libgo/go/big/int.go

   1 // Copyright 2009 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 signed multi-precision integers.
   6
   7 package big
   8
   9 import (
  10         "fmt"
  11         "rand"
  12 )
  13
  14 // An Int represents a signed multi-precision integer.
  15 // The zero value for an Int represents the value 0.
  16 type Int struct {
  17         neg bool // sign
  18         abs nat  // absolute value of the integer
  19 }
  20
  21
  22 var intOne = &Int{false, natOne}
  23
  24
  25 // Sign returns:
  26 //
  27 //      -1 if x <  0
  28 //       0 if x == 0
  29 //      +1 if x >  0
  30 //
  31 func (x *Int) Sign() int {
  32         if len(x.abs) == 0 {
  33                 return 0
  34         }
  35         if x.neg {
  36                 return -1
  37         }
  38         return 1
  39 }
  40
  41
  42 // SetInt64 sets z to x and returns z.
  43 func (z *Int) SetInt64(x int64) *Int {
  44         neg := false
  45         if x < 0 {
  46                 neg = true
  47                 x = -x
  48         }
  49         z.abs = z.abs.setUint64(uint64(x))
  50         z.neg = neg
  51         return z
  52 }
  53
  54
  55 // NewInt allocates and returns a new Int set to x.
  56 func NewInt(x int64) *Int {
  57         return new(Int).SetInt64(x)
  58 }
  59
  60
  61 // Set sets z to x and returns z.
  62 func (z *Int) Set(x *Int) *Int {
  63         z.abs = z.abs.set(x.abs)
  64         z.neg = x.neg
  65         return z
  66 }
  67
  68
  69 // Abs sets z to |x| (the absolute value of x) and returns z.
  70 func (z *Int) Abs(x *Int) *Int {
  71         z.abs = z.abs.set(x.abs)
  72         z.neg = false
  73         return z
  74 }
  75
  76
  77 // Neg sets z to -x and returns z.
  78 func (z *Int) Neg(x *Int) *Int {
  79         z.abs = z.abs.set(x.abs)
  80         z.neg = len(z.abs) > 0 && !x.neg // 0 has no sign
  81         return z
  82 }
  83
  84
  85 // Add sets z to the sum x+y and returns z.
  86 func (z *Int) Add(x, y *Int) *Int {
  87         neg := x.neg
  88         if x.neg == y.neg {
  89                 // x + y == x + y
  90                 // (-x) + (-y) == -(x + y)
  91                 z.abs = z.abs.add(x.abs, y.abs)
  92         } else {
  93                 // x + (-y) == x - y == -(y - x)
  94                 // (-x) + y == y - x == -(x - y)
  95                 if x.abs.cmp(y.abs) >= 0 {
  96                         z.abs = z.abs.sub(x.abs, y.abs)
  97                 } else {
  98                         neg = !neg
  99                         z.abs = z.abs.sub(y.abs, x.abs)
 100                 }
 101         }
 102         z.neg = len(z.abs) > 0 && neg // 0 has no sign
 103         return z
 104 }
 105
 106
 107 // Sub sets z to the difference x-y and returns z.
 108 func (z *Int) Sub(x, y *Int) *Int {
 109         neg := x.neg
 110         if x.neg != y.neg {
 111                 // x - (-y) == x + y
 112                 // (-x) - y == -(x + y)
 113                 z.abs = z.abs.add(x.abs, y.abs)
 114         } else {
 115                 // x - y == x - y == -(y - x)
 116                 // (-x) - (-y) == y - x == -(x - y)
 117                 if x.abs.cmp(y.abs) >= 0 {
 118                         z.abs = z.abs.sub(x.abs, y.abs)
 119                 } else {
 120                         neg = !neg
 121                         z.abs = z.abs.sub(y.abs, x.abs)
 122                 }
 123         }
 124         z.neg = len(z.abs) > 0 && neg // 0 has no sign
 125         return z
 126 }
 127
 128
 129 // Mul sets z to the product x*y and returns z.
 130 func (z *Int) Mul(x, y *Int) *Int {
 131         // x * y == x * y
 132         // x * (-y) == -(x * y)
 133         // (-x) * y == -(x * y)
 134         // (-x) * (-y) == x * y
 135         z.abs = z.abs.mul(x.abs, y.abs)
 136         z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
 137         return z
 138 }
 139
 140
 141 // MulRange sets z to the product of all integers
 142 // in the range [a, b] inclusively and returns z.
 143 // If a > b (empty range), the result is 1.
 144 func (z *Int) MulRange(a, b int64) *Int {
 145         switch {
 146         case a > b:
 147                 return z.SetInt64(1) // empty range
 148         case a <= 0 && b >= 0:
 149                 return z.SetInt64(0) // range includes 0
 150         }
 151         // a <= b && (b < 0 || a > 0)
 152
 153         neg := false
 154         if a < 0 {
 155                 neg = (b-a)&1 == 0
 156                 a, b = -b, -a
 157         }
 158
 159         z.abs = z.abs.mulRange(uint64(a), uint64(b))
 160         z.neg = neg
 161         return z
 162 }
 163
 164
 165 // Binomial sets z to the binomial coefficient of (n, k) and returns z.
 166 func (z *Int) Binomial(n, k int64) *Int {
 167         var a, b Int
 168         a.MulRange(n-k+1, n)
 169         b.MulRange(1, k)
 170         return z.Quo(&a, &b)
 171 }
 172
 173
 174 // Quo sets z to the quotient x/y for y != 0 and returns z.
 175 // If y == 0, a division-by-zero run-time panic occurs.
 176 // See QuoRem for more details.
 177 func (z *Int) Quo(x, y *Int) *Int {
 178         z.abs, _ = z.abs.div(nil, x.abs, y.abs)
 179         z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
 180         return z
 181 }
 182
 183
 184 // Rem sets z to the remainder x%y for y != 0 and returns z.
 185 // If y == 0, a division-by-zero run-time panic occurs.
 186 // See QuoRem for more details.
 187 func (z *Int) Rem(x, y *Int) *Int {
 188         _, z.abs = nat(nil).div(z.abs, x.abs, y.abs)
 189         z.neg = len(z.abs) > 0 && x.neg // 0 has no sign
 190         return z
 191 }
 192
 193
 194 // QuoRem sets z to the quotient x/y and r to the remainder x%y
 195 // and returns the pair (z, r) for y != 0.
 196 // If y == 0, a division-by-zero run-time panic occurs.
 197 //
 198 // QuoRem implements T-division and modulus (like Go):
 199 //
 200 //      q = x/y      with the result truncated to zero
 201 //      r = x - y*q
 202 //
 203 // (See Daan Leijen, ``Division and Modulus for Computer Scientists''.)
 204 //
 205 func (z *Int) QuoRem(x, y, r *Int) (*Int, *Int) {
 206         z.abs, r.abs = z.abs.div(r.abs, x.abs, y.abs)
 207         z.neg, r.neg = len(z.abs) > 0 && x.neg != y.neg, len(r.abs) > 0 && x.neg // 0 has no sign
 208         return z, r
 209 }
 210
 211
 212 // Div sets z to the quotient x/y for y != 0 and returns z.
 213 // If y == 0, a division-by-zero run-time panic occurs.
 214 // See DivMod for more details.
 215 func (z *Int) Div(x, y *Int) *Int {
 216         y_neg := y.neg // z may be an alias for y
 217         var r Int
 218         z.QuoRem(x, y, &r)
 219         if r.neg {
 220                 if y_neg {
 221                         z.Add(z, intOne)
 222                 } else {
 223                         z.Sub(z, intOne)
 224                 }
 225         }
 226         return z
 227 }
 228
 229
 230 // Mod sets z to the modulus x%y for y != 0 and returns z.
 231 // If y == 0, a division-by-zero run-time panic occurs.
 232 // See DivMod for more details.
 233 func (z *Int) Mod(x, y *Int) *Int {
 234         y0 := y // save y
 235         if z == y || alias(z.abs, y.abs) {
 236                 y0 = new(Int).Set(y)
 237         }
 238         var q Int
 239         q.QuoRem(x, y, z)
 240         if z.neg {
 241                 if y0.neg {
 242                         z.Sub(z, y0)
 243                 } else {
 244                         z.Add(z, y0)
 245                 }
 246         }
 247         return z
 248 }
 249
 250
 251 // DivMod sets z to the quotient x div y and m to the modulus x mod y
 252 // and returns the pair (z, m) for y != 0.
 253 // If y == 0, a division-by-zero run-time panic occurs.
 254 //
 255 // DivMod implements Euclidean division and modulus (unlike Go):
 256 //
 257 //      q = x div y  such that
 258 //      m = x - y*q  with 0 <= m < |q|
 259 //
 260 // (See Raymond T. Boute, ``The Euclidean definition of the functions
 261 // div and mod''. ACM Transactions on Programming Languages and
 262 // Systems (TOPLAS), 14(2):127-144, New York, NY, USA, 4/1992.
 263 // ACM press.)
 264 //
 265 func (z *Int) DivMod(x, y, m *Int) (*Int, *Int) {
 266         y0 := y // save y
 267         if z == y || alias(z.abs, y.abs) {
 268                 y0 = new(Int).Set(y)
 269         }
 270         z.QuoRem(x, y, m)
 271         if m.neg {
 272                 if y0.neg {
 273                         z.Add(z, intOne)
 274                         m.Sub(m, y0)
 275                 } else {
 276                         z.Sub(z, intOne)
 277                         m.Add(m, y0)
 278                 }
 279         }
 280         return z, m
 281 }
 282
 283
 284 // Cmp compares x and y and returns:
 285 //
 286 //   -1 if x <  y
 287 //    0 if x == y
 288 //   +1 if x >  y
 289 //
 290 func (x *Int) Cmp(y *Int) (r int) {
 291         // x cmp y == x cmp y
 292         // x cmp (-y) == x
 293         // (-x) cmp y == y
 294         // (-x) cmp (-y) == -(x cmp y)
 295         switch {
 296         case x.neg == y.neg:
 297                 r = x.abs.cmp(y.abs)
 298                 if x.neg {
 299                         r = -r
 300                 }
 301         case x.neg:
 302                 r = -1
 303         default:
 304                 r = 1
 305         }
 306         return
 307 }
 308
 309
 310 func (x *Int) String() string {
 311         s := ""
 312         if x.neg {
 313                 s = "-"
 314         }
 315         return s + x.abs.string(10)
 316 }
 317
 318
 319 func fmtbase(ch int) int {
 320         switch ch {
 321         case 'b':
 322                 return 2
 323         case 'o':
 324                 return 8
 325         case 'd':
 326                 return 10
 327         case 'x':
 328                 return 16
 329         }
 330         return 10
 331 }
 332
 333
 334 // Format is a support routine for fmt.Formatter. It accepts
 335 // the formats 'b' (binary), 'o' (octal), 'd' (decimal) and
 336 // 'x' (hexadecimal).
 337 //
 338 func (x *Int) Format(s fmt.State, ch int) {
 339         if x.neg {
 340                 fmt.Fprint(s, "-")
 341         }
 342         fmt.Fprint(s, x.abs.string(fmtbase(ch)))
 343 }
 344
 345
 346 // Int64 returns the int64 representation of z.
 347 // If z cannot be represented in an int64, the result is undefined.
 348 func (x *Int) Int64() int64 {
 349         if len(x.abs) == 0 {
 350                 return 0
 351         }
 352         v := int64(x.abs[0])
 353         if _W == 32 && len(x.abs) > 1 {
 354                 v |= int64(x.abs[1]) << 32
 355         }
 356         if x.neg {
 357                 v = -v
 358         }
 359         return v
 360 }
 361
 362
 363 // SetString sets z to the value of s, interpreted in the given base,
 364 // and returns z and a boolean indicating success. If SetString fails,
 365 // the value of z is undefined.
 366 //
 367 // If the base argument is 0, the string prefix determines the actual
 368 // conversion base. A prefix of ``0x'' or ``0X'' selects base 16; the
 369 // ``0'' prefix selects base 8, and a ``0b'' or ``0B'' prefix selects
 370 // base 2. Otherwise the selected base is 10.
 371 //
 372 func (z *Int) SetString(s string, base int) (*Int, bool) {
 373         if len(s) == 0 || base < 0 || base == 1 || 16 < base {
 374                 return z, false
 375         }
 376
 377         neg := s[0] == '-'
 378         if neg || s[0] == '+' {
 379                 s = s[1:]
 380                 if len(s) == 0 {
 381                         return z, false
 382                 }
 383         }
 384
 385         var scanned int
 386         z.abs, _, scanned = z.abs.scan(s, base)
 387         if scanned != len(s) {
 388                 return z, false
 389         }
 390         z.neg = len(z.abs) > 0 && neg // 0 has no sign
 391
 392         return z, true
 393 }
 394
 395
 396 // SetBytes interprets b as the bytes of a big-endian, unsigned integer and
 397 // sets z to that value.
 398 func (z *Int) SetBytes(b []byte) *Int {
 399         const s = _S
 400         z.abs = z.abs.make((len(b) + s - 1) / s)
 401
 402         j := 0
 403         for len(b) >= s {
 404                 var w Word
 405
 406                 for i := s; i > 0; i-- {
 407                         w <<= 8
 408                         w |= Word(b[len(b)-i])
 409                 }
 410
 411                 z.abs[j] = w
 412                 j++
 413                 b = b[0 : len(b)-s]
 414         }
 415
 416         if len(b) > 0 {
 417                 var w Word
 418
 419                 for i := len(b); i > 0; i-- {
 420                         w <<= 8
 421                         w |= Word(b[len(b)-i])
 422                 }
 423
 424                 z.abs[j] = w
 425         }
 426
 427         z.abs = z.abs.norm()
 428         z.neg = false
 429         return z
 430 }
 431
 432
 433 // Bytes returns the absolute value of x as a big-endian byte array.
 434 func (z *Int) Bytes() []byte {
 435         const s = _S
 436         b := make([]byte, len(z.abs)*s)
 437
 438         for i, w := range z.abs {
 439                 wordBytes := b[(len(z.abs)-i-1)*s : (len(z.abs)-i)*s]
 440                 for j := s - 1; j >= 0; j-- {
 441                         wordBytes[j] = byte(w)
 442                         w >>= 8
 443                 }
 444         }
 445
 446         i := 0
 447         for i < len(b) && b[i] == 0 {
 448                 i++
 449         }
 450
 451         return b[i:]
 452 }
 453
 454
 455 // BitLen returns the length of the absolute value of z in bits.
 456 // The bit length of 0 is 0.
 457 func (z *Int) BitLen() int {
 458         return z.abs.bitLen()
 459 }
 460
 461
 462 // Exp sets z = x**y mod m. If m is nil, z = x**y.
 463 // See Knuth, volume 2, section 4.6.3.
 464 func (z *Int) Exp(x, y, m *Int) *Int {
 465         if y.neg || len(y.abs) == 0 {
 466                 neg := x.neg
 467                 z.SetInt64(1)
 468                 z.neg = neg
 469                 return z
 470         }
 471
 472         var mWords nat
 473         if m != nil {
 474                 mWords = m.abs
 475         }
 476
 477         z.abs = z.abs.expNN(x.abs, y.abs, mWords)
 478         z.neg = len(z.abs) > 0 && x.neg && y.abs[0]&1 == 1 // 0 has no sign
 479         return z
 480 }
 481
 482
 483 // GcdInt sets d to the greatest common divisor of a and b, which must be
 484 // positive numbers.
 485 // If x and y are not nil, GcdInt sets x and y such that d = a*x + b*y.
 486 // If either a or b is not positive, GcdInt sets d = x = y = 0.
 487 func GcdInt(d, x, y, a, b *Int) {
 488         if a.neg || b.neg {
 489                 d.SetInt64(0)
 490                 if x != nil {
 491                         x.SetInt64(0)
 492                 }
 493                 if y != nil {
 494                         y.SetInt64(0)
 495                 }
 496                 return
 497         }
 498
 499         A := new(Int).Set(a)
 500         B := new(Int).Set(b)
 501
 502         X := new(Int)
 503         Y := new(Int).SetInt64(1)
 504
 505         lastX := new(Int).SetInt64(1)
 506         lastY := new(Int)
 507
 508         q := new(Int)
 509         temp := new(Int)
 510
 511         for len(B.abs) > 0 {
 512                 r := new(Int)
 513                 q, r = q.QuoRem(A, B, r)
 514
 515                 A, B = B, r
 516
 517                 temp.Set(X)
 518                 X.Mul(X, q)
 519                 X.neg = !X.neg
 520                 X.Add(X, lastX)
 521                 lastX.Set(temp)
 522
 523                 temp.Set(Y)
 524                 Y.Mul(Y, q)
 525                 Y.neg = !Y.neg
 526                 Y.Add(Y, lastY)
 527                 lastY.Set(temp)
 528         }
 529
 530         if x != nil {
 531                 *x = *lastX
 532         }
 533
 534         if y != nil {
 535                 *y = *lastY
 536         }
 537
 538         *d = *A
 539 }
 540
 541
 542 // ProbablyPrime performs n Miller-Rabin tests to check whether z is prime.
 543 // If it returns true, z is prime with probability 1 - 1/4^n.
 544 // If it returns false, z is not prime.
 545 func ProbablyPrime(z *Int, n int) bool {
 546         return !z.neg && z.abs.probablyPrime(n)
 547 }
 548
 549
 550 // Rand sets z to a pseudo-random number in [0, n) and returns z.
 551 func (z *Int) Rand(rnd *rand.Rand, n *Int) *Int {
 552         z.neg = false
 553         if n.neg == true || len(n.abs) == 0 {
 554                 z.abs = nil
 555                 return z
 556         }
 557         z.abs = z.abs.random(rnd, n.abs, n.abs.bitLen())
 558         return z
 559 }
 560
 561
 562 // ModInverse sets z to the multiplicative inverse of g in the group ℤ/pℤ (where
 563 // p is a prime) and returns z.
 564 func (z *Int) ModInverse(g, p *Int) *Int {
 565         var d Int
 566         GcdInt(&d, z, nil, g, p)
 567         // x and y are such that g*x + p*y = d. Since p is prime, d = 1. Taking
 568         // that modulo p results in g*x = 1, therefore x is the inverse element.
 569         if z.neg {
 570                 z.Add(z, p)
 571         }
 572         return z
 573 }
 574
 575
 576 // Lsh sets z = x << n and returns z.
 577 func (z *Int) Lsh(x *Int, n uint) *Int {
 578         z.abs = z.abs.shl(x.abs, n)
 579         z.neg = x.neg
 580         return z
 581 }
 582
 583
 584 // Rsh sets z = x >> n and returns z.
 585 func (z *Int) Rsh(x *Int, n uint) *Int {
 586         if x.neg {
 587                 // (-x) >> s == ^(x-1) >> s == ^((x-1) >> s) == -(((x-1) >> s) + 1)
 588                 t := z.abs.sub(x.abs, natOne) // no underflow because |x| > 0
 589                 t = t.shr(t, n)
 590                 z.abs = t.add(t, natOne)
 591                 z.neg = true // z cannot be zero if x is negative
 592                 return z
 593         }
 594
 595         z.abs = z.abs.shr(x.abs, n)
 596         z.neg = false
 597         return z
 598 }
 599
 600
 601 // And sets z = x & y and returns z.
 602 func (z *Int) And(x, y *Int) *Int {
 603         if x.neg == y.neg {
 604                 if x.neg {
 605                         // (-x) & (-y) == ^(x-1) & ^(y-1) == ^((x-1) | (y-1)) == -(((x-1) | (y-1)) + 1)
 606                         x1 := nat{}.sub(x.abs, natOne)
 607                         y1 := nat{}.sub(y.abs, natOne)
 608                         z.abs = z.abs.add(z.abs.or(x1, y1), natOne)
 609                         z.neg = true // z cannot be zero if x and y are negative
 610                         return z
 611                 }
 612
 613                 // x & y == x & y
 614                 z.abs = z.abs.and(x.abs, y.abs)
 615                 z.neg = false
 616                 return z
 617         }
 618
 619         // x.neg != y.neg
 620         if x.neg {
 621                 x, y = y, x // & is symmetric
 622         }
 623
 624         // x & (-y) == x & ^(y-1) == x &^ (y-1)
 625         y1 := nat{}.sub(y.abs, natOne)
 626         z.abs = z.abs.andNot(x.abs, y1)
 627         z.neg = false
 628         return z
 629 }
 630
 631
 632 // AndNot sets z = x &^ y and returns z.
 633 func (z *Int) AndNot(x, y *Int) *Int {
 634         if x.neg == y.neg {
 635                 if x.neg {
 636                         // (-x) &^ (-y) == ^(x-1) &^ ^(y-1) == ^(x-1) & (y-1) == (y-1) &^ (x-1)
 637                         x1 := nat{}.sub(x.abs, natOne)
 638                         y1 := nat{}.sub(y.abs, natOne)
 639                         z.abs = z.abs.andNot(y1, x1)
 640                         z.neg = false
 641                         return z
 642                 }
 643
 644                 // x &^ y == x &^ y
 645                 z.abs = z.abs.andNot(x.abs, y.abs)
 646                 z.neg = false
 647                 return z
 648         }
 649
 650         if x.neg {
 651                 // (-x) &^ y == ^(x-1) &^ y == ^(x-1) & ^y == ^((x-1) | y) == -(((x-1) | y) + 1)
 652                 x1 := nat{}.sub(x.abs, natOne)
 653                 z.abs = z.abs.add(z.abs.or(x1, y.abs), natOne)
 654                 z.neg = true // z cannot be zero if x is negative and y is positive
 655                 return z
 656         }
 657
 658         // x &^ (-y) == x &^ ^(y-1) == x & (y-1)
 659         y1 := nat{}.add(y.abs, natOne)
 660         z.abs = z.abs.and(x.abs, y1)
 661         z.neg = false
 662         return z
 663 }
 664
 665
 666 // Or sets z = x | y and returns z.
 667 func (z *Int) Or(x, y *Int) *Int {
 668         if x.neg == y.neg {
 669                 if x.neg {
 670                         // (-x) | (-y) == ^(x-1) | ^(y-1) == ^((x-1) & (y-1)) == -(((x-1) & (y-1)) + 1)
 671                         x1 := nat{}.sub(x.abs, natOne)
 672                         y1 := nat{}.sub(y.abs, natOne)
 673                         z.abs = z.abs.add(z.abs.and(x1, y1), natOne)
 674                         z.neg = true // z cannot be zero if x and y are negative
 675                         return z
 676                 }
 677
 678                 // x | y == x | y
 679                 z.abs = z.abs.or(x.abs, y.abs)
 680                 z.neg = false
 681                 return z
 682         }
 683
 684         // x.neg != y.neg
 685         if x.neg {
 686                 x, y = y, x // | is symmetric
 687         }
 688
 689         // x | (-y) == x | ^(y-1) == ^((y-1) &^ x) == -(^((y-1) &^ x) + 1)
 690         y1 := nat{}.sub(y.abs, natOne)
 691         z.abs = z.abs.add(z.abs.andNot(y1, x.abs), natOne)
 692         z.neg = true // z cannot be zero if one of x or y is negative
 693         return z
 694 }
 695
 696
 697 // Xor sets z = x ^ y and returns z.
 698 func (z *Int) Xor(x, y *Int) *Int {
 699         if x.neg == y.neg {
 700                 if x.neg {
 701                         // (-x) ^ (-y) == ^(x-1) ^ ^(y-1) == (x-1) ^ (y-1)
 702                         x1 := nat{}.sub(x.abs, natOne)
 703                         y1 := nat{}.sub(y.abs, natOne)
 704                         z.abs = z.abs.xor(x1, y1)
 705                         z.neg = false
 706                         return z
 707                 }
 708
 709                 // x ^ y == x ^ y
 710                 z.abs = z.abs.xor(x.abs, y.abs)
 711                 z.neg = false
 712                 return z
 713         }
 714
 715         // x.neg != y.neg
 716         if x.neg {
 717                 x, y = y, x // ^ is symmetric
 718         }
 719
 720         // x ^ (-y) == x ^ ^(y-1) == ^(x ^ (y-1)) == -((x ^ (y-1)) + 1)
 721         y1 := nat{}.sub(y.abs, natOne)
 722         z.abs = z.abs.add(z.abs.xor(x.abs, y1), natOne)
 723         z.neg = true // z cannot be zero if only one of x or y is negative
 724         return z
 725 }
 726
 727
 728 // Not sets z = ^x and returns z.
 729 func (z *Int) Not(x *Int) *Int {
 730         if x.neg {
 731                 // ^(-x) == ^(^(x-1)) == x-1
 732                 z.abs = z.abs.sub(x.abs, natOne)
 733                 z.neg = false
 734                 return z
 735         }
 736
 737         // ^x == -x-1 == -(x+1)
 738         z.abs = z.abs.add(x.abs, natOne)
 739         z.neg = true // z cannot be zero if x is positive
 740         return z
 741 }