ricow@chromium.org | 55ee807 | 2011-09-08 16:33:10 +0000 | [diff] [blame] | 1 | // Copyright 2011 the V8 project authors. All rights reserved. |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 2 | // Redistribution and use in source and binary forms, with or without |
| 3 | // modification, are permitted provided that the following conditions are |
| 4 | // met: |
| 5 | // |
| 6 | // * Redistributions of source code must retain the above copyright |
| 7 | // notice, this list of conditions and the following disclaimer. |
| 8 | // * Redistributions in binary form must reproduce the above |
| 9 | // copyright notice, this list of conditions and the following |
| 10 | // disclaimer in the documentation and/or other materials provided |
| 11 | // with the distribution. |
| 12 | // * Neither the name of Google Inc. nor the names of its |
| 13 | // contributors may be used to endorse or promote products derived |
| 14 | // from this software without specific prior written permission. |
| 15 | // |
| 16 | // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS |
| 17 | // "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT |
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| 27 | |
ricow@chromium.org | 55ee807 | 2011-09-08 16:33:10 +0000 | [diff] [blame] | 28 | #include "../include/v8stdint.h" |
| 29 | #include "checks.h" |
| 30 | #include "utils.h" |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 31 | |
| 32 | #include "fast-dtoa.h" |
| 33 | |
| 34 | #include "cached-powers.h" |
| 35 | #include "diy-fp.h" |
| 36 | #include "double.h" |
| 37 | |
| 38 | namespace v8 { |
| 39 | namespace internal { |
| 40 | |
| 41 | // The minimal and maximal target exponent define the range of w's binary |
| 42 | // exponent, where 'w' is the result of multiplying the input by a cached power |
| 43 | // of ten. |
| 44 | // |
| 45 | // A different range might be chosen on a different platform, to optimize digit |
| 46 | // generation, but a smaller range requires more powers of ten to be cached. |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 47 | static const int kMinimalTargetExponent = -60; |
| 48 | static const int kMaximalTargetExponent = -32; |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 49 | |
| 50 | |
| 51 | // Adjusts the last digit of the generated number, and screens out generated |
| 52 | // solutions that may be inaccurate. A solution may be inaccurate if it is |
| 53 | // outside the safe interval, or if we ctannot prove that it is closer to the |
| 54 | // input than a neighboring representation of the same length. |
| 55 | // |
| 56 | // Input: * buffer containing the digits of too_high / 10^kappa |
| 57 | // * the buffer's length |
| 58 | // * distance_too_high_w == (too_high - w).f() * unit |
| 59 | // * unsafe_interval == (too_high - too_low).f() * unit |
| 60 | // * rest = (too_high - buffer * 10^kappa).f() * unit |
| 61 | // * ten_kappa = 10^kappa * unit |
| 62 | // * unit = the common multiplier |
| 63 | // Output: returns true if the buffer is guaranteed to contain the closest |
| 64 | // representable number to the input. |
| 65 | // Modifies the generated digits in the buffer to approach (round towards) w. |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 66 | static bool RoundWeed(Vector<char> buffer, |
| 67 | int length, |
| 68 | uint64_t distance_too_high_w, |
| 69 | uint64_t unsafe_interval, |
| 70 | uint64_t rest, |
| 71 | uint64_t ten_kappa, |
| 72 | uint64_t unit) { |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 73 | uint64_t small_distance = distance_too_high_w - unit; |
| 74 | uint64_t big_distance = distance_too_high_w + unit; |
| 75 | // Let w_low = too_high - big_distance, and |
| 76 | // w_high = too_high - small_distance. |
| 77 | // Note: w_low < w < w_high |
| 78 | // |
| 79 | // The real w (* unit) must lie somewhere inside the interval |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 80 | // ]w_low; w_high[ (often written as "(w_low; w_high)") |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 81 | |
| 82 | // Basically the buffer currently contains a number in the unsafe interval |
| 83 | // ]too_low; too_high[ with too_low < w < too_high |
| 84 | // |
| 85 | // too_high - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - |
| 86 | // ^v 1 unit ^ ^ ^ ^ |
| 87 | // boundary_high --------------------- . . . . |
| 88 | // ^v 1 unit . . . . |
| 89 | // - - - - - - - - - - - - - - - - - - - + - - + - - - - - - . . |
| 90 | // . . ^ . . |
| 91 | // . big_distance . . . |
| 92 | // . . . . rest |
| 93 | // small_distance . . . . |
| 94 | // v . . . . |
| 95 | // w_high - - - - - - - - - - - - - - - - - - . . . . |
| 96 | // ^v 1 unit . . . . |
| 97 | // w ---------------------------------------- . . . . |
| 98 | // ^v 1 unit v . . . |
| 99 | // w_low - - - - - - - - - - - - - - - - - - - - - . . . |
| 100 | // . . v |
| 101 | // buffer --------------------------------------------------+-------+-------- |
| 102 | // . . |
| 103 | // safe_interval . |
| 104 | // v . |
| 105 | // - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - . |
| 106 | // ^v 1 unit . |
| 107 | // boundary_low ------------------------- unsafe_interval |
| 108 | // ^v 1 unit v |
| 109 | // too_low - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - |
| 110 | // |
| 111 | // |
| 112 | // Note that the value of buffer could lie anywhere inside the range too_low |
| 113 | // to too_high. |
| 114 | // |
| 115 | // boundary_low, boundary_high and w are approximations of the real boundaries |
| 116 | // and v (the input number). They are guaranteed to be precise up to one unit. |
| 117 | // In fact the error is guaranteed to be strictly less than one unit. |
| 118 | // |
| 119 | // Anything that lies outside the unsafe interval is guaranteed not to round |
| 120 | // to v when read again. |
| 121 | // Anything that lies inside the safe interval is guaranteed to round to v |
| 122 | // when read again. |
| 123 | // If the number inside the buffer lies inside the unsafe interval but not |
| 124 | // inside the safe interval then we simply do not know and bail out (returning |
| 125 | // false). |
| 126 | // |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 127 | // Similarly we have to take into account the imprecision of 'w' when finding |
| 128 | // the closest representation of 'w'. If we have two potential |
| 129 | // representations, and one is closer to both w_low and w_high, then we know |
| 130 | // it is closer to the actual value v. |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 131 | // |
| 132 | // By generating the digits of too_high we got the largest (closest to |
| 133 | // too_high) buffer that is still in the unsafe interval. In the case where |
| 134 | // w_high < buffer < too_high we try to decrement the buffer. |
| 135 | // This way the buffer approaches (rounds towards) w. |
| 136 | // There are 3 conditions that stop the decrementation process: |
| 137 | // 1) the buffer is already below w_high |
| 138 | // 2) decrementing the buffer would make it leave the unsafe interval |
| 139 | // 3) decrementing the buffer would yield a number below w_high and farther |
| 140 | // away than the current number. In other words: |
| 141 | // (buffer{-1} < w_high) && w_high - buffer{-1} > buffer - w_high |
| 142 | // Instead of using the buffer directly we use its distance to too_high. |
| 143 | // Conceptually rest ~= too_high - buffer |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 144 | // We need to do the following tests in this order to avoid over- and |
| 145 | // underflows. |
| 146 | ASSERT(rest <= unsafe_interval); |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 147 | while (rest < small_distance && // Negated condition 1 |
| 148 | unsafe_interval - rest >= ten_kappa && // Negated condition 2 |
| 149 | (rest + ten_kappa < small_distance || // buffer{-1} > w_high |
| 150 | small_distance - rest >= rest + ten_kappa - small_distance)) { |
| 151 | buffer[length - 1]--; |
| 152 | rest += ten_kappa; |
| 153 | } |
| 154 | |
| 155 | // We have approached w+ as much as possible. We now test if approaching w- |
| 156 | // would require changing the buffer. If yes, then we have two possible |
| 157 | // representations close to w, but we cannot decide which one is closer. |
| 158 | if (rest < big_distance && |
| 159 | unsafe_interval - rest >= ten_kappa && |
| 160 | (rest + ten_kappa < big_distance || |
| 161 | big_distance - rest > rest + ten_kappa - big_distance)) { |
| 162 | return false; |
| 163 | } |
| 164 | |
| 165 | // Weeding test. |
| 166 | // The safe interval is [too_low + 2 ulp; too_high - 2 ulp] |
| 167 | // Since too_low = too_high - unsafe_interval this is equivalent to |
| 168 | // [too_high - unsafe_interval + 4 ulp; too_high - 2 ulp] |
| 169 | // Conceptually we have: rest ~= too_high - buffer |
| 170 | return (2 * unit <= rest) && (rest <= unsafe_interval - 4 * unit); |
| 171 | } |
| 172 | |
| 173 | |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 174 | // Rounds the buffer upwards if the result is closer to v by possibly adding |
| 175 | // 1 to the buffer. If the precision of the calculation is not sufficient to |
| 176 | // round correctly, return false. |
| 177 | // The rounding might shift the whole buffer in which case the kappa is |
| 178 | // adjusted. For example "99", kappa = 3 might become "10", kappa = 4. |
| 179 | // |
| 180 | // If 2*rest > ten_kappa then the buffer needs to be round up. |
| 181 | // rest can have an error of +/- 1 unit. This function accounts for the |
| 182 | // imprecision and returns false, if the rounding direction cannot be |
| 183 | // unambiguously determined. |
| 184 | // |
| 185 | // Precondition: rest < ten_kappa. |
| 186 | static bool RoundWeedCounted(Vector<char> buffer, |
| 187 | int length, |
| 188 | uint64_t rest, |
| 189 | uint64_t ten_kappa, |
| 190 | uint64_t unit, |
| 191 | int* kappa) { |
| 192 | ASSERT(rest < ten_kappa); |
| 193 | // The following tests are done in a specific order to avoid overflows. They |
| 194 | // will work correctly with any uint64 values of rest < ten_kappa and unit. |
| 195 | // |
| 196 | // If the unit is too big, then we don't know which way to round. For example |
| 197 | // a unit of 50 means that the real number lies within rest +/- 50. If |
| 198 | // 10^kappa == 40 then there is no way to tell which way to round. |
| 199 | if (unit >= ten_kappa) return false; |
| 200 | // Even if unit is just half the size of 10^kappa we are already completely |
| 201 | // lost. (And after the previous test we know that the expression will not |
| 202 | // over/underflow.) |
| 203 | if (ten_kappa - unit <= unit) return false; |
| 204 | // If 2 * (rest + unit) <= 10^kappa we can safely round down. |
| 205 | if ((ten_kappa - rest > rest) && (ten_kappa - 2 * rest >= 2 * unit)) { |
| 206 | return true; |
| 207 | } |
| 208 | // If 2 * (rest - unit) >= 10^kappa, then we can safely round up. |
| 209 | if ((rest > unit) && (ten_kappa - (rest - unit) <= (rest - unit))) { |
| 210 | // Increment the last digit recursively until we find a non '9' digit. |
| 211 | buffer[length - 1]++; |
| 212 | for (int i = length - 1; i > 0; --i) { |
| 213 | if (buffer[i] != '0' + 10) break; |
| 214 | buffer[i] = '0'; |
| 215 | buffer[i - 1]++; |
| 216 | } |
| 217 | // If the first digit is now '0'+ 10 we had a buffer with all '9's. With the |
| 218 | // exception of the first digit all digits are now '0'. Simply switch the |
| 219 | // first digit to '1' and adjust the kappa. Example: "99" becomes "10" and |
| 220 | // the power (the kappa) is increased. |
| 221 | if (buffer[0] == '0' + 10) { |
| 222 | buffer[0] = '1'; |
| 223 | (*kappa) += 1; |
| 224 | } |
| 225 | return true; |
| 226 | } |
| 227 | return false; |
| 228 | } |
| 229 | |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 230 | |
| 231 | static const uint32_t kTen4 = 10000; |
| 232 | static const uint32_t kTen5 = 100000; |
| 233 | static const uint32_t kTen6 = 1000000; |
| 234 | static const uint32_t kTen7 = 10000000; |
| 235 | static const uint32_t kTen8 = 100000000; |
| 236 | static const uint32_t kTen9 = 1000000000; |
| 237 | |
| 238 | // Returns the biggest power of ten that is less than or equal than the given |
| 239 | // number. We furthermore receive the maximum number of bits 'number' has. |
| 240 | // If number_bits == 0 then 0^-1 is returned |
| 241 | // The number of bits must be <= 32. |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 242 | // Precondition: number < (1 << (number_bits + 1)). |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 243 | static void BiggestPowerTen(uint32_t number, |
| 244 | int number_bits, |
| 245 | uint32_t* power, |
| 246 | int* exponent) { |
| 247 | switch (number_bits) { |
| 248 | case 32: |
| 249 | case 31: |
| 250 | case 30: |
| 251 | if (kTen9 <= number) { |
| 252 | *power = kTen9; |
| 253 | *exponent = 9; |
| 254 | break; |
| 255 | } // else fallthrough |
| 256 | case 29: |
| 257 | case 28: |
| 258 | case 27: |
| 259 | if (kTen8 <= number) { |
| 260 | *power = kTen8; |
| 261 | *exponent = 8; |
| 262 | break; |
| 263 | } // else fallthrough |
| 264 | case 26: |
| 265 | case 25: |
| 266 | case 24: |
| 267 | if (kTen7 <= number) { |
| 268 | *power = kTen7; |
| 269 | *exponent = 7; |
| 270 | break; |
| 271 | } // else fallthrough |
| 272 | case 23: |
| 273 | case 22: |
| 274 | case 21: |
| 275 | case 20: |
| 276 | if (kTen6 <= number) { |
| 277 | *power = kTen6; |
| 278 | *exponent = 6; |
| 279 | break; |
| 280 | } // else fallthrough |
| 281 | case 19: |
| 282 | case 18: |
| 283 | case 17: |
| 284 | if (kTen5 <= number) { |
| 285 | *power = kTen5; |
| 286 | *exponent = 5; |
| 287 | break; |
| 288 | } // else fallthrough |
| 289 | case 16: |
| 290 | case 15: |
| 291 | case 14: |
| 292 | if (kTen4 <= number) { |
| 293 | *power = kTen4; |
| 294 | *exponent = 4; |
| 295 | break; |
| 296 | } // else fallthrough |
| 297 | case 13: |
| 298 | case 12: |
| 299 | case 11: |
| 300 | case 10: |
| 301 | if (1000 <= number) { |
| 302 | *power = 1000; |
| 303 | *exponent = 3; |
| 304 | break; |
| 305 | } // else fallthrough |
| 306 | case 9: |
| 307 | case 8: |
| 308 | case 7: |
| 309 | if (100 <= number) { |
| 310 | *power = 100; |
| 311 | *exponent = 2; |
| 312 | break; |
| 313 | } // else fallthrough |
| 314 | case 6: |
| 315 | case 5: |
| 316 | case 4: |
| 317 | if (10 <= number) { |
| 318 | *power = 10; |
| 319 | *exponent = 1; |
| 320 | break; |
| 321 | } // else fallthrough |
| 322 | case 3: |
| 323 | case 2: |
| 324 | case 1: |
| 325 | if (1 <= number) { |
| 326 | *power = 1; |
| 327 | *exponent = 0; |
| 328 | break; |
| 329 | } // else fallthrough |
| 330 | case 0: |
| 331 | *power = 0; |
| 332 | *exponent = -1; |
| 333 | break; |
| 334 | default: |
| 335 | // Following assignments are here to silence compiler warnings. |
| 336 | *power = 0; |
| 337 | *exponent = 0; |
| 338 | UNREACHABLE(); |
| 339 | } |
| 340 | } |
| 341 | |
| 342 | |
| 343 | // Generates the digits of input number w. |
| 344 | // w is a floating-point number (DiyFp), consisting of a significand and an |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 345 | // exponent. Its exponent is bounded by kMinimalTargetExponent and |
| 346 | // kMaximalTargetExponent. |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 347 | // Hence -60 <= w.e() <= -32. |
| 348 | // |
| 349 | // Returns false if it fails, in which case the generated digits in the buffer |
| 350 | // should not be used. |
| 351 | // Preconditions: |
| 352 | // * low, w and high are correct up to 1 ulp (unit in the last place). That |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 353 | // is, their error must be less than a unit of their last digits. |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 354 | // * low.e() == w.e() == high.e() |
| 355 | // * low < w < high, and taking into account their error: low~ <= high~ |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 356 | // * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 357 | // Postconditions: returns false if procedure fails. |
| 358 | // otherwise: |
| 359 | // * buffer is not null-terminated, but len contains the number of digits. |
| 360 | // * buffer contains the shortest possible decimal digit-sequence |
| 361 | // such that LOW < buffer * 10^kappa < HIGH, where LOW and HIGH are the |
| 362 | // correct values of low and high (without their error). |
| 363 | // * if more than one decimal representation gives the minimal number of |
| 364 | // decimal digits then the one closest to W (where W is the correct value |
| 365 | // of w) is chosen. |
| 366 | // Remark: this procedure takes into account the imprecision of its input |
| 367 | // numbers. If the precision is not enough to guarantee all the postconditions |
| 368 | // then false is returned. This usually happens rarely (~0.5%). |
| 369 | // |
| 370 | // Say, for the sake of example, that |
| 371 | // w.e() == -48, and w.f() == 0x1234567890abcdef |
| 372 | // w's value can be computed by w.f() * 2^w.e() |
| 373 | // We can obtain w's integral digits by simply shifting w.f() by -w.e(). |
| 374 | // -> w's integral part is 0x1234 |
| 375 | // w's fractional part is therefore 0x567890abcdef. |
| 376 | // Printing w's integral part is easy (simply print 0x1234 in decimal). |
| 377 | // In order to print its fraction we repeatedly multiply the fraction by 10 and |
sgjesse@chromium.org | 720dc0b | 2010-05-10 09:25:39 +0000 | [diff] [blame] | 378 | // get each digit. Example the first digit after the point would be computed by |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 379 | // (0x567890abcdef * 10) >> 48. -> 3 |
| 380 | // The whole thing becomes slightly more complicated because we want to stop |
| 381 | // once we have enough digits. That is, once the digits inside the buffer |
| 382 | // represent 'w' we can stop. Everything inside the interval low - high |
| 383 | // represents w. However we have to pay attention to low, high and w's |
| 384 | // imprecision. |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 385 | static bool DigitGen(DiyFp low, |
| 386 | DiyFp w, |
| 387 | DiyFp high, |
| 388 | Vector<char> buffer, |
| 389 | int* length, |
| 390 | int* kappa) { |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 391 | ASSERT(low.e() == w.e() && w.e() == high.e()); |
| 392 | ASSERT(low.f() + 1 <= high.f() - 1); |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 393 | ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent); |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 394 | // low, w and high are imprecise, but by less than one ulp (unit in the last |
| 395 | // place). |
| 396 | // If we remove (resp. add) 1 ulp from low (resp. high) we are certain that |
| 397 | // the new numbers are outside of the interval we want the final |
| 398 | // representation to lie in. |
| 399 | // Inversely adding (resp. removing) 1 ulp from low (resp. high) would yield |
| 400 | // numbers that are certain to lie in the interval. We will use this fact |
| 401 | // later on. |
| 402 | // We will now start by generating the digits within the uncertain |
| 403 | // interval. Later we will weed out representations that lie outside the safe |
| 404 | // interval and thus _might_ lie outside the correct interval. |
| 405 | uint64_t unit = 1; |
| 406 | DiyFp too_low = DiyFp(low.f() - unit, low.e()); |
| 407 | DiyFp too_high = DiyFp(high.f() + unit, high.e()); |
| 408 | // too_low and too_high are guaranteed to lie outside the interval we want the |
| 409 | // generated number in. |
| 410 | DiyFp unsafe_interval = DiyFp::Minus(too_high, too_low); |
| 411 | // We now cut the input number into two parts: the integral digits and the |
| 412 | // fractionals. We will not write any decimal separator though, but adapt |
| 413 | // kappa instead. |
| 414 | // Reminder: we are currently computing the digits (stored inside the buffer) |
| 415 | // such that: too_low < buffer * 10^kappa < too_high |
| 416 | // We use too_high for the digit_generation and stop as soon as possible. |
| 417 | // If we stop early we effectively round down. |
| 418 | DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e()); |
| 419 | // Division by one is a shift. |
| 420 | uint32_t integrals = static_cast<uint32_t>(too_high.f() >> -one.e()); |
| 421 | // Modulo by one is an and. |
| 422 | uint64_t fractionals = too_high.f() & (one.f() - 1); |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 423 | uint32_t divisor; |
| 424 | int divisor_exponent; |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 425 | BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()), |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 426 | &divisor, &divisor_exponent); |
| 427 | *kappa = divisor_exponent + 1; |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 428 | *length = 0; |
| 429 | // Loop invariant: buffer = too_high / 10^kappa (integer division) |
| 430 | // The invariant holds for the first iteration: kappa has been initialized |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 431 | // with the divisor exponent + 1. And the divisor is the biggest power of ten |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 432 | // that is smaller than integrals. |
| 433 | while (*kappa > 0) { |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 434 | int digit = integrals / divisor; |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 435 | buffer[*length] = '0' + digit; |
| 436 | (*length)++; |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 437 | integrals %= divisor; |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 438 | (*kappa)--; |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 439 | // Note that kappa now equals the exponent of the divisor and that the |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 440 | // invariant thus holds again. |
| 441 | uint64_t rest = |
| 442 | (static_cast<uint64_t>(integrals) << -one.e()) + fractionals; |
| 443 | // Invariant: too_high = buffer * 10^kappa + DiyFp(rest, one.e()) |
| 444 | // Reminder: unsafe_interval.e() == one.e() |
| 445 | if (rest < unsafe_interval.f()) { |
| 446 | // Rounding down (by not emitting the remaining digits) yields a number |
| 447 | // that lies within the unsafe interval. |
| 448 | return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f(), |
| 449 | unsafe_interval.f(), rest, |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 450 | static_cast<uint64_t>(divisor) << -one.e(), unit); |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 451 | } |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 452 | divisor /= 10; |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 453 | } |
| 454 | |
| 455 | // The integrals have been generated. We are at the point of the decimal |
| 456 | // separator. In the following loop we simply multiply the remaining digits by |
| 457 | // 10 and divide by one. We just need to pay attention to multiply associated |
| 458 | // data (like the interval or 'unit'), too. |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 459 | // Note that the multiplication by 10 does not overflow, because w.e >= -60 |
| 460 | // and thus one.e >= -60. |
| 461 | ASSERT(one.e() >= -60); |
| 462 | ASSERT(fractionals < one.f()); |
| 463 | ASSERT(V8_2PART_UINT64_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f()); |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 464 | while (true) { |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 465 | fractionals *= 10; |
| 466 | unit *= 10; |
| 467 | unsafe_interval.set_f(unsafe_interval.f() * 10); |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 468 | // Integer division by one. |
| 469 | int digit = static_cast<int>(fractionals >> -one.e()); |
| 470 | buffer[*length] = '0' + digit; |
| 471 | (*length)++; |
| 472 | fractionals &= one.f() - 1; // Modulo by one. |
| 473 | (*kappa)--; |
| 474 | if (fractionals < unsafe_interval.f()) { |
| 475 | return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f() * unit, |
| 476 | unsafe_interval.f(), fractionals, one.f(), unit); |
| 477 | } |
| 478 | } |
| 479 | } |
| 480 | |
| 481 | |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 482 | |
| 483 | // Generates (at most) requested_digits of input number w. |
| 484 | // w is a floating-point number (DiyFp), consisting of a significand and an |
| 485 | // exponent. Its exponent is bounded by kMinimalTargetExponent and |
| 486 | // kMaximalTargetExponent. |
| 487 | // Hence -60 <= w.e() <= -32. |
| 488 | // |
| 489 | // Returns false if it fails, in which case the generated digits in the buffer |
| 490 | // should not be used. |
| 491 | // Preconditions: |
| 492 | // * w is correct up to 1 ulp (unit in the last place). That |
| 493 | // is, its error must be strictly less than a unit of its last digit. |
| 494 | // * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent |
| 495 | // |
| 496 | // Postconditions: returns false if procedure fails. |
| 497 | // otherwise: |
| 498 | // * buffer is not null-terminated, but length contains the number of |
| 499 | // digits. |
| 500 | // * the representation in buffer is the most precise representation of |
| 501 | // requested_digits digits. |
| 502 | // * buffer contains at most requested_digits digits of w. If there are less |
| 503 | // than requested_digits digits then some trailing '0's have been removed. |
| 504 | // * kappa is such that |
| 505 | // w = buffer * 10^kappa + eps with |eps| < 10^kappa / 2. |
| 506 | // |
| 507 | // Remark: This procedure takes into account the imprecision of its input |
| 508 | // numbers. If the precision is not enough to guarantee all the postconditions |
| 509 | // then false is returned. This usually happens rarely, but the failure-rate |
| 510 | // increases with higher requested_digits. |
| 511 | static bool DigitGenCounted(DiyFp w, |
| 512 | int requested_digits, |
| 513 | Vector<char> buffer, |
| 514 | int* length, |
| 515 | int* kappa) { |
| 516 | ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent); |
| 517 | ASSERT(kMinimalTargetExponent >= -60); |
| 518 | ASSERT(kMaximalTargetExponent <= -32); |
| 519 | // w is assumed to have an error less than 1 unit. Whenever w is scaled we |
| 520 | // also scale its error. |
| 521 | uint64_t w_error = 1; |
| 522 | // We cut the input number into two parts: the integral digits and the |
| 523 | // fractional digits. We don't emit any decimal separator, but adapt kappa |
| 524 | // instead. Example: instead of writing "1.2" we put "12" into the buffer and |
| 525 | // increase kappa by 1. |
| 526 | DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e()); |
| 527 | // Division by one is a shift. |
| 528 | uint32_t integrals = static_cast<uint32_t>(w.f() >> -one.e()); |
| 529 | // Modulo by one is an and. |
| 530 | uint64_t fractionals = w.f() & (one.f() - 1); |
| 531 | uint32_t divisor; |
| 532 | int divisor_exponent; |
| 533 | BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()), |
| 534 | &divisor, &divisor_exponent); |
| 535 | *kappa = divisor_exponent + 1; |
| 536 | *length = 0; |
| 537 | |
| 538 | // Loop invariant: buffer = w / 10^kappa (integer division) |
| 539 | // The invariant holds for the first iteration: kappa has been initialized |
| 540 | // with the divisor exponent + 1. And the divisor is the biggest power of ten |
| 541 | // that is smaller than 'integrals'. |
| 542 | while (*kappa > 0) { |
| 543 | int digit = integrals / divisor; |
| 544 | buffer[*length] = '0' + digit; |
| 545 | (*length)++; |
| 546 | requested_digits--; |
| 547 | integrals %= divisor; |
| 548 | (*kappa)--; |
| 549 | // Note that kappa now equals the exponent of the divisor and that the |
| 550 | // invariant thus holds again. |
| 551 | if (requested_digits == 0) break; |
| 552 | divisor /= 10; |
| 553 | } |
| 554 | |
| 555 | if (requested_digits == 0) { |
| 556 | uint64_t rest = |
| 557 | (static_cast<uint64_t>(integrals) << -one.e()) + fractionals; |
| 558 | return RoundWeedCounted(buffer, *length, rest, |
| 559 | static_cast<uint64_t>(divisor) << -one.e(), w_error, |
| 560 | kappa); |
| 561 | } |
| 562 | |
| 563 | // The integrals have been generated. We are at the point of the decimal |
| 564 | // separator. In the following loop we simply multiply the remaining digits by |
| 565 | // 10 and divide by one. We just need to pay attention to multiply associated |
| 566 | // data (the 'unit'), too. |
| 567 | // Note that the multiplication by 10 does not overflow, because w.e >= -60 |
| 568 | // and thus one.e >= -60. |
| 569 | ASSERT(one.e() >= -60); |
| 570 | ASSERT(fractionals < one.f()); |
| 571 | ASSERT(V8_2PART_UINT64_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f()); |
| 572 | while (requested_digits > 0 && fractionals > w_error) { |
| 573 | fractionals *= 10; |
| 574 | w_error *= 10; |
| 575 | // Integer division by one. |
| 576 | int digit = static_cast<int>(fractionals >> -one.e()); |
| 577 | buffer[*length] = '0' + digit; |
| 578 | (*length)++; |
| 579 | requested_digits--; |
| 580 | fractionals &= one.f() - 1; // Modulo by one. |
| 581 | (*kappa)--; |
| 582 | } |
| 583 | if (requested_digits != 0) return false; |
| 584 | return RoundWeedCounted(buffer, *length, fractionals, one.f(), w_error, |
| 585 | kappa); |
| 586 | } |
| 587 | |
| 588 | |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 589 | // Provides a decimal representation of v. |
| 590 | // Returns true if it succeeds, otherwise the result cannot be trusted. |
| 591 | // There will be *length digits inside the buffer (not null-terminated). |
| 592 | // If the function returns true then |
| 593 | // v == (double) (buffer * 10^decimal_exponent). |
| 594 | // The digits in the buffer are the shortest representation possible: no |
| 595 | // 0.09999999999999999 instead of 0.1. The shorter representation will even be |
| 596 | // chosen even if the longer one would be closer to v. |
| 597 | // The last digit will be closest to the actual v. That is, even if several |
| 598 | // digits might correctly yield 'v' when read again, the closest will be |
| 599 | // computed. |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 600 | static bool Grisu3(double v, |
| 601 | Vector<char> buffer, |
| 602 | int* length, |
| 603 | int* decimal_exponent) { |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 604 | DiyFp w = Double(v).AsNormalizedDiyFp(); |
| 605 | // boundary_minus and boundary_plus are the boundaries between v and its |
| 606 | // closest floating-point neighbors. Any number strictly between |
| 607 | // boundary_minus and boundary_plus will round to v when convert to a double. |
| 608 | // Grisu3 will never output representations that lie exactly on a boundary. |
| 609 | DiyFp boundary_minus, boundary_plus; |
| 610 | Double(v).NormalizedBoundaries(&boundary_minus, &boundary_plus); |
| 611 | ASSERT(boundary_plus.e() == w.e()); |
| 612 | DiyFp ten_mk; // Cached power of ten: 10^-k |
| 613 | int mk; // -k |
ager@chromium.org | b61a0d1 | 2010-10-13 08:35:23 +0000 | [diff] [blame] | 614 | int ten_mk_minimal_binary_exponent = |
| 615 | kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize); |
| 616 | int ten_mk_maximal_binary_exponent = |
| 617 | kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize); |
lrn@chromium.org | 303ada7 | 2010-10-27 09:33:13 +0000 | [diff] [blame] | 618 | PowersOfTenCache::GetCachedPowerForBinaryExponentRange( |
| 619 | ten_mk_minimal_binary_exponent, |
| 620 | ten_mk_maximal_binary_exponent, |
| 621 | &ten_mk, &mk); |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 622 | ASSERT((kMinimalTargetExponent <= w.e() + ten_mk.e() + |
| 623 | DiyFp::kSignificandSize) && |
| 624 | (kMaximalTargetExponent >= w.e() + ten_mk.e() + |
| 625 | DiyFp::kSignificandSize)); |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 626 | // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a |
| 627 | // 64 bit significand and ten_mk is thus only precise up to 64 bits. |
| 628 | |
| 629 | // The DiyFp::Times procedure rounds its result, and ten_mk is approximated |
| 630 | // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now |
| 631 | // off by a small amount. |
| 632 | // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w. |
| 633 | // In other words: let f = scaled_w.f() and e = scaled_w.e(), then |
| 634 | // (f-1) * 2^e < w*10^k < (f+1) * 2^e |
| 635 | DiyFp scaled_w = DiyFp::Times(w, ten_mk); |
| 636 | ASSERT(scaled_w.e() == |
| 637 | boundary_plus.e() + ten_mk.e() + DiyFp::kSignificandSize); |
| 638 | // In theory it would be possible to avoid some recomputations by computing |
| 639 | // the difference between w and boundary_minus/plus (a power of 2) and to |
| 640 | // compute scaled_boundary_minus/plus by subtracting/adding from |
| 641 | // scaled_w. However the code becomes much less readable and the speed |
| 642 | // enhancements are not terriffic. |
| 643 | DiyFp scaled_boundary_minus = DiyFp::Times(boundary_minus, ten_mk); |
| 644 | DiyFp scaled_boundary_plus = DiyFp::Times(boundary_plus, ten_mk); |
| 645 | |
| 646 | // DigitGen will generate the digits of scaled_w. Therefore we have |
| 647 | // v == (double) (scaled_w * 10^-mk). |
| 648 | // Set decimal_exponent == -mk and pass it to DigitGen. If scaled_w is not an |
| 649 | // integer than it will be updated. For instance if scaled_w == 1.23 then |
| 650 | // the buffer will be filled with "123" und the decimal_exponent will be |
| 651 | // decreased by 2. |
| 652 | int kappa; |
| 653 | bool result = DigitGen(scaled_boundary_minus, scaled_w, scaled_boundary_plus, |
| 654 | buffer, length, &kappa); |
| 655 | *decimal_exponent = -mk + kappa; |
| 656 | return result; |
| 657 | } |
| 658 | |
| 659 | |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 660 | // The "counted" version of grisu3 (see above) only generates requested_digits |
| 661 | // number of digits. This version does not generate the shortest representation, |
| 662 | // and with enough requested digits 0.1 will at some point print as 0.9999999... |
| 663 | // Grisu3 is too imprecise for real halfway cases (1.5 will not work) and |
| 664 | // therefore the rounding strategy for halfway cases is irrelevant. |
| 665 | static bool Grisu3Counted(double v, |
| 666 | int requested_digits, |
| 667 | Vector<char> buffer, |
| 668 | int* length, |
| 669 | int* decimal_exponent) { |
| 670 | DiyFp w = Double(v).AsNormalizedDiyFp(); |
| 671 | DiyFp ten_mk; // Cached power of ten: 10^-k |
| 672 | int mk; // -k |
ager@chromium.org | b61a0d1 | 2010-10-13 08:35:23 +0000 | [diff] [blame] | 673 | int ten_mk_minimal_binary_exponent = |
| 674 | kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize); |
| 675 | int ten_mk_maximal_binary_exponent = |
| 676 | kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize); |
lrn@chromium.org | 303ada7 | 2010-10-27 09:33:13 +0000 | [diff] [blame] | 677 | PowersOfTenCache::GetCachedPowerForBinaryExponentRange( |
| 678 | ten_mk_minimal_binary_exponent, |
| 679 | ten_mk_maximal_binary_exponent, |
| 680 | &ten_mk, &mk); |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 681 | ASSERT((kMinimalTargetExponent <= w.e() + ten_mk.e() + |
| 682 | DiyFp::kSignificandSize) && |
| 683 | (kMaximalTargetExponent >= w.e() + ten_mk.e() + |
| 684 | DiyFp::kSignificandSize)); |
| 685 | // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a |
| 686 | // 64 bit significand and ten_mk is thus only precise up to 64 bits. |
| 687 | |
| 688 | // The DiyFp::Times procedure rounds its result, and ten_mk is approximated |
| 689 | // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now |
| 690 | // off by a small amount. |
| 691 | // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w. |
| 692 | // In other words: let f = scaled_w.f() and e = scaled_w.e(), then |
| 693 | // (f-1) * 2^e < w*10^k < (f+1) * 2^e |
| 694 | DiyFp scaled_w = DiyFp::Times(w, ten_mk); |
| 695 | |
| 696 | // We now have (double) (scaled_w * 10^-mk). |
| 697 | // DigitGen will generate the first requested_digits digits of scaled_w and |
| 698 | // return together with a kappa such that scaled_w ~= buffer * 10^kappa. (It |
| 699 | // will not always be exactly the same since DigitGenCounted only produces a |
| 700 | // limited number of digits.) |
| 701 | int kappa; |
| 702 | bool result = DigitGenCounted(scaled_w, requested_digits, |
| 703 | buffer, length, &kappa); |
| 704 | *decimal_exponent = -mk + kappa; |
| 705 | return result; |
| 706 | } |
| 707 | |
| 708 | |
kmillikin@chromium.org | 4111b80 | 2010-05-03 10:34:42 +0000 | [diff] [blame] | 709 | bool FastDtoa(double v, |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 710 | FastDtoaMode mode, |
| 711 | int requested_digits, |
kmillikin@chromium.org | 4111b80 | 2010-05-03 10:34:42 +0000 | [diff] [blame] | 712 | Vector<char> buffer, |
kmillikin@chromium.org | 4111b80 | 2010-05-03 10:34:42 +0000 | [diff] [blame] | 713 | int* length, |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 714 | int* decimal_point) { |
sgjesse@chromium.org | 720dc0b | 2010-05-10 09:25:39 +0000 | [diff] [blame] | 715 | ASSERT(v > 0); |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 716 | ASSERT(!Double(v).IsSpecial()); |
| 717 | |
fschneider@chromium.org | c20610a | 2010-09-22 09:44:58 +0000 | [diff] [blame] | 718 | bool result = false; |
| 719 | int decimal_exponent = 0; |
| 720 | switch (mode) { |
| 721 | case FAST_DTOA_SHORTEST: |
| 722 | result = Grisu3(v, buffer, length, &decimal_exponent); |
| 723 | break; |
| 724 | case FAST_DTOA_PRECISION: |
| 725 | result = Grisu3Counted(v, requested_digits, |
| 726 | buffer, length, &decimal_exponent); |
| 727 | break; |
| 728 | default: |
| 729 | UNREACHABLE(); |
| 730 | } |
| 731 | if (result) { |
| 732 | *decimal_point = *length + decimal_exponent; |
| 733 | buffer[*length] = '\0'; |
| 734 | } |
whesse@chromium.org | cec079d | 2010-03-22 14:44:04 +0000 | [diff] [blame] | 735 | return result; |
| 736 | } |
| 737 | |
| 738 | } } // namespace v8::internal |