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1 // Copyright 2010 the V8 project authors. All rights reserved.
2 // Redistribution and use in source and binary forms, with or without
3 // modification, are permitted provided that the following conditions are
4 // met:
5 //
6 //     * Redistributions of source code must retain the above copyright
7 //       notice, this list of conditions and the following disclaimer.
8 //     * Redistributions in binary form must reproduce the above
9 //       copyright notice, this list of conditions and the following
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13 //       contributors may be used to endorse or promote products derived
14 //       from this software without specific prior written permission.
15 //
16 // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
17 // "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
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22 // LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
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25 // (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
26 // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
27 
28 #include <cmath>
29 
30 #include "bignum-dtoa.h"
31 
32 #include "bignum.h"
33 #include "ieee.h"
34 
35 namespace double_conversion {
36 
NormalizedExponent(uint64_t significand,int exponent)37 static int NormalizedExponent(uint64_t significand, int exponent) {
38   DOUBLE_CONVERSION_ASSERT(significand != 0);
39   while ((significand & Double::kHiddenBit) == 0) {
40     significand = significand << 1;
41     exponent = exponent - 1;
42   }
43   return exponent;
44 }
45 
46 
47 // Forward declarations:
48 // Returns an estimation of k such that 10^(k-1) <= v < 10^k.
49 static int EstimatePower(int exponent);
50 // Computes v / 10^estimated_power exactly, as a ratio of two bignums, numerator
51 // and denominator.
52 static void InitialScaledStartValues(uint64_t significand,
53                                      int exponent,
54                                      bool lower_boundary_is_closer,
55                                      int estimated_power,
56                                      bool need_boundary_deltas,
57                                      Bignum* numerator,
58                                      Bignum* denominator,
59                                      Bignum* delta_minus,
60                                      Bignum* delta_plus);
61 // Multiplies numerator/denominator so that its values lies in the range 1-10.
62 // Returns decimal_point s.t.
63 //  v = numerator'/denominator' * 10^(decimal_point-1)
64 //     where numerator' and denominator' are the values of numerator and
65 //     denominator after the call to this function.
66 static void FixupMultiply10(int estimated_power, bool is_even,
67                             int* decimal_point,
68                             Bignum* numerator, Bignum* denominator,
69                             Bignum* delta_minus, Bignum* delta_plus);
70 // Generates digits from the left to the right and stops when the generated
71 // digits yield the shortest decimal representation of v.
72 static void GenerateShortestDigits(Bignum* numerator, Bignum* denominator,
73                                    Bignum* delta_minus, Bignum* delta_plus,
74                                    bool is_even,
75                                    Vector<char> buffer, int* length);
76 // Generates 'requested_digits' after the decimal point.
77 static void BignumToFixed(int requested_digits, int* decimal_point,
78                           Bignum* numerator, Bignum* denominator,
79                           Vector<char> buffer, int* length);
80 // Generates 'count' digits of numerator/denominator.
81 // Once 'count' digits have been produced rounds the result depending on the
82 // remainder (remainders of exactly .5 round upwards). Might update the
83 // decimal_point when rounding up (for example for 0.9999).
84 static void GenerateCountedDigits(int count, int* decimal_point,
85                                   Bignum* numerator, Bignum* denominator,
86                                   Vector<char> buffer, int* length);
87 
88 
BignumDtoa(double v,BignumDtoaMode mode,int requested_digits,Vector<char> buffer,int * length,int * decimal_point)89 void BignumDtoa(double v, BignumDtoaMode mode, int requested_digits,
90                 Vector<char> buffer, int* length, int* decimal_point) {
91   DOUBLE_CONVERSION_ASSERT(v > 0);
92   DOUBLE_CONVERSION_ASSERT(!Double(v).IsSpecial());
93   uint64_t significand;
94   int exponent;
95   bool lower_boundary_is_closer;
96   if (mode == BIGNUM_DTOA_SHORTEST_SINGLE) {
97     float f = static_cast<float>(v);
98     DOUBLE_CONVERSION_ASSERT(f == v);
99     significand = Single(f).Significand();
100     exponent = Single(f).Exponent();
101     lower_boundary_is_closer = Single(f).LowerBoundaryIsCloser();
102   } else {
103     significand = Double(v).Significand();
104     exponent = Double(v).Exponent();
105     lower_boundary_is_closer = Double(v).LowerBoundaryIsCloser();
106   }
107   bool need_boundary_deltas =
108       (mode == BIGNUM_DTOA_SHORTEST || mode == BIGNUM_DTOA_SHORTEST_SINGLE);
109 
110   bool is_even = (significand & 1) == 0;
111   int normalized_exponent = NormalizedExponent(significand, exponent);
112   // estimated_power might be too low by 1.
113   int estimated_power = EstimatePower(normalized_exponent);
114 
115   // Shortcut for Fixed.
116   // The requested digits correspond to the digits after the point. If the
117   // number is much too small, then there is no need in trying to get any
118   // digits.
119   if (mode == BIGNUM_DTOA_FIXED && -estimated_power - 1 > requested_digits) {
120     buffer[0] = '\0';
121     *length = 0;
122     // Set decimal-point to -requested_digits. This is what Gay does.
123     // Note that it should not have any effect anyways since the string is
124     // empty.
125     *decimal_point = -requested_digits;
126     return;
127   }
128 
129   Bignum numerator;
130   Bignum denominator;
131   Bignum delta_minus;
132   Bignum delta_plus;
133   // Make sure the bignum can grow large enough. The smallest double equals
134   // 4e-324. In this case the denominator needs fewer than 324*4 binary digits.
135   // The maximum double is 1.7976931348623157e308 which needs fewer than
136   // 308*4 binary digits.
137   DOUBLE_CONVERSION_ASSERT(Bignum::kMaxSignificantBits >= 324*4);
138   InitialScaledStartValues(significand, exponent, lower_boundary_is_closer,
139                            estimated_power, need_boundary_deltas,
140                            &numerator, &denominator,
141                            &delta_minus, &delta_plus);
142   // We now have v = (numerator / denominator) * 10^estimated_power.
143   FixupMultiply10(estimated_power, is_even, decimal_point,
144                   &numerator, &denominator,
145                   &delta_minus, &delta_plus);
146   // We now have v = (numerator / denominator) * 10^(decimal_point-1), and
147   //  1 <= (numerator + delta_plus) / denominator < 10
148   switch (mode) {
149     case BIGNUM_DTOA_SHORTEST:
150     case BIGNUM_DTOA_SHORTEST_SINGLE:
151       GenerateShortestDigits(&numerator, &denominator,
152                              &delta_minus, &delta_plus,
153                              is_even, buffer, length);
154       break;
155     case BIGNUM_DTOA_FIXED:
156       BignumToFixed(requested_digits, decimal_point,
157                     &numerator, &denominator,
158                     buffer, length);
159       break;
160     case BIGNUM_DTOA_PRECISION:
161       GenerateCountedDigits(requested_digits, decimal_point,
162                             &numerator, &denominator,
163                             buffer, length);
164       break;
165     default:
166       DOUBLE_CONVERSION_UNREACHABLE();
167   }
168   buffer[*length] = '\0';
169 }
170 
171 
172 // The procedure starts generating digits from the left to the right and stops
173 // when the generated digits yield the shortest decimal representation of v. A
174 // decimal representation of v is a number lying closer to v than to any other
175 // double, so it converts to v when read.
176 //
177 // This is true if d, the decimal representation, is between m- and m+, the
178 // upper and lower boundaries. d must be strictly between them if !is_even.
179 //           m- := (numerator - delta_minus) / denominator
180 //           m+ := (numerator + delta_plus) / denominator
181 //
182 // Precondition: 0 <= (numerator+delta_plus) / denominator < 10.
183 //   If 1 <= (numerator+delta_plus) / denominator < 10 then no leading 0 digit
184 //   will be produced. This should be the standard precondition.
GenerateShortestDigits(Bignum * numerator,Bignum * denominator,Bignum * delta_minus,Bignum * delta_plus,bool is_even,Vector<char> buffer,int * length)185 static void GenerateShortestDigits(Bignum* numerator, Bignum* denominator,
186                                    Bignum* delta_minus, Bignum* delta_plus,
187                                    bool is_even,
188                                    Vector<char> buffer, int* length) {
189   // Small optimization: if delta_minus and delta_plus are the same just reuse
190   // one of the two bignums.
191   if (Bignum::Equal(*delta_minus, *delta_plus)) {
192     delta_plus = delta_minus;
193   }
194   *length = 0;
195   for (;;) {
196     uint16_t digit;
197     digit = numerator->DivideModuloIntBignum(*denominator);
198     DOUBLE_CONVERSION_ASSERT(digit <= 9);  // digit is a uint16_t and therefore always positive.
199     // digit = numerator / denominator (integer division).
200     // numerator = numerator % denominator.
201     buffer[(*length)++] = static_cast<char>(digit + '0');
202 
203     // Can we stop already?
204     // If the remainder of the division is less than the distance to the lower
205     // boundary we can stop. In this case we simply round down (discarding the
206     // remainder).
207     // Similarly we test if we can round up (using the upper boundary).
208     bool in_delta_room_minus;
209     bool in_delta_room_plus;
210     if (is_even) {
211       in_delta_room_minus = Bignum::LessEqual(*numerator, *delta_minus);
212     } else {
213       in_delta_room_minus = Bignum::Less(*numerator, *delta_minus);
214     }
215     if (is_even) {
216       in_delta_room_plus =
217           Bignum::PlusCompare(*numerator, *delta_plus, *denominator) >= 0;
218     } else {
219       in_delta_room_plus =
220           Bignum::PlusCompare(*numerator, *delta_plus, *denominator) > 0;
221     }
222     if (!in_delta_room_minus && !in_delta_room_plus) {
223       // Prepare for next iteration.
224       numerator->Times10();
225       delta_minus->Times10();
226       // We optimized delta_plus to be equal to delta_minus (if they share the
227       // same value). So don't multiply delta_plus if they point to the same
228       // object.
229       if (delta_minus != delta_plus) {
230         delta_plus->Times10();
231       }
232     } else if (in_delta_room_minus && in_delta_room_plus) {
233       // Let's see if 2*numerator < denominator.
234       // If yes, then the next digit would be < 5 and we can round down.
235       int compare = Bignum::PlusCompare(*numerator, *numerator, *denominator);
236       if (compare < 0) {
237         // Remaining digits are less than .5. -> Round down (== do nothing).
238       } else if (compare > 0) {
239         // Remaining digits are more than .5 of denominator. -> Round up.
240         // Note that the last digit could not be a '9' as otherwise the whole
241         // loop would have stopped earlier.
242         // We still have an assert here in case the preconditions were not
243         // satisfied.
244         DOUBLE_CONVERSION_ASSERT(buffer[(*length) - 1] != '9');
245         buffer[(*length) - 1]++;
246       } else {
247         // Halfway case.
248         // TODO(floitsch): need a way to solve half-way cases.
249         //   For now let's round towards even (since this is what Gay seems to
250         //   do).
251 
252         if ((buffer[(*length) - 1] - '0') % 2 == 0) {
253           // Round down => Do nothing.
254         } else {
255           DOUBLE_CONVERSION_ASSERT(buffer[(*length) - 1] != '9');
256           buffer[(*length) - 1]++;
257         }
258       }
259       return;
260     } else if (in_delta_room_minus) {
261       // Round down (== do nothing).
262       return;
263     } else {  // in_delta_room_plus
264       // Round up.
265       // Note again that the last digit could not be '9' since this would have
266       // stopped the loop earlier.
267       // We still have an DOUBLE_CONVERSION_ASSERT here, in case the preconditions were not
268       // satisfied.
269       DOUBLE_CONVERSION_ASSERT(buffer[(*length) -1] != '9');
270       buffer[(*length) - 1]++;
271       return;
272     }
273   }
274 }
275 
276 
277 // Let v = numerator / denominator < 10.
278 // Then we generate 'count' digits of d = x.xxxxx... (without the decimal point)
279 // from left to right. Once 'count' digits have been produced we decide wether
280 // to round up or down. Remainders of exactly .5 round upwards. Numbers such
281 // as 9.999999 propagate a carry all the way, and change the
282 // exponent (decimal_point), when rounding upwards.
GenerateCountedDigits(int count,int * decimal_point,Bignum * numerator,Bignum * denominator,Vector<char> buffer,int * length)283 static void GenerateCountedDigits(int count, int* decimal_point,
284                                   Bignum* numerator, Bignum* denominator,
285                                   Vector<char> buffer, int* length) {
286   DOUBLE_CONVERSION_ASSERT(count >= 0);
287   for (int i = 0; i < count - 1; ++i) {
288     uint16_t digit;
289     digit = numerator->DivideModuloIntBignum(*denominator);
290     DOUBLE_CONVERSION_ASSERT(digit <= 9);  // digit is a uint16_t and therefore always positive.
291     // digit = numerator / denominator (integer division).
292     // numerator = numerator % denominator.
293     buffer[i] = static_cast<char>(digit + '0');
294     // Prepare for next iteration.
295     numerator->Times10();
296   }
297   // Generate the last digit.
298   uint16_t digit;
299   digit = numerator->DivideModuloIntBignum(*denominator);
300   if (Bignum::PlusCompare(*numerator, *numerator, *denominator) >= 0) {
301     digit++;
302   }
303   DOUBLE_CONVERSION_ASSERT(digit <= 10);
304   buffer[count - 1] = static_cast<char>(digit + '0');
305   // Correct bad digits (in case we had a sequence of '9's). Propagate the
306   // carry until we hat a non-'9' or til we reach the first digit.
307   for (int i = count - 1; i > 0; --i) {
308     if (buffer[i] != '0' + 10) break;
309     buffer[i] = '0';
310     buffer[i - 1]++;
311   }
312   if (buffer[0] == '0' + 10) {
313     // Propagate a carry past the top place.
314     buffer[0] = '1';
315     (*decimal_point)++;
316   }
317   *length = count;
318 }
319 
320 
321 // Generates 'requested_digits' after the decimal point. It might omit
322 // trailing '0's. If the input number is too small then no digits at all are
323 // generated (ex.: 2 fixed digits for 0.00001).
324 //
325 // Input verifies:  1 <= (numerator + delta) / denominator < 10.
BignumToFixed(int requested_digits,int * decimal_point,Bignum * numerator,Bignum * denominator,Vector<char> buffer,int * length)326 static void BignumToFixed(int requested_digits, int* decimal_point,
327                           Bignum* numerator, Bignum* denominator,
328                           Vector<char> buffer, int* length) {
329   // Note that we have to look at more than just the requested_digits, since
330   // a number could be rounded up. Example: v=0.5 with requested_digits=0.
331   // Even though the power of v equals 0 we can't just stop here.
332   if (-(*decimal_point) > requested_digits) {
333     // The number is definitively too small.
334     // Ex: 0.001 with requested_digits == 1.
335     // Set decimal-point to -requested_digits. This is what Gay does.
336     // Note that it should not have any effect anyways since the string is
337     // empty.
338     *decimal_point = -requested_digits;
339     *length = 0;
340     return;
341   } else if (-(*decimal_point) == requested_digits) {
342     // We only need to verify if the number rounds down or up.
343     // Ex: 0.04 and 0.06 with requested_digits == 1.
344     DOUBLE_CONVERSION_ASSERT(*decimal_point == -requested_digits);
345     // Initially the fraction lies in range (1, 10]. Multiply the denominator
346     // by 10 so that we can compare more easily.
347     denominator->Times10();
348     if (Bignum::PlusCompare(*numerator, *numerator, *denominator) >= 0) {
349       // If the fraction is >= 0.5 then we have to include the rounded
350       // digit.
351       buffer[0] = '1';
352       *length = 1;
353       (*decimal_point)++;
354     } else {
355       // Note that we caught most of similar cases earlier.
356       *length = 0;
357     }
358     return;
359   } else {
360     // The requested digits correspond to the digits after the point.
361     // The variable 'needed_digits' includes the digits before the point.
362     int needed_digits = (*decimal_point) + requested_digits;
363     GenerateCountedDigits(needed_digits, decimal_point,
364                           numerator, denominator,
365                           buffer, length);
366   }
367 }
368 
369 
370 // Returns an estimation of k such that 10^(k-1) <= v < 10^k where
371 // v = f * 2^exponent and 2^52 <= f < 2^53.
372 // v is hence a normalized double with the given exponent. The output is an
373 // approximation for the exponent of the decimal approximation .digits * 10^k.
374 //
375 // The result might undershoot by 1 in which case 10^k <= v < 10^k+1.
376 // Note: this property holds for v's upper boundary m+ too.
377 //    10^k <= m+ < 10^k+1.
378 //   (see explanation below).
379 //
380 // Examples:
381 //  EstimatePower(0)   => 16
382 //  EstimatePower(-52) => 0
383 //
384 // Note: e >= 0 => EstimatedPower(e) > 0. No similar claim can be made for e<0.
EstimatePower(int exponent)385 static int EstimatePower(int exponent) {
386   // This function estimates log10 of v where v = f*2^e (with e == exponent).
387   // Note that 10^floor(log10(v)) <= v, but v <= 10^ceil(log10(v)).
388   // Note that f is bounded by its container size. Let p = 53 (the double's
389   // significand size). Then 2^(p-1) <= f < 2^p.
390   //
391   // Given that log10(v) == log2(v)/log2(10) and e+(len(f)-1) is quite close
392   // to log2(v) the function is simplified to (e+(len(f)-1)/log2(10)).
393   // The computed number undershoots by less than 0.631 (when we compute log3
394   // and not log10).
395   //
396   // Optimization: since we only need an approximated result this computation
397   // can be performed on 64 bit integers. On x86/x64 architecture the speedup is
398   // not really measurable, though.
399   //
400   // Since we want to avoid overshooting we decrement by 1e10 so that
401   // floating-point imprecisions don't affect us.
402   //
403   // Explanation for v's boundary m+: the computation takes advantage of
404   // the fact that 2^(p-1) <= f < 2^p. Boundaries still satisfy this requirement
405   // (even for denormals where the delta can be much more important).
406 
407   const double k1Log10 = 0.30102999566398114;  // 1/lg(10)
408 
409   // For doubles len(f) == 53 (don't forget the hidden bit).
410   const int kSignificandSize = Double::kSignificandSize;
411   double estimate = ceil((exponent + kSignificandSize - 1) * k1Log10 - 1e-10);
412   return static_cast<int>(estimate);
413 }
414 
415 
416 // See comments for InitialScaledStartValues.
InitialScaledStartValuesPositiveExponent(uint64_t significand,int exponent,int estimated_power,bool need_boundary_deltas,Bignum * numerator,Bignum * denominator,Bignum * delta_minus,Bignum * delta_plus)417 static void InitialScaledStartValuesPositiveExponent(
418     uint64_t significand, int exponent,
419     int estimated_power, bool need_boundary_deltas,
420     Bignum* numerator, Bignum* denominator,
421     Bignum* delta_minus, Bignum* delta_plus) {
422   // A positive exponent implies a positive power.
423   DOUBLE_CONVERSION_ASSERT(estimated_power >= 0);
424   // Since the estimated_power is positive we simply multiply the denominator
425   // by 10^estimated_power.
426 
427   // numerator = v.
428   numerator->AssignUInt64(significand);
429   numerator->ShiftLeft(exponent);
430   // denominator = 10^estimated_power.
431   denominator->AssignPowerUInt16(10, estimated_power);
432 
433   if (need_boundary_deltas) {
434     // Introduce a common denominator so that the deltas to the boundaries are
435     // integers.
436     denominator->ShiftLeft(1);
437     numerator->ShiftLeft(1);
438     // Let v = f * 2^e, then m+ - v = 1/2 * 2^e; With the common
439     // denominator (of 2) delta_plus equals 2^e.
440     delta_plus->AssignUInt16(1);
441     delta_plus->ShiftLeft(exponent);
442     // Same for delta_minus. The adjustments if f == 2^p-1 are done later.
443     delta_minus->AssignUInt16(1);
444     delta_minus->ShiftLeft(exponent);
445   }
446 }
447 
448 
449 // See comments for InitialScaledStartValues
InitialScaledStartValuesNegativeExponentPositivePower(uint64_t significand,int exponent,int estimated_power,bool need_boundary_deltas,Bignum * numerator,Bignum * denominator,Bignum * delta_minus,Bignum * delta_plus)450 static void InitialScaledStartValuesNegativeExponentPositivePower(
451     uint64_t significand, int exponent,
452     int estimated_power, bool need_boundary_deltas,
453     Bignum* numerator, Bignum* denominator,
454     Bignum* delta_minus, Bignum* delta_plus) {
455   // v = f * 2^e with e < 0, and with estimated_power >= 0.
456   // This means that e is close to 0 (have a look at how estimated_power is
457   // computed).
458 
459   // numerator = significand
460   //  since v = significand * 2^exponent this is equivalent to
461   //  numerator = v * / 2^-exponent
462   numerator->AssignUInt64(significand);
463   // denominator = 10^estimated_power * 2^-exponent (with exponent < 0)
464   denominator->AssignPowerUInt16(10, estimated_power);
465   denominator->ShiftLeft(-exponent);
466 
467   if (need_boundary_deltas) {
468     // Introduce a common denominator so that the deltas to the boundaries are
469     // integers.
470     denominator->ShiftLeft(1);
471     numerator->ShiftLeft(1);
472     // Let v = f * 2^e, then m+ - v = 1/2 * 2^e; With the common
473     // denominator (of 2) delta_plus equals 2^e.
474     // Given that the denominator already includes v's exponent the distance
475     // to the boundaries is simply 1.
476     delta_plus->AssignUInt16(1);
477     // Same for delta_minus. The adjustments if f == 2^p-1 are done later.
478     delta_minus->AssignUInt16(1);
479   }
480 }
481 
482 
483 // See comments for InitialScaledStartValues
InitialScaledStartValuesNegativeExponentNegativePower(uint64_t significand,int exponent,int estimated_power,bool need_boundary_deltas,Bignum * numerator,Bignum * denominator,Bignum * delta_minus,Bignum * delta_plus)484 static void InitialScaledStartValuesNegativeExponentNegativePower(
485     uint64_t significand, int exponent,
486     int estimated_power, bool need_boundary_deltas,
487     Bignum* numerator, Bignum* denominator,
488     Bignum* delta_minus, Bignum* delta_plus) {
489   // Instead of multiplying the denominator with 10^estimated_power we
490   // multiply all values (numerator and deltas) by 10^-estimated_power.
491 
492   // Use numerator as temporary container for power_ten.
493   Bignum* power_ten = numerator;
494   power_ten->AssignPowerUInt16(10, -estimated_power);
495 
496   if (need_boundary_deltas) {
497     // Since power_ten == numerator we must make a copy of 10^estimated_power
498     // before we complete the computation of the numerator.
499     // delta_plus = delta_minus = 10^estimated_power
500     delta_plus->AssignBignum(*power_ten);
501     delta_minus->AssignBignum(*power_ten);
502   }
503 
504   // numerator = significand * 2 * 10^-estimated_power
505   //  since v = significand * 2^exponent this is equivalent to
506   // numerator = v * 10^-estimated_power * 2 * 2^-exponent.
507   // Remember: numerator has been abused as power_ten. So no need to assign it
508   //  to itself.
509   DOUBLE_CONVERSION_ASSERT(numerator == power_ten);
510   numerator->MultiplyByUInt64(significand);
511 
512   // denominator = 2 * 2^-exponent with exponent < 0.
513   denominator->AssignUInt16(1);
514   denominator->ShiftLeft(-exponent);
515 
516   if (need_boundary_deltas) {
517     // Introduce a common denominator so that the deltas to the boundaries are
518     // integers.
519     numerator->ShiftLeft(1);
520     denominator->ShiftLeft(1);
521     // With this shift the boundaries have their correct value, since
522     // delta_plus = 10^-estimated_power, and
523     // delta_minus = 10^-estimated_power.
524     // These assignments have been done earlier.
525     // The adjustments if f == 2^p-1 (lower boundary is closer) are done later.
526   }
527 }
528 
529 
530 // Let v = significand * 2^exponent.
531 // Computes v / 10^estimated_power exactly, as a ratio of two bignums, numerator
532 // and denominator. The functions GenerateShortestDigits and
533 // GenerateCountedDigits will then convert this ratio to its decimal
534 // representation d, with the required accuracy.
535 // Then d * 10^estimated_power is the representation of v.
536 // (Note: the fraction and the estimated_power might get adjusted before
537 // generating the decimal representation.)
538 //
539 // The initial start values consist of:
540 //  - a scaled numerator: s.t. numerator/denominator == v / 10^estimated_power.
541 //  - a scaled (common) denominator.
542 //  optionally (used by GenerateShortestDigits to decide if it has the shortest
543 //  decimal converting back to v):
544 //  - v - m-: the distance to the lower boundary.
545 //  - m+ - v: the distance to the upper boundary.
546 //
547 // v, m+, m-, and therefore v - m- and m+ - v all share the same denominator.
548 //
549 // Let ep == estimated_power, then the returned values will satisfy:
550 //  v / 10^ep = numerator / denominator.
551 //  v's boundaries m- and m+:
552 //    m- / 10^ep == v / 10^ep - delta_minus / denominator
553 //    m+ / 10^ep == v / 10^ep + delta_plus / denominator
554 //  Or in other words:
555 //    m- == v - delta_minus * 10^ep / denominator;
556 //    m+ == v + delta_plus * 10^ep / denominator;
557 //
558 // Since 10^(k-1) <= v < 10^k    (with k == estimated_power)
559 //  or       10^k <= v < 10^(k+1)
560 //  we then have 0.1 <= numerator/denominator < 1
561 //           or    1 <= numerator/denominator < 10
562 //
563 // It is then easy to kickstart the digit-generation routine.
564 //
565 // The boundary-deltas are only filled if the mode equals BIGNUM_DTOA_SHORTEST
566 // or BIGNUM_DTOA_SHORTEST_SINGLE.
567 
InitialScaledStartValues(uint64_t significand,int exponent,bool lower_boundary_is_closer,int estimated_power,bool need_boundary_deltas,Bignum * numerator,Bignum * denominator,Bignum * delta_minus,Bignum * delta_plus)568 static void InitialScaledStartValues(uint64_t significand,
569                                      int exponent,
570                                      bool lower_boundary_is_closer,
571                                      int estimated_power,
572                                      bool need_boundary_deltas,
573                                      Bignum* numerator,
574                                      Bignum* denominator,
575                                      Bignum* delta_minus,
576                                      Bignum* delta_plus) {
577   if (exponent >= 0) {
578     InitialScaledStartValuesPositiveExponent(
579         significand, exponent, estimated_power, need_boundary_deltas,
580         numerator, denominator, delta_minus, delta_plus);
581   } else if (estimated_power >= 0) {
582     InitialScaledStartValuesNegativeExponentPositivePower(
583         significand, exponent, estimated_power, need_boundary_deltas,
584         numerator, denominator, delta_minus, delta_plus);
585   } else {
586     InitialScaledStartValuesNegativeExponentNegativePower(
587         significand, exponent, estimated_power, need_boundary_deltas,
588         numerator, denominator, delta_minus, delta_plus);
589   }
590 
591   if (need_boundary_deltas && lower_boundary_is_closer) {
592     // The lower boundary is closer at half the distance of "normal" numbers.
593     // Increase the common denominator and adapt all but the delta_minus.
594     denominator->ShiftLeft(1);  // *2
595     numerator->ShiftLeft(1);    // *2
596     delta_plus->ShiftLeft(1);   // *2
597   }
598 }
599 
600 
601 // This routine multiplies numerator/denominator so that its values lies in the
602 // range 1-10. That is after a call to this function we have:
603 //    1 <= (numerator + delta_plus) /denominator < 10.
604 // Let numerator the input before modification and numerator' the argument
605 // after modification, then the output-parameter decimal_point is such that
606 //  numerator / denominator * 10^estimated_power ==
607 //    numerator' / denominator' * 10^(decimal_point - 1)
608 // In some cases estimated_power was too low, and this is already the case. We
609 // then simply adjust the power so that 10^(k-1) <= v < 10^k (with k ==
610 // estimated_power) but do not touch the numerator or denominator.
611 // Otherwise the routine multiplies the numerator and the deltas by 10.
FixupMultiply10(int estimated_power,bool is_even,int * decimal_point,Bignum * numerator,Bignum * denominator,Bignum * delta_minus,Bignum * delta_plus)612 static void FixupMultiply10(int estimated_power, bool is_even,
613                             int* decimal_point,
614                             Bignum* numerator, Bignum* denominator,
615                             Bignum* delta_minus, Bignum* delta_plus) {
616   bool in_range;
617   if (is_even) {
618     // For IEEE doubles half-way cases (in decimal system numbers ending with 5)
619     // are rounded to the closest floating-point number with even significand.
620     in_range = Bignum::PlusCompare(*numerator, *delta_plus, *denominator) >= 0;
621   } else {
622     in_range = Bignum::PlusCompare(*numerator, *delta_plus, *denominator) > 0;
623   }
624   if (in_range) {
625     // Since numerator + delta_plus >= denominator we already have
626     // 1 <= numerator/denominator < 10. Simply update the estimated_power.
627     *decimal_point = estimated_power + 1;
628   } else {
629     *decimal_point = estimated_power;
630     numerator->Times10();
631     if (Bignum::Equal(*delta_minus, *delta_plus)) {
632       delta_minus->Times10();
633       delta_plus->AssignBignum(*delta_minus);
634     } else {
635       delta_minus->Times10();
636       delta_plus->Times10();
637     }
638   }
639 }
640 
641 }  // namespace double_conversion
642