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Source file src/math/big/int_test.go

Documentation: math/big

  // 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 (
  	"bytes"
  	"encoding/hex"
  	"math/rand"
  	"strconv"
  	"strings"
  	"testing"
  	"testing/quick"
  )
  
  func isNormalized(x *Int) bool {
  	if len(x.abs) == 0 {
  		return !x.neg
  	}
  	// len(x.abs) > 0
  	return x.abs[len(x.abs)-1] != 0
  }
  
  type funZZ func(z, x, y *Int) *Int
  type argZZ struct {
  	z, x, y *Int
  }
  
  var sumZZ = []argZZ{
  	{NewInt(0), NewInt(0), NewInt(0)},
  	{NewInt(1), NewInt(1), NewInt(0)},
  	{NewInt(1111111110), NewInt(123456789), NewInt(987654321)},
  	{NewInt(-1), NewInt(-1), NewInt(0)},
  	{NewInt(864197532), NewInt(-123456789), NewInt(987654321)},
  	{NewInt(-1111111110), NewInt(-123456789), NewInt(-987654321)},
  }
  
  var prodZZ = []argZZ{
  	{NewInt(0), NewInt(0), NewInt(0)},
  	{NewInt(0), NewInt(1), NewInt(0)},
  	{NewInt(1), NewInt(1), NewInt(1)},
  	{NewInt(-991 * 991), NewInt(991), NewInt(-991)},
  	// TODO(gri) add larger products
  }
  
  func TestSignZ(t *testing.T) {
  	var zero Int
  	for _, a := range sumZZ {
  		s := a.z.Sign()
  		e := a.z.Cmp(&zero)
  		if s != e {
  			t.Errorf("got %d; want %d for z = %v", s, e, a.z)
  		}
  	}
  }
  
  func TestSetZ(t *testing.T) {
  	for _, a := range sumZZ {
  		var z Int
  		z.Set(a.z)
  		if !isNormalized(&z) {
  			t.Errorf("%v is not normalized", z)
  		}
  		if (&z).Cmp(a.z) != 0 {
  			t.Errorf("got z = %v; want %v", z, a.z)
  		}
  	}
  }
  
  func TestAbsZ(t *testing.T) {
  	var zero Int
  	for _, a := range sumZZ {
  		var z Int
  		z.Abs(a.z)
  		var e Int
  		e.Set(a.z)
  		if e.Cmp(&zero) < 0 {
  			e.Sub(&zero, &e)
  		}
  		if z.Cmp(&e) != 0 {
  			t.Errorf("got z = %v; want %v", z, e)
  		}
  	}
  }
  
  func testFunZZ(t *testing.T, msg string, f funZZ, a argZZ) {
  	var z Int
  	f(&z, a.x, a.y)
  	if !isNormalized(&z) {
  		t.Errorf("%s%v is not normalized", msg, z)
  	}
  	if (&z).Cmp(a.z) != 0 {
  		t.Errorf("%s%+v\n\tgot z = %v; want %v", msg, a, &z, a.z)
  	}
  }
  
  func TestSumZZ(t *testing.T) {
  	AddZZ := func(z, x, y *Int) *Int { return z.Add(x, y) }
  	SubZZ := func(z, x, y *Int) *Int { return z.Sub(x, y) }
  	for _, a := range sumZZ {
  		arg := a
  		testFunZZ(t, "AddZZ", AddZZ, arg)
  
  		arg = argZZ{a.z, a.y, a.x}
  		testFunZZ(t, "AddZZ symmetric", AddZZ, arg)
  
  		arg = argZZ{a.x, a.z, a.y}
  		testFunZZ(t, "SubZZ", SubZZ, arg)
  
  		arg = argZZ{a.y, a.z, a.x}
  		testFunZZ(t, "SubZZ symmetric", SubZZ, arg)
  	}
  }
  
  func TestProdZZ(t *testing.T) {
  	MulZZ := func(z, x, y *Int) *Int { return z.Mul(x, y) }
  	for _, a := range prodZZ {
  		arg := a
  		testFunZZ(t, "MulZZ", MulZZ, arg)
  
  		arg = argZZ{a.z, a.y, a.x}
  		testFunZZ(t, "MulZZ symmetric", MulZZ, arg)
  	}
  }
  
  // mulBytes returns x*y via grade school multiplication. Both inputs
  // and the result are assumed to be in big-endian representation (to
  // match the semantics of Int.Bytes and Int.SetBytes).
  func mulBytes(x, y []byte) []byte {
  	z := make([]byte, len(x)+len(y))
  
  	// multiply
  	k0 := len(z) - 1
  	for j := len(y) - 1; j >= 0; j-- {
  		d := int(y[j])
  		if d != 0 {
  			k := k0
  			carry := 0
  			for i := len(x) - 1; i >= 0; i-- {
  				t := int(z[k]) + int(x[i])*d + carry
  				z[k], carry = byte(t), t>>8
  				k--
  			}
  			z[k] = byte(carry)
  		}
  		k0--
  	}
  
  	// normalize (remove leading 0's)
  	i := 0
  	for i < len(z) && z[i] == 0 {
  		i++
  	}
  
  	return z[i:]
  }
  
  func checkMul(a, b []byte) bool {
  	var x, y, z1 Int
  	x.SetBytes(a)
  	y.SetBytes(b)
  	z1.Mul(&x, &y)
  
  	var z2 Int
  	z2.SetBytes(mulBytes(a, b))
  
  	return z1.Cmp(&z2) == 0
  }
  
  func TestMul(t *testing.T) {
  	if err := quick.Check(checkMul, nil); err != nil {
  		t.Error(err)
  	}
  }
  
  var mulRangesZ = []struct {
  	a, b int64
  	prod string
  }{
  	// entirely positive ranges are covered by mulRangesN
  	{-1, 1, "0"},
  	{-2, -1, "2"},
  	{-3, -2, "6"},
  	{-3, -1, "-6"},
  	{1, 3, "6"},
  	{-10, -10, "-10"},
  	{0, -1, "1"},                      // empty range
  	{-1, -100, "1"},                   // empty range
  	{-1, 1, "0"},                      // range includes 0
  	{-1e9, 0, "0"},                    // range includes 0
  	{-1e9, 1e9, "0"},                  // range includes 0
  	{-10, -1, "3628800"},              // 10!
  	{-20, -2, "-2432902008176640000"}, // -20!
  	{-99, -1,
  		"-933262154439441526816992388562667004907159682643816214685929" +
  			"638952175999932299156089414639761565182862536979208272237582" +
  			"511852109168640000000000000000000000", // -99!
  	},
  }
  
  func TestMulRangeZ(t *testing.T) {
  	var tmp Int
  	// test entirely positive ranges
  	for i, r := range mulRangesN {
  		prod := tmp.MulRange(int64(r.a), int64(r.b)).String()
  		if prod != r.prod {
  			t.Errorf("#%da: got %s; want %s", i, prod, r.prod)
  		}
  	}
  	// test other ranges
  	for i, r := range mulRangesZ {
  		prod := tmp.MulRange(r.a, r.b).String()
  		if prod != r.prod {
  			t.Errorf("#%db: got %s; want %s", i, prod, r.prod)
  		}
  	}
  }
  
  func TestBinomial(t *testing.T) {
  	var z Int
  	for _, test := range []struct {
  		n, k int64
  		want string
  	}{
  		{0, 0, "1"},
  		{0, 1, "0"},
  		{1, 0, "1"},
  		{1, 1, "1"},
  		{1, 10, "0"},
  		{4, 0, "1"},
  		{4, 1, "4"},
  		{4, 2, "6"},
  		{4, 3, "4"},
  		{4, 4, "1"},
  		{10, 1, "10"},
  		{10, 9, "10"},
  		{10, 5, "252"},
  		{11, 5, "462"},
  		{11, 6, "462"},
  		{100, 10, "17310309456440"},
  		{100, 90, "17310309456440"},
  		{1000, 10, "263409560461970212832400"},
  		{1000, 990, "263409560461970212832400"},
  	} {
  		if got := z.Binomial(test.n, test.k).String(); got != test.want {
  			t.Errorf("Binomial(%d, %d) = %s; want %s", test.n, test.k, got, test.want)
  		}
  	}
  }
  
  func BenchmarkBinomial(b *testing.B) {
  	var z Int
  	for i := b.N - 1; i >= 0; i-- {
  		z.Binomial(1000, 990)
  	}
  }
  
  // Examples from the Go Language Spec, section "Arithmetic operators"
  var divisionSignsTests = []struct {
  	x, y int64
  	q, r int64 // T-division
  	d, m int64 // Euclidean division
  }{
  	{5, 3, 1, 2, 1, 2},
  	{-5, 3, -1, -2, -2, 1},
  	{5, -3, -1, 2, -1, 2},
  	{-5, -3, 1, -2, 2, 1},
  	{1, 2, 0, 1, 0, 1},
  	{8, 4, 2, 0, 2, 0},
  }
  
  func TestDivisionSigns(t *testing.T) {
  	for i, test := range divisionSignsTests {
  		x := NewInt(test.x)
  		y := NewInt(test.y)
  		q := NewInt(test.q)
  		r := NewInt(test.r)
  		d := NewInt(test.d)
  		m := NewInt(test.m)
  
  		q1 := new(Int).Quo(x, y)
  		r1 := new(Int).Rem(x, y)
  		if !isNormalized(q1) {
  			t.Errorf("#%d Quo: %v is not normalized", i, *q1)
  		}
  		if !isNormalized(r1) {
  			t.Errorf("#%d Rem: %v is not normalized", i, *r1)
  		}
  		if q1.Cmp(q) != 0 || r1.Cmp(r) != 0 {
  			t.Errorf("#%d QuoRem: got (%s, %s), want (%s, %s)", i, q1, r1, q, r)
  		}
  
  		q2, r2 := new(Int).QuoRem(x, y, new(Int))
  		if !isNormalized(q2) {
  			t.Errorf("#%d Quo: %v is not normalized", i, *q2)
  		}
  		if !isNormalized(r2) {
  			t.Errorf("#%d Rem: %v is not normalized", i, *r2)
  		}
  		if q2.Cmp(q) != 0 || r2.Cmp(r) != 0 {
  			t.Errorf("#%d QuoRem: got (%s, %s), want (%s, %s)", i, q2, r2, q, r)
  		}
  
  		d1 := new(Int).Div(x, y)
  		m1 := new(Int).Mod(x, y)
  		if !isNormalized(d1) {
  			t.Errorf("#%d Div: %v is not normalized", i, *d1)
  		}
  		if !isNormalized(m1) {
  			t.Errorf("#%d Mod: %v is not normalized", i, *m1)
  		}
  		if d1.Cmp(d) != 0 || m1.Cmp(m) != 0 {
  			t.Errorf("#%d DivMod: got (%s, %s), want (%s, %s)", i, d1, m1, d, m)
  		}
  
  		d2, m2 := new(Int).DivMod(x, y, new(Int))
  		if !isNormalized(d2) {
  			t.Errorf("#%d Div: %v is not normalized", i, *d2)
  		}
  		if !isNormalized(m2) {
  			t.Errorf("#%d Mod: %v is not normalized", i, *m2)
  		}
  		if d2.Cmp(d) != 0 || m2.Cmp(m) != 0 {
  			t.Errorf("#%d DivMod: got (%s, %s), want (%s, %s)", i, d2, m2, d, m)
  		}
  	}
  }
  
  func norm(x nat) nat {
  	i := len(x)
  	for i > 0 && x[i-1] == 0 {
  		i--
  	}
  	return x[:i]
  }
  
  func TestBits(t *testing.T) {
  	for _, test := range []nat{
  		nil,
  		{0},
  		{1},
  		{0, 1, 2, 3, 4},
  		{4, 3, 2, 1, 0},
  		{4, 3, 2, 1, 0, 0, 0, 0},
  	} {
  		var z Int
  		z.neg = true
  		got := z.SetBits(test)
  		want := norm(test)
  		if got.abs.cmp(want) != 0 {
  			t.Errorf("SetBits(%v) = %v; want %v", test, got.abs, want)
  		}
  
  		if got.neg {
  			t.Errorf("SetBits(%v): got negative result", test)
  		}
  
  		bits := nat(z.Bits())
  		if bits.cmp(want) != 0 {
  			t.Errorf("%v.Bits() = %v; want %v", z.abs, bits, want)
  		}
  	}
  }
  
  func checkSetBytes(b []byte) bool {
  	hex1 := hex.EncodeToString(new(Int).SetBytes(b).Bytes())
  	hex2 := hex.EncodeToString(b)
  
  	for len(hex1) < len(hex2) {
  		hex1 = "0" + hex1
  	}
  
  	for len(hex1) > len(hex2) {
  		hex2 = "0" + hex2
  	}
  
  	return hex1 == hex2
  }
  
  func TestSetBytes(t *testing.T) {
  	if err := quick.Check(checkSetBytes, nil); err != nil {
  		t.Error(err)
  	}
  }
  
  func checkBytes(b []byte) bool {
  	// trim leading zero bytes since Bytes() won't return them
  	// (was issue 12231)
  	for len(b) > 0 && b[0] == 0 {
  		b = b[1:]
  	}
  	b2 := new(Int).SetBytes(b).Bytes()
  	return bytes.Equal(b, b2)
  }
  
  func TestBytes(t *testing.T) {
  	if err := quick.Check(checkBytes, nil); err != nil {
  		t.Error(err)
  	}
  }
  
  func checkQuo(x, y []byte) bool {
  	u := new(Int).SetBytes(x)
  	v := new(Int).SetBytes(y)
  
  	if len(v.abs) == 0 {
  		return true
  	}
  
  	r := new(Int)
  	q, r := new(Int).QuoRem(u, v, r)
  
  	if r.Cmp(v) >= 0 {
  		return false
  	}
  
  	uprime := new(Int).Set(q)
  	uprime.Mul(uprime, v)
  	uprime.Add(uprime, r)
  
  	return uprime.Cmp(u) == 0
  }
  
  var quoTests = []struct {
  	x, y string
  	q, r string
  }{
  	{
  		"476217953993950760840509444250624797097991362735329973741718102894495832294430498335824897858659711275234906400899559094370964723884706254265559534144986498357",
  		"9353930466774385905609975137998169297361893554149986716853295022578535724979483772383667534691121982974895531435241089241440253066816724367338287092081996",
  		"50911",
  		"1",
  	},
  	{
  		"11510768301994997771168",
  		"1328165573307167369775",
  		"8",
  		"885443715537658812968",
  	},
  }
  
  func TestQuo(t *testing.T) {
  	if err := quick.Check(checkQuo, nil); err != nil {
  		t.Error(err)
  	}
  
  	for i, test := range quoTests {
  		x, _ := new(Int).SetString(test.x, 10)
  		y, _ := new(Int).SetString(test.y, 10)
  		expectedQ, _ := new(Int).SetString(test.q, 10)
  		expectedR, _ := new(Int).SetString(test.r, 10)
  
  		r := new(Int)
  		q, r := new(Int).QuoRem(x, y, r)
  
  		if q.Cmp(expectedQ) != 0 || r.Cmp(expectedR) != 0 {
  			t.Errorf("#%d got (%s, %s) want (%s, %s)", i, q, r, expectedQ, expectedR)
  		}
  	}
  }
  
  func TestQuoStepD6(t *testing.T) {
  	// See Knuth, Volume 2, section 4.3.1, exercise 21. This code exercises
  	// a code path which only triggers 1 in 10^{-19} cases.
  
  	u := &Int{false, nat{0, 0, 1 + 1<<(_W-1), _M ^ (1 << (_W - 1))}}
  	v := &Int{false, nat{5, 2 + 1<<(_W-1), 1 << (_W - 1)}}
  
  	r := new(Int)
  	q, r := new(Int).QuoRem(u, v, r)
  	const expectedQ64 = "18446744073709551613"
  	const expectedR64 = "3138550867693340382088035895064302439801311770021610913807"
  	const expectedQ32 = "4294967293"
  	const expectedR32 = "39614081266355540837921718287"
  	if q.String() != expectedQ64 && q.String() != expectedQ32 ||
  		r.String() != expectedR64 && r.String() != expectedR32 {
  		t.Errorf("got (%s, %s) want (%s, %s) or (%s, %s)", q, r, expectedQ64, expectedR64, expectedQ32, expectedR32)
  	}
  }
  
  func BenchmarkQuoRem(b *testing.B) {
  	x, _ := new(Int).SetString("153980389784927331788354528594524332344709972855165340650588877572729725338415474372475094155672066328274535240275856844648695200875763869073572078279316458648124537905600131008790701752441155668003033945258023841165089852359980273279085783159654751552359397986180318708491098942831252291841441726305535546071", 0)
  	y, _ := new(Int).SetString("7746362281539803897849273317883545285945243323447099728551653406505888775727297253384154743724750941556720663282745352402758568446486952008757638690735720782793164586481245379056001310087907017524411556680030339452580238411650898523599802732790857831596547515523593979861803187084910989428312522918414417263055355460715745539358014631136245887418412633787074173796862711588221766398229333338511838891484974940633857861775630560092874987828057333663969469797013996401149696897591265769095952887917296740109742927689053276850469671231961384715398038978492733178835452859452433234470997285516534065058887757272972533841547437247509415567206632827453524027585684464869520087576386907357207827931645864812453790560013100879070175244115566800303394525802384116508985235998027327908578315965475155235939798618031870849109894283125229184144172630553554607112725169432413343763989564437170644270643461665184965150423819594083121075825", 0)
  	q := new(Int)
  	r := new(Int)
  
  	b.ResetTimer()
  	for i := 0; i < b.N; i++ {
  		q.QuoRem(y, x, r)
  	}
  }
  
  var bitLenTests = []struct {
  	in  string
  	out int
  }{
  	{"-1", 1},
  	{"0", 0},
  	{"1", 1},
  	{"2", 2},
  	{"4", 3},
  	{"0xabc", 12},
  	{"0x8000", 16},
  	{"0x80000000", 32},
  	{"0x800000000000", 48},
  	{"0x8000000000000000", 64},
  	{"0x80000000000000000000", 80},
  	{"-0x4000000000000000000000", 87},
  }
  
  func TestBitLen(t *testing.T) {
  	for i, test := range bitLenTests {
  		x, ok := new(Int).SetString(test.in, 0)
  		if !ok {
  			t.Errorf("#%d test input invalid: %s", i, test.in)
  			continue
  		}
  
  		if n := x.BitLen(); n != test.out {
  			t.Errorf("#%d got %d want %d", i, n, test.out)
  		}
  	}
  }
  
  var expTests = []struct {
  	x, y, m string
  	out     string
  }{
  	// y <= 0
  	{"0", "0", "", "1"},
  	{"1", "0", "", "1"},
  	{"-10", "0", "", "1"},
  	{"1234", "-1", "", "1"},
  
  	// m == 1
  	{"0", "0", "1", "0"},
  	{"1", "0", "1", "0"},
  	{"-10", "0", "1", "0"},
  	{"1234", "-1", "1", "0"},
  
  	// misc
  	{"5", "1", "3", "2"},
  	{"5", "-7", "", "1"},
  	{"-5", "-7", "", "1"},
  	{"5", "0", "", "1"},
  	{"-5", "0", "", "1"},
  	{"5", "1", "", "5"},
  	{"-5", "1", "", "-5"},
  	{"-5", "1", "7", "2"},
  	{"-2", "3", "2", "0"},
  	{"5", "2", "", "25"},
  	{"1", "65537", "2", "1"},
  	{"0x8000000000000000", "2", "", "0x40000000000000000000000000000000"},
  	{"0x8000000000000000", "2", "6719", "4944"},
  	{"0x8000000000000000", "3", "6719", "5447"},
  	{"0x8000000000000000", "1000", "6719", "1603"},
  	{"0x8000000000000000", "1000000", "6719", "3199"},
  	{"0x8000000000000000", "-1000000", "6719", "1"},
  
  	{"0xffffffffffffffffffffffffffffffff", "0x12345678123456781234567812345678123456789", "0x01112222333344445555666677778889", "0x36168FA1DB3AAE6C8CE647E137F97A"},
  
  	{
  		"2938462938472983472983659726349017249287491026512746239764525612965293865296239471239874193284792387498274256129746192347",
  		"298472983472983471903246121093472394872319615612417471234712061",
  		"29834729834729834729347290846729561262544958723956495615629569234729836259263598127342374289365912465901365498236492183464",
  		"23537740700184054162508175125554701713153216681790245129157191391322321508055833908509185839069455749219131480588829346291",
  	},
  	// test case for issue 8822
  	{
  		"11001289118363089646017359372117963499250546375269047542777928006103246876688756735760905680604646624353196869572752623285140408755420374049317646428185270079555372763503115646054602867593662923894140940837479507194934267532831694565516466765025434902348314525627418515646588160955862839022051353653052947073136084780742729727874803457643848197499548297570026926927502505634297079527299004267769780768565695459945235586892627059178884998772989397505061206395455591503771677500931269477503508150175717121828518985901959919560700853226255420793148986854391552859459511723547532575574664944815966793196961286234040892865",
  		"0xB08FFB20760FFED58FADA86DFEF71AD72AA0FA763219618FE022C197E54708BB1191C66470250FCE8879487507CEE41381CA4D932F81C2B3F1AB20B539D50DCD",
  		"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
  		"21484252197776302499639938883777710321993113097987201050501182909581359357618579566746556372589385361683610524730509041328855066514963385522570894839035884713051640171474186548713546686476761306436434146475140156284389181808675016576845833340494848283681088886584219750554408060556769486628029028720727393293111678826356480455433909233520504112074401376133077150471237549474149190242010469539006449596611576612573955754349042329130631128234637924786466585703488460540228477440853493392086251021228087076124706778899179648655221663765993962724699135217212118535057766739392069738618682722216712319320435674779146070442",
  	},
  	{
  		"-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
  		"0xB08FFB20760FFED58FADA86DFEF71AD72AA0FA763219618FE022C197E54708BB1191C66470250FCE8879487507CEE41381CA4D932F81C2B3F1AB20B539D50DCD",
  		"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
  		"21484252197776302499639938883777710321993113097987201050501182909581359357618579566746556372589385361683610524730509041328855066514963385522570894839035884713051640171474186548713546686476761306436434146475140156284389181808675016576845833340494848283681088886584219750554408060556769486628029028720727393293111678826356480455433909233520504112074401376133077150471237549474149190242010469539006449596611576612573955754349042329130631128234637924786466585703488460540228477440853493392086251021228087076124706778899179648655221663765993962724699135217212118535057766739392069738618682722216712319320435674779146070442",
  	},
  
  	// test cases for issue 13907
  	{"0xffffffff00000001", "0xffffffff00000001", "0xffffffff00000001", "0"},
  	{"0xffffffffffffffff00000001", "0xffffffffffffffff00000001", "0xffffffffffffffff00000001", "0"},
  	{"0xffffffffffffffffffffffff00000001", "0xffffffffffffffffffffffff00000001", "0xffffffffffffffffffffffff00000001", "0"},
  	{"0xffffffffffffffffffffffffffffffff00000001", "0xffffffffffffffffffffffffffffffff00000001", "0xffffffffffffffffffffffffffffffff00000001", "0"},
  
  	{
  		"2",
  		"0xB08FFB20760FFED58FADA86DFEF71AD72AA0FA763219618FE022C197E54708BB1191C66470250FCE8879487507CEE41381CA4D932F81C2B3F1AB20B539D50DCD",
  		"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odd
  		"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
  	},
  	{
  		"2",
  		"0xB08FFB20760FFED58FADA86DFEF71AD72AA0FA763219618FE022C197E54708BB1191C66470250FCE8879487507CEE41381CA4D932F81C2B3F1AB20B539D50DCD",
  		"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even
  		"0x7858794B5897C29F4ED0B40913416AB6C48588484E6A45F2ED3E26C941D878E923575AAC434EE2750E6439A6976F9BB4D64CEDB2A53CE8D04DD48CADCDF8E46F22747C6B81C6CEA86C0D873FBF7CEF262BAAC43A522BD7F32F3CDAC52B9337C77B3DCFB3DB3EDD80476331E82F4B1DF8EFDC1220C92656DFC9197BDC1877804E28D928A2A284B8DED506CBA304435C9D0133C246C98A7D890D1DE60CBC53A024361DA83A9B8775019083D22AC6820ED7C3C68F8E801DD4EC779EE0A05C6EB682EF9840D285B838369BA7E148FA27691D524FAEAF7C6ECE2A4B99A294B9F2C241857B5B90CC8BFFCFCF18DFA7D676131D5CD3855A5A3E8EBFA0CDFADB4D198B4A",
  	},
  }
  
  func TestExp(t *testing.T) {
  	for i, test := range expTests {
  		x, ok1 := new(Int).SetString(test.x, 0)
  		y, ok2 := new(Int).SetString(test.y, 0)
  		out, ok3 := new(Int).SetString(test.out, 0)
  
  		var ok4 bool
  		var m *Int
  
  		if len(test.m) == 0 {
  			m, ok4 = nil, true
  		} else {
  			m, ok4 = new(Int).SetString(test.m, 0)
  		}
  
  		if !ok1 || !ok2 || !ok3 || !ok4 {
  			t.Errorf("#%d: error in input", i)
  			continue
  		}
  
  		z1 := new(Int).Exp(x, y, m)
  		if !isNormalized(z1) {
  			t.Errorf("#%d: %v is not normalized", i, *z1)
  		}
  		if z1.Cmp(out) != 0 {
  			t.Errorf("#%d: got %x want %x", i, z1, out)
  		}
  
  		if m == nil {
  			// The result should be the same as for m == 0;
  			// specifically, there should be no div-zero panic.
  			m = &Int{abs: nat{}} // m != nil && len(m.abs) == 0
  			z2 := new(Int).Exp(x, y, m)
  			if z2.Cmp(z1) != 0 {
  				t.Errorf("#%d: got %x want %x", i, z2, z1)
  			}
  		}
  	}
  }
  
  func BenchmarkExp(b *testing.B) {
  	x, _ := new(Int).SetString("11001289118363089646017359372117963499250546375269047542777928006103246876688756735760905680604646624353196869572752623285140408755420374049317646428185270079555372763503115646054602867593662923894140940837479507194934267532831694565516466765025434902348314525627418515646588160955862839022051353653052947073136084780742729727874803457643848197499548297570026926927502505634297079527299004267769780768565695459945235586892627059178884998772989397505061206395455591503771677500931269477503508150175717121828518985901959919560700853226255420793148986854391552859459511723547532575574664944815966793196961286234040892865", 0)
  	y, _ := new(Int).SetString("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
  	n, _ := new(Int).SetString("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
  	out := new(Int)
  	for i := 0; i < b.N; i++ {
  		out.Exp(x, y, n)
  	}
  }
  
  func BenchmarkExp2(b *testing.B) {
  	x, _ := new(Int).SetString("2", 0)
  	y, _ := new(Int).SetString("0xAC6BDB41324A9A9BF166DE5E1389582FAF72B6651987EE07FC3192943DB56050A37329CBB4A099ED8193E0757767A13DD52312AB4B03310DCD7F48A9DA04FD50E8083969EDB767B0CF6095179A163AB3661A05FBD5FAAAE82918A9962F0B93B855F97993EC975EEAA80D740ADBF4FF747359D041D5C33EA71D281E446B14773BCA97B43A23FB801676BD207A436C6481F1D2B9078717461A5B9D32E688F87748544523B524B0D57D5EA77A2775D2ECFA032CFBDBF52FB3786160279004E57AE6AF874E7303CE53299CCC041C7BC308D82A5698F3A8D0C38271AE35F8E9DBFBB694B5C803D89F7AE435DE236D525F54759B65E372FCD68EF20FA7111F9E4AFF72", 0)
  	n, _ := new(Int).SetString("0xAC6BDB41324A9A9BF166DE5E1389582FAF72B6651987EE07FC3192943DB56050A37329CBB4A099ED8193E0757767A13DD52312AB4B03310DCD7F48A9DA04FD50E8083969EDB767B0CF6095179A163AB3661A05FBD5FAAAE82918A9962F0B93B855F97993EC975EEAA80D740ADBF4FF747359D041D5C33EA71D281E446B14773BCA97B43A23FB801676BD207A436C6481F1D2B9078717461A5B9D32E688F87748544523B524B0D57D5EA77A2775D2ECFA032CFBDBF52FB3786160279004E57AE6AF874E7303CE53299CCC041C7BC308D82A5698F3A8D0C38271AE35F8E9DBFBB694B5C803D89F7AE435DE236D525F54759B65E372FCD68EF20FA7111F9E4AFF73", 0)
  	out := new(Int)
  	for i := 0; i < b.N; i++ {
  		out.Exp(x, y, n)
  	}
  }
  
  func checkGcd(aBytes, bBytes []byte) bool {
  	x := new(Int)
  	y := new(Int)
  	a := new(Int).SetBytes(aBytes)
  	b := new(Int).SetBytes(bBytes)
  
  	d := new(Int).GCD(x, y, a, b)
  	x.Mul(x, a)
  	y.Mul(y, b)
  	x.Add(x, y)
  
  	return x.Cmp(d) == 0
  }
  
  var gcdTests = []struct {
  	d, x, y, a, b string
  }{
  	// a <= 0 || b <= 0
  	{"0", "0", "0", "0", "0"},
  	{"0", "0", "0", "0", "7"},
  	{"0", "0", "0", "11", "0"},
  	{"0", "0", "0", "-77", "35"},
  	{"0", "0", "0", "64515", "-24310"},
  	{"0", "0", "0", "-64515", "-24310"},
  
  	{"1", "-9", "47", "120", "23"},
  	{"7", "1", "-2", "77", "35"},
  	{"935", "-3", "8", "64515", "24310"},
  	{"935000000000000000", "-3", "8", "64515000000000000000", "24310000000000000000"},
  	{"1", "-221", "22059940471369027483332068679400581064239780177629666810348940098015901108344", "98920366548084643601728869055592650835572950932266967461790948584315647051443", "991"},
  
  	// test early exit (after one Euclidean iteration) in binaryGCD
  	{"1", "", "", "1", "98920366548084643601728869055592650835572950932266967461790948584315647051443"},
  }
  
  func testGcd(t *testing.T, d, x, y, a, b *Int) {
  	var X *Int
  	if x != nil {
  		X = new(Int)
  	}
  	var Y *Int
  	if y != nil {
  		Y = new(Int)
  	}
  
  	D := new(Int).GCD(X, Y, a, b)
  	if D.Cmp(d) != 0 {
  		t.Errorf("GCD(%s, %s): got d = %s, want %s", a, b, D, d)
  	}
  	if x != nil && X.Cmp(x) != 0 {
  		t.Errorf("GCD(%s, %s): got x = %s, want %s", a, b, X, x)
  	}
  	if y != nil && Y.Cmp(y) != 0 {
  		t.Errorf("GCD(%s, %s): got y = %s, want %s", a, b, Y, y)
  	}
  
  	// binaryGCD requires a > 0 && b > 0
  	if a.Sign() <= 0 || b.Sign() <= 0 {
  		return
  	}
  
  	D.binaryGCD(a, b)
  	if D.Cmp(d) != 0 {
  		t.Errorf("binaryGcd(%s, %s): got d = %s, want %s", a, b, D, d)
  	}
  
  	// check results in presence of aliasing (issue #11284)
  	a2 := new(Int).Set(a)
  	b2 := new(Int).Set(b)
  	a2.binaryGCD(a2, b2) // result is same as 1st argument
  	if a2.Cmp(d) != 0 {
  		t.Errorf("binaryGcd(%s, %s): got d = %s, want %s", a, b, a2, d)
  	}
  
  	a2 = new(Int).Set(a)
  	b2 = new(Int).Set(b)
  	b2.binaryGCD(a2, b2) // result is same as 2nd argument
  	if b2.Cmp(d) != 0 {
  		t.Errorf("binaryGcd(%s, %s): got d = %s, want %s", a, b, b2, d)
  	}
  }
  
  func TestGcd(t *testing.T) {
  	for _, test := range gcdTests {
  		d, _ := new(Int).SetString(test.d, 0)
  		x, _ := new(Int).SetString(test.x, 0)
  		y, _ := new(Int).SetString(test.y, 0)
  		a, _ := new(Int).SetString(test.a, 0)
  		b, _ := new(Int).SetString(test.b, 0)
  
  		testGcd(t, d, nil, nil, a, b)
  		testGcd(t, d, x, nil, a, b)
  		testGcd(t, d, nil, y, a, b)
  		testGcd(t, d, x, y, a, b)
  	}
  
  	if err := quick.Check(checkGcd, nil); err != nil {
  		t.Error(err)
  	}
  }
  
  type intShiftTest struct {
  	in    string
  	shift uint
  	out   string
  }
  
  var rshTests = []intShiftTest{
  	{"0", 0, "0"},
  	{"-0", 0, "0"},
  	{"0", 1, "0"},
  	{"0", 2, "0"},
  	{"1", 0, "1"},
  	{"1", 1, "0"},
  	{"1", 2, "0"},
  	{"2", 0, "2"},
  	{"2", 1, "1"},
  	{"-1", 0, "-1"},
  	{"-1", 1, "-1"},
  	{"-1", 10, "-1"},
  	{"-100", 2, "-25"},
  	{"-100", 3, "-13"},
  	{"-100", 100, "-1"},
  	{"4294967296", 0, "4294967296"},
  	{"4294967296", 1, "2147483648"},
  	{"4294967296", 2, "1073741824"},
  	{"18446744073709551616", 0, "18446744073709551616"},
  	{"18446744073709551616", 1, "9223372036854775808"},
  	{"18446744073709551616", 2, "4611686018427387904"},
  	{"18446744073709551616", 64, "1"},
  	{"340282366920938463463374607431768211456", 64, "18446744073709551616"},
  	{"340282366920938463463374607431768211456", 128, "1"},
  }
  
  func TestRsh(t *testing.T) {
  	for i, test := range rshTests {
  		in, _ := new(Int).SetString(test.in, 10)
  		expected, _ := new(Int).SetString(test.out, 10)
  		out := new(Int).Rsh(in, test.shift)
  
  		if !isNormalized(out) {
  			t.Errorf("#%d: %v is not normalized", i, *out)
  		}
  		if out.Cmp(expected) != 0 {
  			t.Errorf("#%d: got %s want %s", i, out, expected)
  		}
  	}
  }
  
  func TestRshSelf(t *testing.T) {
  	for i, test := range rshTests {
  		z, _ := new(Int).SetString(test.in, 10)
  		expected, _ := new(Int).SetString(test.out, 10)
  		z.Rsh(z, test.shift)
  
  		if !isNormalized(z) {
  			t.Errorf("#%d: %v is not normalized", i, *z)
  		}
  		if z.Cmp(expected) != 0 {
  			t.Errorf("#%d: got %s want %s", i, z, expected)
  		}
  	}
  }
  
  var lshTests = []intShiftTest{
  	{"0", 0, "0"},
  	{"0", 1, "0"},
  	{"0", 2, "0"},
  	{"1", 0, "1"},
  	{"1", 1, "2"},
  	{"1", 2, "4"},
  	{"2", 0, "2"},
  	{"2", 1, "4"},
  	{"2", 2, "8"},
  	{"-87", 1, "-174"},
  	{"4294967296", 0, "4294967296"},
  	{"4294967296", 1, "8589934592"},
  	{"4294967296", 2, "17179869184"},
  	{"18446744073709551616", 0, "18446744073709551616"},
  	{"9223372036854775808", 1, "18446744073709551616"},
  	{"4611686018427387904", 2, "18446744073709551616"},
  	{"1", 64, "18446744073709551616"},
  	{"18446744073709551616", 64, "340282366920938463463374607431768211456"},
  	{"1", 128, "340282366920938463463374607431768211456"},
  }
  
  func TestLsh(t *testing.T) {
  	for i, test := range lshTests {
  		in, _ := new(Int).SetString(test.in, 10)
  		expected, _ := new(Int).SetString(test.out, 10)
  		out := new(Int).Lsh(in, test.shift)
  
  		if !isNormalized(out) {
  			t.Errorf("#%d: %v is not normalized", i, *out)
  		}
  		if out.Cmp(expected) != 0 {
  			t.Errorf("#%d: got %s want %s", i, out, expected)
  		}
  	}
  }
  
  func TestLshSelf(t *testing.T) {
  	for i, test := range lshTests {
  		z, _ := new(Int).SetString(test.in, 10)
  		expected, _ := new(Int).SetString(test.out, 10)
  		z.Lsh(z, test.shift)
  
  		if !isNormalized(z) {
  			t.Errorf("#%d: %v is not normalized", i, *z)
  		}
  		if z.Cmp(expected) != 0 {
  			t.Errorf("#%d: got %s want %s", i, z, expected)
  		}
  	}
  }
  
  func TestLshRsh(t *testing.T) {
  	for i, test := range rshTests {
  		in, _ := new(Int).SetString(test.in, 10)
  		out := new(Int).Lsh(in, test.shift)
  		out = out.Rsh(out, test.shift)
  
  		if !isNormalized(out) {
  			t.Errorf("#%d: %v is not normalized", i, *out)
  		}
  		if in.Cmp(out) != 0 {
  			t.Errorf("#%d: got %s want %s", i, out, in)
  		}
  	}
  	for i, test := range lshTests {
  		in, _ := new(Int).SetString(test.in, 10)
  		out := new(Int).Lsh(in, test.shift)
  		out.Rsh(out, test.shift)
  
  		if !isNormalized(out) {
  			t.Errorf("#%d: %v is not normalized", i, *out)
  		}
  		if in.Cmp(out) != 0 {
  			t.Errorf("#%d: got %s want %s", i, out, in)
  		}
  	}
  }
  
  var int64Tests = []string{
  	// int64
  	"0",
  	"1",
  	"-1",
  	"4294967295",
  	"-4294967295",
  	"4294967296",
  	"-4294967296",
  	"9223372036854775807",
  	"-9223372036854775807",
  	"-9223372036854775808",
  
  	// not int64
  	"0x8000000000000000",
  	"-0x8000000000000001",
  	"38579843757496759476987459679745",
  	"-38579843757496759476987459679745",
  }
  
  func TestInt64(t *testing.T) {
  	for _, s := range int64Tests {
  		var x Int
  		_, ok := x.SetString(s, 0)
  		if !ok {
  			t.Errorf("SetString(%s, 0) failed", s)
  			continue
  		}
  
  		want, err := strconv.ParseInt(s, 0, 64)
  		if err != nil {
  			if err.(*strconv.NumError).Err == strconv.ErrRange {
  				if x.IsInt64() {
  					t.Errorf("IsInt64(%s) succeeded unexpectedly", s)
  				}
  			} else {
  				t.Errorf("ParseInt(%s) failed", s)
  			}
  			continue
  		}
  
  		if !x.IsInt64() {
  			t.Errorf("IsInt64(%s) failed unexpectedly", s)
  		}
  
  		got := x.Int64()
  		if got != want {
  			t.Errorf("Int64(%s) = %d; want %d", s, got, want)
  		}
  	}
  }
  
  var uint64Tests = []string{
  	// uint64
  	"0",
  	"1",
  	"4294967295",
  	"4294967296",
  	"8589934591",
  	"8589934592",
  	"9223372036854775807",
  	"9223372036854775808",
  	"0x08000000000000000",
  
  	// not uint64
  	"0x10000000000000000",
  	"-0x08000000000000000",
  	"-1",
  }
  
  func TestUint64(t *testing.T) {
  	for _, s := range uint64Tests {
  		var x Int
  		_, ok := x.SetString(s, 0)
  		if !ok {
  			t.Errorf("SetString(%s, 0) failed", s)
  			continue
  		}
  
  		want, err := strconv.ParseUint(s, 0, 64)
  		if err != nil {
  			// check for sign explicitly (ErrRange doesn't cover signed input)
  			if s[0] == '-' || err.(*strconv.NumError).Err == strconv.ErrRange {
  				if x.IsUint64() {
  					t.Errorf("IsUint64(%s) succeeded unexpectedly", s)
  				}
  			} else {
  				t.Errorf("ParseUint(%s) failed", s)
  			}
  			continue
  		}
  
  		if !x.IsUint64() {
  			t.Errorf("IsUint64(%s) failed unexpectedly", s)
  		}
  
  		got := x.Uint64()
  		if got != want {
  			t.Errorf("Uint64(%s) = %d; want %d", s, got, want)
  		}
  	}
  }
  
  var bitwiseTests = []struct {
  	x, y                 string
  	and, or, xor, andNot string
  }{
  	{"0x00", "0x00", "0x00", "0x00", "0x00", "0x00"},
  	{"0x00", "0x01", "0x00", "0x01", "0x01", "0x00"},
  	{"0x01", "0x00", "0x00", "0x01", "0x01", "0x01"},
  	{"-0x01", "0x00", "0x00", "-0x01", "-0x01", "-0x01"},
  	{"-0xaf", "-0x50", "-0xf0", "-0x0f", "0xe1", "0x41"},
  	{"0x00", "-0x01", "0x00", "-0x01", "-0x01", "0x00"},
  	{"0x01", "0x01", "0x01", "0x01", "0x00", "0x00"},
  	{"-0x01", "-0x01", "-0x01", "-0x01", "0x00", "0x00"},
  	{"0x07", "0x08", "0x00", "0x0f", "0x0f", "0x07"},
  	{"0x05", "0x0f", "0x05", "0x0f", "0x0a", "0x00"},
  	{"0xff", "-0x0a", "0xf6", "-0x01", "-0xf7", "0x09"},
  	{"0x013ff6", "0x9a4e", "0x1a46", "0x01bffe", "0x01a5b8", "0x0125b0"},
  	{"-0x013ff6", "0x9a4e", "0x800a", "-0x0125b2", "-0x01a5bc", "-0x01c000"},
  	{"-0x013ff6", "-0x9a4e", "-0x01bffe", "-0x1a46", "0x01a5b8", "0x8008"},
  	{
  		"0x1000009dc6e3d9822cba04129bcbe3401",
  		"0xb9bd7d543685789d57cb918e833af352559021483cdb05cc21fd",
  		"0x1000001186210100001000009048c2001",
  		"0xb9bd7d543685789d57cb918e8bfeff7fddb2ebe87dfbbdfe35fd",
  		"0xb9bd7d543685789d57ca918e8ae69d6fcdb2eae87df2b97215fc",
  		"0x8c40c2d8822caa04120b8321400",
  	},
  	{
  		"0x1000009dc6e3d9822cba04129bcbe3401",
  		"-0xb9bd7d543685789d57cb918e833af352559021483cdb05cc21fd",
  		"0x8c40c2d8822caa04120b8321401",
  		"-0xb9bd7d543685789d57ca918e82229142459020483cd2014001fd",
  		"-0xb9bd7d543685789d57ca918e8ae69d6fcdb2eae87df2b97215fe",
  		"0x1000001186210100001000009048c2000",
  	},
  	{
  		"-0x1000009dc6e3d9822cba04129bcbe3401",
  		"-0xb9bd7d543685789d57cb918e833af352559021483cdb05cc21fd",
  		"-0xb9bd7d543685789d57cb918e8bfeff7fddb2ebe87dfbbdfe35fd",
  		"-0x1000001186210100001000009048c2001",
  		"0xb9bd7d543685789d57ca918e8ae69d6fcdb2eae87df2b97215fc",
  		"0xb9bd7d543685789d57ca918e82229142459020483cd2014001fc",
  	},
  }
  
  type bitFun func(z, x, y *Int) *Int
  
  func testBitFun(t *testing.T, msg string, f bitFun, x, y *Int, exp string) {
  	expected := new(Int)
  	expected.SetString(exp, 0)
  
  	out := f(new(Int), x, y)
  	if out.Cmp(expected) != 0 {
  		t.Errorf("%s: got %s want %s", msg, out, expected)
  	}
  }
  
  func testBitFunSelf(t *testing.T, msg string, f bitFun, x, y *Int, exp string) {
  	self := new(Int)
  	self.Set(x)
  	expected := new(Int)
  	expected.SetString(exp, 0)
  
  	self = f(self, self, y)
  	if self.Cmp(expected) != 0 {
  		t.Errorf("%s: got %s want %s", msg, self, expected)
  	}
  }
  
  func altBit(x *Int, i int) uint {
  	z := new(Int).Rsh(x, uint(i))
  	z = z.And(z, NewInt(1))
  	if z.Cmp(new(Int)) != 0 {
  		return 1
  	}
  	return 0
  }
  
  func altSetBit(z *Int, x *Int, i int, b uint) *Int {
  	one := NewInt(1)
  	m := one.Lsh(one, uint(i))
  	switch b {
  	case 1:
  		return z.Or(x, m)
  	case 0:
  		return z.AndNot(x, m)
  	}
  	panic("set bit is not 0 or 1")
  }
  
  func testBitset(t *testing.T, x *Int) {
  	n := x.BitLen()
  	z := new(Int).Set(x)
  	z1 := new(Int).Set(x)
  	for i := 0; i < n+10; i++ {
  		old := z.Bit(i)
  		old1 := altBit(z1, i)
  		if old != old1 {
  			t.Errorf("bitset: inconsistent value for Bit(%s, %d), got %v want %v", z1, i, old, old1)
  		}
  		z := new(Int).SetBit(z, i, 1)
  		z1 := altSetBit(new(Int), z1, i, 1)
  		if z.Bit(i) == 0 {
  			t.Errorf("bitset: bit %d of %s got 0 want 1", i, x)
  		}
  		if z.Cmp(z1) != 0 {
  			t.Errorf("bitset: inconsistent value after SetBit 1, got %s want %s", z, z1)
  		}
  		z.SetBit(z, i, 0)
  		altSetBit(z1, z1, i, 0)
  		if z.Bit(i) != 0 {
  			t.Errorf("bitset: bit %d of %s got 1 want 0", i, x)
  		}
  		if z.Cmp(z1) != 0 {
  			t.Errorf("bitset: inconsistent value after SetBit 0, got %s want %s", z, z1)
  		}
  		altSetBit(z1, z1, i, old)
  		z.SetBit(z, i, old)
  		if z.Cmp(z1) != 0 {
  			t.Errorf("bitset: inconsistent value after SetBit old, got %s want %s", z, z1)
  		}
  	}
  	if z.Cmp(x) != 0 {
  		t.Errorf("bitset: got %s want %s", z, x)
  	}
  }
  
  var bitsetTests = []struct {
  	x string
  	i int
  	b uint
  }{
  	{"0", 0, 0},
  	{"0", 200, 0},
  	{"1", 0, 1},
  	{"1", 1, 0},
  	{"-1", 0, 1},
  	{"-1", 200, 1},
  	{"0x2000000000000000000000000000", 108, 0},
  	{"0x2000000000000000000000000000", 109, 1},
  	{"0x2000000000000000000000000000", 110, 0},
  	{"-0x2000000000000000000000000001", 108, 1},
  	{"-0x2000000000000000000000000001", 109, 0},
  	{"-0x2000000000000000000000000001", 110, 1},
  }
  
  func TestBitSet(t *testing.T) {
  	for _, test := range bitwiseTests {
  		x := new(Int)
  		x.SetString(test.x, 0)
  		testBitset(t, x)
  		x = new(Int)
  		x.SetString(test.y, 0)
  		testBitset(t, x)
  	}
  	for i, test := range bitsetTests {
  		x := new(Int)
  		x.SetString(test.x, 0)
  		b := x.Bit(test.i)
  		if b != test.b {
  			t.Errorf("#%d got %v want %v", i, b, test.b)
  		}
  	}
  	z := NewInt(1)
  	z.SetBit(NewInt(0), 2, 1)
  	if z.Cmp(NewInt(4)) != 0 {
  		t.Errorf("destination leaked into result; got %s want 4", z)
  	}
  }
  
  func BenchmarkBitset(b *testing.B) {
  	z := new(Int)
  	z.SetBit(z, 512, 1)
  	b.ResetTimer()
  	b.StartTimer()
  	for i := b.N - 1; i >= 0; i-- {
  		z.SetBit(z, i&512, 1)
  	}
  }
  
  func BenchmarkBitsetNeg(b *testing.B) {
  	z := NewInt(-1)
  	z.SetBit(z, 512, 0)
  	b.ResetTimer()
  	b.StartTimer()
  	for i := b.N - 1; i >= 0; i-- {
  		z.SetBit(z, i&512, 0)
  	}
  }
  
  func BenchmarkBitsetOrig(b *testing.B) {
  	z := new(Int)
  	altSetBit(z, z, 512, 1)
  	b.ResetTimer()
  	b.StartTimer()
  	for i := b.N - 1; i >= 0; i-- {
  		altSetBit(z, z, i&512, 1)
  	}
  }
  
  func BenchmarkBitsetNegOrig(b *testing.B) {
  	z := NewInt(-1)
  	altSetBit(z, z, 512, 0)
  	b.ResetTimer()
  	b.StartTimer()
  	for i := b.N - 1; i >= 0; i-- {
  		altSetBit(z, z, i&512, 0)
  	}
  }
  
  // tri generates the trinomial 2**(n*2) - 2**n - 1, which is always 3 mod 4 and
  // 7 mod 8, so that 2 is always a quadratic residue.
  func tri(n uint) *Int {
  	x := NewInt(1)
  	x.Lsh(x, n)
  	x2 := new(Int).Lsh(x, n)
  	x2.Sub(x2, x)
  	x2.Sub(x2, intOne)
  	return x2
  }
  
  func BenchmarkModSqrt225_Tonelli(b *testing.B) {
  	p := tri(225)
  	x := NewInt(2)
  	for i := 0; i < b.N; i++ {
  		x.SetUint64(2)
  		x.modSqrtTonelliShanks(x, p)
  	}
  }
  
  func BenchmarkModSqrt224_3Mod4(b *testing.B) {
  	p := tri(225)
  	x := new(Int).SetUint64(2)
  	for i := 0; i < b.N; i++ {
  		x.SetUint64(2)
  		x.modSqrt3Mod4Prime(x, p)
  	}
  }
  
  func BenchmarkModSqrt5430_Tonelli(b *testing.B) {
  	if isRaceBuilder {
  		b.Skip("skipping on race builder")
  	}
  	p := tri(5430)
  	x := new(Int).SetUint64(2)
  	for i := 0; i < b.N; i++ {
  		x.SetUint64(2)
  		x.modSqrtTonelliShanks(x, p)
  	}
  }
  
  func BenchmarkModSqrt5430_3Mod4(b *testing.B) {
  	if isRaceBuilder {
  		b.Skip("skipping on race builder")
  	}
  	p := tri(5430)
  	x := new(Int).SetUint64(2)
  	for i := 0; i < b.N; i++ {
  		x.SetUint64(2)
  		x.modSqrt3Mod4Prime(x, p)
  	}
  }
  
  func TestBitwise(t *testing.T) {
  	x := new(Int)
  	y := new(Int)
  	for _, test := range bitwiseTests {
  		x.SetString(test.x, 0)
  		y.SetString(test.y, 0)
  
  		testBitFun(t, "and", (*Int).And, x, y, test.and)
  		testBitFunSelf(t, "and", (*Int).And, x, y, test.and)
  		testBitFun(t, "andNot", (*Int).AndNot, x, y, test.andNot)
  		testBitFunSelf(t, "andNot", (*Int).AndNot, x, y, test.andNot)
  		testBitFun(t, "or", (*Int).Or, x, y, test.or)
  		testBitFunSelf(t, "or", (*Int).Or, x, y, test.or)
  		testBitFun(t, "xor", (*Int).Xor, x, y, test.xor)
  		testBitFunSelf(t, "xor", (*Int).Xor, x, y, test.xor)
  	}
  }
  
  var notTests = []struct {
  	in  string
  	out string
  }{
  	{"0", "-1"},
  	{"1", "-2"},
  	{"7", "-8"},
  	{"0", "-1"},
  	{"-81910", "81909"},
  	{
  		"298472983472983471903246121093472394872319615612417471234712061",
  		"-298472983472983471903246121093472394872319615612417471234712062",
  	},
  }
  
  func TestNot(t *testing.T) {
  	in := new(Int)
  	out := new(Int)
  	expected := new(Int)
  	for i, test := range notTests {
  		in.SetString(test.in, 10)
  		expected.SetString(test.out, 10)
  		out = out.Not(in)
  		if out.Cmp(expected) != 0 {
  			t.Errorf("#%d: got %s want %s", i, out, expected)
  		}
  		out = out.Not(out)
  		if out.Cmp(in) != 0 {
  			t.Errorf("#%d: got %s want %s", i, out, in)
  		}
  	}
  }
  
  var modInverseTests = []struct {
  	element string
  	modulus string
  }{
  	{"1234567", "458948883992"},
  	{"239487239847", "2410312426921032588552076022197566074856950548502459942654116941958108831682612228890093858261341614673227141477904012196503648957050582631942730706805009223062734745341073406696246014589361659774041027169249453200378729434170325843778659198143763193776859869524088940195577346119843545301547043747207749969763750084308926339295559968882457872412993810129130294592999947926365264059284647209730384947211681434464714438488520940127459844288859336526896320919633919"},
  	{"-10", "13"}, // issue #16984
  }
  
  func TestModInverse(t *testing.T) {
  	var element, modulus, gcd, inverse Int
  	one := NewInt(1)
  	for i, test := range modInverseTests {
  		(&element).SetString(test.element, 10)
  		(&modulus).SetString(test.modulus, 10)
  		(&inverse).ModInverse(&element, &modulus)
  		(&inverse).Mul(&inverse, &element)
  		(&inverse).Mod(&inverse, &modulus)
  		if (&inverse).Cmp(one) != 0 {
  			t.Errorf("#%d: failed (e·e^(-1)=%s)", i, &inverse)
  		}
  	}
  	// exhaustive test for small values
  	for n := 2; n < 100; n++ {
  		(&modulus).SetInt64(int64(n))
  		for x := 1; x < n; x++ {
  			(&element).SetInt64(int64(x))
  			(&gcd).GCD(nil, nil, &element, &modulus)
  			if (&gcd).Cmp(one) != 0 {
  				continue
  			}
  			(&inverse).ModInverse(&element, &modulus)
  			(&inverse).Mul(&inverse, &element)
  			(&inverse).Mod(&inverse, &modulus)
  			if (&inverse).Cmp(one) != 0 {
  				t.Errorf("ModInverse(%d,%d)*%d%%%d=%d, not 1", &element, &modulus, &element, &modulus, &inverse)
  			}
  		}
  	}
  }
  
  // testModSqrt is a helper for TestModSqrt,
  // which checks that ModSqrt can compute a square-root of elt^2.
  func testModSqrt(t *testing.T, elt, mod, sq, sqrt *Int) bool {
  	var sqChk, sqrtChk, sqrtsq Int
  	sq.Mul(elt, elt)
  	sq.Mod(sq, mod)
  	z := sqrt.ModSqrt(sq, mod)
  	if z != sqrt {
  		t.Errorf("ModSqrt returned wrong value %s", z)
  	}
  
  	// test ModSqrt arguments outside the range [0,mod)
  	sqChk.Add(sq, mod)
  	z = sqrtChk.ModSqrt(&sqChk, mod)
  	if z != &sqrtChk || z.Cmp(sqrt) != 0 {
  		t.Errorf("ModSqrt returned inconsistent value %s", z)
  	}
  	sqChk.Sub(sq, mod)
  	z = sqrtChk.ModSqrt(&sqChk, mod)
  	if z != &sqrtChk || z.Cmp(sqrt) != 0 {
  		t.Errorf("ModSqrt returned inconsistent value %s", z)
  	}
  
  	// make sure we actually got a square root
  	if sqrt.Cmp(elt) == 0 {
  		return true // we found the "desired" square root
  	}
  	sqrtsq.Mul(sqrt, sqrt) // make sure we found the "other" one
  	sqrtsq.Mod(&sqrtsq, mod)
  	return sq.Cmp(&sqrtsq) == 0
  }
  
  func TestModSqrt(t *testing.T) {
  	var elt, mod, modx4, sq, sqrt Int
  	r := rand.New(rand.NewSource(9))
  	for i, s := range primes[1:] { // skip 2, use only odd primes
  		mod.SetString(s, 10)
  		modx4.Lsh(&mod, 2)
  
  		// test a few random elements per prime
  		for x := 1; x < 5; x++ {
  			elt.Rand(r, &modx4)
  			elt.Sub(&elt, &mod) // test range [-mod, 3*mod)
  			if !testModSqrt(t, &elt, &mod, &sq, &sqrt) {
  				t.Errorf("#%d: failed (sqrt(e) = %s)", i, &sqrt)
  			}
  		}
  
  		if testing.Short() && i > 2 {
  			break
  		}
  	}
  
  	if testing.Short() {
  		return
  	}
  
  	// exhaustive test for small values
  	for n := 3; n < 100; n++ {
  		mod.SetInt64(int64(n))
  		if !mod.ProbablyPrime(10) {
  			continue
  		}
  		isSquare := make([]bool, n)
  
  		// test all the squares
  		for x := 1; x < n; x++ {
  			elt.SetInt64(int64(x))
  			if !testModSqrt(t, &elt, &mod, &sq, &sqrt) {
  				t.Errorf("#%d: failed (sqrt(%d,%d) = %s)", x, &elt, &mod, &sqrt)
  			}
  			isSquare[sq.Uint64()] = true
  		}
  
  		// test all non-squares
  		for x := 1; x < n; x++ {
  			sq.SetInt64(int64(x))
  			z := sqrt.ModSqrt(&sq, &mod)
  			if !isSquare[x] && z != nil {
  				t.Errorf("#%d: failed (sqrt(%d,%d) = nil)", x, &sqrt, &mod)
  			}
  		}
  	}
  }
  
  func TestJacobi(t *testing.T) {
  	testCases := []struct {
  		x, y   int64
  		result int
  	}{
  		{0, 1, 1},
  		{0, -1, 1},
  		{1, 1, 1},
  		{1, -1, 1},
  		{0, 5, 0},
  		{1, 5, 1},
  		{2, 5, -1},
  		{-2, 5, -1},
  		{2, -5, -1},
  		{-2, -5, 1},
  		{3, 5, -1},
  		{5, 5, 0},
  		{-5, 5, 0},
  		{6, 5, 1},
  		{6, -5, 1},
  		{-6, 5, 1},
  		{-6, -5, -1},
  	}
  
  	var x, y Int
  
  	for i, test := range testCases {
  		x.SetInt64(test.x)
  		y.SetInt64(test.y)
  		expected := test.result
  		actual := Jacobi(&x, &y)
  		if actual != expected {
  			t.Errorf("#%d: Jacobi(%d, %d) = %d, but expected %d", i, test.x, test.y, actual, expected)
  		}
  	}
  }
  
  func TestJacobiPanic(t *testing.T) {
  	const failureMsg = "test failure"
  	defer func() {
  		msg := recover()
  		if msg == nil || msg == failureMsg {
  			panic(msg)
  		}
  		t.Log(msg)
  	}()
  	x := NewInt(1)
  	y := NewInt(2)
  	// Jacobi should panic when the second argument is even.
  	Jacobi(x, y)
  	panic(failureMsg)
  }
  
  func TestIssue2607(t *testing.T) {
  	// This code sequence used to hang.
  	n := NewInt(10)
  	n.Rand(rand.New(rand.NewSource(9)), n)
  }
  
  func TestSqrt(t *testing.T) {
  	root := 0
  	r := new(Int)
  	for i := 0; i < 10000; i++ {
  		if (root+1)*(root+1) <= i {
  			root++
  		}
  		n := NewInt(int64(i))
  		r.SetInt64(-2)
  		r.Sqrt(n)
  		if r.Cmp(NewInt(int64(root))) != 0 {
  			t.Errorf("Sqrt(%v) = %v, want %v", n, r, root)
  		}
  	}
  
  	for i := 0; i < 1000; i += 10 {
  		n, _ := new(Int).SetString("1"+strings.Repeat("0", i), 10)
  		r := new(Int).Sqrt(n)
  		root, _ := new(Int).SetString("1"+strings.Repeat("0", i/2), 10)
  		if r.Cmp(root) != 0 {
  			t.Errorf("Sqrt(1e%d) = %v, want 1e%d", i, r, i/2)
  		}
  	}
  
  	// Test aliasing.
  	r.SetInt64(100)
  	r.Sqrt(r)
  	if r.Int64() != 10 {
  		t.Errorf("Sqrt(100) = %v, want 10 (aliased output)", r.Int64())
  	}
  }
  
  func BenchmarkSqrt(b *testing.B) {
  	n, _ := new(Int).SetString("1"+strings.Repeat("0", 1001), 10)
  	b.ResetTimer()
  	t := new(Int)
  	for i := 0; i < b.N; i++ {
  		t.Sqrt(n)
  	}
  }
  

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