1 : // Copyright 2010 the V8 project authors. All rights reserved.
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3 : // modification, are permitted provided that the following conditions are
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26 : // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
27 :
28 : #include "v8.h"
29 :
30 : #include "fast-dtoa.h"
31 :
32 : #include "cached-powers.h"
33 : #include "diy-fp.h"
34 : #include "double.h"
35 :
36 : namespace v8 {
37 : namespace internal {
38 :
39 : // The minimal and maximal target exponent define the range of w's binary
40 : // exponent, where 'w' is the result of multiplying the input by a cached power
41 : // of ten.
42 : //
43 : // A different range might be chosen on a different platform, to optimize digit
44 : // generation, but a smaller range requires more powers of ten to be cached.
45 : static const int minimal_target_exponent = -60;
46 : static const int maximal_target_exponent = -32;
47 :
48 :
49 : // Adjusts the last digit of the generated number, and screens out generated
50 : // solutions that may be inaccurate. A solution may be inaccurate if it is
51 : // outside the safe interval, or if we ctannot prove that it is closer to the
52 : // input than a neighboring representation of the same length.
53 : //
54 : // Input: * buffer containing the digits of too_high / 10^kappa
55 : // * the buffer's length
56 : // * distance_too_high_w == (too_high - w).f() * unit
57 : // * unsafe_interval == (too_high - too_low).f() * unit
58 : // * rest = (too_high - buffer * 10^kappa).f() * unit
59 : // * ten_kappa = 10^kappa * unit
60 : // * unit = the common multiplier
61 : // Output: returns true if the buffer is guaranteed to contain the closest
62 : // representable number to the input.
63 : // Modifies the generated digits in the buffer to approach (round towards) w.
64 257335 : bool RoundWeed(Vector<char> buffer,
65 : int length,
66 : uint64_t distance_too_high_w,
67 : uint64_t unsafe_interval,
68 : uint64_t rest,
69 : uint64_t ten_kappa,
70 : uint64_t unit) {
71 257335 : uint64_t small_distance = distance_too_high_w - unit;
72 257335 : uint64_t big_distance = distance_too_high_w + unit;
73 : // Let w_low = too_high - big_distance, and
74 : // w_high = too_high - small_distance.
75 : // Note: w_low < w < w_high
76 : //
77 : // The real w (* unit) must lie somewhere inside the interval
78 : // ]w_low; w_low[ (often written as "(w_low; w_low)")
79 :
80 : // Basically the buffer currently contains a number in the unsafe interval
81 : // ]too_low; too_high[ with too_low < w < too_high
82 : //
83 : // too_high - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
84 : // ^v 1 unit ^ ^ ^ ^
85 : // boundary_high --------------------- . . . .
86 : // ^v 1 unit . . . .
87 : // - - - - - - - - - - - - - - - - - - - + - - + - - - - - - . .
88 : // . . ^ . .
89 : // . big_distance . . .
90 : // . . . . rest
91 : // small_distance . . . .
92 : // v . . . .
93 : // w_high - - - - - - - - - - - - - - - - - - . . . .
94 : // ^v 1 unit . . . .
95 : // w ---------------------------------------- . . . .
96 : // ^v 1 unit v . . .
97 : // w_low - - - - - - - - - - - - - - - - - - - - - . . .
98 : // . . v
99 : // buffer --------------------------------------------------+-------+--------
100 : // . .
101 : // safe_interval .
102 : // v .
103 : // - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - .
104 : // ^v 1 unit .
105 : // boundary_low ------------------------- unsafe_interval
106 : // ^v 1 unit v
107 : // too_low - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
108 : //
109 : //
110 : // Note that the value of buffer could lie anywhere inside the range too_low
111 : // to too_high.
112 : //
113 : // boundary_low, boundary_high and w are approximations of the real boundaries
114 : // and v (the input number). They are guaranteed to be precise up to one unit.
115 : // In fact the error is guaranteed to be strictly less than one unit.
116 : //
117 : // Anything that lies outside the unsafe interval is guaranteed not to round
118 : // to v when read again.
119 : // Anything that lies inside the safe interval is guaranteed to round to v
120 : // when read again.
121 : // If the number inside the buffer lies inside the unsafe interval but not
122 : // inside the safe interval then we simply do not know and bail out (returning
123 : // false).
124 : //
125 : // Similarly we have to take into account the imprecision of 'w' when rounding
126 : // the buffer. If we have two potential representations we need to make sure
127 : // that the chosen one is closer to w_low and w_high since v can be anywhere
128 : // between them.
129 : //
130 : // By generating the digits of too_high we got the largest (closest to
131 : // too_high) buffer that is still in the unsafe interval. In the case where
132 : // w_high < buffer < too_high we try to decrement the buffer.
133 : // This way the buffer approaches (rounds towards) w.
134 : // There are 3 conditions that stop the decrementation process:
135 : // 1) the buffer is already below w_high
136 : // 2) decrementing the buffer would make it leave the unsafe interval
137 : // 3) decrementing the buffer would yield a number below w_high and farther
138 : // away than the current number. In other words:
139 : // (buffer{-1} < w_high) && w_high - buffer{-1} > buffer - w_high
140 : // Instead of using the buffer directly we use its distance to too_high.
141 : // Conceptually rest ~= too_high - buffer
142 581554 : while (rest < small_distance && // Negated condition 1
143 : unsafe_interval - rest >= ten_kappa && // Negated condition 2
144 : (rest + ten_kappa < small_distance || // buffer{-1} > w_high
145 : small_distance - rest >= rest + ten_kappa - small_distance)) {
146 66884 : buffer[length - 1]--;
147 66884 : rest += ten_kappa;
148 : }
149 :
150 : // We have approached w+ as much as possible. We now test if approaching w-
151 : // would require changing the buffer. If yes, then we have two possible
152 : // representations close to w, but we cannot decide which one is closer.
153 257335 : if (rest < big_distance &&
154 : unsafe_interval - rest >= ten_kappa &&
155 : (rest + ten_kappa < big_distance ||
156 : big_distance - rest > rest + ten_kappa - big_distance)) {
157 315 : return false;
158 : }
159 :
160 : // Weeding test.
161 : // The safe interval is [too_low + 2 ulp; too_high - 2 ulp]
162 : // Since too_low = too_high - unsafe_interval this is equivalent to
163 : // [too_high - unsafe_interval + 4 ulp; too_high - 2 ulp]
164 : // Conceptually we have: rest ~= too_high - buffer
165 257020 : return (2 * unit <= rest) && (rest <= unsafe_interval - 4 * unit);
166 : }
167 :
168 :
169 :
170 : static const uint32_t kTen4 = 10000;
171 : static const uint32_t kTen5 = 100000;
172 : static const uint32_t kTen6 = 1000000;
173 : static const uint32_t kTen7 = 10000000;
174 : static const uint32_t kTen8 = 100000000;
175 : static const uint32_t kTen9 = 1000000000;
176 :
177 : // Returns the biggest power of ten that is less than or equal than the given
178 : // number. We furthermore receive the maximum number of bits 'number' has.
179 : // If number_bits == 0 then 0^-1 is returned
180 : // The number of bits must be <= 32.
181 : // Precondition: (1 << number_bits) <= number < (1 << (number_bits + 1)).
182 257335 : static void BiggestPowerTen(uint32_t number,
183 : int number_bits,
184 : uint32_t* power,
185 : int* exponent) {
186 257335 : switch (number_bits) {
187 : case 32:
188 : case 31:
189 : case 30:
190 540 : if (kTen9 <= number) {
191 0 : *power = kTen9;
192 0 : *exponent = 9;
193 0 : break;
194 : } // else fallthrough
195 : case 29:
196 : case 28:
197 : case 27:
198 34702 : if (kTen8 <= number) {
199 34150 : *power = kTen8;
200 34150 : *exponent = 8;
201 34150 : break;
202 : } // else fallthrough
203 : case 26:
204 : case 25:
205 : case 24:
206 976 : if (kTen7 <= number) {
207 924 : *power = kTen7;
208 924 : *exponent = 7;
209 924 : break;
210 : } // else fallthrough
211 : case 23:
212 : case 22:
213 : case 21:
214 : case 20:
215 9417 : if (kTen6 <= number) {
216 426 : *power = kTen6;
217 426 : *exponent = 6;
218 426 : break;
219 : } // else fallthrough
220 : case 19:
221 : case 18:
222 : case 17:
223 59992 : if (kTen5 <= number) {
224 59698 : *power = kTen5;
225 59698 : *exponent = 5;
226 59698 : break;
227 : } // else fallthrough
228 : case 16:
229 : case 15:
230 : case 14:
231 46516 : if (kTen4 <= number) {
232 675 : *power = kTen4;
233 675 : *exponent = 4;
234 675 : break;
235 : } // else fallthrough
236 : case 13:
237 : case 12:
238 : case 11:
239 : case 10:
240 92035 : if (1000 <= number) {
241 82309 : *power = 1000;
242 82309 : *exponent = 3;
243 82309 : break;
244 : } // else fallthrough
245 : case 9:
246 : case 8:
247 : case 7:
248 71302 : if (100 <= number) {
249 61427 : *power = 100;
250 61427 : *exponent = 2;
251 61427 : break;
252 : } // else fallthrough
253 : case 6:
254 : case 5:
255 : case 4:
256 17726 : if (10 <= number) {
257 16923 : *power = 10;
258 16923 : *exponent = 1;
259 16923 : break;
260 : } // else fallthrough
261 : case 3:
262 : case 2:
263 : case 1:
264 803 : if (1 <= number) {
265 803 : *power = 1;
266 803 : *exponent = 0;
267 803 : break;
268 : } // else fallthrough
269 : case 0:
270 0 : *power = 0;
271 0 : *exponent = -1;
272 0 : break;
273 : default:
274 : // Following assignments are here to silence compiler warnings.
275 0 : *power = 0;
276 0 : *exponent = 0;
277 0 : UNREACHABLE();
278 : }
279 257335 : }
280 :
281 :
282 : // Generates the digits of input number w.
283 : // w is a floating-point number (DiyFp), consisting of a significand and an
284 : // exponent. Its exponent is bounded by minimal_target_exponent and
285 : // maximal_target_exponent.
286 : // Hence -60 <= w.e() <= -32.
287 : //
288 : // Returns false if it fails, in which case the generated digits in the buffer
289 : // should not be used.
290 : // Preconditions:
291 : // * low, w and high are correct up to 1 ulp (unit in the last place). That
292 : // is, their error must be less that a unit of their last digits.
293 : // * low.e() == w.e() == high.e()
294 : // * low < w < high, and taking into account their error: low~ <= high~
295 : // * minimal_target_exponent <= w.e() <= maximal_target_exponent
296 : // Postconditions: returns false if procedure fails.
297 : // otherwise:
298 : // * buffer is not null-terminated, but len contains the number of digits.
299 : // * buffer contains the shortest possible decimal digit-sequence
300 : // such that LOW < buffer * 10^kappa < HIGH, where LOW and HIGH are the
301 : // correct values of low and high (without their error).
302 : // * if more than one decimal representation gives the minimal number of
303 : // decimal digits then the one closest to W (where W is the correct value
304 : // of w) is chosen.
305 : // Remark: this procedure takes into account the imprecision of its input
306 : // numbers. If the precision is not enough to guarantee all the postconditions
307 : // then false is returned. This usually happens rarely (~0.5%).
308 : //
309 : // Say, for the sake of example, that
310 : // w.e() == -48, and w.f() == 0x1234567890abcdef
311 : // w's value can be computed by w.f() * 2^w.e()
312 : // We can obtain w's integral digits by simply shifting w.f() by -w.e().
313 : // -> w's integral part is 0x1234
314 : // w's fractional part is therefore 0x567890abcdef.
315 : // Printing w's integral part is easy (simply print 0x1234 in decimal).
316 : // In order to print its fraction we repeatedly multiply the fraction by 10 and
317 : // get each digit. Example the first digit after the point would be computed by
318 : // (0x567890abcdef * 10) >> 48. -> 3
319 : // The whole thing becomes slightly more complicated because we want to stop
320 : // once we have enough digits. That is, once the digits inside the buffer
321 : // represent 'w' we can stop. Everything inside the interval low - high
322 : // represents w. However we have to pay attention to low, high and w's
323 : // imprecision.
324 257335 : bool DigitGen(DiyFp low,
325 : DiyFp w,
326 : DiyFp high,
327 : Vector<char> buffer,
328 : int* length,
329 : int* kappa) {
330 257335 : ASSERT(low.e() == w.e() && w.e() == high.e());
331 257335 : ASSERT(low.f() + 1 <= high.f() - 1);
332 257335 : ASSERT(minimal_target_exponent <= w.e() && w.e() <= maximal_target_exponent);
333 : // low, w and high are imprecise, but by less than one ulp (unit in the last
334 : // place).
335 : // If we remove (resp. add) 1 ulp from low (resp. high) we are certain that
336 : // the new numbers are outside of the interval we want the final
337 : // representation to lie in.
338 : // Inversely adding (resp. removing) 1 ulp from low (resp. high) would yield
339 : // numbers that are certain to lie in the interval. We will use this fact
340 : // later on.
341 : // We will now start by generating the digits within the uncertain
342 : // interval. Later we will weed out representations that lie outside the safe
343 : // interval and thus _might_ lie outside the correct interval.
344 257335 : uint64_t unit = 1;
345 257335 : DiyFp too_low = DiyFp(low.f() - unit, low.e());
346 257335 : DiyFp too_high = DiyFp(high.f() + unit, high.e());
347 : // too_low and too_high are guaranteed to lie outside the interval we want the
348 : // generated number in.
349 257335 : DiyFp unsafe_interval = DiyFp::Minus(too_high, too_low);
350 : // We now cut the input number into two parts: the integral digits and the
351 : // fractionals. We will not write any decimal separator though, but adapt
352 : // kappa instead.
353 : // Reminder: we are currently computing the digits (stored inside the buffer)
354 : // such that: too_low < buffer * 10^kappa < too_high
355 : // We use too_high for the digit_generation and stop as soon as possible.
356 : // If we stop early we effectively round down.
357 257335 : DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());
358 : // Division by one is a shift.
359 257335 : uint32_t integrals = static_cast<uint32_t>(too_high.f() >> -one.e());
360 : // Modulo by one is an and.
361 257335 : uint64_t fractionals = too_high.f() & (one.f() - 1);
362 : uint32_t divider;
363 : int divider_exponent;
364 257335 : BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),
365 257335 : ÷r, ÷r_exponent);
366 257335 : *kappa = divider_exponent + 1;
367 257335 : *length = 0;
368 : // Loop invariant: buffer = too_high / 10^kappa (integer division)
369 : // The invariant holds for the first iteration: kappa has been initialized
370 : // with the divider exponent + 1. And the divider is the biggest power of ten
371 : // that is smaller than integrals.
372 1687432 : while (*kappa > 0) {
373 1189631 : int digit = integrals / divider;
374 1189631 : buffer[*length] = '0' + digit;
375 1189631 : (*length)++;
376 1189631 : integrals %= divider;
377 1189631 : (*kappa)--;
378 : // Note that kappa now equals the exponent of the divider and that the
379 : // invariant thus holds again.
380 : uint64_t rest =
381 1189631 : (static_cast<uint64_t>(integrals) << -one.e()) + fractionals;
382 : // Invariant: too_high = buffer * 10^kappa + DiyFp(rest, one.e())
383 : // Reminder: unsafe_interval.e() == one.e()
384 1189631 : if (rest < unsafe_interval.f()) {
385 : // Rounding down (by not emitting the remaining digits) yields a number
386 : // that lies within the unsafe interval.
387 : return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f(),
388 : unsafe_interval.f(), rest,
389 16869 : static_cast<uint64_t>(divider) << -one.e(), unit);
390 : }
391 1172762 : divider /= 10;
392 : }
393 :
394 : // The integrals have been generated. We are at the point of the decimal
395 : // separator. In the following loop we simply multiply the remaining digits by
396 : // 10 and divide by one. We just need to pay attention to multiply associated
397 : // data (like the interval or 'unit'), too.
398 : // Instead of multiplying by 10 we multiply by 5 (cheaper operation) and
399 : // increase its (imaginary) exponent. At the same time we decrease the
400 : // divider's (one's) exponent and shift its significand.
401 : // Basically, if fractionals was a DiyFp (with fractionals.e == one.e):
402 : // fractionals.f *= 10;
403 : // fractionals.f >>= 1; fractionals.e++; // value remains unchanged.
404 : // one.f >>= 1; one.e++; // value remains unchanged.
405 : // and we have again fractionals.e == one.e which allows us to divide
406 : // fractionals.f() by one.f()
407 : // We simply combine the *= 10 and the >>= 1.
408 1612840 : while (true) {
409 1853306 : fractionals *= 5;
410 1853306 : unit *= 5;
411 1853306 : unsafe_interval.set_f(unsafe_interval.f() * 5);
412 1853306 : unsafe_interval.set_e(unsafe_interval.e() + 1); // Will be optimized out.
413 1853306 : one.set_f(one.f() >> 1);
414 1853306 : one.set_e(one.e() + 1);
415 : // Integer division by one.
416 1853306 : int digit = static_cast<int>(fractionals >> -one.e());
417 1853306 : buffer[*length] = '0' + digit;
418 1853306 : (*length)++;
419 1853306 : fractionals &= one.f() - 1; // Modulo by one.
420 1853306 : (*kappa)--;
421 1853306 : if (fractionals < unsafe_interval.f()) {
422 240466 : return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f() * unit,
423 480932 : unsafe_interval.f(), fractionals, one.f(), unit);
424 : }
425 : }
426 : }
427 :
428 :
429 : // Provides a decimal representation of v.
430 : // Returns true if it succeeds, otherwise the result cannot be trusted.
431 : // There will be *length digits inside the buffer (not null-terminated).
432 : // If the function returns true then
433 : // v == (double) (buffer * 10^decimal_exponent).
434 : // The digits in the buffer are the shortest representation possible: no
435 : // 0.09999999999999999 instead of 0.1. The shorter representation will even be
436 : // chosen even if the longer one would be closer to v.
437 : // The last digit will be closest to the actual v. That is, even if several
438 : // digits might correctly yield 'v' when read again, the closest will be
439 : // computed.
440 257335 : bool grisu3(double v, Vector<char> buffer, int* length, int* decimal_exponent) {
441 257335 : DiyFp w = Double(v).AsNormalizedDiyFp();
442 : // boundary_minus and boundary_plus are the boundaries between v and its
443 : // closest floating-point neighbors. Any number strictly between
444 : // boundary_minus and boundary_plus will round to v when convert to a double.
445 : // Grisu3 will never output representations that lie exactly on a boundary.
446 257335 : DiyFp boundary_minus, boundary_plus;
447 257335 : Double(v).NormalizedBoundaries(&boundary_minus, &boundary_plus);
448 257335 : ASSERT(boundary_plus.e() == w.e());
449 257335 : DiyFp ten_mk; // Cached power of ten: 10^-k
450 : int mk; // -k
451 257335 : GetCachedPower(w.e() + DiyFp::kSignificandSize, minimal_target_exponent,
452 257335 : maximal_target_exponent, &mk, &ten_mk);
453 257335 : ASSERT(minimal_target_exponent <= w.e() + ten_mk.e() +
454 : DiyFp::kSignificandSize &&
455 : maximal_target_exponent >= w.e() + ten_mk.e() +
456 257335 : DiyFp::kSignificandSize);
457 : // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a
458 : // 64 bit significand and ten_mk is thus only precise up to 64 bits.
459 :
460 : // The DiyFp::Times procedure rounds its result, and ten_mk is approximated
461 : // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now
462 : // off by a small amount.
463 : // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w.
464 : // In other words: let f = scaled_w.f() and e = scaled_w.e(), then
465 : // (f-1) * 2^e < w*10^k < (f+1) * 2^e
466 257335 : DiyFp scaled_w = DiyFp::Times(w, ten_mk);
467 257335 : ASSERT(scaled_w.e() ==
468 257335 : boundary_plus.e() + ten_mk.e() + DiyFp::kSignificandSize);
469 : // In theory it would be possible to avoid some recomputations by computing
470 : // the difference between w and boundary_minus/plus (a power of 2) and to
471 : // compute scaled_boundary_minus/plus by subtracting/adding from
472 : // scaled_w. However the code becomes much less readable and the speed
473 : // enhancements are not terriffic.
474 257335 : DiyFp scaled_boundary_minus = DiyFp::Times(boundary_minus, ten_mk);
475 257335 : DiyFp scaled_boundary_plus = DiyFp::Times(boundary_plus, ten_mk);
476 :
477 : // DigitGen will generate the digits of scaled_w. Therefore we have
478 : // v == (double) (scaled_w * 10^-mk).
479 : // Set decimal_exponent == -mk and pass it to DigitGen. If scaled_w is not an
480 : // integer than it will be updated. For instance if scaled_w == 1.23 then
481 : // the buffer will be filled with "123" und the decimal_exponent will be
482 : // decreased by 2.
483 : int kappa;
484 : bool result = DigitGen(scaled_boundary_minus, scaled_w, scaled_boundary_plus,
485 257335 : buffer, length, &kappa);
486 257335 : *decimal_exponent = -mk + kappa;
487 257335 : return result;
488 : }
489 :
490 :
491 257335 : bool FastDtoa(double v,
492 : Vector<char> buffer,
493 : int* length,
494 : int* point) {
495 257335 : ASSERT(v > 0);
496 257335 : ASSERT(!Double(v).IsSpecial());
497 :
498 : int decimal_exponent;
499 257335 : bool result = grisu3(v, buffer, length, &decimal_exponent);
500 257335 : *point = *length + decimal_exponent;
501 257335 : buffer[*length] = '\0';
502 257335 : return result;
503 : }
504 :
505 : } } // namespace v8::internal
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