Explain the strategy to avoid huge alignment shifts in _mpd_qadd() in detail.

Stefan Krah · Stefan Krah · commit 5d0d2e2b0463 · 2012-04-18T18:59:56.000+02:00
diff --git a/Modules/_decimal/libmpdec/mpdecimal.c b/Modules/_decimal/libmpdec/mpdecimal.c
@@ -3103,22 +3103,51 @@ _mpd_qaddsub(mpd_t *result, const mpd_t *a, const mpd_t *b, uint8_t sign_b,
             _mpd_ptrswap(&big, &small);
             swap++;
         }
+        /* align the coefficients */
         if (!mpd_iszerocoeff(big)) {
-            /* Test for adjexp(small) + big->digits < adjexp(big), if big-digits > prec
-             * Test for adjexp(small) + prec + 1    < adjexp(big), if big-digits <= prec
-             * If true, the magnitudes of the numbers are so far apart that one can as
-             * well add or subtract 1*10**big->exp. */
             exp = big->exp - 1;
             exp += (big->digits > ctx->prec) ? 0 : big->digits-ctx->prec-1;
             if (mpd_adjexp(small) < exp) {
+                /*
+                 * Avoid huge shifts by substituting a value for small that is
+                 * guaranteed to produce the same results.
+                 *
+                 * adjexp(small) < exp if and only if:
+                 *
+                 *   bdigits <= prec AND
+                 *   bdigits+shift >= prec+2+sdigits AND
+                 *   exp = bexp+bdigits-prec-2
+                 *
+                 *     1234567000000000  ->  bdigits + shift
+                 *     ----------XX1234  ->  sdigits
+                 *     ----------X1      ->  tiny-digits
+                 *     |- prec -|
+                 *     
+                 *      OR
+                 *
+                 *   bdigits > prec AND
+                 *   shift > sdigits AND
+                 *   exp = bexp-1
+                 *
+                 *     1234567892100000  ->  bdigits + shift
+                 *     ----------XX1234  ->  sdigits
+                 *     ----------X1      ->  tiny-digits
+                 *     |- prec -|
+                 *
+                 * If tiny is zero, adding or subtracting is a no-op.
+                 * Otherwise, adding tiny generates a non-zero digit either
+                 * below the rounding digit or the least significant digit
+                 * of big. When subtracting, tiny is in the same position as
+                 * the carry that would be generated by subtracting sdigits.
+                 */
                 mpd_copy_flags(&tiny, small);
                 tiny.exp = exp;
                 tiny.digits = 1;
                 tiny.len = 1;
                 tiny.data[0] = mpd_iszerocoeff(small) ? 0 : 1;
                 small = &tiny;
             }
-            /* this cannot wrap: the difference is positive and <= maxprec+1 */
+            /* This cannot wrap: the difference is positive and <= maxprec */
             shift = big->exp - small->exp;
             if (!mpd_qshiftl(&big_aligned, big, shift, status)) {
                 mpd_seterror(result, MPD_Malloc_error, status);
@@ -3521,7 +3550,7 @@ mpd_qdiv(mpd_t *q, const mpd_t *a, const mpd_t *b,
 /* Internal function. */
 static void
 _mpd_qdivmod(mpd_t *q, mpd_t *r, const mpd_t *a, const mpd_t *b,
-         const mpd_context_t *ctx, uint32_t *status)
+             const mpd_context_t *ctx, uint32_t *status)
 {
     MPD_NEW_STATIC(aligned,0,0,0,0);
     mpd_ssize_t qsize, rsize;
