Source file src/pkg/strconv/extfloat.go
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 "math" 8 9 // An extFloat represents an extended floating-point number, with more 10 // precision than a float64. It does not try to save bits: the 11 // number represented by the structure is mant*(2^exp), with a negative 12 // sign if neg is true. 13 type extFloat struct { 14 mant uint64 15 exp int 16 neg bool 17 } 18 19 // Powers of ten taken from double-conversion library. 20 // http://code.google.com/p/double-conversion/ 21 const ( 22 firstPowerOfTen = -348 23 stepPowerOfTen = 8 24 ) 25 26 var smallPowersOfTen = [...]extFloat{ 27 {1 << 63, -63, false}, // 1 28 {0xa << 60, -60, false}, // 1e1 29 {0x64 << 57, -57, false}, // 1e2 30 {0x3e8 << 54, -54, false}, // 1e3 31 {0x2710 << 50, -50, false}, // 1e4 32 {0x186a0 << 47, -47, false}, // 1e5 33 {0xf4240 << 44, -44, false}, // 1e6 34 {0x989680 << 40, -40, false}, // 1e7 35 } 36 37 var powersOfTen = [...]extFloat{ 38 {0xfa8fd5a0081c0288, -1220, false}, // 10^-348 39 {0xbaaee17fa23ebf76, -1193, false}, // 10^-340 40 {0x8b16fb203055ac76, -1166, false}, // 10^-332 41 {0xcf42894a5dce35ea, -1140, false}, // 10^-324 42 {0x9a6bb0aa55653b2d, -1113, false}, // 10^-316 43 {0xe61acf033d1a45df, -1087, false}, // 10^-308 44 {0xab70fe17c79ac6ca, -1060, false}, // 10^-300 45 {0xff77b1fcbebcdc4f, -1034, false}, // 10^-292 46 {0xbe5691ef416bd60c, -1007, false}, // 10^-284 47 {0x8dd01fad907ffc3c, -980, false}, // 10^-276 48 {0xd3515c2831559a83, -954, false}, // 10^-268 49 {0x9d71ac8fada6c9b5, -927, false}, // 10^-260 50 {0xea9c227723ee8bcb, -901, false}, // 10^-252 51 {0xaecc49914078536d, -874, false}, // 10^-244 52 {0x823c12795db6ce57, -847, false}, // 10^-236 53 {0xc21094364dfb5637, -821, false}, // 10^-228 54 {0x9096ea6f3848984f, -794, false}, // 10^-220 55 {0xd77485cb25823ac7, -768, false}, // 10^-212 56 {0xa086cfcd97bf97f4, -741, false}, // 10^-204 57 {0xef340a98172aace5, -715, false}, // 10^-196 58 {0xb23867fb2a35b28e, -688, false}, // 10^-188 59 {0x84c8d4dfd2c63f3b, -661, false}, // 10^-180 60 {0xc5dd44271ad3cdba, -635, false}, // 10^-172 61 {0x936b9fcebb25c996, -608, false}, // 10^-164 62 {0xdbac6c247d62a584, -582, false}, // 10^-156 63 {0xa3ab66580d5fdaf6, -555, false}, // 10^-148 64 {0xf3e2f893dec3f126, -529, false}, // 10^-140 65 {0xb5b5ada8aaff80b8, -502, false}, // 10^-132 66 {0x87625f056c7c4a8b, -475, false}, // 10^-124 67 {0xc9bcff6034c13053, -449, false}, // 10^-116 68 {0x964e858c91ba2655, -422, false}, // 10^-108 69 {0xdff9772470297ebd, -396, false}, // 10^-100 70 {0xa6dfbd9fb8e5b88f, -369, false}, // 10^-92 71 {0xf8a95fcf88747d94, -343, false}, // 10^-84 72 {0xb94470938fa89bcf, -316, false}, // 10^-76 73 {0x8a08f0f8bf0f156b, -289, false}, // 10^-68 74 {0xcdb02555653131b6, -263, false}, // 10^-60 75 {0x993fe2c6d07b7fac, -236, false}, // 10^-52 76 {0xe45c10c42a2b3b06, -210, false}, // 10^-44 77 {0xaa242499697392d3, -183, false}, // 10^-36 78 {0xfd87b5f28300ca0e, -157, false}, // 10^-28 79 {0xbce5086492111aeb, -130, false}, // 10^-20 80 {0x8cbccc096f5088cc, -103, false}, // 10^-12 81 {0xd1b71758e219652c, -77, false}, // 10^-4 82 {0x9c40000000000000, -50, false}, // 10^4 83 {0xe8d4a51000000000, -24, false}, // 10^12 84 {0xad78ebc5ac620000, 3, false}, // 10^20 85 {0x813f3978f8940984, 30, false}, // 10^28 86 {0xc097ce7bc90715b3, 56, false}, // 10^36 87 {0x8f7e32ce7bea5c70, 83, false}, // 10^44 88 {0xd5d238a4abe98068, 109, false}, // 10^52 89 {0x9f4f2726179a2245, 136, false}, // 10^60 90 {0xed63a231d4c4fb27, 162, false}, // 10^68 91 {0xb0de65388cc8ada8, 189, false}, // 10^76 92 {0x83c7088e1aab65db, 216, false}, // 10^84 93 {0xc45d1df942711d9a, 242, false}, // 10^92 94 {0x924d692ca61be758, 269, false}, // 10^100 95 {0xda01ee641a708dea, 295, false}, // 10^108 96 {0xa26da3999aef774a, 322, false}, // 10^116 97 {0xf209787bb47d6b85, 348, false}, // 10^124 98 {0xb454e4a179dd1877, 375, false}, // 10^132 99 {0x865b86925b9bc5c2, 402, false}, // 10^140 100 {0xc83553c5c8965d3d, 428, false}, // 10^148 101 {0x952ab45cfa97a0b3, 455, false}, // 10^156 102 {0xde469fbd99a05fe3, 481, false}, // 10^164 103 {0xa59bc234db398c25, 508, false}, // 10^172 104 {0xf6c69a72a3989f5c, 534, false}, // 10^180 105 {0xb7dcbf5354e9bece, 561, false}, // 10^188 106 {0x88fcf317f22241e2, 588, false}, // 10^196 107 {0xcc20ce9bd35c78a5, 614, false}, // 10^204 108 {0x98165af37b2153df, 641, false}, // 10^212 109 {0xe2a0b5dc971f303a, 667, false}, // 10^220 110 {0xa8d9d1535ce3b396, 694, false}, // 10^228 111 {0xfb9b7cd9a4a7443c, 720, false}, // 10^236 112 {0xbb764c4ca7a44410, 747, false}, // 10^244 113 {0x8bab8eefb6409c1a, 774, false}, // 10^252 114 {0xd01fef10a657842c, 800, false}, // 10^260 115 {0x9b10a4e5e9913129, 827, false}, // 10^268 116 {0xe7109bfba19c0c9d, 853, false}, // 10^276 117 {0xac2820d9623bf429, 880, false}, // 10^284 118 {0x80444b5e7aa7cf85, 907, false}, // 10^292 119 {0xbf21e44003acdd2d, 933, false}, // 10^300 120 {0x8e679c2f5e44ff8f, 960, false}, // 10^308 121 {0xd433179d9c8cb841, 986, false}, // 10^316 122 {0x9e19db92b4e31ba9, 1013, false}, // 10^324 123 {0xeb96bf6ebadf77d9, 1039, false}, // 10^332 124 {0xaf87023b9bf0ee6b, 1066, false}, // 10^340 125 } 126 127 // floatBits returns the bits of the float64 that best approximates 128 // the extFloat passed as receiver. Overflow is set to true if 129 // the resulting float64 is ±Inf. 130 func (f *extFloat) floatBits() (bits uint64, overflow bool) { 131 flt := &float64info 132 f.Normalize() 133 134 exp := f.exp + 63 135 136 // Exponent too small. 137 if exp < flt.bias+1 { 138 n := flt.bias + 1 - exp 139 f.mant >>= uint(n) 140 exp += n 141 } 142 143 // Extract 1+flt.mantbits bits. 144 mant := f.mant >> (63 - flt.mantbits) 145 if f.mant&(1<<(62-flt.mantbits)) != 0 { 146 // Round up. 147 mant += 1 148 } 149 150 // Rounding might have added a bit; shift down. 151 if mant == 2<<flt.mantbits { 152 mant >>= 1 153 exp++ 154 } 155 156 // Infinities. 157 if exp-flt.bias >= 1<<flt.expbits-1 { 158 goto overflow 159 } 160 161 // Denormalized? 162 if mant&(1<<flt.mantbits) == 0 { 163 exp = flt.bias 164 } 165 goto out 166 167 overflow: 168 // ±Inf 169 mant = 0 170 exp = 1<<flt.expbits - 1 + flt.bias 171 overflow = true 172 173 out: 174 // Assemble bits. 175 bits = mant & (uint64(1)<<flt.mantbits - 1) 176 bits |= uint64((exp-flt.bias)&(1<<flt.expbits-1)) << flt.mantbits 177 if f.neg { 178 bits |= 1 << (flt.mantbits + flt.expbits) 179 } 180 return 181 } 182 183 // Assign sets f to the value of x. 184 func (f *extFloat) Assign(x float64) { 185 if x < 0 { 186 x = -x 187 f.neg = true 188 } 189 x, f.exp = math.Frexp(x) 190 f.mant = uint64(x * float64(1<<64)) 191 f.exp -= 64 192 } 193 194 // AssignComputeBounds sets f to the value of x and returns 195 // lower, upper such that any number in the closed interval 196 // [lower, upper] is converted back to x. 197 func (f *extFloat) AssignComputeBounds(x float64) (lower, upper extFloat) { 198 // Special cases. 199 bits := math.Float64bits(x) 200 flt := &float64info 201 neg := bits>>(flt.expbits+flt.mantbits) != 0 202 expBiased := int(bits>>flt.mantbits) & (1<<flt.expbits - 1) 203 mant := bits & (uint64(1)<<flt.mantbits - 1) 204 205 if expBiased == 0 { 206 // denormalized. 207 f.mant = mant 208 f.exp = 1 + flt.bias - int(flt.mantbits) 209 } else { 210 f.mant = mant | 1<<flt.mantbits 211 f.exp = expBiased + flt.bias - int(flt.mantbits) 212 } 213 f.neg = neg 214 215 upper = extFloat{mant: 2*f.mant + 1, exp: f.exp - 1, neg: f.neg} 216 if mant != 0 || expBiased == 1 { 217 lower = extFloat{mant: 2*f.mant - 1, exp: f.exp - 1, neg: f.neg} 218 } else { 219 lower = extFloat{mant: 4*f.mant - 1, exp: f.exp - 2, neg: f.neg} 220 } 221 return 222 } 223 224 // Normalize normalizes f so that the highest bit of the mantissa is 225 // set, and returns the number by which the mantissa was left-shifted. 226 func (f *extFloat) Normalize() uint { 227 if f.mant == 0 { 228 return 0 229 } 230 exp_before := f.exp 231 for f.mant < (1 << 55) { 232 f.mant <<= 8 233 f.exp -= 8 234 } 235 for f.mant < (1 << 63) { 236 f.mant <<= 1 237 f.exp -= 1 238 } 239 return uint(exp_before - f.exp) 240 } 241 242 // Multiply sets f to the product f*g: the result is correctly rounded, 243 // but not normalized. 244 func (f *extFloat) Multiply(g extFloat) { 245 fhi, flo := f.mant>>32, uint64(uint32(f.mant)) 246 ghi, glo := g.mant>>32, uint64(uint32(g.mant)) 247 248 // Cross products. 249 cross1 := fhi * glo 250 cross2 := flo * ghi 251 252 // f.mant*g.mant is fhi*ghi << 64 + (cross1+cross2) << 32 + flo*glo 253 f.mant = fhi*ghi + (cross1 >> 32) + (cross2 >> 32) 254 rem := uint64(uint32(cross1)) + uint64(uint32(cross2)) + ((flo * glo) >> 32) 255 // Round up. 256 rem += (1 << 31) 257 258 f.mant += (rem >> 32) 259 f.exp = f.exp + g.exp + 64 260 } 261 262 var uint64pow10 = [...]uint64{ 263 1, 1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7, 1e8, 1e9, 264 1e10, 1e11, 1e12, 1e13, 1e14, 1e15, 1e16, 1e17, 1e18, 1e19, 265 } 266 267 // AssignDecimal sets f to an approximate value of the decimal d. It 268 // returns true if the value represented by f is guaranteed to be the 269 // best approximation of d after being rounded to a float64. 270 func (f *extFloat) AssignDecimal(d *decimal) (ok bool) { 271 const uint64digits = 19 272 const errorscale = 8 273 mant10, digits := d.atou64() 274 exp10 := d.dp - digits 275 errors := 0 // An upper bound for error, computed in errorscale*ulp. 276 277 if digits < d.nd { 278 // the decimal number was truncated. 279 errors += errorscale / 2 280 } 281 282 f.mant = mant10 283 f.exp = 0 284 f.neg = d.neg 285 286 // Multiply by powers of ten. 287 i := (exp10 - firstPowerOfTen) / stepPowerOfTen 288 if exp10 < firstPowerOfTen || i >= len(powersOfTen) { 289 return false 290 } 291 adjExp := (exp10 - firstPowerOfTen) % stepPowerOfTen 292 293 // We multiply by exp%step 294 if digits+adjExp <= uint64digits { 295 // We can multiply the mantissa 296 f.mant *= uint64(float64pow10[adjExp]) 297 f.Normalize() 298 } else { 299 f.Normalize() 300 f.Multiply(smallPowersOfTen[adjExp]) 301 errors += errorscale / 2 302 } 303 304 // We multiply by 10 to the exp - exp%step. 305 f.Multiply(powersOfTen[i]) 306 if errors > 0 { 307 errors += 1 308 } 309 errors += errorscale / 2 310 311 // Normalize 312 shift := f.Normalize() 313 errors <<= shift 314 315 // Now f is a good approximation of the decimal. 316 // Check whether the error is too large: that is, if the mantissa 317 // is perturbated by the error, the resulting float64 will change. 318 // The 64 bits mantissa is 1 + 52 bits for float64 + 11 extra bits. 319 // 320 // In many cases the approximation will be good enough. 321 const denormalExp = -1023 - 63 322 flt := &float64info 323 var extrabits uint 324 if f.exp <= denormalExp { 325 extrabits = uint(63 - flt.mantbits + 1 + uint(denormalExp-f.exp)) 326 } else { 327 extrabits = uint(63 - flt.mantbits) 328 } 329 330 halfway := uint64(1) << (extrabits - 1) 331 mant_extra := f.mant & (1<<extrabits - 1) 332 333 // Do a signed comparison here! If the error estimate could make 334 // the mantissa round differently for the conversion to double, 335 // then we can't give a definite answer. 336 if int64(halfway)-int64(errors) < int64(mant_extra) && 337 int64(mant_extra) < int64(halfway)+int64(errors) { 338 return false 339 } 340 return true 341 } 342 343 // Frexp10 is an analogue of math.Frexp for decimal powers. It scales 344 // f by an approximate power of ten 10^-exp, and returns exp10, so 345 // that f*10^exp10 has the same value as the old f, up to an ulp, 346 // as well as the index of 10^-exp in the powersOfTen table. 347 // The arguments expMin and expMax constrain the final value of the 348 // binary exponent of f. 349 func (f *extFloat) frexp10(expMin, expMax int) (exp10, index int) { 350 // it is illegal to call this function with a too restrictive exponent range. 351 if expMax-expMin <= 25 { 352 panic("strconv: invalid exponent range") 353 } 354 // Find power of ten such that x * 10^n has a binary exponent 355 // between expMin and expMax 356 approxExp10 := -(f.exp + 100) * 28 / 93 // log(10)/log(2) is close to 93/28. 357 i := (approxExp10 - firstPowerOfTen) / stepPowerOfTen 358 Loop: 359 for { 360 exp := f.exp + powersOfTen[i].exp + 64 361 switch { 362 case exp < expMin: 363 i++ 364 case exp > expMax: 365 i-- 366 default: 367 break Loop 368 } 369 } 370 // Apply the desired decimal shift on f. It will have exponent 371 // in the desired range. This is multiplication by 10^-exp10. 372 f.Multiply(powersOfTen[i]) 373 374 return -(firstPowerOfTen + i*stepPowerOfTen), i 375 } 376 377 // frexp10Many applies a common shift by a power of ten to a, b, c. 378 func frexp10Many(expMin, expMax int, a, b, c *extFloat) (exp10 int) { 379 exp10, i := c.frexp10(expMin, expMax) 380 a.Multiply(powersOfTen[i]) 381 b.Multiply(powersOfTen[i]) 382 return 383 } 384 385 // ShortestDecimal stores in d the shortest decimal representation of f 386 // which belongs to the open interval (lower, upper), where f is supposed 387 // to lie. It returns false whenever the result is unsure. The implementation 388 // uses the Grisu3 algorithm. 389 func (f *extFloat) ShortestDecimal(d *decimal, lower, upper *extFloat) bool { 390 if f.mant == 0 { 391 d.d[0] = '0' 392 d.nd = 1 393 d.dp = 0 394 d.neg = f.neg 395 } 396 const minExp = -60 397 const maxExp = -32 398 upper.Normalize() 399 // Uniformize exponents. 400 if f.exp > upper.exp { 401 f.mant <<= uint(f.exp - upper.exp) 402 f.exp = upper.exp 403 } 404 if lower.exp > upper.exp { 405 lower.mant <<= uint(lower.exp - upper.exp) 406 lower.exp = upper.exp 407 } 408 409 exp10 := frexp10Many(minExp, maxExp, lower, f, upper) 410 // Take a safety margin due to rounding in frexp10Many, but we lose precision. 411 upper.mant++ 412 lower.mant-- 413 414 // The shortest representation of f is either rounded up or down, but 415 // in any case, it is a truncation of upper. 416 shift := uint(-upper.exp) 417 integer := uint32(upper.mant >> shift) 418 fraction := upper.mant - (uint64(integer) << shift) 419 420 // How far we can go down from upper until the result is wrong. 421 allowance := upper.mant - lower.mant 422 // How far we should go to get a very precise result. 423 targetDiff := upper.mant - f.mant 424 425 // Count integral digits: there are at most 10. 426 var integerDigits int 427 for i, pow := range uint64pow10 { 428 if uint64(integer) >= pow { 429 integerDigits = i + 1 430 } 431 } 432 for i := 0; i < integerDigits; i++ { 433 pow := uint64pow10[integerDigits-i-1] 434 digit := integer / uint32(pow) 435 d.d[i] = byte(digit + '0') 436 integer -= digit * uint32(pow) 437 // evaluate whether we should stop. 438 if currentDiff := uint64(integer)<<shift + fraction; currentDiff < allowance { 439 d.nd = i + 1 440 d.dp = integerDigits + exp10 441 d.neg = f.neg 442 // Sometimes allowance is so large the last digit might need to be 443 // decremented to get closer to f. 444 return adjustLastDigit(d, currentDiff, targetDiff, allowance, pow<<shift, 2) 445 } 446 } 447 d.nd = integerDigits 448 d.dp = d.nd + exp10 449 d.neg = f.neg 450 451 // Compute digits of the fractional part. At each step fraction does not 452 // overflow. The choice of minExp implies that fraction is less than 2^60. 453 var digit int 454 multiplier := uint64(1) 455 for { 456 fraction *= 10 457 multiplier *= 10 458 digit = int(fraction >> shift) 459 d.d[d.nd] = byte(digit + '0') 460 d.nd++ 461 fraction -= uint64(digit) << shift 462 if fraction < allowance*multiplier { 463 // We are in the admissible range. Note that if allowance is about to 464 // overflow, that is, allowance > 2^64/10, the condition is automatically 465 // true due to the limited range of fraction. 466 return adjustLastDigit(d, 467 fraction, targetDiff*multiplier, allowance*multiplier, 468 1<<shift, multiplier*2) 469 } 470 } 471 return false 472 } 473 474 // adjustLastDigit modifies d = x-currentDiff*ε, to get closest to 475 // d = x-targetDiff*ε, without becoming smaller than x-maxDiff*ε. 476 // It assumes that a decimal digit is worth ulpDecimal*ε, and that 477 // all data is known with a error estimate of ulpBinary*ε. 478 func adjustLastDigit(d *decimal, currentDiff, targetDiff, maxDiff, ulpDecimal, ulpBinary uint64) bool { 479 if ulpDecimal < 2*ulpBinary { 480 // Approximation is too wide. 481 return false 482 } 483 for currentDiff+ulpDecimal/2+ulpBinary < targetDiff { 484 d.d[d.nd-1]-- 485 currentDiff += ulpDecimal 486 } 487 if currentDiff+ulpDecimal <= targetDiff+ulpDecimal/2+ulpBinary { 488 // we have two choices, and don't know what to do. 489 return false 490 } 491 if currentDiff < ulpBinary || currentDiff > maxDiff-ulpBinary { 492 // we went too far 493 return false 494 } 495 if d.nd == 1 && d.d[0] == '0' { 496 // the number has actually reached zero. 497 d.nd = 0 498 d.dp = 0 499 } 500 return true 501 }