+bl) explanation.
diff --git a/Objects/longobject.c b/Objects/longobject.c
@@ -1791,6 +1791,13 @@ remaining, and that's obviously plenty to hold 2*shift+2 digits + 2 bits.
 Else (bsize is odd and asize < bsize) ah and al each have at most shift digits,
 so ah+al has at most shift digits + 1 bit, and (ah+al)*(bh+bl) has at most
 2*shift+1 digits + 2 bits, and again 2*shift+2 digits is enough to hold it.
+
+Note that the "lazy" analysis is enough to show that there's always enough
+room to subtract al*bl and ah*bh.  al and bl each have no more than shift
+digits, so al*bl has no more than 2*shift, so there's at least one digit
+to spare in the remaining allocated digits.  The same is true for ah*bh when
+bsize is even.  When bsize is odd, ah*bh has at most 2*shift+2 digits, and
+there are at least that many remaining allocated digits when bsize is odd.
 */
 
 /* b has at least twice the digits of a, and a is big enough that Karatsuba
