src/math/big/float_test.go

// Copyright 2014 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 (
	"flag"
	"fmt"
	"math"
	"strconv"
	"strings"
	"testing"
)

// Verify that ErrNaN implements the error interface.
var _ error = ErrNaN{}

func (x *Float) uint64() uint64 {
	u, acc := x.Uint64()
	if acc != Exact {
		panic(fmt.Sprintf("%s is not a uint64", x.Text('g', 10)))
	}
	return u
}

func (x *Float) int64() int64 {
	i, acc := x.Int64()
	if acc != Exact {
		panic(fmt.Sprintf("%s is not an int64", x.Text('g', 10)))
	}
	return i
}

func TestFloatZeroValue(t *testing.T) {
	// zero (uninitialized) value is a ready-to-use 0.0
	var x Float
	if s := x.Text('f', 1); s != "0.0" {
		t.Errorf("zero value = %s; want 0.0", s)
	}

	// zero value has precision 0
	if prec := x.Prec(); prec != 0 {
		t.Errorf("prec = %d; want 0", prec)
	}

	// zero value can be used in any and all positions of binary operations
	make := func(x int) *Float {
		var f Float
		if x != 0 {
			f.SetInt64(int64(x))
		}
		// x == 0 translates into the zero value
		return &f
	}
	for _, test := range []struct {
		z, x, y, want int
		opname        rune
		op            func(z, x, y *Float) *Float
	}{
		{0, 0, 0, 0, '+', (*Float).Add},
		{0, 1, 2, 3, '+', (*Float).Add},
		{1, 2, 0, 2, '+', (*Float).Add},
		{2, 0, 1, 1, '+', (*Float).Add},

		{0, 0, 0, 0, '-', (*Float).Sub},
		{0, 1, 2, -1, '-', (*Float).Sub},
		{1, 2, 0, 2, '-', (*Float).Sub},
		{2, 0, 1, -1, '-', (*Float).Sub},

		{0, 0, 0, 0, '*', (*Float).Mul},
		{0, 1, 2, 2, '*', (*Float).Mul},
		{1, 2, 0, 0, '*', (*Float).Mul},
		{2, 0, 1, 0, '*', (*Float).Mul},

		// {0, 0, 0, 0, '/', (*Float).Quo}, // panics
		{0, 2, 1, 2, '/', (*Float).Quo},
		{1, 2, 0, 0, '/', (*Float).Quo}, // = +Inf
		{2, 0, 1, 0, '/', (*Float).Quo},
	} {
		z := make(test.z)
		test.op(z, make(test.x), make(test.y))
		got := 0
		if !z.IsInf() {
			got = int(z.int64())
		}
		if got != test.want {
			t.Errorf("%d %c %d = %d; want %d", test.x, test.opname, test.y, got, test.want)
		}
	}

	// TODO(gri) test how precision is set for zero value results
}

func makeFloat(s string) *Float {
	x, _, err := ParseFloat(s, 0, 1000, ToNearestEven)
	if err != nil {
		panic(err)
	}
	return x
}

func TestFloatSetPrec(t *testing.T) {
	for _, test := range []struct {
		x    string
		prec uint
		want string
		acc  Accuracy
	}{
		// prec 0
		{"0", 0, "0", Exact},
		{"-0", 0, "-0", Exact},
		{"-Inf", 0, "-Inf", Exact},
		{"+Inf", 0, "+Inf", Exact},
		{"123", 0, "0", Below},
		{"-123", 0, "-0", Above},

		// prec at upper limit
		{"0", MaxPrec, "0", Exact},
		{"-0", MaxPrec, "-0", Exact},
		{"-Inf", MaxPrec, "-Inf", Exact},
		{"+Inf", MaxPrec, "+Inf", Exact},

		// just a few regular cases - general rounding is tested elsewhere
		{"1.5", 1, "2", Above},
		{"-1.5", 1, "-2", Below},
		{"123", 1e6, "123", Exact},
		{"-123", 1e6, "-123", Exact},
	} {
		x := makeFloat(test.x).SetPrec(test.prec)
		prec := test.prec
		if prec > MaxPrec {
			prec = MaxPrec
		}
		if got := x.Prec(); got != prec {
			t.Errorf("%s.SetPrec(%d).Prec() == %d; want %d", test.x, test.prec, got, prec)
		}
		if got, acc := x.String(), x.Acc(); got != test.want || acc != test.acc {
			t.Errorf("%s.SetPrec(%d) = %s (%s); want %s (%s)", test.x, test.prec, got, acc, test.want, test.acc)
		}
	}
}

func TestFloatMinPrec(t *testing.T) {
	const max = 100
	for _, test := range []struct {
		x    string
		want uint
	}{
		{"0", 0},
		{"-0", 0},
		{"+Inf", 0},
		{"-Inf", 0},
		{"1", 1},
		{"2", 1},
		{"3", 2},
		{"0x8001", 16},
		{"0x8001p-1000", 16},
		{"0x8001p+1000", 16},
		{"0.1", max},
	} {
		x := makeFloat(test.x).SetPrec(max)
		if got := x.MinPrec(); got != test.want {
			t.Errorf("%s.MinPrec() = %d; want %d", test.x, got, test.want)
		}
	}
}

func TestFloatSign(t *testing.T) {
	for _, test := range []struct {
		x string
		s int
	}{
		{"-Inf", -1},
		{"-1", -1},
		{"-0", 0},
		{"+0", 0},
		{"+1", +1},
		{"+Inf", +1},
	} {
		x := makeFloat(test.x)
		s := x.Sign()
		if s != test.s {
			t.Errorf("%s.Sign() = %d; want %d", test.x, s, test.s)
		}
	}
}

// alike(x, y) is like x.Cmp(y) == 0 but also considers the sign of 0 (0 != -0).
func alike(x, y *Float) bool {
	return x.Cmp(y) == 0 && x.Signbit() == y.Signbit()
}

func alike32(x, y float32) bool {
	// we can ignore NaNs
	return x == y && math.Signbit(float64(x)) == math.Signbit(float64(y))

}

func alike64(x, y float64) bool {
	// we can ignore NaNs
	return x == y && math.Signbit(x) == math.Signbit(y)

}

func TestFloatMantExp(t *testing.T) {
	for _, test := range []struct {
		x    string
		mant string
		exp  int
	}{
		{"0", "0", 0},
		{"+0", "0", 0},
		{"-0", "-0", 0},
		{"Inf", "+Inf", 0},
		{"+Inf", "+Inf", 0},
		{"-Inf", "-Inf", 0},
		{"1.5", "0.75", 1},
		{"1.024e3", "0.5", 11},
		{"-0.125", "-0.5", -2},
	} {
		x := makeFloat(test.x)
		mant := makeFloat(test.mant)
		m := new(Float)
		e := x.MantExp(m)
		if !alike(m, mant) || e != test.exp {
			t.Errorf("%s.MantExp() = %s, %d; want %s, %d", test.x, m.Text('g', 10), e, test.mant, test.exp)
		}
	}
}

func TestFloatMantExpAliasing(t *testing.T) {
	x := makeFloat("0.5p10")
	if e := x.MantExp(x); e != 10 {
		t.Fatalf("Float.MantExp aliasing error: got %d; want 10", e)
	}
	if want := makeFloat("0.5"); !alike(x, want) {
		t.Fatalf("Float.MantExp aliasing error: got %s; want %s", x.Text('g', 10), want.Text('g', 10))
	}
}

func TestFloatSetMantExp(t *testing.T) {
	for _, test := range []struct {
		frac string
		exp  int
		z    string
	}{
		{"0", 0, "0"},
		{"+0", 0, "0"},
		{"-0", 0, "-0"},
		{"Inf", 1234, "+Inf"},
		{"+Inf", -1234, "+Inf"},
		{"-Inf", -1234, "-Inf"},
		{"0", MinExp, "0"},
		{"0.25", MinExp, "+0"},    // exponent underflow
		{"-0.25", MinExp, "-0"},   // exponent underflow
		{"1", MaxExp, "+Inf"},     // exponent overflow
		{"2", MaxExp - 1, "+Inf"}, // exponent overflow
		{"0.75", 1, "1.5"},
		{"0.5", 11, "1024"},
		{"-0.5", -2, "-0.125"},
		{"32", 5, "1024"},
		{"1024", -10, "1"},
	} {
		frac := makeFloat(test.frac)
		want := makeFloat(test.z)
		var z Float
		z.SetMantExp(frac, test.exp)
		if !alike(&z, want) {
			t.Errorf("SetMantExp(%s, %d) = %s; want %s", test.frac, test.exp, z.Text('g', 10), test.z)
		}
		// test inverse property
		mant := new(Float)
		if z.SetMantExp(mant, want.MantExp(mant)).Cmp(want) != 0 {
			t.Errorf("Inverse property not satisfied: got %s; want %s", z.Text('g', 10), test.z)
		}
	}
}

func TestFloatPredicates(t *testing.T) {
	for _, test := range []struct {
		x            string
		sign         int
		signbit, inf bool
	}{
		{x: "-Inf", sign: -1, signbit: true, inf: true},
		{x: "-1", sign: -1, signbit: true},
		{x: "-0", signbit: true},
		{x: "0"},
		{x: "1", sign: 1},
		{x: "+Inf", sign: 1, inf: true},
	} {
		x := makeFloat(test.x)
		if got := x.Signbit(); got != test.signbit {
			t.Errorf("(%s).Signbit() = %v; want %v", test.x, got, test.signbit)
		}
		if got := x.Sign(); got != test.sign {
			t.Errorf("(%s).Sign() = %d; want %d", test.x, got, test.sign)
		}
		if got := x.IsInf(); got != test.inf {
			t.Errorf("(%s).IsInf() = %v; want %v", test.x, got, test.inf)
		}
	}
}

func TestFloatIsInt(t *testing.T) {
	for _, test := range []string{
		"0 int",
		"-0 int",
		"1 int",
		"-1 int",
		"0.5",
		"1.23",
		"1.23e1",
		"1.23e2 int",
		"0.000000001e+8",
		"0.000000001e+9 int",
		"1.2345e200 int",
		"Inf",
		"+Inf",
		"-Inf",
	} {
		s := strings.TrimSuffix(test, " int")
		want := s != test
		if got := makeFloat(s).IsInt(); got != want {
			t.Errorf("%s.IsInt() == %t", s, got)
		}
	}
}

func fromBinary(s string) int64 {
	x, err := strconv.ParseInt(s, 2, 64)
	if err != nil {
		panic(err)
	}
	return x
}

func toBinary(x int64) string {
	return strconv.FormatInt(x, 2)
}

func testFloatRound(t *testing.T, x, r int64, prec uint, mode RoundingMode) {
	// verify test data
	var ok bool
	switch mode {
	case ToNearestEven, ToNearestAway:
		ok = true // nothing to do for now
	case ToZero:
		if x < 0 {
			ok = r >= x
		} else {
			ok = r <= x
		}
	case AwayFromZero:
		if x < 0 {
			ok = r <= x
		} else {
			ok = r >= x
		}
	case ToNegativeInf:
		ok = r <= x
	case ToPositiveInf:
		ok = r >= x
	default:
		panic("unreachable")
	}
	if !ok {
		t.Fatalf("incorrect test data for prec = %d, %s: x = %s, r = %s", prec, mode, toBinary(x), toBinary(r))
	}

	// compute expected accuracy
	a := Exact
	switch {
	case r < x:
		a = Below
	case r > x:
		a = Above
	}

	// round
	f := new(Float).SetMode(mode).SetInt64(x).SetPrec(prec)

	// check result
	r1 := f.int64()
	p1 := f.Prec()
	a1 := f.Acc()
	if r1 != r || p1 != prec || a1 != a {
		t.Errorf("round %s (%d bits, %s) incorrect: got %s (%d bits, %s); want %s (%d bits, %s)",
			toBinary(x), prec, mode,
			toBinary(r1), p1, a1,
			toBinary(r), prec, a)
		return
	}

	// g and f should be the same
	// (rounding by SetPrec after SetInt64 using default precision
	// should be the same as rounding by SetInt64 after setting the
	// precision)
	g := new(Float).SetMode(mode).SetPrec(prec).SetInt64(x)
	if !alike(g, f) {
		t.Errorf("round %s (%d bits, %s) not symmetric: got %s and %s; want %s",
			toBinary(x), prec, mode,
			toBinary(g.int64()),
			toBinary(r1),
			toBinary(r),
		)
		return
	}

	// h and f should be the same
	// (repeated rounding should be idempotent)
	h := new(Float).SetMode(mode).SetPrec(prec).Set(f)
	if !alike(h, f) {
		t.Errorf("round %s (%d bits, %s) not idempotent: got %s and %s; want %s",
			toBinary(x), prec, mode,
			toBinary(h.int64()),
			toBinary(r1),
			toBinary(r),
		)
		return
	}
}

// TestFloatRound tests basic rounding.
func TestFloatRound(t *testing.T) {
	for _, test := range []struct {
		prec                        uint
		x, zero, neven, naway, away string // input, results rounded to prec bits
	}{
		{5, "1000", "1000", "1000", "1000", "1000"},
		{5, "1001", "1001", "1001", "1001", "1001"},
		{5, "1010", "1010", "1010", "1010", "1010"},
		{5, "1011", "1011", "1011", "1011", "1011"},
		{5, "1100", "1100", "1100", "1100", "1100"},
		{5, "1101", "1101", "1101", "1101", "1101"},
		{5, "1110", "1110", "1110", "1110", "1110"},
		{5, "1111", "1111", "1111", "1111", "1111"},

		{4, "1000", "1000", "1000", "1000", "1000"},
		{4, "1001", "1001", "1001", "1001", "1001"},
		{4, "1010", "1010", "1010", "1010", "1010"},
		{4, "1011", "1011", "1011", "1011", "1011"},
		{4, "1100", "1100", "1100", "1100", "1100"},
		{4, "1101", "1101", "1101", "1101", "1101"},
		{4, "1110", "1110", "1110", "1110", "1110"},
		{4, "1111", "1111", "1111", "1111", "1111"},

		{3, "1000", "1000", "1000", "1000", "1000"},
		{3, "1001", "1000", "1000", "1010", "1010"},
		{3, "1010", "1010", "1010", "1010", "1010"},
		{3, "1011", "1010", "1100", "1100", "1100"},
		{3, "1100", "1100", "1100", "1100", "1100"},
		{3, "1101", "1100", "1100", "1110", "1110"},
		{3, "1110", "1110", "1110", "1110", "1110"},
		{3, "1111", "1110", "10000", "10000", "10000"},

		{3, "1000001", "1000000", "1000000", "1000000", "1010000"},
		{3, "1001001", "1000000", "1010000", "1010000", "1010000"},
		{3, "1010001", "1010000", "1010000", "1010000", "1100000"},
		{3, "1011001", "1010000", "1100000", "1100000", "1100000"},
		{3, "1100001", "1100000", "1100000", "1100000", "1110000"},
		{3, "1101001", "1100000", "1110000", "1110000", "1110000"},
		{3, "1110001", "1110000", "1110000", "1110000", "10000000"},
		{3, "1111001", "1110000", "10000000", "10000000", "10000000"},

		{2, "1000", "1000", "1000", "1000", "1000"},
		{2, "1001", "1000", "1000", "1000", "1100"},
		{2, "1010", "1000", "1000", "1100", "1100"},
		{2, "1011", "1000", "1100", "1100", "1100"},
		{2, "1100", "1100", "1100", "1100", "1100"},
		{2, "1101", "1100", "1100", "1100", "10000"},
		{2, "1110", "1100", "10000", "10000", "10000"},
		{2, "1111", "1100", "10000", "10000", "10000"},

		{2, "1000001", "1000000", "1000000", "1000000", "1100000"},
		{2, "1001001", "1000000", "1000000", "1000000", "1100000"},
		{2, "1010001", "1000000", "1100000", "1100000", "1100000"},
		{2, "1011001", "1000000", "1100000", "1100000", "1100000"},
		{2, "1100001", "1100000", "1100000", "1100000", "10000000"},
		{2, "1101001", "1100000", "1100000", "1100000", "10000000"},
		{2, "1110001", "1100000", "10000000", "10000000", "10000000"},
		{2, "1111001", "1100000", "10000000", "10000000", "10000000"},

		{1, "1000", "1000", "1000", "1000", "1000"},
		{1, "1001", "1000", "1000", "1000", "10000"},
		{1, "1010", "1000", "1000", "1000", "10000"},
		{1, "1011", "1000", "1000", "1000", "10000"},
		{1, "1100", "1000", "10000", "10000", "10000"},
		{1, "1101", "1000", "10000", "10000", "10000"},
		{1, "1110", "1000", "10000", "10000", "10000"},
		{1, "1111", "1000", "10000", "10000", "10000"},

		{1, "1000001", "1000000", "1000000", "1000000", "10000000"},
		{1, "1001001", "1000000", "1000000", "1000000", "10000000"},
		{1, "1010001", "1000000", "1000000", "1000000", "10000000"},
		{1, "1011001", "1000000", "1000000", "1000000", "10000000"},
		{1, "1100001", "1000000", "10000000", "10000000", "10000000"},
		{1, "1101001", "1000000", "10000000", "10000000", "10000000"},
		{1, "1110001", "1000000", "10000000", "10000000", "10000000"},
		{1, "1111001", "1000000", "10000000", "10000000", "10000000"},
	} {
		x := fromBinary(test.x)
		z := fromBinary(test.zero)
		e := fromBinary(test.neven)
		n := fromBinary(test.naway)
		a := fromBinary(test.away)
		prec := test.prec

		testFloatRound(t, x, z, prec, ToZero)
		testFloatRound(t, x, e, prec, ToNearestEven)
		testFloatRound(t, x, n, prec, ToNearestAway)
		testFloatRound(t, x, a, prec, AwayFromZero)

		testFloatRound(t, x, z, prec, ToNegativeInf)
		testFloatRound(t, x, a, prec, ToPositiveInf)

		testFloatRound(t, -x, -a, prec, ToNegativeInf)
		testFloatRound(t, -x, -z, prec, ToPositiveInf)
	}
}

// TestFloatRound24 tests that rounding a float64 to 24 bits
// matches IEEE-754 rounding to nearest when converting a
// float64 to a float32 (excluding denormal numbers).
func TestFloatRound24(t *testing.T) {
	const x0 = 1<<26 - 0x10 // 11...110000 (26 bits)
	for d := 0; d <= 0x10; d++ {
		x := float64(x0 + d)
		f := new(Float).SetPrec(24).SetFloat64(x)
		got, _ := f.Float32()
		want := float32(x)
		if got != want {
			t.Errorf("Round(%g, 24) = %g; want %g", x, got, want)
		}
	}
}

func TestFloatSetUint64(t *testing.T) {
	for _, want := range []uint64{
		0,
		1,
		2,
		10,
		100,
		1<<32 - 1,
		1 << 32,
		1<<64 - 1,
	} {
		var f Float
		f.SetUint64(want)
		if got := f.uint64(); got != want {
			t.Errorf("got %#x (%s); want %#x", got, f.Text('p', 0), want)
		}
	}

	// test basic rounding behavior (exhaustive rounding testing is done elsewhere)
	const x uint64 = 0x8765432187654321 // 64 bits needed
	for prec := uint(1); prec <= 64; prec++ {
		f := new(Float).SetPrec(prec).SetMode(ToZero).SetUint64(x)
		got := f.uint64()
		want := x &^ (1<<(64-prec) - 1) // cut off (round to zero) low 64-prec bits
		if got != want {
			t.Errorf("got %#x (%s); want %#x", got, f.Text('p', 0), want)
		}
	}
}

func TestFloatSetInt64(t *testing.T) {
	for _, want := range []int64{
		0,
		1,
		2,
		10,
		100,
		1<<32 - 1,
		1 << 32,
		1<<63 - 1,
	} {
		for i := range [2]int{} {
			if i&1 != 0 {
				want = -want
			}
			var f Float
			f.SetInt64(want)
			if got := f.int64(); got != want {
				t.Errorf("got %#x (%s); want %#x", got, f.Text('p', 0), want)
			}
		}
	}

	// test basic rounding behavior (exhaustive rounding testing is done elsewhere)
	const x int64 = 0x7654321076543210 // 63 bits needed
	for prec := uint(1); prec <= 63; prec++ {
		f := new(Float).SetPrec(prec).SetMode(ToZero).SetInt64(x)
		got := f.int64()
		want := x &^ (1<<(63-prec) - 1) // cut off (round to zero) low 63-prec bits
		if got != want {
			t.Errorf("got %#x (%s); want %#x", got, f.Text('p', 0), want)
		}
	}
}

func TestFloatSetFloat64(t *testing.T) {
	for _, want := range []float64{
		0,
		1,
		2,
		12345,
		1e10,
		1e100,
		3.14159265e10,
		2.718281828e-123,
		1.0 / 3,
		math.MaxFloat32,
		math.MaxFloat64,
		math.SmallestNonzeroFloat32,
		math.SmallestNonzeroFloat64,
		math.Inf(-1),
		math.Inf(0),
		-math.Inf(1),
	} {
		for i := range [2]int{} {
			if i&1 != 0 {
				want = -want
			}
			var f Float
			f.SetFloat64(want)
			if got, acc := f.Float64(); got != want || acc != Exact {
				t.Errorf("got %g (%s, %s); want %g (Exact)", got, f.Text('p', 0), acc, want)
			}
		}
	}

	// test basic rounding behavior (exhaustive rounding testing is done elsewhere)
	const x uint64 = 0x8765432143218 // 53 bits needed
	for prec := uint(1); prec <= 52; prec++ {
		f := new(Float).SetPrec(prec).SetMode(ToZero).SetFloat64(float64(x))
		got, _ := f.Float64()
		want := float64(x &^ (1<<(52-prec) - 1)) // cut off (round to zero) low 53-prec bits
		if got != want {
			t.Errorf("got %g (%s); want %g", got, f.Text('p', 0), want)
		}
	}

	// test NaN
	defer func() {
		if p, ok := recover().(ErrNaN); !ok {
			t.Errorf("got %v; want ErrNaN panic", p)
		}
	}()
	var f Float
	f.SetFloat64(math.NaN())
	// should not reach here
	t.Errorf("got %s; want ErrNaN panic", f.Text('p', 0))
}

func TestFloatSetInt(t *testing.T) {
	for _, want := range []string{
		"0",
		"1",
		"-1",
		"1234567890",
		"123456789012345678901234567890",
		"123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890",
	} {
		var x Int
		_, ok := x.SetString(want, 0)
		if !ok {
			t.Errorf("invalid integer %s", want)
			continue
		}
		n := x.BitLen()

		var f Float
		f.SetInt(&x)

		// check precision
		if n < 64 {
			n = 64
		}
		if prec := f.Prec(); prec != uint(n) {
			t.Errorf("got prec = %d; want %d", prec, n)
		}

		// check value
		got := f.Text('g', 100)
		if got != want {
			t.Errorf("got %s (%s); want %s", got, f.Text('p', 0), want)
		}
	}

	// TODO(gri) test basic rounding behavior
}

func TestFloatSetRat(t *testing.T) {
	for _, want := range []string{
		"0",
		"1",
		"-1",
		"1234567890",
		"123456789012345678901234567890",
		"123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890",
		"1.2",
		"3.14159265",
		// TODO(gri) expand
	} {
		var x Rat
		_, ok := x.SetString(want)
		if !ok {
			t.Errorf("invalid fraction %s", want)
			continue
		}
		n := max(x.Num().BitLen(), x.Denom().BitLen())

		var f1, f2 Float
		f2.SetPrec(1000)
		f1.SetRat(&x)
		f2.SetRat(&x)

		// check precision when set automatically
		if n < 64 {
			n = 64
		}
		if prec := f1.Prec(); prec != uint(n) {
			t.Errorf("got prec = %d; want %d", prec, n)
		}

		got := f2.Text('g', 100)
		if got != want {
			t.Errorf("got %s (%s); want %s", got, f2.Text('p', 0), want)
		}
	}
}

func TestFloatSetInf(t *testing.T) {
	var f Float
	for _, test := range []struct {
		signbit bool
		prec    uint
		want    string
	}{
		{false, 0, "+Inf"},
		{true, 0, "-Inf"},
		{false, 10, "+Inf"},
		{true, 30, "-Inf"},
	} {
		x := f.SetPrec(test.prec).SetInf(test.signbit)
		if got := x.String(); got != test.want || x.Prec() != test.prec {
			t.Errorf("SetInf(%v) = %s (prec = %d); want %s (prec = %d)", test.signbit, got, x.Prec(), test.want, test.prec)
		}
	}
}

func TestFloatUint64(t *testing.T) {
	for _, test := range []struct {
		x   string
		out uint64
		acc Accuracy
	}{
		{"-Inf", 0, Above},
		{"-1", 0, Above},
		{"-1e-1000", 0, Above},
		{"-0", 0, Exact},
		{"0", 0, Exact},
		{"1e-1000", 0, Below},
		{"1", 1, Exact},
		{"1.000000000000000000001", 1, Below},
		{"12345.0", 12345, Exact},
		{"12345.000000000000000000001", 12345, Below},
		{"18446744073709551615", 18446744073709551615, Exact},
		{"18446744073709551615.000000000000000000001", math.MaxUint64, Below},
		{"18446744073709551616", math.MaxUint64, Below},
		{"1e10000", math.MaxUint64, Below},
		{"+Inf", math.MaxUint64, Below},
	} {
		x := makeFloat(test.x)
		out, acc := x.Uint64()
		if out != test.out || acc != test.acc {
			t.Errorf("%s: got %d (%s); want %d (%s)", test.x, out, acc, test.out, test.acc)
		}
	}
}

func TestFloatInt64(t *testing.T) {
	for _, test := range []struct {
		x   string
		out int64
		acc Accuracy
	}{
		{"-Inf", math.MinInt64, Above},
		{"-1e10000", math.MinInt64, Above},
		{"-9223372036854775809", math.MinInt64, Above},
		{"-9223372036854775808.000000000000000000001", math.MinInt64, Above},
		{"-9223372036854775808", -9223372036854775808, Exact},
		{"-9223372036854775807.000000000000000000001", -9223372036854775807, Above},
		{"-9223372036854775807", -9223372036854775807, Exact},
		{"-12345.000000000000000000001", -12345, Above},
		{"-12345.0", -12345, Exact},
		{"-1.000000000000000000001", -1, Above},
		{"-1.5", -1, Above},
		{"-1", -1, Exact},
		{"-1e-1000", 0, Above},
		{"0", 0, Exact},
		{"1e-1000", 0, Below},
		{"1", 1, Exact},
		{"1.000000000000000000001", 1, Below},
		{"1.5", 1, Below},
		{"12345.0", 12345, Exact},
		{"12345.000000000000000000001", 12345, Below},
		{"9223372036854775807", 9223372036854775807, Exact},
		{"9223372036854775807.000000000000000000001", math.MaxInt64, Below},
		{"9223372036854775808", math.MaxInt64, Below},
		{"1e10000", math.MaxInt64, Below},
		{"+Inf", math.MaxInt64, Below},
	} {
		x := makeFloat(test.x)
		out, acc := x.Int64()
		if out != test.out || acc != test.acc {
			t.Errorf("%s: got %d (%s); want %d (%s)", test.x, out, acc, test.out, test.acc)
		}
	}
}

func TestFloatFloat32(t *testing.T) {
	for _, test := range []struct {
		x   string
		out float32
		acc Accuracy
	}{
		{"0", 0, Exact},

		// underflow to zero
		{"1e-1000", 0, Below},
		{"0x0.000002p-127", 0, Below},
		{"0x.0000010p-126", 0, Below},

		// denormals
		{"1.401298464e-45", math.SmallestNonzeroFloat32, Above}, // rounded up to smallest denormal
		{"0x.ffffff8p-149", math.SmallestNonzeroFloat32, Above}, // rounded up to smallest denormal
		{"0x.0000018p-126", math.SmallestNonzeroFloat32, Above}, // rounded up to smallest denormal
		{"0x.0000020p-126", math.SmallestNonzeroFloat32, Exact},
		{"0x.8p-148", math.SmallestNonzeroFloat32, Exact},
		{"1p-149", math.SmallestNonzeroFloat32, Exact},
		{"0x.fffffep-126", math.Float32frombits(0x7fffff), Exact}, // largest denormal

		// special denormal cases (see issues 14553, 14651)
		{"0x0.0000001p-126", math.Float32frombits(0x00000000), Below}, // underflow to zero
		{"0x0.0000008p-126", math.Float32frombits(0x00000000), Below}, // underflow to zero
		{"0x0.0000010p-126", math.Float32frombits(0x00000000), Below}, // rounded down to even
		{"0x0.0000011p-126", math.Float32frombits(0x00000001), Above}, // rounded up to smallest denormal
		{"0x0.0000018p-126", math.Float32frombits(0x00000001), Above}, // rounded up to smallest denormal

		{"0x1.0000000p-149", math.Float32frombits(0x00000001), Exact}, // smallest denormal
		{"0x0.0000020p-126", math.Float32frombits(0x00000001), Exact}, // smallest denormal
		{"0x0.fffffe0p-126", math.Float32frombits(0x007fffff), Exact}, // largest denormal
		{"0x1.0000000p-126", math.Float32frombits(0x00800000), Exact}, // smallest normal

		{"0x0.8p-149", math.Float32frombits(0x000000000), Below}, // rounded down to even
		{"0x0.9p-149", math.Float32frombits(0x000000001), Above}, // rounded up to smallest denormal
		{"0x0.ap-149", math.Float32frombits(0x000000001), Above}, // rounded up to smallest denormal
		{"0x0.bp-149", math.Float32frombits(0x000000001), Above}, // rounded up to smallest denormal
		{"0x0.cp-149", math.Float32frombits(0x000000001), Above}, // rounded up to smallest denormal

		{"0x1.0p-149", math.Float32frombits(0x000000001), Exact}, // smallest denormal
		{"0x1.7p-149", math.Float32frombits(0x000000001), Below},
		{"0x1.8p-149", math.Float32frombits(0x000000002), Above},
		{"0x1.9p-149", math.Float32frombits(0x000000002), Above},

		{"0x2.0p-149", math.Float32frombits(0x000000002), Exact},
		{"0x2.8p-149", math.Float32frombits(0x000000002), Below}, // rounded down to even
		{"0x2.9p-149", math.Float32frombits(0x000000003), Above},

		{"0x3.0p-149", math.Float32frombits(0x000000003), Exact},
		{"0x3.7p-149", math.Float32frombits(0x000000003), Below},
		{"0x3.8p-149", math.Float32frombits(0x000000004), Above}, // rounded up to even

		{"0x4.0p-149", math.Float32frombits(0x000000004), Exact},
		{"0x4.8p-149", math.Float32frombits(0x000000004), Below}, // rounded down to even
		{"0x4.9p-149", math.Float32frombits(0x000000005), Above},

		// specific case from issue 14553
		{"0x7.7p-149", math.Float32frombits(0x000000007), Below},
		{"0x7.8p-149", math.Float32frombits(0x000000008), Above},
		{"0x7.9p-149", math.Float32frombits(0x000000008), Above},

		// normals
		{"0x.ffffffp-126", math.Float32frombits(0x00800000), Above}, // rounded up to smallest normal
		{"1p-126", math.Float32frombits(0x00800000), Exact},         // smallest normal
		{"0x1.fffffep-126", math.Float32frombits(0x00ffffff), Exact},
		{"0x1.ffffffp-126", math.Float32frombits(0x01000000), Above}, // rounded up
		{"1", 1, Exact},
		{"1.000000000000000000001", 1, Below},
		{"12345.0", 12345, Exact},
		{"12345.000000000000000000001", 12345, Below},
		{"0x1.fffffe0p127", math.MaxFloat32, Exact},
		{"0x1.fffffe8p127", math.MaxFloat32, Below},

		// overflow
		{"0x1.ffffff0p127", float32(math.Inf(+1)), Above},
		{"0x1p128", float32(math.Inf(+1)), Above},
		{"1e10000", float32(math.Inf(+1)), Above},
		{"0x1.ffffff0p2147483646", float32(math.Inf(+1)), Above}, // overflow in rounding

		// inf
		{"Inf", float32(math.Inf(+1)), Exact},
	} {
		for i := 0; i < 2; i++ {
			// test both signs
			tx, tout, tacc := test.x, test.out, test.acc
			if i != 0 {
				tx = "-" + tx
				tout = -tout
				tacc = -tacc
			}

			// conversion should match strconv where syntax is agreeable
			if f, err := strconv.ParseFloat(tx, 32); err == nil && !alike32(float32(f), tout) {
				t.Errorf("%s: got %g; want %g (incorrect test data)", tx, f, tout)
			}

			x := makeFloat(tx)
			out, acc := x.Float32()
			if !alike32(out, tout) || acc != tacc {
				t.Errorf("%s: got %g (%#08x, %s); want %g (%#08x, %s)", tx, out, math.Float32bits(out), acc, test.out, math.Float32bits(test.out), tacc)
			}

			// test that x.SetFloat64(float64(f)).Float32() == f
			var x2 Float
			out2, acc2 := x2.SetFloat64(float64(out)).Float32()
			if !alike32(out2, out) || acc2 != Exact {
				t.Errorf("idempotency test: got %g (%s); want %g (Exact)", out2, acc2, out)
			}
		}
	}
}

func TestFloatFloat64(t *testing.T) {
	const smallestNormalFloat64 = 2.2250738585072014e-308 // 1p-1022
	for _, test := range []struct {
		x   string
		out float64
		acc Accuracy
	}{
		{"0", 0, Exact},

		// underflow to zero
		{"1e-1000", 0, Below},
		{"0x0.0000000000001p-1023", 0, Below},
		{"0x0.00000000000008p-1022", 0, Below},

		// denormals
		{"0x0.0000000000000cp-1022", math.SmallestNonzeroFloat64, Above}, // rounded up to smallest denormal
		{"0x0.00000000000010p-1022", math.SmallestNonzeroFloat64, Exact}, // smallest denormal
		{"0x.8p-1073", math.SmallestNonzeroFloat64, Exact},
		{"1p-1074", math.SmallestNonzeroFloat64, Exact},
		{"0x.fffffffffffffp-1022", math.Float64frombits(0x000fffffffffffff), Exact}, // largest denormal

		// special denormal cases (see issues 14553, 14651)
		{"0x0.00000000000001p-1022", math.Float64frombits(0x00000000000000000), Below}, // underflow to zero
		{"0x0.00000000000004p-1022", math.Float64frombits(0x00000000000000000), Below}, // underflow to zero
		{"0x0.00000000000008p-1022", math.Float64frombits(0x00000000000000000), Below}, // rounded down to even
		{"0x0.00000000000009p-1022", math.Float64frombits(0x00000000000000001), Above}, // rounded up to smallest denormal
		{"0x0.0000000000000ap-1022", math.Float64frombits(0x00000000000000001), Above}, // rounded up to smallest denormal

		{"0x0.8p-1074", math.Float64frombits(0x00000000000000000), Below}, // rounded down to even
		{"0x0.9p-1074", math.Float64frombits(0x00000000000000001), Above}, // rounded up to smallest denormal
		{"0x0.ap-1074", math.Float64frombits(0x00000000000000001), Above}, // rounded up to smallest denormal
		{"0x0.bp-1074", math.Float64frombits(0x00000000000000001), Above}, // rounded up to smallest denormal
		{"0x0.cp-1074", math.Float64frombits(0x00000000000000001), Above}, // rounded up to smallest denormal

		{"0x1.0p-1074", math.Float64frombits(0x00000000000000001), Exact},
		{"0x1.7p-1074", math.Float64frombits(0x00000000000000001), Below},
		{"0x1.8p-1074", math.Float64frombits(0x00000000000000002), Above},
		{"0x1.9p-1074", math.Float64frombits(0x00000000000000002), Above},

		{"0x2.0p-1074", math.Float64frombits(0x00000000000000002), Exact},
		{"0x2.8p-1074", math.Float64frombits(0x00000000000000002), Below}, // rounded down to even
		{"0x2.9p-1074", math.Float64frombits(0x00000000000000003), Above},

		{"0x3.0p-1074", math.Float64frombits(0x00000000000000003), Exact},
		{"0x3.7p-1074", math.Float64frombits(0x00000000000000003), Below},
		{"0x3.8p-1074", math.Float64frombits(0x00000000000000004), Above}, // rounded up to even

		{"0x4.0p-1074", math.Float64frombits(0x00000000000000004), Exact},
		{"0x4.8p-1074", math.Float64frombits(0x00000000000000004), Below}, // rounded down to even
		{"0x4.9p-1074", math.Float64frombits(0x00000000000000005), Above},

		// normals
		{"0x.fffffffffffff8p-1022", math.Float64frombits(0x0010000000000000), Above}, // rounded up to smallest normal
		{"1p-1022", math.Float64frombits(0x0010000000000000), Exact},                 // smallest normal
		{"1", 1, Exact},
		{"1.000000000000000000001", 1, Below},
		{"12345.0", 12345, Exact},
		{"12345.000000000000000000001", 12345, Below},
		{"0x1.fffffffffffff0p1023", math.MaxFloat64, Exact},
		{"0x1.fffffffffffff4p1023", math.MaxFloat64, Below},

		// overflow
		{"0x1.fffffffffffff8p1023", math.Inf(+1), Above},
		{"0x1p1024", math.Inf(+1), Above},
		{"1e10000", math.Inf(+1), Above},
		{"0x1.fffffffffffff8p2147483646", math.Inf(+1), Above}, // overflow in rounding
		{"Inf", math.Inf(+1), Exact},

		// selected denormalized values that were handled incorrectly in the past
		{"0x.fffffffffffffp-1022", smallestNormalFloat64 - math.SmallestNonzeroFloat64, Exact},
		{"4503599627370495p-1074", smallestNormalFloat64 - math.SmallestNonzeroFloat64, Exact},

		// http://www.exploringbinary.com/php-hangs-on-numeric-value-2-2250738585072011e-308/
		{"2.2250738585072011e-308", 2.225073858507201e-308, Below},
		// http://www.exploringbinary.com/java-hangs-when-converting-2-2250738585072012e-308/
		{"2.2250738585072012e-308", 2.2250738585072014e-308, Above},
	} {
		for i := 0; i < 2; i++ {
			// test both signs
			tx, tout, tacc := test.x, test.out, test.acc
			if i != 0 {
				tx = "-" + tx
				tout = -tout
				tacc = -tacc
			}

			// conversion should match strconv where syntax is agreeable
			if f, err := strconv.ParseFloat(tx, 64); err == nil && !alike64(f, tout) {
				t.Errorf("%s: got %g; want %g (incorrect test data)", tx, f, tout)
			}

			x := makeFloat(tx)
			out, acc := x.Float64()
			if !alike64(out, tout) || acc != tacc {
				t.Errorf("%s: got %g (%#016x, %s); want %g (%#016x, %s)", tx, out, math.Float64bits(out), acc, test.out, math.Float64bits(test.out), tacc)
			}

			// test that x.SetFloat64(f).Float64() == f
			var x2 Float
			out2, acc2 := x2.SetFloat64(out).Float64()
			if !alike64(out2, out) || acc2 != Exact {
				t.Errorf("idempotency test: got %g (%s); want %g (Exact)", out2, acc2, out)
			}
		}
	}
}

func TestFloatInt(t *testing.T) {
	for _, test := range []struct {
		x    string
		want string
		acc  Accuracy
	}{
		{"0", "0", Exact},
		{"+0", "0", Exact},
		{"-0", "0", Exact},
		{"Inf", "nil", Below},
		{"+Inf", "nil", Below},
		{"-Inf", "nil", Above},
		{"1", "1", Exact},
		{"-1", "-1", Exact},
		{"1.23", "1", Below},
		{"-1.23", "-1", Above},
		{"123e-2", "1", Below},
		{"123e-3", "0", Below},
		{"123e-4", "0", Below},
		{"1e-1000", "0", Below},
		{"-1e-1000", "0", Above},
		{"1e+10", "10000000000", Exact},
		{"1e+100", "10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000", Exact},
	} {
		x := makeFloat(test.x)
		res, acc := x.Int(nil)
		got := "nil"
		if res != nil {
			got = res.String()
		}
		if got != test.want || acc != test.acc {
			t.Errorf("%s: got %s (%s); want %s (%s)", test.x, got, acc, test.want, test.acc)
		}
	}

	// check that supplied *Int is used
	for _, f := range []string{"0", "1", "-1", "1234"} {
		x := makeFloat(f)
		i := new(Int)
		if res, _ := x.Int(i); res != i {
			t.Errorf("(%s).Int is not using supplied *Int", f)
		}
	}
}

func TestFloatRat(t *testing.T) {
	for _, test := range []struct {
		x, want string
		acc     Accuracy
	}{
		{"0", "0/1", Exact},
		{"+0", "0/1", Exact},
		{"-0", "0/1", Exact},
		{"Inf", "nil", Below},
		{"+Inf", "nil", Below},
		{"-Inf", "nil", Above},
		{"1", "1/1", Exact},
		{"-1", "-1/1", Exact},
		{"1.25", "5/4", Exact},
		{"-1.25", "-5/4", Exact},
		{"1e10", "10000000000/1", Exact},
		{"1p10", "1024/1", Exact},
		{"-1p-10", "-1/1024", Exact},
		{"3.14159265", "7244019449799623199/2305843009213693952", Exact},
	} {
		x := makeFloat(test.x).SetPrec(64)
		res, acc := x.Rat(nil)
		got := "nil"
		if res != nil {
			got = res.String()
		}
		if got != test.want {
			t.Errorf("%s: got %s; want %s", test.x, got, test.want)
			continue
		}
		if acc != test.acc {
			t.Errorf("%s: got %s; want %s", test.x, acc, test.acc)
			continue
		}

		// inverse conversion
		if res != nil {
			got := new(Float).SetPrec(64).SetRat(res)
			if got.Cmp(x) != 0 {
				t.Errorf("%s: got %s; want %s", test.x, got, x)
			}
		}
	}

	// check that supplied *Rat is used
	for _, f := range []string{"0", "1", "-1", "1234"} {
		x := makeFloat(f)
		r := new(Rat)
		if res, _ := x.Rat(r); res != r {
			t.Errorf("(%s).Rat is not using supplied *Rat", f)
		}
	}
}

func TestFloatAbs(t *testing.T) {
	for _, test := range []string{
		"0",
		"1",
		"1234",
		"1.23e-2",
		"1e-1000",
		"1e1000",
		"Inf",
	} {
		p := makeFloat(test)
		a := new(Float).Abs(p)
		if !alike(a, p) {
			t.Errorf("%s: got %s; want %s", test, a.Text('g', 10), test)
		}

		n := makeFloat("-" + test)
		a.Abs(n)
		if !alike(a, p) {
			t.Errorf("-%s: got %s; want %s", test, a.Text('g', 10), test)
		}
	}
}

func TestFloatNeg(t *testing.T) {
	for _, test := range []string{
		"0",
		"1",
		"1234",
		"1.23e-2",
		"1e-1000",
		"1e1000",
		"Inf",
	} {
		p1 := makeFloat(test)
		n1 := makeFloat("-" + test)
		n2 := new(Float).Neg(p1)
		p2 := new(Float).Neg(n2)
		if !alike(n2, n1) {
			t.Errorf("%s: got %s; want %s", test, n2.Text('g', 10), n1.Text('g', 10))
		}
		if !alike(p2, p1) {
			t.Errorf("%s: got %s; want %s", test, p2.Text('g', 10), p1.Text('g', 10))
		}
	}
}

func TestFloatInc(t *testing.T) {
	const n = 10
	for _, prec := range precList {
		if 1<<prec < n {
			continue // prec must be large enough to hold all numbers from 0 to n
		}
		var x, one Float
		x.SetPrec(prec)
		one.SetInt64(1)
		for i := 0; i < n; i++ {
			x.Add(&x, &one)
		}
		if x.Cmp(new(Float).SetInt64(n)) != 0 {
			t.Errorf("prec = %d: got %s; want %d", prec, &x, n)
		}
	}
}

// Selected precisions with which to run various tests.
var precList = [...]uint{1, 2, 5, 8, 10, 16, 23, 24, 32, 50, 53, 64, 100, 128, 500, 511, 512, 513, 1000, 10000}

// Selected bits with which to run various tests.
// Each entry is a list of bits representing a floating-point number (see fromBits).
var bitsList = [...]Bits{
	{},           // = 0
	{0},          // = 1
	{1},          // = 2
	{-1},         // = 1/2
	{10},         // = 2**10 == 1024
	{-10},        // = 2**-10 == 1/1024
	{100, 10, 1}, // = 2**100 + 2**10 + 2**1
	{0, -1, -2, -10},
	// TODO(gri) add more test cases
}

// TestFloatAdd tests Float.Add/Sub by comparing the result of a "manual"
// addition/subtraction of arguments represented by Bits values with the
// respective Float addition/subtraction for a variety of precisions
// and rounding modes.
func TestFloatAdd(t *testing.T) {
	for _, xbits := range bitsList {
		for _, ybits := range bitsList {
			// exact values
			x := xbits.Float()
			y := ybits.Float()
			zbits := xbits.add(ybits)
			z := zbits.Float()

			for i, mode := range [...]RoundingMode{ToZero, ToNearestEven, AwayFromZero} {
				for _, prec := range precList {
					got := new(Float).SetPrec(prec).SetMode(mode)
					got.Add(x, y)
					want := zbits.round(prec, mode)
					if got.Cmp(want) != 0 {
						t.Errorf("i = %d, prec = %d, %s:\n\t     %s %v\n\t+    %s %v\n\t=    %s\n\twant %s",
							i, prec, mode, x, xbits, y, ybits, got, want)
					}

					got.Sub(z, x)
					want = ybits.round(prec, mode)
					if got.Cmp(want) != 0 {
						t.Errorf("i = %d, prec = %d, %s:\n\t     %s %v\n\t-    %s %v\n\t=    %s\n\twant %s",
							i, prec, mode, z, zbits, x, xbits, got, want)
					}
				}
			}
		}
	}
}

// TestFloatAdd32 tests that Float.Add/Sub of numbers with
// 24bit mantissa behaves like float32 addition/subtraction
// (excluding denormal numbers).
func TestFloatAdd32(t *testing.T) {
	// chose base such that we cross the mantissa precision limit
	const base = 1<<26 - 0x10 // 11...110000 (26 bits)
	for d := 0; d <= 0x10; d++ {
		for i := range [2]int{} {
			x0, y0 := float64(base), float64(d)
			if i&1 != 0 {
				x0, y0 = y0, x0
			}

			x := NewFloat(x0)
			y := NewFloat(y0)
			z := new(Float).SetPrec(24)

			z.Add(x, y)
			got, acc := z.Float32()
			want := float32(y0) + float32(x0)
			if got != want || acc != Exact {
				t.Errorf("d = %d: %g + %g = %g (%s); want %g (Exact)", d, x0, y0, got, acc, want)
			}

			z.Sub(z, y)
			got, acc = z.Float32()
			want = float32(want) - float32(y0)
			if got != want || acc != Exact {
				t.Errorf("d = %d: %g - %g = %g (%s); want %g (Exact)", d, x0+y0, y0, got, acc, want)
			}
		}
	}
}

// TestFloatAdd64 tests that Float.Add/Sub of numbers with
// 53bit mantissa behaves like float64 addition/subtraction.
func TestFloatAdd64(t *testing.T) {
	// chose base such that we cross the mantissa precision limit
	const base = 1<<55 - 0x10 // 11...110000 (55 bits)
	for d := 0; d <= 0x10; d++ {
		for i := range [2]int{} {
			x0, y0 := float64(base), float64(d)
			if i&1 != 0 {
				x0, y0 = y0, x0
			}

			x := NewFloat(x0)
			y := NewFloat(y0)
			z := new(Float).SetPrec(53)

			z.Add(x, y)
			got, acc := z.Float64()
			want := x0 + y0
			if got != want || acc != Exact {
				t.Errorf("d = %d: %g + %g = %g (%s); want %g (Exact)", d, x0, y0, got, acc, want)
			}

			z.Sub(z, y)
			got, acc = z.Float64()
			want -= y0
			if got != want || acc != Exact {
				t.Errorf("d = %d: %g - %g = %g (%s); want %g (Exact)", d, x0+y0, y0, got, acc, want)
			}
		}
	}
}

func TestIssue20490(t *testing.T) {
	var tests = []struct {
		a, b float64
	}{
		{4, 1},
		{-4, 1},
		{4, -1},
		{-4, -1},
	}

	for _, test := range tests {
		a, b := NewFloat(test.a), NewFloat(test.b)
		diff := new(Float).Sub(a, b)
		b.Sub(a, b)
		if b.Cmp(diff) != 0 {
			t.Errorf("got %g - %g = %g; want %g\n", a, NewFloat(test.b), b, diff)
		}

		b = NewFloat(test.b)
		sum := new(Float).Add(a, b)
		b.Add(a, b)
		if b.Cmp(sum) != 0 {
			t.Errorf("got %g + %g = %g; want %g\n", a, NewFloat(test.b), b, sum)
		}

	}
}

// TestFloatMul tests Float.Mul/Quo by comparing the result of a "manual"
// multiplication/division of arguments represented by Bits values with the
// respective Float multiplication/division for a variety of precisions
// and rounding modes.
func TestFloatMul(t *testing.T) {
	for _, xbits := range bitsList {
		for _, ybits := range bitsList {
			// exact values
			x := xbits.Float()
			y := ybits.Float()
			zbits := xbits.mul(ybits)
			z := zbits.Float()

			for i, mode := range [...]RoundingMode{ToZero, ToNearestEven, AwayFromZero} {
				for _, prec := range precList {
					got := new(Float).SetPrec(prec).SetMode(mode)
					got.Mul(x, y)
					want := zbits.round(prec, mode)
					if got.Cmp(want) != 0 {
						t.Errorf("i = %d, prec = %d, %s:\n\t     %v %v\n\t*    %v %v\n\t=    %v\n\twant %v",
							i, prec, mode, x, xbits, y, ybits, got, want)
					}

					if x.Sign() == 0 {
						continue // ignore div-0 case (not invertable)
					}
					got.Quo(z, x)
					want = ybits.round(prec, mode)
					if got.Cmp(want) != 0 {
						t.Errorf("i = %d, prec = %d, %s:\n\t     %v %v\n\t/    %v %v\n\t=    %v\n\twant %v",
							i, prec, mode, z, zbits, x, xbits, got, want)
					}
				}
			}
		}
	}
}

// TestFloatMul64 tests that Float.Mul/Quo of numbers with
// 53bit mantissa behaves like float64 multiplication/division.
func TestFloatMul64(t *testing.T) {
	for _, test := range []struct {
		x, y float64
	}{
		{0, 0},
		{0, 1},
		{1, 1},
		{1, 1.5},
		{1.234, 0.5678},
		{2.718281828, 3.14159265358979},
		{2.718281828e10, 3.14159265358979e-32},
		{1.0 / 3, 1e200},
	} {
		for i := range [8]int{} {
			x0, y0 := test.x, test.y
			if i&1 != 0 {
				x0 = -x0
			}
			if i&2 != 0 {
				y0 = -y0
			}
			if i&4 != 0 {
				x0, y0 = y0, x0
			}

			x := NewFloat(x0)
			y := NewFloat(y0)
			z := new(Float).SetPrec(53)

			z.Mul(x, y)
			got, _ := z.Float64()
			want := x0 * y0
			if got != want {
				t.Errorf("%g * %g = %g; want %g", x0, y0, got, want)
			}

			if y0 == 0 {
				continue // avoid division-by-zero
			}
			z.Quo(z, y)
			got, _ = z.Float64()
			want /= y0
			if got != want {
				t.Errorf("%g / %g = %g; want %g", x0*y0, y0, got, want)
			}
		}
	}
}

func TestIssue6866(t *testing.T) {
	for _, prec := range precList {
		two := new(Float).SetPrec(prec).SetInt64(2)
		one := new(Float).SetPrec(prec).SetInt64(1)
		three := new(Float).SetPrec(prec).SetInt64(3)
		msix := new(Float).SetPrec(prec).SetInt64(-6)
		psix := new(Float).SetPrec(prec).SetInt64(+6)

		p := new(Float).SetPrec(prec)
		z1 := new(Float).SetPrec(prec)
		z2 := new(Float).SetPrec(prec)

		// z1 = 2 + 1.0/3*-6
		p.Quo(one, three)
		p.Mul(p, msix)
		z1.Add(two, p)

		// z2 = 2 - 1.0/3*+6
		p.Quo(one, three)
		p.Mul(p, psix)
		z2.Sub(two, p)

		if z1.Cmp(z2) != 0 {
			t.Fatalf("prec %d: got z1 = %v != z2 = %v; want z1 == z2\n", prec, z1, z2)
		}
		if z1.Sign() != 0 {
			t.Errorf("prec %d: got z1 = %v; want 0", prec, z1)
		}
		if z2.Sign() != 0 {
			t.Errorf("prec %d: got z2 = %v; want 0", prec, z2)
		}
	}
}

func TestFloatQuo(t *testing.T) {
	// TODO(gri) make the test vary these precisions
	preci := 200 // precision of integer part
	precf := 20  // precision of fractional part

	for i := 0; i < 8; i++ {
		// compute accurate (not rounded) result z
		bits := Bits{preci - 1}
		if i&3 != 0 {
			bits = append(bits, 0)
		}
		if i&2 != 0 {
			bits = append(bits, -1)
		}
		if i&1 != 0 {
			bits = append(bits, -precf)
		}
		z := bits.Float()

		// compute accurate x as z*y
		y := NewFloat(3.14159265358979323e123)

		x := new(Float).SetPrec(z.Prec() + y.Prec()).SetMode(ToZero)
		x.Mul(z, y)

		// leave for debugging
		// fmt.Printf("x = %s\ny = %s\nz = %s\n", x, y, z)

		if got := x.Acc(); got != Exact {
			t.Errorf("got acc = %s; want exact", got)
		}

		// round accurate z for a variety of precisions and
		// modes and compare against result of x / y.
		for _, mode := range [...]RoundingMode{ToZero, ToNearestEven, AwayFromZero} {
			for d := -5; d < 5; d++ {
				prec := uint(preci + d)
				got := new(Float).SetPrec(prec).SetMode(mode).Quo(x, y)
				want := bits.round(prec, mode)
				if got.Cmp(want) != 0 {
					t.Errorf("i = %d, prec = %d, %s:\n\t     %s\n\t/    %s\n\t=    %s\n\twant %s",
						i, prec, mode, x, y, got, want)
				}
			}
		}
	}
}

var long = flag.Bool("long", false, "run very long tests")

// TestFloatQuoSmoke tests all divisions x/y for values x, y in the range [-n, +n];
// it serves as a smoke test for basic correctness of division.
func TestFloatQuoSmoke(t *testing.T) {
	n := 10
	if *long {
		n = 1000
	}

	const dprec = 3         // max. precision variation
	const prec = 10 + dprec // enough bits to hold n precisely
	for x := -n; x <= n; x++ {
		for y := -n; y < n; y++ {
			if y == 0 {
				continue
			}

			a := float64(x)
			b := float64(y)
			c := a / b

			// vary operand precision (only ok as long as a, b can be represented correctly)
			for ad := -dprec; ad <= dprec; ad++ {
				for bd := -dprec; bd <= dprec; bd++ {
					A := new(Float).SetPrec(uint(prec + ad)).SetFloat64(a)
					B := new(Float).SetPrec(uint(prec + bd)).SetFloat64(b)
					C := new(Float).SetPrec(53).Quo(A, B) // C has float64 mantissa width

					cc, acc := C.Float64()
					if cc != c {
						t.Errorf("%g/%g = %s; want %.5g\n", a, b, C.Text('g', 5), c)
						continue
					}
					if acc != Exact {
						t.Errorf("%g/%g got %s result; want exact result", a, b, acc)
					}
				}
			}
		}
	}
}

// TestFloatArithmeticSpecialValues tests that Float operations produce the
// correct results for combinations of zero (±0), finite (±1 and ±2.71828),
// and infinite (±Inf) operands.
func TestFloatArithmeticSpecialValues(t *testing.T) {
	zero := 0.0
	args := []float64{math.Inf(-1), -2.71828, -1, -zero, zero, 1, 2.71828, math.Inf(1)}
	xx := new(Float)
	yy := new(Float)
	got := new(Float)
	want := new(Float)
	for i := 0; i < 4; i++ {
		for _, x := range args {
			xx.SetFloat64(x)
			// check conversion is correct
			// (no need to do this for y, since we see exactly the
			// same values there)
			if got, acc := xx.Float64(); got != x || acc != Exact {
				t.Errorf("Float(%g) == %g (%s)", x, got, acc)
			}
			for _, y := range args {
				yy.SetFloat64(y)
				var (
					op string
					z  float64
					f  func(z, x, y *Float) *Float
				)
				switch i {
				case 0:
					op = "+"
					z = x + y
					f = (*Float).Add
				case 1:
					op = "-"
					z = x - y
					f = (*Float).Sub
				case 2:
					op = "*"
					z = x * y
					f = (*Float).Mul
				case 3:
					op = "/"
					z = x / y
					f = (*Float).Quo
				default:
					panic("unreachable")
				}
				var errnan bool // set if execution of f panicked with ErrNaN
				// protect execution of f
				func() {
					defer func() {
						if p := recover(); p != nil {
							_ = p.(ErrNaN) // re-panic if not ErrNaN
							errnan = true
						}
					}()
					f(got, xx, yy)
				}()
				if math.IsNaN(z) {
					if !errnan {
						t.Errorf("%5g %s %5g = %5s; want ErrNaN panic", x, op, y, got)
					}
					continue
				}
				if errnan {
					t.Errorf("%5g %s %5g panicked with ErrNan; want %5s", x, op, y, want)
					continue
				}
				want.SetFloat64(z)
				if !alike(got, want) {
					t.Errorf("%5g %s %5g = %5s; want %5s", x, op, y, got, want)
				}
			}
		}
	}
}

func TestFloatArithmeticOverflow(t *testing.T) {
	for _, test := range []struct {
		prec       uint
		mode       RoundingMode
		op         byte
		x, y, want string
		acc        Accuracy
	}{
		{4, ToNearestEven, '+', "0", "0", "0", Exact},                   // smoke test
		{4, ToNearestEven, '+', "0x.8p+0", "0x.8p+0", "0x.8p+1", Exact}, // smoke test

		{4, ToNearestEven, '+', "0", "0x.8p2147483647", "0x.8p+2147483647", Exact},
		{4, ToNearestEven, '+', "0x.8p2147483500", "0x.8p2147483647", "0x.8p+2147483647", Below}, // rounded to zero
		{4, ToNearestEven, '+', "0x.8p2147483647", "0x.8p2147483647", "+Inf", Above},             // exponent overflow in +
		{4, ToNearestEven, '+', "-0x.8p2147483647", "-0x.8p2147483647", "-Inf", Below},           // exponent overflow in +
		{4, ToNearestEven, '-', "-0x.8p2147483647", "0x.8p2147483647", "-Inf", Below},            // exponent overflow in -

		{4, ToZero, '+', "0x.fp2147483647", "0x.8p2147483643", "0x.fp+2147483647", Below}, // rounded to zero
		{4, ToNearestEven, '+', "0x.fp2147483647", "0x.8p2147483643", "+Inf", Above},      // exponent overflow in rounding
		{4, AwayFromZero, '+', "0x.fp2147483647", "0x.8p2147483643", "+Inf", Above},       // exponent overflow in rounding

		{4, AwayFromZero, '-', "-0x.fp2147483647", "0x.8p2147483644", "-Inf", Below},        // exponent overflow in rounding
		{4, ToNearestEven, '-', "-0x.fp2147483647", "0x.8p2147483643", "-Inf", Below},       // exponent overflow in rounding
		{4, ToZero, '-', "-0x.fp2147483647", "0x.8p2147483643", "-0x.fp+2147483647", Above}, // rounded to zero

		{4, ToNearestEven, '+', "0", "0x.8p-2147483648", "0x.8p-2147483648", Exact},
		{4, ToNearestEven, '+', "0x.8p-2147483648", "0x.8p-2147483648", "0x.8p-2147483647", Exact},

		{4, ToNearestEven, '*', "1", "0x.8p2147483647", "0x.8p+2147483647", Exact},
		{4, ToNearestEven, '*', "2", "0x.8p2147483647", "+Inf", Above},  // exponent overflow in *
		{4, ToNearestEven, '*', "-2", "0x.8p2147483647", "-Inf", Below}, // exponent overflow in *

		{4, ToNearestEven, '/', "0.5", "0x.8p2147483647", "0x.8p-2147483646", Exact},
		{4, ToNearestEven, '/', "0x.8p+0", "0x.8p2147483647", "0x.8p-2147483646", Exact},
		{4, ToNearestEven, '/', "0x.8p-1", "0x.8p2147483647", "0x.8p-2147483647", Exact},
		{4, ToNearestEven, '/', "0x.8p-2", "0x.8p2147483647", "0x.8p-2147483648", Exact},
		{4, ToNearestEven, '/', "0x.8p-3", "0x.8p2147483647", "0", Below}, // exponent underflow in /
	} {
		x := makeFloat(test.x)
		y := makeFloat(test.y)
		z := new(Float).SetPrec(test.prec).SetMode(test.mode)
		switch test.op {
		case '+':
			z.Add(x, y)
		case '-':
			z.Sub(x, y)
		case '*':
			z.Mul(x, y)
		case '/':
			z.Quo(x, y)
		default:
			panic("unreachable")
		}
		if got := z.Text('p', 0); got != test.want || z.Acc() != test.acc {
			t.Errorf(
				"prec = %d (%s): %s %c %s = %s (%s); want %s (%s)",
				test.prec, test.mode, x.Text('p', 0), test.op, y.Text('p', 0), got, z.Acc(), test.want, test.acc,
			)
		}
	}
}

// TODO(gri) Add tests that check correctness in the presence of aliasing.

// For rounding modes ToNegativeInf and ToPositiveInf, rounding is affected
// by the sign of the value to be rounded. Test that rounding happens after
// the sign of a result has been set.
// This test uses specific values that are known to fail if rounding is
// "factored" out before setting the result sign.
func TestFloatArithmeticRounding(t *testing.T) {
	for _, test := range []struct {
		mode       RoundingMode
		prec       uint
		x, y, want int64
		op         byte
	}{
		{ToZero, 3, -0x8, -0x1, -0x8, '+'},
		{AwayFromZero, 3, -0x8, -0x1, -0xa, '+'},
		{ToNegativeInf, 3, -0x8, -0x1, -0xa, '+'},

		{ToZero, 3, -0x8, 0x1, -0x8, '-'},
		{AwayFromZero, 3, -0x8, 0x1, -0xa, '-'},
		{ToNegativeInf, 3, -0x8, 0x1, -0xa, '-'},

		{ToZero, 3, -0x9, 0x1, -0x8, '*'},
		{AwayFromZero, 3, -0x9, 0x1, -0xa, '*'},
		{ToNegativeInf, 3, -0x9, 0x1, -0xa, '*'},

		{ToZero, 3, -0x9, 0x1, -0x8, '/'},
		{AwayFromZero, 3, -0x9, 0x1, -0xa, '/'},
		{ToNegativeInf, 3, -0x9, 0x1, -0xa, '/'},
	} {
		var x, y, z Float
		x.SetInt64(test.x)
		y.SetInt64(test.y)
		z.SetPrec(test.prec).SetMode(test.mode)
		switch test.op {
		case '+':
			z.Add(&x, &y)
		case '-':
			z.Sub(&x, &y)
		case '*':
			z.Mul(&x, &y)
		case '/':
			z.Quo(&x, &y)
		default:
			panic("unreachable")
		}
		if got, acc := z.Int64(); got != test.want || acc != Exact {
			t.Errorf("%s, %d bits: %d %c %d = %d (%s); want %d (Exact)",
				test.mode, test.prec, test.x, test.op, test.y, got, acc, test.want,
			)
		}
	}
}

// TestFloatCmpSpecialValues tests that Cmp produces the correct results for
// combinations of zero (±0), finite (±1 and ±2.71828), and infinite (±Inf)
// operands.
func TestFloatCmpSpecialValues(t *testing.T) {
	zero := 0.0
	args := []float64{math.Inf(-1), -2.71828, -1, -zero, zero, 1, 2.71828, math.Inf(1)}
	xx := new(Float)
	yy := new(Float)
	for i := 0; i < 4; i++ {
		for _, x := range args {
			xx.SetFloat64(x)
			// check conversion is correct
			// (no need to do this for y, since we see exactly the
			// same values there)
			if got, acc := xx.Float64(); got != x || acc != Exact {
				t.Errorf("Float(%g) == %g (%s)", x, got, acc)
			}
			for _, y := range args {
				yy.SetFloat64(y)
				got := xx.Cmp(yy)
				want := 0
				switch {
				case x < y:
					want = -1
				case x > y:
					want = +1
				}
				if got != want {
					t.Errorf("(%g).Cmp(%g) = %v; want %v", x, y, got, want)
				}
			}
		}
	}
}

func BenchmarkFloatAdd(b *testing.B) {
	x := new(Float)
	y := new(Float)
	z := new(Float)

	for _, prec := range []uint{10, 1e2, 1e3, 1e4, 1e5} {
		x.SetPrec(prec).SetRat(NewRat(1, 3))
		y.SetPrec(prec).SetRat(NewRat(1, 6))
		z.SetPrec(prec)

		b.Run(fmt.Sprintf("%v", prec), func(b *testing.B) {
			b.ReportAllocs()
			for i := 0; i < b.N; i++ {
				z.Add(x, y)
			}
		})
	}
}

func BenchmarkFloatSub(b *testing.B) {
	x := new(Float)
	y := new(Float)
	z := new(Float)

	for _, prec := range []uint{10, 1e2, 1e3, 1e4, 1e5} {
		x.SetPrec(prec).SetRat(NewRat(1, 3))
		y.SetPrec(prec).SetRat(NewRat(1, 6))
		z.SetPrec(prec)

		b.Run(fmt.Sprintf("%v", prec), func(b *testing.B) {
			b.ReportAllocs()
			for i := 0; i < b.N; i++ {
				z.Sub(x, y)
			}
		})
	}
}