1// Copyright 2012 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 <stdarg.h>
6#include <cmath>
7
8#include "src/v8.h"
9
10#include "src/bignum.h"
11#include "src/cached-powers.h"
12#include "src/double.h"
13#include "src/globals.h"
14#include "src/strtod.h"
15#include "src/utils.h"
16
17namespace v8 {
18namespace internal {
19
20// 2^53 = 9007199254740992.
21// Any integer with at most 15 decimal digits will hence fit into a double
22// (which has a 53bit significand) without loss of precision.
23static const int kMaxExactDoubleIntegerDecimalDigits = 15;
24// 2^64 = 18446744073709551616 > 10^19
25static const int kMaxUint64DecimalDigits = 19;
26
27// Max double: 1.7976931348623157 x 10^308
28// Min non-zero double: 4.9406564584124654 x 10^-324
29// Any x >= 10^309 is interpreted as +infinity.
30// Any x <= 10^-324 is interpreted as 0.
31// Note that 2.5e-324 (despite being smaller than the min double) will be read
32// as non-zero (equal to the min non-zero double).
33static const int kMaxDecimalPower = 309;
34static const int kMinDecimalPower = -324;
35
36// 2^64 = 18446744073709551616
37static const uint64_t kMaxUint64 = V8_2PART_UINT64_C(0xFFFFFFFF, FFFFFFFF);
38
39
40static const double exact_powers_of_ten[] = {
41  1.0,  // 10^0
42  10.0,
43  100.0,
44  1000.0,
45  10000.0,
46  100000.0,
47  1000000.0,
48  10000000.0,
49  100000000.0,
50  1000000000.0,
51  10000000000.0,  // 10^10
52  100000000000.0,
53  1000000000000.0,
54  10000000000000.0,
55  100000000000000.0,
56  1000000000000000.0,
57  10000000000000000.0,
58  100000000000000000.0,
59  1000000000000000000.0,
60  10000000000000000000.0,
61  100000000000000000000.0,  // 10^20
62  1000000000000000000000.0,
63  // 10^22 = 0x21e19e0c9bab2400000 = 0x878678326eac9 * 2^22
64  10000000000000000000000.0
65};
66static const int kExactPowersOfTenSize = arraysize(exact_powers_of_ten);
67
68// Maximum number of significant digits in the decimal representation.
69// In fact the value is 772 (see conversions.cc), but to give us some margin
70// we round up to 780.
71static const int kMaxSignificantDecimalDigits = 780;
72
73static Vector<const char> TrimLeadingZeros(Vector<const char> buffer) {
74  for (int i = 0; i < buffer.length(); i++) {
75    if (buffer[i] != '0') {
76      return buffer.SubVector(i, buffer.length());
77    }
78  }
79  return Vector<const char>(buffer.start(), 0);
80}
81
82
83static Vector<const char> TrimTrailingZeros(Vector<const char> buffer) {
84  for (int i = buffer.length() - 1; i >= 0; --i) {
85    if (buffer[i] != '0') {
86      return buffer.SubVector(0, i + 1);
87    }
88  }
89  return Vector<const char>(buffer.start(), 0);
90}
91
92
93static void TrimToMaxSignificantDigits(Vector<const char> buffer,
94                                       int exponent,
95                                       char* significant_buffer,
96                                       int* significant_exponent) {
97  for (int i = 0; i < kMaxSignificantDecimalDigits - 1; ++i) {
98    significant_buffer[i] = buffer[i];
99  }
100  // The input buffer has been trimmed. Therefore the last digit must be
101  // different from '0'.
102  DCHECK(buffer[buffer.length() - 1] != '0');
103  // Set the last digit to be non-zero. This is sufficient to guarantee
104  // correct rounding.
105  significant_buffer[kMaxSignificantDecimalDigits - 1] = '1';
106  *significant_exponent =
107      exponent + (buffer.length() - kMaxSignificantDecimalDigits);
108}
109
110
111// Reads digits from the buffer and converts them to a uint64.
112// Reads in as many digits as fit into a uint64.
113// When the string starts with "1844674407370955161" no further digit is read.
114// Since 2^64 = 18446744073709551616 it would still be possible read another
115// digit if it was less or equal than 6, but this would complicate the code.
116static uint64_t ReadUint64(Vector<const char> buffer,
117                           int* number_of_read_digits) {
118  uint64_t result = 0;
119  int i = 0;
120  while (i < buffer.length() && result <= (kMaxUint64 / 10 - 1)) {
121    int digit = buffer[i++] - '0';
122    DCHECK(0 <= digit && digit <= 9);
123    result = 10 * result + digit;
124  }
125  *number_of_read_digits = i;
126  return result;
127}
128
129
130// Reads a DiyFp from the buffer.
131// The returned DiyFp is not necessarily normalized.
132// If remaining_decimals is zero then the returned DiyFp is accurate.
133// Otherwise it has been rounded and has error of at most 1/2 ulp.
134static void ReadDiyFp(Vector<const char> buffer,
135                      DiyFp* result,
136                      int* remaining_decimals) {
137  int read_digits;
138  uint64_t significand = ReadUint64(buffer, &read_digits);
139  if (buffer.length() == read_digits) {
140    *result = DiyFp(significand, 0);
141    *remaining_decimals = 0;
142  } else {
143    // Round the significand.
144    if (buffer[read_digits] >= '5') {
145      significand++;
146    }
147    // Compute the binary exponent.
148    int exponent = 0;
149    *result = DiyFp(significand, exponent);
150    *remaining_decimals = buffer.length() - read_digits;
151  }
152}
153
154
155static bool DoubleStrtod(Vector<const char> trimmed,
156                         int exponent,
157                         double* result) {
158#if (V8_TARGET_ARCH_IA32 || V8_TARGET_ARCH_X87 || defined(USE_SIMULATOR)) && \
159    !defined(_MSC_VER)
160  // On x86 the floating-point stack can be 64 or 80 bits wide. If it is
161  // 80 bits wide (as is the case on Linux) then double-rounding occurs and the
162  // result is not accurate.
163  // We know that Windows32 with MSVC, unlike with MinGW32, uses 64 bits and is
164  // therefore accurate.
165  // Note that the ARM and MIPS simulators are compiled for 32bits. They
166  // therefore exhibit the same problem.
167  return false;
168#endif
169  if (trimmed.length() <= kMaxExactDoubleIntegerDecimalDigits) {
170    int read_digits;
171    // The trimmed input fits into a double.
172    // If the 10^exponent (resp. 10^-exponent) fits into a double too then we
173    // can compute the result-double simply by multiplying (resp. dividing) the
174    // two numbers.
175    // This is possible because IEEE guarantees that floating-point operations
176    // return the best possible approximation.
177    if (exponent < 0 && -exponent < kExactPowersOfTenSize) {
178      // 10^-exponent fits into a double.
179      *result = static_cast<double>(ReadUint64(trimmed, &read_digits));
180      DCHECK(read_digits == trimmed.length());
181      *result /= exact_powers_of_ten[-exponent];
182      return true;
183    }
184    if (0 <= exponent && exponent < kExactPowersOfTenSize) {
185      // 10^exponent fits into a double.
186      *result = static_cast<double>(ReadUint64(trimmed, &read_digits));
187      DCHECK(read_digits == trimmed.length());
188      *result *= exact_powers_of_ten[exponent];
189      return true;
190    }
191    int remaining_digits =
192        kMaxExactDoubleIntegerDecimalDigits - trimmed.length();
193    if ((0 <= exponent) &&
194        (exponent - remaining_digits < kExactPowersOfTenSize)) {
195      // The trimmed string was short and we can multiply it with
196      // 10^remaining_digits. As a result the remaining exponent now fits
197      // into a double too.
198      *result = static_cast<double>(ReadUint64(trimmed, &read_digits));
199      DCHECK(read_digits == trimmed.length());
200      *result *= exact_powers_of_ten[remaining_digits];
201      *result *= exact_powers_of_ten[exponent - remaining_digits];
202      return true;
203    }
204  }
205  return false;
206}
207
208
209// Returns 10^exponent as an exact DiyFp.
210// The given exponent must be in the range [1; kDecimalExponentDistance[.
211static DiyFp AdjustmentPowerOfTen(int exponent) {
212  DCHECK(0 < exponent);
213  DCHECK(exponent < PowersOfTenCache::kDecimalExponentDistance);
214  // Simply hardcode the remaining powers for the given decimal exponent
215  // distance.
216  DCHECK(PowersOfTenCache::kDecimalExponentDistance == 8);
217  switch (exponent) {
218    case 1: return DiyFp(V8_2PART_UINT64_C(0xa0000000, 00000000), -60);
219    case 2: return DiyFp(V8_2PART_UINT64_C(0xc8000000, 00000000), -57);
220    case 3: return DiyFp(V8_2PART_UINT64_C(0xfa000000, 00000000), -54);
221    case 4: return DiyFp(V8_2PART_UINT64_C(0x9c400000, 00000000), -50);
222    case 5: return DiyFp(V8_2PART_UINT64_C(0xc3500000, 00000000), -47);
223    case 6: return DiyFp(V8_2PART_UINT64_C(0xf4240000, 00000000), -44);
224    case 7: return DiyFp(V8_2PART_UINT64_C(0x98968000, 00000000), -40);
225    default:
226      UNREACHABLE();
227      return DiyFp(0, 0);
228  }
229}
230
231
232// If the function returns true then the result is the correct double.
233// Otherwise it is either the correct double or the double that is just below
234// the correct double.
235static bool DiyFpStrtod(Vector<const char> buffer,
236                        int exponent,
237                        double* result) {
238  DiyFp input;
239  int remaining_decimals;
240  ReadDiyFp(buffer, &input, &remaining_decimals);
241  // Since we may have dropped some digits the input is not accurate.
242  // If remaining_decimals is different than 0 than the error is at most
243  // .5 ulp (unit in the last place).
244  // We don't want to deal with fractions and therefore keep a common
245  // denominator.
246  const int kDenominatorLog = 3;
247  const int kDenominator = 1 << kDenominatorLog;
248  // Move the remaining decimals into the exponent.
249  exponent += remaining_decimals;
250  int64_t error = (remaining_decimals == 0 ? 0 : kDenominator / 2);
251
252  int old_e = input.e();
253  input.Normalize();
254  error <<= old_e - input.e();
255
256  DCHECK(exponent <= PowersOfTenCache::kMaxDecimalExponent);
257  if (exponent < PowersOfTenCache::kMinDecimalExponent) {
258    *result = 0.0;
259    return true;
260  }
261  DiyFp cached_power;
262  int cached_decimal_exponent;
263  PowersOfTenCache::GetCachedPowerForDecimalExponent(exponent,
264                                                     &cached_power,
265                                                     &cached_decimal_exponent);
266
267  if (cached_decimal_exponent != exponent) {
268    int adjustment_exponent = exponent - cached_decimal_exponent;
269    DiyFp adjustment_power = AdjustmentPowerOfTen(adjustment_exponent);
270    input.Multiply(adjustment_power);
271    if (kMaxUint64DecimalDigits - buffer.length() >= adjustment_exponent) {
272      // The product of input with the adjustment power fits into a 64 bit
273      // integer.
274      DCHECK(DiyFp::kSignificandSize == 64);
275    } else {
276      // The adjustment power is exact. There is hence only an error of 0.5.
277      error += kDenominator / 2;
278    }
279  }
280
281  input.Multiply(cached_power);
282  // The error introduced by a multiplication of a*b equals
283  //   error_a + error_b + error_a*error_b/2^64 + 0.5
284  // Substituting a with 'input' and b with 'cached_power' we have
285  //   error_b = 0.5  (all cached powers have an error of less than 0.5 ulp),
286  //   error_ab = 0 or 1 / kDenominator > error_a*error_b/ 2^64
287  int error_b = kDenominator / 2;
288  int error_ab = (error == 0 ? 0 : 1);  // We round up to 1.
289  int fixed_error = kDenominator / 2;
290  error += error_b + error_ab + fixed_error;
291
292  old_e = input.e();
293  input.Normalize();
294  error <<= old_e - input.e();
295
296  // See if the double's significand changes if we add/subtract the error.
297  int order_of_magnitude = DiyFp::kSignificandSize + input.e();
298  int effective_significand_size =
299      Double::SignificandSizeForOrderOfMagnitude(order_of_magnitude);
300  int precision_digits_count =
301      DiyFp::kSignificandSize - effective_significand_size;
302  if (precision_digits_count + kDenominatorLog >= DiyFp::kSignificandSize) {
303    // This can only happen for very small denormals. In this case the
304    // half-way multiplied by the denominator exceeds the range of an uint64.
305    // Simply shift everything to the right.
306    int shift_amount = (precision_digits_count + kDenominatorLog) -
307        DiyFp::kSignificandSize + 1;
308    input.set_f(input.f() >> shift_amount);
309    input.set_e(input.e() + shift_amount);
310    // We add 1 for the lost precision of error, and kDenominator for
311    // the lost precision of input.f().
312    error = (error >> shift_amount) + 1 + kDenominator;
313    precision_digits_count -= shift_amount;
314  }
315  // We use uint64_ts now. This only works if the DiyFp uses uint64_ts too.
316  DCHECK(DiyFp::kSignificandSize == 64);
317  DCHECK(precision_digits_count < 64);
318  uint64_t one64 = 1;
319  uint64_t precision_bits_mask = (one64 << precision_digits_count) - 1;
320  uint64_t precision_bits = input.f() & precision_bits_mask;
321  uint64_t half_way = one64 << (precision_digits_count - 1);
322  precision_bits *= kDenominator;
323  half_way *= kDenominator;
324  DiyFp rounded_input(input.f() >> precision_digits_count,
325                      input.e() + precision_digits_count);
326  if (precision_bits >= half_way + error) {
327    rounded_input.set_f(rounded_input.f() + 1);
328  }
329  // If the last_bits are too close to the half-way case than we are too
330  // inaccurate and round down. In this case we return false so that we can
331  // fall back to a more precise algorithm.
332
333  *result = Double(rounded_input).value();
334  if (half_way - error < precision_bits && precision_bits < half_way + error) {
335    // Too imprecise. The caller will have to fall back to a slower version.
336    // However the returned number is guaranteed to be either the correct
337    // double, or the next-lower double.
338    return false;
339  } else {
340    return true;
341  }
342}
343
344
345// Returns the correct double for the buffer*10^exponent.
346// The variable guess should be a close guess that is either the correct double
347// or its lower neighbor (the nearest double less than the correct one).
348// Preconditions:
349//   buffer.length() + exponent <= kMaxDecimalPower + 1
350//   buffer.length() + exponent > kMinDecimalPower
351//   buffer.length() <= kMaxDecimalSignificantDigits
352static double BignumStrtod(Vector<const char> buffer,
353                           int exponent,
354                           double guess) {
355  if (guess == V8_INFINITY) {
356    return guess;
357  }
358
359  DiyFp upper_boundary = Double(guess).UpperBoundary();
360
361  DCHECK(buffer.length() + exponent <= kMaxDecimalPower + 1);
362  DCHECK(buffer.length() + exponent > kMinDecimalPower);
363  DCHECK(buffer.length() <= kMaxSignificantDecimalDigits);
364  // Make sure that the Bignum will be able to hold all our numbers.
365  // Our Bignum implementation has a separate field for exponents. Shifts will
366  // consume at most one bigit (< 64 bits).
367  // ln(10) == 3.3219...
368  DCHECK(((kMaxDecimalPower + 1) * 333 / 100) < Bignum::kMaxSignificantBits);
369  Bignum input;
370  Bignum boundary;
371  input.AssignDecimalString(buffer);
372  boundary.AssignUInt64(upper_boundary.f());
373  if (exponent >= 0) {
374    input.MultiplyByPowerOfTen(exponent);
375  } else {
376    boundary.MultiplyByPowerOfTen(-exponent);
377  }
378  if (upper_boundary.e() > 0) {
379    boundary.ShiftLeft(upper_boundary.e());
380  } else {
381    input.ShiftLeft(-upper_boundary.e());
382  }
383  int comparison = Bignum::Compare(input, boundary);
384  if (comparison < 0) {
385    return guess;
386  } else if (comparison > 0) {
387    return Double(guess).NextDouble();
388  } else if ((Double(guess).Significand() & 1) == 0) {
389    // Round towards even.
390    return guess;
391  } else {
392    return Double(guess).NextDouble();
393  }
394}
395
396
397double Strtod(Vector<const char> buffer, int exponent) {
398  Vector<const char> left_trimmed = TrimLeadingZeros(buffer);
399  Vector<const char> trimmed = TrimTrailingZeros(left_trimmed);
400  exponent += left_trimmed.length() - trimmed.length();
401  if (trimmed.length() == 0) return 0.0;
402  if (trimmed.length() > kMaxSignificantDecimalDigits) {
403    char significant_buffer[kMaxSignificantDecimalDigits];
404    int significant_exponent;
405    TrimToMaxSignificantDigits(trimmed, exponent,
406                               significant_buffer, &significant_exponent);
407    return Strtod(Vector<const char>(significant_buffer,
408                                     kMaxSignificantDecimalDigits),
409                  significant_exponent);
410  }
411  if (exponent + trimmed.length() - 1 >= kMaxDecimalPower) return V8_INFINITY;
412  if (exponent + trimmed.length() <= kMinDecimalPower) return 0.0;
413
414  double guess;
415  if (DoubleStrtod(trimmed, exponent, &guess) ||
416      DiyFpStrtod(trimmed, exponent, &guess)) {
417    return guess;
418  }
419  return BignumStrtod(trimmed, exponent, guess);
420}
421
422} }  // namespace v8::internal
423