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Source file src/math/big/float.go

Documentation: math/big

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