/* -*-c-*-
*
- * $Id: mpx-kmul.c,v 1.1 1999/12/10 23:23:51 mdw Exp $
+ * $Id: mpx-kmul.c,v 1.4 2000/06/17 11:42:11 mdw Exp $
*
* Karatsuba's multiplication algorithm
*
/*----- Revision history --------------------------------------------------*
*
* $Log: mpx-kmul.c,v $
+ * Revision 1.4 2000/06/17 11:42:11 mdw
+ * Moved the Karatsuba macros into a separate file for better sharing.
+ * Fixed some comments.
+ *
+ * Revision 1.3 1999/12/13 15:35:01 mdw
+ * Simplify and improve.
+ *
+ * Revision 1.2 1999/12/11 10:58:02 mdw
+ * Remove tweakable comments.
+ *
* Revision 1.1 1999/12/10 23:23:51 mdw
* Karatsuba-Ofman multiplication algorithm.
*
/*----- Header files ------------------------------------------------------*/
+#include <assert.h>
#include <stdio.h>
#include "mpx.h"
+#include "mpx-kmac.h"
/*----- Tweakables --------------------------------------------------------*/
-/* --- @KARATSUBA_CUTOFF@ --- *
- *
- * If either of the arguments to @mpx_kmul@ contains this number of words or
- * fewer, the job is dumped out to @mpx_umul@ instead. Reduce the size when
- * testing, to ensure better coverage.
- */
-
#ifdef TEST_RIG
# undef KARATSUBA_CUTOFF
# define KARATSUBA_CUTOFF 2
#endif
-/*----- Addition macros ---------------------------------------------------*/
-
-#define UADD(dv, av, avl) do { \
- mpw *_dv = (dv); \
- const mpw *_av = (av), *_avl = (avl); \
- mpw _c = 0; \
- \
- while (_av < _avl) { \
- mpw _a, _b; \
- mpd _x; \
- _a = *_av++; \
- _b = *_dv; \
- _x = (mpd)_a + (mpd)_b + _c; \
- *_dv++ = MPW(_x); \
- _c = _x >> MPW_BITS; \
- } \
- while (_c) { \
- mpd _x = (mpd)*_dv + (mpd)_c; \
- *_dv++ = MPW(_x); \
- _c = _x >> MPW_BITS; \
- } \
-} while (0)
-
-#define UADD2(dv, dvl, av, avl, bv, bvl) do { \
- mpw *_dv = (dv), *_dvl = (dvl); \
- const mpw *_av = (av), *_avl = (avl); \
- const mpw *_bv = (bv), *_bvl = (bvl); \
- mpw _c = 0; \
- \
- while (_av < _avl || _bv < _bvl) { \
- mpw _a, _b; \
- mpd _x; \
- _a = (_av < _avl) ? *_av++ : 0; \
- _b = (_bv < _bvl) ? *_bv++ : 0; \
- _x = (mpd)_a + (mpd)_b + _c; \
- *_dv++ = MPW(_x); \
- _c = _x >> MPW_BITS; \
- } \
- *_dv++ = _c; \
- while (_dv < _dvl) \
- *_dv++ = 0; \
-} while (0)
-
-#define USUB(dv, av, avl) do { \
- mpw *_dv = (dv); \
- const mpw *_av = (av), *_avl = (avl); \
- mpw _c = 0; \
- \
- while (_av < _avl) { \
- mpw _a, _b; \
- mpd _x; \
- _a = *_av++; \
- _b = *_dv; \
- _x = (mpd)_b - (mpd)_a - _c; \
- *_dv++ = MPW(_x); \
- if (_x >> MPW_BITS) \
- _c = 1; \
- else \
- _c = 0; \
- } \
- while (_c) { \
- mpd _x = (mpd)*_dv - (mpd)_c; \
- *_dv++ = MPW(_x); \
- if (_x >> MPW_BITS) \
- _c = 1; \
- else \
- _c = 0; \
- } \
-} while (0)
-
/*----- Main code ---------------------------------------------------------*/
/* --- @mpx_kmul@ --- *
* The destination must be twice as large as the larger
* argument. The scratch space must be twice as large as the
* larger argument, plus the magic number @KARATSUBA_SLOP@.
- * (Actually, a number of words proportional to the depth of
- * recursion, but since recusion is strongly bounded by memory,
- * I can replace it with a constant as long as it's `big
- * enough'.)
*/
void mpx_kmul(mpw *dv, mpw *dvl,
/* --- How the algorithm works --- *
*
- * Let %$A = xb + y$% and %$B = ub + v$%. Then, simply by expanding, %$AB
- * = x u b^2 + b(x v + y u) + y v$%. That's not helped any, because I've
- * got four multiplications, each four times easier than the one I started
- * with. However, note that I can rewrite the coefficient of %$b$% as
- * %$xv + yu = (x + y)(u + v) - xu - yv$%. The terms %$xu$% and %$yv$%
+ * Let %$A = xb + y$% and %$B = ub + v$%. Then, simply by expanding,
+ * %$AB = x u b^2 + b(x v + y u) + y v$%. That's not helped any, because
+ * I've got four multiplications, each four times easier than the one I
+ * started with. However, note that I can rewrite the coefficient of %$b$%
+ * as %$xv + yu = (x + y)(u + v) - xu - yv$%. The terms %$xu$% and %$yv$%
* I've already calculated, and that leaves only one more multiplication to
* do. So now I have three multiplications, each four times easier, and
* that's a win.
avm = avl;
}
- /* --- Sort out the middle term --- *
- *
- * I'm going to keep track of the carry by hand rather than pass it down to
- * the next level, because it means multiplication by one or zero, which I
- * can do easily myself.
- */
+ assert(((void)"Destination too small for Karatsuba multiply",
+ dvl - dv >= 4 * m));
+ assert(((void)"Not enough workspace for Karatsuba multiply",
+ svl - sv >= 4 * m));
+
+ /* --- Sort out the middle term --- */
{
- unsigned f = 0;
- enum {
- carry_a = 1,
- carry_b = 2
- };
-
- mpw *bsv = sv + m, *ssv = bsv + m;
- mpw *rdv = dv + m, *rdvl = rdv + 2 * m;
-
- UADD2(sv, bsv + 1, av, avm, avm, avl);
- if (*bsv)
- f |= carry_a;
- UADD2(bsv, ssv + 1, bv, bvm, bvm, bvl);
- if (*ssv)
- f |= carry_b;
- MPX_ZERO(dv, rdv);
+ mpw *bsv = sv + m + 1, *ssv = bsv + m + 1;
+ mpw *rdv = dv + m, *rdvl = rdv + 2 * (m + 2);
+
+ UADD2(sv, bsv, av, avm, avm, avl);
+ UADD2(bsv, ssv, bv, bvm, bvm, bvl);
if (m > KARATSUBA_CUTOFF)
mpx_kmul(rdv, rdvl, sv, bsv, bsv, ssv, ssv, svl);
else
mpx_umul(rdv, rdvl, sv, bsv, bsv, ssv);
- MPX_ZERO(rdvl, dvl);
- rdv += m; rdvl += m;
- if (f & carry_b)
- UADD(rdv, sv, bsv);
- if (f & carry_a)
- UADD(rdv, bsv, ssv);
- if (!(~f & (carry_a | carry_b)))
- MPX_UADDN(rdv + m, rdvl, 1);
}
/* --- Sort out the other two terms --- */
{
- mpw *ssv = sv + 2 * m;
+ mpw *svm = sv + m, *svn = svm + m, *ssv = svn + 4;
mpw *tdv = dv + m;
mpw *rdv = tdv + m;
- if (m > KARATSUBA_CUTOFF)
- mpx_kmul(sv, ssv, avm, avl, bvm, bvl, ssv, svl);
- else
- mpx_umul(sv, ssv, avm, avl, bvm, bvl);
- UADD(rdv, sv, ssv);
- USUB(tdv, sv, ssv);
-
+ if (avl == avm || bvl == bvm)
+ MPX_ZERO(rdv + m + 1, dvl);
+ else {
+ if (m > KARATSUBA_CUTOFF)
+ mpx_kmul(sv, ssv, avm, avl, bvm, bvl, ssv, svl);
+ else
+ mpx_umul(sv, ssv, avm, avl, bvm, bvl);
+ MPX_COPY(rdv + m + 1, dvl, svm + 1, svn);
+ UADD(rdv, sv, svm + 1);
+ USUB(tdv, sv, svn);
+ }
+
if (m > KARATSUBA_CUTOFF)
mpx_kmul(sv, ssv, av, avm, bv, bvm, ssv, svl);
else
mpx_umul(sv, ssv, av, avm, bv, bvm);
- USUB(tdv, sv, ssv);
- UADD(dv, sv, ssv);
+ MPX_COPY(dv, tdv, sv, svm);
+ USUB(tdv, sv, svn);
+ UADD(tdv, svm, svn);
}
}
projects / u / mdw / catacomb / blobdiff