1/* 2 * Copyright (c) 2015, Oracle and/or its affiliates. All rights reserved. 3 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. 4 * 5 * This code is free software; you can redistribute it and/or modify it 6 * under the terms of the GNU General Public License version 2 only, as 7 * published by the Free Software Foundation. Oracle designates this 8 * particular file as subject to the "Classpath" exception as provided 9 * by Oracle in the LICENSE file that accompanied this code. 10 * 11 * This code is distributed in the hope that it will be useful, but WITHOUT 12 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or 13 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License 14 * version 2 for more details (a copy is included in the LICENSE file that 15 * accompanied this code). 16 * 17 * You should have received a copy of the GNU General Public License version 18 * 2 along with this work; if not, write to the Free Software Foundation, 19 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. 20 * 21 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA 22 * or visit www.oracle.com if you need additional information or have any 23 * questions. 24 */ 25 26// This file is available under and governed by the GNU General Public 27// License version 2 only, as published by the Free Software Foundation. 28// However, the following notice accompanied the original version of this 29// file: 30// 31// Copyright 2010 the V8 project authors. All rights reserved. 32// Redistribution and use in source and binary forms, with or without 33// modification, are permitted provided that the following conditions are 34// met: 35// 36// * Redistributions of source code must retain the above copyright 37// notice, this list of conditions and the following disclaimer. 38// * Redistributions in binary form must reproduce the above 39// copyright notice, this list of conditions and the following 40// disclaimer in the documentation and/or other materials provided 41// with the distribution. 42// * Neither the name of Google Inc. nor the names of its 43// contributors may be used to endorse or promote products derived 44// from this software without specific prior written permission. 45// 46// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS 47// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT 48// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR 49// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT 50// OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, 51// SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT 52// LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, 53// DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY 54// THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT 55// (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE 56// OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. 57 58package jdk.nashorn.internal.runtime.doubleconv; 59 60// Fast Dtoa implementation supporting shortest and precision modes. Does not 61// work for all numbers so BugnumDtoa is used as fallback. 62class FastDtoa { 63 64 // FastDtoa will produce at most kFastDtoaMaximalLength digits. This does not 65 // include the terminating '\0' character. 66 static final int kFastDtoaMaximalLength = 17; 67 68 // The minimal and maximal target exponent define the range of w's binary 69 // exponent, where 'w' is the result of multiplying the input by a cached power 70 // of ten. 71 // 72 // A different range might be chosen on a different platform, to optimize digit 73 // generation, but a smaller range requires more powers of ten to be cached. 74 static final int kMinimalTargetExponent = -60; 75 static final int kMaximalTargetExponent = -32; 76 77 78 // Adjusts the last digit of the generated number, and screens out generated 79 // solutions that may be inaccurate. A solution may be inaccurate if it is 80 // outside the safe interval, or if we cannot prove that it is closer to the 81 // input than a neighboring representation of the same length. 82 // 83 // Input: * buffer containing the digits of too_high / 10^kappa 84 // * distance_too_high_w == (too_high - w).f() * unit 85 // * unsafe_interval == (too_high - too_low).f() * unit 86 // * rest = (too_high - buffer * 10^kappa).f() * unit 87 // * ten_kappa = 10^kappa * unit 88 // * unit = the common multiplier 89 // Output: returns true if the buffer is guaranteed to contain the closest 90 // representable number to the input. 91 // Modifies the generated digits in the buffer to approach (round towards) w. 92 static boolean roundWeed(final DtoaBuffer buffer, 93 final long distance_too_high_w, 94 final long unsafe_interval, 95 long rest, 96 final long ten_kappa, 97 final long unit) { 98 final long small_distance = distance_too_high_w - unit; 99 final long big_distance = distance_too_high_w + unit; 100 // Let w_low = too_high - big_distance, and 101 // w_high = too_high - small_distance. 102 // Note: w_low < w < w_high 103 // 104 // The real w (* unit) must lie somewhere inside the interval 105 // ]w_low; w_high[ (often written as "(w_low; w_high)") 106 107 // Basically the buffer currently contains a number in the unsafe interval 108 // ]too_low; too_high[ with too_low < w < too_high 109 // 110 // too_high - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 111 // ^v 1 unit ^ ^ ^ ^ 112 // boundary_high --------------------- . . . . 113 // ^v 1 unit . . . . 114 // - - - - - - - - - - - - - - - - - - - + - - + - - - - - - . . 115 // . . ^ . . 116 // . big_distance . . . 117 // . . . . rest 118 // small_distance . . . . 119 // v . . . . 120 // w_high - - - - - - - - - - - - - - - - - - . . . . 121 // ^v 1 unit . . . . 122 // w ---------------------------------------- . . . . 123 // ^v 1 unit v . . . 124 // w_low - - - - - - - - - - - - - - - - - - - - - . . . 125 // . . v 126 // buffer --------------------------------------------------+-------+-------- 127 // . . 128 // safe_interval . 129 // v . 130 // - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - . 131 // ^v 1 unit . 132 // boundary_low ------------------------- unsafe_interval 133 // ^v 1 unit v 134 // too_low - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 135 // 136 // 137 // Note that the value of buffer could lie anywhere inside the range too_low 138 // to too_high. 139 // 140 // boundary_low, boundary_high and w are approximations of the real boundaries 141 // and v (the input number). They are guaranteed to be precise up to one unit. 142 // In fact the error is guaranteed to be strictly less than one unit. 143 // 144 // Anything that lies outside the unsafe interval is guaranteed not to round 145 // to v when read again. 146 // Anything that lies inside the safe interval is guaranteed to round to v 147 // when read again. 148 // If the number inside the buffer lies inside the unsafe interval but not 149 // inside the safe interval then we simply do not know and bail out (returning 150 // false). 151 // 152 // Similarly we have to take into account the imprecision of 'w' when finding 153 // the closest representation of 'w'. If we have two potential 154 // representations, and one is closer to both w_low and w_high, then we know 155 // it is closer to the actual value v. 156 // 157 // By generating the digits of too_high we got the largest (closest to 158 // too_high) buffer that is still in the unsafe interval. In the case where 159 // w_high < buffer < too_high we try to decrement the buffer. 160 // This way the buffer approaches (rounds towards) w. 161 // There are 3 conditions that stop the decrementation process: 162 // 1) the buffer is already below w_high 163 // 2) decrementing the buffer would make it leave the unsafe interval 164 // 3) decrementing the buffer would yield a number below w_high and farther 165 // away than the current number. In other words: 166 // (buffer{-1} < w_high) && w_high - buffer{-1} > buffer - w_high 167 // Instead of using the buffer directly we use its distance to too_high. 168 // Conceptually rest ~= too_high - buffer 169 // We need to do the following tests in this order to avoid over- and 170 // underflows. 171 assert (Long.compareUnsigned(rest, unsafe_interval) <= 0); 172 while (Long.compareUnsigned(rest, small_distance) < 0 && // Negated condition 1 173 Long.compareUnsigned(unsafe_interval - rest, ten_kappa) >= 0 && // Negated condition 2 174 (Long.compareUnsigned(rest + ten_kappa, small_distance) < 0 || // buffer{-1} > w_high 175 Long.compareUnsigned(small_distance - rest, rest + ten_kappa - small_distance) >= 0)) { 176 buffer.chars[buffer.length - 1]--; 177 rest += ten_kappa; 178 } 179 180 // We have approached w+ as much as possible. We now test if approaching w- 181 // would require changing the buffer. If yes, then we have two possible 182 // representations close to w, but we cannot decide which one is closer. 183 if (Long.compareUnsigned(rest, big_distance) < 0 && 184 Long.compareUnsigned(unsafe_interval - rest, ten_kappa) >= 0 && 185 (Long.compareUnsigned(rest + ten_kappa, big_distance) < 0 || 186 Long.compareUnsigned(big_distance - rest, rest + ten_kappa - big_distance) > 0)) { 187 return false; 188 } 189 190 // Weeding test. 191 // The safe interval is [too_low + 2 ulp; too_high - 2 ulp] 192 // Since too_low = too_high - unsafe_interval this is equivalent to 193 // [too_high - unsafe_interval + 4 ulp; too_high - 2 ulp] 194 // Conceptually we have: rest ~= too_high - buffer 195 return Long.compareUnsigned(2 * unit, rest) <= 0 && Long.compareUnsigned(rest, unsafe_interval - 4 * unit) <= 0; 196 } 197 198 // Rounds the buffer upwards if the result is closer to v by possibly adding 199 // 1 to the buffer. If the precision of the calculation is not sufficient to 200 // round correctly, return false. 201 // The rounding might shift the whole buffer in which case the kappa is 202 // adjusted. For example "99", kappa = 3 might become "10", kappa = 4. 203 // 204 // If 2*rest > ten_kappa then the buffer needs to be round up. 205 // rest can have an error of +/- 1 unit. This function accounts for the 206 // imprecision and returns false, if the rounding direction cannot be 207 // unambiguously determined. 208 // 209 // Precondition: rest < ten_kappa. 210 // Changed return type to int to let caller know they should increase kappa (return value 2) 211 static int roundWeedCounted(final char[] buffer, 212 final int length, 213 final long rest, 214 final long ten_kappa, 215 final long unit) { 216 assert(Long.compareUnsigned(rest, ten_kappa) < 0); 217 // The following tests are done in a specific order to avoid overflows. They 218 // will work correctly with any uint64 values of rest < ten_kappa and unit. 219 // 220 // If the unit is too big, then we don't know which way to round. For example 221 // a unit of 50 means that the real number lies within rest +/- 50. If 222 // 10^kappa == 40 then there is no way to tell which way to round. 223 if (Long.compareUnsigned(unit, ten_kappa) >= 0) return 0; 224 // Even if unit is just half the size of 10^kappa we are already completely 225 // lost. (And after the previous test we know that the expression will not 226 // over/underflow.) 227 if (Long.compareUnsigned(ten_kappa - unit, unit) <= 0) return 0; 228 // If 2 * (rest + unit) <= 10^kappa we can safely round down. 229 if (Long.compareUnsigned(ten_kappa - rest, rest) > 0 && Long.compareUnsigned(ten_kappa - 2 * rest, 2 * unit) >= 0) { 230 return 1; 231 } 232 // If 2 * (rest - unit) >= 10^kappa, then we can safely round up. 233 if (Long.compareUnsigned(rest, unit) > 0 && Long.compareUnsigned(ten_kappa - (rest - unit), (rest - unit)) <= 0) { 234 // Increment the last digit recursively until we find a non '9' digit. 235 buffer[length - 1]++; 236 for (int i = length - 1; i > 0; --i) { 237 if (buffer[i] != '0' + 10) break; 238 buffer[i] = '0'; 239 buffer[i - 1]++; 240 } 241 // If the first digit is now '0'+ 10 we had a buffer with all '9's. With the 242 // exception of the first digit all digits are now '0'. Simply switch the 243 // first digit to '1' and adjust the kappa. Example: "99" becomes "10" and 244 // the power (the kappa) is increased. 245 if (buffer[0] == '0' + 10) { 246 buffer[0] = '1'; 247 // Return value of 2 tells caller to increase (*kappa) += 1 248 return 2; 249 } 250 return 1; 251 } 252 return 0; 253 } 254 255 // Returns the biggest power of ten that is less than or equal to the given 256 // number. We furthermore receive the maximum number of bits 'number' has. 257 // 258 // Returns power == 10^(exponent_plus_one-1) such that 259 // power <= number < power * 10. 260 // If number_bits == 0 then 0^(0-1) is returned. 261 // The number of bits must be <= 32. 262 // Precondition: number < (1 << (number_bits + 1)). 263 264 // Inspired by the method for finding an integer log base 10 from here: 265 // http://graphics.stanford.edu/~seander/bithacks.html#IntegerLog10 266 static final int kSmallPowersOfTen[] = 267 {0, 1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 268 1000000000}; 269 270 // Returns the biggest power of ten that is less than or equal than the given 271 // number. We furthermore receive the maximum number of bits 'number' has. 272 // If number_bits == 0 then 0^-1 is returned 273 // The number of bits must be <= 32. 274 // Precondition: (1 << number_bits) <= number < (1 << (number_bits + 1)). 275 static long biggestPowerTen(final int number, 276 final int number_bits) { 277 final int power, exponent_plus_one; 278 assert ((number & 0xFFFFFFFFL) < (1l << (number_bits + 1))); 279 // 1233/4096 is approximately 1/lg(10). 280 int exponent_plus_one_guess = ((number_bits + 1) * 1233 >>> 12); 281 // We increment to skip over the first entry in the kPowersOf10 table. 282 // Note: kPowersOf10[i] == 10^(i-1). 283 exponent_plus_one_guess++; 284 // We don't have any guarantees that 2^number_bits <= number. 285 if (number < kSmallPowersOfTen[exponent_plus_one_guess]) { 286 exponent_plus_one_guess--; 287 } 288 power = kSmallPowersOfTen[exponent_plus_one_guess]; 289 exponent_plus_one = exponent_plus_one_guess; 290 291 return ((long) power << 32) | (long) exponent_plus_one; 292 } 293 294 // Generates the digits of input number w. 295 // w is a floating-point number (DiyFp), consisting of a significand and an 296 // exponent. Its exponent is bounded by kMinimalTargetExponent and 297 // kMaximalTargetExponent. 298 // Hence -60 <= w.e() <= -32. 299 // 300 // Returns false if it fails, in which case the generated digits in the buffer 301 // should not be used. 302 // Preconditions: 303 // * low, w and high are correct up to 1 ulp (unit in the last place). That 304 // is, their error must be less than a unit of their last digits. 305 // * low.e() == w.e() == high.e() 306 // * low < w < high, and taking into account their error: low~ <= high~ 307 // * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent 308 // Postconditions: returns false if procedure fails. 309 // otherwise: 310 // * buffer is not null-terminated, but len contains the number of digits. 311 // * buffer contains the shortest possible decimal digit-sequence 312 // such that LOW < buffer * 10^kappa < HIGH, where LOW and HIGH are the 313 // correct values of low and high (without their error). 314 // * if more than one decimal representation gives the minimal number of 315 // decimal digits then the one closest to W (where W is the correct value 316 // of w) is chosen. 317 // Remark: this procedure takes into account the imprecision of its input 318 // numbers. If the precision is not enough to guarantee all the postconditions 319 // then false is returned. This usually happens rarely (~0.5%). 320 // 321 // Say, for the sake of example, that 322 // w.e() == -48, and w.f() == 0x1234567890abcdef 323 // w's value can be computed by w.f() * 2^w.e() 324 // We can obtain w's integral digits by simply shifting w.f() by -w.e(). 325 // -> w's integral part is 0x1234 326 // w's fractional part is therefore 0x567890abcdef. 327 // Printing w's integral part is easy (simply print 0x1234 in decimal). 328 // In order to print its fraction we repeatedly multiply the fraction by 10 and 329 // get each digit. Example the first digit after the point would be computed by 330 // (0x567890abcdef * 10) >> 48. -> 3 331 // The whole thing becomes slightly more complicated because we want to stop 332 // once we have enough digits. That is, once the digits inside the buffer 333 // represent 'w' we can stop. Everything inside the interval low - high 334 // represents w. However we have to pay attention to low, high and w's 335 // imprecision. 336 static boolean digitGen(final DiyFp low, 337 final DiyFp w, 338 final DiyFp high, 339 final DtoaBuffer buffer, 340 final int mk) { 341 assert(low.e() == w.e() && w.e() == high.e()); 342 assert Long.compareUnsigned(low.f() + 1, high.f() - 1) <= 0; 343 assert(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent); 344 // low, w and high are imprecise, but by less than one ulp (unit in the last 345 // place). 346 // If we remove (resp. add) 1 ulp from low (resp. high) we are certain that 347 // the new numbers are outside of the interval we want the final 348 // representation to lie in. 349 // Inversely adding (resp. removing) 1 ulp from low (resp. high) would yield 350 // numbers that are certain to lie in the interval. We will use this fact 351 // later on. 352 // We will now start by generating the digits within the uncertain 353 // interval. Later we will weed out representations that lie outside the safe 354 // interval and thus _might_ lie outside the correct interval. 355 long unit = 1; 356 final DiyFp too_low = new DiyFp(low.f() - unit, low.e()); 357 final DiyFp too_high = new DiyFp(high.f() + unit, high.e()); 358 // too_low and too_high are guaranteed to lie outside the interval we want the 359 // generated number in. 360 final DiyFp unsafe_interval = DiyFp.minus(too_high, too_low); 361 // We now cut the input number into two parts: the integral digits and the 362 // fractionals. We will not write any decimal separator though, but adapt 363 // kappa instead. 364 // Reminder: we are currently computing the digits (stored inside the buffer) 365 // such that: too_low < buffer * 10^kappa < too_high 366 // We use too_high for the digit_generation and stop as soon as possible. 367 // If we stop early we effectively round down. 368 final DiyFp one = new DiyFp(1l << -w.e(), w.e()); 369 // Division by one is a shift. 370 int integrals = (int)(too_high.f() >>> -one.e()); 371 // Modulo by one is an and. 372 long fractionals = too_high.f() & (one.f() - 1); 373 int divisor; 374 final int divisor_exponent_plus_one; 375 final long result = biggestPowerTen(integrals, DiyFp.kSignificandSize - (-one.e())); 376 divisor = (int) (result >>> 32); 377 divisor_exponent_plus_one = (int) result; 378 int kappa = divisor_exponent_plus_one; 379 // Loop invariant: buffer = too_high / 10^kappa (integer division) 380 // The invariant holds for the first iteration: kappa has been initialized 381 // with the divisor exponent + 1. And the divisor is the biggest power of ten 382 // that is smaller than integrals. 383 while (kappa > 0) { 384 final int digit = integrals / divisor; 385 assert (digit <= 9); 386 buffer.append((char) ('0' + digit)); 387 integrals %= divisor; 388 kappa--; 389 // Note that kappa now equals the exponent of the divisor and that the 390 // invariant thus holds again. 391 final long rest = 392 ((long) integrals << -one.e()) + fractionals; 393 // Invariant: too_high = buffer * 10^kappa + DiyFp(rest, one.e()) 394 // Reminder: unsafe_interval.e() == one.e() 395 if (Long.compareUnsigned(rest, unsafe_interval.f()) < 0) { 396 // Rounding down (by not emitting the remaining digits) yields a number 397 // that lies within the unsafe interval. 398 buffer.decimalPoint = buffer.length - mk + kappa; 399 return roundWeed(buffer, DiyFp.minus(too_high, w).f(), 400 unsafe_interval.f(), rest, 401 (long) divisor << -one.e(), unit); 402 } 403 divisor /= 10; 404 } 405 406 // The integrals have been generated. We are at the point of the decimal 407 // separator. In the following loop we simply multiply the remaining digits by 408 // 10 and divide by one. We just need to pay attention to multiply associated 409 // data (like the interval or 'unit'), too. 410 // Note that the multiplication by 10 does not overflow, because w.e >= -60 411 // and thus one.e >= -60. 412 assert (one.e() >= -60); 413 assert (fractionals < one.f()); 414 assert (Long.compareUnsigned(Long.divideUnsigned(0xFFFFFFFFFFFFFFFFL, 10), one.f()) >= 0); 415 for (;;) { 416 fractionals *= 10; 417 unit *= 10; 418 unsafe_interval.setF(unsafe_interval.f() * 10); 419 // Integer division by one. 420 final int digit = (int) (fractionals >>> -one.e()); 421 assert (digit <= 9); 422 buffer.append((char) ('0' + digit)); 423 fractionals &= one.f() - 1; // Modulo by one. 424 kappa--; 425 if (Long.compareUnsigned(fractionals, unsafe_interval.f()) < 0) { 426 buffer.decimalPoint = buffer.length - mk + kappa; 427 return roundWeed(buffer, DiyFp.minus(too_high, w).f() * unit, 428 unsafe_interval.f(), fractionals, one.f(), unit); 429 } 430 } 431 } 432 433 // Generates (at most) requested_digits digits of input number w. 434 // w is a floating-point number (DiyFp), consisting of a significand and an 435 // exponent. Its exponent is bounded by kMinimalTargetExponent and 436 // kMaximalTargetExponent. 437 // Hence -60 <= w.e() <= -32. 438 // 439 // Returns false if it fails, in which case the generated digits in the buffer 440 // should not be used. 441 // Preconditions: 442 // * w is correct up to 1 ulp (unit in the last place). That 443 // is, its error must be strictly less than a unit of its last digit. 444 // * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent 445 // 446 // Postconditions: returns false if procedure fails. 447 // otherwise: 448 // * buffer is not null-terminated, but length contains the number of 449 // digits. 450 // * the representation in buffer is the most precise representation of 451 // requested_digits digits. 452 // * buffer contains at most requested_digits digits of w. If there are less 453 // than requested_digits digits then some trailing '0's have been removed. 454 // * kappa is such that 455 // w = buffer * 10^kappa + eps with |eps| < 10^kappa / 2. 456 // 457 // Remark: This procedure takes into account the imprecision of its input 458 // numbers. If the precision is not enough to guarantee all the postconditions 459 // then false is returned. This usually happens rarely, but the failure-rate 460 // increases with higher requested_digits. 461 static boolean digitGenCounted(final DiyFp w, 462 int requested_digits, 463 final DtoaBuffer buffer, 464 final int mk) { 465 assert (kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent); 466 assert (kMinimalTargetExponent >= -60); 467 assert (kMaximalTargetExponent <= -32); 468 // w is assumed to have an error less than 1 unit. Whenever w is scaled we 469 // also scale its error. 470 long w_error = 1; 471 // We cut the input number into two parts: the integral digits and the 472 // fractional digits. We don't emit any decimal separator, but adapt kappa 473 // instead. Example: instead of writing "1.2" we put "12" into the buffer and 474 // increase kappa by 1. 475 final DiyFp one = new DiyFp(1l << -w.e(), w.e()); 476 // Division by one is a shift. 477 int integrals = (int) (w.f() >>> -one.e()); 478 // Modulo by one is an and. 479 long fractionals = w.f() & (one.f() - 1); 480 int divisor; 481 final int divisor_exponent_plus_one; 482 final long biggestPower = biggestPowerTen(integrals, DiyFp.kSignificandSize - (-one.e())); 483 divisor = (int) (biggestPower >>> 32); 484 divisor_exponent_plus_one = (int) biggestPower; 485 int kappa = divisor_exponent_plus_one; 486 487 // Loop invariant: buffer = w / 10^kappa (integer division) 488 // The invariant holds for the first iteration: kappa has been initialized 489 // with the divisor exponent + 1. And the divisor is the biggest power of ten 490 // that is smaller than 'integrals'. 491 while (kappa > 0) { 492 final int digit = integrals / divisor; 493 assert (digit <= 9); 494 buffer.append((char) ('0' + digit)); 495 requested_digits--; 496 integrals %= divisor; 497 kappa--; 498 // Note that kappa now equals the exponent of the divisor and that the 499 // invariant thus holds again. 500 if (requested_digits == 0) break; 501 divisor /= 10; 502 } 503 504 if (requested_digits == 0) { 505 final long rest = 506 ((long) (integrals) << -one.e()) + fractionals; 507 final int result = roundWeedCounted(buffer.chars, buffer.length, rest, 508 (long) divisor << -one.e(), w_error); 509 buffer.decimalPoint = buffer.length - mk + kappa + (result == 2 ? 1 : 0); 510 return result > 0; 511 } 512 513 // The integrals have been generated. We are at the decimalPoint of the decimal 514 // separator. In the following loop we simply multiply the remaining digits by 515 // 10 and divide by one. We just need to pay attention to multiply associated 516 // data (the 'unit'), too. 517 // Note that the multiplication by 10 does not overflow, because w.e >= -60 518 // and thus one.e >= -60. 519 assert (one.e() >= -60); 520 assert (fractionals < one.f()); 521 assert (Long.compareUnsigned(Long.divideUnsigned(0xFFFFFFFFFFFFFFFFL, 10), one.f()) >= 0); 522 while (requested_digits > 0 && fractionals > w_error) { 523 fractionals *= 10; 524 w_error *= 10; 525 // Integer division by one. 526 final int digit = (int) (fractionals >>> -one.e()); 527 assert (digit <= 9); 528 buffer.append((char) ('0' + digit)); 529 requested_digits--; 530 fractionals &= one.f() - 1; // Modulo by one. 531 kappa--; 532 } 533 if (requested_digits != 0) return false; 534 final int result = roundWeedCounted(buffer.chars, buffer.length, fractionals, one.f(), w_error); 535 buffer.decimalPoint = buffer.length - mk + kappa + (result == 2 ? 1 : 0); 536 return result > 0; 537 } 538 539 540 // Provides a decimal representation of v. 541 // Returns true if it succeeds, otherwise the result cannot be trusted. 542 // There will be *length digits inside the buffer (not null-terminated). 543 // If the function returns true then 544 // v == (double) (buffer * 10^decimal_exponent). 545 // The digits in the buffer are the shortest representation possible: no 546 // 0.09999999999999999 instead of 0.1. The shorter representation will even be 547 // chosen even if the longer one would be closer to v. 548 // The last digit will be closest to the actual v. That is, even if several 549 // digits might correctly yield 'v' when read again, the closest will be 550 // computed. 551 static boolean grisu3(final double v, final DtoaBuffer buffer) { 552 final long d64 = IeeeDouble.doubleToLong(v); 553 final DiyFp w = IeeeDouble.asNormalizedDiyFp(d64); 554 // boundary_minus and boundary_plus are the boundaries between v and its 555 // closest floating-point neighbors. Any number strictly between 556 // boundary_minus and boundary_plus will round to v when convert to a double. 557 // Grisu3 will never output representations that lie exactly on a boundary. 558 final DiyFp boundary_minus = new DiyFp(), boundary_plus = new DiyFp(); 559 IeeeDouble.normalizedBoundaries(d64, boundary_minus, boundary_plus); 560 assert(boundary_plus.e() == w.e()); 561 final DiyFp ten_mk = new DiyFp(); // Cached power of ten: 10^-k 562 final int mk; // -k 563 final int ten_mk_minimal_binary_exponent = 564 kMinimalTargetExponent - (w.e() + DiyFp.kSignificandSize); 565 final int ten_mk_maximal_binary_exponent = 566 kMaximalTargetExponent - (w.e() + DiyFp.kSignificandSize); 567 mk = CachedPowers.getCachedPowerForBinaryExponentRange( 568 ten_mk_minimal_binary_exponent, 569 ten_mk_maximal_binary_exponent, 570 ten_mk); 571 assert(kMinimalTargetExponent <= w.e() + ten_mk.e() + 572 DiyFp.kSignificandSize && 573 kMaximalTargetExponent >= w.e() + ten_mk.e() + 574 DiyFp.kSignificandSize); 575 // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a 576 // 64 bit significand and ten_mk is thus only precise up to 64 bits. 577 578 // The DiyFp::Times procedure rounds its result, and ten_mk is approximated 579 // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now 580 // off by a small amount. 581 // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w. 582 // In other words: let f = scaled_w.f() and e = scaled_w.e(), then 583 // (f-1) * 2^e < w*10^k < (f+1) * 2^e 584 final DiyFp scaled_w = DiyFp.times(w, ten_mk); 585 assert(scaled_w.e() == 586 boundary_plus.e() + ten_mk.e() + DiyFp.kSignificandSize); 587 // In theory it would be possible to avoid some recomputations by computing 588 // the difference between w and boundary_minus/plus (a power of 2) and to 589 // compute scaled_boundary_minus/plus by subtracting/adding from 590 // scaled_w. However the code becomes much less readable and the speed 591 // enhancements are not terriffic. 592 final DiyFp scaled_boundary_minus = DiyFp.times(boundary_minus, ten_mk); 593 final DiyFp scaled_boundary_plus = DiyFp.times(boundary_plus, ten_mk); 594 595 // DigitGen will generate the digits of scaled_w. Therefore we have 596 // v == (double) (scaled_w * 10^-mk). 597 // Set decimal_exponent == -mk and pass it to DigitGen. If scaled_w is not an 598 // integer than it will be updated. For instance if scaled_w == 1.23 then 599 // the buffer will be filled with "123" und the decimal_exponent will be 600 // decreased by 2. 601 final boolean result = digitGen(scaled_boundary_minus, scaled_w, scaled_boundary_plus, 602 buffer, mk); 603 return result; 604 } 605 606 // The "counted" version of grisu3 (see above) only generates requested_digits 607 // number of digits. This version does not generate the shortest representation, 608 // and with enough requested digits 0.1 will at some point print as 0.9999999... 609 // Grisu3 is too imprecise for real halfway cases (1.5 will not work) and 610 // therefore the rounding strategy for halfway cases is irrelevant. 611 static boolean grisu3Counted(final double v, 612 final int requested_digits, 613 final DtoaBuffer buffer) { 614 final long d64 = IeeeDouble.doubleToLong(v); 615 final DiyFp w = IeeeDouble.asNormalizedDiyFp(d64); 616 final DiyFp ten_mk = new DiyFp(); // Cached power of ten: 10^-k 617 final int mk; // -k 618 final int ten_mk_minimal_binary_exponent = 619 kMinimalTargetExponent - (w.e() + DiyFp.kSignificandSize); 620 final int ten_mk_maximal_binary_exponent = 621 kMaximalTargetExponent - (w.e() + DiyFp.kSignificandSize); 622 mk = CachedPowers.getCachedPowerForBinaryExponentRange( 623 ten_mk_minimal_binary_exponent, 624 ten_mk_maximal_binary_exponent, 625 ten_mk); 626 assert ((kMinimalTargetExponent <= w.e() + ten_mk.e() + 627 DiyFp.kSignificandSize) && 628 (kMaximalTargetExponent >= w.e() + ten_mk.e() + 629 DiyFp.kSignificandSize)); 630 // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a 631 // 64 bit significand and ten_mk is thus only precise up to 64 bits. 632 633 // The DiyFp::Times procedure rounds its result, and ten_mk is approximated 634 // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now 635 // off by a small amount. 636 // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w. 637 // In other words: let f = scaled_w.f() and e = scaled_w.e(), then 638 // (f-1) * 2^e < w*10^k < (f+1) * 2^e 639 final DiyFp scaled_w = DiyFp.times(w, ten_mk); 640 641 // We now have (double) (scaled_w * 10^-mk). 642 // DigitGen will generate the first requested_digits digits of scaled_w and 643 // return together with a kappa such that scaled_w ~= buffer * 10^kappa. (It 644 // will not always be exactly the same since DigitGenCounted only produces a 645 // limited number of digits.) 646 final boolean result = digitGenCounted(scaled_w, requested_digits, 647 buffer, mk); 648 return result; 649 } 650 651}