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+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements Newton-Raphson division and uses
+// it as an additional test case for bignum.
+//
+// Division of x/y is achieved by computing r = 1/y to
+// obtain the quotient q = x*r = x*(1/y) = x/y. The
+// reciprocal r is the solution for f(x) = 1/x - y and
+// the solution is approximated through iteration. The
+// iteration does not require division.
+
+package bignum
+
+import "testing"
+
+
+// An fpNat is a Natural scaled by a power of two
+// (an unsigned floating point representation). The
+// value of an fpNat x is x.m * 2^x.e .
+//
+type fpNat struct {
+ m Natural
+ e int
+}
+
+
+// sub computes x - y.
+func (x fpNat) sub(y fpNat) fpNat {
+ switch d := x.e - y.e; {
+ case d < 0:
+ return fpNat{x.m.Sub(y.m.Shl(uint(-d))), x.e}
+ case d > 0:
+ return fpNat{x.m.Shl(uint(d)).Sub(y.m), y.e}
+ }
+ return fpNat{x.m.Sub(y.m), x.e}
+}
+
+
+// mul2 computes x*2.
+func (x fpNat) mul2() fpNat { return fpNat{x.m, x.e + 1} }
+
+
+// mul computes x*y.
+func (x fpNat) mul(y fpNat) fpNat { return fpNat{x.m.Mul(y.m), x.e + y.e} }
+
+
+// mant computes the (possibly truncated) Natural representation
+// of an fpNat x.
+//
+func (x fpNat) mant() Natural {
+ switch {
+ case x.e > 0:
+ return x.m.Shl(uint(x.e))
+ case x.e < 0:
+ return x.m.Shr(uint(-x.e))
+ }
+ return x.m
+}
+
+
+// nrDivEst computes an estimate of the quotient q = x0/y0 and returns q.
+// q may be too small (usually by 1).
+//
+func nrDivEst(x0, y0 Natural) Natural {
+ if y0.IsZero() {
+ panic("division by zero")
+ return nil
+ }
+ // y0 > 0
+
+ if y0.Cmp(Nat(1)) == 0 {
+ return x0
+ }
+ // y0 > 1
+
+ switch d := x0.Cmp(y0); {
+ case d < 0:
+ return Nat(0)
+ case d == 0:
+ return Nat(1)
+ }
+ // x0 > y0 > 1
+
+ // Determine maximum result length.
+ maxLen := int(x0.Log2() - y0.Log2() + 1)
+
+ // In the following, each number x is represented
+ // as a mantissa x.m and an exponent x.e such that
+ // x = xm * 2^x.e.
+ x := fpNat{x0, 0}
+ y := fpNat{y0, 0}
+
+ // Determine a scale factor f = 2^e such that
+ // 0.5 <= y/f == y*(2^-e) < 1.0
+ // and scale y accordingly.
+ e := int(y.m.Log2()) + 1
+ y.e -= e
+
+ // t1
+ var c = 2.9142
+ const n = 14
+ t1 := fpNat{Nat(uint64(c * (1 << n))), -n}
+
+ // Compute initial value r0 for the reciprocal of y/f.
+ // r0 = t1 - 2*y
+ r := t1.sub(y.mul2())
+ two := fpNat{Nat(2), 0}
+
+ // Newton-Raphson iteration
+ p := Nat(0)
+ for i := 0; ; i++ {
+ // check if we are done
+ // TODO: Need to come up with a better test here
+ // as it will reduce computation time significantly.
+ // q = x*r/f
+ q := x.mul(r)
+ q.e -= e
+ res := q.mant()
+ if res.Cmp(p) == 0 {
+ return res
+ }
+ p = res
+
+ // r' = r*(2 - y*r)
+ r = r.mul(two.sub(y.mul(r)))
+
+ // reduce mantissa size
+ // TODO: Find smaller bound as it will reduce
+ // computation time massively.
+ d := int(r.m.Log2()+1) - maxLen
+ if d > 0 {
+ r = fpNat{r.m.Shr(uint(d)), r.e + d}
+ }
+ }
+
+ panic("unreachable")
+ return nil
+}
+
+
+func nrdiv(x, y Natural) (q, r Natural) {
+ q = nrDivEst(x, y)
+ r = x.Sub(y.Mul(q))
+ // if r is too large, correct q and r
+ // (usually one iteration)
+ for r.Cmp(y) >= 0 {
+ q = q.Add(Nat(1))
+ r = r.Sub(y)
+ }
+ return
+}
+
+
+func div(t *testing.T, x, y Natural) {
+ q, r := nrdiv(x, y)
+ qx, rx := x.DivMod(y)
+ if q.Cmp(qx) != 0 {
+ t.Errorf("x = %s, y = %s, got q = %s, want q = %s", x, y, q, qx)
+ }
+ if r.Cmp(rx) != 0 {
+ t.Errorf("x = %s, y = %s, got r = %s, want r = %s", x, y, r, rx)
+ }
+}
+
+
+func idiv(t *testing.T, x0, y0 uint64) { div(t, Nat(x0), Nat(y0)) }
+
+
+func TestNRDiv(t *testing.T) {
+ idiv(t, 17, 18)
+ idiv(t, 17, 17)
+ idiv(t, 17, 1)
+ idiv(t, 17, 16)
+ idiv(t, 17, 10)
+ idiv(t, 17, 9)
+ idiv(t, 17, 8)
+ idiv(t, 17, 5)
+ idiv(t, 17, 3)
+ idiv(t, 1025, 512)
+ idiv(t, 7489595, 2)
+ idiv(t, 5404679459, 78495)
+ idiv(t, 7484890589595, 7484890589594)
+ div(t, Fact(100), Fact(91))
+ div(t, Fact(1000), Fact(991))
+ //div(t, Fact(10000), Fact(9991)); // takes too long - disabled for now
+}