Black Lives Matter. Support the Equal Justice Initiative.

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  
   326  	if z.form == finite {
   327  		// 0 < |mant| < +Inf
   328  		z.setExpAndRound(int64(z.exp)+int64(exp), 0)
   329  	}
   330  	return z
   331  }
   332  
   333  // Signbit reports whether x is negative or negative zero.
   334  func (x *Float) Signbit() bool {
   335  	return x.neg
   336  }
   337  
   338  // IsInf reports whether x is +Inf or -Inf.
   339  func (x *Float) IsInf() bool {
   340  	return x.form == inf
   341  }
   342  
   343  // IsInt reports whether x is an integer.
   344  // ±Inf values are not integers.
   345  func (x *Float) IsInt() bool {
   346  	if debugFloat {
   347  		x.validate()
   348  	}
   349  	// special cases
   350  	if x.form != finite {
   351  		return x.form == zero
   352  	}
   353  	// x.form == finite
   354  	if x.exp <= 0 {
   355  		return false
   356  	}
   357  	// x.exp > 0
   358  	return x.prec <= uint32(x.exp) || x.MinPrec() <= uint(x.exp) // not enough bits for fractional mantissa
   359  }
   360  
   361  // debugging support
   362  func (x *Float) validate() {
   363  	if !debugFloat {
   364  		// avoid performance bugs
   365  		panic("validate called but debugFloat is not set")
   366  	}
   367  	if x.form != finite {
   368  		return
   369  	}
   370  	m := len(x.mant)
   371  	if m == 0 {
   372  		panic("nonzero finite number with empty mantissa")
   373  	}
   374  	const msb = 1 << (_W - 1)
   375  	if x.mant[m-1]&msb == 0 {
   376  		panic(fmt.Sprintf("msb not set in last word %#x of %s", x.mant[m-1], x.Text('p', 0)))
   377  	}
   378  	if x.prec == 0 {
   379  		panic("zero precision finite number")
   380  	}
   381  }
   382  
   383  // round rounds z according to z.mode to z.prec bits and sets z.acc accordingly.
   384  // sbit must be 0 or 1 and summarizes any "sticky bit" information one might
   385  // have before calling round. z's mantissa must be normalized (with the msb set)
   386  // or empty.
   387  //
   388  // CAUTION: The rounding modes ToNegativeInf, ToPositiveInf are affected by the
   389  // sign of z. For correct rounding, the sign of z must be set correctly before
   390  // calling round.
   391  func (z *Float) round(sbit uint) {
   392  	if debugFloat {
   393  		z.validate()
   394  	}
   395  
   396  	z.acc = Exact
   397  	if z.form != finite {
   398  		// ±0 or ±Inf => nothing left to do
   399  		return
   400  	}
   401  	// z.form == finite && len(z.mant) > 0
   402  	// m > 0 implies z.prec > 0 (checked by validate)
   403  
   404  	m := uint32(len(z.mant)) // present mantissa length in words
   405  	bits := m * _W           // present mantissa bits; bits > 0
   406  	if bits <= z.prec {
   407  		// mantissa fits => nothing to do
   408  		return
   409  	}
   410  	// bits > z.prec
   411  
   412  	// Rounding is based on two bits: the rounding bit (rbit) and the
   413  	// sticky bit (sbit). The rbit is the bit immediately before the
   414  	// z.prec leading mantissa bits (the "0.5"). The sbit is set if any
   415  	// of the bits before the rbit are set (the "0.25", "0.125", etc.):
   416  	//
   417  	//   rbit  sbit  => "fractional part"
   418  	//
   419  	//   0     0        == 0
   420  	//   0     1        >  0  , < 0.5
   421  	//   1     0        == 0.5
   422  	//   1     1        >  0.5, < 1.0
   423  
   424  	// bits > z.prec: mantissa too large => round
   425  	r := uint(bits - z.prec - 1) // rounding bit position; r >= 0
   426  	rbit := z.mant.bit(r) & 1    // rounding bit; be safe and ensure it's a single bit
   427  	// The sticky bit is only needed for rounding ToNearestEven
   428  	// or when the rounding bit is zero. Avoid computation otherwise.
   429  	if sbit == 0 && (rbit == 0 || z.mode == ToNearestEven) {
   430  		sbit = z.mant.sticky(r)
   431  	}
   432  	sbit &= 1 // be safe and ensure it's a single bit
   433  
   434  	// cut off extra words
   435  	n := (z.prec + (_W - 1)) / _W // mantissa length in words for desired precision
   436  	if m > n {
   437  		copy(z.mant, z.mant[m-n:]) // move n last words to front
   438  		z.mant = z.mant[:n]
   439  	}
   440  
   441  	// determine number of trailing zero bits (ntz) and compute lsb mask of mantissa's least-significant word
   442  	ntz := n*_W - z.prec // 0 <= ntz < _W
   443  	lsb := Word(1) << ntz
   444  
   445  	// round if result is inexact
   446  	if rbit|sbit != 0 {
   447  		// Make rounding decision: The result mantissa is truncated ("rounded down")
   448  		// by default. Decide if we need to increment, or "round up", the (unsigned)
   449  		// mantissa.
   450  		inc := false
   451  		switch z.mode {
   452  		case ToNegativeInf:
   453  			inc = z.neg
   454  		case ToZero:
   455  			// nothing to do
   456  		case ToNearestEven:
   457  			inc = rbit != 0 && (sbit != 0 || z.mant[0]&lsb != 0)
   458  		case ToNearestAway:
   459  			inc = rbit != 0
   460  		case AwayFromZero:
   461  			inc = true
   462  		case ToPositiveInf:
   463  			inc = !z.neg
   464  		default:
   465  			panic("unreachable")
   466  		}
   467  
   468  		// A positive result (!z.neg) is Above the exact result if we increment,
   469  		// and it's Below if we truncate (Exact results require no rounding).
   470  		// For a negative result (z.neg) it is exactly the opposite.
   471  		z.acc = makeAcc(inc != z.neg)
   472  
   473  		if inc {
   474  			// add 1 to mantissa
   475  			if addVW(z.mant, z.mant, lsb) != 0 {
   476  				// mantissa overflow => adjust exponent
   477  				if z.exp >= MaxExp {
   478  					// exponent overflow
   479  					z.form = inf
   480  					return
   481  				}
   482  				z.exp++
   483  				// adjust mantissa: divide by 2 to compensate for exponent adjustment
   484  				shrVU(z.mant, z.mant, 1)
   485  				// set msb == carry == 1 from the mantissa overflow above
   486  				const msb = 1 << (_W - 1)
   487  				z.mant[n-1] |= msb
   488  			}
   489  		}
   490  	}
   491  
   492  	// zero out trailing bits in least-significant word
   493  	z.mant[0] &^= lsb - 1
   494  
   495  	if debugFloat {
   496  		z.validate()
   497  	}
   498  }
   499  
   500  func (z *Float) setBits64(neg bool, x uint64) *Float {
   501  	if z.prec == 0 {
   502  		z.prec = 64
   503  	}
   504  	z.acc = Exact
   505  	z.neg = neg
   506  	if x == 0 {
   507  		z.form = zero
   508  		return z
   509  	}
   510  	// x != 0
   511  	z.form = finite
   512  	s := bits.LeadingZeros64(x)
   513  	z.mant = z.mant.setUint64(x << uint(s))
   514  	z.exp = int32(64 - s) // always fits
   515  	if z.prec < 64 {
   516  		z.round(0)
   517  	}
   518  	return z
   519  }
   520  
   521  // SetUint64 sets z to the (possibly rounded) value of x and returns z.
   522  // If z's precision is 0, it is changed to 64 (and rounding will have
   523  // no effect).
   524  func (z *Float) SetUint64(x uint64) *Float {
   525  	return z.setBits64(false, x)
   526  }
   527  
   528  // SetInt64 sets z to the (possibly rounded) value of x and returns z.
   529  // If z's precision is 0, it is changed to 64 (and rounding will have
   530  // no effect).
   531  func (z *Float) SetInt64(x int64) *Float {
   532  	u := x
   533  	if u < 0 {
   534  		u = -u
   535  	}
   536  	// We cannot simply call z.SetUint64(uint64(u)) and change
   537  	// the sign afterwards because the sign affects rounding.
   538  	return z.setBits64(x < 0, uint64(u))
   539  }
   540  
   541  // SetFloat64 sets z to the (possibly rounded) value of x and returns z.
   542  // If z's precision is 0, it is changed to 53 (and rounding will have
   543  // no effect). SetFloat64 panics with ErrNaN if x is a NaN.
   544  func (z *Float) SetFloat64(x float64) *Float {
   545  	if z.prec == 0 {
   546  		z.prec = 53
   547  	}
   548  	if math.IsNaN(x) {
   549  		panic(ErrNaN{"Float.SetFloat64(NaN)"})
   550  	}
   551  	z.acc = Exact
   552  	z.neg = math.Signbit(x) // handle -0, -Inf correctly
   553  	if x == 0 {
   554  		z.form = zero
   555  		return z
   556  	}
   557  	if math.IsInf(x, 0) {
   558  		z.form = inf
   559  		return z
   560  	}
   561  	// normalized x != 0
   562  	z.form = finite
   563  	fmant, exp := math.Frexp(x) // get normalized mantissa
   564  	z.mant = z.mant.setUint64(1<<63 | math.Float64bits(fmant)<<11)
   565  	z.exp = int32(exp) // always fits
   566  	if z.prec < 53 {
   567  		z.round(0)
   568  	}
   569  	return z
   570  }
   571  
   572  // fnorm normalizes mantissa m by shifting it to the left
   573  // such that the msb of the most-significant word (msw) is 1.
   574  // It returns the shift amount. It assumes that len(m) != 0.
   575  func fnorm(m nat) int64 {
   576  	if debugFloat && (len(m) == 0 || m[len(m)-1] == 0) {
   577  		panic("msw of mantissa is 0")
   578  	}
   579  	s := nlz(m[len(m)-1])
   580  	if s > 0 {
   581  		c := shlVU(m, m, s)
   582  		if debugFloat && c != 0 {
   583  			panic("nlz or shlVU incorrect")
   584  		}
   585  	}
   586  	return int64(s)
   587  }
   588  
   589  // SetInt sets z to the (possibly rounded) value of x and returns z.
   590  // If z's precision is 0, it is changed to the larger of x.BitLen()
   591  // or 64 (and rounding will have no effect).
   592  func (z *Float) SetInt(x *Int) *Float {
   593  	// TODO(gri) can be more efficient if z.prec > 0
   594  	// but small compared to the size of x, or if there
   595  	// are many trailing 0's.
   596  	bits := uint32(x.BitLen())
   597  	if z.prec == 0 {
   598  		z.prec = umax32(bits, 64)
   599  	}
   600  	z.acc = Exact
   601  	z.neg = x.neg
   602  	if len(x.abs) == 0 {
   603  		z.form = zero
   604  		return z
   605  	}
   606  	// x != 0
   607  	z.mant = z.mant.set(x.abs)
   608  	fnorm(z.mant)
   609  	z.setExpAndRound(int64(bits), 0)
   610  	return z
   611  }
   612  
   613  // SetRat sets z to the (possibly rounded) value of x and returns z.
   614  // If z's precision is 0, it is changed to the largest of a.BitLen(),
   615  // b.BitLen(), or 64; with x = a/b.
   616  func (z *Float) SetRat(x *Rat) *Float {
   617  	if x.IsInt() {
   618  		return z.SetInt(x.Num())
   619  	}
   620  	var a, b Float
   621  	a.SetInt(x.Num())
   622  	b.SetInt(x.Denom())
   623  	if z.prec == 0 {
   624  		z.prec = umax32(a.prec, b.prec)
   625  	}
   626  	return z.Quo(&a, &b)
   627  }
   628  
   629  // SetInf sets z to the infinite Float -Inf if signbit is
   630  // set, or +Inf if signbit is not set, and returns z. The
   631  // precision of z is unchanged and the result is always
   632  // Exact.
   633  func (z *Float) SetInf(signbit bool) *Float {
   634  	z.acc = Exact
   635  	z.form = inf
   636  	z.neg = signbit
   637  	return z
   638  }
   639  
   640  // Set sets z to the (possibly rounded) value of x and returns z.
   641  // If z's precision is 0, it is changed to the precision of x
   642  // before setting z (and rounding will have no effect).
   643  // Rounding is performed according to z's precision and rounding
   644  // mode; and z's accuracy reports the result error relative to the
   645  // exact (not rounded) result.
   646  func (z *Float) Set(x *Float) *Float {
   647  	if debugFloat {
   648  		x.validate()
   649  	}
   650  	z.acc = Exact
   651  	if z != x {
   652  		z.form = x.form
   653  		z.neg = x.neg
   654  		if x.form == finite {
   655  			z.exp = x.exp
   656  			z.mant = z.mant.set(x.mant)
   657  		}
   658  		if z.prec == 0 {
   659  			z.prec = x.prec
   660  		} else if z.prec < x.prec {
   661  			z.round(0)
   662  		}
   663  	}
   664  	return z
   665  }
   666  
   667  // Copy sets z to x, with the same precision, rounding mode, and
   668  // accuracy as x, and returns z. x is not changed even if z and
   669  // x are the same.
   670  func (z *Float) Copy(x *Float) *Float {
   671  	if debugFloat {
   672  		x.validate()
   673  	}
   674  	if z != x {
   675  		z.prec = x.prec
   676  		z.mode = x.mode
   677  		z.acc = x.acc
   678  		z.form = x.form
   679  		z.neg = x.neg
   680  		if z.form == finite {
   681  			z.mant = z.mant.set(x.mant)
   682  			z.exp = x.exp
   683  		}
   684  	}
   685  	return z
   686  }
   687  
   688  // msb32 returns the 32 most significant bits of x.
   689  func msb32(x nat) uint32 {
   690  	i := len(x) - 1
   691  	if i < 0 {
   692  		return 0
   693  	}
   694  	if debugFloat && x[i]&(1<<(_W-1)) == 0 {
   695  		panic("x not normalized")
   696  	}
   697  	switch _W {
   698  	case 32:
   699  		return uint32(x[i])
   700  	case 64:
   701  		return uint32(x[i] >> 32)
   702  	}
   703  	panic("unreachable")
   704  }
   705  
   706  // msb64 returns the 64 most significant bits of x.
   707  func msb64(x nat) uint64 {
   708  	i := len(x) - 1
   709  	if i < 0 {
   710  		return 0
   711  	}
   712  	if debugFloat && x[i]&(1<<(_W-1)) == 0 {
   713  		panic("x not normalized")
   714  	}
   715  	switch _W {
   716  	case 32:
   717  		v := uint64(x[i]) << 32
   718  		if i > 0 {
   719  			v |= uint64(x[i-1])
   720  		}
   721  		return v
   722  	case 64:
   723  		return uint64(x[i])
   724  	}
   725  	panic("unreachable")
   726  }
   727  
   728  // Uint64 returns the unsigned integer resulting from truncating x
   729  // towards zero. If 0 <= x <= math.MaxUint64, the result is Exact
   730  // if x is an integer and Below otherwise.
   731  // The result is (0, Above) for x < 0, and (math.MaxUint64, Below)
   732  // for x > math.MaxUint64.
   733  func (x *Float) Uint64() (uint64, Accuracy) {
   734  	if debugFloat {
   735  		x.validate()
   736  	}
   737  
   738  	switch x.form {
   739  	case finite:
   740  		if x.neg {
   741  			return 0, Above
   742  		}
   743  		// 0 < x < +Inf
   744  		if x.exp <= 0 {
   745  			// 0 < x < 1
   746  			return 0, Below
   747  		}
   748  		// 1 <= x < Inf
   749  		if x.exp <= 64 {
   750  			// u = trunc(x) fits into a uint64
   751  			u := msb64(x.mant) >> (64 - uint32(x.exp))
   752  			if x.MinPrec() <= 64 {
   753  				return u, Exact
   754  			}
   755  			return u, Below // x truncated
   756  		}
   757  		// x too large
   758  		return math.MaxUint64, Below
   759  
   760  	case zero:
   761  		return 0, Exact
   762  
   763  	case inf:
   764  		if x.neg {
   765  			return 0, Above
   766  		}
   767  		return math.MaxUint64, Below
   768  	}
   769  
   770  	panic("unreachable")
   771  }
   772  
   773  // Int64 returns the integer resulting from truncating x towards zero.
   774  // If math.MinInt64 <= x <= math.MaxInt64, the result is Exact if x is
   775  // an integer, and Above (x < 0) or Below (x > 0) otherwise.
   776  // The result is (math.MinInt64, Above) for x < math.MinInt64,
   777  // and (math.MaxInt64, Below) for x > math.MaxInt64.
   778  func (x *Float) Int64() (int64, Accuracy) {
   779  	if debugFloat {
   780  		x.validate()
   781  	}
   782  
   783  	switch x.form {
   784  	case finite:
   785  		// 0 < |x| < +Inf
   786  		acc := makeAcc(x.neg)
   787  		if x.exp <= 0 {
   788  			// 0 < |x| < 1
   789  			return 0, acc
   790  		}
   791  		// x.exp > 0
   792  
   793  		// 1 <= |x| < +Inf
   794  		if x.exp <= 63 {
   795  			// i = trunc(x) fits into an int64 (excluding math.MinInt64)
   796  			i := int64(msb64(x.mant) >> (64 - uint32(x.exp)))
   797  			if x.neg {
   798  				i = -i
   799  			}
   800  			if x.MinPrec() <= uint(x.exp) {
   801  				return i, Exact
   802  			}
   803  			return i, acc // x truncated
   804  		}
   805  		if x.neg {
   806  			// check for special case x == math.MinInt64 (i.e., x == -(0.5 << 64))
   807  			if x.exp == 64 && x.MinPrec() == 1 {
   808  				acc = Exact
   809  			}
   810  			return math.MinInt64, acc
   811  		}
   812  		// x too large
   813  		return math.MaxInt64, Below
   814  
   815  	case zero:
   816  		return 0, Exact
   817  
   818  	case inf:
   819  		if x.neg {
   820  			return math.MinInt64, Above
   821  		}
   822  		return math.MaxInt64, Below
   823  	}
   824  
   825  	panic("unreachable")
   826  }
   827  
   828  // Float32 returns the float32 value nearest to x. If x is too small to be
   829  // represented by a float32 (|x| < math.SmallestNonzeroFloat32), the result
   830  // is (0, Below) or (-0, Above), respectively, depending on the sign of x.
   831  // If x is too large to be represented by a float32 (|x| > math.MaxFloat32),
   832  // the result is (+Inf, Above) or (-Inf, Below), depending on the sign of x.
   833  func (x *Float) Float32() (float32, Accuracy) {
   834  	if debugFloat {
   835  		x.validate()
   836  	}
   837  
   838  	switch x.form {
   839  	case finite:
   840  		// 0 < |x| < +Inf
   841  
   842  		const (
   843  			fbits = 32                //        float size
   844  			mbits = 23                //        mantissa size (excluding implicit msb)
   845  			ebits = fbits - mbits - 1 //     8  exponent size
   846  			bias  = 1<<(ebits-1) - 1  //   127  exponent bias
   847  			dmin  = 1 - bias - mbits  //  -149  smallest unbiased exponent (denormal)
   848  			emin  = 1 - bias          //  -126  smallest unbiased exponent (normal)
   849  			emax  = bias              //   127  largest unbiased exponent (normal)
   850  		)
   851  
   852  		// Float mantissa m is 0.5 <= m < 1.0; compute exponent e for float32 mantissa.
   853  		e := x.exp - 1 // exponent for normal mantissa m with 1.0 <= m < 2.0
   854  
   855  		// Compute precision p for float32 mantissa.
   856  		// If the exponent is too small, we have a denormal number before
   857  		// rounding and fewer than p mantissa bits of precision available
   858  		// (the exponent remains fixed but the mantissa gets shifted right).
   859  		p := mbits + 1 // precision of normal float
   860  		if e < emin {
   861  			// recompute precision
   862  			p = mbits + 1 - emin + int(e)
   863  			// If p == 0, the mantissa of x is shifted so much to the right
   864  			// that its msb falls immediately to the right of the float32
   865  			// mantissa space. In other words, if the smallest denormal is
   866  			// considered "1.0", for p == 0, the mantissa value m is >= 0.5.
   867  			// If m > 0.5, it is rounded up to 1.0; i.e., the smallest denormal.
   868  			// If m == 0.5, it is rounded down to even, i.e., 0.0.
   869  			// If p < 0, the mantissa value m is <= "0.25" which is never rounded up.
   870  			if p < 0 /* m <= 0.25 */ || p == 0 && x.mant.sticky(uint(len(x.mant))*_W-1) == 0 /* m == 0.5 */ {
   871  				// underflow to ±0
   872  				if x.neg {
   873  					var z float32
   874  					return -z, Above
   875  				}
   876  				return 0.0, Below
   877  			}
   878  			// otherwise, round up
   879  			// We handle p == 0 explicitly because it's easy and because
   880  			// Float.round doesn't support rounding to 0 bits of precision.
   881  			if p == 0 {
   882  				if x.neg {
   883  					return -math.SmallestNonzeroFloat32, Below
   884  				}
   885  				return math.SmallestNonzeroFloat32, Above
   886  			}
   887  		}
   888  		// p > 0
   889  
   890  		// round
   891  		var r Float
   892  		r.prec = uint32(p)
   893  		r.Set(x)
   894  		e = r.exp - 1
   895  
   896  		// Rounding may have caused r to overflow to ±Inf
   897  		// (rounding never causes underflows to 0).
   898  		// If the exponent is too large, also overflow to ±Inf.
   899  		if r.form == inf || e > emax {
   900  			// overflow
   901  			if x.neg {
   902  				return float32(math.Inf(-1)), Below
   903  			}
   904  			return float32(math.Inf(+1)), Above
   905  		}
   906  		// e <= emax
   907  
   908  		// Determine sign, biased exponent, and mantissa.
   909  		var sign, bexp, mant uint32
   910  		if x.neg {
   911  			sign = 1 << (fbits - 1)
   912  		}
   913  
   914  		// Rounding may have caused a denormal number to
   915  		// become normal. Check again.
   916  		if e < emin {
   917  			// denormal number: recompute precision
   918  			// Since rounding may have at best increased precision
   919  			// and we have eliminated p <= 0 early, we know p > 0.
   920  			// bexp == 0 for denormals
   921  			p = mbits + 1 - emin + int(e)
   922  			mant = msb32(r.mant) >> uint(fbits-p)
   923  		} else {
   924  			// normal number: emin <= e <= emax
   925  			bexp = uint32(e+bias) << mbits
   926  			mant = msb32(r.mant) >> ebits & (1<<mbits - 1) // cut off msb (implicit 1 bit)
   927  		}
   928  
   929  		return math.Float32frombits(sign | bexp | mant), r.acc
   930  
   931  	case zero:
   932  		if x.neg {
   933  			var z float32
   934  			return -z, Exact
   935  		}
   936  		return 0.0, Exact
   937  
   938  	case inf:
   939  		if x.neg {
   940  			return float32(math.Inf(-1)), Exact
   941  		}
   942  		return float32(math.Inf(+1)), Exact
   943  	}
   944  
   945  	panic("unreachable")
   946  }
   947  
   948  // Float64 returns the float64 value nearest to x. If x is too small to be
   949  // represented by a float64 (|x| < math.SmallestNonzeroFloat64), the result
   950  // is (0, Below) or (-0, Above), respectively, depending on the sign of x.
   951  // If x is too large to be represented by a float64 (|x| > math.MaxFloat64),
   952  // the result is (+Inf, Above) or (-Inf, Below), depending on the sign of x.
   953  func (x *Float) Float64() (float64, Accuracy) {
   954  	if debugFloat {
   955  		x.validate()
   956  	}
   957  
   958  	switch x.form {
   959  	case finite:
   960  		// 0 < |x| < +Inf
   961  
   962  		const (
   963  			fbits = 64                //        float size
   964  			mbits = 52                //        mantissa size (excluding implicit msb)
   965  			ebits = fbits - mbits - 1 //    11  exponent size
   966  			bias  = 1<<(ebits-1) - 1  //  1023  exponent bias
   967  			dmin  = 1 - bias - mbits  // -1074  smallest unbiased exponent (denormal)
   968  			emin  = 1 - bias          // -1022  smallest unbiased exponent (normal)
   969  			emax  = bias              //  1023  largest unbiased exponent (normal)
   970  		)
   971  
   972  		// Float mantissa m is 0.5 <= m < 1.0; compute exponent e for float64 mantissa.
   973  		e := x.exp - 1 // exponent for normal mantissa m with 1.0 <= m < 2.0
   974  
   975  		// Compute precision p for float64 mantissa.
   976  		// If the exponent is too small, we have a denormal number before
   977  		// rounding and fewer than p mantissa bits of precision available
   978  		// (the exponent remains fixed but the mantissa gets shifted right).
   979  		p := mbits + 1 // precision of normal float
   980  		if e < emin {
   981  			// recompute precision
   982  			p = mbits + 1 - emin + int(e)
   983  			// If p == 0, the mantissa of x is shifted so much to the right
   984  			// that its msb falls immediately to the right of the float64
   985  			// mantissa space. In other words, if the smallest denormal is
   986  			// considered "1.0", for p == 0, the mantissa value m is >= 0.5.
   987  			// If m > 0.5, it is rounded up to 1.0; i.e., the smallest denormal.
   988  			// If m == 0.5, it is rounded down to even, i.e., 0.0.
   989  			// If p < 0, the mantissa value m is <= "0.25" which is never rounded up.
   990  			if p < 0 /* m <= 0.25 */ || p == 0 && x.mant.sticky(uint(len(x.mant))*_W-1) == 0 /* m == 0.5 */ {
   991  				// underflow to ±0
   992  				if x.neg {
   993  					var z float64
   994  					return -z, Above
   995  				}
   996  				return 0.0, Below
   997  			}
   998  			// otherwise, round up
   999  			// We handle p == 0 explicitly because it's easy and because
  1000  			// Float.round doesn't support rounding to 0 bits of precision.
  1001  			if p == 0 {
  1002  				if x.neg {
  1003  					return -math.SmallestNonzeroFloat64, Below
  1004  				}
  1005  				return math.SmallestNonzeroFloat64, Above
  1006  			}
  1007  		}
  1008  		// p > 0
  1009  
  1010  		// round
  1011  		var r Float
  1012  		r.prec = uint32(p)
  1013  		r.Set(x)
  1014  		e = r.exp - 1
  1015  
  1016  		// Rounding may have caused r to overflow to ±Inf
  1017  		// (rounding never causes underflows to 0).
  1018  		// If the exponent is too large, also overflow to ±Inf.
  1019  		if r.form == inf || e > emax {
  1020  			// overflow
  1021  			if x.neg {
  1022  				return math.Inf(-1), Below
  1023  			}
  1024  			return math.Inf(+1), Above
  1025  		}
  1026  		// e <= emax
  1027  
  1028  		// Determine sign, biased exponent, and mantissa.
  1029  		var sign, bexp, mant uint64
  1030  		if x.neg {
  1031  			sign = 1 << (fbits - 1)
  1032  		}
  1033  
  1034  		// Rounding may have caused a denormal number to
  1035  		// become normal. Check again.
  1036  		if e < emin {
  1037  			// denormal number: recompute precision
  1038  			// Since rounding may have at best increased precision
  1039  			// and we have eliminated p <= 0 early, we know p > 0.
  1040  			// bexp == 0 for denormals
  1041  			p = mbits + 1 - emin + int(e)
  1042  			mant = msb64(r.mant) >> uint(fbits-p)
  1043  		} else {
  1044  			// normal number: emin <= e <= emax
  1045  			bexp = uint64(e+bias) << mbits
  1046  			mant = msb64(r.mant) >> ebits & (1<<mbits - 1) // cut off msb (implicit 1 bit)
  1047  		}
  1048  
  1049  		return math.Float64frombits(sign | bexp | mant), r.acc
  1050  
  1051  	case zero:
  1052  		if x.neg {
  1053  			var z float64
  1054  			return -z, Exact
  1055  		}
  1056  		return 0.0, Exact
  1057  
  1058  	case inf:
  1059  		if x.neg {
  1060  			return math.Inf(-1), Exact
  1061  		}
  1062  		return math.Inf(+1), Exact
  1063  	}
  1064  
  1065  	panic("unreachable")
  1066  }
  1067  
  1068  // Int returns the result of truncating x towards zero;
  1069  // or nil if x is an infinity.
  1070  // The result is Exact if x.IsInt(); otherwise it is Below
  1071  // for x > 0, and Above for x < 0.
  1072  // If a non-nil *Int argument z is provided, Int stores
  1073  // the result in z instead of allocating a new Int.
  1074  func (x *Float) Int(z *Int) (*Int, Accuracy) {
  1075  	if debugFloat {
  1076  		x.validate()
  1077  	}
  1078  
  1079  	if z == nil && x.form <= finite {
  1080  		z = new(Int)
  1081  	}
  1082  
  1083  	switch x.form {
  1084  	case finite:
  1085  		// 0 < |x| < +Inf
  1086  		acc := makeAcc(x.neg)
  1087  		if x.exp <= 0 {
  1088  			// 0 < |x| < 1
  1089  			return z.SetInt64(0), acc
  1090  		}
  1091  		// x.exp > 0
  1092  
  1093  		// 1 <= |x| < +Inf
  1094  		// determine minimum required precision for x
  1095  		allBits := uint(len(x.mant)) * _W
  1096  		exp := uint(x.exp)
  1097  		if x.MinPrec() <= exp {
  1098  			acc = Exact
  1099  		}
  1100  		// shift mantissa as needed
  1101  		if z == nil {
  1102  			z = new(Int)
  1103  		}
  1104  		z.neg = x.neg
  1105  		switch {
  1106  		case exp > allBits:
  1107  			z.abs = z.abs.shl(x.mant, exp-allBits)
  1108  		default:
  1109  			z.abs = z.abs.set(x.mant)
  1110  		case exp < allBits:
  1111  			z.abs = z.abs.shr(x.mant, allBits-exp)
  1112  		}
  1113  		return z, acc
  1114  
  1115  	case zero:
  1116  		return z.SetInt64(0), Exact
  1117  
  1118  	case inf:
  1119  		return nil, makeAcc(x.neg)
  1120  	}
  1121  
  1122  	panic("unreachable")
  1123  }
  1124  
  1125  // Rat returns the rational number corresponding to x;
  1126  // or nil if x is an infinity.
  1127  // The result is Exact if x is not an Inf.
  1128  // If a non-nil *Rat argument z is provided, Rat stores
  1129  // the result in z instead of allocating a new Rat.
  1130  func (x *Float) Rat(z *Rat) (*Rat, Accuracy) {
  1131  	if debugFloat {
  1132  		x.validate()
  1133  	}
  1134  
  1135  	if z == nil && x.form <= finite {
  1136  		z = new(Rat)
  1137  	}
  1138  
  1139  	switch x.form {
  1140  	case finite:
  1141  		// 0 < |x| < +Inf
  1142  		allBits := int32(len(x.mant)) * _W
  1143  		// build up numerator and denominator
  1144  		z.a.neg = x.neg
  1145  		switch {
  1146  		case x.exp > allBits:
  1147  			z.a.abs = z.a.abs.shl(x.mant, uint(x.exp-allBits))
  1148  			z.b.abs = z.b.abs[:0] // == 1 (see Rat)
  1149  			// z already in normal form
  1150  		default:
  1151  			z.a.abs = z.a.abs.set(x.mant)
  1152  			z.b.abs = z.b.abs[:0] // == 1 (see Rat)
  1153  			// z already in normal form
  1154  		case x.exp < allBits:
  1155  			z.a.abs = z.a.abs.set(x.mant)
  1156  			t := z.b.abs.setUint64(1)
  1157  			z.b.abs = t.shl(t, uint(allBits-x.exp))
  1158  			z.norm()
  1159  		}
  1160  		return z, Exact
  1161  
  1162  	case zero:
  1163  		return z.SetInt64(0), Exact
  1164  
  1165  	case inf:
  1166  		return nil, makeAcc(x.neg)
  1167  	}
  1168  
  1169  	panic("unreachable")
  1170  }
  1171  
  1172  // Abs sets z to the (possibly rounded) value |x| (the absolute value of x)
  1173  // and returns z.
  1174  func (z *Float) Abs(x *Float) *Float {
  1175  	z.Set(x)
  1176  	z.neg = false
  1177  	return z
  1178  }
  1179  
  1180  // Neg sets z to the (possibly rounded) value of x with its sign negated,
  1181  // and returns z.
  1182  func (z *Float) Neg(x *Float) *Float {
  1183  	z.Set(x)
  1184  	z.neg = !z.neg
  1185  	return z
  1186  }
  1187  
  1188  func validateBinaryOperands(x, y *Float) {
  1189  	if !debugFloat {
  1190  		// avoid performance bugs
  1191  		panic("validateBinaryOperands called but debugFloat is not set")
  1192  	}
  1193  	if len(x.mant) == 0 {
  1194  		panic("empty mantissa for x")
  1195  	}
  1196  	if len(y.mant) == 0 {
  1197  		panic("empty mantissa for y")
  1198  	}
  1199  }
  1200  
  1201  // z = x + y, ignoring signs of x and y for the addition
  1202  // but using the sign of z for rounding the result.
  1203  // x and y must have a non-empty mantissa and valid exponent.
  1204  func (z *Float) uadd(x, y *Float) {
  1205  	// Note: This implementation requires 2 shifts most of the
  1206  	// time. It is also inefficient if exponents or precisions
  1207  	// differ by wide margins. The following article describes
  1208  	// an efficient (but much more complicated) implementation
  1209  	// compatible with the internal representation used here:
  1210  	//
  1211  	// Vincent Lefèvre: "The Generic Multiple-Precision Floating-
  1212  	// Point Addition With Exact Rounding (as in the MPFR Library)"
  1213  	// http://www.vinc17.net/research/papers/rnc6.pdf
  1214  
  1215  	if debugFloat {
  1216  		validateBinaryOperands(x, y)
  1217  	}
  1218  
  1219  	// compute exponents ex, ey for mantissa with "binary point"
  1220  	// on the right (mantissa.0) - use int64 to avoid overflow
  1221  	ex := int64(x.exp) - int64(len(x.mant))*_W
  1222  	ey := int64(y.exp) - int64(len(y.mant))*_W
  1223  
  1224  	al := alias(z.mant, x.mant) || alias(z.mant, y.mant)
  1225  
  1226  	// TODO(gri) having a combined add-and-shift primitive
  1227  	//           could make this code significantly faster
  1228  	switch {
  1229  	case ex < ey:
  1230  		if al {
  1231  			t := nat(nil).shl(y.mant, uint(ey-ex))
  1232  			z.mant = z.mant.add(x.mant, t)
  1233  		} else {
  1234  			z.mant = z.mant.shl(y.mant, uint(ey-ex))
  1235  			z.mant = z.mant.add(x.mant, z.mant)
  1236  		}
  1237  	default:
  1238  		// ex == ey, no shift needed
  1239  		z.mant = z.mant.add(x.mant, y.mant)
  1240  	case ex > ey:
  1241  		if al {
  1242  			t := nat(nil).shl(x.mant, uint(ex-ey))
  1243  			z.mant = z.mant.add(t, y.mant)
  1244  		} else {
  1245  			z.mant = z.mant.shl(x.mant, uint(ex-ey))
  1246  			z.mant = z.mant.add(z.mant, y.mant)
  1247  		}
  1248  		ex = ey
  1249  	}
  1250  	// len(z.mant) > 0
  1251  
  1252  	z.setExpAndRound(ex+int64(len(z.mant))*_W-fnorm(z.mant), 0)
  1253  }
  1254  
  1255  // z = x - y for |x| > |y|, ignoring signs of x and y for the subtraction
  1256  // but using the sign of z for rounding the result.
  1257  // x and y must have a non-empty mantissa and valid exponent.
  1258  func (z *Float) usub(x, y *Float) {
  1259  	// This code is symmetric to uadd.
  1260  	// We have not factored the common code out because
  1261  	// eventually uadd (and usub) should be optimized
  1262  	// by special-casing, and the code will diverge.
  1263  
  1264  	if debugFloat {
  1265  		validateBinaryOperands(x, y)
  1266  	}
  1267  
  1268  	ex := int64(x.exp) - int64(len(x.mant))*_W
  1269  	ey := int64(y.exp) - int64(len(y.mant))*_W
  1270  
  1271  	al := alias(z.mant, x.mant) || alias(z.mant, y.mant)
  1272  
  1273  	switch {
  1274  	case ex < ey:
  1275  		if al {
  1276  			t := nat(nil).shl(y.mant, uint(ey-ex))
  1277  			z.mant = t.sub(x.mant, t)
  1278  		} else {
  1279  			z.mant = z.mant.shl(y.mant, uint(ey-ex))
  1280  			z.mant = z.mant.sub(x.mant, z.mant)
  1281  		}
  1282  	default:
  1283  		// ex == ey, no shift needed
  1284  		z.mant = z.mant.sub(x.mant, y.mant)
  1285  	case ex > ey:
  1286  		if al {
  1287  			t := nat(nil).shl(x.mant, uint(ex-ey))
  1288  			z.mant = t.sub(t, y.mant)
  1289  		} else {
  1290  			z.mant = z.mant.shl(x.mant, uint(ex-ey))
  1291  			z.mant = z.mant.sub(z.mant, y.mant)
  1292  		}
  1293  		ex = ey
  1294  	}
  1295  
  1296  	// operands may have canceled each other out
  1297  	if len(z.mant) == 0 {
  1298  		z.acc = Exact
  1299  		z.form = zero
  1300  		z.neg = false
  1301  		return
  1302  	}
  1303  	// len(z.mant) > 0
  1304  
  1305  	z.setExpAndRound(ex+int64(len(z.mant))*_W-fnorm(z.mant), 0)
  1306  }
  1307  
  1308  // z = x * y, ignoring signs of x and y for the multiplication
  1309  // but using the sign of z for rounding the result.
  1310  // x and y must have a non-empty mantissa and valid exponent.
  1311  func (z *Float) umul(x, y *Float) {
  1312  	if debugFloat {
  1313  		validateBinaryOperands(x, y)
  1314  	}
  1315  
  1316  	// Note: This is doing too much work if the precision
  1317  	// of z is less than the sum of the precisions of x
  1318  	// and y which is often the case (e.g., if all floats
  1319  	// have the same precision).
  1320  	// TODO(gri) Optimize this for the common case.
  1321  
  1322  	e := int64(x.exp) + int64(y.exp)
  1323  	if x == y {
  1324  		z.mant = z.mant.sqr(x.mant)
  1325  	} else {
  1326  		z.mant = z.mant.mul(x.mant, y.mant)
  1327  	}
  1328  	z.setExpAndRound(e-fnorm(z.mant), 0)
  1329  }
  1330  
  1331  // z = x / y, ignoring signs of x and y for the division
  1332  // but using the sign of z for rounding the result.
  1333  // x and y must have a non-empty mantissa and valid exponent.
  1334  func (z *Float) uquo(x, y *Float) {
  1335  	if debugFloat {
  1336  		validateBinaryOperands(x, y)
  1337  	}
  1338  
  1339  	// mantissa length in words for desired result precision + 1
  1340  	// (at least one extra bit so we get the rounding bit after
  1341  	// the division)
  1342  	n := int(z.prec/_W) + 1
  1343  
  1344  	// compute adjusted x.mant such that we get enough result precision
  1345  	xadj := x.mant
  1346  	if d := n - len(x.mant) + len(y.mant); d > 0 {
  1347  		// d extra words needed => add d "0 digits" to x
  1348  		xadj = make(nat, len(x.mant)+d)
  1349  		copy(xadj[d:], x.mant)
  1350  	}
  1351  	// TODO(gri): If we have too many digits (d < 0), we should be able
  1352  	// to shorten x for faster division. But we must be extra careful
  1353  	// with rounding in that case.
  1354  
  1355  	// Compute d before division since there may be aliasing of x.mant
  1356  	// (via xadj) or y.mant with z.mant.
  1357  	d := len(xadj) - len(y.mant)
  1358  
  1359  	// divide
  1360  	var r nat
  1361  	z.mant, r = z.mant.div(nil, xadj, y.mant)
  1362  	e := int64(x.exp) - int64(y.exp) - int64(d-len(z.mant))*_W
  1363  
  1364  	// The result is long enough to include (at least) the rounding bit.
  1365  	// If there's a non-zero remainder, the corresponding fractional part
  1366  	// (if it were computed), would have a non-zero sticky bit (if it were
  1367  	// zero, it couldn't have a non-zero remainder).
  1368  	var sbit uint
  1369  	if len(r) > 0 {
  1370  		sbit = 1
  1371  	}
  1372  
  1373  	z.setExpAndRound(e-fnorm(z.mant), sbit)
  1374  }
  1375  
  1376  // ucmp returns -1, 0, or +1, depending on whether
  1377  // |x| < |y|, |x| == |y|, or |x| > |y|.
  1378  // x and y must have a non-empty mantissa and valid exponent.
  1379  func (x *Float) ucmp(y *Float) int {
  1380  	if debugFloat {
  1381  		validateBinaryOperands(x, y)
  1382  	}
  1383  
  1384  	switch {
  1385  	case x.exp < y.exp:
  1386  		return -1
  1387  	case x.exp > y.exp:
  1388  		return +1
  1389  	}
  1390  	// x.exp == y.exp
  1391  
  1392  	// compare mantissas
  1393  	i := len(x.mant)
  1394  	j := len(y.mant)
  1395  	for i > 0 || j > 0 {
  1396  		var xm, ym Word
  1397  		if i > 0 {
  1398  			i--
  1399  			xm = x.mant[i]
  1400  		}
  1401  		if j > 0 {
  1402  			j--
  1403  			ym = y.mant[j]
  1404  		}
  1405  		switch {
  1406  		case xm < ym:
  1407  			return -1
  1408  		case xm > ym:
  1409  			return +1
  1410  		}
  1411  	}
  1412  
  1413  	return 0
  1414  }
  1415  
  1416  // Handling of sign bit as defined by IEEE 754-2008, section 6.3:
  1417  //
  1418  // When neither the inputs nor result are NaN, the sign of a product or
  1419  // quotient is the exclusive OR of the operands’ signs; the sign of a sum,
  1420  // or of a difference x−y regarded as a sum x+(−y), differs from at most
  1421  // one of the addends’ signs; and the sign of the result of conversions,
  1422  // the quantize operation, the roundToIntegral operations, and the
  1423  // roundToIntegralExact (see 5.3.1) is the sign of the first or only operand.
  1424  // These rules shall apply even when operands or results are zero or infinite.
  1425  //
  1426  // When the sum of two operands with opposite signs (or the difference of
  1427  // two operands with like signs) is exactly zero, the sign of that sum (or
  1428  // difference) shall be +0 in all rounding-direction attributes except
  1429  // roundTowardNegative; under that attribute, the sign of an exact zero
  1430  // sum (or difference) shall be −0. However, x+x = x−(−x) retains the same
  1431  // sign as x even when x is zero.
  1432  //
  1433  // See also: https://play.golang.org/p/RtH3UCt5IH
  1434  
  1435  // Add sets z to the rounded sum x+y and returns z. If z's precision is 0,
  1436  // it is changed to the larger of x's or y's precision before the operation.
  1437  // Rounding is performed according to z's precision and rounding mode; and
  1438  // z's accuracy reports the result error relative to the exact (not rounded)
  1439  // result. Add panics with ErrNaN if x and y are infinities with opposite
  1440  // signs. The value of z is undefined in that case.
  1441  func (z *Float) Add(x, y *Float) *Float {
  1442  	if debugFloat {
  1443  		x.validate()
  1444  		y.validate()
  1445  	}
  1446  
  1447  	if z.prec == 0 {
  1448  		z.prec = umax32(x.prec, y.prec)
  1449  	}
  1450  
  1451  	if x.form == finite && y.form == finite {
  1452  		// x + y (common case)
  1453  
  1454  		// Below we set z.neg = x.neg, and when z aliases y this will
  1455  		// change the y operand's sign. This is fine, because if an
  1456  		// operand aliases the receiver it'll be overwritten, but we still
  1457  		// want the original x.neg and y.neg values when we evaluate
  1458  		// x.neg != y.neg, so we need to save y.neg before setting z.neg.
  1459  		yneg := y.neg
  1460  
  1461  		z.neg = x.neg
  1462  		if x.neg == yneg {
  1463  			// x + y == x + y
  1464  			// (-x) + (-y) == -(x + y)
  1465  			z.uadd(x, y)
  1466  		} else {
  1467  			// x + (-y) == x - y == -(y - x)
  1468  			// (-x) + y == y - x == -(x - y)
  1469  			if x.ucmp(y) > 0 {
  1470  				z.usub(x, y)
  1471  			} else {
  1472  				z.neg = !z.neg
  1473  				z.usub(y, x)
  1474  			}
  1475  		}
  1476  		if z.form == zero && z.mode == ToNegativeInf && z.acc == Exact {
  1477  			z.neg = true
  1478  		}
  1479  		return z
  1480  	}
  1481  
  1482  	if x.form == inf && y.form == inf && x.neg != y.neg {
  1483  		// +Inf + -Inf
  1484  		// -Inf + +Inf
  1485  		// value of z is undefined but make sure it's valid
  1486  		z.acc = Exact
  1487  		z.form = zero
  1488  		z.neg = false
  1489  		panic(ErrNaN{"addition of infinities with opposite signs"})
  1490  	}
  1491  
  1492  	if x.form == zero && y.form == zero {
  1493  		// ±0 + ±0
  1494  		z.acc = Exact
  1495  		z.form = zero
  1496  		z.neg = x.neg && y.neg // -0 + -0 == -0
  1497  		return z
  1498  	}
  1499  
  1500  	if x.form == inf || y.form == zero {
  1501  		// ±Inf + y
  1502  		// x + ±0
  1503  		return z.Set(x)
  1504  	}
  1505  
  1506  	// ±0 + y
  1507  	// x + ±Inf
  1508  	return z.Set(y)
  1509  }
  1510  
  1511  // Sub sets z to the rounded difference x-y and returns z.
  1512  // Precision, rounding, and accuracy reporting are as for Add.
  1513  // Sub panics with ErrNaN if x and y are infinities with equal
  1514  // signs. The value of z is undefined in that case.
  1515  func (z *Float) Sub(x, y *Float) *Float {
  1516  	if debugFloat {
  1517  		x.validate()
  1518  		y.validate()
  1519  	}
  1520  
  1521  	if z.prec == 0 {
  1522  		z.prec = umax32(x.prec, y.prec)
  1523  	}
  1524  
  1525  	if x.form == finite && y.form == finite {
  1526  		// x - y (common case)
  1527  		yneg := y.neg
  1528  		z.neg = x.neg
  1529  		if x.neg != yneg {
  1530  			// x - (-y) == x + y
  1531  			// (-x) - y == -(x + y)
  1532  			z.uadd(x, y)
  1533  		} else {
  1534  			// x - y == x - y == -(y - x)
  1535  			// (-x) - (-y) == y - x == -(x - y)
  1536  			if x.ucmp(y) > 0 {
  1537  				z.usub(x, y)
  1538  			} else {
  1539  				z.neg = !z.neg
  1540  				z.usub(y, x)
  1541  			}
  1542  		}
  1543  		if z.form == zero && z.mode == ToNegativeInf && z.acc == Exact {
  1544  			z.neg = true
  1545  		}
  1546  		return z
  1547  	}
  1548  
  1549  	if x.form == inf && y.form == inf && x.neg == y.neg {
  1550  		// +Inf - +Inf
  1551  		// -Inf - -Inf
  1552  		// value of z is undefined but make sure it's valid
  1553  		z.acc = Exact
  1554  		z.form = zero
  1555  		z.neg = false
  1556  		panic(ErrNaN{"subtraction of infinities with equal signs"})
  1557  	}
  1558  
  1559  	if x.form == zero && y.form == zero {
  1560  		// ±0 - ±0
  1561  		z.acc = Exact
  1562  		z.form = zero
  1563  		z.neg = x.neg && !y.neg // -0 - +0 == -0
  1564  		return z
  1565  	}
  1566  
  1567  	if x.form == inf || y.form == zero {
  1568  		// ±Inf - y
  1569  		// x - ±0
  1570  		return z.Set(x)
  1571  	}
  1572  
  1573  	// ±0 - y
  1574  	// x - ±Inf
  1575  	return z.Neg(y)
  1576  }
  1577  
  1578  // Mul sets z to the rounded product x*y and returns z.
  1579  // Precision, rounding, and accuracy reporting are as for Add.
  1580  // Mul panics with ErrNaN if one operand is zero and the other
  1581  // operand an infinity. The value of z is undefined in that case.
  1582  func (z *Float) Mul(x, y *Float) *Float {
  1583  	if debugFloat {
  1584  		x.validate()
  1585  		y.validate()
  1586  	}
  1587  
  1588  	if z.prec == 0 {
  1589  		z.prec = umax32(x.prec, y.prec)
  1590  	}
  1591  
  1592  	z.neg = x.neg != y.neg
  1593  
  1594  	if x.form == finite && y.form == finite {
  1595  		// x * y (common case)
  1596  		z.umul(x, y)
  1597  		return z
  1598  	}
  1599  
  1600  	z.acc = Exact
  1601  	if x.form == zero && y.form == inf || x.form == inf && y.form == zero {
  1602  		// ±0 * ±Inf
  1603  		// ±Inf * ±0
  1604  		// value of z is undefined but make sure it's valid
  1605  		z.form = zero
  1606  		z.neg = false
  1607  		panic(ErrNaN{"multiplication of zero with infinity"})
  1608  	}
  1609  
  1610  	if x.form == inf || y.form == inf {
  1611  		// ±Inf * y
  1612  		// x * ±Inf
  1613  		z.form = inf
  1614  		return z
  1615  	}
  1616  
  1617  	// ±0 * y
  1618  	// x * ±0
  1619  	z.form = zero
  1620  	return z
  1621  }
  1622  
  1623  // Quo sets z to the rounded quotient x/y and returns z.
  1624  // Precision, rounding, and accuracy reporting are as for Add.
  1625  // Quo panics with ErrNaN if both operands are zero or infinities.
  1626  // The value of z is undefined in that case.
  1627  func (z *Float) Quo(x, y *Float) *Float {
  1628  	if debugFloat {
  1629  		x.validate()
  1630  		y.validate()
  1631  	}
  1632  
  1633  	if z.prec == 0 {
  1634  		z.prec = umax32(x.prec, y.prec)
  1635  	}
  1636  
  1637  	z.neg = x.neg != y.neg
  1638  
  1639  	if x.form == finite && y.form == finite {
  1640  		// x / y (common case)
  1641  		z.uquo(x, y)
  1642  		return z
  1643  	}
  1644  
  1645  	z.acc = Exact
  1646  	if x.form == zero && y.form == zero || x.form == inf && y.form == inf {
  1647  		// ±0 / ±0
  1648  		// ±Inf / ±Inf
  1649  		// value of z is undefined but make sure it's valid
  1650  		z.form = zero
  1651  		z.neg = false
  1652  		panic(ErrNaN{"division of zero by zero or infinity by infinity"})
  1653  	}
  1654  
  1655  	if x.form == zero || y.form == inf {
  1656  		// ±0 / y
  1657  		// x / ±Inf
  1658  		z.form = zero
  1659  		return z
  1660  	}
  1661  
  1662  	// x / ±0
  1663  	// ±Inf / y
  1664  	z.form = inf
  1665  	return z
  1666  }
  1667  
  1668  // Cmp compares x and y and returns:
  1669  //
  1670  //   -1 if x <  y
  1671  //    0 if x == y (incl. -0 == 0, -Inf == -Inf, and +Inf == +Inf)
  1672  //   +1 if x >  y
  1673  //
  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  //
  1709  func (x *Float) ord() int {
  1710  	var m int
  1711  	switch x.form {
  1712  	case finite:
  1713  		m = 1
  1714  	case zero:
  1715  		return 0
  1716  	case inf:
  1717  		m = 2
  1718  	}
  1719  	if x.neg {
  1720  		m = -m
  1721  	}
  1722  	return m
  1723  }
  1724  
  1725  func umax32(x, y uint32) uint32 {
  1726  	if x > y {
  1727  		return x
  1728  	}
  1729  	return y
  1730  }
  1731  

View as plain text