Daily bump.

[official-gcc.git] / libgo / go / math / big / nat_test.go

// 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.

package big

import (
        "fmt"
        "runtime"
        "strings"
        "testing"
)

var cmpTests = []struct {
        x, y nat
        r    int
}{
        {nil, nil, 0},
        {nil, nat(nil), 0},
        {nat(nil), nil, 0},
        {nat(nil), nat(nil), 0},
        {nat{0}, nat{0}, 0},
        {nat{0}, nat{1}, -1},
        {nat{1}, nat{0}, 1},
        {nat{1}, nat{1}, 0},
        {nat{0, _M}, nat{1}, 1},
        {nat{1}, nat{0, _M}, -1},
        {nat{1, _M}, nat{0, _M}, 1},
        {nat{0, _M}, nat{1, _M}, -1},
        {nat{16, 571956, 8794, 68}, nat{837, 9146, 1, 754489}, -1},
        {nat{34986, 41, 105, 1957}, nat{56, 7458, 104, 1957}, 1},
}

func TestCmp(t *testing.T) {
        for i, a := range cmpTests {
                r := a.x.cmp(a.y)
                if r != a.r {
                        t.Errorf("#%d got r = %v; want %v", i, r, a.r)
                }
        }
}

type funNN func(z, x, y nat) nat
type argNN struct {
        z, x, y nat
}

var sumNN = []argNN{
        {},
        {nat{1}, nil, nat{1}},
        {nat{1111111110}, nat{123456789}, nat{987654321}},
        {nat{0, 0, 0, 1}, nil, nat{0, 0, 0, 1}},
        {nat{0, 0, 0, 1111111110}, nat{0, 0, 0, 123456789}, nat{0, 0, 0, 987654321}},
        {nat{0, 0, 0, 1}, nat{0, 0, _M}, nat{0, 0, 1}},
}

var prodNN = []argNN{
        {},
        {nil, nil, nil},
        {nil, nat{991}, nil},
        {nat{991}, nat{991}, nat{1}},
        {nat{991 * 991}, nat{991}, nat{991}},
        {nat{0, 0, 991 * 991}, nat{0, 991}, nat{0, 991}},
        {nat{1 * 991, 2 * 991, 3 * 991, 4 * 991}, nat{1, 2, 3, 4}, nat{991}},
        {nat{4, 11, 20, 30, 20, 11, 4}, nat{1, 2, 3, 4}, nat{4, 3, 2, 1}},
        // 3^100 * 3^28 = 3^128
        {
                natFromString("11790184577738583171520872861412518665678211592275841109096961"),
                natFromString("515377520732011331036461129765621272702107522001"),
                natFromString("22876792454961"),
        },
        // z = 111....1 (70000 digits)
        // x = 10^(99*700) + ... + 10^1400 + 10^700 + 1
        // y = 111....1 (700 digits, larger than Karatsuba threshold on 32-bit and 64-bit)
        {
                natFromString(strings.Repeat("1", 70000)),
                natFromString("1" + strings.Repeat(strings.Repeat("0", 699)+"1", 99)),
                natFromString(strings.Repeat("1", 700)),
        },
        // z = 111....1 (20000 digits)
        // x = 10^10000 + 1
        // y = 111....1 (10000 digits)
        {
                natFromString(strings.Repeat("1", 20000)),
                natFromString("1" + strings.Repeat("0", 9999) + "1"),
                natFromString(strings.Repeat("1", 10000)),
        },
}

func natFromString(s string) nat {
        x, _, _, err := nat(nil).scan(strings.NewReader(s), 0, false)
        if err != nil {
                panic(err)
        }
        return x
}

func TestSet(t *testing.T) {
        for _, a := range sumNN {
                z := nat(nil).set(a.z)
                if z.cmp(a.z) != 0 {
                        t.Errorf("got z = %v; want %v", z, a.z)
                }
        }
}

func testFunNN(t *testing.T, msg string, f funNN, a argNN) {
        z := f(nil, a.x, a.y)
        if z.cmp(a.z) != 0 {
                t.Errorf("%s%+v\n\tgot z = %v; want %v", msg, a, z, a.z)
        }
}

func TestFunNN(t *testing.T) {
        for _, a := range sumNN {
                arg := a
                testFunNN(t, "add", nat.add, arg)

                arg = argNN{a.z, a.y, a.x}
                testFunNN(t, "add symmetric", nat.add, arg)

                arg = argNN{a.x, a.z, a.y}
                testFunNN(t, "sub", nat.sub, arg)

                arg = argNN{a.y, a.z, a.x}
                testFunNN(t, "sub symmetric", nat.sub, arg)
        }

        for _, a := range prodNN {
                arg := a
                testFunNN(t, "mul", nat.mul, arg)

                arg = argNN{a.z, a.y, a.x}
                testFunNN(t, "mul symmetric", nat.mul, arg)
        }
}

var mulRangesN = []struct {
        a, b uint64
        prod string
}{
        {0, 0, "0"},
        {1, 1, "1"},
        {1, 2, "2"},
        {1, 3, "6"},
        {10, 10, "10"},
        {0, 100, "0"},
        {0, 1e9, "0"},
        {1, 0, "1"},                    // empty range
        {100, 1, "1"},                  // empty range
        {1, 10, "3628800"},             // 10!
        {1, 20, "2432902008176640000"}, // 20!
        {1, 100,
                "933262154439441526816992388562667004907159682643816214685929" +
                        "638952175999932299156089414639761565182862536979208272237582" +
                        "51185210916864000000000000000000000000", // 100!
        },
}

func TestMulRangeN(t *testing.T) {
        for i, r := range mulRangesN {
                prod := string(nat(nil).mulRange(r.a, r.b).utoa(10))
                if prod != r.prod {
                        t.Errorf("#%d: got %s; want %s", i, prod, r.prod)
                }
        }
}

// allocBytes returns the number of bytes allocated by invoking f.
func allocBytes(f func()) uint64 {
        var stats runtime.MemStats
        runtime.ReadMemStats(&stats)
        t := stats.TotalAlloc
        f()
        runtime.ReadMemStats(&stats)
        return stats.TotalAlloc - t
}

// TestMulUnbalanced tests that multiplying numbers of different lengths
// does not cause deep recursion and in turn allocate too much memory.
// Test case for issue 3807.
func TestMulUnbalanced(t *testing.T) {
        defer runtime.GOMAXPROCS(runtime.GOMAXPROCS(1))
        x := rndNat(50000)
        y := rndNat(40)
        allocSize := allocBytes(func() {
                nat(nil).mul(x, y)
        })
        inputSize := uint64(len(x)+len(y)) * _S
        if ratio := allocSize / uint64(inputSize); ratio > 10 {
                t.Errorf("multiplication uses too much memory (%d > %d times the size of inputs)", allocSize, ratio)
        }
}

// rndNat returns a random nat value >= 0 of (usually) n words in length.
// In extremely unlikely cases it may be smaller than n words if the top-
// most words are 0.
func rndNat(n int) nat {
        return nat(rndV(n)).norm()
}

// rndNat1 is like rndNat but the result is guaranteed to be > 0.
func rndNat1(n int) nat {
        x := nat(rndV(n)).norm()
        if len(x) == 0 {
                x.setWord(1)
        }
        return x
}

func BenchmarkMul(b *testing.B) {
        mulx := rndNat(1e4)
        muly := rndNat(1e4)
        b.ResetTimer()
        for i := 0; i < b.N; i++ {
                var z nat
                z.mul(mulx, muly)
        }
}

func benchmarkNatMul(b *testing.B, nwords int) {
        x := rndNat(nwords)
        y := rndNat(nwords)
        var z nat
        b.ResetTimer()
        for i := 0; i < b.N; i++ {
                z.mul(x, y)
        }
}

var mulBenchSizes = []int{10, 100, 1000, 10000, 100000}

func BenchmarkNatMul(b *testing.B) {
        for _, n := range mulBenchSizes {
                if isRaceBuilder && n > 1e3 {
                        continue
                }
                b.Run(fmt.Sprintf("%d", n), func(b *testing.B) {
                        benchmarkNatMul(b, n)
                })
        }
}

func TestNLZ(t *testing.T) {
        var x Word = _B >> 1
        for i := 0; i <= _W; i++ {
                if int(nlz(x)) != i {
                        t.Errorf("failed at %x: got %d want %d", x, nlz(x), i)
                }
                x >>= 1
        }
}

type shiftTest struct {
        in    nat
        shift uint
        out   nat
}

var leftShiftTests = []shiftTest{
        {nil, 0, nil},
        {nil, 1, nil},
        {natOne, 0, natOne},
        {natOne, 1, natTwo},
        {nat{1 << (_W - 1)}, 1, nat{0}},
        {nat{1 << (_W - 1), 0}, 1, nat{0, 1}},
}

func TestShiftLeft(t *testing.T) {
        for i, test := range leftShiftTests {
                var z nat
                z = z.shl(test.in, test.shift)
                for j, d := range test.out {
                        if j >= len(z) || z[j] != d {
                                t.Errorf("#%d: got: %v want: %v", i, z, test.out)
                                break
                        }
                }
        }
}

var rightShiftTests = []shiftTest{
        {nil, 0, nil},
        {nil, 1, nil},
        {natOne, 0, natOne},
        {natOne, 1, nil},
        {natTwo, 1, natOne},
        {nat{0, 1}, 1, nat{1 << (_W - 1)}},
        {nat{2, 1, 1}, 1, nat{1<<(_W-1) + 1, 1 << (_W - 1)}},
}

func TestShiftRight(t *testing.T) {
        for i, test := range rightShiftTests {
                var z nat
                z = z.shr(test.in, test.shift)
                for j, d := range test.out {
                        if j >= len(z) || z[j] != d {
                                t.Errorf("#%d: got: %v want: %v", i, z, test.out)
                                break
                        }
                }
        }
}

func BenchmarkZeroShifts(b *testing.B) {
        x := rndNat(800)

        b.Run("Shl", func(b *testing.B) {
                for i := 0; i < b.N; i++ {
                        var z nat
                        z.shl(x, 0)
                }
        })
        b.Run("ShlSame", func(b *testing.B) {
                for i := 0; i < b.N; i++ {
                        x.shl(x, 0)
                }
        })

        b.Run("Shr", func(b *testing.B) {
                for i := 0; i < b.N; i++ {
                        var z nat
                        z.shr(x, 0)
                }
        })
        b.Run("ShrSame", func(b *testing.B) {
                for i := 0; i < b.N; i++ {
                        x.shr(x, 0)
                }
        })
}

type modWTest struct {
        in       string
        dividend string
        out      string
}

var modWTests32 = []modWTest{
        {"23492635982634928349238759823742", "252341", "220170"},
}

var modWTests64 = []modWTest{
        {"6527895462947293856291561095690465243862946", "524326975699234", "375066989628668"},
}

func runModWTests(t *testing.T, tests []modWTest) {
        for i, test := range tests {
                in, _ := new(Int).SetString(test.in, 10)
                d, _ := new(Int).SetString(test.dividend, 10)
                out, _ := new(Int).SetString(test.out, 10)

                r := in.abs.modW(d.abs[0])
                if r != out.abs[0] {
                        t.Errorf("#%d failed: got %d want %s", i, r, out)
                }
        }
}

func TestModW(t *testing.T) {
        if _W >= 32 {
                runModWTests(t, modWTests32)
        }
        if _W >= 64 {
                runModWTests(t, modWTests64)
        }
}

var montgomeryTests = []struct {
        x, y, m      string
        k0           uint64
        out32, out64 string
}{
        {
                "0xffffffffffffffffffffffffffffffffffffffffffffffffe",
                "0xffffffffffffffffffffffffffffffffffffffffffffffffe",
                "0xfffffffffffffffffffffffffffffffffffffffffffffffff",
                1,
                "0x1000000000000000000000000000000000000000000",
                "0x10000000000000000000000000000000000",
        },
        {
                "0x000000000ffffff5",
                "0x000000000ffffff0",
                "0x0000000010000001",
                0xff0000000fffffff,
                "0x000000000bfffff4",
                "0x0000000003400001",
        },
        {
                "0x0000000080000000",
                "0x00000000ffffffff",
                "0x1000000000000001",
                0xfffffffffffffff,
                "0x0800000008000001",
                "0x0800000008000001",
        },
        {
                "0x0000000080000000",
                "0x0000000080000000",
                "0xffffffff00000001",
                0xfffffffeffffffff,
                "0xbfffffff40000001",
                "0xbfffffff40000001",
        },
        {
                "0x0000000080000000",
                "0x0000000080000000",
                "0x00ffffff00000001",
                0xfffffeffffffff,
                "0xbfffff40000001",
                "0xbfffff40000001",
        },
        {
                "0x0000000080000000",
                "0x0000000080000000",
                "0x0000ffff00000001",
                0xfffeffffffff,
                "0xbfff40000001",
                "0xbfff40000001",
        },
        {
                "0x3321ffffffffffffffffffffffffffff00000000000022222623333333332bbbb888c0",
                "0x3321ffffffffffffffffffffffffffff00000000000022222623333333332bbbb888c0",
                "0x33377fffffffffffffffffffffffffffffffffffffffffffff0000000000022222eee1",
                0xdecc8f1249812adf,
                "0x04eb0e11d72329dc0915f86784820fc403275bf2f6620a20e0dd344c5cd0875e50deb5",
                "0x0d7144739a7d8e11d72329dc0915f86784820fc403275bf2f61ed96f35dd34dbb3d6a0",
        },
        {
                "0x10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000ffffffffffffffffffffffffffffffff00000000000022222223333333333444444444",
                "0x10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000ffffffffffffffffffffffffffffffff999999999999999aaabbbbbbbbcccccccccccc",
                "0xffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff33377fffffffffffffffffffffffffffffffffffffffffffff0000000000022222eee1",
                0xdecc8f1249812adf,
                "0x5c0d52f451aec609b15da8e5e5626c4eaa88723bdeac9d25ca9b961269400410ca208a16af9c2fb07d7a11c7772cba02c22f9711078d51a3797eb18e691295293284d988e349fa6deba46b25a4ecd9f715",
                "0x92fcad4b5c0d52f451aec609b15da8e5e5626c4eaa88723bdeac9d25ca9b961269400410ca208a16af9c2fb07d799c32fe2f3cc5422f9711078d51a3797eb18e691295293284d8f5e69caf6decddfe1df6",
        },
}

func TestMontgomery(t *testing.T) {
        one := NewInt(1)
        _B := new(Int).Lsh(one, _W)
        for i, test := range montgomeryTests {
                x := natFromString(test.x)
                y := natFromString(test.y)
                m := natFromString(test.m)
                for len(x) < len(m) {
                        x = append(x, 0)
                }
                for len(y) < len(m) {
                        y = append(y, 0)
                }

                if x.cmp(m) > 0 {
                        _, r := nat(nil).div(nil, x, m)
                        t.Errorf("#%d: x > m (0x%s > 0x%s; use 0x%s)", i, x.utoa(16), m.utoa(16), r.utoa(16))
                }
                if y.cmp(m) > 0 {
                        _, r := nat(nil).div(nil, x, m)
                        t.Errorf("#%d: y > m (0x%s > 0x%s; use 0x%s)", i, y.utoa(16), m.utoa(16), r.utoa(16))
                }

                var out nat
                if _W == 32 {
                        out = natFromString(test.out32)
                } else {
                        out = natFromString(test.out64)
                }

                // t.Logf("#%d: len=%d\n", i, len(m))

                // check output in table
                xi := &Int{abs: x}
                yi := &Int{abs: y}
                mi := &Int{abs: m}
                p := new(Int).Mod(new(Int).Mul(xi, new(Int).Mul(yi, new(Int).ModInverse(new(Int).Lsh(one, uint(len(m))*_W), mi))), mi)
                if out.cmp(p.abs.norm()) != 0 {
                        t.Errorf("#%d: out in table=0x%s, computed=0x%s", i, out.utoa(16), p.abs.norm().utoa(16))
                }

                // check k0 in table
                k := new(Int).Mod(&Int{abs: m}, _B)
                k = new(Int).Sub(_B, k)
                k = new(Int).Mod(k, _B)
                k0 := Word(new(Int).ModInverse(k, _B).Uint64())
                if k0 != Word(test.k0) {
                        t.Errorf("#%d: k0 in table=%#x, computed=%#x\n", i, test.k0, k0)
                }

                // check montgomery with correct k0 produces correct output
                z := nat(nil).montgomery(x, y, m, k0, len(m))
                z = z.norm()
                if z.cmp(out) != 0 {
                        t.Errorf("#%d: got 0x%s want 0x%s", i, z.utoa(16), out.utoa(16))
                }
        }
}

var expNNTests = []struct {
        x, y, m string
        out     string
}{
        {"0", "0", "0", "1"},
        {"0", "0", "1", "0"},
        {"1", "1", "1", "0"},
        {"2", "1", "1", "0"},
        {"2", "2", "1", "0"},
        {"10", "100000000000", "1", "0"},
        {"0x8000000000000000", "2", "", "0x40000000000000000000000000000000"},
        {"0x8000000000000000", "2", "6719", "4944"},
        {"0x8000000000000000", "3", "6719", "5447"},
        {"0x8000000000000000", "1000", "6719", "1603"},
        {"0x8000000000000000", "1000000", "6719", "3199"},
        {
                "2938462938472983472983659726349017249287491026512746239764525612965293865296239471239874193284792387498274256129746192347",
                "298472983472983471903246121093472394872319615612417471234712061",
                "29834729834729834729347290846729561262544958723956495615629569234729836259263598127342374289365912465901365498236492183464",
                "23537740700184054162508175125554701713153216681790245129157191391322321508055833908509185839069455749219131480588829346291",
        },
        {
                "11521922904531591643048817447554701904414021819823889996244743037378330903763518501116638828335352811871131385129455853417360623007349090150042001944696604737499160174391019030572483602867266711107136838523916077674888297896995042968746762200926853379",
                "426343618817810911523",
                "444747819283133684179",
                "42",
        },
}

func TestExpNN(t *testing.T) {
        for i, test := range expNNTests {
                x := natFromString(test.x)
                y := natFromString(test.y)
                out := natFromString(test.out)

                var m nat
                if len(test.m) > 0 {
                        m = natFromString(test.m)
                }

                z := nat(nil).expNN(x, y, m)
                if z.cmp(out) != 0 {
                        t.Errorf("#%d got %s want %s", i, z.utoa(10), out.utoa(10))
                }
        }
}

func BenchmarkExp3Power(b *testing.B) {
        const x = 3
        for _, y := range []Word{
                0x10, 0x40, 0x100, 0x400, 0x1000, 0x4000, 0x10000, 0x40000, 0x100000, 0x400000,
        } {
                b.Run(fmt.Sprintf("%#x", y), func(b *testing.B) {
                        var z nat
                        for i := 0; i < b.N; i++ {
                                z.expWW(x, y)
                        }
                })
        }
}

func fibo(n int) nat {
        switch n {
        case 0:
                return nil
        case 1:
                return nat{1}
        }
        f0 := fibo(0)
        f1 := fibo(1)
        var f2 nat
        for i := 1; i < n; i++ {
                f2 = f2.add(f0, f1)
                f0, f1, f2 = f1, f2, f0
        }
        return f1
}

var fiboNums = []string{
        "0",
        "55",
        "6765",
        "832040",
        "102334155",
        "12586269025",
        "1548008755920",
        "190392490709135",
        "23416728348467685",
        "2880067194370816120",
        "354224848179261915075",
}

func TestFibo(t *testing.T) {
        for i, want := range fiboNums {
                n := i * 10
                got := string(fibo(n).utoa(10))
                if got != want {
                        t.Errorf("fibo(%d) failed: got %s want %s", n, got, want)
                }
        }
}

func BenchmarkFibo(b *testing.B) {
        for i := 0; i < b.N; i++ {
                fibo(1e0)
                fibo(1e1)
                fibo(1e2)
                fibo(1e3)
                fibo(1e4)
                fibo(1e5)
        }
}

var bitTests = []struct {
        x    string
        i    uint
        want uint
}{
        {"0", 0, 0},
        {"0", 1, 0},
        {"0", 1000, 0},

        {"0x1", 0, 1},
        {"0x10", 0, 0},
        {"0x10", 3, 0},
        {"0x10", 4, 1},
        {"0x10", 5, 0},

        {"0x8000000000000000", 62, 0},
        {"0x8000000000000000", 63, 1},
        {"0x8000000000000000", 64, 0},

        {"0x3" + strings.Repeat("0", 32), 127, 0},
        {"0x3" + strings.Repeat("0", 32), 128, 1},
        {"0x3" + strings.Repeat("0", 32), 129, 1},
        {"0x3" + strings.Repeat("0", 32), 130, 0},
}

func TestBit(t *testing.T) {
        for i, test := range bitTests {
                x := natFromString(test.x)
                if got := x.bit(test.i); got != test.want {
                        t.Errorf("#%d: %s.bit(%d) = %v; want %v", i, test.x, test.i, got, test.want)
                }
        }
}

var stickyTests = []struct {
        x    string
        i    uint
        want uint
}{
        {"0", 0, 0},
        {"0", 1, 0},
        {"0", 1000, 0},

        {"0x1", 0, 0},
        {"0x1", 1, 1},

        {"0x1350", 0, 0},
        {"0x1350", 4, 0},
        {"0x1350", 5, 1},

        {"0x8000000000000000", 63, 0},
        {"0x8000000000000000", 64, 1},

        {"0x1" + strings.Repeat("0", 100), 400, 0},
        {"0x1" + strings.Repeat("0", 100), 401, 1},
}

func TestSticky(t *testing.T) {
        for i, test := range stickyTests {
                x := natFromString(test.x)
                if got := x.sticky(test.i); got != test.want {
                        t.Errorf("#%d: %s.sticky(%d) = %v; want %v", i, test.x, test.i, got, test.want)
                }
                if test.want == 1 {
                        // all subsequent i's should also return 1
                        for d := uint(1); d <= 3; d++ {
                                if got := x.sticky(test.i + d); got != 1 {
                                        t.Errorf("#%d: %s.sticky(%d) = %v; want %v", i, test.x, test.i+d, got, 1)
                                }
                        }
                }
        }
}

func testSqr(t *testing.T, x nat) {
        got := make(nat, 2*len(x))
        want := make(nat, 2*len(x))
        got = got.sqr(x)
        want = want.mul(x, x)
        if got.cmp(want) != 0 {
                t.Errorf("basicSqr(%v), got %v, want %v", x, got, want)
        }
}

func TestSqr(t *testing.T) {
        for _, a := range prodNN {
                if a.x != nil {
                        testSqr(t, a.x)
                }
                if a.y != nil {
                        testSqr(t, a.y)
                }
                if a.z != nil {
                        testSqr(t, a.z)
                }
        }
}

func benchmarkNatSqr(b *testing.B, nwords int) {
        x := rndNat(nwords)
        var z nat
        b.ResetTimer()
        for i := 0; i < b.N; i++ {
                z.sqr(x)
        }
}

var sqrBenchSizes = []int{
        1, 2, 3, 5, 8, 10, 20, 30, 50, 80,
        100, 200, 300, 500, 800,
        1000, 10000, 100000,
}

func BenchmarkNatSqr(b *testing.B) {
        for _, n := range sqrBenchSizes {
                if isRaceBuilder && n > 1e3 {
                        continue
                }
                b.Run(fmt.Sprintf("%d", n), func(b *testing.B) {
                        benchmarkNatSqr(b, n)
                })
        }
}

func BenchmarkNatSetBytes(b *testing.B) {
        const maxLength = 128
        lengths := []int{
                // No remainder:
                8, 24, maxLength,
                // With remainder:
                7, 23, maxLength - 1,
        }
        n := make(nat, maxLength/_W) // ensure n doesn't need to grow during the test
        buf := make([]byte, maxLength)
        for _, l := range lengths {
                b.Run(fmt.Sprint(l), func(b *testing.B) {
                        for i := 0; i < b.N; i++ {
                                n.setBytes(buf[:l])
                        }
                })
        }
}

func TestNatDiv(t *testing.T) {
        sizes := []int{
                1, 2, 5, 8, 15, 25, 40, 65, 100,
                200, 500, 800, 1500, 2500, 4000, 6500, 10000,
        }
        for _, i := range sizes {
                for _, j := range sizes {
                        a := rndNat1(i)
                        b := rndNat1(j)
                        // the test requires b >= 2
                        if len(b) == 1 && b[0] == 1 {
                                b[0] = 2
                        }
                        // choose a remainder c < b
                        c := rndNat1(len(b))
                        if len(c) == len(b) && c[len(c)-1] >= b[len(b)-1] {
                                c[len(c)-1] = 0
                                c = c.norm()
                        }
                        // compute x = a*b+c
                        x := nat(nil).mul(a, b)
                        x = x.add(x, c)

                        var q, r nat
                        q, r = q.div(r, x, b)
                        if q.cmp(a) != 0 {
                                t.Fatalf("wrong quotient: got %s; want %s for %s/%s", q.utoa(10), a.utoa(10), x.utoa(10), b.utoa(10))
                        }
                        if r.cmp(c) != 0 {
                                t.Fatalf("wrong remainder: got %s; want %s for %s/%s", r.utoa(10), c.utoa(10), x.utoa(10), b.utoa(10))
                        }
                }
        }
}

// TestIssue37499 triggers the edge case of divBasic where
// the inaccurate estimate of the first word's quotient
// happens at the very beginning of the loop.
func TestIssue37499(t *testing.T) {
        // Choose u and v such that v is slightly larger than u >> N.
        // This tricks divBasic into choosing 1 as the first word
        // of the quotient. This works in both 32-bit and 64-bit settings.
        u := natFromString("0x2b6c385a05be027f5c22005b63c42a1165b79ff510e1706b39f8489c1d28e57bb5ba4ef9fd9387a3e344402c0a453381")
        v := natFromString("0x2b6c385a05be027f5c22005b63c42a1165b79ff510e1706c")

        q := nat(nil).make(8)
        q.divBasic(u, v)
        q = q.norm()
        if s := string(q.utoa(16)); s != "fffffffffffffffffffffffffffffffffffffffffffffffb" {
                t.Fatalf("incorrect quotient: %s", s)
        }
}

// TestIssue42552 triggers an edge case of recursive division
// where the first division loop is never entered, and correcting
// the remainder takes exactly two iterations in the final loop.
func TestIssue42552(t *testing.T) {
        u := natFromString("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")
        v := natFromString("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")
        q := nat(nil).make(16)
        q.div(q, u, v)
}