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27
28#include <cmath>
29
30#include <double-conversion/fixed-dtoa.h>
31#include <double-conversion/ieee.h>
32
33namespace double_conversion {
34
35// Represents a 128bit type. This class should be replaced by a native type on
36// platforms that support 128bit integers.
37class UInt128 {
38 public:
39 UInt128() : high_bits_(0), low_bits_(0) { }
40 UInt128(uint64_t high, uint64_t low) : high_bits_(high), low_bits_(low) { }
41
42 void Multiply(uint32_t multiplicand) {
43 uint64_t accumulator;
44
45 accumulator = (low_bits_ & kMask32) * multiplicand;
46 uint32_t part = static_cast<uint32_t>(accumulator & kMask32);
47 accumulator >>= 32;
48 accumulator = accumulator + (low_bits_ >> 32) * multiplicand;
49 low_bits_ = (accumulator << 32) + part;
50 accumulator >>= 32;
51 accumulator = accumulator + (high_bits_ & kMask32) * multiplicand;
52 part = static_cast<uint32_t>(accumulator & kMask32);
53 accumulator >>= 32;
54 accumulator = accumulator + (high_bits_ >> 32) * multiplicand;
55 high_bits_ = (accumulator << 32) + part;
56 ASSERT((accumulator >> 32) == 0);
57 }
58
59 void Shift(int shift_amount) {
60 ASSERT(-64 <= shift_amount && shift_amount <= 64);
61 if (shift_amount == 0) {
62 return;
63 } else if (shift_amount == -64) {
64 high_bits_ = low_bits_;
65 low_bits_ = 0;
66 } else if (shift_amount == 64) {
67 low_bits_ = high_bits_;
68 high_bits_ = 0;
69 } else if (shift_amount <= 0) {
70 high_bits_ <<= -shift_amount;
71 high_bits_ += low_bits_ >> (64 + shift_amount);
72 low_bits_ <<= -shift_amount;
73 } else {
74 low_bits_ >>= shift_amount;
75 low_bits_ += high_bits_ << (64 - shift_amount);
76 high_bits_ >>= shift_amount;
77 }
78 }
79
80 // Modifies *this to *this MOD (2^power).
81 // Returns *this DIV (2^power).
82 int DivModPowerOf2(int power) {
83 if (power >= 64) {
84 int result = static_cast<int>(high_bits_ >> (power - 64));
85 high_bits_ -= static_cast<uint64_t>(result) << (power - 64);
86 return result;
87 } else {
88 uint64_t part_low = low_bits_ >> power;
89 uint64_t part_high = high_bits_ << (64 - power);
90 int result = static_cast<int>(part_low + part_high);
91 high_bits_ = 0;
92 low_bits_ -= part_low << power;
93 return result;
94 }
95 }
96
97 bool IsZero() const {
98 return high_bits_ == 0 && low_bits_ == 0;
99 }
100
101 int BitAt(int position) const {
102 if (position >= 64) {
103 return static_cast<int>(high_bits_ >> (position - 64)) & 1;
104 } else {
105 return static_cast<int>(low_bits_ >> position) & 1;
106 }
107 }
108
109 private:
110 static const uint64_t kMask32 = 0xFFFFFFFF;
111 // Value == (high_bits_ << 64) + low_bits_
112 uint64_t high_bits_;
113 uint64_t low_bits_;
114};
115
116
117static const int kDoubleSignificandSize = 53; // Includes the hidden bit.
118
119
120static void FillDigits32FixedLength(uint32_t number, int requested_length,
121 Vector<char> buffer, int* length) {
122 for (int i = requested_length - 1; i >= 0; --i) {
123 buffer[(*length) + i] = '0' + number % 10;
124 number /= 10;
125 }
126 *length += requested_length;
127}
128
129
130static void FillDigits32(uint32_t number, Vector<char> buffer, int* length) {
131 int number_length = 0;
132 // We fill the digits in reverse order and exchange them afterwards.
133 while (number != 0) {
134 int digit = number % 10;
135 number /= 10;
136 buffer[(*length) + number_length] = static_cast<char>('0' + digit);
137 number_length++;
138 }
139 // Exchange the digits.
140 int i = *length;
141 int j = *length + number_length - 1;
142 while (i < j) {
143 char tmp = buffer[i];
144 buffer[i] = buffer[j];
145 buffer[j] = tmp;
146 i++;
147 j--;
148 }
149 *length += number_length;
150}
151
152
153static void FillDigits64FixedLength(uint64_t number,
154 Vector<char> buffer, int* length) {
155 const uint32_t kTen7 = 10000000;
156 // For efficiency cut the number into 3 uint32_t parts, and print those.
157 uint32_t part2 = static_cast<uint32_t>(number % kTen7);
158 number /= kTen7;
159 uint32_t part1 = static_cast<uint32_t>(number % kTen7);
160 uint32_t part0 = static_cast<uint32_t>(number / kTen7);
161
162 FillDigits32FixedLength(part0, 3, buffer, length);
163 FillDigits32FixedLength(part1, 7, buffer, length);
164 FillDigits32FixedLength(part2, 7, buffer, length);
165}
166
167
168static void FillDigits64(uint64_t number, Vector<char> buffer, int* length) {
169 const uint32_t kTen7 = 10000000;
170 // For efficiency cut the number into 3 uint32_t parts, and print those.
171 uint32_t part2 = static_cast<uint32_t>(number % kTen7);
172 number /= kTen7;
173 uint32_t part1 = static_cast<uint32_t>(number % kTen7);
174 uint32_t part0 = static_cast<uint32_t>(number / kTen7);
175
176 if (part0 != 0) {
177 FillDigits32(part0, buffer, length);
178 FillDigits32FixedLength(part1, 7, buffer, length);
179 FillDigits32FixedLength(part2, 7, buffer, length);
180 } else if (part1 != 0) {
181 FillDigits32(part1, buffer, length);
182 FillDigits32FixedLength(part2, 7, buffer, length);
183 } else {
184 FillDigits32(part2, buffer, length);
185 }
186}
187
188
189static void RoundUp(Vector<char> buffer, int* length, int* decimal_point) {
190 // An empty buffer represents 0.
191 if (*length == 0) {
192 buffer[0] = '1';
193 *decimal_point = 1;
194 *length = 1;
195 return;
196 }
197 // Round the last digit until we either have a digit that was not '9' or until
198 // we reached the first digit.
199 buffer[(*length) - 1]++;
200 for (int i = (*length) - 1; i > 0; --i) {
201 if (buffer[i] != '0' + 10) {
202 return;
203 }
204 buffer[i] = '0';
205 buffer[i - 1]++;
206 }
207 // If the first digit is now '0' + 10, we would need to set it to '0' and add
208 // a '1' in front. However we reach the first digit only if all following
209 // digits had been '9' before rounding up. Now all trailing digits are '0' and
210 // we simply switch the first digit to '1' and update the decimal-point
211 // (indicating that the point is now one digit to the right).
212 if (buffer[0] == '0' + 10) {
213 buffer[0] = '1';
214 (*decimal_point)++;
215 }
216}
217
218
219// The given fractionals number represents a fixed-point number with binary
220// point at bit (-exponent).
221// Preconditions:
222// -128 <= exponent <= 0.
223// 0 <= fractionals * 2^exponent < 1
224// The buffer holds the result.
225// The function will round its result. During the rounding-process digits not
226// generated by this function might be updated, and the decimal-point variable
227// might be updated. If this function generates the digits 99 and the buffer
228// already contained "199" (thus yielding a buffer of "19999") then a
229// rounding-up will change the contents of the buffer to "20000".
230static void FillFractionals(uint64_t fractionals, int exponent,
231 int fractional_count, Vector<char> buffer,
232 int* length, int* decimal_point) {
233 ASSERT(-128 <= exponent && exponent <= 0);
234 // 'fractionals' is a fixed-point number, with binary point at bit
235 // (-exponent). Inside the function the non-converted remainder of fractionals
236 // is a fixed-point number, with binary point at bit 'point'.
237 if (-exponent <= 64) {
238 // One 64 bit number is sufficient.
239 ASSERT(fractionals >> 56 == 0);
240 int point = -exponent;
241 for (int i = 0; i < fractional_count; ++i) {
242 if (fractionals == 0) break;
243 // Instead of multiplying by 10 we multiply by 5 and adjust the point
244 // location. This way the fractionals variable will not overflow.
245 // Invariant at the beginning of the loop: fractionals < 2^point.
246 // Initially we have: point <= 64 and fractionals < 2^56
247 // After each iteration the point is decremented by one.
248 // Note that 5^3 = 125 < 128 = 2^7.
249 // Therefore three iterations of this loop will not overflow fractionals
250 // (even without the subtraction at the end of the loop body). At this
251 // time point will satisfy point <= 61 and therefore fractionals < 2^point
252 // and any further multiplication of fractionals by 5 will not overflow.
253 fractionals *= 5;
254 point--;
255 int digit = static_cast<int>(fractionals >> point);
256 ASSERT(digit <= 9);
257 buffer[*length] = static_cast<char>('0' + digit);
258 (*length)++;
259 fractionals -= static_cast<uint64_t>(digit) << point;
260 }
261 // If the first bit after the point is set we have to round up.
262 ASSERT(fractionals == 0 || point - 1 >= 0);
263 if ((fractionals != 0) && ((fractionals >> (point - 1)) & 1) == 1) {
264 RoundUp(buffer, length, decimal_point);
265 }
266 } else { // We need 128 bits.
267 ASSERT(64 < -exponent && -exponent <= 128);
268 UInt128 fractionals128 = UInt128(fractionals, 0);
269 fractionals128.Shift(-exponent - 64);
270 int point = 128;
271 for (int i = 0; i < fractional_count; ++i) {
272 if (fractionals128.IsZero()) break;
273 // As before: instead of multiplying by 10 we multiply by 5 and adjust the
274 // point location.
275 // This multiplication will not overflow for the same reasons as before.
276 fractionals128.Multiply(5);
277 point--;
278 int digit = fractionals128.DivModPowerOf2(point);
279 ASSERT(digit <= 9);
280 buffer[*length] = static_cast<char>('0' + digit);
281 (*length)++;
282 }
283 if (fractionals128.BitAt(point - 1) == 1) {
284 RoundUp(buffer, length, decimal_point);
285 }
286 }
287}
288
289
290// Removes leading and trailing zeros.
291// If leading zeros are removed then the decimal point position is adjusted.
292static void TrimZeros(Vector<char> buffer, int* length, int* decimal_point) {
293 while (*length > 0 && buffer[(*length) - 1] == '0') {
294 (*length)--;
295 }
296 int first_non_zero = 0;
297 while (first_non_zero < *length && buffer[first_non_zero] == '0') {
298 first_non_zero++;
299 }
300 if (first_non_zero != 0) {
301 for (int i = first_non_zero; i < *length; ++i) {
302 buffer[i - first_non_zero] = buffer[i];
303 }
304 *length -= first_non_zero;
305 *decimal_point -= first_non_zero;
306 }
307}
308
309
310bool FastFixedDtoa(double v,
311 int fractional_count,
312 Vector<char> buffer,
313 int* length,
314 int* decimal_point) {
315 const uint32_t kMaxUInt32 = 0xFFFFFFFF;
316 uint64_t significand = Double(v).Significand();
317 int exponent = Double(v).Exponent();
318 // v = significand * 2^exponent (with significand a 53bit integer).
319 // If the exponent is larger than 20 (i.e. we may have a 73bit number) then we
320 // don't know how to compute the representation. 2^73 ~= 9.5*10^21.
321 // If necessary this limit could probably be increased, but we don't need
322 // more.
323 if (exponent > 20) return false;
324 if (fractional_count > 20) return false;
325 *length = 0;
326 // At most kDoubleSignificandSize bits of the significand are non-zero.
327 // Given a 64 bit integer we have 11 0s followed by 53 potentially non-zero
328 // bits: 0..11*..0xxx..53*..xx
329 if (exponent + kDoubleSignificandSize > 64) {
330 // The exponent must be > 11.
331 //
332 // We know that v = significand * 2^exponent.
333 // And the exponent > 11.
334 // We simplify the task by dividing v by 10^17.
335 // The quotient delivers the first digits, and the remainder fits into a 64
336 // bit number.
337 // Dividing by 10^17 is equivalent to dividing by 5^17*2^17.
338 const uint64_t kFive17 = UINT64_2PART_C(0xB1, A2BC2EC5); // 5^17
339 uint64_t divisor = kFive17;
340 int divisor_power = 17;
341 uint64_t dividend = significand;
342 uint32_t quotient;
343 uint64_t remainder;
344 // Let v = f * 2^e with f == significand and e == exponent.
345 // Then need q (quotient) and r (remainder) as follows:
346 // v = q * 10^17 + r
347 // f * 2^e = q * 10^17 + r
348 // f * 2^e = q * 5^17 * 2^17 + r
349 // If e > 17 then
350 // f * 2^(e-17) = q * 5^17 + r/2^17
351 // else
352 // f = q * 5^17 * 2^(17-e) + r/2^e
353 if (exponent > divisor_power) {
354 // We only allow exponents of up to 20 and therefore (17 - e) <= 3
355 dividend <<= exponent - divisor_power;
356 quotient = static_cast<uint32_t>(dividend / divisor);
357 remainder = (dividend % divisor) << divisor_power;
358 } else {
359 divisor <<= divisor_power - exponent;
360 quotient = static_cast<uint32_t>(dividend / divisor);
361 remainder = (dividend % divisor) << exponent;
362 }
363 FillDigits32(quotient, buffer, length);
364 FillDigits64FixedLength(remainder, buffer, length);
365 *decimal_point = *length;
366 } else if (exponent >= 0) {
367 // 0 <= exponent <= 11
368 significand <<= exponent;
369 FillDigits64(significand, buffer, length);
370 *decimal_point = *length;
371 } else if (exponent > -kDoubleSignificandSize) {
372 // We have to cut the number.
373 uint64_t integrals = significand >> -exponent;
374 uint64_t fractionals = significand - (integrals << -exponent);
375 if (integrals > kMaxUInt32) {
376 FillDigits64(integrals, buffer, length);
377 } else {
378 FillDigits32(static_cast<uint32_t>(integrals), buffer, length);
379 }
380 *decimal_point = *length;
381 FillFractionals(fractionals, exponent, fractional_count,
382 buffer, length, decimal_point);
383 } else if (exponent < -128) {
384 // This configuration (with at most 20 digits) means that all digits must be
385 // 0.
386 ASSERT(fractional_count <= 20);
387 buffer[0] = '\0';
388 *length = 0;
389 *decimal_point = -fractional_count;
390 } else {
391 *decimal_point = 0;
392 FillFractionals(significand, exponent, fractional_count,
393 buffer, length, decimal_point);
394 }
395 TrimZeros(buffer, length, decimal_point);
396 buffer[*length] = '\0';
397 if ((*length) == 0) {
398 // The string is empty and the decimal_point thus has no importance. Mimick
399 // Gay's dtoa and and set it to -fractional_count.
400 *decimal_point = -fractional_count;
401 }
402 return true;
403}
404
405} // namespace double_conversion
406