1// Copyright 2011 the V8 project authors. All rights reserved.
2// Use of this source code is governed by a BSD-style license that can be
3// found in the LICENSE file.
4
5#include "src/bignum-dtoa.h"
6
7#include <cmath>
8
9#include "src/base/logging.h"
10#include "src/bignum.h"
11#include "src/double.h"
12#include "src/utils.h"
13
14namespace v8 {
15namespace internal {
16
17static int NormalizedExponent(uint64_t significand, int exponent) {
18 DCHECK_NE(significand, 0);
19 while ((significand & Double::kHiddenBit) == 0) {
20 significand = significand << 1;
21 exponent = exponent - 1;
22 }
23 return exponent;
24}
25
26
27// Forward declarations:
28// Returns an estimation of k such that 10^(k-1) <= v < 10^k.
29static int EstimatePower(int exponent);
30// Computes v / 10^estimated_power exactly, as a ratio of two bignums, numerator
31// and denominator.
32static void InitialScaledStartValues(double v,
33 int estimated_power,
34 bool need_boundary_deltas,
35 Bignum* numerator,
36 Bignum* denominator,
37 Bignum* delta_minus,
38 Bignum* delta_plus);
39// Multiplies numerator/denominator so that its values lies in the range 1-10.
40// Returns decimal_point s.t.
41// v = numerator'/denominator' * 10^(decimal_point-1)
42// where numerator' and denominator' are the values of numerator and
43// denominator after the call to this function.
44static void FixupMultiply10(int estimated_power, bool is_even,
45 int* decimal_point,
46 Bignum* numerator, Bignum* denominator,
47 Bignum* delta_minus, Bignum* delta_plus);
48// Generates digits from the left to the right and stops when the generated
49// digits yield the shortest decimal representation of v.
50static void GenerateShortestDigits(Bignum* numerator, Bignum* denominator,
51 Bignum* delta_minus, Bignum* delta_plus,
52 bool is_even,
53 Vector<char> buffer, int* length);
54// Generates 'requested_digits' after the decimal point.
55static void BignumToFixed(int requested_digits, int* decimal_point,
56 Bignum* numerator, Bignum* denominator,
57 Vector<char>(buffer), int* length);
58// Generates 'count' digits of numerator/denominator.
59// Once 'count' digits have been produced rounds the result depending on the
60// remainder (remainders of exactly .5 round upwards). Might update the
61// decimal_point when rounding up (for example for 0.9999).
62static void GenerateCountedDigits(int count, int* decimal_point,
63 Bignum* numerator, Bignum* denominator,
64 Vector<char>(buffer), int* length);
65
66
67void BignumDtoa(double v, BignumDtoaMode mode, int requested_digits,
68 Vector<char> buffer, int* length, int* decimal_point) {
69 DCHECK_GT(v, 0);
70 DCHECK(!Double(v).IsSpecial());
71 uint64_t significand = Double(v).Significand();
72 bool is_even = (significand & 1) == 0;
73 int exponent = Double(v).Exponent();
74 int normalized_exponent = NormalizedExponent(significand, exponent);
75 // estimated_power might be too low by 1.
76 int estimated_power = EstimatePower(normalized_exponent);
77
78 // Shortcut for Fixed.
79 // The requested digits correspond to the digits after the point. If the
80 // number is much too small, then there is no need in trying to get any
81 // digits.
82 if (mode == BIGNUM_DTOA_FIXED && -estimated_power - 1 > requested_digits) {
83 buffer[0] = '\0';
84 *length = 0;
85 // Set decimal-point to -requested_digits. This is what Gay does.
86 // Note that it should not have any effect anyways since the string is
87 // empty.
88 *decimal_point = -requested_digits;
89 return;
90 }
91
92 Bignum numerator;
93 Bignum denominator;
94 Bignum delta_minus;
95 Bignum delta_plus;
96 // Make sure the bignum can grow large enough. The smallest double equals
97 // 4e-324. In this case the denominator needs fewer than 324*4 binary digits.
98 // The maximum double is 1.7976931348623157e308 which needs fewer than
99 // 308*4 binary digits.
100 DCHECK_GE(Bignum::kMaxSignificantBits, 324 * 4);
101 bool need_boundary_deltas = (mode == BIGNUM_DTOA_SHORTEST);
102 InitialScaledStartValues(v, estimated_power, need_boundary_deltas,
103 &numerator, &denominator,
104 &delta_minus, &delta_plus);
105 // We now have v = (numerator / denominator) * 10^estimated_power.
106 FixupMultiply10(estimated_power, is_even, decimal_point,
107 &numerator, &denominator,
108 &delta_minus, &delta_plus);
109 // We now have v = (numerator / denominator) * 10^(decimal_point-1), and
110 // 1 <= (numerator + delta_plus) / denominator < 10
111 switch (mode) {
112 case BIGNUM_DTOA_SHORTEST:
113 GenerateShortestDigits(&numerator, &denominator,
114 &delta_minus, &delta_plus,
115 is_even, buffer, length);
116 break;
117 case BIGNUM_DTOA_FIXED:
118 BignumToFixed(requested_digits, decimal_point,
119 &numerator, &denominator,
120 buffer, length);
121 break;
122 case BIGNUM_DTOA_PRECISION:
123 GenerateCountedDigits(requested_digits, decimal_point,
124 &numerator, &denominator,
125 buffer, length);
126 break;
127 default:
128 UNREACHABLE();
129 }
130 buffer[*length] = '\0';
131}
132
133
134// The procedure starts generating digits from the left to the right and stops
135// when the generated digits yield the shortest decimal representation of v. A
136// decimal representation of v is a number lying closer to v than to any other
137// double, so it converts to v when read.
138//
139// This is true if d, the decimal representation, is between m- and m+, the
140// upper and lower boundaries. d must be strictly between them if !is_even.
141// m- := (numerator - delta_minus) / denominator
142// m+ := (numerator + delta_plus) / denominator
143//
144// Precondition: 0 <= (numerator+delta_plus) / denominator < 10.
145// If 1 <= (numerator+delta_plus) / denominator < 10 then no leading 0 digit
146// will be produced. This should be the standard precondition.
147static void GenerateShortestDigits(Bignum* numerator, Bignum* denominator,
148 Bignum* delta_minus, Bignum* delta_plus,
149 bool is_even,
150 Vector<char> buffer, int* length) {
151 // Small optimization: if delta_minus and delta_plus are the same just reuse
152 // one of the two bignums.
153 if (Bignum::Equal(*delta_minus, *delta_plus)) {
154 delta_plus = delta_minus;
155 }
156 *length = 0;
157 while (true) {
158 uint16_t digit;
159 digit = numerator->DivideModuloIntBignum(*denominator);
160 DCHECK_LE(digit, 9); // digit is a uint16_t and therefore always positive.
161 // digit = numerator / denominator (integer division).
162 // numerator = numerator % denominator.
163 buffer[(*length)++] = digit + '0';
164
165 // Can we stop already?
166 // If the remainder of the division is less than the distance to the lower
167 // boundary we can stop. In this case we simply round down (discarding the
168 // remainder).
169 // Similarly we test if we can round up (using the upper boundary).
170 bool in_delta_room_minus;
171 bool in_delta_room_plus;
172 if (is_even) {
173 in_delta_room_minus = Bignum::LessEqual(*numerator, *delta_minus);
174 } else {
175 in_delta_room_minus = Bignum::Less(*numerator, *delta_minus);
176 }
177 if (is_even) {
178 in_delta_room_plus =
179 Bignum::PlusCompare(*numerator, *delta_plus, *denominator) >= 0;
180 } else {
181 in_delta_room_plus =
182 Bignum::PlusCompare(*numerator, *delta_plus, *denominator) > 0;
183 }
184 if (!in_delta_room_minus && !in_delta_room_plus) {
185 // Prepare for next iteration.
186 numerator->Times10();
187 delta_minus->Times10();
188 // We optimized delta_plus to be equal to delta_minus (if they share the
189 // same value). So don't multiply delta_plus if they point to the same
190 // object.
191 if (delta_minus != delta_plus) {
192 delta_plus->Times10();
193 }
194 } else if (in_delta_room_minus && in_delta_room_plus) {
195 // Let's see if 2*numerator < denominator.
196 // If yes, then the next digit would be < 5 and we can round down.
197 int compare = Bignum::PlusCompare(*numerator, *numerator, *denominator);
198 if (compare < 0) {
199 // Remaining digits are less than .5. -> Round down (== do nothing).
200 } else if (compare > 0) {
201 // Remaining digits are more than .5 of denominator. -> Round up.
202 // Note that the last digit could not be a '9' as otherwise the whole
203 // loop would have stopped earlier.
204 // We still have an assert here in case the preconditions were not
205 // satisfied.
206 DCHECK_NE(buffer[(*length) - 1], '9');
207 buffer[(*length) - 1]++;
208 } else {
209 // Halfway case.
210 // TODO(floitsch): need a way to solve half-way cases.
211 // For now let's round towards even (since this is what Gay seems to
212 // do).
213
214 if ((buffer[(*length) - 1] - '0') % 2 == 0) {
215 // Round down => Do nothing.
216 } else {
217 DCHECK_NE(buffer[(*length) - 1], '9');
218 buffer[(*length) - 1]++;
219 }
220 }
221 return;
222 } else if (in_delta_room_minus) {
223 // Round down (== do nothing).
224 return;
225 } else { // in_delta_room_plus
226 // Round up.
227 // Note again that the last digit could not be '9' since this would have
228 // stopped the loop earlier.
229 // We still have an DCHECK here, in case the preconditions were not
230 // satisfied.
231 DCHECK_NE(buffer[(*length) - 1], '9');
232 buffer[(*length) - 1]++;
233 return;
234 }
235 }
236}
237
238
239// Let v = numerator / denominator < 10.
240// Then we generate 'count' digits of d = x.xxxxx... (without the decimal point)
241// from left to right. Once 'count' digits have been produced we decide wether
242// to round up or down. Remainders of exactly .5 round upwards. Numbers such
243// as 9.999999 propagate a carry all the way, and change the
244// exponent (decimal_point), when rounding upwards.
245static void GenerateCountedDigits(int count, int* decimal_point,
246 Bignum* numerator, Bignum* denominator,
247 Vector<char>(buffer), int* length) {
248 DCHECK_GE(count, 0);
249 for (int i = 0; i < count - 1; ++i) {
250 uint16_t digit;
251 digit = numerator->DivideModuloIntBignum(*denominator);
252 DCHECK_LE(digit, 9); // digit is a uint16_t and therefore always positive.
253 // digit = numerator / denominator (integer division).
254 // numerator = numerator % denominator.
255 buffer[i] = digit + '0';
256 // Prepare for next iteration.
257 numerator->Times10();
258 }
259 // Generate the last digit.
260 uint16_t digit;
261 digit = numerator->DivideModuloIntBignum(*denominator);
262 if (Bignum::PlusCompare(*numerator, *numerator, *denominator) >= 0) {
263 digit++;
264 }
265 buffer[count - 1] = digit + '0';
266 // Correct bad digits (in case we had a sequence of '9's). Propagate the
267 // carry until we hat a non-'9' or til we reach the first digit.
268 for (int i = count - 1; i > 0; --i) {
269 if (buffer[i] != '0' + 10) break;
270 buffer[i] = '0';
271 buffer[i - 1]++;
272 }
273 if (buffer[0] == '0' + 10) {
274 // Propagate a carry past the top place.
275 buffer[0] = '1';
276 (*decimal_point)++;
277 }
278 *length = count;
279}
280
281
282// Generates 'requested_digits' after the decimal point. It might omit
283// trailing '0's. If the input number is too small then no digits at all are
284// generated (ex.: 2 fixed digits for 0.00001).
285//
286// Input verifies: 1 <= (numerator + delta) / denominator < 10.
287static void BignumToFixed(int requested_digits, int* decimal_point,
288 Bignum* numerator, Bignum* denominator,
289 Vector<char>(buffer), int* length) {
290 // Note that we have to look at more than just the requested_digits, since
291 // a number could be rounded up. Example: v=0.5 with requested_digits=0.
292 // Even though the power of v equals 0 we can't just stop here.
293 if (-(*decimal_point) > requested_digits) {
294 // The number is definitively too small.
295 // Ex: 0.001 with requested_digits == 1.
296 // Set decimal-point to -requested_digits. This is what Gay does.
297 // Note that it should not have any effect anyways since the string is
298 // empty.
299 *decimal_point = -requested_digits;
300 *length = 0;
301 return;
302 } else if (-(*decimal_point) == requested_digits) {
303 // We only need to verify if the number rounds down or up.
304 // Ex: 0.04 and 0.06 with requested_digits == 1.
305 DCHECK(*decimal_point == -requested_digits);
306 // Initially the fraction lies in range (1, 10]. Multiply the denominator
307 // by 10 so that we can compare more easily.
308 denominator->Times10();
309 if (Bignum::PlusCompare(*numerator, *numerator, *denominator) >= 0) {
310 // If the fraction is >= 0.5 then we have to include the rounded
311 // digit.
312 buffer[0] = '1';
313 *length = 1;
314 (*decimal_point)++;
315 } else {
316 // Note that we caught most of similar cases earlier.
317 *length = 0;
318 }
319 return;
320 } else {
321 // The requested digits correspond to the digits after the point.
322 // The variable 'needed_digits' includes the digits before the point.
323 int needed_digits = (*decimal_point) + requested_digits;
324 GenerateCountedDigits(needed_digits, decimal_point,
325 numerator, denominator,
326 buffer, length);
327 }
328}
329
330
331// Returns an estimation of k such that 10^(k-1) <= v < 10^k where
332// v = f * 2^exponent and 2^52 <= f < 2^53.
333// v is hence a normalized double with the given exponent. The output is an
334// approximation for the exponent of the decimal approimation .digits * 10^k.
335//
336// The result might undershoot by 1 in which case 10^k <= v < 10^k+1.
337// Note: this property holds for v's upper boundary m+ too.
338// 10^k <= m+ < 10^k+1.
339// (see explanation below).
340//
341// Examples:
342// EstimatePower(0) => 16
343// EstimatePower(-52) => 0
344//
345// Note: e >= 0 => EstimatedPower(e) > 0. No similar claim can be made for e<0.
346static int EstimatePower(int exponent) {
347 // This function estimates log10 of v where v = f*2^e (with e == exponent).
348 // Note that 10^floor(log10(v)) <= v, but v <= 10^ceil(log10(v)).
349 // Note that f is bounded by its container size. Let p = 53 (the double's
350 // significand size). Then 2^(p-1) <= f < 2^p.
351 //
352 // Given that log10(v) == log2(v)/log2(10) and e+(len(f)-1) is quite close
353 // to log2(v) the function is simplified to (e+(len(f)-1)/log2(10)).
354 // The computed number undershoots by less than 0.631 (when we compute log3
355 // and not log10).
356 //
357 // Optimization: since we only need an approximated result this computation
358 // can be performed on 64 bit integers. On x86/x64 architecture the speedup is
359 // not really measurable, though.
360 //
361 // Since we want to avoid overshooting we decrement by 1e10 so that
362 // floating-point imprecisions don't affect us.
363 //
364 // Explanation for v's boundary m+: the computation takes advantage of
365 // the fact that 2^(p-1) <= f < 2^p. Boundaries still satisfy this requirement
366 // (even for denormals where the delta can be much more important).
367
368 const double k1Log10 = 0.30102999566398114; // 1/lg(10)
369
370 // For doubles len(f) == 53 (don't forget the hidden bit).
371 const int kSignificandSize = 53;
372 double estimate =
373 std::ceil((exponent + kSignificandSize - 1) * k1Log10 - 1e-10);
374 return static_cast<int>(estimate);
375}
376
377
378// See comments for InitialScaledStartValues.
379static void InitialScaledStartValuesPositiveExponent(
380 double v, int estimated_power, bool need_boundary_deltas,
381 Bignum* numerator, Bignum* denominator,
382 Bignum* delta_minus, Bignum* delta_plus) {
383 // A positive exponent implies a positive power.
384 DCHECK_GE(estimated_power, 0);
385 // Since the estimated_power is positive we simply multiply the denominator
386 // by 10^estimated_power.
387
388 // numerator = v.
389 numerator->AssignUInt64(Double(v).Significand());
390 numerator->ShiftLeft(Double(v).Exponent());
391 // denominator = 10^estimated_power.
392 denominator->AssignPowerUInt16(10, estimated_power);
393
394 if (need_boundary_deltas) {
395 // Introduce a common denominator so that the deltas to the boundaries are
396 // integers.
397 denominator->ShiftLeft(1);
398 numerator->ShiftLeft(1);
399 // Let v = f * 2^e, then m+ - v = 1/2 * 2^e; With the common
400 // denominator (of 2) delta_plus equals 2^e.
401 delta_plus->AssignUInt16(1);
402 delta_plus->ShiftLeft(Double(v).Exponent());
403 // Same for delta_minus (with adjustments below if f == 2^p-1).
404 delta_minus->AssignUInt16(1);
405 delta_minus->ShiftLeft(Double(v).Exponent());
406
407 // If the significand (without the hidden bit) is 0, then the lower
408 // boundary is closer than just half a ulp (unit in the last place).
409 // There is only one exception: if the next lower number is a denormal then
410 // the distance is 1 ulp. This cannot be the case for exponent >= 0 (but we
411 // have to test it in the other function where exponent < 0).
412 uint64_t v_bits = Double(v).AsUint64();
413 if ((v_bits & Double::kSignificandMask) == 0) {
414 // The lower boundary is closer at half the distance of "normal" numbers.
415 // Increase the common denominator and adapt all but the delta_minus.
416 denominator->ShiftLeft(1); // *2
417 numerator->ShiftLeft(1); // *2
418 delta_plus->ShiftLeft(1); // *2
419 }
420 }
421}
422
423
424// See comments for InitialScaledStartValues
425static void InitialScaledStartValuesNegativeExponentPositivePower(
426 double v, int estimated_power, bool need_boundary_deltas,
427 Bignum* numerator, Bignum* denominator,
428 Bignum* delta_minus, Bignum* delta_plus) {
429 uint64_t significand = Double(v).Significand();
430 int exponent = Double(v).Exponent();
431 // v = f * 2^e with e < 0, and with estimated_power >= 0.
432 // This means that e is close to 0 (have a look at how estimated_power is
433 // computed).
434
435 // numerator = significand
436 // since v = significand * 2^exponent this is equivalent to
437 // numerator = v * / 2^-exponent
438 numerator->AssignUInt64(significand);
439 // denominator = 10^estimated_power * 2^-exponent (with exponent < 0)
440 denominator->AssignPowerUInt16(10, estimated_power);
441 denominator->ShiftLeft(-exponent);
442
443 if (need_boundary_deltas) {
444 // Introduce a common denominator so that the deltas to the boundaries are
445 // integers.
446 denominator->ShiftLeft(1);
447 numerator->ShiftLeft(1);
448 // Let v = f * 2^e, then m+ - v = 1/2 * 2^e; With the common
449 // denominator (of 2) delta_plus equals 2^e.
450 // Given that the denominator already includes v's exponent the distance
451 // to the boundaries is simply 1.
452 delta_plus->AssignUInt16(1);
453 // Same for delta_minus (with adjustments below if f == 2^p-1).
454 delta_minus->AssignUInt16(1);
455
456 // If the significand (without the hidden bit) is 0, then the lower
457 // boundary is closer than just one ulp (unit in the last place).
458 // There is only one exception: if the next lower number is a denormal
459 // then the distance is 1 ulp. Since the exponent is close to zero
460 // (otherwise estimated_power would have been negative) this cannot happen
461 // here either.
462 uint64_t v_bits = Double(v).AsUint64();
463 if ((v_bits & Double::kSignificandMask) == 0) {
464 // The lower boundary is closer at half the distance of "normal" numbers.
465 // Increase the denominator and adapt all but the delta_minus.
466 denominator->ShiftLeft(1); // *2
467 numerator->ShiftLeft(1); // *2
468 delta_plus->ShiftLeft(1); // *2
469 }
470 }
471}
472
473
474// See comments for InitialScaledStartValues
475static void InitialScaledStartValuesNegativeExponentNegativePower(
476 double v, int estimated_power, bool need_boundary_deltas,
477 Bignum* numerator, Bignum* denominator,
478 Bignum* delta_minus, Bignum* delta_plus) {
479 const uint64_t kMinimalNormalizedExponent =
480 V8_2PART_UINT64_C(0x00100000, 00000000);
481 uint64_t significand = Double(v).Significand();
482 int exponent = Double(v).Exponent();
483 // Instead of multiplying the denominator with 10^estimated_power we
484 // multiply all values (numerator and deltas) by 10^-estimated_power.
485
486 // Use numerator as temporary container for power_ten.
487 Bignum* power_ten = numerator;
488 power_ten->AssignPowerUInt16(10, -estimated_power);
489
490 if (need_boundary_deltas) {
491 // Since power_ten == numerator we must make a copy of 10^estimated_power
492 // before we complete the computation of the numerator.
493 // delta_plus = delta_minus = 10^estimated_power
494 delta_plus->AssignBignum(*power_ten);
495 delta_minus->AssignBignum(*power_ten);
496 }
497
498 // numerator = significand * 2 * 10^-estimated_power
499 // since v = significand * 2^exponent this is equivalent to
500 // numerator = v * 10^-estimated_power * 2 * 2^-exponent.
501 // Remember: numerator has been abused as power_ten. So no need to assign it
502 // to itself.
503 DCHECK(numerator == power_ten);
504 numerator->MultiplyByUInt64(significand);
505
506 // denominator = 2 * 2^-exponent with exponent < 0.
507 denominator->AssignUInt16(1);
508 denominator->ShiftLeft(-exponent);
509
510 if (need_boundary_deltas) {
511 // Introduce a common denominator so that the deltas to the boundaries are
512 // integers.
513 numerator->ShiftLeft(1);
514 denominator->ShiftLeft(1);
515 // With this shift the boundaries have their correct value, since
516 // delta_plus = 10^-estimated_power, and
517 // delta_minus = 10^-estimated_power.
518 // These assignments have been done earlier.
519
520 // The special case where the lower boundary is twice as close.
521 // This time we have to look out for the exception too.
522 uint64_t v_bits = Double(v).AsUint64();
523 if ((v_bits & Double::kSignificandMask) == 0 &&
524 // The only exception where a significand == 0 has its boundaries at
525 // "normal" distances:
526 (v_bits & Double::kExponentMask) != kMinimalNormalizedExponent) {
527 numerator->ShiftLeft(1); // *2
528 denominator->ShiftLeft(1); // *2
529 delta_plus->ShiftLeft(1); // *2
530 }
531 }
532}
533
534
535// Let v = significand * 2^exponent.
536// Computes v / 10^estimated_power exactly, as a ratio of two bignums, numerator
537// and denominator. The functions GenerateShortestDigits and
538// GenerateCountedDigits will then convert this ratio to its decimal
539// representation d, with the required accuracy.
540// Then d * 10^estimated_power is the representation of v.
541// (Note: the fraction and the estimated_power might get adjusted before
542// generating the decimal representation.)
543//
544// The initial start values consist of:
545// - a scaled numerator: s.t. numerator/denominator == v / 10^estimated_power.
546// - a scaled (common) denominator.
547// optionally (used by GenerateShortestDigits to decide if it has the shortest
548// decimal converting back to v):
549// - v - m-: the distance to the lower boundary.
550// - m+ - v: the distance to the upper boundary.
551//
552// v, m+, m-, and therefore v - m- and m+ - v all share the same denominator.
553//
554// Let ep == estimated_power, then the returned values will satisfy:
555// v / 10^ep = numerator / denominator.
556// v's boundarys m- and m+:
557// m- / 10^ep == v / 10^ep - delta_minus / denominator
558// m+ / 10^ep == v / 10^ep + delta_plus / denominator
559// Or in other words:
560// m- == v - delta_minus * 10^ep / denominator;
561// m+ == v + delta_plus * 10^ep / denominator;
562//
563// Since 10^(k-1) <= v < 10^k (with k == estimated_power)
564// or 10^k <= v < 10^(k+1)
565// we then have 0.1 <= numerator/denominator < 1
566// or 1 <= numerator/denominator < 10
567//
568// It is then easy to kickstart the digit-generation routine.
569//
570// The boundary-deltas are only filled if need_boundary_deltas is set.
571static void InitialScaledStartValues(double v,
572 int estimated_power,
573 bool need_boundary_deltas,
574 Bignum* numerator,
575 Bignum* denominator,
576 Bignum* delta_minus,
577 Bignum* delta_plus) {
578 if (Double(v).Exponent() >= 0) {
579 InitialScaledStartValuesPositiveExponent(
580 v, estimated_power, need_boundary_deltas,
581 numerator, denominator, delta_minus, delta_plus);
582 } else if (estimated_power >= 0) {
583 InitialScaledStartValuesNegativeExponentPositivePower(
584 v, estimated_power, need_boundary_deltas,
585 numerator, denominator, delta_minus, delta_plus);
586 } else {
587 InitialScaledStartValuesNegativeExponentNegativePower(
588 v, estimated_power, need_boundary_deltas,
589 numerator, denominator, delta_minus, delta_plus);
590 }
591}
592
593
594// This routine multiplies numerator/denominator so that its values lies in the
595// range 1-10. That is after a call to this function we have:
596// 1 <= (numerator + delta_plus) /denominator < 10.
597// Let numerator the input before modification and numerator' the argument
598// after modification, then the output-parameter decimal_point is such that
599// numerator / denominator * 10^estimated_power ==
600// numerator' / denominator' * 10^(decimal_point - 1)
601// In some cases estimated_power was too low, and this is already the case. We
602// then simply adjust the power so that 10^(k-1) <= v < 10^k (with k ==
603// estimated_power) but do not touch the numerator or denominator.
604// Otherwise the routine multiplies the numerator and the deltas by 10.
605static void FixupMultiply10(int estimated_power, bool is_even,
606 int* decimal_point,
607 Bignum* numerator, Bignum* denominator,
608 Bignum* delta_minus, Bignum* delta_plus) {
609 bool in_range;
610 if (is_even) {
611 // For IEEE doubles half-way cases (in decimal system numbers ending with 5)
612 // are rounded to the closest floating-point number with even significand.
613 in_range = Bignum::PlusCompare(*numerator, *delta_plus, *denominator) >= 0;
614 } else {
615 in_range = Bignum::PlusCompare(*numerator, *delta_plus, *denominator) > 0;
616 }
617 if (in_range) {
618 // Since numerator + delta_plus >= denominator we already have
619 // 1 <= numerator/denominator < 10. Simply update the estimated_power.
620 *decimal_point = estimated_power + 1;
621 } else {
622 *decimal_point = estimated_power;
623 numerator->Times10();
624 if (Bignum::Equal(*delta_minus, *delta_plus)) {
625 delta_minus->Times10();
626 delta_plus->AssignBignum(*delta_minus);
627 } else {
628 delta_minus->Times10();
629 delta_plus->Times10();
630 }
631 }
632}
633
634} // namespace internal
635} // namespace v8
636