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 <stdint.h>
6
7#include <cmath>
8
9#include "src/base/logging.h"
10#include "src/utils.h"
11
12#include "src/double.h"
13#include "src/fixed-dtoa.h"
14
15namespace v8 {
16namespace internal {
17
18// Represents a 128bit type. This class should be replaced by a native type on
19// platforms that support 128bit integers.
20class UInt128 {
21 public:
22 UInt128() : high_bits_(0), low_bits_(0) { }
23 UInt128(uint64_t high, uint64_t low) : high_bits_(high), low_bits_(low) { }
24
25 void Multiply(uint32_t multiplicand) {
26 uint64_t accumulator;
27
28 accumulator = (low_bits_ & kMask32) * multiplicand;
29 uint32_t part = static_cast<uint32_t>(accumulator & kMask32);
30 accumulator >>= 32;
31 accumulator = accumulator + (low_bits_ >> 32) * multiplicand;
32 low_bits_ = (accumulator << 32) + part;
33 accumulator >>= 32;
34 accumulator = accumulator + (high_bits_ & kMask32) * multiplicand;
35 part = static_cast<uint32_t>(accumulator & kMask32);
36 accumulator >>= 32;
37 accumulator = accumulator + (high_bits_ >> 32) * multiplicand;
38 high_bits_ = (accumulator << 32) + part;
39 DCHECK_EQ(accumulator >> 32, 0);
40 }
41
42 void Shift(int shift_amount) {
43 DCHECK(-64 <= shift_amount && shift_amount <= 64);
44 if (shift_amount == 0) {
45 return;
46 } else if (shift_amount == -64) {
47 high_bits_ = low_bits_;
48 low_bits_ = 0;
49 } else if (shift_amount == 64) {
50 low_bits_ = high_bits_;
51 high_bits_ = 0;
52 } else if (shift_amount <= 0) {
53 high_bits_ <<= -shift_amount;
54 high_bits_ += low_bits_ >> (64 + shift_amount);
55 low_bits_ <<= -shift_amount;
56 } else {
57 low_bits_ >>= shift_amount;
58 low_bits_ += high_bits_ << (64 - shift_amount);
59 high_bits_ >>= shift_amount;
60 }
61 }
62
63 // Modifies *this to *this MOD (2^power).
64 // Returns *this DIV (2^power).
65 int DivModPowerOf2(int power) {
66 if (power >= 64) {
67 int result = static_cast<int>(high_bits_ >> (power - 64));
68 high_bits_ -= static_cast<uint64_t>(result) << (power - 64);
69 return result;
70 } else {
71 uint64_t part_low = low_bits_ >> power;
72 uint64_t part_high = high_bits_ << (64 - power);
73 int result = static_cast<int>(part_low + part_high);
74 high_bits_ = 0;
75 low_bits_ -= part_low << power;
76 return result;
77 }
78 }
79
80 bool IsZero() const {
81 return high_bits_ == 0 && low_bits_ == 0;
82 }
83
84 int BitAt(int position) {
85 if (position >= 64) {
86 return static_cast<int>(high_bits_ >> (position - 64)) & 1;
87 } else {
88 return static_cast<int>(low_bits_ >> position) & 1;
89 }
90 }
91
92 private:
93 static const uint64_t kMask32 = 0xFFFFFFFF;
94 // Value == (high_bits_ << 64) + low_bits_
95 uint64_t high_bits_;
96 uint64_t low_bits_;
97};
98
99
100static const int kDoubleSignificandSize = 53; // Includes the hidden bit.
101
102
103static void FillDigits32FixedLength(uint32_t number, int requested_length,
104 Vector<char> buffer, int* length) {
105 for (int i = requested_length - 1; i >= 0; --i) {
106 buffer[(*length) + i] = '0' + number % 10;
107 number /= 10;
108 }
109 *length += requested_length;
110}
111
112
113static void FillDigits32(uint32_t number, Vector<char> buffer, int* length) {
114 int number_length = 0;
115 // We fill the digits in reverse order and exchange them afterwards.
116 while (number != 0) {
117 int digit = number % 10;
118 number /= 10;
119 buffer[(*length) + number_length] = '0' + digit;
120 number_length++;
121 }
122 // Exchange the digits.
123 int i = *length;
124 int j = *length + number_length - 1;
125 while (i < j) {
126 char tmp = buffer[i];
127 buffer[i] = buffer[j];
128 buffer[j] = tmp;
129 i++;
130 j--;
131 }
132 *length += number_length;
133}
134
135
136static void FillDigits64FixedLength(uint64_t number, int requested_length,
137 Vector<char> buffer, int* length) {
138 const uint32_t kTen7 = 10000000;
139 // For efficiency cut the number into 3 uint32_t parts, and print those.
140 uint32_t part2 = static_cast<uint32_t>(number % kTen7);
141 number /= kTen7;
142 uint32_t part1 = static_cast<uint32_t>(number % kTen7);
143 uint32_t part0 = static_cast<uint32_t>(number / kTen7);
144
145 FillDigits32FixedLength(part0, 3, buffer, length);
146 FillDigits32FixedLength(part1, 7, buffer, length);
147 FillDigits32FixedLength(part2, 7, buffer, length);
148}
149
150
151static void FillDigits64(uint64_t number, Vector<char> buffer, int* length) {
152 const uint32_t kTen7 = 10000000;
153 // For efficiency cut the number into 3 uint32_t parts, and print those.
154 uint32_t part2 = static_cast<uint32_t>(number % kTen7);
155 number /= kTen7;
156 uint32_t part1 = static_cast<uint32_t>(number % kTen7);
157 uint32_t part0 = static_cast<uint32_t>(number / kTen7);
158
159 if (part0 != 0) {
160 FillDigits32(part0, buffer, length);
161 FillDigits32FixedLength(part1, 7, buffer, length);
162 FillDigits32FixedLength(part2, 7, buffer, length);
163 } else if (part1 != 0) {
164 FillDigits32(part1, buffer, length);
165 FillDigits32FixedLength(part2, 7, buffer, length);
166 } else {
167 FillDigits32(part2, buffer, length);
168 }
169}
170
171static void DtoaRoundUp(Vector<char> buffer, int* length, int* decimal_point) {
172 // An empty buffer represents 0.
173 if (*length == 0) {
174 buffer[0] = '1';
175 *decimal_point = 1;
176 *length = 1;
177 return;
178 }
179 // Round the last digit until we either have a digit that was not '9' or until
180 // we reached the first digit.
181 buffer[(*length) - 1]++;
182 for (int i = (*length) - 1; i > 0; --i) {
183 if (buffer[i] != '0' + 10) {
184 return;
185 }
186 buffer[i] = '0';
187 buffer[i - 1]++;
188 }
189 // If the first digit is now '0' + 10, we would need to set it to '0' and add
190 // a '1' in front. However we reach the first digit only if all following
191 // digits had been '9' before rounding up. Now all trailing digits are '0' and
192 // we simply switch the first digit to '1' and update the decimal-point
193 // (indicating that the point is now one digit to the right).
194 if (buffer[0] == '0' + 10) {
195 buffer[0] = '1';
196 (*decimal_point)++;
197 }
198}
199
200
201// The given fractionals number represents a fixed-point number with binary
202// point at bit (-exponent).
203// Preconditions:
204// -128 <= exponent <= 0.
205// 0 <= fractionals * 2^exponent < 1
206// The buffer holds the result.
207// The function will round its result. During the rounding-process digits not
208// generated by this function might be updated, and the decimal-point variable
209// might be updated. If this function generates the digits 99 and the buffer
210// already contained "199" (thus yielding a buffer of "19999") then a
211// rounding-up will change the contents of the buffer to "20000".
212static void FillFractionals(uint64_t fractionals, int exponent,
213 int fractional_count, Vector<char> buffer,
214 int* length, int* decimal_point) {
215 DCHECK(-128 <= exponent && exponent <= 0);
216 // 'fractionals' is a fixed-point number, with binary point at bit
217 // (-exponent). Inside the function the non-converted remainder of fractionals
218 // is a fixed-point number, with binary point at bit 'point'.
219 if (-exponent <= 64) {
220 // One 64 bit number is sufficient.
221 DCHECK_EQ(fractionals >> 56, 0);
222 int point = -exponent;
223 for (int i = 0; i < fractional_count; ++i) {
224 if (fractionals == 0) break;
225 // Instead of multiplying by 10 we multiply by 5 and adjust the point
226 // location. This way the fractionals variable will not overflow.
227 // Invariant at the beginning of the loop: fractionals < 2^point.
228 // Initially we have: point <= 64 and fractionals < 2^56
229 // After each iteration the point is decremented by one.
230 // Note that 5^3 = 125 < 128 = 2^7.
231 // Therefore three iterations of this loop will not overflow fractionals
232 // (even without the subtraction at the end of the loop body). At this
233 // time point will satisfy point <= 61 and therefore fractionals < 2^point
234 // and any further multiplication of fractionals by 5 will not overflow.
235 fractionals *= 5;
236 point--;
237 int digit = static_cast<int>(fractionals >> point);
238 buffer[*length] = '0' + digit;
239 (*length)++;
240 fractionals -= static_cast<uint64_t>(digit) << point;
241 }
242 // If the first bit after the point is set we have to round up.
243 if (point > 0 && ((fractionals >> (point - 1)) & 1) == 1) {
244 DtoaRoundUp(buffer, length, decimal_point);
245 }
246 } else { // We need 128 bits.
247 DCHECK(64 < -exponent && -exponent <= 128);
248 UInt128 fractionals128 = UInt128(fractionals, 0);
249 fractionals128.Shift(-exponent - 64);
250 int point = 128;
251 for (int i = 0; i < fractional_count; ++i) {
252 if (fractionals128.IsZero()) break;
253 // As before: instead of multiplying by 10 we multiply by 5 and adjust the
254 // point location.
255 // This multiplication will not overflow for the same reasons as before.
256 fractionals128.Multiply(5);
257 point--;
258 int digit = fractionals128.DivModPowerOf2(point);
259 buffer[*length] = '0' + digit;
260 (*length)++;
261 }
262 if (fractionals128.BitAt(point - 1) == 1) {
263 DtoaRoundUp(buffer, length, decimal_point);
264 }
265 }
266}
267
268
269// Removes leading and trailing zeros.
270// If leading zeros are removed then the decimal point position is adjusted.
271static void TrimZeros(Vector<char> buffer, int* length, int* decimal_point) {
272 while (*length > 0 && buffer[(*length) - 1] == '0') {
273 (*length)--;
274 }
275 int first_non_zero = 0;
276 while (first_non_zero < *length && buffer[first_non_zero] == '0') {
277 first_non_zero++;
278 }
279 if (first_non_zero != 0) {
280 for (int i = first_non_zero; i < *length; ++i) {
281 buffer[i - first_non_zero] = buffer[i];
282 }
283 *length -= first_non_zero;
284 *decimal_point -= first_non_zero;
285 }
286}
287
288
289bool FastFixedDtoa(double v,
290 int fractional_count,
291 Vector<char> buffer,
292 int* length,
293 int* decimal_point) {
294 const uint32_t kMaxUInt32 = 0xFFFFFFFF;
295 uint64_t significand = Double(v).Significand();
296 int exponent = Double(v).Exponent();
297 // v = significand * 2^exponent (with significand a 53bit integer).
298 // If the exponent is larger than 20 (i.e. we may have a 73bit number) then we
299 // don't know how to compute the representation. 2^73 ~= 9.5*10^21.
300 // If necessary this limit could probably be increased, but we don't need
301 // more.
302 if (exponent > 20) return false;
303 if (fractional_count > 20) return false;
304 *length = 0;
305 // At most kDoubleSignificandSize bits of the significand are non-zero.
306 // Given a 64 bit integer we have 11 0s followed by 53 potentially non-zero
307 // bits: 0..11*..0xxx..53*..xx
308 if (exponent + kDoubleSignificandSize > 64) {
309 // The exponent must be > 11.
310 //
311 // We know that v = significand * 2^exponent.
312 // And the exponent > 11.
313 // We simplify the task by dividing v by 10^17.
314 // The quotient delivers the first digits, and the remainder fits into a 64
315 // bit number.
316 // Dividing by 10^17 is equivalent to dividing by 5^17*2^17.
317 const uint64_t kFive17 = V8_2PART_UINT64_C(0xB1, A2BC2EC5); // 5^17
318 uint64_t divisor = kFive17;
319 int divisor_power = 17;
320 uint64_t dividend = significand;
321 uint32_t quotient;
322 uint64_t remainder;
323 // Let v = f * 2^e with f == significand and e == exponent.
324 // Then need q (quotient) and r (remainder) as follows:
325 // v = q * 10^17 + r
326 // f * 2^e = q * 10^17 + r
327 // f * 2^e = q * 5^17 * 2^17 + r
328 // If e > 17 then
329 // f * 2^(e-17) = q * 5^17 + r/2^17
330 // else
331 // f = q * 5^17 * 2^(17-e) + r/2^e
332 if (exponent > divisor_power) {
333 // We only allow exponents of up to 20 and therefore (17 - e) <= 3
334 dividend <<= exponent - divisor_power;
335 quotient = static_cast<uint32_t>(dividend / divisor);
336 remainder = (dividend % divisor) << divisor_power;
337 } else {
338 divisor <<= divisor_power - exponent;
339 quotient = static_cast<uint32_t>(dividend / divisor);
340 remainder = (dividend % divisor) << exponent;
341 }
342 FillDigits32(quotient, buffer, length);
343 FillDigits64FixedLength(remainder, divisor_power, buffer, length);
344 *decimal_point = *length;
345 } else if (exponent >= 0) {
346 // 0 <= exponent <= 11
347 significand <<= exponent;
348 FillDigits64(significand, buffer, length);
349 *decimal_point = *length;
350 } else if (exponent > -kDoubleSignificandSize) {
351 // We have to cut the number.
352 uint64_t integrals = significand >> -exponent;
353 uint64_t fractionals = significand - (integrals << -exponent);
354 if (integrals > kMaxUInt32) {
355 FillDigits64(integrals, buffer, length);
356 } else {
357 FillDigits32(static_cast<uint32_t>(integrals), buffer, length);
358 }
359 *decimal_point = *length;
360 FillFractionals(fractionals, exponent, fractional_count,
361 buffer, length, decimal_point);
362 } else if (exponent < -128) {
363 // This configuration (with at most 20 digits) means that all digits must be
364 // 0.
365 DCHECK_LE(fractional_count, 20);
366 buffer[0] = '\0';
367 *length = 0;
368 *decimal_point = -fractional_count;
369 } else {
370 *decimal_point = 0;
371 FillFractionals(significand, exponent, fractional_count,
372 buffer, length, decimal_point);
373 }
374 TrimZeros(buffer, length, decimal_point);
375 buffer[*length] = '\0';
376 if ((*length) == 0) {
377 // The string is empty and the decimal_point thus has no importance. Mimick
378 // Gay's dtoa and and set it to -fractional_count.
379 *decimal_point = -fractional_count;
380 }
381 return true;
382}
383
384} // namespace internal
385} // namespace v8
386