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
58 package 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.
62 class 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 }