The divmod function for large numbers now has an ACL2 proof. Related changes:

Stefan Krah · Stefan Krah · commit 3c23a87e58ba · 2012-04-20T19:59:20.000+02:00
1) Rename _mpd_qbarrett_divmod into _mpd_base_ndivmod: The function is
     only marginally related to either Barrett's algorithm or to the version
     in Hasselstrom's paper.

  2) In places where the proof assumes exact operations, use new versions of
     add/sub/multiply that set NaN/Invalid_operation if this condition is
     not met. According to the proof this cannot happen, so this should be
     regarded as an extra safety net.

  3) Raise Division_impossible for operands with a number of digits greater
     than MPD_MAX_PREC. This facilitates the audit of the function and can
     practically only occur in the 32-bit version under conditions where
     a MemoryError is already imminent.

  4) Use _mpd_qmul() in places where the result can exceed MPD_MAX_PREC in
     a well defined manner.

  5) Test for mpd_isspecial(qq) in a place where the addition of one
     can theoretically trigger a Malloc_error.

  6) Remove redundant code in _mpd_qdivmod().

  7) Add many comments.
diff --git a/Modules/_decimal/libmpdec/mpdecimal.c b/Modules/_decimal/libmpdec/mpdecimal.c
@@ -105,8 +105,8 @@ static void _mpd_qadd(mpd_t *result, const mpd_t *a, const mpd_t *b,
                       const mpd_context_t *ctx, uint32_t *status);
 static inline void _mpd_qmul(mpd_t *result, const mpd_t *a, const mpd_t *b,
                              const mpd_context_t *ctx, uint32_t *status);
-static void _mpd_qbarrett_divmod(mpd_t *q, mpd_t *r, const mpd_t *a,
-                                 const mpd_t *b, uint32_t *status);
+static void _mpd_base_ndivmod(mpd_t *q, mpd_t *r, const mpd_t *a,
+                              const mpd_t *b, uint32_t *status);
 static inline void _mpd_qpow_uint(mpd_t *result, mpd_t *base, mpd_uint_t exp,
                 uint8_t resultsign, const mpd_context_t *ctx, uint32_t *status);
 
@@ -3256,6 +3256,20 @@ mpd_qadd(mpd_t *result, const mpd_t *a, const mpd_t *b,
     mpd_qfinalize(result, ctx, status);
 }
 
+/* Add a and b. Set NaN/Invalid_operation if the result is inexact. */
+static void
+_mpd_qadd_exact(mpd_t *result, const mpd_t *a, const mpd_t *b,
+                const mpd_context_t *ctx, uint32_t *status)
+{
+    uint32_t workstatus = 0;
+
+    mpd_qadd(result, a, b, ctx, &workstatus);
+    *status |= workstatus;
+    if (workstatus & (MPD_Inexact|MPD_Rounded|MPD_Clamped)) {
+        mpd_seterror(result, MPD_Invalid_operation, status);
+    }
+}
+
 /* Subtract b from a. */
 void
 mpd_qsub(mpd_t *result, const mpd_t *a, const mpd_t *b,
@@ -3273,6 +3287,20 @@ mpd_qsub(mpd_t *result, const mpd_t *a, const mpd_t *b,
     mpd_qfinalize(result, ctx, status);
 }
 
+/* Subtract b from a. Set NaN/Invalid_operation if the result is inexact. */
+static void
+_mpd_qsub_exact(mpd_t *result, const mpd_t *a, const mpd_t *b,
+                const mpd_context_t *ctx, uint32_t *status)
+{
+    uint32_t workstatus = 0;
+
+    mpd_qsub(result, a, b, ctx, &workstatus);
+    *status |= workstatus;
+    if (workstatus & (MPD_Inexact|MPD_Rounded|MPD_Clamped)) {
+        mpd_seterror(result, MPD_Invalid_operation, status);
+    }
+}
+
 /* Add decimal and mpd_ssize_t. */
 void
 mpd_qadd_ssize(mpd_t *result, const mpd_t *a, mpd_ssize_t b,
@@ -3500,7 +3528,7 @@ _mpd_qdiv(int action, mpd_t *q, const mpd_t *a, const mpd_t *b,
     }
     else {
         MPD_NEW_STATIC(r,0,0,0,0);
-        _mpd_qbarrett_divmod(q, &r, a, b, status);
+        _mpd_base_ndivmod(q, &r, a, b, status);
         if (mpd_isspecial(q) || mpd_isspecial(&r)) {
             mpd_del(&r);
             goto finish;
@@ -3652,14 +3680,10 @@ _mpd_qdivmod(mpd_t *q, mpd_t *r, const mpd_t *a, const mpd_t *b,
         }
     }
     else {
-        _mpd_qbarrett_divmod(q, r, a, b, status);
+        _mpd_base_ndivmod(q, r, a, b, status);
         if (mpd_isspecial(q) || mpd_isspecial(r)) {
             goto nanresult;
         }
-        if (mpd_isinfinite(q) || q->digits > ctx->prec) {
-            *status |= MPD_Division_impossible;
-            goto nanresult;
-        }
         qsize = q->len;
         rsize = r->len;
     }
@@ -5369,6 +5393,20 @@ mpd_qmul(mpd_t *result, const mpd_t *a, const mpd_t *b,
     mpd_qfinalize(result, ctx, status);
 }
 
+/* Multiply a and b. Set NaN/Invalid_operation if the result is inexact. */
+static void
+_mpd_qmul_exact(mpd_t *result, const mpd_t *a, const mpd_t *b,
+                const mpd_context_t *ctx, uint32_t *status)
+{
+    uint32_t workstatus = 0;
+
+    mpd_qmul(result, a, b, ctx, &workstatus);
+    *status |= workstatus;
+    if (workstatus & (MPD_Inexact|MPD_Rounded|MPD_Clamped)) {
+        mpd_seterror(result, MPD_Invalid_operation, status);
+    }
+}
+
 /* Multiply decimal and mpd_ssize_t. */
 void
 mpd_qmul_ssize(mpd_t *result, const mpd_t *a, mpd_ssize_t b,
@@ -6691,11 +6729,18 @@ recpr_schedule_prec(mpd_ssize_t klist[MPD_MAX_PREC_LOG2],
     return i-1;
 }
 
-/* Initial approximation for the reciprocal. */
+/*
+ * Initial approximation for the reciprocal:
+ *    k_0 := MPD_RDIGITS-2
+ *    z_0 := 10**(-k_0) * floor(10**(2*k_0 + 2) / floor(v * 10**(k_0 + 2)))
+ * Absolute error:
+ *    |1/v - z_0| < 10**(-k_0)
+ * ACL2 proof: maxerror-inverse-approx
+ */
 static void
 _mpd_qreciprocal_approx(mpd_t *z, const mpd_t *v, uint32_t *status)
 {
-    mpd_uint_t p10data[2] = {0, mpd_pow10[MPD_RDIGITS-2]}; /* 10**(2*MPD_RDIGITS-2) */
+    mpd_uint_t p10data[2] = {0, mpd_pow10[MPD_RDIGITS-2]};
     mpd_uint_t dummy, word;
     int n;
 
@@ -6714,7 +6759,12 @@ _mpd_qreciprocal_approx(mpd_t *z, const mpd_t *v, uint32_t *status)
     mpd_setdigits(z);
 }
 
-/* Reciprocal, calculated with Newton's Method. Assumption: result != a. */
+/*
+ * Reciprocal, calculated with Newton's Method. Assumption: result != a.
+ * NOTE: The comments in the function show that certain operations are
+ * exact. The proof for the maximum error is too long to fit in here.
+ * ACL2 proof: maxerror-inverse-complete
+ */
 static void
 _mpd_qreciprocal(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx,
                  uint32_t *status)
@@ -6738,32 +6788,43 @@ _mpd_qreciprocal(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx,
     adj = v->digits + v->exp;
     v->exp = -v->digits;
 
-    /* initial approximation */
+    /* Initial approximation */
     _mpd_qreciprocal_approx(z, v, status);
 
     mpd_maxcontext(&varcontext);
     mpd_maxcontext(&maxcontext);
-    varcontext.round = MPD_ROUND_TRUNC;
-    maxcontext.round = MPD_ROUND_TRUNC;
+    varcontext.round = maxcontext.round = MPD_ROUND_TRUNC;
+    varcontext.emax = maxcontext.emax = MPD_MAX_EMAX + 100;
+    varcontext.emin = maxcontext.emin = MPD_MIN_EMIN - 100;
+    maxcontext.prec = MPD_MAX_PREC + 100;
 
-    maxprec = (v->digits > ctx->prec) ? v->digits : ctx->prec;
+    maxprec = ctx->prec;
     maxprec += 2;
     initprec = MPD_RDIGITS-3;
 
     i = recpr_schedule_prec(klist, maxprec, initprec);
     for (; i >= 0; i--) {
-        mpd_qmul(&s, z, z, &maxcontext, status);
+         /* Loop invariant: z->digits <= klist[i]+7 */
+         /* Let s := z**2, exact result */
+        _mpd_qmul_exact(&s, z, z, &maxcontext, status);
         varcontext.prec = 2*klist[i] + 5;
         if (v->digits > varcontext.prec) {
+            /* Let t := v, truncated to n >= 2*k+5 fraction digits */
             mpd_qshiftr(&t, v, v->digits-varcontext.prec, status);
             t.exp = -varcontext.prec;
+            /* Let t := trunc(v)*s, truncated to n >= 2*k+1 fraction digits */
             mpd_qmul(&t, &t, &s, &varcontext, status);
         }
-        else {
+        else { /* v->digits <= 2*k+5 */
+            /* Let t := v*s, truncated to n >= 2*k+1 fraction digits */
             mpd_qmul(&t, v, &s, &varcontext, status);
         }
-        mpd_qmul(&s, z, &two, &maxcontext, status);
-        mpd_qsub(z, &s, &t, &maxcontext, status);
+        /* Let s := 2*z, exact result */
+        _mpd_qmul_exact(&s, z, &two, &maxcontext, status);
+        /* s.digits < t.digits <= 2*k+5, |adjexp(s)-adjexp(t)| <= 1,
+         * so the subtraction generates at most 2*k+6 <= klist[i+1]+7
+         * digits. The loop invariant is preserved. */
+        _mpd_qsub_exact(z, &s, &t, &maxcontext, status);
     }
 
     if (!mpd_isspecial(z)) {
@@ -6777,22 +6838,29 @@ _mpd_qreciprocal(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx,
 }
 
 /*
- * Integer division with remainder of the coefficients: coeff(a) / coeff(b).
- * This function is for large numbers where it is faster to divide by
- * multiplying the dividend by the reciprocal of the divisor.
- * The inexact result is fixed by a small loop, which should not take
- * more than 2 iterations.
+ * Internal function for large numbers:
+ *
+ *     q, r = divmod(coeff(a), coeff(b))
+ *
+ * Strategy: Multiply the dividend by the reciprocal of the divisor. The
+ * inexact result is fixed by a small loop, using at most 2 iterations.
+ *
+ * ACL2 proofs:
+ * ------------
+ *    1) q is a natural number.  (ndivmod-quotient-natp)
+ *    2) r is a natural number.  (ndivmod-remainder-natp)
+ *    3) a = q * b + r           (ndivmod-q*b+r==a)
+ *    4) r < b                   (ndivmod-remainder-<-b)
  */
 static void
-_mpd_qbarrett_divmod(mpd_t *q, mpd_t *r, const mpd_t *a, const mpd_t *b,
-                     uint32_t *status)
+_mpd_base_ndivmod(mpd_t *q, mpd_t *r, const mpd_t *a, const mpd_t *b,
+                  uint32_t *status)
 {
     mpd_context_t workctx;
     mpd_t *qq = q, *rr = r;
     mpd_t aa, bb;
     int k;
 
-    mpd_maxcontext(&workctx);
     _mpd_copy_shared(&aa, a);
     _mpd_copy_shared(&bb, b);
 
@@ -6814,41 +6882,68 @@ _mpd_qbarrett_divmod(mpd_t *q, mpd_t *r, const mpd_t *a, const mpd_t *b,
         }
     }
 
-    /* maximum length of q + 3 digits */
-    workctx.prec = aa.digits - bb.digits + 1 + 3;
-    /* we get the reciprocal with precision maxlen(q) + 3 */
+    mpd_maxcontext(&workctx);
+
+    /* Let prec := adigits - bdigits + 4 */
+    workctx.prec = a->digits - b->digits + 1 + 3;
+    if (a->digits > MPD_MAX_PREC || workctx.prec > MPD_MAX_PREC) {
+        *status |= MPD_Division_impossible;
+        goto nanresult;
+    }
+
+    /* Let x := _mpd_qreciprocal(b, prec) 
+     * Then x is bounded by:
+     *    1) 1/b - 10**(-prec - bdigits) < x < 1/b + 10**(-prec - bdigits)
+     *    2) 1/b - 10**(-adigits - 4) < x < 1/b + 10**(-adigits - 4)
+     */
     _mpd_qreciprocal(rr, &bb, &workctx, &workctx.status);
 
-    mpd_qmul(qq, &aa, rr, &workctx, &workctx.status);
+    /* Get an estimate for the quotient. Let q := a * x
+     * Then q is bounded by:
+     *    3) a/b - 10**-4 < q < a/b + 10**-4
+     */
+    _mpd_qmul(qq, &aa, rr, &workctx, &workctx.status);
+    /* Truncate q to an integer:
+     *    4) a/b - 2 < trunc(q) < a/b + 1
+     */
     mpd_qtrunc(qq, qq, &workctx, &workctx.status);
 
     workctx.prec = aa.digits + 3;
-    /* get the remainder */
-    mpd_qmul(rr, &bb, qq, &workctx, &workctx.status);
-    mpd_qsub(rr, &aa, rr, &workctx, &workctx.status);
+    workctx.emax = MPD_MAX_EMAX + 3;
+    workctx.emin = MPD_MIN_EMIN - 3;
+    /* Multiply the estimate for q by b:
+     *    5) a - 2 * b < trunc(q) * b < a + b
+     */
+    _mpd_qmul(rr, &bb, qq, &workctx, &workctx.status);
+    /* Get the estimate for r such that a = q * b + r. */
+    _mpd_qsub_exact(rr, &aa, rr, &workctx, &workctx.status);
 
-    /* Fix the result. Algorithm from: Karl Hasselstrom, Fast Division of Large Integers */
+    /* Fix the result. At this point -b < r < 2*b, so the correction loop
+       takes at most one iteration. */
     for (k = 0;; k++) {
-        if (mpd_isspecial(rr)) {
+        if (mpd_isspecial(qq) || mpd_isspecial(rr)) {
             *status |= (workctx.status&MPD_Errors);
             goto nanresult;
         }
-        if (k > 2) {
-            mpd_err_warn("libmpdec: internal error in "          /* GCOV_NOT_REACHED */
-                         "_mpd_qbarrett_divmod: please report"); /* GCOV_NOT_REACHED */
-            *status |= MPD_Invalid_operation;                    /* GCOV_NOT_REACHED */
-            goto nanresult;                                      /* GCOV_NOT_REACHED */
+        if (k > 2) { /* Allow two iterations despite the proof. */
+            mpd_err_warn("libmpdec: internal error in "       /* GCOV_NOT_REACHED */
+                         "_mpd_base_ndivmod: please report"); /* GCOV_NOT_REACHED */
+            *status |= MPD_Invalid_operation;                 /* GCOV_NOT_REACHED */
+            goto nanresult;                                   /* GCOV_NOT_REACHED */
         }
+        /* r < 0 */
         else if (_mpd_cmp(&zero, rr) == 1) {
-            mpd_qadd(rr, rr, &bb, &workctx, &workctx.status);
-            mpd_qadd(qq, qq, &minus_one, &workctx, &workctx.status);
+            _mpd_qadd_exact(rr, rr, &bb, &workctx, &workctx.status);
+            _mpd_qadd_exact(qq, qq, &minus_one, &workctx, &workctx.status);
         }
+        /* 0 <= r < b */
         else if (_mpd_cmp(rr, &bb) == -1) {
             break;
         }
+        /* r >= b */
         else {
-            mpd_qsub(rr, rr, &bb, &workctx, &workctx.status);
-            mpd_qadd(qq, qq, &one, &workctx, &workctx.status);
+            _mpd_qsub_exact(rr, rr, &bb, &workctx, &workctx.status);
+            _mpd_qadd_exact(qq, qq, &one, &workctx, &workctx.status);
         }
     }
 
