math/mpreduce.[ch]: Extend the domain to all positive integers.

[u/mdw/catacomb] / math / mpreduce.c

diff --git a/math/mpreduce.c b/math/mpreduce.c
index cc74bc2..b148dd5 100644 (file)
--- a/math/mpreduce.c
+++ b/math/mpreduce.c
@@ -81,7 +81,10 @@ DA_DECL(instr_v, mpreduce_instr);
  * Arguments:  @gfreduce *r@ = structure to fill in
  *             @mp *x@ = an integer
  *
- * Returns:    Zero if successful; nonzero on failure.
+ * Returns:    Zero if successful; nonzero on failure.  The current
+ *             algorithm always succeeds when given positive @x@.  Earlier
+ *             versions used to fail on particular kinds of integers, but
+ *             this is guaranteed not to happen any more.
  *
  * Use:                Initializes a context structure for reduction.
  */
@@ -196,18 +199,27 @@ int mpreduce_create(mpreduce *r, mp *p)
     }
   }
 
-  /* --- This doesn't always work --- *
+  /* --- Fix up wrong-sided decompositions --- *
    *
-   * If %$d \ge 2^{n-1}$% then the above recurrence will output a subtraction
-   * as the final instruction, which may sometimes underflow.  (It interprets
-   * such numbers as being in the form %$2^{n-1} + d$%.)  This is clearly
-   * bad, so detect the situation and fail gracefully.
+   * At this point, we haven't actually finished up the state machine
+   * properly.  We stopped scanning just after bit %$n - 1$% -- the most
+   * significant one, which we know in advance must be set (since @x@ is
+   * strictly positive).  Therefore we are either in state @X@ or @Z1@.  In
+   * the former case, we have nothing to do.  In the latter, there are two
+   * subcases to deal with.  If there are no other instructions, then @x@ is
+   * a perfect power of two, and %$d = 0$%, so again there is nothing to do.
+   *
+   * In the remaining case, we have decomposed @x@ as %$2^{n-1} + d$%, for
+   * some positive %$d%, which is unfortuante: if we're asked to reduce
+   * %$2^n$%, say, we'll end up with %$-d$% (or would do, if we weren't
+   * sticking to unsigned arithmetic for good performance).  So instead, we
+   * rewrite this as %$2^n - 2^{n-1} + d$% and everything will be good.
    */
 
-  if (DA_LEN(&iv) && (DA(&iv)[DA_LEN(&iv) - 1].op & ~1u) == MPRI_SUB) {
-    mp_drop(r->p);
-    DA_DESTROY(&iv);
-    return (-1);
+  if (st == Z1 && DA_LEN(&iv)) {
+    w = 1;
+    b = (bb + d)%MPW_BITS;
+    INSTR(MPRI_ADD | !!b, w, b);
   }
 
 #undef INSTR