1 // Copyright 2015 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 nat-to-string conversion functions. 6 7 package big 8 9 import ( 10 "errors" 11 "fmt" 12 "io" 13 "math" 14 "math/bits" 15 "sync" 16 ) 17 18 const digits = "0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ" 19 20 // Note: MaxBase = len(digits), but it must remain an untyped rune constant 21 // for API compatibility. 22 23 // MaxBase is the largest number base accepted for string conversions. 24 const MaxBase = 10 + ('z' - 'a' + 1) + ('Z' - 'A' + 1) 25 const maxBaseSmall = 10 + ('z' - 'a' + 1) 26 27 // maxPow returns (b**n, n) such that b**n is the largest power b**n <= _M. 28 // For instance maxPow(10) == (1e19, 19) for 19 decimal digits in a 64bit Word. 29 // In other words, at most n digits in base b fit into a Word. 30 // TODO(gri) replace this with a table, generated at build time. 31 func maxPow(b Word) (p Word, n int) { 32 p, n = b, 1 // assuming b <= _M 33 for max := _M / b; p <= max; { 34 // p == b**n && p <= max 35 p *= b 36 n++ 37 } 38 // p == b**n && p <= _M 39 return 40 } 41 42 // pow returns x**n for n > 0, and 1 otherwise. 43 func pow(x Word, n int) (p Word) { 44 // n == sum of bi * 2**i, for 0 <= i < imax, and bi is 0 or 1 45 // thus x**n == product of x**(2**i) for all i where bi == 1 46 // (Russian Peasant Method for exponentiation) 47 p = 1 48 for n > 0 { 49 if n&1 != 0 { 50 p *= x 51 } 52 x *= x 53 n >>= 1 54 } 55 return 56 } 57 58 // scan errors 59 var ( 60 errNoDigits = errors.New("number has no digits") 61 errInvalSep = errors.New("'_' must separate successive digits") 62 ) 63 64 // scan scans the number corresponding to the longest possible prefix 65 // from r representing an unsigned number in a given conversion base. 66 // scan returns the corresponding natural number res, the actual base b, 67 // a digit count, and a read or syntax error err, if any. 68 // 69 // For base 0, an underscore character ``_'' may appear between a base 70 // prefix and an adjacent digit, and between successive digits; such 71 // underscores do not change the value of the number, or the returned 72 // digit count. Incorrect placement of underscores is reported as an 73 // error if there are no other errors. If base != 0, underscores are 74 // not recognized and thus terminate scanning like any other character 75 // that is not a valid radix point or digit. 76 // 77 // number = mantissa | prefix pmantissa . 78 // prefix = "0" [ "b" | "B" | "o" | "O" | "x" | "X" ] . 79 // mantissa = digits "." [ digits ] | digits | "." digits . 80 // pmantissa = [ "_" ] digits "." [ digits ] | [ "_" ] digits | "." digits . 81 // digits = digit { [ "_" ] digit } . 82 // digit = "0" ... "9" | "a" ... "z" | "A" ... "Z" . 83 // 84 // Unless fracOk is set, the base argument must be 0 or a value between 85 // 2 and MaxBase. If fracOk is set, the base argument must be one of 86 // 0, 2, 8, 10, or 16. Providing an invalid base argument leads to a run- 87 // time panic. 88 // 89 // For base 0, the number prefix determines the actual base: A prefix of 90 // ``0b'' or ``0B'' selects base 2, ``0o'' or ``0O'' selects base 8, and 91 // ``0x'' or ``0X'' selects base 16. If fracOk is false, a ``0'' prefix 92 // (immediately followed by digits) selects base 8 as well. Otherwise, 93 // the selected base is 10 and no prefix is accepted. 94 // 95 // If fracOk is set, a period followed by a fractional part is permitted. 96 // The result value is computed as if there were no period present; and 97 // the count value is used to determine the fractional part. 98 // 99 // For bases <= 36, lower and upper case letters are considered the same: 100 // The letters 'a' to 'z' and 'A' to 'Z' represent digit values 10 to 35. 101 // For bases > 36, the upper case letters 'A' to 'Z' represent the digit 102 // values 36 to 61. 103 // 104 // A result digit count > 0 corresponds to the number of (non-prefix) digits 105 // parsed. A digit count <= 0 indicates the presence of a period (if fracOk 106 // is set, only), and -count is the number of fractional digits found. 107 // In this case, the actual value of the scanned number is res * b**count. 108 // 109 func (z nat) scan(r io.ByteScanner, base int, fracOk bool) (res nat, b, count int, err error) { 110 // reject invalid bases 111 baseOk := base == 0 || 112 !fracOk && 2 <= base && base <= MaxBase || 113 fracOk && (base == 2 || base == 8 || base == 10 || base == 16) 114 if !baseOk { 115 panic(fmt.Sprintf("invalid number base %d", base)) 116 } 117 118 // prev encodes the previously seen char: it is one 119 // of '_', '0' (a digit), or '.' (anything else). A 120 // valid separator '_' may only occur after a digit 121 // and if base == 0. 122 prev := '.' 123 invalSep := false 124 125 // one char look-ahead 126 ch, err := r.ReadByte() 127 128 // determine actual base 129 b, prefix := base, 0 130 if base == 0 { 131 // actual base is 10 unless there's a base prefix 132 b = 10 133 if err == nil && ch == '0' { 134 prev = '0' 135 count = 1 136 ch, err = r.ReadByte() 137 if err == nil { 138 // possibly one of 0b, 0B, 0o, 0O, 0x, 0X 139 switch ch { 140 case 'b', 'B': 141 b, prefix = 2, 'b' 142 case 'o', 'O': 143 b, prefix = 8, 'o' 144 case 'x', 'X': 145 b, prefix = 16, 'x' 146 default: 147 if !fracOk { 148 b, prefix = 8, '0' 149 } 150 } 151 if prefix != 0 { 152 count = 0 // prefix is not counted 153 if prefix != '0' { 154 ch, err = r.ReadByte() 155 } 156 } 157 } 158 } 159 } 160 161 // convert string 162 // Algorithm: Collect digits in groups of at most n digits in di 163 // and then use mulAddWW for every such group to add them to the 164 // result. 165 z = z[:0] 166 b1 := Word(b) 167 bn, n := maxPow(b1) // at most n digits in base b1 fit into Word 168 di := Word(0) // 0 <= di < b1**i < bn 169 i := 0 // 0 <= i < n 170 dp := -1 // position of decimal point 171 for err == nil { 172 if ch == '.' && fracOk { 173 fracOk = false 174 if prev == '_' { 175 invalSep = true 176 } 177 prev = '.' 178 dp = count 179 } else if ch == '_' && base == 0 { 180 if prev != '0' { 181 invalSep = true 182 } 183 prev = '_' 184 } else { 185 // convert rune into digit value d1 186 var d1 Word 187 switch { 188 case '0' <= ch && ch <= '9': 189 d1 = Word(ch - '0') 190 case 'a' <= ch && ch <= 'z': 191 d1 = Word(ch - 'a' + 10) 192 case 'A' <= ch && ch <= 'Z': 193 if b <= maxBaseSmall { 194 d1 = Word(ch - 'A' + 10) 195 } else { 196 d1 = Word(ch - 'A' + maxBaseSmall) 197 } 198 default: 199 d1 = MaxBase + 1 200 } 201 if d1 >= b1 { 202 r.UnreadByte() // ch does not belong to number anymore 203 break 204 } 205 prev = '0' 206 count++ 207 208 // collect d1 in di 209 di = di*b1 + d1 210 i++ 211 212 // if di is "full", add it to the result 213 if i == n { 214 z = z.mulAddWW(z, bn, di) 215 di = 0 216 i = 0 217 } 218 } 219 220 ch, err = r.ReadByte() 221 } 222 223 if err == io.EOF { 224 err = nil 225 } 226 227 // other errors take precedence over invalid separators 228 if err == nil && (invalSep || prev == '_') { 229 err = errInvalSep 230 } 231 232 if count == 0 { 233 // no digits found 234 if prefix == '0' { 235 // there was only the octal prefix 0 (possibly followed by separators and digits > 7); 236 // interpret as decimal 0 237 return z[:0], 10, 1, err 238 } 239 err = errNoDigits // fall through; result will be 0 240 } 241 242 // add remaining digits to result 243 if i > 0 { 244 z = z.mulAddWW(z, pow(b1, i), di) 245 } 246 res = z.norm() 247 248 // adjust count for fraction, if any 249 if dp >= 0 { 250 // 0 <= dp <= count 251 count = dp - count 252 } 253 254 return 255 } 256 257 // utoa converts x to an ASCII representation in the given base; 258 // base must be between 2 and MaxBase, inclusive. 259 func (x nat) utoa(base int) []byte { 260 return x.itoa(false, base) 261 } 262 263 // itoa is like utoa but it prepends a '-' if neg && x != 0. 264 func (x nat) itoa(neg bool, base int) []byte { 265 if base < 2 || base > MaxBase { 266 panic("invalid base") 267 } 268 269 // x == 0 270 if len(x) == 0 { 271 return []byte("0") 272 } 273 // len(x) > 0 274 275 // allocate buffer for conversion 276 i := int(float64(x.bitLen())/math.Log2(float64(base))) + 1 // off by 1 at most 277 if neg { 278 i++ 279 } 280 s := make([]byte, i) 281 282 // convert power of two and non power of two bases separately 283 if b := Word(base); b == b&-b { 284 // shift is base b digit size in bits 285 shift := uint(bits.TrailingZeros(uint(b))) // shift > 0 because b >= 2 286 mask := Word(1<<shift - 1) 287 w := x[0] // current word 288 nbits := uint(_W) // number of unprocessed bits in w 289 290 // convert less-significant words (include leading zeros) 291 for k := 1; k < len(x); k++ { 292 // convert full digits 293 for nbits >= shift { 294 i-- 295 s[i] = digits[w&mask] 296 w >>= shift 297 nbits -= shift 298 } 299 300 // convert any partial leading digit and advance to next word 301 if nbits == 0 { 302 // no partial digit remaining, just advance 303 w = x[k] 304 nbits = _W 305 } else { 306 // partial digit in current word w (== x[k-1]) and next word x[k] 307 w |= x[k] << nbits 308 i-- 309 s[i] = digits[w&mask] 310 311 // advance 312 w = x[k] >> (shift - nbits) 313 nbits = _W - (shift - nbits) 314 } 315 } 316 317 // convert digits of most-significant word w (omit leading zeros) 318 for w != 0 { 319 i-- 320 s[i] = digits[w&mask] 321 w >>= shift 322 } 323 324 } else { 325 bb, ndigits := maxPow(b) 326 327 // construct table of successive squares of bb*leafSize to use in subdivisions 328 // result (table != nil) <=> (len(x) > leafSize > 0) 329 table := divisors(len(x), b, ndigits, bb) 330 331 // preserve x, create local copy for use by convertWords 332 q := nat(nil).set(x) 333 334 // convert q to string s in base b 335 q.convertWords(s, b, ndigits, bb, table) 336 337 // strip leading zeros 338 // (x != 0; thus s must contain at least one non-zero digit 339 // and the loop will terminate) 340 i = 0 341 for s[i] == '0' { 342 i++ 343 } 344 } 345 346 if neg { 347 i-- 348 s[i] = '-' 349 } 350 351 return s[i:] 352 } 353 354 // Convert words of q to base b digits in s. If q is large, it is recursively "split in half" 355 // by nat/nat division using tabulated divisors. Otherwise, it is converted iteratively using 356 // repeated nat/Word division. 357 // 358 // The iterative method processes n Words by n divW() calls, each of which visits every Word in the 359 // incrementally shortened q for a total of n + (n-1) + (n-2) ... + 2 + 1, or n(n+1)/2 divW()'s. 360 // Recursive conversion divides q by its approximate square root, yielding two parts, each half 361 // the size of q. Using the iterative method on both halves means 2 * (n/2)(n/2 + 1)/2 divW()'s 362 // plus the expensive long div(). Asymptotically, the ratio is favorable at 1/2 the divW()'s, and 363 // is made better by splitting the subblocks recursively. Best is to split blocks until one more 364 // split would take longer (because of the nat/nat div()) than the twice as many divW()'s of the 365 // iterative approach. This threshold is represented by leafSize. Benchmarking of leafSize in the 366 // range 2..64 shows that values of 8 and 16 work well, with a 4x speedup at medium lengths and 367 // ~30x for 20000 digits. Use nat_test.go's BenchmarkLeafSize tests to optimize leafSize for 368 // specific hardware. 369 // 370 func (q nat) convertWords(s []byte, b Word, ndigits int, bb Word, table []divisor) { 371 // split larger blocks recursively 372 if table != nil { 373 // len(q) > leafSize > 0 374 var r nat 375 index := len(table) - 1 376 for len(q) > leafSize { 377 // find divisor close to sqrt(q) if possible, but in any case < q 378 maxLength := q.bitLen() // ~= log2 q, or at of least largest possible q of this bit length 379 minLength := maxLength >> 1 // ~= log2 sqrt(q) 380 for index > 0 && table[index-1].nbits > minLength { 381 index-- // desired 382 } 383 if table[index].nbits >= maxLength && table[index].bbb.cmp(q) >= 0 { 384 index-- 385 if index < 0 { 386 panic("internal inconsistency") 387 } 388 } 389 390 // split q into the two digit number (q'*bbb + r) to form independent subblocks 391 q, r = q.div(r, q, table[index].bbb) 392 393 // convert subblocks and collect results in s[:h] and s[h:] 394 h := len(s) - table[index].ndigits 395 r.convertWords(s[h:], b, ndigits, bb, table[0:index]) 396 s = s[:h] // == q.convertWords(s, b, ndigits, bb, table[0:index+1]) 397 } 398 } 399 400 // having split any large blocks now process the remaining (small) block iteratively 401 i := len(s) 402 var r Word 403 if b == 10 { 404 // hard-coding for 10 here speeds this up by 1.25x (allows for / and % by constants) 405 for len(q) > 0 { 406 // extract least significant, base bb "digit" 407 q, r = q.divW(q, bb) 408 for j := 0; j < ndigits && i > 0; j++ { 409 i-- 410 // avoid % computation since r%10 == r - int(r/10)*10; 411 // this appears to be faster for BenchmarkString10000Base10 412 // and smaller strings (but a bit slower for larger ones) 413 t := r / 10 414 s[i] = '0' + byte(r-t*10) 415 r = t 416 } 417 } 418 } else { 419 for len(q) > 0 { 420 // extract least significant, base bb "digit" 421 q, r = q.divW(q, bb) 422 for j := 0; j < ndigits && i > 0; j++ { 423 i-- 424 s[i] = digits[r%b] 425 r /= b 426 } 427 } 428 } 429 430 // prepend high-order zeros 431 for i > 0 { // while need more leading zeros 432 i-- 433 s[i] = '0' 434 } 435 } 436 437 // Split blocks greater than leafSize Words (or set to 0 to disable recursive conversion) 438 // Benchmark and configure leafSize using: go test -bench="Leaf" 439 // 8 and 16 effective on 3.0 GHz Xeon "Clovertown" CPU (128 byte cache lines) 440 // 8 and 16 effective on 2.66 GHz Core 2 Duo "Penryn" CPU 441 var leafSize int = 8 // number of Word-size binary values treat as a monolithic block 442 443 type divisor struct { 444 bbb nat // divisor 445 nbits int // bit length of divisor (discounting leading zeros) ~= log2(bbb) 446 ndigits int // digit length of divisor in terms of output base digits 447 } 448 449 var cacheBase10 struct { 450 sync.Mutex 451 table [64]divisor // cached divisors for base 10 452 } 453 454 // expWW computes x**y 455 func (z nat) expWW(x, y Word) nat { 456 return z.expNN(nat(nil).setWord(x), nat(nil).setWord(y), nil) 457 } 458 459 // construct table of powers of bb*leafSize to use in subdivisions 460 func divisors(m int, b Word, ndigits int, bb Word) []divisor { 461 // only compute table when recursive conversion is enabled and x is large 462 if leafSize == 0 || m <= leafSize { 463 return nil 464 } 465 466 // determine k where (bb**leafSize)**(2**k) >= sqrt(x) 467 k := 1 468 for words := leafSize; words < m>>1 && k < len(cacheBase10.table); words <<= 1 { 469 k++ 470 } 471 472 // reuse and extend existing table of divisors or create new table as appropriate 473 var table []divisor // for b == 10, table overlaps with cacheBase10.table 474 if b == 10 { 475 cacheBase10.Lock() 476 table = cacheBase10.table[0:k] // reuse old table for this conversion 477 } else { 478 table = make([]divisor, k) // create new table for this conversion 479 } 480 481 // extend table 482 if table[k-1].ndigits == 0 { 483 // add new entries as needed 484 var larger nat 485 for i := 0; i < k; i++ { 486 if table[i].ndigits == 0 { 487 if i == 0 { 488 table[0].bbb = nat(nil).expWW(bb, Word(leafSize)) 489 table[0].ndigits = ndigits * leafSize 490 } else { 491 table[i].bbb = nat(nil).sqr(table[i-1].bbb) 492 table[i].ndigits = 2 * table[i-1].ndigits 493 } 494 495 // optimization: exploit aggregated extra bits in macro blocks 496 larger = nat(nil).set(table[i].bbb) 497 for mulAddVWW(larger, larger, b, 0) == 0 { 498 table[i].bbb = table[i].bbb.set(larger) 499 table[i].ndigits++ 500 } 501 502 table[i].nbits = table[i].bbb.bitLen() 503 } 504 } 505 } 506 507 if b == 10 { 508 cacheBase10.Unlock() 509 } 510 511 return table 512 } 513