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fixed-dtoa.cc
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27 
28 #include <cmath>
29 
30 #include "../include/v8stdint.h"
31 #include "checks.h"
32 #include "utils.h"
33 
34 #include "double.h"
35 #include "fixed-dtoa.h"
36 
37 namespace v8 {
38 namespace internal {
39 
40 // Represents a 128bit type. This class should be replaced by a native type on
41 // platforms that support 128bit integers.
42 class UInt128 {
43  public:
44  UInt128() : high_bits_(0), low_bits_(0) { }
45  UInt128(uint64_t high, uint64_t low) : high_bits_(high), low_bits_(low) { }
46 
47  void Multiply(uint32_t multiplicand) {
48  uint64_t accumulator;
49 
50  accumulator = (low_bits_ & kMask32) * multiplicand;
51  uint32_t part = static_cast<uint32_t>(accumulator & kMask32);
52  accumulator >>= 32;
53  accumulator = accumulator + (low_bits_ >> 32) * multiplicand;
54  low_bits_ = (accumulator << 32) + part;
55  accumulator >>= 32;
56  accumulator = accumulator + (high_bits_ & kMask32) * multiplicand;
57  part = static_cast<uint32_t>(accumulator & kMask32);
58  accumulator >>= 32;
59  accumulator = accumulator + (high_bits_ >> 32) * multiplicand;
60  high_bits_ = (accumulator << 32) + part;
61  ASSERT((accumulator >> 32) == 0);
62  }
63 
64  void Shift(int shift_amount) {
65  ASSERT(-64 <= shift_amount && shift_amount <= 64);
66  if (shift_amount == 0) {
67  return;
68  } else if (shift_amount == -64) {
69  high_bits_ = low_bits_;
70  low_bits_ = 0;
71  } else if (shift_amount == 64) {
72  low_bits_ = high_bits_;
73  high_bits_ = 0;
74  } else if (shift_amount <= 0) {
75  high_bits_ <<= -shift_amount;
76  high_bits_ += low_bits_ >> (64 + shift_amount);
77  low_bits_ <<= -shift_amount;
78  } else {
79  low_bits_ >>= shift_amount;
80  low_bits_ += high_bits_ << (64 - shift_amount);
81  high_bits_ >>= shift_amount;
82  }
83  }
84 
85  // Modifies *this to *this MOD (2^power).
86  // Returns *this DIV (2^power).
87  int DivModPowerOf2(int power) {
88  if (power >= 64) {
89  int result = static_cast<int>(high_bits_ >> (power - 64));
90  high_bits_ -= static_cast<uint64_t>(result) << (power - 64);
91  return result;
92  } else {
93  uint64_t part_low = low_bits_ >> power;
94  uint64_t part_high = high_bits_ << (64 - power);
95  int result = static_cast<int>(part_low + part_high);
96  high_bits_ = 0;
97  low_bits_ -= part_low << power;
98  return result;
99  }
100  }
101 
102  bool IsZero() const {
103  return high_bits_ == 0 && low_bits_ == 0;
104  }
105 
106  int BitAt(int position) {
107  if (position >= 64) {
108  return static_cast<int>(high_bits_ >> (position - 64)) & 1;
109  } else {
110  return static_cast<int>(low_bits_ >> position) & 1;
111  }
112  }
113 
114  private:
115  static const uint64_t kMask32 = 0xFFFFFFFF;
116  // Value == (high_bits_ << 64) + low_bits_
117  uint64_t high_bits_;
118  uint64_t low_bits_;
119 };
120 
121 
122 static const int kDoubleSignificandSize = 53; // Includes the hidden bit.
123 
124 
125 static void FillDigits32FixedLength(uint32_t number, int requested_length,
126  Vector<char> buffer, int* length) {
127  for (int i = requested_length - 1; i >= 0; --i) {
128  buffer[(*length) + i] = '0' + number % 10;
129  number /= 10;
130  }
131  *length += requested_length;
132 }
133 
134 
135 static void FillDigits32(uint32_t number, Vector<char> buffer, int* length) {
136  int number_length = 0;
137  // We fill the digits in reverse order and exchange them afterwards.
138  while (number != 0) {
139  int digit = number % 10;
140  number /= 10;
141  buffer[(*length) + number_length] = '0' + digit;
142  number_length++;
143  }
144  // Exchange the digits.
145  int i = *length;
146  int j = *length + number_length - 1;
147  while (i < j) {
148  char tmp = buffer[i];
149  buffer[i] = buffer[j];
150  buffer[j] = tmp;
151  i++;
152  j--;
153  }
154  *length += number_length;
155 }
156 
157 
158 static void FillDigits64FixedLength(uint64_t number, int requested_length,
159  Vector<char> buffer, int* length) {
160  const uint32_t kTen7 = 10000000;
161  // For efficiency cut the number into 3 uint32_t parts, and print those.
162  uint32_t part2 = static_cast<uint32_t>(number % kTen7);
163  number /= kTen7;
164  uint32_t part1 = static_cast<uint32_t>(number % kTen7);
165  uint32_t part0 = static_cast<uint32_t>(number / kTen7);
166 
167  FillDigits32FixedLength(part0, 3, buffer, length);
168  FillDigits32FixedLength(part1, 7, buffer, length);
169  FillDigits32FixedLength(part2, 7, buffer, length);
170 }
171 
172 
173 static void FillDigits64(uint64_t number, Vector<char> buffer, int* length) {
174  const uint32_t kTen7 = 10000000;
175  // For efficiency cut the number into 3 uint32_t parts, and print those.
176  uint32_t part2 = static_cast<uint32_t>(number % kTen7);
177  number /= kTen7;
178  uint32_t part1 = static_cast<uint32_t>(number % kTen7);
179  uint32_t part0 = static_cast<uint32_t>(number / kTen7);
180 
181  if (part0 != 0) {
182  FillDigits32(part0, buffer, length);
183  FillDigits32FixedLength(part1, 7, buffer, length);
184  FillDigits32FixedLength(part2, 7, buffer, length);
185  } else if (part1 != 0) {
186  FillDigits32(part1, buffer, length);
187  FillDigits32FixedLength(part2, 7, buffer, length);
188  } else {
189  FillDigits32(part2, buffer, length);
190  }
191 }
192 
193 
194 static void RoundUp(Vector<char> buffer, int* length, int* decimal_point) {
195  // An empty buffer represents 0.
196  if (*length == 0) {
197  buffer[0] = '1';
198  *decimal_point = 1;
199  *length = 1;
200  return;
201  }
202  // Round the last digit until we either have a digit that was not '9' or until
203  // we reached the first digit.
204  buffer[(*length) - 1]++;
205  for (int i = (*length) - 1; i > 0; --i) {
206  if (buffer[i] != '0' + 10) {
207  return;
208  }
209  buffer[i] = '0';
210  buffer[i - 1]++;
211  }
212  // If the first digit is now '0' + 10, we would need to set it to '0' and add
213  // a '1' in front. However we reach the first digit only if all following
214  // digits had been '9' before rounding up. Now all trailing digits are '0' and
215  // we simply switch the first digit to '1' and update the decimal-point
216  // (indicating that the point is now one digit to the right).
217  if (buffer[0] == '0' + 10) {
218  buffer[0] = '1';
219  (*decimal_point)++;
220  }
221 }
222 
223 
224 // The given fractionals number represents a fixed-point number with binary
225 // point at bit (-exponent).
226 // Preconditions:
227 // -128 <= exponent <= 0.
228 // 0 <= fractionals * 2^exponent < 1
229 // The buffer holds the result.
230 // The function will round its result. During the rounding-process digits not
231 // generated by this function might be updated, and the decimal-point variable
232 // might be updated. If this function generates the digits 99 and the buffer
233 // already contained "199" (thus yielding a buffer of "19999") then a
234 // rounding-up will change the contents of the buffer to "20000".
235 static void FillFractionals(uint64_t fractionals, int exponent,
236  int fractional_count, Vector<char> buffer,
237  int* length, int* decimal_point) {
238  ASSERT(-128 <= exponent && exponent <= 0);
239  // 'fractionals' is a fixed-point number, with binary point at bit
240  // (-exponent). Inside the function the non-converted remainder of fractionals
241  // is a fixed-point number, with binary point at bit 'point'.
242  if (-exponent <= 64) {
243  // One 64 bit number is sufficient.
244  ASSERT(fractionals >> 56 == 0);
245  int point = -exponent;
246  for (int i = 0; i < fractional_count; ++i) {
247  if (fractionals == 0) break;
248  // Instead of multiplying by 10 we multiply by 5 and adjust the point
249  // location. This way the fractionals variable will not overflow.
250  // Invariant at the beginning of the loop: fractionals < 2^point.
251  // Initially we have: point <= 64 and fractionals < 2^56
252  // After each iteration the point is decremented by one.
253  // Note that 5^3 = 125 < 128 = 2^7.
254  // Therefore three iterations of this loop will not overflow fractionals
255  // (even without the subtraction at the end of the loop body). At this
256  // time point will satisfy point <= 61 and therefore fractionals < 2^point
257  // and any further multiplication of fractionals by 5 will not overflow.
258  fractionals *= 5;
259  point--;
260  int digit = static_cast<int>(fractionals >> point);
261  buffer[*length] = '0' + digit;
262  (*length)++;
263  fractionals -= static_cast<uint64_t>(digit) << point;
264  }
265  // If the first bit after the point is set we have to round up.
266  if (((fractionals >> (point - 1)) & 1) == 1) {
267  RoundUp(buffer, length, decimal_point);
268  }
269  } else { // We need 128 bits.
270  ASSERT(64 < -exponent && -exponent <= 128);
271  UInt128 fractionals128 = UInt128(fractionals, 0);
272  fractionals128.Shift(-exponent - 64);
273  int point = 128;
274  for (int i = 0; i < fractional_count; ++i) {
275  if (fractionals128.IsZero()) break;
276  // As before: instead of multiplying by 10 we multiply by 5 and adjust the
277  // point location.
278  // This multiplication will not overflow for the same reasons as before.
279  fractionals128.Multiply(5);
280  point--;
281  int digit = fractionals128.DivModPowerOf2(point);
282  buffer[*length] = '0' + digit;
283  (*length)++;
284  }
285  if (fractionals128.BitAt(point - 1) == 1) {
286  RoundUp(buffer, length, decimal_point);
287  }
288  }
289 }
290 
291 
292 // Removes leading and trailing zeros.
293 // If leading zeros are removed then the decimal point position is adjusted.
294 static void TrimZeros(Vector<char> buffer, int* length, int* decimal_point) {
295  while (*length > 0 && buffer[(*length) - 1] == '0') {
296  (*length)--;
297  }
298  int first_non_zero = 0;
299  while (first_non_zero < *length && buffer[first_non_zero] == '0') {
300  first_non_zero++;
301  }
302  if (first_non_zero != 0) {
303  for (int i = first_non_zero; i < *length; ++i) {
304  buffer[i - first_non_zero] = buffer[i];
305  }
306  *length -= first_non_zero;
307  *decimal_point -= first_non_zero;
308  }
309 }
310 
311 
312 bool FastFixedDtoa(double v,
313  int fractional_count,
314  Vector<char> buffer,
315  int* length,
316  int* decimal_point) {
317  const uint32_t kMaxUInt32 = 0xFFFFFFFF;
318  uint64_t significand = Double(v).Significand();
319  int exponent = Double(v).Exponent();
320  // v = significand * 2^exponent (with significand a 53bit integer).
321  // If the exponent is larger than 20 (i.e. we may have a 73bit number) then we
322  // don't know how to compute the representation. 2^73 ~= 9.5*10^21.
323  // If necessary this limit could probably be increased, but we don't need
324  // more.
325  if (exponent > 20) return false;
326  if (fractional_count > 20) return false;
327  *length = 0;
328  // At most kDoubleSignificandSize bits of the significand are non-zero.
329  // Given a 64 bit integer we have 11 0s followed by 53 potentially non-zero
330  // bits: 0..11*..0xxx..53*..xx
331  if (exponent + kDoubleSignificandSize > 64) {
332  // The exponent must be > 11.
333  //
334  // We know that v = significand * 2^exponent.
335  // And the exponent > 11.
336  // We simplify the task by dividing v by 10^17.
337  // The quotient delivers the first digits, and the remainder fits into a 64
338  // bit number.
339  // Dividing by 10^17 is equivalent to dividing by 5^17*2^17.
340  const uint64_t kFive17 = V8_2PART_UINT64_C(0xB1, A2BC2EC5); // 5^17
341  uint64_t divisor = kFive17;
342  int divisor_power = 17;
343  uint64_t dividend = significand;
344  uint32_t quotient;
345  uint64_t remainder;
346  // Let v = f * 2^e with f == significand and e == exponent.
347  // Then need q (quotient) and r (remainder) as follows:
348  // v = q * 10^17 + r
349  // f * 2^e = q * 10^17 + r
350  // f * 2^e = q * 5^17 * 2^17 + r
351  // If e > 17 then
352  // f * 2^(e-17) = q * 5^17 + r/2^17
353  // else
354  // f = q * 5^17 * 2^(17-e) + r/2^e
355  if (exponent > divisor_power) {
356  // We only allow exponents of up to 20 and therefore (17 - e) <= 3
357  dividend <<= exponent - divisor_power;
358  quotient = static_cast<uint32_t>(dividend / divisor);
359  remainder = (dividend % divisor) << divisor_power;
360  } else {
361  divisor <<= divisor_power - exponent;
362  quotient = static_cast<uint32_t>(dividend / divisor);
363  remainder = (dividend % divisor) << exponent;
364  }
365  FillDigits32(quotient, buffer, length);
366  FillDigits64FixedLength(remainder, divisor_power, buffer, length);
367  *decimal_point = *length;
368  } else if (exponent >= 0) {
369  // 0 <= exponent <= 11
370  significand <<= exponent;
371  FillDigits64(significand, buffer, length);
372  *decimal_point = *length;
373  } else if (exponent > -kDoubleSignificandSize) {
374  // We have to cut the number.
375  uint64_t integrals = significand >> -exponent;
376  uint64_t fractionals = significand - (integrals << -exponent);
377  if (integrals > kMaxUInt32) {
378  FillDigits64(integrals, buffer, length);
379  } else {
380  FillDigits32(static_cast<uint32_t>(integrals), buffer, length);
381  }
382  *decimal_point = *length;
383  FillFractionals(fractionals, exponent, fractional_count,
384  buffer, length, decimal_point);
385  } else if (exponent < -128) {
386  // This configuration (with at most 20 digits) means that all digits must be
387  // 0.
388  ASSERT(fractional_count <= 20);
389  buffer[0] = '\0';
390  *length = 0;
391  *decimal_point = -fractional_count;
392  } else {
393  *decimal_point = 0;
394  FillFractionals(significand, exponent, fractional_count,
395  buffer, length, decimal_point);
396  }
397  TrimZeros(buffer, length, decimal_point);
398  buffer[*length] = '\0';
399  if ((*length) == 0) {
400  // The string is empty and the decimal_point thus has no importance. Mimick
401  // Gay's dtoa and and set it to -fractional_count.
402  *decimal_point = -fractional_count;
403  }
404  return true;
405 }
406 
407 } } // namespace v8::internal
uint64_t Significand() const
Definition: double.h:110
int DivModPowerOf2(int power)
Definition: fixed-dtoa.cc:87
void Multiply(uint32_t multiplicand)
Definition: fixed-dtoa.cc:47
bool IsZero() const
Definition: fixed-dtoa.cc:102
int Exponent() const
Definition: double.h:101
#define ASSERT(condition)
Definition: checks.h:329
#define V8_2PART_UINT64_C(a, b)
Definition: globals.h:226
void Shift(int shift_amount)
Definition: fixed-dtoa.cc:64
int BitAt(int position)
Definition: fixed-dtoa.cc:106
const uint32_t kMaxUInt32
Definition: globals.h:259
bool FastFixedDtoa(double v, int fractional_count, Vector< char > buffer, int *length, int *decimal_point)
Definition: fixed-dtoa.cc:312
UInt128(uint64_t high, uint64_t low)
Definition: fixed-dtoa.cc:45