[PATCH v4] Don't spin around multiplying zeroes in __mul

[Siddhesh Poyarekar <siddhesh at redhat dot com>]

Hi,

I've updated the patch to fix the incorrect comments I had written.
No additional performance improvement due to this ;)

Siddhesh

	* sysdeps/ieee754/dbl-64/mpa.c (__mul): Don't bother with zero
	values in the mantissa.

diff --git a/sysdeps/ieee754/dbl-64/mpa.c b/sysdeps/ieee754/dbl-64/mpa.c
index ede8ed1..1137e94 100644
--- a/sysdeps/ieee754/dbl-64/mpa.c
+++ b/sysdeps/ieee754/dbl-64/mpa.c
@@ -602,7 +602,7 @@ void
 SECTION
 __mul (const mp_no *x, const mp_no *y, mp_no *z, int p)
 {
-  int i, j, k, k2;
+  int i, j, k, ip, ip2;
   double u, zk;
 
   /* Is z=0?  */
@@ -612,11 +612,51 @@ __mul (const mp_no *x, const mp_no *y, mp_no *z, int p)
       return;
     }
 
-  /* Multiply, add and carry.  */
-  k2 = (__glibc_unlikely (p < 3)) ? p + p : p + 3;
-  zk = Z[k2] = ZERO;
+  /* We need not iterate through all X's and Y's since it's pointless to
+     multiply zeroes.  Here, both are zero...  */
+  for (ip2 = p; ip2 > 0; ip2--)
+    if (X[ip2] != ZERO || Y[ip2] != ZERO)
+      break;
 
-  for (k = k2; k > p; k--)
+  /* ... and here, at least one of them is still zero.  */
+  for (ip = ip2; ip > 0; ip--)
+    if (X[ip] * Y[ip] != ZERO)
+      break;
+
+  /* The product looks like this for p = 3 (as an example):
+
+
+				a1    a2    a3
+		 x		b1    b2    b3
+		 -----------------------------
+			     a1*b3 a2*b3 a3*b3
+		       a1*b2 a2*b2 a3*b2
+		 a1*b1 a2*b1 a3*b1
+
+     So our K needs to ideally be P*2, but we're limiting ourselves to P + 3
+     for P >= 3.  We compute the above digits in two parts; the last P-1
+     digits and then the first P digits.  The last P-1 digits are a sum of
+     products of the input digits from P to P-k where K is 0 for the least
+     significant digit and increases as we go towards the left.  The product
+     term is of the form X[k]*X[P-k] as can be seen in the above example.
+
+     The first P digits are also a sum of products with the same product term,
+     except that the sum is from 1 to k.  This is also evident from the above
+     example.
+
+     Another thing that becomes evident is that only the most significant
+     ip+ip2 digits of the result are non-zero, where ip and ip2 are the
+     'internal precision' of the input numbers, i.e. digits after ip and ip2
+     are all ZERO.  */
+
+  k = (__glibc_unlikely (p < 3)) ? p + p : p + 3;
+
+  while (k > ip + ip2 + 1)
+    Z[k--] = ZERO;
+
+  zk = Z[k] = ZERO;
+
+  while (k > p)
     {
       for (i = k - p, j = p; i < p + 1; i++, j--)
 	zk += X[i] * Y[j];
@@ -624,10 +664,11 @@ __mul (const mp_no *x, const mp_no *y, mp_no *z, int p)
       u = (zk + CUTTER) - CUTTER;
       if (u > zk)
 	u -= RADIX;
-      Z[k] = zk - u;
+      Z[k--] = zk - u;
       zk = u * RADIXI;
     }
 
+  /* Part two.  */
   while (k > 1)
     {
       for (i = 1, j = k - 1; i < k; i++, j--)
@@ -636,9 +677,8 @@ __mul (const mp_no *x, const mp_no *y, mp_no *z, int p)
       u = (zk + CUTTER) - CUTTER;
       if (u > zk)
 	u -= RADIX;
-      Z[k] = zk - u;
+      Z[k--] = zk - u;
       zk = u * RADIXI;
-      k--;
     }
   Z[k] = zk;