float.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.

// This file implements multi-precision floating-point numbers.
// Like in the GNU MPFR library (http://www.mpfr.org/), operands
// can be of mixed precision. Unlike MPFR, the rounding mode is
// not specified with each operation, but with each operand. The
// rounding mode of the result operand determines the rounding
// mode of an operation. This is a from-scratch implementation.

package big

import (
	"fmt"
	"math"
)

const debugFloat = false // enable for debugging

// A nonzero finite Float represents a multi-precision floating point number
//
//   sign × mantissa × 2**exponent
//
// with 0.5 <= mantissa < 1.0, and MinExp <= exponent <= MaxExp.
// A Float may also be zero (+0, -0) or infinite (+Inf, -Inf).
// All Floats are ordered, and the ordering of two Floats x and y
// is defined by x.Cmp(y).
//
// Each Float value also has a precision, rounding mode, and accuracy.
// The precision is the maximum number of mantissa bits available to
// represent the value. The rounding mode specifies how a result should
// be rounded to fit into the mantissa bits, and accuracy describes the
// rounding error with respect to the exact result.
//
// Unless specified otherwise, all operations (including setters) that
// specify a *Float variable for the result (usually via the receiver
// with the exception of MantExp), round the numeric result according
// to the precision and rounding mode of the result variable.
//
// If the provided result precision is 0 (see below), it is set to the
// precision of the argument with the largest precision value before any
// rounding takes place, and the rounding mode remains unchanged. Thus,
// uninitialized Floats provided as result arguments will have their
// precision set to a reasonable value determined by the operands and
// their mode is the zero value for RoundingMode (ToNearestEven).
//
// By setting the desired precision to 24 or 53 and using matching rounding
// mode (typically ToNearestEven), Float operations produce the same results
// as the corresponding float32 or float64 IEEE-754 arithmetic for operands
// that correspond to normal (i.e., not denormal) float32 or float64 numbers.
// Exponent underflow and overflow lead to a 0 or an Infinity for different
// values than IEEE-754 because Float exponents have a much larger range.
//
// The zero (uninitialized) value for a Float is ready to use and represents
// the number +0.0 exactly, with precision 0 and rounding mode ToNearestEven.
//
type Float struct {
	prec uint32
	mode RoundingMode
	acc  Accuracy
	form form
	neg  bool
	mant nat
	exp  int32
}

// An ErrNaN panic is raised by a Float operation that would lead to
// a NaN under IEEE-754 rules. An ErrNaN implements the error interface.
type ErrNaN struct {
	msg string
}

func (err ErrNaN) Error() string {
	return err.msg
}

// NewFloat allocates and returns a new Float set to x,
// with precision 53 and rounding mode ToNearestEven.
// NewFloat panics with ErrNaN if x is a NaN.
func NewFloat(x float64) *Float {
	if math.IsNaN(x) {
		panic(ErrNaN{"NewFloat(NaN)"})
	}
	return new(Float).SetFloat64(x)
}

// Exponent and precision limits.
const (
	MaxExp  = math.MaxInt32  // largest supported exponent
	MinExp  = math.MinInt32  // smallest supported exponent
	MaxPrec = math.MaxUint32 // largest (theoretically) supported precision; likely memory-limited
)

// Internal representation: The mantissa bits x.mant of a nonzero finite
// Float x are stored in a nat slice long enough to hold up to x.prec bits;
// the slice may (but doesn't have to) be shorter if the mantissa contains
// trailing 0 bits. x.mant is normalized if the msb of x.mant == 1 (i.e.,
// the msb is shifted all the way "to the left"). Thus, if the mantissa has
// trailing 0 bits or x.prec is not a multiple of the the Word size _W,
// x.mant[0] has trailing zero bits. The msb of the mantissa corresponds
// to the value 0.5; the exponent x.exp shifts the binary point as needed.
//
// A zero or non-finite Float x ignores x.mant and x.exp.
//
// x                 form      neg      mant         exp
// ----------------------------------------------------------
// ±0                zero      sign     -            -
// 0 < |x| < +Inf    finite    sign     mantissa     exponent
// ±Inf              inf       sign     -            -

// A form value describes the internal representation.
type form byte

// The form value order is relevant - do not change!
const (
	zero form = iota
	finite
	inf
)

// RoundingMode determines how a Float value is rounded to the
// desired precision. Rounding may change the Float value; the
// rounding error is described by the Float's Accuracy.
type RoundingMode byte

// These constants define supported rounding modes.
const (
	ToNearestEven RoundingMode = iota // == IEEE 754-2008 roundTiesToEven
	ToNearestAway                     // == IEEE 754-2008 roundTiesToAway
	ToZero                            // == IEEE 754-2008 roundTowardZero
	AwayFromZero                      // no IEEE 754-2008 equivalent
	ToNegativeInf                     // == IEEE 754-2008 roundTowardNegative
	ToPositiveInf                     // == IEEE 754-2008 roundTowardPositive
)

//go:generate stringer -type=RoundingMode

// Accuracy describes the rounding error produced by the most recent
// operation that generated a Float value, relative to the exact value.
type Accuracy int8

// Constants describing the Accuracy of a Float.
const (
	Below Accuracy = -1
	Exact Accuracy = 0
	Above Accuracy = +1
)

//go:generate stringer -type=Accuracy

// SetPrec sets z's precision to prec and returns the (possibly) rounded
// value of z. Rounding occurs according to z's rounding mode if the mantissa
// cannot be represented in prec bits without loss of precision.
// SetPrec(0) maps all finite values to ±0; infinite values remain unchanged.
// If prec > MaxPrec, it is set to MaxPrec.
func (z *Float) SetPrec(prec uint) *Float {
	z.acc = Exact // optimistically assume no rounding is needed

	// special case
	if prec == 0 {
		z.prec = 0
		if z.form == finite {
			// truncate z to 0
			z.acc = makeAcc(z.neg)
			z.form = zero
		}
		return z
	}

	// general case
	if prec > MaxPrec {
		prec = MaxPrec
	}
	old := z.prec
	z.prec = uint32(prec)
	if z.prec < old {
		z.round(0)
	}
	return z
}

func makeAcc(above bool) Accuracy {
	if above {
		return Above
	}
	return Below
}

// SetMode sets z's rounding mode to mode and returns an exact z.
// z remains unchanged otherwise.
// z.SetMode(z.Mode()) is a cheap way to set z's accuracy to Exact.
func (z *Float) SetMode(mode RoundingMode) *Float {
	z.mode = mode
	z.acc = Exact
	return z
}

// Prec returns the mantissa precision of x in bits.
// The result may be 0 for |x| == 0 and |x| == Inf.
func (x *Float) Prec() uint {
	return uint(x.prec)
}

// MinPrec returns the minimum precision required to represent x exactly
// (i.e., the smallest prec before x.SetPrec(prec) would start rounding x).
// The result is 0 for |x| == 0 and |x| == Inf.
func (x *Float) MinPrec() uint {
	if x.form != finite {
		return 0
	}
	return uint(len(x.mant))*_W - x.mant.trailingZeroBits()
}

// Mode returns the rounding mode of x.
func (x *Float) Mode() RoundingMode {
	return x.mode
}

// Acc returns the accuracy of x produced by the most recent operation.
func (x *Float) Acc() Accuracy {
	return x.acc
}

// Sign returns:
//
//	-1 if x <   0
//	 0 if x is ±0
//	+1 if x >   0
//
func (x *Float) Sign() int {
	if debugFloat {
		x.validate()
	}
	if x.form == zero {
		return 0
	}
	if x.neg {
		return -1
	}
	return 1
}

// MantExp breaks x into its mantissa and exponent components
// and returns the exponent. If a non-nil mant argument is
// provided its value is set to the mantissa of x, with the
// same precision and rounding mode as x. The components
// satisfy x == mant × 2**exp, with 0.5 <= |mant| < 1.0.
// Calling MantExp with a nil argument is an efficient way to
// get the exponent of the receiver.
//
// Special cases are:
//
//	(  ±0).MantExp(mant) = 0, with mant set to   ±0
//	(±Inf).MantExp(mant) = 0, with mant set to ±Inf
//
// x and mant may be the same in which case x is set to its
// mantissa value.
func (x *Float) MantExp(mant *Float) (exp int) {
	if debugFloat {
		x.validate()
	}
	if x.form == finite {
		exp = int(x.exp)
	}
	if mant != nil {
		mant.Copy(x)
		if mant.form == finite {
			mant.exp = 0
		}
	}
	return
}

func (z *Float) setExpAndRound(exp int64, sbit uint) {
	if exp < MinExp {
		// underflow
		z.acc = makeAcc(z.neg)
		z.form = zero
		return
	}

	if exp > MaxExp {
		// overflow
		z.acc = makeAcc(!z.neg)
		z.form = inf
		return
	}

	z.form = finite
	z.exp = int32(exp)
	z.round(sbit)
}

// SetMantExp sets z to mant × 2**exp and and returns z.
// The result z has the same precision and rounding mode
// as mant. SetMantExp is an inverse of MantExp but does
// not require 0.5 <= |mant| < 1.0. Specifically:
//
//	mant := new(Float)
//	new(Float).SetMantExp(mant, x.MantExp(mant)).Cmp(x) == 0
//
// Special cases are:
//
//	z.SetMantExp(  ±0, exp) =   ±0
//	z.SetMantExp(±Inf, exp) = ±Inf
//
// z and mant may be the same in which case z's exponent
// is set to exp.
func (z *Float) SetMantExp(mant *Float, exp int) *Float {
	if debugFloat {
		z.validate()
		mant.validate()
	}
	z.Copy(mant)
	if z.form != finite {
		return z
	}
	z.setExpAndRound(int64(z.exp)+int64(exp), 0)
	return z
}

// Signbit returns true if x is negative or negative zero.
func (x *Float) Signbit() bool {
	return x.neg
}

// IsInf reports whether x is +Inf or -Inf.
func (x *Float) IsInf() bool {
	return x.form == inf
}

// IsInt reports whether x is an integer.
// ±Inf values are not integers.
func (x *Float) IsInt() bool {
	if debugFloat {
		x.validate()
	}
	// special cases
	if x.form != finite {
		return x.form == zero
	}
	// x.form == finite
	if x.exp <= 0 {
		return false
	}
	// x.exp > 0
	return x.prec <= uint32(x.exp) || x.MinPrec() <= uint(x.exp) // not enough bits for fractional mantissa
}

// debugging support
func (x *Float) validate() {
	if !debugFloat {
		// avoid performance bugs
		panic("validate called but debugFloat is not set")
	}
	if x.form != finite {
		return
	}
	m := len(x.mant)
	if m == 0 {
		panic("nonzero finite number with empty mantissa")
	}
	const msb = 1 << (_W - 1)
	if x.mant[m-1]&msb == 0 {
		panic(fmt.Sprintf("msb not set in last word %#x of %s", x.mant[m-1], x.Text('p', 0)))
	}
	if x.prec == 0 {
		panic("zero precision finite number")
	}
}

// round rounds z according to z.mode to z.prec bits and sets z.acc accordingly.
// sbit must be 0 or 1 and summarizes any "sticky bit" information one might
// have before calling round. z's mantissa must be normalized (with the msb set)
// or empty.
//
// CAUTION: The rounding modes ToNegativeInf, ToPositiveInf are affected by the
// sign of z. For correct rounding, the sign of z must be set correctly before
// calling round.
func (z *Float) round(sbit uint) {
	if debugFloat {
		z.validate()
	}

	z.acc = Exact
	if z.form != finite {
		// ±0 or ±Inf => nothing left to do
		return
	}
	// z.form == finite && len(z.mant) > 0
	// m > 0 implies z.prec > 0 (checked by validate)

	m := uint32(len(z.mant)) // present mantissa length in words
	bits := m * _W           // present mantissa bits; bits > 0
	if bits <= z.prec {
		// mantissa fits => nothing to do
		return
	}
	// bits > z.prec

	// Rounding is based on two bits: the rounding bit (rbit) and the
	// sticky bit (sbit). The rbit is the bit immediately before the
	// z.prec leading mantissa bits (the "0.5"). The sbit is set if any
	// of the bits before the rbit are set (the "0.25", "0.125", etc.):
	//
	//   rbit  sbit  => "fractional part"
	//
	//   0     0        == 0
	//   0     1        >  0  , < 0.5
	//   1     0        == 0.5
	//   1     1        >  0.5, < 1.0

	// bits > z.prec: mantissa too large => round
	r := uint(bits - z.prec - 1) // rounding bit position; r >= 0
	rbit := z.mant.bit(r) & 1    // rounding bit; be safe and ensure it's a single bit
	if sbit == 0 {
		// TODO(gri) if rbit != 0 we don't need to compute sbit for some rounding modes (optimization)
		sbit = z.mant.sticky(r)
	}
	sbit &= 1 // be safe and ensure it's a single bit

	// cut off extra words
	n := (z.prec + (_W - 1)) / _W // mantissa length in words for desired precision
	if m > n {
		copy(z.mant, z.mant[m-n:]) // move n last words to front
		z.mant = z.mant[:n]
	}

	// determine number of trailing zero bits (ntz) and compute lsb mask of mantissa's least-significant word
	ntz := n*_W - z.prec // 0 <= ntz < _W
	lsb := Word(1) << ntz

	// round if result is inexact
	if rbit|sbit != 0 {
		// Make rounding decision: The result mantissa is truncated ("rounded down")
		// by default. Decide if we need to increment, or "round up", the (unsigned)
		// mantissa.
		inc := false
		switch z.mode {
		case ToNegativeInf:
			inc = z.neg
		case ToZero:
			// nothing to do
		case ToNearestEven:
			inc = rbit != 0 && (sbit != 0 || z.mant[0]&lsb != 0)
		case ToNearestAway:
			inc = rbit != 0
		case AwayFromZero:
			inc = true
		case ToPositiveInf:
			inc = !z.neg
		default:
			panic("unreachable")
		}

		// A positive result (!z.neg) is Above the exact result if we increment,
		// and it's Below if we truncate (Exact results require no rounding).
		// For a negative result (z.neg) it is exactly the opposite.
		z.acc = makeAcc(inc != z.neg)

		if inc {
			// add 1 to mantissa
			if addVW(z.mant, z.mant, lsb) != 0 {
				// mantissa overflow => adjust exponent
				if z.exp >= MaxExp {
					// exponent overflow
					z.form = inf
					return
				}
				z.exp++
				// adjust mantissa: divide by 2 to compensate for exponent adjustment
				shrVU(z.mant, z.mant, 1)
				// set msb == carry == 1 from the mantissa overflow above
				const msb = 1 << (_W - 1)
				z.mant[n-1] |= msb
			}
		}
	}

	// zero out trailing bits in least-significant word
	z.mant[0] &^= lsb - 1

	if debugFloat {
		z.validate()
	}
}

func (z *Float) setBits64(neg bool, x uint64) *Float {
	if z.prec == 0 {
		z.prec = 64
	}
	z.acc = Exact
	z.neg = neg
	if x == 0 {
		z.form = zero
		return z
	}
	// x != 0
	z.form = finite
	s := nlz64(x)
	z.mant = z.mant.setUint64(x << s)
	z.exp = int32(64 - s) // always fits
	if z.prec < 64 {
		z.round(0)
	}
	return z
}

// SetUint64 sets z to the (possibly rounded) value of x and returns z.
// If z's precision is 0, it is changed to 64 (and rounding will have
// no effect).
func (z *Float) SetUint64(x uint64) *Float {
	return z.setBits64(false, x)
}

// SetInt64 sets z to the (possibly rounded) value of x and returns z.
// If z's precision is 0, it is changed to 64 (and rounding will have
// no effect).
func (z *Float) SetInt64(x int64) *Float {
	u := x
	if u < 0 {
		u = -u
	}
	// We cannot simply call z.SetUint64(uint64(u)) and change
	// the sign afterwards because the sign affects rounding.
	return z.setBits64(x < 0, uint64(u))
}

// SetFloat64 sets z to the (possibly rounded) value of x and returns z.
// If z's precision is 0, it is changed to 53 (and rounding will have
// no effect). SetFloat64 panics with ErrNaN if x is a NaN.
func (z *Float) SetFloat64(x float64) *Float {
	if z.prec == 0 {
		z.prec = 53
	}
	if math.IsNaN(x) {
		panic(ErrNaN{"Float.SetFloat64(NaN)"})
	}
	z.acc = Exact
	z.neg = math.Signbit(x) // handle -0, -Inf correctly
	if x == 0 {
		z.form = zero
		return z
	}
	if math.IsInf(x, 0) {
		z.form = inf
		return z
	}
	// normalized x != 0
	z.form = finite
	fmant, exp := math.Frexp(x) // get normalized mantissa
	z.mant = z.mant.setUint64(1<<63 | math.Float64bits(fmant)<<11)
	z.exp = int32(exp) // always fits
	if z.prec < 53 {
		z.round(0)
	}
	return z
}

// fnorm normalizes mantissa m by shifting it to the left
// such that the msb of the most-significant word (msw) is 1.
// It returns the shift amount. It assumes that len(m) != 0.
func fnorm(m nat) int64 {
	if debugFloat && (len(m) == 0 || m[len(m)-1] == 0) {
		panic("msw of mantissa is 0")
	}
	s := nlz(m[len(m)-1])
	if s > 0 {
		c := shlVU(m, m, s)
		if debugFloat && c != 0 {
			panic("nlz or shlVU incorrect")
		}
	}
	return int64(s)
}

// SetInt sets z to the (possibly rounded) value of x and returns z.
// If z's precision is 0, it is changed to the larger of x.BitLen()
// or 64 (and rounding will have no effect).
func (z *Float) SetInt(x *Int) *Float {
	// TODO(gri) can be more efficient if z.prec > 0
	// but small compared to the size of x, or if there
	// are many trailing 0's.
	bits := uint32(x.BitLen())
	if z.prec == 0 {
		z.prec = umax32(bits, 64)
	}
	z.acc = Exact
	z.neg = x.neg
	if len(x.abs) == 0 {
		z.form = zero
		return z
	}
	// x != 0
	z.mant = z.mant.set(x.abs)
	fnorm(z.mant)
	z.setExpAndRound(int64(bits), 0)
	return z
}

// SetRat sets z to the (possibly rounded) value of x and returns z.
// If z's precision is 0, it is changed to the largest of a.BitLen(),
// b.BitLen(), or 64; with x = a/b.
func (z *Float) SetRat(x *Rat) *Float {
	if x.IsInt() {
		return z.SetInt(x.Num())
	}
	var a, b Float
	a.SetInt(x.Num())
	b.SetInt(x.Denom())
	if z.prec == 0 {
		z.prec = umax32(a.prec, b.prec)
	}
	return z.Quo(&a, &b)
}

// SetInf sets z to the infinite Float -Inf if signbit is
// set, or +Inf if signbit is not set, and returns z. The
// precision of z is unchanged and the result is always
// Exact.
func (z *Float) SetInf(signbit bool) *Float {
	z.acc = Exact
	z.form = inf
	z.neg = signbit
	return z
}

// Set sets z to the (possibly rounded) value of x and returns z.
// If z's precision is 0, it is changed to the precision of x
// before setting z (and rounding will have no effect).
// Rounding is performed according to z's precision and rounding
// mode; and z's accuracy reports the result error relative to the
// exact (not rounded) result.
func (z *Float) Set(x *Float) *Float {
	if debugFloat {
		x.validate()
	}
	z.acc = Exact
	if z != x {
		z.form = x.form
		z.neg = x.neg
		if x.form == finite {
			z.exp = x.exp
			z.mant = z.mant.set(x.mant)
		}
		if z.prec == 0 {
			z.prec = x.prec
		} else if z.prec < x.prec {
			z.round(0)
		}
	}
	return z
}

// Copy sets z to x, with the same precision, rounding mode, and
// accuracy as x, and returns z. x is not changed even if z and
// x are the same.
func (z *Float) Copy(x *Float) *Float {
	if debugFloat {
		x.validate()
	}
	if z != x {
		z.prec = x.prec
		z.mode = x.mode
		z.acc = x.acc
		z.form = x.form
		z.neg = x.neg
		if z.form == finite {
			z.mant = z.mant.set(x.mant)
			z.exp = x.exp
		}
	}
	return z
}

// msb32 returns the 32 most significant bits of x.
func msb32(x nat) uint32 {
	i := len(x) - 1
	if i < 0 {
		return 0
	}
	if debugFloat && x[i]&(1<<(_W-1)) == 0 {
		panic("x not normalized")
	}
	switch _W {
	case 32:
		return uint32(x[i])
	case 64:
		return uint32(x[i] >> 32)
	}
	panic("unreachable")
}

// msb64 returns the 64 most significant bits of x.
func msb64(x nat) uint64 {
	i := len(x) - 1
	if i < 0 {
		return 0
	}
	if debugFloat && x[i]&(1<<(_W-1)) == 0 {
		panic("x not normalized")
	}
	switch _W {
	case 32:
		v := uint64(x[i]) << 32
		if i > 0 {
			v |= uint64(x[i-1])
		}
		return v
	case 64:
		return uint64(x[i])
	}
	panic("unreachable")
}

// Uint64 returns the unsigned integer resulting from truncating x
// towards zero. If 0 <= x <= math.MaxUint64, the result is Exact
// if x is an integer and Below otherwise.
// The result is (0, Above) for x < 0, and (math.MaxUint64, Below)
// for x > math.MaxUint64.
func (x *Float) Uint64() (uint64, Accuracy) {
	if debugFloat {
		x.validate()
	}

	switch x.form {
	case finite:
		if x.neg {
			return 0, Above
		}
		// 0 < x < +Inf
		if x.exp <= 0 {
			// 0 < x < 1
			return 0, Below
		}
		// 1 <= x < Inf
		if x.exp <= 64 {
			// u = trunc(x) fits into a uint64
			u := msb64(x.mant) >> (64 - uint32(x.exp))
			if x.MinPrec() <= 64 {
				return u, Exact
			}
			return u, Below // x truncated
		}
		// x too large
		return math.MaxUint64, Below

	case zero:
		return 0, Exact

	case inf:
		if x.neg {
			return 0, Above
		}
		return math.MaxUint64, Below
	}

	panic("unreachable")
}

// Int64 returns the integer resulting from truncating x towards zero.
// If math.MinInt64 <= x <= math.MaxInt64, the result is Exact if x is
// an integer, and Above (x < 0) or Below (x > 0) otherwise.
// The result is (math.MinInt64, Above) for x < math.MinInt64,
// and (math.MaxInt64, Below) for x > math.MaxInt64.
func (x *Float) Int64() (int64, Accuracy) {
	if debugFloat {
		x.validate()
	}

	switch x.form {
	case finite:
		// 0 < |x| < +Inf
		acc := makeAcc(x.neg)
		if x.exp <= 0 {
			// 0 < |x| < 1
			return 0, acc
		}
		// x.exp > 0

		// 1 <= |x| < +Inf
		if x.exp <= 63 {
			// i = trunc(x) fits into an int64 (excluding math.MinInt64)
			i := int64(msb64(x.mant) >> (64 - uint32(x.exp)))
			if x.neg {
				i = -i
			}
			if x.MinPrec() <= uint(x.exp) {
				return i, Exact
			}
			return i, acc // x truncated
		}
		if x.neg {
			// check for special case x == math.MinInt64 (i.e., x == -(0.5 << 64))
			if x.exp == 64 && x.MinPrec() == 1 {
				acc = Exact
			}
			return math.MinInt64, acc
		}
		// x too large
		return math.MaxInt64, Below

	case zero:
		return 0, Exact

	case inf:
		if x.neg {
			return math.MinInt64, Above
		}
		return math.MaxInt64, Below
	}

	panic("unreachable")
}

// Float32 returns the float32 value nearest to x. If x is too small to be
// represented by a float32 (|x| < math.SmallestNonzeroFloat32), the result
// is (0, Below) or (-0, Above), respectively, depending on the sign of x.
// If x is too large to be represented by a float32 (|x| > math.MaxFloat32),
// the result is (+Inf, Above) or (-Inf, Below), depending on the sign of x.
func (x *Float) Float32() (float32, Accuracy) {
	if debugFloat {
		x.validate()
	}

	switch x.form {
	case finite:
		// 0 < |x| < +Inf

		const (
			fbits = 32                //        float size
			mbits = 23                //        mantissa size (excluding implicit msb)
			ebits = fbits - mbits - 1 //     8  exponent size
			bias  = 1<<(ebits-1) - 1  //   127  exponent bias
			dmin  = 1 - bias - mbits  //  -149  smallest unbiased exponent (denormal)
			emin  = 1 - bias          //  -126  smallest unbiased exponent (normal)
			emax  = bias              //   127  largest unbiased exponent (normal)
		)

		// Float mantissa m is 0.5 <= m < 1.0; compute exponent e for float32 mantissa.
		e := x.exp - 1 // exponent for normal mantissa m with 1.0 <= m < 2.0

		// Compute precision p for float32 mantissa.
		// If the exponent is too small, we have a denormal number before
		// rounding and fewer than p mantissa bits of precision available
		// (the exponent remains fixed but the mantissa gets shifted right).
		p := mbits + 1 // precision of normal float
		if e < emin {
			// recompute precision
			p = mbits + 1 - emin + int(e)
			// If p == 0, the mantissa of x is shifted so much to the right
			// that its msb falls immediately to the right of the float32
			// mantissa space. In other words, if the smallest denormal is
			// considered "1.0", for p == 0, the mantissa value m is >= 0.5.
			// If m > 0.5, it is rounded up to 1.0; i.e., the smallest denormal.
			// If m == 0.5, it is rounded down to even, i.e., 0.0.
			// If p < 0, the mantissa value m is <= "0.25" which is never rounded up.
			if p < 0 /* m <= 0.25 */ || p == 0 && x.mant.sticky(uint(len(x.mant))*_W-1) == 0 /* m == 0.5 */ {
				// underflow to ±0
				if x.neg {
					var z float32
					return -z, Above
				}
				return 0.0, Below
			}
			// otherwise, round up
			// We handle p == 0 explicitly because it's easy and because
			// Float.round doesn't support rounding to 0 bits of precision.
			if p == 0 {
				if x.neg {
					return -math.SmallestNonzeroFloat32, Below
				}
				return math.SmallestNonzeroFloat32, Above
			}
		}
		// p > 0

		// round
		var r Float
		r.prec = uint32(p)
		r.Set(x)
		e = r.exp - 1

		// Rounding may have caused r to overflow to ±Inf
		// (rounding never causes underflows to 0).
		// If the exponent is too large, also overflow to ±Inf.
		if r.form == inf || e > emax {
			// overflow
			if x.neg {
				return float32(math.Inf(-1)), Below
			}
			return float32(math.Inf(+1)), Above
		}
		// e <= emax

		// Determine sign, biased exponent, and mantissa.
		var sign, bexp, mant uint32
		if x.neg {
			sign = 1 << (fbits - 1)
		}

		// Rounding may have caused a denormal number to
		// become normal. Check again.
		if e < emin {
			// denormal number: recompute precision
			// Since rounding may have at best increased precision
			// and we have eliminated p <= 0 early, we know p > 0.
			// bexp == 0 for denormals
			p = mbits + 1 - emin + int(e)
			mant = msb32(r.mant) >> uint(fbits-p)
		} else {
			// normal number: emin <= e <= emax
			bexp = uint32(e+bias) << mbits
			mant = msb32(r.mant) >> ebits & (1<<mbits - 1) // cut off msb (implicit 1 bit)
		}

		return math.Float32frombits(sign | bexp | mant), r.acc

	case zero:
		if x.neg {
			var z float32
			return -z, Exact
		}
		return 0.0, Exact

	case inf:
		if x.neg {
			return float32(math.Inf(-1)), Exact
		}
		return float32(math.Inf(+1)), Exact
	}

	panic("unreachable")
}

// Float64 returns the float64 value nearest to x. If x is too small to be
// represented by a float64 (|x| < math.SmallestNonzeroFloat64), the result
// is (0, Below) or (-0, Above), respectively, depending on the sign of x.
// If x is too large to be represented by a float64 (|x| > math.MaxFloat64),
// the result is (+Inf, Above) or (-Inf, Below), depending on the sign of x.
func (x *Float) Float64() (float64, Accuracy) {
	if debugFloat {
		x.validate()
	}

	switch x.form {
	case finite:
		// 0 < |x| < +Inf

		const (
			fbits = 64                //        float size
			mbits = 52                //        mantissa size (excluding implicit msb)
			ebits = fbits - mbits - 1 //    11  exponent size
			bias  = 1<<(ebits-1) - 1  //  1023  exponent bias
			dmin  = 1 - bias - mbits  // -1074  smallest unbiased exponent (denormal)
			emin  = 1 - bias          // -1022  smallest unbiased exponent (normal)
			emax  = bias              //  1023  largest unbiased exponent (normal)
		)

		// Float mantissa m is 0.5 <= m < 1.0; compute exponent e for float64 mantissa.
		e := x.exp - 1 // exponent for normal mantissa m with 1.0 <= m < 2.0

		// Compute precision p for float64 mantissa.
		// If the exponent is too small, we have a denormal number before
		// rounding and fewer than p mantissa bits of precision available
		// (the exponent remains fixed but the mantissa gets shifted right).
		p := mbits + 1 // precision of normal float
		if e < emin {
			// recompute precision
			p = mbits + 1 - emin + int(e)
			// If p == 0, the mantissa of x is shifted so much to the right
			// that its msb falls immediately to the right of the float64
			// mantissa space. In other words, if the smallest denormal is
			// considered "1.0", for p == 0, the mantissa value m is >= 0.5.
			// If m > 0.5, it is rounded up to 1.0; i.e., the smallest denormal.
			// If m == 0.5, it is rounded down to even, i.e., 0.0.
			// If p < 0, the mantissa value m is <= "0.25" which is never rounded up.
			if p < 0 /* m <= 0.25 */ || p == 0 && x.mant.sticky(uint(len(x.mant))*_W-1) == 0 /* m == 0.5 */ {
				// underflow to ±0
				if x.neg {
					var z float64
					return -z, Above
				}
				return 0.0, Below
			}
			// otherwise, round up
			// We handle p == 0 explicitly because it's easy and because
			// Float.round doesn't support rounding to 0 bits of precision.
			if p == 0 {
				if x.neg {
					return -math.SmallestNonzeroFloat64, Below
				}
				return math.SmallestNonzeroFloat64, Above
			}
		}
		// p > 0

		// round
		var r Float
		r.prec = uint32(p)
		r.Set(x)
		e = r.exp - 1

		// Rounding may have caused r to overflow to ±Inf
		// (rounding never causes underflows to 0).
		// If the exponent is too large, also overflow to ±Inf.
		if r.form == inf || e > emax {
			// overflow
			if x.neg {
				return float64(math.Inf(-1)), Below
			}
			return float64(math.Inf(+1)), Above
		}
		// e <= emax

		// Determine sign, biased exponent, and mantissa.
		var sign, bexp, mant uint64
		if x.neg {
			sign = 1 << (fbits - 1)
		}

		// Rounding may have caused a denormal number to
		// become normal. Check again.
		if e < emin {
			// denormal number: recompute precision
			// Since rounding may have at best increased precision
			// and we have eliminated p <= 0 early, we know p > 0.
			// bexp == 0 for denormals
			p = mbits + 1 - emin + int(e)
			mant = msb64(r.mant) >> uint(fbits-p)
		} else {
			// normal number: emin <= e <= emax
			bexp = uint64(e+bias) << mbits
			mant = msb64(r.mant) >> ebits & (1<<mbits - 1) // cut off msb (implicit 1 bit)
		}

		return math.Float64frombits(sign | bexp | mant), r.acc

	case zero:
		if x.neg {
			var z float64
			return -z, Exact
		}
		return 0.0, Exact

	case inf:
		if x.neg {
			return math.Inf(-1), Exact
		}
		return math.Inf(+1), Exact
	}

	panic("unreachable")
}

// Int returns the result of truncating x towards zero;
// or nil if x is an infinity.
// The result is Exact if x.IsInt(); otherwise it is Below
// for x > 0, and Above for x < 0.
// If a non-nil *Int argument z is provided, Int stores
// the result in z instead of allocating a new Int.
func (x *Float) Int(z *Int) (*Int, Accuracy) {
	if debugFloat {
		x.validate()
	}

	if z == nil && x.form <= finite {
		z = new(Int)
	}

	switch x.form {
	case finite:
		// 0 < |x| < +Inf
		acc := makeAcc(x.neg)
		if x.exp <= 0 {
			// 0 < |x| < 1
			return z.SetInt64(0), acc
		}
		// x.exp > 0

		// 1 <= |x| < +Inf
		// determine minimum required precision for x
		allBits := uint(len(x.mant)) * _W
		exp := uint(x.exp)
		if x.MinPrec() <= exp {
			acc = Exact
		}
		// shift mantissa as needed
		if z == nil {
			z = new(Int)
		}
		z.neg = x.neg
		switch {
		case exp > allBits:
			z.abs = z.abs.shl(x.mant, exp-allBits)
		default:
			z.abs = z.abs.set(x.mant)
		case exp < allBits:
			z.abs = z.abs.shr(x.mant, allBits-exp)
		}
		return z, acc

	case zero:
		return z.SetInt64(0), Exact

	case inf:
		return nil, makeAcc(x.neg)
	}

	panic("unreachable")
}

// Rat returns the rational number corresponding to x;
// or nil if x is an infinity.
// The result is Exact if x is not an Inf.
// If a non-nil *Rat argument z is provided, Rat stores
// the result in z instead of allocating a new Rat.
func (x *Float) Rat(z *Rat) (*Rat, Accuracy) {
	if debugFloat {
		x.validate()
	}

	if z == nil && x.form <= finite {
		z = new(Rat)
	}

	switch x.form {
	case finite:
		// 0 < |x| < +Inf
		allBits := int32(len(x.mant)) * _W
		// build up numerator and denominator
		z.a.neg = x.neg
		switch {
		case x.exp > allBits:
			z.a.abs = z.a.abs.shl(x.mant, uint(x.exp-allBits))
			z.b.abs = z.b.abs[:0] // == 1 (see Rat)
			// z already in normal form
		default:
			z.a.abs = z.a.abs.set(x.mant)
			z.b.abs = z.b.abs[:0] // == 1 (see Rat)
			// z already in normal form
		case x.exp < allBits:
			z.a.abs = z.a.abs.set(x.mant)
			t := z.b.abs.setUint64(1)
			z.b.abs = t.shl(t, uint(allBits-x.exp))
			z.norm()
		}
		return z, Exact

	case zero:
		return z.SetInt64(0), Exact

	case inf:
		return nil, makeAcc(x.neg)
	}

	panic("unreachable")
}

// Abs sets z to the (possibly rounded) value |x| (the absolute value of x)
// and returns z.
func (z *Float) Abs(x *Float) *Float {
	z.Set(x)
	z.neg = false
	return z
}

// Neg sets z to the (possibly rounded) value of x with its sign negated,
// and returns z.
func (z *Float) Neg(x *Float) *Float {
	z.Set(x)
	z.neg = !z.neg
	return z
}

func validateBinaryOperands(x, y *Float) {
	if !debugFloat {
		// avoid performance bugs
		panic("validateBinaryOperands called but debugFloat is not set")
	}
	if len(x.mant) == 0 {
		panic("empty mantissa for x")
	}
	if len(y.mant) == 0 {
		panic("empty mantissa for y")
	}
}

// z = x + y, ignoring signs of x and y for the addition
// but using the sign of z for rounding the result.
// x and y must have a non-empty mantissa and valid exponent.
func (z *Float) uadd(x, y *Float) {
	// Note: This implementation requires 2 shifts most of the
	// time. It is also inefficient if exponents or precisions
	// differ by wide margins. The following article describes
	// an efficient (but much more complicated) implementation
	// compatible with the internal representation used here:
	//
	// Vincent Lefèvre: "The Generic Multiple-Precision Floating-
	// Point Addition With Exact Rounding (as in the MPFR Library)"
	// http://www.vinc17.net/research/papers/rnc6.pdf

	if debugFloat {
		validateBinaryOperands(x, y)
	}

	// compute exponents ex, ey for mantissa with "binary point"
	// on the right (mantissa.0) - use int64 to avoid overflow
	ex := int64(x.exp) - int64(len(x.mant))*_W
	ey := int64(y.exp) - int64(len(y.mant))*_W

	// TODO(gri) having a combined add-and-shift primitive
	//           could make this code significantly faster
	switch {
	case ex < ey:
		// cannot re-use z.mant w/o testing for aliasing
		t := nat(nil).shl(y.mant, uint(ey-ex))
		z.mant = z.mant.add(x.mant, t)
	default:
		// ex == ey, no shift needed
		z.mant = z.mant.add(x.mant, y.mant)
	case ex > ey:
		// cannot re-use z.mant w/o testing for aliasing
		t := nat(nil).shl(x.mant, uint(ex-ey))
		z.mant = z.mant.add(t, y.mant)
		ex = ey
	}
	// len(z.mant) > 0

	z.setExpAndRound(ex+int64(len(z.mant))*_W-fnorm(z.mant), 0)
}

// z = x - y for |x| > |y|, ignoring signs of x and y for the subtraction
// but using the sign of z for rounding the result.
// x and y must have a non-empty mantissa and valid exponent.
func (z *Float) usub(x, y *Float) {
	// This code is symmetric to uadd.
	// We have not factored the common code out because
	// eventually uadd (and usub) should be optimized
	// by special-casing, and the code will diverge.

	if debugFloat {
		validateBinaryOperands(x, y)
	}

	ex := int64(x.exp) - int64(len(x.mant))*_W
	ey := int64(y.exp) - int64(len(y.mant))*_W

	switch {
	case ex < ey:
		// cannot re-use z.mant w/o testing for aliasing
		t := nat(nil).shl(y.mant, uint(ey-ex))
		z.mant = t.sub(x.mant, t)
	default:
		// ex == ey, no shift needed
		z.mant = z.mant.sub(x.mant, y.mant)
	case ex > ey:
		// cannot re-use z.mant w/o testing for aliasing
		t := nat(nil).shl(x.mant, uint(ex-ey))
		z.mant = t.sub(t, y.mant)
		ex = ey
	}

	// operands may have canceled each other out
	if len(z.mant) == 0 {
		z.acc = Exact
		z.form = zero
		z.neg = false
		return
	}
	// len(z.mant) > 0

	z.setExpAndRound(ex+int64(len(z.mant))*_W-fnorm(z.mant), 0)
}

// z = x * y, ignoring signs of x and y for the multiplication
// but using the sign of z for rounding the result.
// x and y must have a non-empty mantissa and valid exponent.
func (z *Float) umul(x, y *Float) {
	if debugFloat {
		validateBinaryOperands(x, y)
	}

	// Note: This is doing too much work if the precision
	// of z is less than the sum of the precisions of x
	// and y which is often the case (e.g., if all floats
	// have the same precision).
	// TODO(gri) Optimize this for the common case.

	e := int64(x.exp) + int64(y.exp)
	z.mant = z.mant.mul(x.mant, y.mant)

	z.setExpAndRound(e-fnorm(z.mant), 0)
}

// z = x / y, ignoring signs of x and y for the division
// but using the sign of z for rounding the result.
// x and y must have a non-empty mantissa and valid exponent.
func (z *Float) uquo(x, y *Float) {
	if debugFloat {
		validateBinaryOperands(x, y)
	}

	// mantissa length in words for desired result precision + 1
	// (at least one extra bit so we get the rounding bit after
	// the division)
	n := int(z.prec/_W) + 1

	// compute adjusted x.mant such that we get enough result precision
	xadj := x.mant
	if d := n - len(x.mant) + len(y.mant); d > 0 {
		// d extra words needed => add d "0 digits" to x
		xadj = make(nat, len(x.mant)+d)
		copy(xadj[d:], x.mant)
	}
	// TODO(gri): If we have too many digits (d < 0), we should be able
	// to shorten x for faster division. But we must be extra careful
	// with rounding in that case.

	// Compute d before division since there may be aliasing of x.mant
	// (via xadj) or y.mant with z.mant.
	d := len(xadj) - len(y.mant)

	// divide
	var r nat
	z.mant, r = z.mant.div(nil, xadj, y.mant)
	e := int64(x.exp) - int64(y.exp) - int64(d-len(z.mant))*_W

	// The result is long enough to include (at least) the rounding bit.
	// If there's a non-zero remainder, the corresponding fractional part
	// (if it were computed), would have a non-zero sticky bit (if it were
	// zero, it couldn't have a non-zero remainder).
	var sbit uint
	if len(r) > 0 {
		sbit = 1
	}

	z.setExpAndRound(e-fnorm(z.mant), sbit)
}

// ucmp returns -1, 0, or +1, depending on whether
// |x| < |y|, |x| == |y|, or |x| > |y|.
// x and y must have a non-empty mantissa and valid exponent.
func (x *Float) ucmp(y *Float) int {
	if debugFloat {
		validateBinaryOperands(x, y)
	}

	switch {
	case x.exp < y.exp:
		return -1
	case x.exp > y.exp:
		return +1
	}
	// x.exp == y.exp

	// compare mantissas
	i := len(x.mant)
	j := len(y.mant)
	for i > 0 || j > 0 {
		var xm, ym Word
		if i > 0 {
			i--
			xm = x.mant[i]
		}
		if j > 0 {
			j--
			ym = y.mant[j]
		}
		switch {
		case xm < ym:
			return -1
		case xm > ym:
			return +1
		}
	}

	return 0
}

// Handling of sign bit as defined by IEEE 754-2008, section 6.3:
//
// When neither the inputs nor result are NaN, the sign of a product or
// quotient is the exclusive OR of the operands’ signs; the sign of a sum,
// or of a difference x−y regarded as a sum x+(−y), differs from at most
// one of the addends’ signs; and the sign of the result of conversions,
// the quantize operation, the roundToIntegral operations, and the
// roundToIntegralExact (see 5.3.1) is the sign of the first or only operand.
// These rules shall apply even when operands or results are zero or infinite.
//
// When the sum of two operands with opposite signs (or the difference of
// two operands with like signs) is exactly zero, the sign of that sum (or
// difference) shall be +0 in all rounding-direction attributes except
// roundTowardNegative; under that attribute, the sign of an exact zero
// sum (or difference) shall be −0. However, x+x = x−(−x) retains the same
// sign as x even when x is zero.
//
// See also: https://play.golang.org/p/RtH3UCt5IH

// Add sets z to the rounded sum x+y and returns z. If z's precision is 0,
// it is changed to the larger of x's or y's precision before the operation.
// Rounding is performed according to z's precision and rounding mode; and
// z's accuracy reports the result error relative to the exact (not rounded)
// result. Add panics with ErrNaN if x and y are infinities with opposite
// signs. The value of z is undefined in that case.
//
// BUG(gri) When rounding ToNegativeInf, the sign of Float values rounded to 0 is incorrect.
func (z *Float) Add(x, y *Float) *Float {
	if debugFloat {
		x.validate()
		y.validate()
	}

	if z.prec == 0 {
		z.prec = umax32(x.prec, y.prec)
	}

	if x.form == finite && y.form == finite {
		// x + y (common case)
		z.neg = x.neg
		if x.neg == y.neg {
			// x + y == x + y
			// (-x) + (-y) == -(x + y)
			z.uadd(x, y)
		} else {
			// x + (-y) == x - y == -(y - x)
			// (-x) + y == y - x == -(x - y)
			if x.ucmp(y) > 0 {
				z.usub(x, y)
			} else {
				z.neg = !z.neg
				z.usub(y, x)
			}
		}
		return z
	}

	if x.form == inf && y.form == inf && x.neg != y.neg {
		// +Inf + -Inf
		// -Inf + +Inf
		// value of z is undefined but make sure it's valid
		z.acc = Exact
		z.form = zero
		z.neg = false
		panic(ErrNaN{"addition of infinities with opposite signs"})
	}

	if x.form == zero && y.form == zero {
		// ±0 + ±0
		z.acc = Exact
		z.form = zero
		z.neg = x.neg && y.neg // -0 + -0 == -0
		return z
	}

	if x.form == inf || y.form == zero {
		// ±Inf + y
		// x + ±0
		return z.Set(x)
	}

	// ±0 + y
	// x + ±Inf
	return z.Set(y)
}

// Sub sets z to the rounded difference x-y and returns z.
// Precision, rounding, and accuracy reporting are as for Add.
// Sub panics with ErrNaN if x and y are infinities with equal
// signs. The value of z is undefined in that case.
func (z *Float) Sub(x, y *Float) *Float {
	if debugFloat {
		x.validate()
		y.validate()
	}

	if z.prec == 0 {
		z.prec = umax32(x.prec, y.prec)
	}

	if x.form == finite && y.form == finite {
		// x - y (common case)
		z.neg = x.neg
		if x.neg != y.neg {
			// x - (-y) == x + y
			// (-x) - y == -(x + y)
			z.uadd(x, y)
		} else {
			// x - y == x - y == -(y - x)
			// (-x) - (-y) == y - x == -(x - y)
			if x.ucmp(y) > 0 {
				z.usub(x, y)
			} else {
				z.neg = !z.neg
				z.usub(y, x)
			}
		}
		return z
	}

	if x.form == inf && y.form == inf && x.neg == y.neg {
		// +Inf - +Inf
		// -Inf - -Inf
		// value of z is undefined but make sure it's valid
		z.acc = Exact
		z.form = zero
		z.neg = false
		panic(ErrNaN{"subtraction of infinities with equal signs"})
	}

	if x.form == zero && y.form == zero {
		// ±0 - ±0
		z.acc = Exact
		z.form = zero
		z.neg = x.neg && !y.neg // -0 - +0 == -0
		return z
	}

	if x.form == inf || y.form == zero {
		// ±Inf - y
		// x - ±0
		return z.Set(x)
	}

	// ±0 - y
	// x - ±Inf
	return z.Neg(y)
}

// Mul sets z to the rounded product x*y and returns z.
// Precision, rounding, and accuracy reporting are as for Add.
// Mul panics with ErrNaN if one operand is zero and the other
// operand an infinity. The value of z is undefined in that case.
func (z *Float) Mul(x, y *Float) *Float {
	if debugFloat {
		x.validate()
		y.validate()
	}

	if z.prec == 0 {
		z.prec = umax32(x.prec, y.prec)
	}

	z.neg = x.neg != y.neg

	if x.form == finite && y.form == finite {
		// x * y (common case)
		z.umul(x, y)
		return z
	}

	z.acc = Exact
	if x.form == zero && y.form == inf || x.form == inf && y.form == zero {
		// ±0 * ±Inf
		// ±Inf * ±0
		// value of z is undefined but make sure it's valid
		z.form = zero
		z.neg = false
		panic(ErrNaN{"multiplication of zero with infinity"})
	}

	if x.form == inf || y.form == inf {
		// ±Inf * y
		// x * ±Inf
		z.form = inf
		return z
	}

	// ±0 * y
	// x * ±0
	z.form = zero
	return z
}

// Quo sets z to the rounded quotient x/y and returns z.
// Precision, rounding, and accuracy reporting are as for Add.
// Quo panics with ErrNaN if both operands are zero or infinities.
// The value of z is undefined in that case.
func (z *Float) Quo(x, y *Float) *Float {
	if debugFloat {
		x.validate()
		y.validate()
	}

	if z.prec == 0 {
		z.prec = umax32(x.prec, y.prec)
	}

	z.neg = x.neg != y.neg

	if x.form == finite && y.form == finite {
		// x / y (common case)
		z.uquo(x, y)
		return z
	}

	z.acc = Exact
	if x.form == zero && y.form == zero || x.form == inf && y.form == inf {
		// ±0 / ±0
		// ±Inf / ±Inf
		// value of z is undefined but make sure it's valid
		z.form = zero
		z.neg = false
		panic(ErrNaN{"division of zero by zero or infinity by infinity"})
	}

	if x.form == zero || y.form == inf {
		// ±0 / y
		// x / ±Inf
		z.form = zero
		return z
	}

	// x / ±0
	// ±Inf / y
	z.form = inf
	return z
}

// Cmp compares x and y and returns:
//
//   -1 if x <  y
//    0 if x == y (incl. -0 == 0, -Inf == -Inf, and +Inf == +Inf)
//   +1 if x >  y
//
func (x *Float) Cmp(y *Float) int {
	if debugFloat {
		x.validate()
		y.validate()
	}

	mx := x.ord()
	my := y.ord()
	switch {
	case mx < my:
		return -1
	case mx > my:
		return +1
	}
	// mx == my

	// only if |mx| == 1 we have to compare the mantissae
	switch mx {
	case -1:
		return y.ucmp(x)
	case +1:
		return x.ucmp(y)
	}

	return 0
}

// ord classifies x and returns:
//
//	-2 if -Inf == x
//	-1 if -Inf < x < 0
//	 0 if x == 0 (signed or unsigned)
//	+1 if 0 < x < +Inf
//	+2 if x == +Inf
//
func (x *Float) ord() int {
	var m int
	switch x.form {
	case finite:
		m = 1
	case zero:
		return 0
	case inf:
		m = 2
	}
	if x.neg {
		m = -m
	}
	return m
}

func umax32(x, y uint32) uint32 {
	if x > y {
		return x
	}
	return y
}