value.go

     1  // Copyright 2013 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  // Package constant implements Values representing untyped
     6  // Go constants and their corresponding operations.
     7  //
     8  // A special Unknown value may be used when a value
     9  // is unknown due to an error. Operations on unknown
    10  // values produce unknown values unless specified
    11  // otherwise.
    12  package constant
    13  
    14  import (
    15  	"fmt"
    16  	"go/token"
    17  	"math"
    18  	"math/big"
    19  	"math/bits"
    20  	"strconv"
    21  	"strings"
    22  	"sync"
    23  	"unicode/utf8"
    24  )
    25  
    26  //go:generate stringer -type Kind
    27  
    28  // Kind specifies the kind of value represented by a [Value].
    29  type Kind int
    30  
    31  const (
    32  	// unknown values
    33  	Unknown Kind = iota
    34  
    35  	// non-numeric values
    36  	Bool
    37  	String
    38  
    39  	// numeric values
    40  	Int
    41  	Float
    42  	Complex
    43  )
    44  
    45  // A Value represents the value of a Go constant.
    46  type Value interface {
    47  	// Kind returns the value kind.
    48  	Kind() Kind
    49  
    50  	// String returns a short, quoted (human-readable) form of the value.
    51  	// For numeric values, the result may be an approximation;
    52  	// for String values the result may be a shortened string.
    53  	// Use ExactString for a string representing a value exactly.
    54  	String() string
    55  
    56  	// ExactString returns an exact, quoted (human-readable) form of the value.
    57  	// If the Value is of Kind String, use StringVal to obtain the unquoted string.
    58  	ExactString() string
    59  
    60  	// Prevent external implementations.
    61  	implementsValue()
    62  }
    63  
    64  // ----------------------------------------------------------------------------
    65  // Implementations
    66  
    67  // Maximum supported mantissa precision.
    68  // The spec requires at least 256 bits; typical implementations use 512 bits.
    69  const prec = 512
    70  
    71  // TODO(gri) Consider storing "error" information in an unknownVal so clients
    72  // can provide better error messages. For instance, if a number is
    73  // too large (incl. infinity), that could be recorded in unknownVal.
    74  // See also #20583 and #42695 for use cases.
    75  
    76  // Representation of values:
    77  //
    78  // Values of Int and Float Kind have two different representations each: int64Val
    79  // and intVal, and ratVal and floatVal. When possible, the "smaller", respectively
    80  // more precise (for Floats) representation is chosen. However, once a Float value
    81  // is represented as a floatVal, any subsequent results remain floatVals (unless
    82  // explicitly converted); i.e., no attempt is made to convert a floatVal back into
    83  // a ratVal. The reasoning is that all representations but floatVal are mathematically
    84  // exact, but once that precision is lost (by moving to floatVal), moving back to
    85  // a different representation implies a precision that's not actually there.
    86  
    87  type (
    88  	unknownVal struct{}
    89  	boolVal    bool
    90  	stringVal  struct {
    91  		// Lazy value: either a string (l,r==nil) or an addition (l,r!=nil).
    92  		mu   sync.Mutex
    93  		s    string
    94  		l, r *stringVal
    95  	}
    96  	int64Val   int64                    // Int values representable as an int64
    97  	intVal     struct{ val *big.Int }   // Int values not representable as an int64
    98  	ratVal     struct{ val *big.Rat }   // Float values representable as a fraction
    99  	floatVal   struct{ val *big.Float } // Float values not representable as a fraction
   100  	complexVal struct{ re, im Value }
   101  )
   102  
   103  func (unknownVal) Kind() Kind { return Unknown }
   104  func (boolVal) Kind() Kind    { return Bool }
   105  func (*stringVal) Kind() Kind { return String }
   106  func (int64Val) Kind() Kind   { return Int }
   107  func (intVal) Kind() Kind     { return Int }
   108  func (ratVal) Kind() Kind     { return Float }
   109  func (floatVal) Kind() Kind   { return Float }
   110  func (complexVal) Kind() Kind { return Complex }
   111  
   112  func (unknownVal) String() string { return "unknown" }
   113  func (x boolVal) String() string  { return strconv.FormatBool(bool(x)) }
   114  
   115  // String returns a possibly shortened quoted form of the String value.
   116  func (x *stringVal) String() string {
   117  	const maxLen = 72 // a reasonable length
   118  	s := strconv.Quote(x.string())
   119  	if utf8.RuneCountInString(s) > maxLen {
   120  		// The string without the enclosing quotes is greater than maxLen-2 runes
   121  		// long. Remove the last 3 runes (including the closing '"') by keeping
   122  		// only the first maxLen-3 runes; then add "...".
   123  		i := 0
   124  		for n := 0; n < maxLen-3; n++ {
   125  			_, size := utf8.DecodeRuneInString(s[i:])
   126  			i += size
   127  		}
   128  		s = s[:i] + "..."
   129  	}
   130  	return s
   131  }
   132  
   133  // string constructs and returns the actual string literal value.
   134  // If x represents an addition, then it rewrites x to be a single
   135  // string, to speed future calls. This lazy construction avoids
   136  // building different string values for all subpieces of a large
   137  // concatenation. See golang.org/issue/23348.
   138  func (x *stringVal) string() string {
   139  	x.mu.Lock()
   140  	if x.l != nil {
   141  		x.s = strings.Join(reverse(x.appendReverse(nil)), "")
   142  		x.l = nil
   143  		x.r = nil
   144  	}
   145  	s := x.s
   146  	x.mu.Unlock()
   147  
   148  	return s
   149  }
   150  
   151  // reverse reverses x in place and returns it.
   152  func reverse(x []string) []string {
   153  	n := len(x)
   154  	for i := 0; i+i < n; i++ {
   155  		x[i], x[n-1-i] = x[n-1-i], x[i]
   156  	}
   157  	return x
   158  }
   159  
   160  // appendReverse appends to list all of x's subpieces, but in reverse,
   161  // and returns the result. Appending the reversal allows processing
   162  // the right side in a recursive call and the left side in a loop.
   163  // Because a chain like a + b + c + d + e is actually represented
   164  // as ((((a + b) + c) + d) + e), the left-side loop avoids deep recursion.
   165  // x must be locked.
   166  func (x *stringVal) appendReverse(list []string) []string {
   167  	y := x
   168  	for y.r != nil {
   169  		y.r.mu.Lock()
   170  		list = y.r.appendReverse(list)
   171  		y.r.mu.Unlock()
   172  
   173  		l := y.l
   174  		if y != x {
   175  			y.mu.Unlock()
   176  		}
   177  		l.mu.Lock()
   178  		y = l
   179  	}
   180  	s := y.s
   181  	if y != x {
   182  		y.mu.Unlock()
   183  	}
   184  	return append(list, s)
   185  }
   186  
   187  func (x int64Val) String() string { return strconv.FormatInt(int64(x), 10) }
   188  func (x intVal) String() string   { return x.val.String() }
   189  func (x ratVal) String() string   { return rtof(x).String() }
   190  
   191  // String returns a decimal approximation of the Float value.
   192  func (x floatVal) String() string {
   193  	f := x.val
   194  
   195  	// Don't try to convert infinities (will not terminate).
   196  	if f.IsInf() {
   197  		return f.String()
   198  	}
   199  
   200  	// Use exact fmt formatting if in float64 range (common case):
   201  	// proceed if f doesn't underflow to 0 or overflow to inf.
   202  	if x, _ := f.Float64(); f.Sign() == 0 == (x == 0) && !math.IsInf(x, 0) {
   203  		s := fmt.Sprintf("%.6g", x)
   204  		if !f.IsInt() && strings.IndexByte(s, '.') < 0 {
   205  			// f is not an integer, but its string representation
   206  			// doesn't reflect that. Use more digits. See issue 56220.
   207  			s = fmt.Sprintf("%g", x)
   208  		}
   209  		return s
   210  	}
   211  
   212  	// Out of float64 range. Do approximate manual to decimal
   213  	// conversion to avoid precise but possibly slow Float
   214  	// formatting.
   215  	// f = mant * 2**exp
   216  	var mant big.Float
   217  	exp := f.MantExp(&mant) // 0.5 <= |mant| < 1.0
   218  
   219  	// approximate float64 mantissa m and decimal exponent d
   220  	// f ~ m * 10**d
   221  	m, _ := mant.Float64()                     // 0.5 <= |m| < 1.0
   222  	d := float64(exp) * (math.Ln2 / math.Ln10) // log_10(2)
   223  
   224  	// adjust m for truncated (integer) decimal exponent e
   225  	e := int64(d)
   226  	m *= math.Pow(10, d-float64(e))
   227  
   228  	// ensure 1 <= |m| < 10
   229  	switch am := math.Abs(m); {
   230  	case am < 1-0.5e-6:
   231  		// The %.6g format below rounds m to 5 digits after the
   232  		// decimal point. Make sure that m*10 < 10 even after
   233  		// rounding up: m*10 + 0.5e-5 < 10 => m < 1 - 0.5e6.
   234  		m *= 10
   235  		e--
   236  	case am >= 10:
   237  		m /= 10
   238  		e++
   239  	}
   240  
   241  	return fmt.Sprintf("%.6ge%+d", m, e)
   242  }
   243  
   244  func (x complexVal) String() string { return fmt.Sprintf("(%s + %si)", x.re, x.im) }
   245  
   246  func (x unknownVal) ExactString() string { return x.String() }
   247  func (x boolVal) ExactString() string    { return x.String() }
   248  func (x *stringVal) ExactString() string { return strconv.Quote(x.string()) }
   249  func (x int64Val) ExactString() string   { return x.String() }
   250  func (x intVal) ExactString() string     { return x.String() }
   251  
   252  func (x ratVal) ExactString() string {
   253  	r := x.val
   254  	if r.IsInt() {
   255  		return r.Num().String()
   256  	}
   257  	return r.String()
   258  }
   259  
   260  func (x floatVal) ExactString() string { return x.val.Text('p', 0) }
   261  
   262  func (x complexVal) ExactString() string {
   263  	return fmt.Sprintf("(%s + %si)", x.re.ExactString(), x.im.ExactString())
   264  }
   265  
   266  func (unknownVal) implementsValue() {}
   267  func (boolVal) implementsValue()    {}
   268  func (*stringVal) implementsValue() {}
   269  func (int64Val) implementsValue()   {}
   270  func (ratVal) implementsValue()     {}
   271  func (intVal) implementsValue()     {}
   272  func (floatVal) implementsValue()   {}
   273  func (complexVal) implementsValue() {}
   274  
   275  func newInt() *big.Int     { return new(big.Int) }
   276  func newRat() *big.Rat     { return new(big.Rat) }
   277  func newFloat() *big.Float { return new(big.Float).SetPrec(prec) }
   278  
   279  func i64toi(x int64Val) intVal   { return intVal{newInt().SetInt64(int64(x))} }
   280  func i64tor(x int64Val) ratVal   { return ratVal{newRat().SetInt64(int64(x))} }
   281  func i64tof(x int64Val) floatVal { return floatVal{newFloat().SetInt64(int64(x))} }
   282  func itor(x intVal) ratVal       { return ratVal{newRat().SetInt(x.val)} }
   283  func itof(x intVal) floatVal     { return floatVal{newFloat().SetInt(x.val)} }
   284  func rtof(x ratVal) floatVal     { return floatVal{newFloat().SetRat(x.val)} }
   285  func vtoc(x Value) complexVal    { return complexVal{x, int64Val(0)} }
   286  
   287  func makeInt(x *big.Int) Value {
   288  	if x.IsInt64() {
   289  		return int64Val(x.Int64())
   290  	}
   291  	return intVal{x}
   292  }
   293  
   294  func makeRat(x *big.Rat) Value {
   295  	a := x.Num()
   296  	b := x.Denom()
   297  	if smallInt(a) && smallInt(b) {
   298  		// ok to remain fraction
   299  		return ratVal{x}
   300  	}
   301  	// components too large => switch to float
   302  	return floatVal{newFloat().SetRat(x)}
   303  }
   304  
   305  var floatVal0 = floatVal{newFloat()}
   306  
   307  func makeFloat(x *big.Float) Value {
   308  	// convert -0
   309  	if x.Sign() == 0 {
   310  		return floatVal0
   311  	}
   312  	if x.IsInf() {
   313  		return unknownVal{}
   314  	}
   315  	// No attempt is made to "go back" to ratVal, even if possible,
   316  	// to avoid providing the illusion of a mathematically exact
   317  	// representation.
   318  	return floatVal{x}
   319  }
   320  
   321  func makeComplex(re, im Value) Value {
   322  	if re.Kind() == Unknown || im.Kind() == Unknown {
   323  		return unknownVal{}
   324  	}
   325  	return complexVal{re, im}
   326  }
   327  
   328  func makeFloatFromLiteral(lit string) Value {
   329  	if f, ok := newFloat().SetString(lit); ok {
   330  		if smallFloat(f) {
   331  			// ok to use rationals
   332  			if f.Sign() == 0 {
   333  				// Issue 20228: If the float underflowed to zero, parse just "0".
   334  				// Otherwise, lit might contain a value with a large negative exponent,
   335  				// such as -6e-1886451601. As a float, that will underflow to 0,
   336  				// but it'll take forever to parse as a Rat.
   337  				lit = "0"
   338  			}
   339  			if r, ok := newRat().SetString(lit); ok {
   340  				return ratVal{r}
   341  			}
   342  		}
   343  		// otherwise use floats
   344  		return makeFloat(f)
   345  	}
   346  	return nil
   347  }
   348  
   349  // Permit fractions with component sizes up to maxExp
   350  // before switching to using floating-point numbers.
   351  const maxExp = 4 << 10
   352  
   353  // smallInt reports whether x would lead to "reasonably"-sized fraction
   354  // if converted to a *big.Rat.
   355  func smallInt(x *big.Int) bool {
   356  	return x.BitLen() < maxExp
   357  }
   358  
   359  // smallFloat64 reports whether x would lead to "reasonably"-sized fraction
   360  // if converted to a *big.Rat.
   361  func smallFloat64(x float64) bool {
   362  	if math.IsInf(x, 0) {
   363  		return false
   364  	}
   365  	_, e := math.Frexp(x)
   366  	return -maxExp < e && e < maxExp
   367  }
   368  
   369  // smallFloat reports whether x would lead to "reasonably"-sized fraction
   370  // if converted to a *big.Rat.
   371  func smallFloat(x *big.Float) bool {
   372  	if x.IsInf() {
   373  		return false
   374  	}
   375  	e := x.MantExp(nil)
   376  	return -maxExp < e && e < maxExp
   377  }
   378  
   379  // ----------------------------------------------------------------------------
   380  // Factories
   381  
   382  // MakeUnknown returns the [Unknown] value.
   383  func MakeUnknown() Value { return unknownVal{} }
   384  
   385  // MakeBool returns the [Bool] value for b.
   386  func MakeBool(b bool) Value { return boolVal(b) }
   387  
   388  // MakeString returns the [String] value for s.
   389  func MakeString(s string) Value {
   390  	if s == "" {
   391  		return &emptyString // common case
   392  	}
   393  	return &stringVal{s: s}
   394  }
   395  
   396  var emptyString stringVal
   397  
   398  // MakeInt64 returns the [Int] value for x.
   399  func MakeInt64(x int64) Value { return int64Val(x) }
   400  
   401  // MakeUint64 returns the [Int] value for x.
   402  func MakeUint64(x uint64) Value {
   403  	if x < 1<<63 {
   404  		return int64Val(int64(x))
   405  	}
   406  	return intVal{newInt().SetUint64(x)}
   407  }
   408  
   409  // MakeFloat64 returns the [Float] value for x.
   410  // If x is -0.0, the result is 0.0.
   411  // If x is not finite, the result is an [Unknown].
   412  func MakeFloat64(x float64) Value {
   413  	if math.IsInf(x, 0) || math.IsNaN(x) {
   414  		return unknownVal{}
   415  	}
   416  	if smallFloat64(x) {
   417  		return ratVal{newRat().SetFloat64(x + 0)} // convert -0 to 0
   418  	}
   419  	return floatVal{newFloat().SetFloat64(x + 0)}
   420  }
   421  
   422  // MakeFromLiteral returns the corresponding integer, floating-point,
   423  // imaginary, character, or string value for a Go literal string. The
   424  // tok value must be one of [token.INT], [token.FLOAT], [token.IMAG],
   425  // [token.CHAR], or [token.STRING]. The final argument must be zero.
   426  // If the literal string syntax is invalid, the result is an [Unknown].
   427  func MakeFromLiteral(lit string, tok token.Token, zero uint) Value {
   428  	if zero != 0 {
   429  		panic("MakeFromLiteral called with non-zero last argument")
   430  	}
   431  
   432  	switch tok {
   433  	case token.INT:
   434  		if x, err := strconv.ParseInt(lit, 0, 64); err == nil {
   435  			return int64Val(x)
   436  		}
   437  		if x, ok := newInt().SetString(lit, 0); ok {
   438  			return intVal{x}
   439  		}
   440  
   441  	case token.FLOAT:
   442  		if x := makeFloatFromLiteral(lit); x != nil {
   443  			return x
   444  		}
   445  
   446  	case token.IMAG:
   447  		if n := len(lit); n > 0 && lit[n-1] == 'i' {
   448  			if im := makeFloatFromLiteral(lit[:n-1]); im != nil {
   449  				return makeComplex(int64Val(0), im)
   450  			}
   451  		}
   452  
   453  	case token.CHAR:
   454  		if n := len(lit); n >= 2 {
   455  			if code, _, _, err := strconv.UnquoteChar(lit[1:n-1], '\''); err == nil {
   456  				return MakeInt64(int64(code))
   457  			}
   458  		}
   459  
   460  	case token.STRING:
   461  		if s, err := strconv.Unquote(lit); err == nil {
   462  			return MakeString(s)
   463  		}
   464  
   465  	default:
   466  		panic(fmt.Sprintf("%v is not a valid token", tok))
   467  	}
   468  
   469  	return unknownVal{}
   470  }
   471  
   472  // ----------------------------------------------------------------------------
   473  // Accessors
   474  //
   475  // For unknown arguments the result is the zero value for the respective
   476  // accessor type, except for Sign, where the result is 1.
   477  
   478  // BoolVal returns the Go boolean value of x, which must be a [Bool] or an [Unknown].
   479  // If x is [Unknown], the result is false.
   480  func BoolVal(x Value) bool {
   481  	switch x := x.(type) {
   482  	case boolVal:
   483  		return bool(x)
   484  	case unknownVal:
   485  		return false
   486  	default:
   487  		panic(fmt.Sprintf("%v not a Bool", x))
   488  	}
   489  }
   490  
   491  // StringVal returns the Go string value of x, which must be a [String] or an [Unknown].
   492  // If x is [Unknown], the result is "".
   493  func StringVal(x Value) string {
   494  	switch x := x.(type) {
   495  	case *stringVal:
   496  		return x.string()
   497  	case unknownVal:
   498  		return ""
   499  	default:
   500  		panic(fmt.Sprintf("%v not a String", x))
   501  	}
   502  }
   503  
   504  // Int64Val returns the Go int64 value of x and whether the result is exact;
   505  // x must be an [Int] or an [Unknown]. If the result is not exact, its value is undefined.
   506  // If x is [Unknown], the result is (0, false).
   507  func Int64Val(x Value) (int64, bool) {
   508  	switch x := x.(type) {
   509  	case int64Val:
   510  		return int64(x), true
   511  	case intVal:
   512  		return x.val.Int64(), false // not an int64Val and thus not exact
   513  	case unknownVal:
   514  		return 0, false
   515  	default:
   516  		panic(fmt.Sprintf("%v not an Int", x))
   517  	}
   518  }
   519  
   520  // Uint64Val returns the Go uint64 value of x and whether the result is exact;
   521  // x must be an [Int] or an [Unknown]. If the result is not exact, its value is undefined.
   522  // If x is [Unknown], the result is (0, false).
   523  func Uint64Val(x Value) (uint64, bool) {
   524  	switch x := x.(type) {
   525  	case int64Val:
   526  		return uint64(x), x >= 0
   527  	case intVal:
   528  		return x.val.Uint64(), x.val.IsUint64()
   529  	case unknownVal:
   530  		return 0, false
   531  	default:
   532  		panic(fmt.Sprintf("%v not an Int", x))
   533  	}
   534  }
   535  
   536  // Float32Val is like [Float64Val] but for float32 instead of float64.
   537  func Float32Val(x Value) (float32, bool) {
   538  	switch x := x.(type) {
   539  	case int64Val:
   540  		f := float32(x)
   541  		return f, int64Val(f) == x
   542  	case intVal:
   543  		f, acc := newFloat().SetInt(x.val).Float32()
   544  		return f, acc == big.Exact
   545  	case ratVal:
   546  		return x.val.Float32()
   547  	case floatVal:
   548  		f, acc := x.val.Float32()
   549  		return f, acc == big.Exact
   550  	case unknownVal:
   551  		return 0, false
   552  	default:
   553  		panic(fmt.Sprintf("%v not a Float", x))
   554  	}
   555  }
   556  
   557  // Float64Val returns the nearest Go float64 value of x and whether the result is exact;
   558  // x must be numeric or an [Unknown], but not [Complex]. For values too small (too close to 0)
   559  // to represent as float64, [Float64Val] silently underflows to 0. The result sign always
   560  // matches the sign of x, even for 0.
   561  // If x is [Unknown], the result is (0, false).
   562  func Float64Val(x Value) (float64, bool) {
   563  	switch x := x.(type) {
   564  	case int64Val:
   565  		f := float64(int64(x))
   566  		return f, int64Val(f) == x
   567  	case intVal:
   568  		f, acc := newFloat().SetInt(x.val).Float64()
   569  		return f, acc == big.Exact
   570  	case ratVal:
   571  		return x.val.Float64()
   572  	case floatVal:
   573  		f, acc := x.val.Float64()
   574  		return f, acc == big.Exact
   575  	case unknownVal:
   576  		return 0, false
   577  	default:
   578  		panic(fmt.Sprintf("%v not a Float", x))
   579  	}
   580  }
   581  
   582  // Val returns the underlying value for a given constant. Since it returns an
   583  // interface, it is up to the caller to type assert the result to the expected
   584  // type. The possible dynamic return types are:
   585  //
   586  //	x Kind             type of result
   587  //	-----------------------------------------
   588  //	Bool               bool
   589  //	String             string
   590  //	Int                int64 or *big.Int
   591  //	Float              *big.Float or *big.Rat
   592  //	everything else    nil
   593  func Val(x Value) any {
   594  	switch x := x.(type) {
   595  	case boolVal:
   596  		return bool(x)
   597  	case *stringVal:
   598  		return x.string()
   599  	case int64Val:
   600  		return int64(x)
   601  	case intVal:
   602  		return x.val
   603  	case ratVal:
   604  		return x.val
   605  	case floatVal:
   606  		return x.val
   607  	default:
   608  		return nil
   609  	}
   610  }
   611  
   612  // Make returns the [Value] for x.
   613  //
   614  //	type of x        result Kind
   615  //	----------------------------
   616  //	bool             Bool
   617  //	string           String
   618  //	int64            Int
   619  //	*big.Int         Int
   620  //	*big.Float       Float
   621  //	*big.Rat         Float
   622  //	anything else    Unknown
   623  func Make(x any) Value {
   624  	switch x := x.(type) {
   625  	case bool:
   626  		return boolVal(x)
   627  	case string:
   628  		return &stringVal{s: x}
   629  	case int64:
   630  		return int64Val(x)
   631  	case *big.Int:
   632  		return makeInt(x)
   633  	case *big.Rat:
   634  		return makeRat(x)
   635  	case *big.Float:
   636  		return makeFloat(x)
   637  	default:
   638  		return unknownVal{}
   639  	}
   640  }
   641  
   642  // BitLen returns the number of bits required to represent
   643  // the absolute value x in binary representation; x must be an [Int] or an [Unknown].
   644  // If x is [Unknown], the result is 0.
   645  func BitLen(x Value) int {
   646  	switch x := x.(type) {
   647  	case int64Val:
   648  		u := uint64(x)
   649  		if x < 0 {
   650  			u = uint64(-x)
   651  		}
   652  		return 64 - bits.LeadingZeros64(u)
   653  	case intVal:
   654  		return x.val.BitLen()
   655  	case unknownVal:
   656  		return 0
   657  	default:
   658  		panic(fmt.Sprintf("%v not an Int", x))
   659  	}
   660  }
   661  
   662  // Sign returns -1, 0, or 1 depending on whether x < 0, x == 0, or x > 0;
   663  // x must be numeric or [Unknown]. For complex values x, the sign is 0 if x == 0,
   664  // otherwise it is != 0. If x is [Unknown], the result is 1.
   665  func Sign(x Value) int {
   666  	switch x := x.(type) {
   667  	case int64Val:
   668  		switch {
   669  		case x < 0:
   670  			return -1
   671  		case x > 0:
   672  			return 1
   673  		}
   674  		return 0
   675  	case intVal:
   676  		return x.val.Sign()
   677  	case ratVal:
   678  		return x.val.Sign()
   679  	case floatVal:
   680  		return x.val.Sign()
   681  	case complexVal:
   682  		return Sign(x.re) | Sign(x.im)
   683  	case unknownVal:
   684  		return 1 // avoid spurious division by zero errors
   685  	default:
   686  		panic(fmt.Sprintf("%v not numeric", x))
   687  	}
   688  }
   689  
   690  // ----------------------------------------------------------------------------
   691  // Support for assembling/disassembling numeric values
   692  
   693  const (
   694  	// Compute the size of a Word in bytes.
   695  	_m       = ^big.Word(0)
   696  	_log     = _m>>8&1 + _m>>16&1 + _m>>32&1
   697  	wordSize = 1 << _log
   698  )
   699  
   700  // Bytes returns the bytes for the absolute value of x in little-
   701  // endian binary representation; x must be an [Int].
   702  func Bytes(x Value) []byte {
   703  	var t intVal
   704  	switch x := x.(type) {
   705  	case int64Val:
   706  		t = i64toi(x)
   707  	case intVal:
   708  		t = x
   709  	default:
   710  		panic(fmt.Sprintf("%v not an Int", x))
   711  	}
   712  
   713  	words := t.val.Bits()
   714  	bytes := make([]byte, len(words)*wordSize)
   715  
   716  	i := 0
   717  	for _, w := range words {
   718  		for j := 0; j < wordSize; j++ {
   719  			bytes[i] = byte(w)
   720  			w >>= 8
   721  			i++
   722  		}
   723  	}
   724  	// remove leading 0's
   725  	for i > 0 && bytes[i-1] == 0 {
   726  		i--
   727  	}
   728  
   729  	return bytes[:i]
   730  }
   731  
   732  // MakeFromBytes returns the [Int] value given the bytes of its little-endian
   733  // binary representation. An empty byte slice argument represents 0.
   734  func MakeFromBytes(bytes []byte) Value {
   735  	words := make([]big.Word, (len(bytes)+(wordSize-1))/wordSize)
   736  
   737  	i := 0
   738  	var w big.Word
   739  	var s uint
   740  	for _, b := range bytes {
   741  		w |= big.Word(b) << s
   742  		if s += 8; s == wordSize*8 {
   743  			words[i] = w
   744  			i++
   745  			w = 0
   746  			s = 0
   747  		}
   748  	}
   749  	// store last word
   750  	if i < len(words) {
   751  		words[i] = w
   752  		i++
   753  	}
   754  	// remove leading 0's
   755  	for i > 0 && words[i-1] == 0 {
   756  		i--
   757  	}
   758  
   759  	return makeInt(newInt().SetBits(words[:i]))
   760  }
   761  
   762  // Num returns the numerator of x; x must be [Int], [Float], or [Unknown].
   763  // If x is [Unknown], or if it is too large or small to represent as a
   764  // fraction, the result is [Unknown]. Otherwise the result is an [Int]
   765  // with the same sign as x.
   766  func Num(x Value) Value {
   767  	switch x := x.(type) {
   768  	case int64Val, intVal:
   769  		return x
   770  	case ratVal:
   771  		return makeInt(x.val.Num())
   772  	case floatVal:
   773  		if smallFloat(x.val) {
   774  			r, _ := x.val.Rat(nil)
   775  			return makeInt(r.Num())
   776  		}
   777  	case unknownVal:
   778  		break
   779  	default:
   780  		panic(fmt.Sprintf("%v not Int or Float", x))
   781  	}
   782  	return unknownVal{}
   783  }
   784  
   785  // Denom returns the denominator of x; x must be [Int], [Float], or [Unknown].
   786  // If x is [Unknown], or if it is too large or small to represent as a
   787  // fraction, the result is [Unknown]. Otherwise the result is an [Int] >= 1.
   788  func Denom(x Value) Value {
   789  	switch x := x.(type) {
   790  	case int64Val, intVal:
   791  		return int64Val(1)
   792  	case ratVal:
   793  		return makeInt(x.val.Denom())
   794  	case floatVal:
   795  		if smallFloat(x.val) {
   796  			r, _ := x.val.Rat(nil)
   797  			return makeInt(r.Denom())
   798  		}
   799  	case unknownVal:
   800  		break
   801  	default:
   802  		panic(fmt.Sprintf("%v not Int or Float", x))
   803  	}
   804  	return unknownVal{}
   805  }
   806  
   807  // MakeImag returns the [Complex] value x*i;
   808  // x must be [Int], [Float], or [Unknown].
   809  // If x is [Unknown], the result is [Unknown].
   810  func MakeImag(x Value) Value {
   811  	switch x.(type) {
   812  	case unknownVal:
   813  		return x
   814  	case int64Val, intVal, ratVal, floatVal:
   815  		return makeComplex(int64Val(0), x)
   816  	default:
   817  		panic(fmt.Sprintf("%v not Int or Float", x))
   818  	}
   819  }
   820  
   821  // Real returns the real part of x, which must be a numeric or unknown value.
   822  // If x is [Unknown], the result is [Unknown].
   823  func Real(x Value) Value {
   824  	switch x := x.(type) {
   825  	case unknownVal, int64Val, intVal, ratVal, floatVal:
   826  		return x
   827  	case complexVal:
   828  		return x.re
   829  	default:
   830  		panic(fmt.Sprintf("%v not numeric", x))
   831  	}
   832  }
   833  
   834  // Imag returns the imaginary part of x, which must be a numeric or unknown value.
   835  // If x is [Unknown], the result is [Unknown].
   836  func Imag(x Value) Value {
   837  	switch x := x.(type) {
   838  	case unknownVal:
   839  		return x
   840  	case int64Val, intVal, ratVal, floatVal:
   841  		return int64Val(0)
   842  	case complexVal:
   843  		return x.im
   844  	default:
   845  		panic(fmt.Sprintf("%v not numeric", x))
   846  	}
   847  }
   848  
   849  // ----------------------------------------------------------------------------
   850  // Numeric conversions
   851  
   852  // ToInt converts x to an [Int] value if x is representable as an [Int].
   853  // Otherwise it returns an [Unknown].
   854  func ToInt(x Value) Value {
   855  	switch x := x.(type) {
   856  	case int64Val, intVal:
   857  		return x
   858  
   859  	case ratVal:
   860  		if x.val.IsInt() {
   861  			return makeInt(x.val.Num())
   862  		}
   863  
   864  	case floatVal:
   865  		// avoid creation of huge integers
   866  		// (Existing tests require permitting exponents of at least 1024;
   867  		// allow any value that would also be permissible as a fraction.)
   868  		if smallFloat(x.val) {
   869  			i := newInt()
   870  			if _, acc := x.val.Int(i); acc == big.Exact {
   871  				return makeInt(i)
   872  			}
   873  
   874  			// If we can get an integer by rounding up or down,
   875  			// assume x is not an integer because of rounding
   876  			// errors in prior computations.
   877  
   878  			const delta = 4 // a small number of bits > 0
   879  			var t big.Float
   880  			t.SetPrec(prec - delta)
   881  
   882  			// try rounding down a little
   883  			t.SetMode(big.ToZero)
   884  			t.Set(x.val)
   885  			if _, acc := t.Int(i); acc == big.Exact {
   886  				return makeInt(i)
   887  			}
   888  
   889  			// try rounding up a little
   890  			t.SetMode(big.AwayFromZero)
   891  			t.Set(x.val)
   892  			if _, acc := t.Int(i); acc == big.Exact {
   893  				return makeInt(i)
   894  			}
   895  		}
   896  
   897  	case complexVal:
   898  		if re := ToFloat(x); re.Kind() == Float {
   899  			return ToInt(re)
   900  		}
   901  	}
   902  
   903  	return unknownVal{}
   904  }
   905  
   906  // ToFloat converts x to a [Float] value if x is representable as a [Float].
   907  // Otherwise it returns an [Unknown].
   908  func ToFloat(x Value) Value {
   909  	switch x := x.(type) {
   910  	case int64Val:
   911  		return i64tor(x) // x is always a small int
   912  	case intVal:
   913  		if smallInt(x.val) {
   914  			return itor(x)
   915  		}
   916  		return itof(x)
   917  	case ratVal, floatVal:
   918  		return x
   919  	case complexVal:
   920  		if Sign(x.im) == 0 {
   921  			return ToFloat(x.re)
   922  		}
   923  	}
   924  	return unknownVal{}
   925  }
   926  
   927  // ToComplex converts x to a [Complex] value if x is representable as a [Complex].
   928  // Otherwise it returns an [Unknown].
   929  func ToComplex(x Value) Value {
   930  	switch x := x.(type) {
   931  	case int64Val, intVal, ratVal, floatVal:
   932  		return vtoc(x)
   933  	case complexVal:
   934  		return x
   935  	}
   936  	return unknownVal{}
   937  }
   938  
   939  // ----------------------------------------------------------------------------
   940  // Operations
   941  
   942  // is32bit reports whether x can be represented using 32 bits.
   943  func is32bit(x int64) bool {
   944  	const s = 32
   945  	return -1<<(s-1) <= x && x <= 1<<(s-1)-1
   946  }
   947  
   948  // is63bit reports whether x can be represented using 63 bits.
   949  func is63bit(x int64) bool {
   950  	const s = 63
   951  	return -1<<(s-1) <= x && x <= 1<<(s-1)-1
   952  }
   953  
   954  // UnaryOp returns the result of the unary expression op y.
   955  // The operation must be defined for the operand.
   956  // If prec > 0 it specifies the ^ (xor) result size in bits.
   957  // If y is [Unknown], the result is [Unknown].
   958  func UnaryOp(op token.Token, y Value, prec uint) Value {
   959  	switch op {
   960  	case token.ADD:
   961  		switch y.(type) {
   962  		case unknownVal, int64Val, intVal, ratVal, floatVal, complexVal:
   963  			return y
   964  		}
   965  
   966  	case token.SUB:
   967  		switch y := y.(type) {
   968  		case unknownVal:
   969  			return y
   970  		case int64Val:
   971  			if z := -y; z != y {
   972  				return z // no overflow
   973  			}
   974  			return makeInt(newInt().Neg(big.NewInt(int64(y))))
   975  		case intVal:
   976  			return makeInt(newInt().Neg(y.val))
   977  		case ratVal:
   978  			return makeRat(newRat().Neg(y.val))
   979  		case floatVal:
   980  			return makeFloat(newFloat().Neg(y.val))
   981  		case complexVal:
   982  			re := UnaryOp(token.SUB, y.re, 0)
   983  			im := UnaryOp(token.SUB, y.im, 0)
   984  			return makeComplex(re, im)
   985  		}
   986  
   987  	case token.XOR:
   988  		z := newInt()
   989  		switch y := y.(type) {
   990  		case unknownVal:
   991  			return y
   992  		case int64Val:
   993  			z.Not(big.NewInt(int64(y)))
   994  		case intVal:
   995  			z.Not(y.val)
   996  		default:
   997  			goto Error
   998  		}
   999  		// For unsigned types, the result will be negative and
  1000  		// thus "too large": We must limit the result precision
  1001  		// to the type's precision.
  1002  		if prec > 0 {
  1003  			z.AndNot(z, newInt().Lsh(big.NewInt(-1), prec)) // z &^= (-1)<<prec
  1004  		}
  1005  		return makeInt(z)
  1006  
  1007  	case token.NOT:
  1008  		switch y := y.(type) {
  1009  		case unknownVal:
  1010  			return y
  1011  		case boolVal:
  1012  			return !y
  1013  		}
  1014  	}
  1015  
  1016  Error:
  1017  	panic(fmt.Sprintf("invalid unary operation %s%v", op, y))
  1018  }
  1019  
  1020  func ord(x Value) int {
  1021  	switch x.(type) {
  1022  	default:
  1023  		// force invalid value into "x position" in match
  1024  		// (don't panic here so that callers can provide a better error message)
  1025  		return -1
  1026  	case unknownVal:
  1027  		return 0
  1028  	case boolVal, *stringVal:
  1029  		return 1
  1030  	case int64Val:
  1031  		return 2
  1032  	case intVal:
  1033  		return 3
  1034  	case ratVal:
  1035  		return 4
  1036  	case floatVal:
  1037  		return 5
  1038  	case complexVal:
  1039  		return 6
  1040  	}
  1041  }
  1042  
  1043  // match returns the matching representation (same type) with the
  1044  // smallest complexity for two values x and y. If one of them is
  1045  // numeric, both of them must be numeric. If one of them is Unknown
  1046  // or invalid (say, nil) both results are that value.
  1047  func match(x, y Value) (_, _ Value) {
  1048  	switch ox, oy := ord(x), ord(y); {
  1049  	case ox < oy:
  1050  		x, y = match0(x, y)
  1051  	case ox > oy:
  1052  		y, x = match0(y, x)
  1053  	}
  1054  	return x, y
  1055  }
  1056  
  1057  // match0 must only be called by match.
  1058  // Invariant: ord(x) < ord(y)
  1059  func match0(x, y Value) (_, _ Value) {
  1060  	// Prefer to return the original x and y arguments when possible,
  1061  	// to avoid unnecessary heap allocations.
  1062  
  1063  	switch y.(type) {
  1064  	case intVal:
  1065  		switch x1 := x.(type) {
  1066  		case int64Val:
  1067  			return i64toi(x1), y
  1068  		}
  1069  	case ratVal:
  1070  		switch x1 := x.(type) {
  1071  		case int64Val:
  1072  			return i64tor(x1), y
  1073  		case intVal:
  1074  			return itor(x1), y
  1075  		}
  1076  	case floatVal:
  1077  		switch x1 := x.(type) {
  1078  		case int64Val:
  1079  			return i64tof(x1), y
  1080  		case intVal:
  1081  			return itof(x1), y
  1082  		case ratVal:
  1083  			return rtof(x1), y
  1084  		}
  1085  	case complexVal:
  1086  		switch x1 := x.(type) {
  1087  		case int64Val, intVal, ratVal, floatVal:
  1088  			return vtoc(x1), y
  1089  		}
  1090  	}
  1091  
  1092  	// force unknown and invalid values into "x position" in callers of match
  1093  	// (don't panic here so that callers can provide a better error message)
  1094  	return x, x
  1095  }
  1096  
  1097  // BinaryOp returns the result of the binary expression x op y.
  1098  // The operation must be defined for the operands. If one of the
  1099  // operands is [Unknown], the result is [Unknown].
  1100  // BinaryOp doesn't handle comparisons or shifts; use [Compare]
  1101  // or [Shift] instead.
  1102  //
  1103  // To force integer division of [Int] operands, use op == [token.QUO_ASSIGN]
  1104  // instead of [token.QUO]; the result is guaranteed to be [Int] in this case.
  1105  // Division by zero leads to a run-time panic.
  1106  func BinaryOp(x_ Value, op token.Token, y_ Value) Value {
  1107  	x, y := match(x_, y_)
  1108  
  1109  	switch x := x.(type) {
  1110  	case unknownVal:
  1111  		return x
  1112  
  1113  	case boolVal:
  1114  		y := y.(boolVal)
  1115  		switch op {
  1116  		case token.LAND:
  1117  			return x && y
  1118  		case token.LOR:
  1119  			return x || y
  1120  		}
  1121  
  1122  	case int64Val:
  1123  		a := int64(x)
  1124  		b := int64(y.(int64Val))
  1125  		var c int64
  1126  		switch op {
  1127  		case token.ADD:
  1128  			if !is63bit(a) || !is63bit(b) {
  1129  				return makeInt(newInt().Add(big.NewInt(a), big.NewInt(b)))
  1130  			}
  1131  			c = a + b
  1132  		case token.SUB:
  1133  			if !is63bit(a) || !is63bit(b) {
  1134  				return makeInt(newInt().Sub(big.NewInt(a), big.NewInt(b)))
  1135  			}
  1136  			c = a - b
  1137  		case token.MUL:
  1138  			if !is32bit(a) || !is32bit(b) {
  1139  				return makeInt(newInt().Mul(big.NewInt(a), big.NewInt(b)))
  1140  			}
  1141  			c = a * b
  1142  		case token.QUO:
  1143  			return makeRat(big.NewRat(a, b))
  1144  		case token.QUO_ASSIGN: // force integer division
  1145  			c = a / b
  1146  		case token.REM:
  1147  			c = a % b
  1148  		case token.AND:
  1149  			c = a & b
  1150  		case token.OR:
  1151  			c = a | b
  1152  		case token.XOR:
  1153  			c = a ^ b
  1154  		case token.AND_NOT:
  1155  			c = a &^ b
  1156  		default:
  1157  			goto Error
  1158  		}
  1159  		return int64Val(c)
  1160  
  1161  	case intVal:
  1162  		a := x.val
  1163  		b := y.(intVal).val
  1164  		c := newInt()
  1165  		switch op {
  1166  		case token.ADD:
  1167  			c.Add(a, b)
  1168  		case token.SUB:
  1169  			c.Sub(a, b)
  1170  		case token.MUL:
  1171  			c.Mul(a, b)
  1172  		case token.QUO:
  1173  			return makeRat(newRat().SetFrac(a, b))
  1174  		case token.QUO_ASSIGN: // force integer division
  1175  			c.Quo(a, b)
  1176  		case token.REM:
  1177  			c.Rem(a, b)
  1178  		case token.AND:
  1179  			c.And(a, b)
  1180  		case token.OR:
  1181  			c.Or(a, b)
  1182  		case token.XOR:
  1183  			c.Xor(a, b)
  1184  		case token.AND_NOT:
  1185  			c.AndNot(a, b)
  1186  		default:
  1187  			goto Error
  1188  		}
  1189  		return makeInt(c)
  1190  
  1191  	case ratVal:
  1192  		a := x.val
  1193  		b := y.(ratVal).val
  1194  		c := newRat()
  1195  		switch op {
  1196  		case token.ADD:
  1197  			c.Add(a, b)
  1198  		case token.SUB:
  1199  			c.Sub(a, b)
  1200  		case token.MUL:
  1201  			c.Mul(a, b)
  1202  		case token.QUO:
  1203  			c.Quo(a, b)
  1204  		default:
  1205  			goto Error
  1206  		}
  1207  		return makeRat(c)
  1208  
  1209  	case floatVal:
  1210  		a := x.val
  1211  		b := y.(floatVal).val
  1212  		c := newFloat()
  1213  		switch op {
  1214  		case token.ADD:
  1215  			c.Add(a, b)
  1216  		case token.SUB:
  1217  			c.Sub(a, b)
  1218  		case token.MUL:
  1219  			c.Mul(a, b)
  1220  		case token.QUO:
  1221  			c.Quo(a, b)
  1222  		default:
  1223  			goto Error
  1224  		}
  1225  		return makeFloat(c)
  1226  
  1227  	case complexVal:
  1228  		y := y.(complexVal)
  1229  		a, b := x.re, x.im
  1230  		c, d := y.re, y.im
  1231  		var re, im Value
  1232  		switch op {
  1233  		case token.ADD:
  1234  			// (a+c) + i(b+d)
  1235  			re = add(a, c)
  1236  			im = add(b, d)
  1237  		case token.SUB:
  1238  			// (a-c) + i(b-d)
  1239  			re = sub(a, c)
  1240  			im = sub(b, d)
  1241  		case token.MUL:
  1242  			// (ac-bd) + i(bc+ad)
  1243  			ac := mul(a, c)
  1244  			bd := mul(b, d)
  1245  			bc := mul(b, c)
  1246  			ad := mul(a, d)
  1247  			re = sub(ac, bd)
  1248  			im = add(bc, ad)
  1249  		case token.QUO:
  1250  			// (ac+bd)/s + i(bc-ad)/s, with s = cc + dd
  1251  			ac := mul(a, c)
  1252  			bd := mul(b, d)
  1253  			bc := mul(b, c)
  1254  			ad := mul(a, d)
  1255  			cc := mul(c, c)
  1256  			dd := mul(d, d)
  1257  			s := add(cc, dd)
  1258  			re = add(ac, bd)
  1259  			re = quo(re, s)
  1260  			im = sub(bc, ad)
  1261  			im = quo(im, s)
  1262  		default:
  1263  			goto Error
  1264  		}
  1265  		return makeComplex(re, im)
  1266  
  1267  	case *stringVal:
  1268  		if op == token.ADD {
  1269  			return &stringVal{l: x, r: y.(*stringVal)}
  1270  		}
  1271  	}
  1272  
  1273  Error:
  1274  	panic(fmt.Sprintf("invalid binary operation %v %s %v", x_, op, y_))
  1275  }
  1276  
  1277  func add(x, y Value) Value { return BinaryOp(x, token.ADD, y) }
  1278  func sub(x, y Value) Value { return BinaryOp(x, token.SUB, y) }
  1279  func mul(x, y Value) Value { return BinaryOp(x, token.MUL, y) }
  1280  func quo(x, y Value) Value { return BinaryOp(x, token.QUO, y) }
  1281  
  1282  // Shift returns the result of the shift expression x op s
  1283  // with op == [token.SHL] or [token.SHR] (<< or >>). x must be
  1284  // an [Int] or an [Unknown]. If x is [Unknown], the result is x.
  1285  func Shift(x Value, op token.Token, s uint) Value {
  1286  	switch x := x.(type) {
  1287  	case unknownVal:
  1288  		return x
  1289  
  1290  	case int64Val:
  1291  		if s == 0 {
  1292  			return x
  1293  		}
  1294  		switch op {
  1295  		case token.SHL:
  1296  			z := i64toi(x).val
  1297  			return makeInt(z.Lsh(z, s))
  1298  		case token.SHR:
  1299  			return x >> s
  1300  		}
  1301  
  1302  	case intVal:
  1303  		if s == 0 {
  1304  			return x
  1305  		}
  1306  		z := newInt()
  1307  		switch op {
  1308  		case token.SHL:
  1309  			return makeInt(z.Lsh(x.val, s))
  1310  		case token.SHR:
  1311  			return makeInt(z.Rsh(x.val, s))
  1312  		}
  1313  	}
  1314  
  1315  	panic(fmt.Sprintf("invalid shift %v %s %d", x, op, s))
  1316  }
  1317  
  1318  func cmpZero(x int, op token.Token) bool {
  1319  	switch op {
  1320  	case token.EQL:
  1321  		return x == 0
  1322  	case token.NEQ:
  1323  		return x != 0
  1324  	case token.LSS:
  1325  		return x < 0
  1326  	case token.LEQ:
  1327  		return x <= 0
  1328  	case token.GTR:
  1329  		return x > 0
  1330  	case token.GEQ:
  1331  		return x >= 0
  1332  	}
  1333  	panic(fmt.Sprintf("invalid comparison %v %s 0", x, op))
  1334  }
  1335  
  1336  // Compare returns the result of the comparison x op y.
  1337  // The comparison must be defined for the operands.
  1338  // If one of the operands is [Unknown], the result is
  1339  // false.
  1340  func Compare(x_ Value, op token.Token, y_ Value) bool {
  1341  	x, y := match(x_, y_)
  1342  
  1343  	switch x := x.(type) {
  1344  	case unknownVal:
  1345  		return false
  1346  
  1347  	case boolVal:
  1348  		y := y.(boolVal)
  1349  		switch op {
  1350  		case token.EQL:
  1351  			return x == y
  1352  		case token.NEQ:
  1353  			return x != y
  1354  		}
  1355  
  1356  	case int64Val:
  1357  		y := y.(int64Val)
  1358  		switch op {
  1359  		case token.EQL:
  1360  			return x == y
  1361  		case token.NEQ:
  1362  			return x != y
  1363  		case token.LSS:
  1364  			return x < y
  1365  		case token.LEQ:
  1366  			return x <= y
  1367  		case token.GTR:
  1368  			return x > y
  1369  		case token.GEQ:
  1370  			return x >= y
  1371  		}
  1372  
  1373  	case intVal:
  1374  		return cmpZero(x.val.Cmp(y.(intVal).val), op)
  1375  
  1376  	case ratVal:
  1377  		return cmpZero(x.val.Cmp(y.(ratVal).val), op)
  1378  
  1379  	case floatVal:
  1380  		return cmpZero(x.val.Cmp(y.(floatVal).val), op)
  1381  
  1382  	case complexVal:
  1383  		y := y.(complexVal)
  1384  		re := Compare(x.re, token.EQL, y.re)
  1385  		im := Compare(x.im, token.EQL, y.im)
  1386  		switch op {
  1387  		case token.EQL:
  1388  			return re && im
  1389  		case token.NEQ:
  1390  			return !re || !im
  1391  		}
  1392  
  1393  	case *stringVal:
  1394  		xs := x.string()
  1395  		ys := y.(*stringVal).string()
  1396  		switch op {
  1397  		case token.EQL:
  1398  			return xs == ys
  1399  		case token.NEQ:
  1400  			return xs != ys
  1401  		case token.LSS:
  1402  			return xs < ys
  1403  		case token.LEQ:
  1404  			return xs <= ys
  1405  		case token.GTR:
  1406  			return xs > ys
  1407  		case token.GEQ:
  1408  			return xs >= ys
  1409  		}
  1410  	}
  1411  
  1412  	panic(fmt.Sprintf("invalid comparison %v %s %v", x_, op, y_))
  1413  }
  1414
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