Imported Upstream version 1.4upstream/1.4

1 files changed, 771 insertions, 0 deletions

diff --git a/src/math/big/nat_test.go b/src/math/big/nat_test.go
new file mode 100644
index 000000000..a2ae53385
--- /dev/null
+++ b/src/math/big/nat_test.go
@@ -0,0 +1,771 @@
+// 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 (
+	"io"
+	"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)
+	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 := nat(nil).mulRange(r.a, r.b).decimalString()
+		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)
+	}
+}
+
+func rndNat(n int) nat {
+	return nat(rndV(n)).norm()
+}
+
+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 toString(x nat, charset string) string {
+	base := len(charset)
+
+	// special cases
+	switch {
+	case base < 2:
+		panic("illegal base")
+	case len(x) == 0:
+		return string(charset[0])
+	}
+
+	// allocate buffer for conversion
+	i := x.bitLen()/log2(Word(base)) + 1 // +1: round up
+	s := make([]byte, i)
+
+	// don't destroy x
+	q := nat(nil).set(x)
+
+	// convert
+	for len(q) > 0 {
+		i--
+		var r Word
+		q, r = q.divW(q, Word(base))
+		s[i] = charset[r]
+	}
+
+	return string(s[i:])
+}
+
+var strTests = []struct {
+	x nat    // nat value to be converted
+	c string // conversion charset
+	s string // expected result
+}{
+	{nil, "01", "0"},
+	{nat{1}, "01", "1"},
+	{nat{0xc5}, "01", "11000101"},
+	{nat{03271}, lowercaseDigits[0:8], "3271"},
+	{nat{10}, lowercaseDigits[0:10], "10"},
+	{nat{1234567890}, uppercaseDigits[0:10], "1234567890"},
+	{nat{0xdeadbeef}, lowercaseDigits[0:16], "deadbeef"},
+	{nat{0xdeadbeef}, uppercaseDigits[0:16], "DEADBEEF"},
+	{nat{0x229be7}, lowercaseDigits[0:17], "1a2b3c"},
+	{nat{0x309663e6}, uppercaseDigits[0:32], "O9COV6"},
+}
+
+func TestString(t *testing.T) {
+	for _, a := range strTests {
+		s := a.x.string(a.c)
+		if s != a.s {
+			t.Errorf("string%+v\n\tgot s = %s; want %s", a, s, a.s)
+		}
+
+		x, b, err := nat(nil).scan(strings.NewReader(a.s), len(a.c))
+		if x.cmp(a.x) != 0 {
+			t.Errorf("scan%+v\n\tgot z = %v; want %v", a, x, a.x)
+		}
+		if b != len(a.c) {
+			t.Errorf("scan%+v\n\tgot b = %d; want %d", a, b, len(a.c))
+		}
+		if err != nil {
+			t.Errorf("scan%+v\n\tgot error = %s", a, err)
+		}
+	}
+}
+
+var natScanTests = []struct {
+	s    string // string to be scanned
+	base int    // input base
+	x    nat    // expected nat
+	b    int    // expected base
+	ok   bool   // expected success
+	next rune   // next character (or 0, if at EOF)
+}{
+	// error: illegal base
+	{base: -1},
+	{base: 1},
+	{base: 37},
+
+	// error: no mantissa
+	{},
+	{s: "?"},
+	{base: 10},
+	{base: 36},
+	{s: "?", base: 10},
+	{s: "0x"},
+	{s: "345", base: 2},
+
+	// no errors
+	{"0", 0, nil, 10, true, 0},
+	{"0", 10, nil, 10, true, 0},
+	{"0", 36, nil, 36, true, 0},
+	{"1", 0, nat{1}, 10, true, 0},
+	{"1", 10, nat{1}, 10, true, 0},
+	{"0 ", 0, nil, 10, true, ' '},
+	{"08", 0, nil, 10, true, '8'},
+	{"018", 0, nat{1}, 8, true, '8'},
+	{"0b1", 0, nat{1}, 2, true, 0},
+	{"0b11000101", 0, nat{0xc5}, 2, true, 0},
+	{"03271", 0, nat{03271}, 8, true, 0},
+	{"10ab", 0, nat{10}, 10, true, 'a'},
+	{"1234567890", 0, nat{1234567890}, 10, true, 0},
+	{"xyz", 36, nat{(33*36+34)*36 + 35}, 36, true, 0},
+	{"xyz?", 36, nat{(33*36+34)*36 + 35}, 36, true, '?'},
+	{"0x", 16, nil, 16, true, 'x'},
+	{"0xdeadbeef", 0, nat{0xdeadbeef}, 16, true, 0},
+	{"0XDEADBEEF", 0, nat{0xdeadbeef}, 16, true, 0},
+}
+
+func TestScanBase(t *testing.T) {
+	for _, a := range natScanTests {
+		r := strings.NewReader(a.s)
+		x, b, err := nat(nil).scan(r, a.base)
+		if err == nil && !a.ok {
+			t.Errorf("scan%+v\n\texpected error", a)
+		}
+		if err != nil {
+			if a.ok {
+				t.Errorf("scan%+v\n\tgot error = %s", a, err)
+			}
+			continue
+		}
+		if x.cmp(a.x) != 0 {
+			t.Errorf("scan%+v\n\tgot z = %v; want %v", a, x, a.x)
+		}
+		if b != a.b {
+			t.Errorf("scan%+v\n\tgot b = %d; want %d", a, b, a.base)
+		}
+		next, _, err := r.ReadRune()
+		if err == io.EOF {
+			next = 0
+			err = nil
+		}
+		if err == nil && next != a.next {
+			t.Errorf("scan%+v\n\tgot next = %q; want %q", a, next, a.next)
+		}
+	}
+}
+
+var pi = "3" +
+	"14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651" +
+	"32823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461" +
+	"28475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920" +
+	"96282925409171536436789259036001133053054882046652138414695194151160943305727036575959195309218611738193261179" +
+	"31051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798" +
+	"60943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901" +
+	"22495343014654958537105079227968925892354201995611212902196086403441815981362977477130996051870721134999999837" +
+	"29780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083" +
+	"81420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909" +
+	"21642019893809525720106548586327886593615338182796823030195203530185296899577362259941389124972177528347913151" +
+	"55748572424541506959508295331168617278558890750983817546374649393192550604009277016711390098488240128583616035" +
+	"63707660104710181942955596198946767837449448255379774726847104047534646208046684259069491293313677028989152104" +
+	"75216205696602405803815019351125338243003558764024749647326391419927260426992279678235478163600934172164121992" +
+	"45863150302861829745557067498385054945885869269956909272107975093029553211653449872027559602364806654991198818" +
+	"34797753566369807426542527862551818417574672890977772793800081647060016145249192173217214772350141441973568548" +
+	"16136115735255213347574184946843852332390739414333454776241686251898356948556209921922218427255025425688767179" +
+	"04946016534668049886272327917860857843838279679766814541009538837863609506800642251252051173929848960841284886" +
+	"26945604241965285022210661186306744278622039194945047123713786960956364371917287467764657573962413890865832645" +
+	"99581339047802759009946576407895126946839835259570982582262052248940772671947826848260147699090264013639443745" +
+	"53050682034962524517493996514314298091906592509372216964615157098583874105978859597729754989301617539284681382" +
+	"68683868942774155991855925245953959431049972524680845987273644695848653836736222626099124608051243884390451244" +
+	"13654976278079771569143599770012961608944169486855584840635342207222582848864815845602850601684273945226746767" +
+	"88952521385225499546667278239864565961163548862305774564980355936345681743241125150760694794510965960940252288" +
+	"79710893145669136867228748940560101503308617928680920874760917824938589009714909675985261365549781893129784821" +
+	"68299894872265880485756401427047755513237964145152374623436454285844479526586782105114135473573952311342716610" +
+	"21359695362314429524849371871101457654035902799344037420073105785390621983874478084784896833214457138687519435" +
+	"06430218453191048481005370614680674919278191197939952061419663428754440643745123718192179998391015919561814675" +
+	"14269123974894090718649423196156794520809514655022523160388193014209376213785595663893778708303906979207734672" +
+	"21825625996615014215030680384477345492026054146659252014974428507325186660021324340881907104863317346496514539" +
+	"05796268561005508106658796998163574736384052571459102897064140110971206280439039759515677157700420337869936007" +
+	"23055876317635942187312514712053292819182618612586732157919841484882916447060957527069572209175671167229109816" +
+	"90915280173506712748583222871835209353965725121083579151369882091444210067510334671103141267111369908658516398" +
+	"31501970165151168517143765761835155650884909989859982387345528331635507647918535893226185489632132933089857064" +
+	"20467525907091548141654985946163718027098199430992448895757128289059232332609729971208443357326548938239119325" +
+	"97463667305836041428138830320382490375898524374417029132765618093773444030707469211201913020330380197621101100" +
+	"44929321516084244485963766983895228684783123552658213144957685726243344189303968642624341077322697802807318915" +
+	"44110104468232527162010526522721116603966655730925471105578537634668206531098965269186205647693125705863566201" +
+	"85581007293606598764861179104533488503461136576867532494416680396265797877185560845529654126654085306143444318" +
+	"58676975145661406800700237877659134401712749470420562230538994561314071127000407854733269939081454664645880797" +
+	"27082668306343285878569830523580893306575740679545716377525420211495576158140025012622859413021647155097925923" +
+	"09907965473761255176567513575178296664547791745011299614890304639947132962107340437518957359614589019389713111" +
+	"79042978285647503203198691514028708085990480109412147221317947647772622414254854540332157185306142288137585043" +
+	"06332175182979866223717215916077166925474873898665494945011465406284336639379003976926567214638530673609657120" +
+	"91807638327166416274888800786925602902284721040317211860820419000422966171196377921337575114959501566049631862" +
+	"94726547364252308177036751590673502350728354056704038674351362222477158915049530984448933309634087807693259939" +
+	"78054193414473774418426312986080998886874132604721569516239658645730216315981931951673538129741677294786724229" +
+	"24654366800980676928238280689964004824354037014163149658979409243237896907069779422362508221688957383798623001" +
+	"59377647165122893578601588161755782973523344604281512627203734314653197777416031990665541876397929334419521541" +
+	"34189948544473456738316249934191318148092777710386387734317720754565453220777092120190516609628049092636019759" +
+	"88281613323166636528619326686336062735676303544776280350450777235547105859548702790814356240145171806246436267" +
+	"94561275318134078330336254232783944975382437205835311477119926063813346776879695970309833913077109870408591337"
+
+// Test case for BenchmarkScanPi.
+func TestScanPi(t *testing.T) {
+	var x nat
+	z, _, err := x.scan(strings.NewReader(pi), 10)
+	if err != nil {
+		t.Errorf("scanning pi: %s", err)
+	}
+	if s := z.decimalString(); s != pi {
+		t.Errorf("scanning pi: got %s", s)
+	}
+}
+
+func TestScanPiParallel(t *testing.T) {
+	const n = 2
+	c := make(chan int)
+	for i := 0; i < n; i++ {
+		go func() {
+			TestScanPi(t)
+			c <- 0
+		}()
+	}
+	for i := 0; i < n; i++ {
+		<-c
+	}
+}
+
+func BenchmarkScanPi(b *testing.B) {
+	for i := 0; i < b.N; i++ {
+		var x nat
+		x.scan(strings.NewReader(pi), 10)
+	}
+}
+
+func BenchmarkStringPiParallel(b *testing.B) {
+	var x nat
+	x, _, _ = x.scan(strings.NewReader(pi), 0)
+	if x.decimalString() != pi {
+		panic("benchmark incorrect: conversion failed")
+	}
+	b.RunParallel(func(pb *testing.PB) {
+		for pb.Next() {
+			x.decimalString()
+		}
+	})
+}
+
+func BenchmarkScan10Base2(b *testing.B)     { ScanHelper(b, 2, 10, 10) }
+func BenchmarkScan100Base2(b *testing.B)    { ScanHelper(b, 2, 10, 100) }
+func BenchmarkScan1000Base2(b *testing.B)   { ScanHelper(b, 2, 10, 1000) }
+func BenchmarkScan10000Base2(b *testing.B)  { ScanHelper(b, 2, 10, 10000) }
+func BenchmarkScan100000Base2(b *testing.B) { ScanHelper(b, 2, 10, 100000) }
+
+func BenchmarkScan10Base8(b *testing.B)     { ScanHelper(b, 8, 10, 10) }
+func BenchmarkScan100Base8(b *testing.B)    { ScanHelper(b, 8, 10, 100) }
+func BenchmarkScan1000Base8(b *testing.B)   { ScanHelper(b, 8, 10, 1000) }
+func BenchmarkScan10000Base8(b *testing.B)  { ScanHelper(b, 8, 10, 10000) }
+func BenchmarkScan100000Base8(b *testing.B) { ScanHelper(b, 8, 10, 100000) }
+
+func BenchmarkScan10Base10(b *testing.B)     { ScanHelper(b, 10, 10, 10) }
+func BenchmarkScan100Base10(b *testing.B)    { ScanHelper(b, 10, 10, 100) }
+func BenchmarkScan1000Base10(b *testing.B)   { ScanHelper(b, 10, 10, 1000) }
+func BenchmarkScan10000Base10(b *testing.B)  { ScanHelper(b, 10, 10, 10000) }
+func BenchmarkScan100000Base10(b *testing.B) { ScanHelper(b, 10, 10, 100000) }
+
+func BenchmarkScan10Base16(b *testing.B)     { ScanHelper(b, 16, 10, 10) }
+func BenchmarkScan100Base16(b *testing.B)    { ScanHelper(b, 16, 10, 100) }
+func BenchmarkScan1000Base16(b *testing.B)   { ScanHelper(b, 16, 10, 1000) }
+func BenchmarkScan10000Base16(b *testing.B)  { ScanHelper(b, 16, 10, 10000) }
+func BenchmarkScan100000Base16(b *testing.B) { ScanHelper(b, 16, 10, 100000) }
+
+func ScanHelper(b *testing.B, base int, x, y Word) {
+	b.StopTimer()
+	var z nat
+	z = z.expWW(x, y)
+
+	var s string
+	s = z.string(lowercaseDigits[0:base])
+	if t := toString(z, lowercaseDigits[0:base]); t != s {
+		b.Fatalf("scanning: got %s; want %s", s, t)
+	}
+	b.StartTimer()
+
+	for i := 0; i < b.N; i++ {
+		z.scan(strings.NewReader(s), base)
+	}
+}
+
+func BenchmarkString10Base2(b *testing.B)     { StringHelper(b, 2, 10, 10) }
+func BenchmarkString100Base2(b *testing.B)    { StringHelper(b, 2, 10, 100) }
+func BenchmarkString1000Base2(b *testing.B)   { StringHelper(b, 2, 10, 1000) }
+func BenchmarkString10000Base2(b *testing.B)  { StringHelper(b, 2, 10, 10000) }
+func BenchmarkString100000Base2(b *testing.B) { StringHelper(b, 2, 10, 100000) }
+
+func BenchmarkString10Base8(b *testing.B)     { StringHelper(b, 8, 10, 10) }
+func BenchmarkString100Base8(b *testing.B)    { StringHelper(b, 8, 10, 100) }
+func BenchmarkString1000Base8(b *testing.B)   { StringHelper(b, 8, 10, 1000) }
+func BenchmarkString10000Base8(b *testing.B)  { StringHelper(b, 8, 10, 10000) }
+func BenchmarkString100000Base8(b *testing.B) { StringHelper(b, 8, 10, 100000) }
+
+func BenchmarkString10Base10(b *testing.B)     { StringHelper(b, 10, 10, 10) }
+func BenchmarkString100Base10(b *testing.B)    { StringHelper(b, 10, 10, 100) }
+func BenchmarkString1000Base10(b *testing.B)   { StringHelper(b, 10, 10, 1000) }
+func BenchmarkString10000Base10(b *testing.B)  { StringHelper(b, 10, 10, 10000) }
+func BenchmarkString100000Base10(b *testing.B) { StringHelper(b, 10, 10, 100000) }
+
+func BenchmarkString10Base16(b *testing.B)     { StringHelper(b, 16, 10, 10) }
+func BenchmarkString100Base16(b *testing.B)    { StringHelper(b, 16, 10, 100) }
+func BenchmarkString1000Base16(b *testing.B)   { StringHelper(b, 16, 10, 1000) }
+func BenchmarkString10000Base16(b *testing.B)  { StringHelper(b, 16, 10, 10000) }
+func BenchmarkString100000Base16(b *testing.B) { StringHelper(b, 16, 10, 100000) }
+
+func StringHelper(b *testing.B, base int, x, y Word) {
+	b.StopTimer()
+	var z nat
+	z = z.expWW(x, y)
+	z.string(lowercaseDigits[0:base]) // warm divisor cache
+	b.StartTimer()
+
+	for i := 0; i < b.N; i++ {
+		_ = z.string(lowercaseDigits[0:base])
+	}
+}
+
+func BenchmarkLeafSize0(b *testing.B)  { LeafSizeHelper(b, 10, 0) } // test without splitting
+func BenchmarkLeafSize1(b *testing.B)  { LeafSizeHelper(b, 10, 1) }
+func BenchmarkLeafSize2(b *testing.B)  { LeafSizeHelper(b, 10, 2) }
+func BenchmarkLeafSize3(b *testing.B)  { LeafSizeHelper(b, 10, 3) }
+func BenchmarkLeafSize4(b *testing.B)  { LeafSizeHelper(b, 10, 4) }
+func BenchmarkLeafSize5(b *testing.B)  { LeafSizeHelper(b, 10, 5) }
+func BenchmarkLeafSize6(b *testing.B)  { LeafSizeHelper(b, 10, 6) }
+func BenchmarkLeafSize7(b *testing.B)  { LeafSizeHelper(b, 10, 7) }
+func BenchmarkLeafSize8(b *testing.B)  { LeafSizeHelper(b, 10, 8) }
+func BenchmarkLeafSize9(b *testing.B)  { LeafSizeHelper(b, 10, 9) }
+func BenchmarkLeafSize10(b *testing.B) { LeafSizeHelper(b, 10, 10) }
+func BenchmarkLeafSize11(b *testing.B) { LeafSizeHelper(b, 10, 11) }
+func BenchmarkLeafSize12(b *testing.B) { LeafSizeHelper(b, 10, 12) }
+func BenchmarkLeafSize13(b *testing.B) { LeafSizeHelper(b, 10, 13) }
+func BenchmarkLeafSize14(b *testing.B) { LeafSizeHelper(b, 10, 14) }
+func BenchmarkLeafSize15(b *testing.B) { LeafSizeHelper(b, 10, 15) }
+func BenchmarkLeafSize16(b *testing.B) { LeafSizeHelper(b, 10, 16) }
+func BenchmarkLeafSize32(b *testing.B) { LeafSizeHelper(b, 10, 32) } // try some large lengths
+func BenchmarkLeafSize64(b *testing.B) { LeafSizeHelper(b, 10, 64) }
+
+func LeafSizeHelper(b *testing.B, base Word, size int) {
+	b.StopTimer()
+	originalLeafSize := leafSize
+	resetTable(cacheBase10.table[:])
+	leafSize = size
+	b.StartTimer()
+
+	for d := 1; d <= 10000; d *= 10 {
+		b.StopTimer()
+		var z nat
+		z = z.expWW(base, Word(d))            // build target number
+		_ = z.string(lowercaseDigits[0:base]) // warm divisor cache
+		b.StartTimer()
+
+		for i := 0; i < b.N; i++ {
+			_ = z.string(lowercaseDigits[0:base])
+		}
+	}
+
+	b.StopTimer()
+	resetTable(cacheBase10.table[:])
+	leafSize = originalLeafSize
+	b.StartTimer()
+}
+
+func resetTable(table []divisor) {
+	if table != nil && table[0].bbb != nil {
+		for i := 0; i < len(table); i++ {
+			table[i].bbb = nil
+			table[i].nbits = 0
+			table[i].ndigits = 0
+		}
+	}
+}
+
+func TestStringPowers(t *testing.T) {
+	var b, p Word
+	for b = 2; b <= 16; b++ {
+		for p = 0; p <= 512; p++ {
+			x := nat(nil).expWW(b, p)
+			xs := x.string(lowercaseDigits[0:b])
+			xs2 := toString(x, lowercaseDigits[0:b])
+			if xs != xs2 {
+				t.Errorf("failed at %d ** %d in base %d: %s != %s", b, p, b, xs, xs2)
+			}
+		}
+		if b >= 3 && testing.Short() {
+			break
+		}
+	}
+}
+
+func TestLeadingZeros(t *testing.T) {
+	var x Word = _B >> 1
+	for i := 0; i <= _W; i++ {
+		if int(leadingZeros(x)) != i {
+			t.Errorf("failed at %x: got %d want %d", x, leadingZeros(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
+			}
+		}
+	}
+}
+
+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)
+	}
+}
+
+func TestTrailingZeroBits(t *testing.T) {
+	x := Word(1)
+	for i := uint(0); i <= _W; i++ {
+		n := trailingZeroBits(x)
+		if n != i%_W {
+			t.Errorf("got trailingZeroBits(%#x) = %d; want %d", x, n, i%_W)
+		}
+		x <<= 1
+	}
+
+	y := nat(nil).set(natOne)
+	for i := uint(0); i <= 3*_W; i++ {
+		n := y.trailingZeroBits()
+		if n != i {
+			t.Errorf("got 0x%s.trailingZeroBits() = %d; want %d", y.string(lowercaseDigits[0:16]), n, i)
+		}
+		y = y.shl(y, 1)
+	}
+}
+
+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",
+	},
+}
+
+func TestExpNN(t *testing.T) {
+	for i, test := range expNNTests {
+		x, _, _ := nat(nil).scan(strings.NewReader(test.x), 0)
+		y, _, _ := nat(nil).scan(strings.NewReader(test.y), 0)
+		out, _, _ := nat(nil).scan(strings.NewReader(test.out), 0)
+
+		var m nat
+
+		if len(test.m) > 0 {
+			m, _, _ = nat(nil).scan(strings.NewReader(test.m), 0)
+		}
+
+		z := nat(nil).expNN(x, y, m)
+		if z.cmp(out) != 0 {
+			t.Errorf("#%d got %s want %s", i, z.decimalString(), out.decimalString())
+		}
+	}
+}
+
+func ExpHelper(b *testing.B, x, y Word) {
+	var z nat
+	for i := 0; i < b.N; i++ {
+		z.expWW(x, y)
+	}
+}
+
+func BenchmarkExp3Power0x10(b *testing.B)     { ExpHelper(b, 3, 0x10) }
+func BenchmarkExp3Power0x40(b *testing.B)     { ExpHelper(b, 3, 0x40) }
+func BenchmarkExp3Power0x100(b *testing.B)    { ExpHelper(b, 3, 0x100) }
+func BenchmarkExp3Power0x400(b *testing.B)    { ExpHelper(b, 3, 0x400) }
+func BenchmarkExp3Power0x1000(b *testing.B)   { ExpHelper(b, 3, 0x1000) }
+func BenchmarkExp3Power0x4000(b *testing.B)   { ExpHelper(b, 3, 0x4000) }
+func BenchmarkExp3Power0x10000(b *testing.B)  { ExpHelper(b, 3, 0x10000) }
+func BenchmarkExp3Power0x40000(b *testing.B)  { ExpHelper(b, 3, 0x40000) }
+func BenchmarkExp3Power0x100000(b *testing.B) { ExpHelper(b, 3, 0x100000) }
+func BenchmarkExp3Power0x400000(b *testing.B) { ExpHelper(b, 3, 0x400000) }