[Ada] Minor fixes mostly in comments of runtime arithmetic unit

Multiple comments in functions Double_Divide and Scaled_Divide were
incorrect. Now fixed.

Also change the expression (if Zhi /= 0 then Ylo * Zhi else 0) to the
simpler equivalent (Ylo * Zhi) in Double_Divide.

Also add a comment explaining why the implementation of Algorithm D for
multiple-precision division from the 2nd Edition of The Art of Computer
Programming does not suffer from two bugs discovered on that version.

There is no impact on execution, hence no test.

2019-09-17  Yannick Moy  <moy@adacore.com>

gcc/ada/

	* libgnat/s-arit64.adb (Double_Divide): Simplify needlessly
	complex computation. Fix comments.
	(Scaled_Divide): Fix comments. Explain why implementation does
	not suffer from bugs in Algorithm D from 2nd Edition of TAOCP.

From-SVN: r275792

2 files changed, 27 insertions, 6 deletions

diff --git a/gcc/ada/ChangeLog b/gcc/ada/ChangeLog
index 9a07751..0d7fc8f 100644
--- a/gcc/ada/ChangeLog
+++ b/gcc/ada/ChangeLog
@@ -1,5 +1,12 @@
 2019-09-17  Yannick Moy  <moy@adacore.com>
 
+	* libgnat/s-arit64.adb (Double_Divide): Simplify needlessly
+	complex computation. Fix comments.
+	(Scaled_Divide): Fix comments. Explain why implementation does
+	not suffer from bugs in Algorithm D from 2nd Edition of TAOCP.
+
+2019-09-17  Yannick Moy  <moy@adacore.com>
+
 	* libgnat/s-arit64.adb (Scaled_Divide): Add protection against
 	undesirable wrap-around.
 
diff --git a/gcc/ada/libgnat/s-arit64.adb b/gcc/ada/libgnat/s-arit64.adb
index 6773dd8..b5a5ac4 100644
--- a/gcc/ada/libgnat/s-arit64.adb
+++ b/gcc/ada/libgnat/s-arit64.adb
@@ -161,7 +161,7 @@ package body System.Arith_64 is
          end if;
 
       else
-         T2 := (if Zhi /= 0 then Ylo * Zhi else 0);
+         T2 := Ylo * Zhi;
       end if;
 
       T1 := Ylo * Zlo;
@@ -179,7 +179,7 @@ package body System.Arith_64 is
 
       Den_Pos := (Y < 0) = (Z < 0);
 
-      --  Check overflow case of largest negative number divided by 1
+      --  Check overflow case of largest negative number divided by -1
 
       if X = Int64'First and then Du = 1 and then not Den_Pos then
          Raise_Error;
@@ -404,15 +404,16 @@ package body System.Arith_64 is
             Ru := T2 rem Zlo;
          end if;
 
-      --  If divisor is double digit and too large, raise error
+      --  If divisor is double digit and dividend is too large, raise error
 
       elsif (D (1) & D (2)) >= Zu then
          Raise_Error;
 
       --  This is the complex case where we definitely have a double digit
       --  divisor and a dividend of at least three digits. We use the classical
-      --  multiple division algorithm (see section (4.3.1) of Knuth's "The Art
-      --  of Computer Programming", Vol. 2 for a description (algorithm D).
+      --  multiple-precision division algorithm (see section (4.3.1) of Knuth's
+      --  "The Art of Computer Programming", Vol. 2 for a description
+      --  (algorithm D).
 
       else
          --  First normalize the divisor so that it has the leading bit on.
@@ -450,7 +451,7 @@ package body System.Arith_64 is
 
          --  Note that when we scale up the dividend, it still fits in four
          --  digits, since we already tested for overflow, and scaling does
-         --  not change the invariant that (D (1) & D (2)) >= Zu.
+         --  not change the invariant that (D (1) & D (2)) < Zu.
 
          T1 := Shift_Left (D (1) & D (2), Scale);
          D (1) := Hi (T1);
@@ -485,6 +486,19 @@ package body System.Arith_64 is
 
             --  Adjust quotient digit if it was too high
 
+            --  We use the version of the algorithm in the 2nd Edition of
+            --  "The Art of Computer Programming". This had a bug not
+            --  discovered till 1995, see Vol 2 errata:
+            --     http://www-cs-faculty.stanford.edu/~uno/err2-2e.ps.gz.
+            --  Under rare circumstances the expression in the test could
+            --  overflow. This version was further corrected in 2005, see
+            --  Vol 2 errata:
+            --     http://www-cs-faculty.stanford.edu/~uno/all2-pre.ps.gz.
+            --  This implementation is not impacted by these bugs, due to the
+            --  use of a word-size comparison done in function Le3 instead of
+            --  a comparison on two-word integer quantities in the original
+            --  algorithm.
+
             loop
                exit when Le3 (S1, S2, S3, D (J + 1), D (J + 2), D (J + 3));
                Qd (J + 1) := Qd (J + 1) - 1;