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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
10 : // disclaimer in the documentation and/or other materials provided
11 : // with the distribution.
12 : // * Neither the name of Google Inc. nor the names of its
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
18 : // LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
19 : // A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
20 : // OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
21 : // SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
22 : // LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
23 : // DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
24 : // THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
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 <math.h>
29 :
30 : #include "fixed-dtoa.h"
31 : #include "ieee.h"
32 :
33 : namespace double_conversion {
34 :
35 : // Represents a 128bit type. This class should be replaced by a native type on
36 : // platforms that support 128bit integers.
37 : class UInt128 {
38 : public:
39 : UInt128() : high_bits_(0), low_bits_(0) { }
40 0 : UInt128(uint64_t high, uint64_t low) : high_bits_(high), low_bits_(low) { }
41 :
42 0 : void Multiply(uint32_t multiplicand) {
43 : uint64_t accumulator;
44 :
45 0 : accumulator = (low_bits_ & kMask32) * multiplicand;
46 0 : uint32_t part = static_cast<uint32_t>(accumulator & kMask32);
47 0 : accumulator >>= 32;
48 0 : accumulator = accumulator + (low_bits_ >> 32) * multiplicand;
49 0 : low_bits_ = (accumulator << 32) + part;
50 0 : accumulator >>= 32;
51 0 : accumulator = accumulator + (high_bits_ & kMask32) * multiplicand;
52 0 : part = static_cast<uint32_t>(accumulator & kMask32);
53 0 : accumulator >>= 32;
54 0 : accumulator = accumulator + (high_bits_ >> 32) * multiplicand;
55 0 : high_bits_ = (accumulator << 32) + part;
56 0 : ASSERT((accumulator >> 32) == 0);
57 0 : }
58 :
59 0 : void Shift(int shift_amount) {
60 0 : ASSERT(-64 <= shift_amount && shift_amount <= 64);
61 0 : if (shift_amount == 0) {
62 0 : return;
63 0 : } else if (shift_amount == -64) {
64 0 : high_bits_ = low_bits_;
65 0 : low_bits_ = 0;
66 0 : } else if (shift_amount == 64) {
67 0 : low_bits_ = high_bits_;
68 0 : high_bits_ = 0;
69 0 : } else if (shift_amount <= 0) {
70 0 : high_bits_ <<= -shift_amount;
71 0 : high_bits_ += low_bits_ >> (64 + shift_amount);
72 0 : low_bits_ <<= -shift_amount;
73 : } else {
74 0 : low_bits_ >>= shift_amount;
75 0 : low_bits_ += high_bits_ << (64 - shift_amount);
76 0 : high_bits_ >>= shift_amount;
77 : }
78 : }
79 :
80 : // Modifies *this to *this MOD (2^power).
81 : // Returns *this DIV (2^power).
82 0 : int DivModPowerOf2(int power) {
83 0 : if (power >= 64) {
84 0 : int result = static_cast<int>(high_bits_ >> (power - 64));
85 0 : high_bits_ -= static_cast<uint64_t>(result) << (power - 64);
86 0 : return result;
87 : } else {
88 0 : uint64_t part_low = low_bits_ >> power;
89 0 : uint64_t part_high = high_bits_ << (64 - power);
90 0 : int result = static_cast<int>(part_low + part_high);
91 0 : high_bits_ = 0;
92 0 : low_bits_ -= part_low << power;
93 0 : return result;
94 : }
95 : }
96 :
97 0 : bool IsZero() const {
98 0 : return high_bits_ == 0 && low_bits_ == 0;
99 : }
100 :
101 0 : int BitAt(int position) const {
102 0 : if (position >= 64) {
103 0 : return static_cast<int>(high_bits_ >> (position - 64)) & 1;
104 : } else {
105 0 : 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 :
117 : static const int kDoubleSignificandSize = 53; // Includes the hidden bit.
118 :
119 :
120 0 : static void FillDigits32FixedLength(uint32_t number, int requested_length,
121 : Vector<char> buffer, int* length) {
122 0 : for (int i = requested_length - 1; i >= 0; --i) {
123 0 : buffer[(*length) + i] = '0' + number % 10;
124 0 : number /= 10;
125 : }
126 0 : *length += requested_length;
127 0 : }
128 :
129 :
130 0 : static void FillDigits32(uint32_t number, Vector<char> buffer, int* length) {
131 0 : int number_length = 0;
132 : // We fill the digits in reverse order and exchange them afterwards.
133 0 : while (number != 0) {
134 0 : int digit = number % 10;
135 0 : number /= 10;
136 0 : buffer[(*length) + number_length] = static_cast<char>('0' + digit);
137 0 : number_length++;
138 : }
139 : // Exchange the digits.
140 0 : int i = *length;
141 0 : int j = *length + number_length - 1;
142 0 : while (i < j) {
143 0 : char tmp = buffer[i];
144 0 : buffer[i] = buffer[j];
145 0 : buffer[j] = tmp;
146 0 : i++;
147 0 : j--;
148 : }
149 0 : *length += number_length;
150 0 : }
151 :
152 :
153 0 : static void FillDigits64FixedLength(uint64_t number,
154 : Vector<char> buffer, int* length) {
155 0 : const uint32_t kTen7 = 10000000;
156 : // For efficiency cut the number into 3 uint32_t parts, and print those.
157 0 : uint32_t part2 = static_cast<uint32_t>(number % kTen7);
158 0 : number /= kTen7;
159 0 : uint32_t part1 = static_cast<uint32_t>(number % kTen7);
160 0 : uint32_t part0 = static_cast<uint32_t>(number / kTen7);
161 :
162 0 : FillDigits32FixedLength(part0, 3, buffer, length);
163 0 : FillDigits32FixedLength(part1, 7, buffer, length);
164 0 : FillDigits32FixedLength(part2, 7, buffer, length);
165 0 : }
166 :
167 :
168 0 : static void FillDigits64(uint64_t number, Vector<char> buffer, int* length) {
169 0 : const uint32_t kTen7 = 10000000;
170 : // For efficiency cut the number into 3 uint32_t parts, and print those.
171 0 : uint32_t part2 = static_cast<uint32_t>(number % kTen7);
172 0 : number /= kTen7;
173 0 : uint32_t part1 = static_cast<uint32_t>(number % kTen7);
174 0 : uint32_t part0 = static_cast<uint32_t>(number / kTen7);
175 :
176 0 : if (part0 != 0) {
177 0 : FillDigits32(part0, buffer, length);
178 0 : FillDigits32FixedLength(part1, 7, buffer, length);
179 0 : FillDigits32FixedLength(part2, 7, buffer, length);
180 0 : } else if (part1 != 0) {
181 0 : FillDigits32(part1, buffer, length);
182 0 : FillDigits32FixedLength(part2, 7, buffer, length);
183 : } else {
184 0 : FillDigits32(part2, buffer, length);
185 : }
186 0 : }
187 :
188 :
189 0 : static void RoundUp(Vector<char> buffer, int* length, int* decimal_point) {
190 : // An empty buffer represents 0.
191 0 : if (*length == 0) {
192 0 : buffer[0] = '1';
193 0 : *decimal_point = 1;
194 0 : *length = 1;
195 0 : 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 0 : buffer[(*length) - 1]++;
200 0 : for (int i = (*length) - 1; i > 0; --i) {
201 0 : if (buffer[i] != '0' + 10) {
202 0 : return;
203 : }
204 0 : buffer[i] = '0';
205 0 : 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 0 : if (buffer[0] == '0' + 10) {
213 0 : buffer[0] = '1';
214 0 : (*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".
230 0 : static void FillFractionals(uint64_t fractionals, int exponent,
231 : int fractional_count, Vector<char> buffer,
232 : int* length, int* decimal_point) {
233 0 : 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 0 : if (-exponent <= 64) {
238 : // One 64 bit number is sufficient.
239 0 : ASSERT(fractionals >> 56 == 0);
240 0 : int point = -exponent;
241 0 : for (int i = 0; i < fractional_count; ++i) {
242 0 : 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 0 : fractionals *= 5;
254 0 : point--;
255 0 : int digit = static_cast<int>(fractionals >> point);
256 0 : ASSERT(digit <= 9);
257 0 : buffer[*length] = static_cast<char>('0' + digit);
258 0 : (*length)++;
259 0 : fractionals -= static_cast<uint64_t>(digit) << point;
260 : }
261 : // If the first bit after the point is set we have to round up.
262 0 : if (((fractionals >> (point - 1)) & 1) == 1) {
263 0 : RoundUp(buffer, length, decimal_point);
264 : }
265 : } else { // We need 128 bits.
266 0 : ASSERT(64 < -exponent && -exponent <= 128);
267 0 : UInt128 fractionals128 = UInt128(fractionals, 0);
268 0 : fractionals128.Shift(-exponent - 64);
269 0 : int point = 128;
270 0 : for (int i = 0; i < fractional_count; ++i) {
271 0 : if (fractionals128.IsZero()) break;
272 : // As before: instead of multiplying by 10 we multiply by 5 and adjust the
273 : // point location.
274 : // This multiplication will not overflow for the same reasons as before.
275 0 : fractionals128.Multiply(5);
276 0 : point--;
277 0 : int digit = fractionals128.DivModPowerOf2(point);
278 0 : ASSERT(digit <= 9);
279 0 : buffer[*length] = static_cast<char>('0' + digit);
280 0 : (*length)++;
281 : }
282 0 : if (fractionals128.BitAt(point - 1) == 1) {
283 0 : RoundUp(buffer, length, decimal_point);
284 : }
285 : }
286 0 : }
287 :
288 :
289 : // Removes leading and trailing zeros.
290 : // If leading zeros are removed then the decimal point position is adjusted.
291 0 : static void TrimZeros(Vector<char> buffer, int* length, int* decimal_point) {
292 0 : while (*length > 0 && buffer[(*length) - 1] == '0') {
293 0 : (*length)--;
294 : }
295 0 : int first_non_zero = 0;
296 0 : while (first_non_zero < *length && buffer[first_non_zero] == '0') {
297 0 : first_non_zero++;
298 : }
299 0 : if (first_non_zero != 0) {
300 0 : for (int i = first_non_zero; i < *length; ++i) {
301 0 : buffer[i - first_non_zero] = buffer[i];
302 : }
303 0 : *length -= first_non_zero;
304 0 : *decimal_point -= first_non_zero;
305 : }
306 0 : }
307 :
308 :
309 0 : bool FastFixedDtoa(double v,
310 : int fractional_count,
311 : Vector<char> buffer,
312 : int* length,
313 : int* decimal_point) {
314 0 : const uint32_t kMaxUInt32 = 0xFFFFFFFF;
315 0 : uint64_t significand = Double(v).Significand();
316 0 : int exponent = Double(v).Exponent();
317 : // v = significand * 2^exponent (with significand a 53bit integer).
318 : // If the exponent is larger than 20 (i.e. we may have a 73bit number) then we
319 : // don't know how to compute the representation. 2^73 ~= 9.5*10^21.
320 : // If necessary this limit could probably be increased, but we don't need
321 : // more.
322 0 : if (exponent > 20) return false;
323 0 : if (fractional_count > 20) return false;
324 0 : *length = 0;
325 : // At most kDoubleSignificandSize bits of the significand are non-zero.
326 : // Given a 64 bit integer we have 11 0s followed by 53 potentially non-zero
327 : // bits: 0..11*..0xxx..53*..xx
328 0 : if (exponent + kDoubleSignificandSize > 64) {
329 : // The exponent must be > 11.
330 : //
331 : // We know that v = significand * 2^exponent.
332 : // And the exponent > 11.
333 : // We simplify the task by dividing v by 10^17.
334 : // The quotient delivers the first digits, and the remainder fits into a 64
335 : // bit number.
336 : // Dividing by 10^17 is equivalent to dividing by 5^17*2^17.
337 0 : const uint64_t kFive17 = UINT64_2PART_C(0xB1, A2BC2EC5); // 5^17
338 0 : uint64_t divisor = kFive17;
339 0 : int divisor_power = 17;
340 0 : uint64_t dividend = significand;
341 : uint32_t quotient;
342 : uint64_t remainder;
343 : // Let v = f * 2^e with f == significand and e == exponent.
344 : // Then need q (quotient) and r (remainder) as follows:
345 : // v = q * 10^17 + r
346 : // f * 2^e = q * 10^17 + r
347 : // f * 2^e = q * 5^17 * 2^17 + r
348 : // If e > 17 then
349 : // f * 2^(e-17) = q * 5^17 + r/2^17
350 : // else
351 : // f = q * 5^17 * 2^(17-e) + r/2^e
352 0 : if (exponent > divisor_power) {
353 : // We only allow exponents of up to 20 and therefore (17 - e) <= 3
354 0 : dividend <<= exponent - divisor_power;
355 0 : quotient = static_cast<uint32_t>(dividend / divisor);
356 0 : remainder = (dividend % divisor) << divisor_power;
357 : } else {
358 0 : divisor <<= divisor_power - exponent;
359 0 : quotient = static_cast<uint32_t>(dividend / divisor);
360 0 : remainder = (dividend % divisor) << exponent;
361 : }
362 0 : FillDigits32(quotient, buffer, length);
363 0 : FillDigits64FixedLength(remainder, buffer, length);
364 0 : *decimal_point = *length;
365 0 : } else if (exponent >= 0) {
366 : // 0 <= exponent <= 11
367 0 : significand <<= exponent;
368 0 : FillDigits64(significand, buffer, length);
369 0 : *decimal_point = *length;
370 0 : } else if (exponent > -kDoubleSignificandSize) {
371 : // We have to cut the number.
372 0 : uint64_t integrals = significand >> -exponent;
373 0 : uint64_t fractionals = significand - (integrals << -exponent);
374 0 : if (integrals > kMaxUInt32) {
375 0 : FillDigits64(integrals, buffer, length);
376 : } else {
377 0 : FillDigits32(static_cast<uint32_t>(integrals), buffer, length);
378 : }
379 0 : *decimal_point = *length;
380 : FillFractionals(fractionals, exponent, fractional_count,
381 0 : buffer, length, decimal_point);
382 0 : } else if (exponent < -128) {
383 : // This configuration (with at most 20 digits) means that all digits must be
384 : // 0.
385 0 : ASSERT(fractional_count <= 20);
386 0 : buffer[0] = '\0';
387 0 : *length = 0;
388 0 : *decimal_point = -fractional_count;
389 : } else {
390 0 : *decimal_point = 0;
391 : FillFractionals(significand, exponent, fractional_count,
392 0 : buffer, length, decimal_point);
393 : }
394 0 : TrimZeros(buffer, length, decimal_point);
395 0 : buffer[*length] = '\0';
396 0 : if ((*length) == 0) {
397 : // The string is empty and the decimal_point thus has no importance. Mimick
398 : // Gay's dtoa and and set it to -fractional_count.
399 0 : *decimal_point = -fractional_count;
400 : }
401 0 : return true;
402 : }
403 :
404 : } // namespace double_conversion
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