1 // Copyright 2011 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 strconv 6 7 import ( 8 "math/bits" 9 ) 10 11 // An extFloat represents an extended floating-point number, with more 12 // precision than a float64. It does not try to save bits: the 13 // number represented by the structure is mant*(2^exp), with a negative 14 // sign if neg is true. 15 type extFloat struct { 16 mant uint64 17 exp int 18 neg bool 19 } 20 21 // Powers of ten taken from double-conversion library. 22 // https://code.google.com/p/double-conversion/ 23 const ( 24 firstPowerOfTen = -348 25 stepPowerOfTen = 8 26 ) 27 28 var smallPowersOfTen = [...]extFloat{ 29 {1 << 63, -63, false}, // 1 30 {0xa << 60, -60, false}, // 1e1 31 {0x64 << 57, -57, false}, // 1e2 32 {0x3e8 << 54, -54, false}, // 1e3 33 {0x2710 << 50, -50, false}, // 1e4 34 {0x186a0 << 47, -47, false}, // 1e5 35 {0xf4240 << 44, -44, false}, // 1e6 36 {0x989680 << 40, -40, false}, // 1e7 37 } 38 39 var powersOfTen = [...]extFloat{ 40 {0xfa8fd5a0081c0288, -1220, false}, // 10^-348 41 {0xbaaee17fa23ebf76, -1193, false}, // 10^-340 42 {0x8b16fb203055ac76, -1166, false}, // 10^-332 43 {0xcf42894a5dce35ea, -1140, false}, // 10^-324 44 {0x9a6bb0aa55653b2d, -1113, false}, // 10^-316 45 {0xe61acf033d1a45df, -1087, false}, // 10^-308 46 {0xab70fe17c79ac6ca, -1060, false}, // 10^-300 47 {0xff77b1fcbebcdc4f, -1034, false}, // 10^-292 48 {0xbe5691ef416bd60c, -1007, false}, // 10^-284 49 {0x8dd01fad907ffc3c, -980, false}, // 10^-276 50 {0xd3515c2831559a83, -954, false}, // 10^-268 51 {0x9d71ac8fada6c9b5, -927, false}, // 10^-260 52 {0xea9c227723ee8bcb, -901, false}, // 10^-252 53 {0xaecc49914078536d, -874, false}, // 10^-244 54 {0x823c12795db6ce57, -847, false}, // 10^-236 55 {0xc21094364dfb5637, -821, false}, // 10^-228 56 {0x9096ea6f3848984f, -794, false}, // 10^-220 57 {0xd77485cb25823ac7, -768, false}, // 10^-212 58 {0xa086cfcd97bf97f4, -741, false}, // 10^-204 59 {0xef340a98172aace5, -715, false}, // 10^-196 60 {0xb23867fb2a35b28e, -688, false}, // 10^-188 61 {0x84c8d4dfd2c63f3b, -661, false}, // 10^-180 62 {0xc5dd44271ad3cdba, -635, false}, // 10^-172 63 {0x936b9fcebb25c996, -608, false}, // 10^-164 64 {0xdbac6c247d62a584, -582, false}, // 10^-156 65 {0xa3ab66580d5fdaf6, -555, false}, // 10^-148 66 {0xf3e2f893dec3f126, -529, false}, // 10^-140 67 {0xb5b5ada8aaff80b8, -502, false}, // 10^-132 68 {0x87625f056c7c4a8b, -475, false}, // 10^-124 69 {0xc9bcff6034c13053, -449, false}, // 10^-116 70 {0x964e858c91ba2655, -422, false}, // 10^-108 71 {0xdff9772470297ebd, -396, false}, // 10^-100 72 {0xa6dfbd9fb8e5b88f, -369, false}, // 10^-92 73 {0xf8a95fcf88747d94, -343, false}, // 10^-84 74 {0xb94470938fa89bcf, -316, false}, // 10^-76 75 {0x8a08f0f8bf0f156b, -289, false}, // 10^-68 76 {0xcdb02555653131b6, -263, false}, // 10^-60 77 {0x993fe2c6d07b7fac, -236, false}, // 10^-52 78 {0xe45c10c42a2b3b06, -210, false}, // 10^-44 79 {0xaa242499697392d3, -183, false}, // 10^-36 80 {0xfd87b5f28300ca0e, -157, false}, // 10^-28 81 {0xbce5086492111aeb, -130, false}, // 10^-20 82 {0x8cbccc096f5088cc, -103, false}, // 10^-12 83 {0xd1b71758e219652c, -77, false}, // 10^-4 84 {0x9c40000000000000, -50, false}, // 10^4 85 {0xe8d4a51000000000, -24, false}, // 10^12 86 {0xad78ebc5ac620000, 3, false}, // 10^20 87 {0x813f3978f8940984, 30, false}, // 10^28 88 {0xc097ce7bc90715b3, 56, false}, // 10^36 89 {0x8f7e32ce7bea5c70, 83, false}, // 10^44 90 {0xd5d238a4abe98068, 109, false}, // 10^52 91 {0x9f4f2726179a2245, 136, false}, // 10^60 92 {0xed63a231d4c4fb27, 162, false}, // 10^68 93 {0xb0de65388cc8ada8, 189, false}, // 10^76 94 {0x83c7088e1aab65db, 216, false}, // 10^84 95 {0xc45d1df942711d9a, 242, false}, // 10^92 96 {0x924d692ca61be758, 269, false}, // 10^100 97 {0xda01ee641a708dea, 295, false}, // 10^108 98 {0xa26da3999aef774a, 322, false}, // 10^116 99 {0xf209787bb47d6b85, 348, false}, // 10^124 100 {0xb454e4a179dd1877, 375, false}, // 10^132 101 {0x865b86925b9bc5c2, 402, false}, // 10^140 102 {0xc83553c5c8965d3d, 428, false}, // 10^148 103 {0x952ab45cfa97a0b3, 455, false}, // 10^156 104 {0xde469fbd99a05fe3, 481, false}, // 10^164 105 {0xa59bc234db398c25, 508, false}, // 10^172 106 {0xf6c69a72a3989f5c, 534, false}, // 10^180 107 {0xb7dcbf5354e9bece, 561, false}, // 10^188 108 {0x88fcf317f22241e2, 588, false}, // 10^196 109 {0xcc20ce9bd35c78a5, 614, false}, // 10^204 110 {0x98165af37b2153df, 641, false}, // 10^212 111 {0xe2a0b5dc971f303a, 667, false}, // 10^220 112 {0xa8d9d1535ce3b396, 694, false}, // 10^228 113 {0xfb9b7cd9a4a7443c, 720, false}, // 10^236 114 {0xbb764c4ca7a44410, 747, false}, // 10^244 115 {0x8bab8eefb6409c1a, 774, false}, // 10^252 116 {0xd01fef10a657842c, 800, false}, // 10^260 117 {0x9b10a4e5e9913129, 827, false}, // 10^268 118 {0xe7109bfba19c0c9d, 853, false}, // 10^276 119 {0xac2820d9623bf429, 880, false}, // 10^284 120 {0x80444b5e7aa7cf85, 907, false}, // 10^292 121 {0xbf21e44003acdd2d, 933, false}, // 10^300 122 {0x8e679c2f5e44ff8f, 960, false}, // 10^308 123 {0xd433179d9c8cb841, 986, false}, // 10^316 124 {0x9e19db92b4e31ba9, 1013, false}, // 10^324 125 {0xeb96bf6ebadf77d9, 1039, false}, // 10^332 126 {0xaf87023b9bf0ee6b, 1066, false}, // 10^340 127 } 128 129 // floatBits returns the bits of the float64 that best approximates 130 // the extFloat passed as receiver. Overflow is set to true if 131 // the resulting float64 is ±Inf. 132 func (f *extFloat) floatBits(flt *floatInfo) (bits uint64, overflow bool) { 133 f.Normalize() 134 135 exp := f.exp + 63 136 137 // Exponent too small. 138 if exp < flt.bias+1 { 139 n := flt.bias + 1 - exp 140 f.mant >>= uint(n) 141 exp += n 142 } 143 144 // Extract 1+flt.mantbits bits from the 64-bit mantissa. 145 mant := f.mant >> (63 - flt.mantbits) 146 if f.mant&(1<<(62-flt.mantbits)) != 0 { 147 // Round up. 148 mant += 1 149 } 150 151 // Rounding might have added a bit; shift down. 152 if mant == 2<<flt.mantbits { 153 mant >>= 1 154 exp++ 155 } 156 157 // Infinities. 158 if exp-flt.bias >= 1<<flt.expbits-1 { 159 // ±Inf 160 mant = 0 161 exp = 1<<flt.expbits - 1 + flt.bias 162 overflow = true 163 } else if mant&(1<<flt.mantbits) == 0 { 164 // Denormalized? 165 exp = flt.bias 166 } 167 // Assemble bits. 168 bits = mant & (uint64(1)<<flt.mantbits - 1) 169 bits |= uint64((exp-flt.bias)&(1<<flt.expbits-1)) << flt.mantbits 170 if f.neg { 171 bits |= 1 << (flt.mantbits + flt.expbits) 172 } 173 return 174 } 175 176 // AssignComputeBounds sets f to the floating point value 177 // defined by mant, exp and precision given by flt. It returns 178 // lower, upper such that any number in the closed interval 179 // [lower, upper] is converted back to the same floating point number. 180 func (f *extFloat) AssignComputeBounds(mant uint64, exp int, neg bool, flt *floatInfo) (lower, upper extFloat) { 181 f.mant = mant 182 f.exp = exp - int(flt.mantbits) 183 f.neg = neg 184 if f.exp <= 0 && mant == (mant>>uint(-f.exp))<<uint(-f.exp) { 185 // An exact integer 186 f.mant >>= uint(-f.exp) 187 f.exp = 0 188 return *f, *f 189 } 190 expBiased := exp - flt.bias 191 192 upper = extFloat{mant: 2*f.mant + 1, exp: f.exp - 1, neg: f.neg} 193 if mant != 1<<flt.mantbits || expBiased == 1 { 194 lower = extFloat{mant: 2*f.mant - 1, exp: f.exp - 1, neg: f.neg} 195 } else { 196 lower = extFloat{mant: 4*f.mant - 1, exp: f.exp - 2, neg: f.neg} 197 } 198 return 199 } 200 201 // Normalize normalizes f so that the highest bit of the mantissa is 202 // set, and returns the number by which the mantissa was left-shifted. 203 func (f *extFloat) Normalize() uint { 204 // bits.LeadingZeros64 would return 64 205 if f.mant == 0 { 206 return 0 207 } 208 shift := bits.LeadingZeros64(f.mant) 209 f.mant <<= uint(shift) 210 f.exp -= shift 211 return uint(shift) 212 } 213 214 // Multiply sets f to the product f*g: the result is correctly rounded, 215 // but not normalized. 216 func (f *extFloat) Multiply(g extFloat) { 217 fhi, flo := f.mant>>32, uint64(uint32(f.mant)) 218 ghi, glo := g.mant>>32, uint64(uint32(g.mant)) 219 220 // Cross products. 221 cross1 := fhi * glo 222 cross2 := flo * ghi 223 224 // f.mant*g.mant is fhi*ghi << 64 + (cross1+cross2) << 32 + flo*glo 225 f.mant = fhi*ghi + (cross1 >> 32) + (cross2 >> 32) 226 rem := uint64(uint32(cross1)) + uint64(uint32(cross2)) + ((flo * glo) >> 32) 227 // Round up. 228 rem += (1 << 31) 229 230 f.mant += (rem >> 32) 231 f.exp = f.exp + g.exp + 64 232 } 233 234 var uint64pow10 = [...]uint64{ 235 1, 1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7, 1e8, 1e9, 236 1e10, 1e11, 1e12, 1e13, 1e14, 1e15, 1e16, 1e17, 1e18, 1e19, 237 } 238 239 // AssignDecimal sets f to an approximate value mantissa*10^exp. It 240 // reports whether the value represented by f is guaranteed to be the 241 // best approximation of d after being rounded to a float64 or 242 // float32 depending on flt. 243 func (f *extFloat) AssignDecimal(mantissa uint64, exp10 int, neg bool, trunc bool, flt *floatInfo) (ok bool) { 244 const uint64digits = 19 245 const errorscale = 8 246 errors := 0 // An upper bound for error, computed in errorscale*ulp. 247 if trunc { 248 // the decimal number was truncated. 249 errors += errorscale / 2 250 } 251 252 f.mant = mantissa 253 f.exp = 0 254 f.neg = neg 255 256 // Multiply by powers of ten. 257 i := (exp10 - firstPowerOfTen) / stepPowerOfTen 258 if exp10 < firstPowerOfTen || i >= len(powersOfTen) { 259 return false 260 } 261 adjExp := (exp10 - firstPowerOfTen) % stepPowerOfTen 262 263 // We multiply by exp%step 264 if adjExp < uint64digits && mantissa < uint64pow10[uint64digits-adjExp] { 265 // We can multiply the mantissa exactly. 266 f.mant *= uint64pow10[adjExp] 267 f.Normalize() 268 } else { 269 f.Normalize() 270 f.Multiply(smallPowersOfTen[adjExp]) 271 errors += errorscale / 2 272 } 273 274 // We multiply by 10 to the exp - exp%step. 275 f.Multiply(powersOfTen[i]) 276 if errors > 0 { 277 errors += 1 278 } 279 errors += errorscale / 2 280 281 // Normalize 282 shift := f.Normalize() 283 errors <<= shift 284 285 // Now f is a good approximation of the decimal. 286 // Check whether the error is too large: that is, if the mantissa 287 // is perturbated by the error, the resulting float64 will change. 288 // The 64 bits mantissa is 1 + 52 bits for float64 + 11 extra bits. 289 // 290 // In many cases the approximation will be good enough. 291 denormalExp := flt.bias - 63 292 var extrabits uint 293 if f.exp <= denormalExp { 294 // f.mant * 2^f.exp is smaller than 2^(flt.bias+1). 295 extrabits = 63 - flt.mantbits + 1 + uint(denormalExp-f.exp) 296 } else { 297 extrabits = 63 - flt.mantbits 298 } 299 300 halfway := uint64(1) << (extrabits - 1) 301 mant_extra := f.mant & (1<<extrabits - 1) 302 303 // Do a signed comparison here! If the error estimate could make 304 // the mantissa round differently for the conversion to double, 305 // then we can't give a definite answer. 306 if int64(halfway)-int64(errors) < int64(mant_extra) && 307 int64(mant_extra) < int64(halfway)+int64(errors) { 308 return false 309 } 310 return true 311 } 312 313 // Frexp10 is an analogue of math.Frexp for decimal powers. It scales 314 // f by an approximate power of ten 10^-exp, and returns exp10, so 315 // that f*10^exp10 has the same value as the old f, up to an ulp, 316 // as well as the index of 10^-exp in the powersOfTen table. 317 func (f *extFloat) frexp10() (exp10, index int) { 318 // The constants expMin and expMax constrain the final value of the 319 // binary exponent of f. We want a small integral part in the result 320 // because finding digits of an integer requires divisions, whereas 321 // digits of the fractional part can be found by repeatedly multiplying 322 // by 10. 323 const expMin = -60 324 const expMax = -32 325 // Find power of ten such that x * 10^n has a binary exponent 326 // between expMin and expMax. 327 approxExp10 := ((expMin+expMax)/2 - f.exp) * 28 / 93 // log(10)/log(2) is close to 93/28. 328 i := (approxExp10 - firstPowerOfTen) / stepPowerOfTen 329 Loop: 330 for { 331 exp := f.exp + powersOfTen[i].exp + 64 332 switch { 333 case exp < expMin: 334 i++ 335 case exp > expMax: 336 i-- 337 default: 338 break Loop 339 } 340 } 341 // Apply the desired decimal shift on f. It will have exponent 342 // in the desired range. This is multiplication by 10^-exp10. 343 f.Multiply(powersOfTen[i]) 344 345 return -(firstPowerOfTen + i*stepPowerOfTen), i 346 } 347 348 // frexp10Many applies a common shift by a power of ten to a, b, c. 349 func frexp10Many(a, b, c *extFloat) (exp10 int) { 350 exp10, i := c.frexp10() 351 a.Multiply(powersOfTen[i]) 352 b.Multiply(powersOfTen[i]) 353 return 354 } 355 356 // FixedDecimal stores in d the first n significant digits 357 // of the decimal representation of f. It returns false 358 // if it cannot be sure of the answer. 359 func (f *extFloat) FixedDecimal(d *decimalSlice, n int) bool { 360 if f.mant == 0 { 361 d.nd = 0 362 d.dp = 0 363 d.neg = f.neg 364 return true 365 } 366 if n == 0 { 367 panic("strconv: internal error: extFloat.FixedDecimal called with n == 0") 368 } 369 // Multiply by an appropriate power of ten to have a reasonable 370 // number to process. 371 f.Normalize() 372 exp10, _ := f.frexp10() 373 374 shift := uint(-f.exp) 375 integer := uint32(f.mant >> shift) 376 fraction := f.mant - (uint64(integer) << shift) 377 ε := uint64(1) // ε is the uncertainty we have on the mantissa of f. 378 379 // Write exactly n digits to d. 380 needed := n // how many digits are left to write. 381 integerDigits := 0 // the number of decimal digits of integer. 382 pow10 := uint64(1) // the power of ten by which f was scaled. 383 for i, pow := 0, uint64(1); i < 20; i++ { 384 if pow > uint64(integer) { 385 integerDigits = i 386 break 387 } 388 pow *= 10 389 } 390 rest := integer 391 if integerDigits > needed { 392 // the integral part is already large, trim the last digits. 393 pow10 = uint64pow10[integerDigits-needed] 394 integer /= uint32(pow10) 395 rest -= integer * uint32(pow10) 396 } else { 397 rest = 0 398 } 399 400 // Write the digits of integer: the digits of rest are omitted. 401 var buf [32]byte 402 pos := len(buf) 403 for v := integer; v > 0; { 404 v1 := v / 10 405 v -= 10 * v1 406 pos-- 407 buf[pos] = byte(v + '0') 408 v = v1 409 } 410 for i := pos; i < len(buf); i++ { 411 d.d[i-pos] = buf[i] 412 } 413 nd := len(buf) - pos 414 d.nd = nd 415 d.dp = integerDigits + exp10 416 needed -= nd 417 418 if needed > 0 { 419 if rest != 0 || pow10 != 1 { 420 panic("strconv: internal error, rest != 0 but needed > 0") 421 } 422 // Emit digits for the fractional part. Each time, 10*fraction 423 // fits in a uint64 without overflow. 424 for needed > 0 { 425 fraction *= 10 426 ε *= 10 // the uncertainty scales as we multiply by ten. 427 if 2*ε > 1<<shift { 428 // the error is so large it could modify which digit to write, abort. 429 return false 430 } 431 digit := fraction >> shift 432 d.d[nd] = byte(digit + '0') 433 fraction -= digit << shift 434 nd++ 435 needed-- 436 } 437 d.nd = nd 438 } 439 440 // We have written a truncation of f (a numerator / 10^d.dp). The remaining part 441 // can be interpreted as a small number (< 1) to be added to the last digit of the 442 // numerator. 443 // 444 // If rest > 0, the amount is: 445 // (rest<<shift | fraction) / (pow10 << shift) 446 // fraction being known with a ±ε uncertainty. 447 // The fact that n > 0 guarantees that pow10 << shift does not overflow a uint64. 448 // 449 // If rest = 0, pow10 == 1 and the amount is 450 // fraction / (1 << shift) 451 // fraction being known with a ±ε uncertainty. 452 // 453 // We pass this information to the rounding routine for adjustment. 454 455 ok := adjustLastDigitFixed(d, uint64(rest)<<shift|fraction, pow10, shift, ε) 456 if !ok { 457 return false 458 } 459 // Trim trailing zeros. 460 for i := d.nd - 1; i >= 0; i-- { 461 if d.d[i] != '0' { 462 d.nd = i + 1 463 break 464 } 465 } 466 return true 467 } 468 469 // adjustLastDigitFixed assumes d contains the representation of the integral part 470 // of some number, whose fractional part is num / (den << shift). The numerator 471 // num is only known up to an uncertainty of size ε, assumed to be less than 472 // (den << shift)/2. 473 // 474 // It will increase the last digit by one to account for correct rounding, typically 475 // when the fractional part is greater than 1/2, and will return false if ε is such 476 // that no correct answer can be given. 477 func adjustLastDigitFixed(d *decimalSlice, num, den uint64, shift uint, ε uint64) bool { 478 if num > den<<shift { 479 panic("strconv: num > den<<shift in adjustLastDigitFixed") 480 } 481 if 2*ε > den<<shift { 482 panic("strconv: ε > (den<<shift)/2") 483 } 484 if 2*(num+ε) < den<<shift { 485 return true 486 } 487 if 2*(num-ε) > den<<shift { 488 // increment d by 1. 489 i := d.nd - 1 490 for ; i >= 0; i-- { 491 if d.d[i] == '9' { 492 d.nd-- 493 } else { 494 break 495 } 496 } 497 if i < 0 { 498 d.d[0] = '1' 499 d.nd = 1 500 d.dp++ 501 } else { 502 d.d[i]++ 503 } 504 return true 505 } 506 return false 507 } 508 509 // ShortestDecimal stores in d the shortest decimal representation of f 510 // which belongs to the open interval (lower, upper), where f is supposed 511 // to lie. It returns false whenever the result is unsure. The implementation 512 // uses the Grisu3 algorithm. 513 func (f *extFloat) ShortestDecimal(d *decimalSlice, lower, upper *extFloat) bool { 514 if f.mant == 0 { 515 d.nd = 0 516 d.dp = 0 517 d.neg = f.neg 518 return true 519 } 520 if f.exp == 0 && *lower == *f && *lower == *upper { 521 // an exact integer. 522 var buf [24]byte 523 n := len(buf) - 1 524 for v := f.mant; v > 0; { 525 v1 := v / 10 526 v -= 10 * v1 527 buf[n] = byte(v + '0') 528 n-- 529 v = v1 530 } 531 nd := len(buf) - n - 1 532 for i := 0; i < nd; i++ { 533 d.d[i] = buf[n+1+i] 534 } 535 d.nd, d.dp = nd, nd 536 for d.nd > 0 && d.d[d.nd-1] == '0' { 537 d.nd-- 538 } 539 if d.nd == 0 { 540 d.dp = 0 541 } 542 d.neg = f.neg 543 return true 544 } 545 upper.Normalize() 546 // Uniformize exponents. 547 if f.exp > upper.exp { 548 f.mant <<= uint(f.exp - upper.exp) 549 f.exp = upper.exp 550 } 551 if lower.exp > upper.exp { 552 lower.mant <<= uint(lower.exp - upper.exp) 553 lower.exp = upper.exp 554 } 555 556 exp10 := frexp10Many(lower, f, upper) 557 // Take a safety margin due to rounding in frexp10Many, but we lose precision. 558 upper.mant++ 559 lower.mant-- 560 561 // The shortest representation of f is either rounded up or down, but 562 // in any case, it is a truncation of upper. 563 shift := uint(-upper.exp) 564 integer := uint32(upper.mant >> shift) 565 fraction := upper.mant - (uint64(integer) << shift) 566 567 // How far we can go down from upper until the result is wrong. 568 allowance := upper.mant - lower.mant 569 // How far we should go to get a very precise result. 570 targetDiff := upper.mant - f.mant 571 572 // Count integral digits: there are at most 10. 573 var integerDigits int 574 for i, pow := 0, uint64(1); i < 20; i++ { 575 if pow > uint64(integer) { 576 integerDigits = i 577 break 578 } 579 pow *= 10 580 } 581 for i := 0; i < integerDigits; i++ { 582 pow := uint64pow10[integerDigits-i-1] 583 digit := integer / uint32(pow) 584 d.d[i] = byte(digit + '0') 585 integer -= digit * uint32(pow) 586 // evaluate whether we should stop. 587 if currentDiff := uint64(integer)<<shift + fraction; currentDiff < allowance { 588 d.nd = i + 1 589 d.dp = integerDigits + exp10 590 d.neg = f.neg 591 // Sometimes allowance is so large the last digit might need to be 592 // decremented to get closer to f. 593 return adjustLastDigit(d, currentDiff, targetDiff, allowance, pow<<shift, 2) 594 } 595 } 596 d.nd = integerDigits 597 d.dp = d.nd + exp10 598 d.neg = f.neg 599 600 // Compute digits of the fractional part. At each step fraction does not 601 // overflow. The choice of minExp implies that fraction is less than 2^60. 602 var digit int 603 multiplier := uint64(1) 604 for { 605 fraction *= 10 606 multiplier *= 10 607 digit = int(fraction >> shift) 608 d.d[d.nd] = byte(digit + '0') 609 d.nd++ 610 fraction -= uint64(digit) << shift 611 if fraction < allowance*multiplier { 612 // We are in the admissible range. Note that if allowance is about to 613 // overflow, that is, allowance > 2^64/10, the condition is automatically 614 // true due to the limited range of fraction. 615 return adjustLastDigit(d, 616 fraction, targetDiff*multiplier, allowance*multiplier, 617 1<<shift, multiplier*2) 618 } 619 } 620 } 621 622 // adjustLastDigit modifies d = x-currentDiff*ε, to get closest to 623 // d = x-targetDiff*ε, without becoming smaller than x-maxDiff*ε. 624 // It assumes that a decimal digit is worth ulpDecimal*ε, and that 625 // all data is known with an error estimate of ulpBinary*ε. 626 func adjustLastDigit(d *decimalSlice, currentDiff, targetDiff, maxDiff, ulpDecimal, ulpBinary uint64) bool { 627 if ulpDecimal < 2*ulpBinary { 628 // Approximation is too wide. 629 return false 630 } 631 for currentDiff+ulpDecimal/2+ulpBinary < targetDiff { 632 d.d[d.nd-1]-- 633 currentDiff += ulpDecimal 634 } 635 if currentDiff+ulpDecimal <= targetDiff+ulpDecimal/2+ulpBinary { 636 // we have two choices, and don't know what to do. 637 return false 638 } 639 if currentDiff < ulpBinary || currentDiff > maxDiff-ulpBinary { 640 // we went too far 641 return false 642 } 643 if d.nd == 1 && d.d[0] == '0' { 644 // the number has actually reached zero. 645 d.nd = 0 646 d.dp = 0 647 } 648 return true 649 } 650
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