Source file src/math/big/float.go

     1  // Copyright 2014 The Go Authors. All rights reserved.
     2  // Use of this source code is governed by a BSD-style
     3  // license that can be found in the LICENSE file.
     4  
     5  // This file implements multi-precision floating-point numbers.
     6  // Like in the GNU MPFR library (https://www.mpfr.org/), operands
     7  // can be of mixed precision. Unlike MPFR, the rounding mode is
     8  // not specified with each operation, but with each operand. The
     9  // rounding mode of the result operand determines the rounding
    10  // mode of an operation. This is a from-scratch implementation.
    11  
    12  package big
    13  
    14  import (
    15  	"fmt"
    16  	"math"
    17  	"math/bits"
    18  )
    19  
    20  const debugFloat = false // enable for debugging
    21  
    22  // A nonzero finite Float represents a multi-precision floating point number
    23  //
    24  //	sign × mantissa × 2**exponent
    25  //
    26  // with 0.5 <= mantissa < 1.0, and MinExp <= exponent <= MaxExp.
    27  // A Float may also be zero (+0, -0) or infinite (+Inf, -Inf).
    28  // All Floats are ordered, and the ordering of two Floats x and y
    29  // is defined by x.Cmp(y).
    30  //
    31  // Each Float value also has a precision, rounding mode, and accuracy.
    32  // The precision is the maximum number of mantissa bits available to
    33  // represent the value. The rounding mode specifies how a result should
    34  // be rounded to fit into the mantissa bits, and accuracy describes the
    35  // rounding error with respect to the exact result.
    36  //
    37  // Unless specified otherwise, all operations (including setters) that
    38  // specify a *Float variable for the result (usually via the receiver
    39  // with the exception of MantExp), round the numeric result according
    40  // to the precision and rounding mode of the result variable.
    41  //
    42  // If the provided result precision is 0 (see below), it is set to the
    43  // precision of the argument with the largest precision value before any
    44  // rounding takes place, and the rounding mode remains unchanged. Thus,
    45  // uninitialized Floats provided as result arguments will have their
    46  // precision set to a reasonable value determined by the operands, and
    47  // their mode is the zero value for RoundingMode (ToNearestEven).
    48  //
    49  // By setting the desired precision to 24 or 53 and using matching rounding
    50  // mode (typically ToNearestEven), Float operations produce the same results
    51  // as the corresponding float32 or float64 IEEE-754 arithmetic for operands
    52  // that correspond to normal (i.e., not denormal) float32 or float64 numbers.
    53  // Exponent underflow and overflow lead to a 0 or an Infinity for different
    54  // values than IEEE-754 because Float exponents have a much larger range.
    55  //
    56  // The zero (uninitialized) value for a Float is ready to use and represents
    57  // the number +0.0 exactly, with precision 0 and rounding mode ToNearestEven.
    58  //
    59  // Operations always take pointer arguments (*Float) rather
    60  // than Float values, and each unique Float value requires
    61  // its own unique *Float pointer. To "copy" a Float value,
    62  // an existing (or newly allocated) Float must be set to
    63  // a new value using the Float.Set method; shallow copies
    64  // of Floats are not supported and may lead to errors.
    65  type Float struct {
    66  	prec uint32
    67  	mode RoundingMode
    68  	acc  Accuracy
    69  	form form
    70  	neg  bool
    71  	mant nat
    72  	exp  int32
    73  }
    74  
    75  // An ErrNaN panic is raised by a Float operation that would lead to
    76  // a NaN under IEEE-754 rules. An ErrNaN implements the error interface.
    77  type ErrNaN struct {
    78  	msg string
    79  }
    80  
    81  func (err ErrNaN) Error() string {
    82  	return err.msg
    83  }
    84  
    85  // NewFloat allocates and returns a new Float set to x,
    86  // with precision 53 and rounding mode ToNearestEven.
    87  // NewFloat panics with ErrNaN if x is a NaN.
    88  func NewFloat(x float64) *Float {
    89  	if math.IsNaN(x) {
    90  		panic(ErrNaN{"NewFloat(NaN)"})
    91  	}
    92  	return new(Float).SetFloat64(x)
    93  }
    94  
    95  // Exponent and precision limits.
    96  const (
    97  	MaxExp  = math.MaxInt32  // largest supported exponent
    98  	MinExp  = math.MinInt32  // smallest supported exponent
    99  	MaxPrec = math.MaxUint32 // largest (theoretically) supported precision; likely memory-limited
   100  )
   101  
   102  // Internal representation: The mantissa bits x.mant of a nonzero finite
   103  // Float x are stored in a nat slice long enough to hold up to x.prec bits;
   104  // the slice may (but doesn't have to) be shorter if the mantissa contains
   105  // trailing 0 bits. x.mant is normalized if the msb of x.mant == 1 (i.e.,
   106  // the msb is shifted all the way "to the left"). Thus, if the mantissa has
   107  // trailing 0 bits or x.prec is not a multiple of the Word size _W,
   108  // x.mant[0] has trailing zero bits. The msb of the mantissa corresponds
   109  // to the value 0.5; the exponent x.exp shifts the binary point as needed.
   110  //
   111  // A zero or non-finite Float x ignores x.mant and x.exp.
   112  //
   113  // x                 form      neg      mant         exp
   114  // ----------------------------------------------------------
   115  // ±0                zero      sign     -            -
   116  // 0 < |x| < +Inf    finite    sign     mantissa     exponent
   117  // ±Inf              inf       sign     -            -
   118  
   119  // A form value describes the internal representation.
   120  type form byte
   121  
   122  // The form value order is relevant - do not change!
   123  const (
   124  	zero form = iota
   125  	finite
   126  	inf
   127  )
   128  
   129  // RoundingMode determines how a Float value is rounded to the
   130  // desired precision. Rounding may change the Float value; the
   131  // rounding error is described by the Float's Accuracy.
   132  type RoundingMode byte
   133  
   134  // These constants define supported rounding modes.
   135  const (
   136  	ToNearestEven RoundingMode = iota // == IEEE 754-2008 roundTiesToEven
   137  	ToNearestAway                     // == IEEE 754-2008 roundTiesToAway
   138  	ToZero                            // == IEEE 754-2008 roundTowardZero
   139  	AwayFromZero                      // no IEEE 754-2008 equivalent
   140  	ToNegativeInf                     // == IEEE 754-2008 roundTowardNegative
   141  	ToPositiveInf                     // == IEEE 754-2008 roundTowardPositive
   142  )
   143  
   144  //go:generate stringer -type=RoundingMode
   145  
   146  // Accuracy describes the rounding error produced by the most recent
   147  // operation that generated a Float value, relative to the exact value.
   148  type Accuracy int8
   149  
   150  // Constants describing the Accuracy of a Float.
   151  const (
   152  	Below Accuracy = -1
   153  	Exact Accuracy = 0
   154  	Above Accuracy = +1
   155  )
   156  
   157  //go:generate stringer -type=Accuracy
   158  
   159  // SetPrec sets z's precision to prec and returns the (possibly) rounded
   160  // value of z. Rounding occurs according to z's rounding mode if the mantissa
   161  // cannot be represented in prec bits without loss of precision.
   162  // SetPrec(0) maps all finite values to ±0; infinite values remain unchanged.
   163  // If prec > MaxPrec, it is set to MaxPrec.
   164  func (z *Float) SetPrec(prec uint) *Float {
   165  	z.acc = Exact // optimistically assume no rounding is needed
   166  
   167  	// special case
   168  	if prec == 0 {
   169  		z.prec = 0
   170  		if z.form == finite {
   171  			// truncate z to 0
   172  			z.acc = makeAcc(z.neg)
   173  			z.form = zero
   174  		}
   175  		return z
   176  	}
   177  
   178  	// general case
   179  	if prec > MaxPrec {
   180  		prec = MaxPrec
   181  	}
   182  	old := z.prec
   183  	z.prec = uint32(prec)
   184  	if z.prec < old {
   185  		z.round(0)
   186  	}
   187  	return z
   188  }
   189  
   190  func makeAcc(above bool) Accuracy {
   191  	if above {
   192  		return Above
   193  	}
   194  	return Below
   195  }
   196  
   197  // SetMode sets z's rounding mode to mode and returns an exact z.
   198  // z remains unchanged otherwise.
   199  // z.SetMode(z.Mode()) is a cheap way to set z's accuracy to Exact.
   200  func (z *Float) SetMode(mode RoundingMode) *Float {
   201  	z.mode = mode
   202  	z.acc = Exact
   203  	return z
   204  }
   205  
   206  // Prec returns the mantissa precision of x in bits.
   207  // The result may be 0 for |x| == 0 and |x| == Inf.
   208  func (x *Float) Prec() uint {
   209  	return uint(x.prec)
   210  }
   211  
   212  // MinPrec returns the minimum precision required to represent x exactly
   213  // (i.e., the smallest prec before x.SetPrec(prec) would start rounding x).
   214  // The result is 0 for |x| == 0 and |x| == Inf.
   215  func (x *Float) MinPrec() uint {
   216  	if x.form != finite {
   217  		return 0
   218  	}
   219  	return uint(len(x.mant))*_W - x.mant.trailingZeroBits()
   220  }
   221  
   222  // Mode returns the rounding mode of x.
   223  func (x *Float) Mode() RoundingMode {
   224  	return x.mode
   225  }
   226  
   227  // Acc returns the accuracy of x produced by the most recent
   228  // operation, unless explicitly documented otherwise by that
   229  // operation.
   230  func (x *Float) Acc() Accuracy {
   231  	return x.acc
   232  }
   233  
   234  // Sign returns:
   235  //
   236  //	-1 if x <   0
   237  //	 0 if x is ±0
   238  //	+1 if x >   0
   239  func (x *Float) Sign() int {
   240  	if debugFloat {
   241  		x.validate()
   242  	}
   243  	if x.form == zero {
   244  		return 0
   245  	}
   246  	if x.neg {
   247  		return -1
   248  	}
   249  	return 1
   250  }
   251  
   252  // MantExp breaks x into its mantissa and exponent components
   253  // and returns the exponent. If a non-nil mant argument is
   254  // provided its value is set to the mantissa of x, with the
   255  // same precision and rounding mode as x. The components
   256  // satisfy x == mant × 2**exp, with 0.5 <= |mant| < 1.0.
   257  // Calling MantExp with a nil argument is an efficient way to
   258  // get the exponent of the receiver.
   259  //
   260  // Special cases are:
   261  //
   262  //	(  ±0).MantExp(mant) = 0, with mant set to   ±0
   263  //	(±Inf).MantExp(mant) = 0, with mant set to ±Inf
   264  //
   265  // x and mant may be the same in which case x is set to its
   266  // mantissa value.
   267  func (x *Float) MantExp(mant *Float) (exp int) {
   268  	if debugFloat {
   269  		x.validate()
   270  	}
   271  	if x.form == finite {
   272  		exp = int(x.exp)
   273  	}
   274  	if mant != nil {
   275  		mant.Copy(x)
   276  		if mant.form == finite {
   277  			mant.exp = 0
   278  		}
   279  	}
   280  	return
   281  }
   282  
   283  func (z *Float) setExpAndRound(exp int64, sbit uint) {
   284  	if exp < MinExp {
   285  		// underflow
   286  		z.acc = makeAcc(z.neg)
   287  		z.form = zero
   288  		return
   289  	}
   290  
   291  	if exp > MaxExp {
   292  		// overflow
   293  		z.acc = makeAcc(!z.neg)
   294  		z.form = inf
   295  		return
   296  	}
   297  
   298  	z.form = finite
   299  	z.exp = int32(exp)
   300  	z.round(sbit)
   301  }
   302  
   303  // SetMantExp sets z to mant × 2**exp and returns z.
   304  // The result z has the same precision and rounding mode
   305  // as mant. SetMantExp is an inverse of MantExp but does
   306  // not require 0.5 <= |mant| < 1.0. Specifically, for a
   307  // given x of type *Float, SetMantExp relates to MantExp
   308  // as follows:
   309  //
   310  //	mant := new(Float)
   311  //	new(Float).SetMantExp(mant, x.MantExp(mant)).Cmp(x) == 0
   312  //
   313  // Special cases are:
   314  //
   315  //	z.SetMantExp(  ±0, exp) =   ±0
   316  //	z.SetMantExp(±Inf, exp) = ±Inf
   317  //
   318  // z and mant may be the same in which case z's exponent
   319  // is set to exp.
   320  func (z *Float) SetMantExp(mant *Float, exp int) *Float {
   321  	if debugFloat {
   322  		z.validate()
   323  		mant.validate()
   324  	}
   325  	z.Copy(mant)
   326  
   327  	if z.form == finite {
   328  		// 0 < |mant| < +Inf
   329  		z.setExpAndRound(int64(z.exp)+int64(exp), 0)
   330  	}
   331  	return z
   332  }
   333  
   334  // Signbit reports whether x is negative or negative zero.
   335  func (x *Float) Signbit() bool {
   336  	return x.neg
   337  }
   338  
   339  // IsInf reports whether x is +Inf or -Inf.
   340  func (x *Float) IsInf() bool {
   341  	return x.form == inf
   342  }
   343  
   344  // IsInt reports whether x is an integer.
   345  // ±Inf values are not integers.
   346  func (x *Float) IsInt() bool {
   347  	if debugFloat {
   348  		x.validate()
   349  	}
   350  	// special cases
   351  	if x.form != finite {
   352  		return x.form == zero
   353  	}
   354  	// x.form == finite
   355  	if x.exp <= 0 {
   356  		return false
   357  	}
   358  	// x.exp > 0
   359  	return x.prec <= uint32(x.exp) || x.MinPrec() <= uint(x.exp) // not enough bits for fractional mantissa
   360  }
   361  
   362  // debugging support
   363  func (x *Float) validate() {
   364  	if !debugFloat {
   365  		// avoid performance bugs
   366  		panic("validate called but debugFloat is not set")
   367  	}
   368  	if x.form != finite {
   369  		return
   370  	}
   371  	m := len(x.mant)
   372  	if m == 0 {
   373  		panic("nonzero finite number with empty mantissa")
   374  	}
   375  	const msb = 1 << (_W - 1)
   376  	if x.mant[m-1]&msb == 0 {
   377  		panic(fmt.Sprintf("msb not set in last word %#x of %s", x.mant[m-1], x.Text('p', 0)))
   378  	}
   379  	if x.prec == 0 {
   380  		panic("zero precision finite number")
   381  	}
   382  }
   383  
   384  // round rounds z according to z.mode to z.prec bits and sets z.acc accordingly.
   385  // sbit must be 0 or 1 and summarizes any "sticky bit" information one might
   386  // have before calling round. z's mantissa must be normalized (with the msb set)
   387  // or empty.
   388  //
   389  // CAUTION: The rounding modes ToNegativeInf, ToPositiveInf are affected by the
   390  // sign of z. For correct rounding, the sign of z must be set correctly before
   391  // calling round.
   392  func (z *Float) round(sbit uint) {
   393  	if debugFloat {
   394  		z.validate()
   395  	}
   396  
   397  	z.acc = Exact
   398  	if z.form != finite {
   399  		// ±0 or ±Inf => nothing left to do
   400  		return
   401  	}
   402  	// z.form == finite && len(z.mant) > 0
   403  	// m > 0 implies z.prec > 0 (checked by validate)
   404  
   405  	m := uint32(len(z.mant)) // present mantissa length in words
   406  	bits := m * _W           // present mantissa bits; bits > 0
   407  	if bits <= z.prec {
   408  		// mantissa fits => nothing to do
   409  		return
   410  	}
   411  	// bits > z.prec
   412  
   413  	// Rounding is based on two bits: the rounding bit (rbit) and the
   414  	// sticky bit (sbit). The rbit is the bit immediately before the
   415  	// z.prec leading mantissa bits (the "0.5"). The sbit is set if any
   416  	// of the bits before the rbit are set (the "0.25", "0.125", etc.):
   417  	//
   418  	//   rbit  sbit  => "fractional part"
   419  	//
   420  	//   0     0        == 0
   421  	//   0     1        >  0  , < 0.5
   422  	//   1     0        == 0.5
   423  	//   1     1        >  0.5, < 1.0
   424  
   425  	// bits > z.prec: mantissa too large => round
   426  	r := uint(bits - z.prec - 1) // rounding bit position; r >= 0
   427  	rbit := z.mant.bit(r) & 1    // rounding bit; be safe and ensure it's a single bit
   428  	// The sticky bit is only needed for rounding ToNearestEven
   429  	// or when the rounding bit is zero. Avoid computation otherwise.
   430  	if sbit == 0 && (rbit == 0 || z.mode == ToNearestEven) {
   431  		sbit = z.mant.sticky(r)
   432  	}
   433  	sbit &= 1 // be safe and ensure it's a single bit
   434  
   435  	// cut off extra words
   436  	n := (z.prec + (_W - 1)) / _W // mantissa length in words for desired precision
   437  	if m > n {
   438  		copy(z.mant, z.mant[m-n:]) // move n last words to front
   439  		z.mant = z.mant[:n]
   440  	}
   441  
   442  	// determine number of trailing zero bits (ntz) and compute lsb mask of mantissa's least-significant word
   443  	ntz := n*_W - z.prec // 0 <= ntz < _W
   444  	lsb := Word(1) << ntz
   445  
   446  	// round if result is inexact
   447  	if rbit|sbit != 0 {
   448  		// Make rounding decision: The result mantissa is truncated ("rounded down")
   449  		// by default. Decide if we need to increment, or "round up", the (unsigned)
   450  		// mantissa.
   451  		inc := false
   452  		switch z.mode {
   453  		case ToNegativeInf:
   454  			inc = z.neg
   455  		case ToZero:
   456  			// nothing to do
   457  		case ToNearestEven:
   458  			inc = rbit != 0 && (sbit != 0 || z.mant[0]&lsb != 0)
   459  		case ToNearestAway:
   460  			inc = rbit != 0
   461  		case AwayFromZero:
   462  			inc = true
   463  		case ToPositiveInf:
   464  			inc = !z.neg
   465  		default:
   466  			panic("unreachable")
   467  		}
   468  
   469  		// A positive result (!z.neg) is Above the exact result if we increment,
   470  		// and it's Below if we truncate (Exact results require no rounding).
   471  		// For a negative result (z.neg) it is exactly the opposite.
   472  		z.acc = makeAcc(inc != z.neg)
   473  
   474  		if inc {
   475  			// add 1 to mantissa
   476  			if addVW(z.mant, z.mant, lsb) != 0 {
   477  				// mantissa overflow => adjust exponent
   478  				if z.exp >= MaxExp {
   479  					// exponent overflow
   480  					z.form = inf
   481  					return
   482  				}
   483  				z.exp++
   484  				// adjust mantissa: divide by 2 to compensate for exponent adjustment
   485  				shrVU(z.mant, z.mant, 1)
   486  				// set msb == carry == 1 from the mantissa overflow above
   487  				const msb = 1 << (_W - 1)
   488  				z.mant[n-1] |= msb
   489  			}
   490  		}
   491  	}
   492  
   493  	// zero out trailing bits in least-significant word
   494  	z.mant[0] &^= lsb - 1
   495  
   496  	if debugFloat {
   497  		z.validate()
   498  	}
   499  }
   500  
   501  func (z *Float) setBits64(neg bool, x uint64) *Float {
   502  	if z.prec == 0 {
   503  		z.prec = 64
   504  	}
   505  	z.acc = Exact
   506  	z.neg = neg
   507  	if x == 0 {
   508  		z.form = zero
   509  		return z
   510  	}
   511  	// x != 0
   512  	z.form = finite
   513  	s := bits.LeadingZeros64(x)
   514  	z.mant = z.mant.setUint64(x << uint(s))
   515  	z.exp = int32(64 - s) // always fits
   516  	if z.prec < 64 {
   517  		z.round(0)
   518  	}
   519  	return z
   520  }
   521  
   522  // SetUint64 sets z to the (possibly rounded) value of x and returns z.
   523  // If z's precision is 0, it is changed to 64 (and rounding will have
   524  // no effect).
   525  func (z *Float) SetUint64(x uint64) *Float {
   526  	return z.setBits64(false, x)
   527  }
   528  
   529  // SetInt64 sets z to the (possibly rounded) value of x and returns z.
   530  // If z's precision is 0, it is changed to 64 (and rounding will have
   531  // no effect).
   532  func (z *Float) SetInt64(x int64) *Float {
   533  	u := x
   534  	if u < 0 {
   535  		u = -u
   536  	}
   537  	// We cannot simply call z.SetUint64(uint64(u)) and change
   538  	// the sign afterwards because the sign affects rounding.
   539  	return z.setBits64(x < 0, uint64(u))
   540  }
   541  
   542  // SetFloat64 sets z to the (possibly rounded) value of x and returns z.
   543  // If z's precision is 0, it is changed to 53 (and rounding will have
   544  // no effect). SetFloat64 panics with ErrNaN if x is a NaN.
   545  func (z *Float) SetFloat64(x float64) *Float {
   546  	if z.prec == 0 {
   547  		z.prec = 53
   548  	}
   549  	if math.IsNaN(x) {
   550  		panic(ErrNaN{"Float.SetFloat64(NaN)"})
   551  	}
   552  	z.acc = Exact
   553  	z.neg = math.Signbit(x) // handle -0, -Inf correctly
   554  	if x == 0 {
   555  		z.form = zero
   556  		return z
   557  	}
   558  	if math.IsInf(x, 0) {
   559  		z.form = inf
   560  		return z
   561  	}
   562  	// normalized x != 0
   563  	z.form = finite
   564  	fmant, exp := math.Frexp(x) // get normalized mantissa
   565  	z.mant = z.mant.setUint64(1<<63 | math.Float64bits(fmant)<<11)
   566  	z.exp = int32(exp) // always fits
   567  	if z.prec < 53 {
   568  		z.round(0)
   569  	}
   570  	return z
   571  }
   572  
   573  // fnorm normalizes mantissa m by shifting it to the left
   574  // such that the msb of the most-significant word (msw) is 1.
   575  // It returns the shift amount. It assumes that len(m) != 0.
   576  func fnorm(m nat) int64 {
   577  	if debugFloat && (len(m) == 0 || m[len(m)-1] == 0) {
   578  		panic("msw of mantissa is 0")
   579  	}
   580  	s := nlz(m[len(m)-1])
   581  	if s > 0 {
   582  		c := shlVU(m, m, s)
   583  		if debugFloat && c != 0 {
   584  			panic("nlz or shlVU incorrect")
   585  		}
   586  	}
   587  	return int64(s)
   588  }
   589  
   590  // SetInt sets z to the (possibly rounded) value of x and returns z.
   591  // If z's precision is 0, it is changed to the larger of x.BitLen()
   592  // or 64 (and rounding will have no effect).
   593  func (z *Float) SetInt(x *Int) *Float {
   594  	// TODO(gri) can be more efficient if z.prec > 0
   595  	// but small compared to the size of x, or if there
   596  	// are many trailing 0's.
   597  	bits := uint32(x.BitLen())
   598  	if z.prec == 0 {
   599  		z.prec = umax32(bits, 64)
   600  	}
   601  	z.acc = Exact
   602  	z.neg = x.neg
   603  	if len(x.abs) == 0 {
   604  		z.form = zero
   605  		return z
   606  	}
   607  	// x != 0
   608  	z.mant = z.mant.set(x.abs)
   609  	fnorm(z.mant)
   610  	z.setExpAndRound(int64(bits), 0)
   611  	return z
   612  }
   613  
   614  // SetRat sets z to the (possibly rounded) value of x and returns z.
   615  // If z's precision is 0, it is changed to the largest of a.BitLen(),
   616  // b.BitLen(), or 64; with x = a/b.
   617  func (z *Float) SetRat(x *Rat) *Float {
   618  	if x.IsInt() {
   619  		return z.SetInt(x.Num())
   620  	}
   621  	var a, b Float
   622  	a.SetInt(x.Num())
   623  	b.SetInt(x.Denom())
   624  	if z.prec == 0 {
   625  		z.prec = umax32(a.prec, b.prec)
   626  	}
   627  	return z.Quo(&a, &b)
   628  }
   629  
   630  // SetInf sets z to the infinite Float -Inf if signbit is
   631  // set, or +Inf if signbit is not set, and returns z. The
   632  // precision of z is unchanged and the result is always
   633  // Exact.
   634  func (z *Float) SetInf(signbit bool) *Float {
   635  	z.acc = Exact
   636  	z.form = inf
   637  	z.neg = signbit
   638  	return z
   639  }
   640  
   641  // Set sets z to the (possibly rounded) value of x and returns z.
   642  // If z's precision is 0, it is changed to the precision of x
   643  // before setting z (and rounding will have no effect).
   644  // Rounding is performed according to z's precision and rounding
   645  // mode; and z's accuracy reports the result error relative to the
   646  // exact (not rounded) result.
   647  func (z *Float) Set(x *Float) *Float {
   648  	if debugFloat {
   649  		x.validate()
   650  	}
   651  	z.acc = Exact
   652  	if z != x {
   653  		z.form = x.form
   654  		z.neg = x.neg
   655  		if x.form == finite {
   656  			z.exp = x.exp
   657  			z.mant = z.mant.set(x.mant)
   658  		}
   659  		if z.prec == 0 {
   660  			z.prec = x.prec
   661  		} else if z.prec < x.prec {
   662  			z.round(0)
   663  		}
   664  	}
   665  	return z
   666  }
   667  
   668  // Copy sets z to x, with the same precision, rounding mode, and
   669  // accuracy as x, and returns z. x is not changed even if z and
   670  // x are the same.
   671  func (z *Float) Copy(x *Float) *Float {
   672  	if debugFloat {
   673  		x.validate()
   674  	}
   675  	if z != x {
   676  		z.prec = x.prec
   677  		z.mode = x.mode
   678  		z.acc = x.acc
   679  		z.form = x.form
   680  		z.neg = x.neg
   681  		if z.form == finite {
   682  			z.mant = z.mant.set(x.mant)
   683  			z.exp = x.exp
   684  		}
   685  	}
   686  	return z
   687  }
   688  
   689  // msb32 returns the 32 most significant bits of x.
   690  func msb32(x nat) uint32 {
   691  	i := len(x) - 1
   692  	if i < 0 {
   693  		return 0
   694  	}
   695  	if debugFloat && x[i]&(1<<(_W-1)) == 0 {
   696  		panic("x not normalized")
   697  	}
   698  	switch _W {
   699  	case 32:
   700  		return uint32(x[i])
   701  	case 64:
   702  		return uint32(x[i] >> 32)
   703  	}
   704  	panic("unreachable")
   705  }
   706  
   707  // msb64 returns the 64 most significant bits of x.
   708  func msb64(x nat) uint64 {
   709  	i := len(x) - 1
   710  	if i < 0 {
   711  		return 0
   712  	}
   713  	if debugFloat && x[i]&(1<<(_W-1)) == 0 {
   714  		panic("x not normalized")
   715  	}
   716  	switch _W {
   717  	case 32:
   718  		v := uint64(x[i]) << 32
   719  		if i > 0 {
   720  			v |= uint64(x[i-1])
   721  		}
   722  		return v
   723  	case 64:
   724  		return uint64(x[i])
   725  	}
   726  	panic("unreachable")
   727  }
   728  
   729  // Uint64 returns the unsigned integer resulting from truncating x
   730  // towards zero. If 0 <= x <= math.MaxUint64, the result is Exact
   731  // if x is an integer and Below otherwise.
   732  // The result is (0, Above) for x < 0, and (math.MaxUint64, Below)
   733  // for x > math.MaxUint64.
   734  func (x *Float) Uint64() (uint64, Accuracy) {
   735  	if debugFloat {
   736  		x.validate()
   737  	}
   738  
   739  	switch x.form {
   740  	case finite:
   741  		if x.neg {
   742  			return 0, Above
   743  		}
   744  		// 0 < x < +Inf
   745  		if x.exp <= 0 {
   746  			// 0 < x < 1
   747  			return 0, Below
   748  		}
   749  		// 1 <= x < Inf
   750  		if x.exp <= 64 {
   751  			// u = trunc(x) fits into a uint64
   752  			u := msb64(x.mant) >> (64 - uint32(x.exp))
   753  			if x.MinPrec() <= 64 {
   754  				return u, Exact
   755  			}
   756  			return u, Below // x truncated
   757  		}
   758  		// x too large
   759  		return math.MaxUint64, Below
   760  
   761  	case zero:
   762  		return 0, Exact
   763  
   764  	case inf:
   765  		if x.neg {
   766  			return 0, Above
   767  		}
   768  		return math.MaxUint64, Below
   769  	}
   770  
   771  	panic("unreachable")
   772  }
   773  
   774  // Int64 returns the integer resulting from truncating x towards zero.
   775  // If math.MinInt64 <= x <= math.MaxInt64, the result is Exact if x is
   776  // an integer, and Above (x < 0) or Below (x > 0) otherwise.
   777  // The result is (math.MinInt64, Above) for x < math.MinInt64,
   778  // and (math.MaxInt64, Below) for x > math.MaxInt64.
   779  func (x *Float) Int64() (int64, Accuracy) {
   780  	if debugFloat {
   781  		x.validate()
   782  	}
   783  
   784  	switch x.form {
   785  	case finite:
   786  		// 0 < |x| < +Inf
   787  		acc := makeAcc(x.neg)
   788  		if x.exp <= 0 {
   789  			// 0 < |x| < 1
   790  			return 0, acc
   791  		}
   792  		// x.exp > 0
   793  
   794  		// 1 <= |x| < +Inf
   795  		if x.exp <= 63 {
   796  			// i = trunc(x) fits into an int64 (excluding math.MinInt64)
   797  			i := int64(msb64(x.mant) >> (64 - uint32(x.exp)))
   798  			if x.neg {
   799  				i = -i
   800  			}
   801  			if x.MinPrec() <= uint(x.exp) {
   802  				return i, Exact
   803  			}
   804  			return i, acc // x truncated
   805  		}
   806  		if x.neg {
   807  			// check for special case x == math.MinInt64 (i.e., x == -(0.5 << 64))
   808  			if x.exp == 64 && x.MinPrec() == 1 {
   809  				acc = Exact
   810  			}
   811  			return math.MinInt64, acc
   812  		}
   813  		// x too large
   814  		return math.MaxInt64, Below
   815  
   816  	case zero:
   817  		return 0, Exact
   818  
   819  	case inf:
   820  		if x.neg {
   821  			return math.MinInt64, Above
   822  		}
   823  		return math.MaxInt64, Below
   824  	}
   825  
   826  	panic("unreachable")
   827  }
   828  
   829  // Float32 returns the float32 value nearest to x. If x is too small to be
   830  // represented by a float32 (|x| < math.SmallestNonzeroFloat32), the result
   831  // is (0, Below) or (-0, Above), respectively, depending on the sign of x.
   832  // If x is too large to be represented by a float32 (|x| > math.MaxFloat32),
   833  // the result is (+Inf, Above) or (-Inf, Below), depending on the sign of x.
   834  func (x *Float) Float32() (float32, Accuracy) {
   835  	if debugFloat {
   836  		x.validate()
   837  	}
   838  
   839  	switch x.form {
   840  	case finite:
   841  		// 0 < |x| < +Inf
   842  
   843  		const (
   844  			fbits = 32                //        float size
   845  			mbits = 23                //        mantissa size (excluding implicit msb)
   846  			ebits = fbits - mbits - 1 //     8  exponent size
   847  			bias  = 1<<(ebits-1) - 1  //   127  exponent bias
   848  			dmin  = 1 - bias - mbits  //  -149  smallest unbiased exponent (denormal)
   849  			emin  = 1 - bias          //  -126  smallest unbiased exponent (normal)
   850  			emax  = bias              //   127  largest unbiased exponent (normal)
   851  		)
   852  
   853  		// Float mantissa m is 0.5 <= m < 1.0; compute exponent e for float32 mantissa.
   854  		e := x.exp - 1 // exponent for normal mantissa m with 1.0 <= m < 2.0
   855  
   856  		// Compute precision p for float32 mantissa.
   857  		// If the exponent is too small, we have a denormal number before
   858  		// rounding and fewer than p mantissa bits of precision available
   859  		// (the exponent remains fixed but the mantissa gets shifted right).
   860  		p := mbits + 1 // precision of normal float
   861  		if e < emin {
   862  			// recompute precision
   863  			p = mbits + 1 - emin + int(e)
   864  			// If p == 0, the mantissa of x is shifted so much to the right
   865  			// that its msb falls immediately to the right of the float32
   866  			// mantissa space. In other words, if the smallest denormal is
   867  			// considered "1.0", for p == 0, the mantissa value m is >= 0.5.
   868  			// If m > 0.5, it is rounded up to 1.0; i.e., the smallest denormal.
   869  			// If m == 0.5, it is rounded down to even, i.e., 0.0.
   870  			// If p < 0, the mantissa value m is <= "0.25" which is never rounded up.
   871  			if p < 0 /* m <= 0.25 */ || p == 0 && x.mant.sticky(uint(len(x.mant))*_W-1) == 0 /* m == 0.5 */ {
   872  				// underflow to ±0
   873  				if x.neg {
   874  					var z float32
   875  					return -z, Above
   876  				}
   877  				return 0.0, Below
   878  			}
   879  			// otherwise, round up
   880  			// We handle p == 0 explicitly because it's easy and because
   881  			// Float.round doesn't support rounding to 0 bits of precision.
   882  			if p == 0 {
   883  				if x.neg {
   884  					return -math.SmallestNonzeroFloat32, Below
   885  				}
   886  				return math.SmallestNonzeroFloat32, Above
   887  			}
   888  		}
   889  		// p > 0
   890  
   891  		// round
   892  		var r Float
   893  		r.prec = uint32(p)
   894  		r.Set(x)
   895  		e = r.exp - 1
   896  
   897  		// Rounding may have caused r to overflow to ±Inf
   898  		// (rounding never causes underflows to 0).
   899  		// If the exponent is too large, also overflow to ±Inf.
   900  		if r.form == inf || e > emax {
   901  			// overflow
   902  			if x.neg {
   903  				return float32(math.Inf(-1)), Below
   904  			}
   905  			return float32(math.Inf(+1)), Above
   906  		}
   907  		// e <= emax
   908  
   909  		// Determine sign, biased exponent, and mantissa.
   910  		var sign, bexp, mant uint32
   911  		if x.neg {
   912  			sign = 1 << (fbits - 1)
   913  		}
   914  
   915  		// Rounding may have caused a denormal number to
   916  		// become normal. Check again.
   917  		if e < emin {
   918  			// denormal number: recompute precision
   919  			// Since rounding may have at best increased precision
   920  			// and we have eliminated p <= 0 early, we know p > 0.
   921  			// bexp == 0 for denormals
   922  			p = mbits + 1 - emin + int(e)
   923  			mant = msb32(r.mant) >> uint(fbits-p)
   924  		} else {
   925  			// normal number: emin <= e <= emax
   926  			bexp = uint32(e+bias) << mbits
   927  			mant = msb32(r.mant) >> ebits & (1<<mbits - 1) // cut off msb (implicit 1 bit)
   928  		}
   929  
   930  		return math.Float32frombits(sign | bexp | mant), r.acc
   931  
   932  	case zero:
   933  		if x.neg {
   934  			var z float32
   935  			return -z, Exact
   936  		}
   937  		return 0.0, Exact
   938  
   939  	case inf:
   940  		if x.neg {
   941  			return float32(math.Inf(-1)), Exact
   942  		}
   943  		return float32(math.Inf(+1)), Exact
   944  	}
   945  
   946  	panic("unreachable")
   947  }
   948  
   949  // Float64 returns the float64 value nearest to x. If x is too small to be
   950  // represented by a float64 (|x| < math.SmallestNonzeroFloat64), the result
   951  // is (0, Below) or (-0, Above), respectively, depending on the sign of x.
   952  // If x is too large to be represented by a float64 (|x| > math.MaxFloat64),
   953  // the result is (+Inf, Above) or (-Inf, Below), depending on the sign of x.
   954  func (x *Float) Float64() (float64, Accuracy) {
   955  	if debugFloat {
   956  		x.validate()
   957  	}
   958  
   959  	switch x.form {
   960  	case finite:
   961  		// 0 < |x| < +Inf
   962  
   963  		const (
   964  			fbits = 64                //        float size
   965  			mbits = 52                //        mantissa size (excluding implicit msb)
   966  			ebits = fbits - mbits - 1 //    11  exponent size
   967  			bias  = 1<<(ebits-1) - 1  //  1023  exponent bias
   968  			dmin  = 1 - bias - mbits  // -1074  smallest unbiased exponent (denormal)
   969  			emin  = 1 - bias          // -1022  smallest unbiased exponent (normal)
   970  			emax  = bias              //  1023  largest unbiased exponent (normal)
   971  		)
   972  
   973  		// Float mantissa m is 0.5 <= m < 1.0; compute exponent e for float64 mantissa.
   974  		e := x.exp - 1 // exponent for normal mantissa m with 1.0 <= m < 2.0
   975  
   976  		// Compute precision p for float64 mantissa.
   977  		// If the exponent is too small, we have a denormal number before
   978  		// rounding and fewer than p mantissa bits of precision available
   979  		// (the exponent remains fixed but the mantissa gets shifted right).
   980  		p := mbits + 1 // precision of normal float
   981  		if e < emin {
   982  			// recompute precision
   983  			p = mbits + 1 - emin + int(e)
   984  			// If p == 0, the mantissa of x is shifted so much to the right
   985  			// that its msb falls immediately to the right of the float64
   986  			// mantissa space. In other words, if the smallest denormal is
   987  			// considered "1.0", for p == 0, the mantissa value m is >= 0.5.
   988  			// If m > 0.5, it is rounded up to 1.0; i.e., the smallest denormal.
   989  			// If m == 0.5, it is rounded down to even, i.e., 0.0.
   990  			// If p < 0, the mantissa value m is <= "0.25" which is never rounded up.
   991  			if p < 0 /* m <= 0.25 */ || p == 0 && x.mant.sticky(uint(len(x.mant))*_W-1) == 0 /* m == 0.5 */ {
   992  				// underflow to ±0
   993  				if x.neg {
   994  					var z float64
   995  					return -z, Above
   996  				}
   997  				return 0.0, Below
   998  			}
   999  			// otherwise, round up
  1000  			// We handle p == 0 explicitly because it's easy and because
  1001  			// Float.round doesn't support rounding to 0 bits of precision.
  1002  			if p == 0 {
  1003  				if x.neg {
  1004  					return -math.SmallestNonzeroFloat64, Below
  1005  				}
  1006  				return math.SmallestNonzeroFloat64, Above
  1007  			}
  1008  		}
  1009  		// p > 0
  1010  
  1011  		// round
  1012  		var r Float
  1013  		r.prec = uint32(p)
  1014  		r.Set(x)
  1015  		e = r.exp - 1
  1016  
  1017  		// Rounding may have caused r to overflow to ±Inf
  1018  		// (rounding never causes underflows to 0).
  1019  		// If the exponent is too large, also overflow to ±Inf.
  1020  		if r.form == inf || e > emax {
  1021  			// overflow
  1022  			if x.neg {
  1023  				return math.Inf(-1), Below
  1024  			}
  1025  			return math.Inf(+1), Above
  1026  		}
  1027  		// e <= emax
  1028  
  1029  		// Determine sign, biased exponent, and mantissa.
  1030  		var sign, bexp, mant uint64
  1031  		if x.neg {
  1032  			sign = 1 << (fbits - 1)
  1033  		}
  1034  
  1035  		// Rounding may have caused a denormal number to
  1036  		// become normal. Check again.
  1037  		if e < emin {
  1038  			// denormal number: recompute precision
  1039  			// Since rounding may have at best increased precision
  1040  			// and we have eliminated p <= 0 early, we know p > 0.
  1041  			// bexp == 0 for denormals
  1042  			p = mbits + 1 - emin + int(e)
  1043  			mant = msb64(r.mant) >> uint(fbits-p)
  1044  		} else {
  1045  			// normal number: emin <= e <= emax
  1046  			bexp = uint64(e+bias) << mbits
  1047  			mant = msb64(r.mant) >> ebits & (1<<mbits - 1) // cut off msb (implicit 1 bit)
  1048  		}
  1049  
  1050  		return math.Float64frombits(sign | bexp | mant), r.acc
  1051  
  1052  	case zero:
  1053  		if x.neg {
  1054  			var z float64
  1055  			return -z, Exact
  1056  		}
  1057  		return 0.0, Exact
  1058  
  1059  	case inf:
  1060  		if x.neg {
  1061  			return math.Inf(-1), Exact
  1062  		}
  1063  		return math.Inf(+1), Exact
  1064  	}
  1065  
  1066  	panic("unreachable")
  1067  }
  1068  
  1069  // Int returns the result of truncating x towards zero;
  1070  // or nil if x is an infinity.
  1071  // The result is Exact if x.IsInt(); otherwise it is Below
  1072  // for x > 0, and Above for x < 0.
  1073  // If a non-nil *Int argument z is provided, Int stores
  1074  // the result in z instead of allocating a new Int.
  1075  func (x *Float) Int(z *Int) (*Int, Accuracy) {
  1076  	if debugFloat {
  1077  		x.validate()
  1078  	}
  1079  
  1080  	if z == nil && x.form <= finite {
  1081  		z = new(Int)
  1082  	}
  1083  
  1084  	switch x.form {
  1085  	case finite:
  1086  		// 0 < |x| < +Inf
  1087  		acc := makeAcc(x.neg)
  1088  		if x.exp <= 0 {
  1089  			// 0 < |x| < 1
  1090  			return z.SetInt64(0), acc
  1091  		}
  1092  		// x.exp > 0
  1093  
  1094  		// 1 <= |x| < +Inf
  1095  		// determine minimum required precision for x
  1096  		allBits := uint(len(x.mant)) * _W
  1097  		exp := uint(x.exp)
  1098  		if x.MinPrec() <= exp {
  1099  			acc = Exact
  1100  		}
  1101  		// shift mantissa as needed
  1102  		if z == nil {
  1103  			z = new(Int)
  1104  		}
  1105  		z.neg = x.neg
  1106  		switch {
  1107  		case exp > allBits:
  1108  			z.abs = z.abs.shl(x.mant, exp-allBits)
  1109  		default:
  1110  			z.abs = z.abs.set(x.mant)
  1111  		case exp < allBits:
  1112  			z.abs = z.abs.shr(x.mant, allBits-exp)
  1113  		}
  1114  		return z, acc
  1115  
  1116  	case zero:
  1117  		return z.SetInt64(0), Exact
  1118  
  1119  	case inf:
  1120  		return nil, makeAcc(x.neg)
  1121  	}
  1122  
  1123  	panic("unreachable")
  1124  }
  1125  
  1126  // Rat returns the rational number corresponding to x;
  1127  // or nil if x is an infinity.
  1128  // The result is Exact if x is not an Inf.
  1129  // If a non-nil *Rat argument z is provided, Rat stores
  1130  // the result in z instead of allocating a new Rat.
  1131  func (x *Float) Rat(z *Rat) (*Rat, Accuracy) {
  1132  	if debugFloat {
  1133  		x.validate()
  1134  	}
  1135  
  1136  	if z == nil && x.form <= finite {
  1137  		z = new(Rat)
  1138  	}
  1139  
  1140  	switch x.form {
  1141  	case finite:
  1142  		// 0 < |x| < +Inf
  1143  		allBits := int32(len(x.mant)) * _W
  1144  		// build up numerator and denominator
  1145  		z.a.neg = x.neg
  1146  		switch {
  1147  		case x.exp > allBits:
  1148  			z.a.abs = z.a.abs.shl(x.mant, uint(x.exp-allBits))
  1149  			z.b.abs = z.b.abs[:0] // == 1 (see Rat)
  1150  			// z already in normal form
  1151  		default:
  1152  			z.a.abs = z.a.abs.set(x.mant)
  1153  			z.b.abs = z.b.abs[:0] // == 1 (see Rat)
  1154  			// z already in normal form
  1155  		case x.exp < allBits:
  1156  			z.a.abs = z.a.abs.set(x.mant)
  1157  			t := z.b.abs.setUint64(1)
  1158  			z.b.abs = t.shl(t, uint(allBits-x.exp))
  1159  			z.norm()
  1160  		}
  1161  		return z, Exact
  1162  
  1163  	case zero:
  1164  		return z.SetInt64(0), Exact
  1165  
  1166  	case inf:
  1167  		return nil, makeAcc(x.neg)
  1168  	}
  1169  
  1170  	panic("unreachable")
  1171  }
  1172  
  1173  // Abs sets z to the (possibly rounded) value |x| (the absolute value of x)
  1174  // and returns z.
  1175  func (z *Float) Abs(x *Float) *Float {
  1176  	z.Set(x)
  1177  	z.neg = false
  1178  	return z
  1179  }
  1180  
  1181  // Neg sets z to the (possibly rounded) value of x with its sign negated,
  1182  // and returns z.
  1183  func (z *Float) Neg(x *Float) *Float {
  1184  	z.Set(x)
  1185  	z.neg = !z.neg
  1186  	return z
  1187  }
  1188  
  1189  func validateBinaryOperands(x, y *Float) {
  1190  	if !debugFloat {
  1191  		// avoid performance bugs
  1192  		panic("validateBinaryOperands called but debugFloat is not set")
  1193  	}
  1194  	if len(x.mant) == 0 {
  1195  		panic("empty mantissa for x")
  1196  	}
  1197  	if len(y.mant) == 0 {
  1198  		panic("empty mantissa for y")
  1199  	}
  1200  }
  1201  
  1202  // z = x + y, ignoring signs of x and y for the addition
  1203  // but using the sign of z for rounding the result.
  1204  // x and y must have a non-empty mantissa and valid exponent.
  1205  func (z *Float) uadd(x, y *Float) {
  1206  	// Note: This implementation requires 2 shifts most of the
  1207  	// time. It is also inefficient if exponents or precisions
  1208  	// differ by wide margins. The following article describes
  1209  	// an efficient (but much more complicated) implementation
  1210  	// compatible with the internal representation used here:
  1211  	//
  1212  	// Vincent Lefèvre: "The Generic Multiple-Precision Floating-
  1213  	// Point Addition With Exact Rounding (as in the MPFR Library)"
  1214  	// http://www.vinc17.net/research/papers/rnc6.pdf
  1215  
  1216  	if debugFloat {
  1217  		validateBinaryOperands(x, y)
  1218  	}
  1219  
  1220  	// compute exponents ex, ey for mantissa with "binary point"
  1221  	// on the right (mantissa.0) - use int64 to avoid overflow
  1222  	ex := int64(x.exp) - int64(len(x.mant))*_W
  1223  	ey := int64(y.exp) - int64(len(y.mant))*_W
  1224  
  1225  	al := alias(z.mant, x.mant) || alias(z.mant, y.mant)
  1226  
  1227  	// TODO(gri) having a combined add-and-shift primitive
  1228  	//           could make this code significantly faster
  1229  	switch {
  1230  	case ex < ey:
  1231  		if al {
  1232  			t := nat(nil).shl(y.mant, uint(ey-ex))
  1233  			z.mant = z.mant.add(x.mant, t)
  1234  		} else {
  1235  			z.mant = z.mant.shl(y.mant, uint(ey-ex))
  1236  			z.mant = z.mant.add(x.mant, z.mant)
  1237  		}
  1238  	default:
  1239  		// ex == ey, no shift needed
  1240  		z.mant = z.mant.add(x.mant, y.mant)
  1241  	case ex > ey:
  1242  		if al {
  1243  			t := nat(nil).shl(x.mant, uint(ex-ey))
  1244  			z.mant = z.mant.add(t, y.mant)
  1245  		} else {
  1246  			z.mant = z.mant.shl(x.mant, uint(ex-ey))
  1247  			z.mant = z.mant.add(z.mant, y.mant)
  1248  		}
  1249  		ex = ey
  1250  	}
  1251  	// len(z.mant) > 0
  1252  
  1253  	z.setExpAndRound(ex+int64(len(z.mant))*_W-fnorm(z.mant), 0)
  1254  }
  1255  
  1256  // z = x - y for |x| > |y|, ignoring signs of x and y for the subtraction
  1257  // but using the sign of z for rounding the result.
  1258  // x and y must have a non-empty mantissa and valid exponent.
  1259  func (z *Float) usub(x, y *Float) {
  1260  	// This code is symmetric to uadd.
  1261  	// We have not factored the common code out because
  1262  	// eventually uadd (and usub) should be optimized
  1263  	// by special-casing, and the code will diverge.
  1264  
  1265  	if debugFloat {
  1266  		validateBinaryOperands(x, y)
  1267  	}
  1268  
  1269  	ex := int64(x.exp) - int64(len(x.mant))*_W
  1270  	ey := int64(y.exp) - int64(len(y.mant))*_W
  1271  
  1272  	al := alias(z.mant, x.mant) || alias(z.mant, y.mant)
  1273  
  1274  	switch {
  1275  	case ex < ey:
  1276  		if al {
  1277  			t := nat(nil).shl(y.mant, uint(ey-ex))
  1278  			z.mant = t.sub(x.mant, t)
  1279  		} else {
  1280  			z.mant = z.mant.shl(y.mant, uint(ey-ex))
  1281  			z.mant = z.mant.sub(x.mant, z.mant)
  1282  		}
  1283  	default:
  1284  		// ex == ey, no shift needed
  1285  		z.mant = z.mant.sub(x.mant, y.mant)
  1286  	case ex > ey:
  1287  		if al {
  1288  			t := nat(nil).shl(x.mant, uint(ex-ey))
  1289  			z.mant = t.sub(t, y.mant)
  1290  		} else {
  1291  			z.mant = z.mant.shl(x.mant, uint(ex-ey))
  1292  			z.mant = z.mant.sub(z.mant, y.mant)
  1293  		}
  1294  		ex = ey
  1295  	}
  1296  
  1297  	// operands may have canceled each other out
  1298  	if len(z.mant) == 0 {
  1299  		z.acc = Exact
  1300  		z.form = zero
  1301  		z.neg = false
  1302  		return
  1303  	}
  1304  	// len(z.mant) > 0
  1305  
  1306  	z.setExpAndRound(ex+int64(len(z.mant))*_W-fnorm(z.mant), 0)
  1307  }
  1308  
  1309  // z = x * y, ignoring signs of x and y for the multiplication
  1310  // but using the sign of z for rounding the result.
  1311  // x and y must have a non-empty mantissa and valid exponent.
  1312  func (z *Float) umul(x, y *Float) {
  1313  	if debugFloat {
  1314  		validateBinaryOperands(x, y)
  1315  	}
  1316  
  1317  	// Note: This is doing too much work if the precision
  1318  	// of z is less than the sum of the precisions of x
  1319  	// and y which is often the case (e.g., if all floats
  1320  	// have the same precision).
  1321  	// TODO(gri) Optimize this for the common case.
  1322  
  1323  	e := int64(x.exp) + int64(y.exp)
  1324  	if x == y {
  1325  		z.mant = z.mant.sqr(x.mant)
  1326  	} else {
  1327  		z.mant = z.mant.mul(x.mant, y.mant)
  1328  	}
  1329  	z.setExpAndRound(e-fnorm(z.mant), 0)
  1330  }
  1331  
  1332  // z = x / y, ignoring signs of x and y for the division
  1333  // but using the sign of z for rounding the result.
  1334  // x and y must have a non-empty mantissa and valid exponent.
  1335  func (z *Float) uquo(x, y *Float) {
  1336  	if debugFloat {
  1337  		validateBinaryOperands(x, y)
  1338  	}
  1339  
  1340  	// mantissa length in words for desired result precision + 1
  1341  	// (at least one extra bit so we get the rounding bit after
  1342  	// the division)
  1343  	n := int(z.prec/_W) + 1
  1344  
  1345  	// compute adjusted x.mant such that we get enough result precision
  1346  	xadj := x.mant
  1347  	if d := n - len(x.mant) + len(y.mant); d > 0 {
  1348  		// d extra words needed => add d "0 digits" to x
  1349  		xadj = make(nat, len(x.mant)+d)
  1350  		copy(xadj[d:], x.mant)
  1351  	}
  1352  	// TODO(gri): If we have too many digits (d < 0), we should be able
  1353  	// to shorten x for faster division. But we must be extra careful
  1354  	// with rounding in that case.
  1355  
  1356  	// Compute d before division since there may be aliasing of x.mant
  1357  	// (via xadj) or y.mant with z.mant.
  1358  	d := len(xadj) - len(y.mant)
  1359  
  1360  	// divide
  1361  	var r nat
  1362  	z.mant, r = z.mant.div(nil, xadj, y.mant)
  1363  	e := int64(x.exp) - int64(y.exp) - int64(d-len(z.mant))*_W
  1364  
  1365  	// The result is long enough to include (at least) the rounding bit.
  1366  	// If there's a non-zero remainder, the corresponding fractional part
  1367  	// (if it were computed), would have a non-zero sticky bit (if it were
  1368  	// zero, it couldn't have a non-zero remainder).
  1369  	var sbit uint
  1370  	if len(r) > 0 {
  1371  		sbit = 1
  1372  	}
  1373  
  1374  	z.setExpAndRound(e-fnorm(z.mant), sbit)
  1375  }
  1376  
  1377  // ucmp returns -1, 0, or +1, depending on whether
  1378  // |x| < |y|, |x| == |y|, or |x| > |y|.
  1379  // x and y must have a non-empty mantissa and valid exponent.
  1380  func (x *Float) ucmp(y *Float) int {
  1381  	if debugFloat {
  1382  		validateBinaryOperands(x, y)
  1383  	}
  1384  
  1385  	switch {
  1386  	case x.exp < y.exp:
  1387  		return -1
  1388  	case x.exp > y.exp:
  1389  		return +1
  1390  	}
  1391  	// x.exp == y.exp
  1392  
  1393  	// compare mantissas
  1394  	i := len(x.mant)
  1395  	j := len(y.mant)
  1396  	for i > 0 || j > 0 {
  1397  		var xm, ym Word
  1398  		if i > 0 {
  1399  			i--
  1400  			xm = x.mant[i]
  1401  		}
  1402  		if j > 0 {
  1403  			j--
  1404  			ym = y.mant[j]
  1405  		}
  1406  		switch {
  1407  		case xm < ym:
  1408  			return -1
  1409  		case xm > ym:
  1410  			return +1
  1411  		}
  1412  	}
  1413  
  1414  	return 0
  1415  }
  1416  
  1417  // Handling of sign bit as defined by IEEE 754-2008, section 6.3:
  1418  //
  1419  // When neither the inputs nor result are NaN, the sign of a product or
  1420  // quotient is the exclusive OR of the operands’ signs; the sign of a sum,
  1421  // or of a difference x−y regarded as a sum x+(−y), differs from at most
  1422  // one of the addends’ signs; and the sign of the result of conversions,
  1423  // the quantize operation, the roundToIntegral operations, and the
  1424  // roundToIntegralExact (see 5.3.1) is the sign of the first or only operand.
  1425  // These rules shall apply even when operands or results are zero or infinite.
  1426  //
  1427  // When the sum of two operands with opposite signs (or the difference of
  1428  // two operands with like signs) is exactly zero, the sign of that sum (or
  1429  // difference) shall be +0 in all rounding-direction attributes except
  1430  // roundTowardNegative; under that attribute, the sign of an exact zero
  1431  // sum (or difference) shall be −0. However, x+x = x−(−x) retains the same
  1432  // sign as x even when x is zero.
  1433  //
  1434  // See also: https://play.golang.org/p/RtH3UCt5IH
  1435  
  1436  // Add sets z to the rounded sum x+y and returns z. If z's precision is 0,
  1437  // it is changed to the larger of x's or y's precision before the operation.
  1438  // Rounding is performed according to z's precision and rounding mode; and
  1439  // z's accuracy reports the result error relative to the exact (not rounded)
  1440  // result. Add panics with ErrNaN if x and y are infinities with opposite
  1441  // signs. The value of z is undefined in that case.
  1442  func (z *Float) Add(x, y *Float) *Float {
  1443  	if debugFloat {
  1444  		x.validate()
  1445  		y.validate()
  1446  	}
  1447  
  1448  	if z.prec == 0 {
  1449  		z.prec = umax32(x.prec, y.prec)
  1450  	}
  1451  
  1452  	if x.form == finite && y.form == finite {
  1453  		// x + y (common case)
  1454  
  1455  		// Below we set z.neg = x.neg, and when z aliases y this will
  1456  		// change the y operand's sign. This is fine, because if an
  1457  		// operand aliases the receiver it'll be overwritten, but we still
  1458  		// want the original x.neg and y.neg values when we evaluate
  1459  		// x.neg != y.neg, so we need to save y.neg before setting z.neg.
  1460  		yneg := y.neg
  1461  
  1462  		z.neg = x.neg
  1463  		if x.neg == yneg {
  1464  			// x + y == x + y
  1465  			// (-x) + (-y) == -(x + y)
  1466  			z.uadd(x, y)
  1467  		} else {
  1468  			// x + (-y) == x - y == -(y - x)
  1469  			// (-x) + y == y - x == -(x - y)
  1470  			if x.ucmp(y) > 0 {
  1471  				z.usub(x, y)
  1472  			} else {
  1473  				z.neg = !z.neg
  1474  				z.usub(y, x)
  1475  			}
  1476  		}
  1477  		if z.form == zero && z.mode == ToNegativeInf && z.acc == Exact {
  1478  			z.neg = true
  1479  		}
  1480  		return z
  1481  	}
  1482  
  1483  	if x.form == inf && y.form == inf && x.neg != y.neg {
  1484  		// +Inf + -Inf
  1485  		// -Inf + +Inf
  1486  		// value of z is undefined but make sure it's valid
  1487  		z.acc = Exact
  1488  		z.form = zero
  1489  		z.neg = false
  1490  		panic(ErrNaN{"addition of infinities with opposite signs"})
  1491  	}
  1492  
  1493  	if x.form == zero && y.form == zero {
  1494  		// ±0 + ±0
  1495  		z.acc = Exact
  1496  		z.form = zero
  1497  		z.neg = x.neg && y.neg // -0 + -0 == -0
  1498  		return z
  1499  	}
  1500  
  1501  	if x.form == inf || y.form == zero {
  1502  		// ±Inf + y
  1503  		// x + ±0
  1504  		return z.Set(x)
  1505  	}
  1506  
  1507  	// ±0 + y
  1508  	// x + ±Inf
  1509  	return z.Set(y)
  1510  }
  1511  
  1512  // Sub sets z to the rounded difference x-y and returns z.
  1513  // Precision, rounding, and accuracy reporting are as for Add.
  1514  // Sub panics with ErrNaN if x and y are infinities with equal
  1515  // signs. The value of z is undefined in that case.
  1516  func (z *Float) Sub(x, y *Float) *Float {
  1517  	if debugFloat {
  1518  		x.validate()
  1519  		y.validate()
  1520  	}
  1521  
  1522  	if z.prec == 0 {
  1523  		z.prec = umax32(x.prec, y.prec)
  1524  	}
  1525  
  1526  	if x.form == finite && y.form == finite {
  1527  		// x - y (common case)
  1528  		yneg := y.neg
  1529  		z.neg = x.neg
  1530  		if x.neg != yneg {
  1531  			// x - (-y) == x + y
  1532  			// (-x) - y == -(x + y)
  1533  			z.uadd(x, y)
  1534  		} else {
  1535  			// x - y == x - y == -(y - x)
  1536  			// (-x) - (-y) == y - x == -(x - y)
  1537  			if x.ucmp(y) > 0 {
  1538  				z.usub(x, y)
  1539  			} else {
  1540  				z.neg = !z.neg
  1541  				z.usub(y, x)
  1542  			}
  1543  		}
  1544  		if z.form == zero && z.mode == ToNegativeInf && z.acc == Exact {
  1545  			z.neg = true
  1546  		}
  1547  		return z
  1548  	}
  1549  
  1550  	if x.form == inf && y.form == inf && x.neg == y.neg {
  1551  		// +Inf - +Inf
  1552  		// -Inf - -Inf
  1553  		// value of z is undefined but make sure it's valid
  1554  		z.acc = Exact
  1555  		z.form = zero
  1556  		z.neg = false
  1557  		panic(ErrNaN{"subtraction of infinities with equal signs"})
  1558  	}
  1559  
  1560  	if x.form == zero && y.form == zero {
  1561  		// ±0 - ±0
  1562  		z.acc = Exact
  1563  		z.form = zero
  1564  		z.neg = x.neg && !y.neg // -0 - +0 == -0
  1565  		return z
  1566  	}
  1567  
  1568  	if x.form == inf || y.form == zero {
  1569  		// ±Inf - y
  1570  		// x - ±0
  1571  		return z.Set(x)
  1572  	}
  1573  
  1574  	// ±0 - y
  1575  	// x - ±Inf
  1576  	return z.Neg(y)
  1577  }
  1578  
  1579  // Mul sets z to the rounded product x*y and returns z.
  1580  // Precision, rounding, and accuracy reporting are as for Add.
  1581  // Mul panics with ErrNaN if one operand is zero and the other
  1582  // operand an infinity. The value of z is undefined in that case.
  1583  func (z *Float) Mul(x, y *Float) *Float {
  1584  	if debugFloat {
  1585  		x.validate()
  1586  		y.validate()
  1587  	}
  1588  
  1589  	if z.prec == 0 {
  1590  		z.prec = umax32(x.prec, y.prec)
  1591  	}
  1592  
  1593  	z.neg = x.neg != y.neg
  1594  
  1595  	if x.form == finite && y.form == finite {
  1596  		// x * y (common case)
  1597  		z.umul(x, y)
  1598  		return z
  1599  	}
  1600  
  1601  	z.acc = Exact
  1602  	if x.form == zero && y.form == inf || x.form == inf && y.form == zero {
  1603  		// ±0 * ±Inf
  1604  		// ±Inf * ±0
  1605  		// value of z is undefined but make sure it's valid
  1606  		z.form = zero
  1607  		z.neg = false
  1608  		panic(ErrNaN{"multiplication of zero with infinity"})
  1609  	}
  1610  
  1611  	if x.form == inf || y.form == inf {
  1612  		// ±Inf * y
  1613  		// x * ±Inf
  1614  		z.form = inf
  1615  		return z
  1616  	}
  1617  
  1618  	// ±0 * y
  1619  	// x * ±0
  1620  	z.form = zero
  1621  	return z
  1622  }
  1623  
  1624  // Quo sets z to the rounded quotient x/y and returns z.
  1625  // Precision, rounding, and accuracy reporting are as for Add.
  1626  // Quo panics with ErrNaN if both operands are zero or infinities.
  1627  // The value of z is undefined in that case.
  1628  func (z *Float) Quo(x, y *Float) *Float {
  1629  	if debugFloat {
  1630  		x.validate()
  1631  		y.validate()
  1632  	}
  1633  
  1634  	if z.prec == 0 {
  1635  		z.prec = umax32(x.prec, y.prec)
  1636  	}
  1637  
  1638  	z.neg = x.neg != y.neg
  1639  
  1640  	if x.form == finite && y.form == finite {
  1641  		// x / y (common case)
  1642  		z.uquo(x, y)
  1643  		return z
  1644  	}
  1645  
  1646  	z.acc = Exact
  1647  	if x.form == zero && y.form == zero || x.form == inf && y.form == inf {
  1648  		// ±0 / ±0
  1649  		// ±Inf / ±Inf
  1650  		// value of z is undefined but make sure it's valid
  1651  		z.form = zero
  1652  		z.neg = false
  1653  		panic(ErrNaN{"division of zero by zero or infinity by infinity"})
  1654  	}
  1655  
  1656  	if x.form == zero || y.form == inf {
  1657  		// ±0 / y
  1658  		// x / ±Inf
  1659  		z.form = zero
  1660  		return z
  1661  	}
  1662  
  1663  	// x / ±0
  1664  	// ±Inf / y
  1665  	z.form = inf
  1666  	return z
  1667  }
  1668  
  1669  // Cmp compares x and y and returns:
  1670  //
  1671  //	-1 if x <  y
  1672  //	 0 if x == y (incl. -0 == 0, -Inf == -Inf, and +Inf == +Inf)
  1673  //	+1 if x >  y
  1674  func (x *Float) Cmp(y *Float) int {
  1675  	if debugFloat {
  1676  		x.validate()
  1677  		y.validate()
  1678  	}
  1679  
  1680  	mx := x.ord()
  1681  	my := y.ord()
  1682  	switch {
  1683  	case mx < my:
  1684  		return -1
  1685  	case mx > my:
  1686  		return +1
  1687  	}
  1688  	// mx == my
  1689  
  1690  	// only if |mx| == 1 we have to compare the mantissae
  1691  	switch mx {
  1692  	case -1:
  1693  		return y.ucmp(x)
  1694  	case +1:
  1695  		return x.ucmp(y)
  1696  	}
  1697  
  1698  	return 0
  1699  }
  1700  
  1701  // ord classifies x and returns:
  1702  //
  1703  //	-2 if -Inf == x
  1704  //	-1 if -Inf < x < 0
  1705  //	 0 if x == 0 (signed or unsigned)
  1706  //	+1 if 0 < x < +Inf
  1707  //	+2 if x == +Inf
  1708  func (x *Float) ord() int {
  1709  	var m int
  1710  	switch x.form {
  1711  	case finite:
  1712  		m = 1
  1713  	case zero:
  1714  		return 0
  1715  	case inf:
  1716  		m = 2
  1717  	}
  1718  	if x.neg {
  1719  		m = -m
  1720  	}
  1721  	return m
  1722  }
  1723  
  1724  func umax32(x, y uint32) uint32 {
  1725  	if x > y {
  1726  		return x
  1727  	}
  1728  	return y
  1729  }
  1730  

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