Adding upstream version 1.16.10.upstream/1.16.10upstream

Signed-off-by: Daniel Baumann <daniel.baumann@progress-linux.org>

1 files changed, 1858 insertions, 0 deletions

diff --git a/src/math/big/float_test.go b/src/math/big/float_test.go
new file mode 100644
index 0000000..7d6bf03
--- /dev/null
+++ b/src/math/big/float_test.go
@@ -0,0 +1,1858 @@
+// 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},
+
+		// https://www.exploringbinary.com/php-hangs-on-numeric-value-2-2250738585072011e-308/
+		{"2.2250738585072011e-308", 2.225073858507201e-308, Below},
+		// https://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)
+					}
+				}
+			}
+		}
+	}
+}
+
+// TestFloatAddRoundZero tests Float.Add/Sub rounding when the result is exactly zero.
+// x + (-x) or x - x for non-zero x should be +0 in all cases except when
+// the rounding mode is ToNegativeInf in which case it should be -0.
+func TestFloatAddRoundZero(t *testing.T) {
+	for _, mode := range [...]RoundingMode{ToNearestEven, ToNearestAway, ToZero, AwayFromZero, ToPositiveInf, ToNegativeInf} {
+		x := NewFloat(5.0)
+		y := new(Float).Neg(x)
+		want := NewFloat(0.0)
+		if mode == ToNegativeInf {
+			want.Neg(want)
+		}
+		got := new(Float).SetMode(mode)
+		got.Add(x, y)
+		if got.Cmp(want) != 0 || got.neg != (mode == ToNegativeInf) {
+			t.Errorf("%s:\n\t     %v\n\t+    %v\n\t=    %v\n\twant %v",
+				mode, x, y, got, want)
+		}
+		got.Sub(x, x)
+		if got.Cmp(want) != 0 || got.neg != (mode == ToNegativeInf) {
+			t.Errorf("%v:\n\t     %v\n\t-    %v\n\t=    %v\n\twant %v",
+				mode, x, x, 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)
+			}
+		})
+	}
+}