import (
	"fmt"
	"math"
)

// A Rat represents a quotient a/b of arbitrary precision.
// The zero value for a Rat represents the value 0.
//
// Operations always take pointer arguments (*Rat) rather
// than Rat values, and each unique Rat value requires
// its own unique *Rat pointer. To "copy" a Rat value,
// an existing (or newly allocated) Rat must be set to
// a new value using the [Rat.Set] method; shallow copies

func i2tor(u, v int64) *rat {
	g := gcd(u, v)
	r := new(rat)
	if v > 0 {
		r.num = u / g
		r.den = v / g
	} else {
		r.num = -u / g
		r.den = -v / g
	}
	return r
}

func itor(u int64) *rat {
	return i2tor(u, 1)
}

var zero *rat
var one *rat

// End mark and end test

var finis *rat

func end(u *rat) int64 {
	if u.den == 0 {
		return 1

// when simply accessing the Rat value.
func TestIssue34919(t *testing.T) {
	for _, acc := range []struct {
		name string
		f    func(*Rat)
	}{
		{"Float32", func(x *Rat) { x.Float32() }},
		{"Float64", func(x *Rat) { x.Float64() }},
		{"Inv", func(x *Rat) { new(Rat).Inv(x) }},
		{"Sign", func(x *Rat) { x.Sign() }},
		{"IsInt", func(x *Rat) { x.IsInt() }},
		{"Num", func(x *Rat) { x.Num() }},

The following numeric types are supported:

	Int    signed integers
	Rat    rational numbers
	Float  floating-point numbers

The zero value for an [Int], [Rat], or [Float] correspond to 0. Thus, new
values can be declared in the usual ways and denote 0 without further
initialization:

	var x Int        // &x is an *Int of value 0
	var r = &Rat{}   // r is a *Rat of value 0
	y := new(Float)  // y is a *Float of value 0

func newInverseHilbert(n int) *matrix {
	a := newMatrix(n, n)
	for i := 0; i < n; i++ {
		for j := 0; j < n; j++ {
			x1 := new(Rat).SetInt64(int64(i + j + 1))
			x2 := new(Rat).SetInt(new(Int).Binomial(int64(n+i), int64(n-j-1)))
			x3 := new(Rat).SetInt(new(Int).Binomial(int64(n+j), int64(n-i-1)))
			x4 := new(Rat).SetInt(new(Int).Binomial(int64(i+j), int64(i)))

			x1.Mul(x1, x2)
			x1.Mul(x1, x3)
			x1.Mul(x1, x4)
			x1.Mul(x1, x4)

func i2tor(u, v int64) rat {
	g := gcd(u, v)
	var r rat
	if v > 0 {
		r.num = u / g
		r.den = v / g
	} else {
		r.num = -u / g
		r.den = -v / g
	}
	return r
}

func itor(u int64) rat {
	return i2tor(u, 1)
}

var zero rat
var one rat

// End mark and end test

var finis rat

func end(u rat) int64 {
	if u.den == 0 {
		return 1
	}

}

// GobDecode implements the [encoding/gob.GobDecoder] interface.
func (z *Rat) GobDecode(buf []byte) error {
	if len(buf) == 0 {
		// Other side sent a nil or default value.
		*z = Rat{}
		return nil
	}
	if len(buf) < 5 {
		return errors.New("Rat.GobDecode: buffer too small")
	}
	b := buf[0]
	if b>>1 != ratGobVersion {
		return fmt.Errorf("Rat.GobDecode: encoding version %d not supported", b>>1)
	}
	const j = 1 + 4

	for _, test := range encodingTests {
		medium.Reset() // empty buffer for each test case (in case of failures)
		var tx Rat
		tx.SetString(test + ".14159265")
		if err := enc.Encode(&tx); err != nil {
			t.Errorf("encoding of %s failed: %s", &tx, err)
			continue
		}
		var rx Rat
		if err := dec.Decode(&rx); err != nil {
			t.Errorf("decoding of %s failed: %s", &tx, err)
			continue
		}
		if rx.Cmp(&tx) != 0 {

//	(n-1)/3 * 2   if   n mod 3 == 1
func recur(n, lim int64) *big.Rat {
	term := new(big.Rat)
	if n%3 != 1 {
		term.SetInt64(1)
	} else {
		term.SetInt64((n - 1) / 3 * 2)
	}

	if n > lim {
		return term
	}

	// Directly initialize frac as the fractional
	// inverse of the result of recur.
	frac := new(big.Rat).Inv(recur(n+1, lim))

	return term.Add(term, frac)
}

			}
		}
	}
}

func TestRatSetStringZero(t *testing.T) {
	got, _ := new(Rat).SetString("0")
	want := new(Rat).SetInt64(0)
	if !reflect.DeepEqual(got, want) {
		t.Errorf("got %#+v, want %#+v", got, want)
	}
}

func TestRatScan(t *testing.T) {
	var buf bytes.Buffer
	for i, test := range setStringTests {
		x := new(Rat)
		buf.Reset()
		buf.WriteString(test.in)

		_, err := fmt.Fscanf(&buf, "%v", x)