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Issue 2700123003: DO NOT COMMIT: Results of running old (current) clang-format on Blink (Closed)
Patch Set: Created 3 years, 10 months ago
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1 // Copyright 2010 the V8 project authors. All rights reserved. 1 // Copyright 2010 the V8 project authors. All rights reserved.
2 // Redistribution and use in source and binary forms, with or without 2 // Redistribution and use in source and binary forms, with or without
3 // modification, are permitted provided that the following conditions are 3 // modification, are permitted provided that the following conditions are
4 // met: 4 // met:
5 // 5 //
6 // * Redistributions of source code must retain the above copyright 6 // * Redistributions of source code must retain the above copyright
7 // notice, this list of conditions and the following disclaimer. 7 // notice, this list of conditions and the following disclaimer.
8 // * Redistributions in binary form must reproduce the above 8 // * Redistributions in binary form must reproduce the above
9 // copyright notice, this list of conditions and the following 9 // copyright notice, this list of conditions and the following
10 // disclaimer in the documentation and/or other materials provided 10 // disclaimer in the documentation and/or other materials provided
(...skipping 17 matching lines...) Expand all
28 #include "fast-dtoa.h" 28 #include "fast-dtoa.h"
29 29
30 #include "cached-powers.h" 30 #include "cached-powers.h"
31 #include "diy-fp.h" 31 #include "diy-fp.h"
32 #include "double.h" 32 #include "double.h"
33 33
34 namespace WTF { 34 namespace WTF {
35 35
36 namespace double_conversion { 36 namespace double_conversion {
37 37
38 // The minimal and maximal target exponent define the range of w's binary 38 // The minimal and maximal target exponent define the range of w's binary
39 // exponent, where 'w' is the result of multiplying the input by a cached po wer 39 // exponent, where 'w' is the result of multiplying the input by a cached power
40 // of ten. 40 // of ten.
41 // 41 //
42 // A different range might be chosen on a different platform, to optimize di git 42 // A different range might be chosen on a different platform, to optimize digit
43 // generation, but a smaller range requires more powers of ten to be cached. 43 // generation, but a smaller range requires more powers of ten to be cached.
44 static const int kMinimalTargetExponent = -60; 44 static const int kMinimalTargetExponent = -60;
45 static const int kMaximalTargetExponent = -32; 45 static const int kMaximalTargetExponent = -32;
46 46
47 47 // Adjusts the last digit of the generated number, and screens out generated
48 // Adjusts the last digit of the generated number, and screens out generated 48 // solutions that may be inaccurate. A solution may be inaccurate if it is
49 // solutions that may be inaccurate. A solution may be inaccurate if it is 49 // outside the safe interval, or if we cannot prove that it is closer to the
50 // outside the safe interval, or if we cannot prove that it is closer to the 50 // input than a neighboring representation of the same length.
51 // input than a neighboring representation of the same length. 51 //
52 // 52 // Input: * buffer containing the digits of too_high / 10^kappa
53 // Input: * buffer containing the digits of too_high / 10^kappa 53 // * the buffer's length
54 // * the buffer's length 54 // * distance_too_high_w == (too_high - w).f() * unit
55 // * distance_too_high_w == (too_high - w).f() * unit 55 // * unsafe_interval == (too_high - too_low).f() * unit
56 // * unsafe_interval == (too_high - too_low).f() * unit 56 // * rest = (too_high - buffer * 10^kappa).f() * unit
57 // * rest = (too_high - buffer * 10^kappa).f() * unit 57 // * ten_kappa = 10^kappa * unit
58 // * ten_kappa = 10^kappa * unit 58 // * unit = the common multiplier
59 // * unit = the common multiplier 59 // Output: returns true if the buffer is guaranteed to contain the closest
60 // Output: returns true if the buffer is guaranteed to contain the closest 60 // representable number to the input.
61 // representable number to the input. 61 // Modifies the generated digits in the buffer to approach (round towards) w.
62 // Modifies the generated digits in the buffer to approach (round towards) w. 62 static bool RoundWeed(Vector<char> buffer,
63 static bool RoundWeed(Vector<char> buffer, 63 int length,
64 int length, 64 uint64_t distance_too_high_w,
65 uint64_t distance_too_high_w, 65 uint64_t unsafe_interval,
66 uint64_t unsafe_interval, 66 uint64_t rest,
67 uint64_t rest, 67 uint64_t ten_kappa,
68 uint64_t ten_kappa, 68 uint64_t unit) {
69 uint64_t unit) { 69 uint64_t small_distance = distance_too_high_w - unit;
70 uint64_t small_distance = distance_too_high_w - unit; 70 uint64_t big_distance = distance_too_high_w + unit;
71 uint64_t big_distance = distance_too_high_w + unit; 71 // Let w_low = too_high - big_distance, and
72 // Let w_low = too_high - big_distance, and 72 // w_high = too_high - small_distance.
73 // w_high = too_high - small_distance. 73 // Note: w_low < w < w_high
74 // Note: w_low < w < w_high 74 //
75 // 75 // The real w (* unit) must lie somewhere inside the interval
76 // The real w (* unit) must lie somewhere inside the interval 76 // ]w_low; w_high[ (often written as "(w_low; w_high)")
77 // ]w_low; w_high[ (often written as "(w_low; w_high)") 77
78 78 // Basically the buffer currently contains a number in the unsafe interval
79 // Basically the buffer currently contains a number in the unsafe interv al 79 // ]too_low; too_high[ with too_low < w < too_high
80 // ]too_low; too_high[ with too_low < w < too_high 80 //
81 // 81 // too_high - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
82 // too_high - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 82 // ^v 1 unit ^ ^ ^ ^
83 // ^v 1 unit ^ ^ ^ ^ 83 // boundary_high --------------------- . . . .
84 // boundary_high --------------------- . . . . 84 // ^v 1 unit . . . .
85 // ^v 1 unit . . . . 85 // - - - - - - - - - - - - - - - - - - - + - - + - - - - - - . .
86 // - - - - - - - - - - - - - - - - - - - + - - + - - - - - - . . 86 // . . ^ . .
87 // . . ^ . . 87 // . big_distance . . .
88 // . big_distance . . . 88 // . . . . rest
89 // . . . . rest 89 // small_distance . . . .
90 // small_distance . . . . 90 // v . . . .
91 // v . . . . 91 // w_high - - - - - - - - - - - - - - - - - - . . . .
92 // w_high - - - - - - - - - - - - - - - - - - . . . . 92 // ^v 1 unit . . . .
93 // ^v 1 unit . . . . 93 // w ---------------------------------------- . . . .
94 // w ---------------------------------------- . . . . 94 // ^v 1 unit v . . .
95 // ^v 1 unit v . . . 95 // w_low - - - - - - - - - - - - - - - - - - - - - . . .
96 // w_low - - - - - - - - - - - - - - - - - - - - - . . . 96 // . . v
97 // . . v 97 // buffer --------------------------------------------------+-------+--------
98 // buffer --------------------------------------------------+-------+-- ------ 98 // . .
99 // . . 99 // safe_interval .
100 // safe_interval . 100 // v .
101 // v . 101 // - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - .
102 // - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - . 102 // ^v 1 unit .
103 // ^v 1 unit . 103 // boundary_low ------------------------- unsafe_interval
104 // boundary_low ------------------------- unsafe_in terval 104 // ^v 1 unit v
105 // ^v 1 unit v 105 // too_low - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
106 // too_low - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 106 //
107 // 107 //
108 // 108 // Note that the value of buffer could lie anywhere inside the range too_low
109 // Note that the value of buffer could lie anywhere inside the range too _low 109 // to too_high.
110 // to too_high. 110 //
111 // 111 // boundary_low, boundary_high and w are approximations of the real boundaries
112 // boundary_low, boundary_high and w are approximations of the real boun daries 112 // and v (the input number). They are guaranteed to be precise up to one unit.
113 // and v (the input number). They are guaranteed to be precise up to one unit. 113 // In fact the error is guaranteed to be strictly less than one unit.
114 // In fact the error is guaranteed to be strictly less than one unit. 114 //
115 // 115 // Anything that lies outside the unsafe interval is guaranteed not to round
116 // Anything that lies outside the unsafe interval is guaranteed not to r ound 116 // to v when read again.
117 // to v when read again. 117 // Anything that lies inside the safe interval is guaranteed to round to v
118 // Anything that lies inside the safe interval is guaranteed to round to v 118 // when read again.
119 // when read again. 119 // If the number inside the buffer lies inside the unsafe interval but not
120 // If the number inside the buffer lies inside the unsafe interval but n ot 120 // inside the safe interval then we simply do not know and bail out (returning
121 // inside the safe interval then we simply do not know and bail out (ret urning 121 // false).
122 // false). 122 //
123 // 123 // Similarly we have to take into account the imprecision of 'w' when finding
124 // Similarly we have to take into account the imprecision of 'w' when fi nding 124 // the closest representation of 'w'. If we have two potential
125 // the closest representation of 'w'. If we have two potential 125 // representations, and one is closer to both w_low and w_high, then we know
126 // representations, and one is closer to both w_low and w_high, then we know 126 // it is closer to the actual value v.
127 // it is closer to the actual value v. 127 //
128 // 128 // By generating the digits of too_high we got the largest (closest to
129 // By generating the digits of too_high we got the largest (closest to 129 // too_high) buffer that is still in the unsafe interval. In the case where
130 // too_high) buffer that is still in the unsafe interval. In the case wh ere 130 // w_high < buffer < too_high we try to decrement the buffer.
131 // w_high < buffer < too_high we try to decrement the buffer. 131 // This way the buffer approaches (rounds towards) w.
132 // This way the buffer approaches (rounds towards) w. 132 // There are 3 conditions that stop the decrementation process:
133 // There are 3 conditions that stop the decrementation process: 133 // 1) the buffer is already below w_high
134 // 1) the buffer is already below w_high 134 // 2) decrementing the buffer would make it leave the unsafe interval
135 // 2) decrementing the buffer would make it leave the unsafe interval 135 // 3) decrementing the buffer would yield a number below w_high and farther
136 // 3) decrementing the buffer would yield a number below w_high and fa rther 136 // away than the current number. In other words:
137 // away than the current number. In other words: 137 // (buffer{-1} < w_high) && w_high - buffer{-1} > buffer - w_high
138 // (buffer{-1} < w_high) && w_high - buffer{-1} > buffer - w_high 138 // Instead of using the buffer directly we use its distance to too_high.
139 // Instead of using the buffer directly we use its distance to too_high. 139 // Conceptually rest ~= too_high - buffer
140 // Conceptually rest ~= too_high - buffer 140 // We need to do the following tests in this order to avoid over- and
141 // We need to do the following tests in this order to avoid over- and 141 // underflows.
142 // underflows. 142 ASSERT(rest <= unsafe_interval);
143 ASSERT(rest <= unsafe_interval); 143 while (rest < small_distance && // Negated condition 1
144 while (rest < small_distance && // Negated condition 1 144 unsafe_interval - rest >= ten_kappa && // Negated condition 2
145 unsafe_interval - rest >= ten_kappa && // Negated condition 2 145 (rest + ten_kappa < small_distance || // buffer{-1} > w_high
146 (rest + ten_kappa < small_distance || // buffer{-1} > w_high 146 small_distance - rest >= rest + ten_kappa - small_distance)) {
147 small_distance - rest >= rest + ten_kappa - small_distance)) { 147 buffer[length - 1]--;
148 buffer[length - 1]--; 148 rest += ten_kappa;
149 rest += ten_kappa; 149 }
150 } 150
151 151 // We have approached w+ as much as possible. We now test if approaching w-
152 // We have approached w+ as much as possible. We now test if approaching w- 152 // would require changing the buffer. If yes, then we have two possible
153 // would require changing the buffer. If yes, then we have two possible 153 // representations close to w, but we cannot decide which one is closer.
154 // representations close to w, but we cannot decide which one is closer. 154 if (rest < big_distance && unsafe_interval - rest >= ten_kappa &&
155 if (rest < big_distance && 155 (rest + ten_kappa < big_distance ||
156 unsafe_interval - rest >= ten_kappa && 156 big_distance - rest > rest + ten_kappa - big_distance)) {
157 (rest + ten_kappa < big_distance || 157 return false;
158 big_distance - rest > rest + ten_kappa - big_distance)) { 158 }
159 return false; 159
160 } 160 // Weeding test.
161 161 // The safe interval is [too_low + 2 ulp; too_high - 2 ulp]
162 // Weeding test. 162 // Since too_low = too_high - unsafe_interval this is equivalent to
163 // The safe interval is [too_low + 2 ulp; too_high - 2 ulp] 163 // [too_high - unsafe_interval + 4 ulp; too_high - 2 ulp]
164 // Since too_low = too_high - unsafe_interval this is equivalent to 164 // Conceptually we have: rest ~= too_high - buffer
165 // [too_high - unsafe_interval + 4 ulp; too_high - 2 ulp] 165 return (2 * unit <= rest) && (rest <= unsafe_interval - 4 * unit);
166 // Conceptually we have: rest ~= too_high - buffer 166 }
167 return (2 * unit <= rest) && (rest <= unsafe_interval - 4 * unit); 167
168 // Rounds the buffer upwards if the result is closer to v by possibly adding
169 // 1 to the buffer. If the precision of the calculation is not sufficient to
170 // round correctly, return false.
171 // The rounding might shift the whole buffer in which case the kappa is
172 // adjusted. For example "99", kappa = 3 might become "10", kappa = 4.
173 //
174 // If 2*rest > ten_kappa then the buffer needs to be round up.
175 // rest can have an error of +/- 1 unit. This function accounts for the
176 // imprecision and returns false, if the rounding direction cannot be
177 // unambiguously determined.
178 //
179 // Precondition: rest < ten_kappa.
180 static bool RoundWeedCounted(Vector<char> buffer,
181 int length,
182 uint64_t rest,
183 uint64_t ten_kappa,
184 uint64_t unit,
185 int* kappa) {
186 ASSERT(rest < ten_kappa);
187 // The following tests are done in a specific order to avoid overflows. They
188 // will work correctly with any uint64 values of rest < ten_kappa and unit.
189 //
190 // If the unit is too big, then we don't know which way to round. For example
191 // a unit of 50 means that the real number lies within rest +/- 50. If
192 // 10^kappa == 40 then there is no way to tell which way to round.
193 if (unit >= ten_kappa)
194 return false;
195 // Even if unit is just half the size of 10^kappa we are already completely
196 // lost. (And after the previous test we know that the expression will not
197 // over/underflow.)
198 if (ten_kappa - unit <= unit)
199 return false;
200 // If 2 * (rest + unit) <= 10^kappa we can safely round down.
201 if ((ten_kappa - rest > rest) && (ten_kappa - 2 * rest >= 2 * unit)) {
202 return true;
203 }
204 // If 2 * (rest - unit) >= 10^kappa, then we can safely round up.
205 if ((rest > unit) && (ten_kappa - (rest - unit) <= (rest - unit))) {
206 // Increment the last digit recursively until we find a non '9' digit.
207 buffer[length - 1]++;
208 for (int i = length - 1; i > 0; --i) {
209 if (buffer[i] != '0' + 10)
210 break;
211 buffer[i] = '0';
212 buffer[i - 1]++;
168 } 213 }
169 214 // If the first digit is now '0'+ 10 we had a buffer with all '9's. With the
170 215 // exception of the first digit all digits are now '0'. Simply switch the
171 // Rounds the buffer upwards if the result is closer to v by possibly adding 216 // first digit to '1' and adjust the kappa. Example: "99" becomes "10" and
172 // 1 to the buffer. If the precision of the calculation is not sufficient to 217 // the power (the kappa) is increased.
173 // round correctly, return false. 218 if (buffer[0] == '0' + 10) {
174 // The rounding might shift the whole buffer in which case the kappa is 219 buffer[0] = '1';
175 // adjusted. For example "99", kappa = 3 might become "10", kappa = 4. 220 (*kappa) += 1;
176 //
177 // If 2*rest > ten_kappa then the buffer needs to be round up.
178 // rest can have an error of +/- 1 unit. This function accounts for the
179 // imprecision and returns false, if the rounding direction cannot be
180 // unambiguously determined.
181 //
182 // Precondition: rest < ten_kappa.
183 static bool RoundWeedCounted(Vector<char> buffer,
184 int length,
185 uint64_t rest,
186 uint64_t ten_kappa,
187 uint64_t unit,
188 int* kappa) {
189 ASSERT(rest < ten_kappa);
190 // The following tests are done in a specific order to avoid overflows. They
191 // will work correctly with any uint64 values of rest < ten_kappa and un it.
192 //
193 // If the unit is too big, then we don't know which way to round. For ex ample
194 // a unit of 50 means that the real number lies within rest +/- 50. If
195 // 10^kappa == 40 then there is no way to tell which way to round.
196 if (unit >= ten_kappa) return false;
197 // Even if unit is just half the size of 10^kappa we are already complet ely
198 // lost. (And after the previous test we know that the expression will n ot
199 // over/underflow.)
200 if (ten_kappa - unit <= unit) return false;
201 // If 2 * (rest + unit) <= 10^kappa we can safely round down.
202 if ((ten_kappa - rest > rest) && (ten_kappa - 2 * rest >= 2 * unit)) {
203 return true;
204 }
205 // If 2 * (rest - unit) >= 10^kappa, then we can safely round up.
206 if ((rest > unit) && (ten_kappa - (rest - unit) <= (rest - unit))) {
207 // Increment the last digit recursively until we find a non '9' digi t.
208 buffer[length - 1]++;
209 for (int i = length - 1; i > 0; --i) {
210 if (buffer[i] != '0' + 10) break;
211 buffer[i] = '0';
212 buffer[i - 1]++;
213 }
214 // If the first digit is now '0'+ 10 we had a buffer with all '9's. With the
215 // exception of the first digit all digits are now '0'. Simply switc h the
216 // first digit to '1' and adjust the kappa. Example: "99" becomes "1 0" and
217 // the power (the kappa) is increased.
218 if (buffer[0] == '0' + 10) {
219 buffer[0] = '1';
220 (*kappa) += 1;
221 }
222 return true;
223 }
224 return false;
225 } 221 }
226 222 return true;
227 223 }
228 static const uint32_t kTen4 = 10000; 224 return false;
229 static const uint32_t kTen5 = 100000; 225 }
230 static const uint32_t kTen6 = 1000000; 226
231 static const uint32_t kTen7 = 10000000; 227 static const uint32_t kTen4 = 10000;
232 static const uint32_t kTen8 = 100000000; 228 static const uint32_t kTen5 = 100000;
233 static const uint32_t kTen9 = 1000000000; 229 static const uint32_t kTen6 = 1000000;
234 230 static const uint32_t kTen7 = 10000000;
235 // Returns the biggest power of ten that is less than or equal to the given 231 static const uint32_t kTen8 = 100000000;
236 // number. We furthermore receive the maximum number of bits 'number' has. 232 static const uint32_t kTen9 = 1000000000;
237 // If number_bits == 0 then 0^-1 is returned 233
238 // The number of bits must be <= 32. 234 // Returns the biggest power of ten that is less than or equal to the given
239 // Precondition: number < (1 << (number_bits + 1)). 235 // number. We furthermore receive the maximum number of bits 'number' has.
240 static void BiggestPowerTen(uint32_t number, 236 // If number_bits == 0 then 0^-1 is returned
241 int number_bits, 237 // The number of bits must be <= 32.
242 uint32_t* power, 238 // Precondition: number < (1 << (number_bits + 1)).
243 int* exponent) { 239 static void BiggestPowerTen(uint32_t number,
244 ASSERT(number < (uint32_t)(1 << (number_bits + 1))); 240 int number_bits,
245 241 uint32_t* power,
246 switch (number_bits) { 242 int* exponent) {
247 case 32: 243 ASSERT(number < (uint32_t)(1 << (number_bits + 1)));
248 case 31: 244
249 case 30: 245 switch (number_bits) {
250 if (kTen9 <= number) { 246 case 32:
251 *power = kTen9; 247 case 31:
252 *exponent = 9; 248 case 30:
253 break; 249 if (kTen9 <= number) {
254 } // else fallthrough 250 *power = kTen9;
255 case 29: 251 *exponent = 9;
256 case 28: 252 break;
257 case 27: 253 } // else fallthrough
258 if (kTen8 <= number) { 254 case 29:
259 *power = kTen8; 255 case 28:
260 *exponent = 8; 256 case 27:
261 break; 257 if (kTen8 <= number) {
262 } // else fallthrough 258 *power = kTen8;
263 case 26: 259 *exponent = 8;
264 case 25: 260 break;
265 case 24: 261 } // else fallthrough
266 if (kTen7 <= number) { 262 case 26:
267 *power = kTen7; 263 case 25:
268 *exponent = 7; 264 case 24:
269 break; 265 if (kTen7 <= number) {
270 } // else fallthrough 266 *power = kTen7;
271 case 23: 267 *exponent = 7;
272 case 22: 268 break;
273 case 21: 269 } // else fallthrough
274 case 20: 270 case 23:
275 if (kTen6 <= number) { 271 case 22:
276 *power = kTen6; 272 case 21:
277 *exponent = 6; 273 case 20:
278 break; 274 if (kTen6 <= number) {
279 } // else fallthrough 275 *power = kTen6;
280 case 19: 276 *exponent = 6;
281 case 18: 277 break;
282 case 17: 278 } // else fallthrough
283 if (kTen5 <= number) { 279 case 19:
284 *power = kTen5; 280 case 18:
285 *exponent = 5; 281 case 17:
286 break; 282 if (kTen5 <= number) {
287 } // else fallthrough 283 *power = kTen5;
288 case 16: 284 *exponent = 5;
289 case 15: 285 break;
290 case 14: 286 } // else fallthrough
291 if (kTen4 <= number) { 287 case 16:
292 *power = kTen4; 288 case 15:
293 *exponent = 4; 289 case 14:
294 break; 290 if (kTen4 <= number) {
295 } // else fallthrough 291 *power = kTen4;
296 case 13: 292 *exponent = 4;
297 case 12: 293 break;
298 case 11: 294 } // else fallthrough
299 case 10: 295 case 13:
300 if (1000 <= number) { 296 case 12:
301 *power = 1000; 297 case 11:
302 *exponent = 3; 298 case 10:
303 break; 299 if (1000 <= number) {
304 } // else fallthrough 300 *power = 1000;
305 case 9: 301 *exponent = 3;
306 case 8: 302 break;
307 case 7: 303 } // else fallthrough
308 if (100 <= number) { 304 case 9:
309 *power = 100; 305 case 8:
310 *exponent = 2; 306 case 7:
311 break; 307 if (100 <= number) {
312 } // else fallthrough 308 *power = 100;
313 case 6: 309 *exponent = 2;
314 case 5: 310 break;
315 case 4: 311 } // else fallthrough
316 if (10 <= number) { 312 case 6:
317 *power = 10; 313 case 5:
318 *exponent = 1; 314 case 4:
319 break; 315 if (10 <= number) {
320 } // else fallthrough 316 *power = 10;
321 case 3: 317 *exponent = 1;
322 case 2: 318 break;
323 case 1: 319 } // else fallthrough
324 if (1 <= number) { 320 case 3:
325 *power = 1; 321 case 2:
326 *exponent = 0; 322 case 1:
327 break; 323 if (1 <= number) {
328 } // else fallthrough 324 *power = 1;
329 case 0: 325 *exponent = 0;
330 *power = 0; 326 break;
331 *exponent = -1; 327 } // else fallthrough
332 break; 328 case 0:
333 default: 329 *power = 0;
334 // Following assignments are here to silence compiler warnings. 330 *exponent = -1;
335 *power = 0; 331 break;
336 *exponent = 0; 332 default:
337 UNREACHABLE(); 333 // Following assignments are here to silence compiler warnings.
338 } 334 *power = 0;
335 *exponent = 0;
336 UNREACHABLE();
337 }
338 }
339
340 // Generates the digits of input number w.
341 // w is a floating-point number (DiyFp), consisting of a significand and an
342 // exponent. Its exponent is bounded by kMinimalTargetExponent and
343 // kMaximalTargetExponent.
344 // Hence -60 <= w.e() <= -32.
345 //
346 // Returns false if it fails, in which case the generated digits in the buffer
347 // should not be used.
348 // Preconditions:
349 // * low, w and high are correct up to 1 ulp (unit in the last place). That
350 // is, their error must be less than a unit of their last digits.
351 // * low.e() == w.e() == high.e()
352 // * low < w < high, and taking into account their error: low~ <= high~
353 // * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent
354 // Postconditions: returns false if procedure fails.
355 // otherwise:
356 // * buffer is not null-terminated, but len contains the number of digits.
357 // * buffer contains the shortest possible decimal digit-sequence
358 // such that LOW < buffer * 10^kappa < HIGH, where LOW and HIGH are the
359 // correct values of low and high (without their error).
360 // * if more than one decimal representation gives the minimal number of
361 // decimal digits then the one closest to W (where W is the correct value
362 // of w) is chosen.
363 // Remark: this procedure takes into account the imprecision of its input
364 // numbers. If the precision is not enough to guarantee all the postconditions
365 // then false is returned. This usually happens rarely (~0.5%).
366 //
367 // Say, for the sake of example, that
368 // w.e() == -48, and w.f() == 0x1234567890abcdef
369 // w's value can be computed by w.f() * 2^w.e()
370 // We can obtain w's integral digits by simply shifting w.f() by -w.e().
371 // -> w's integral part is 0x1234
372 // w's fractional part is therefore 0x567890abcdef.
373 // Printing w's integral part is easy (simply print 0x1234 in decimal).
374 // In order to print its fraction we repeatedly multiply the fraction by 10 and
375 // get each digit. Example the first digit after the point would be computed by
376 // (0x567890abcdef * 10) >> 48. -> 3
377 // The whole thing becomes slightly more complicated because we want to stop
378 // once we have enough digits. That is, once the digits inside the buffer
379 // represent 'w' we can stop. Everything inside the interval low - high
380 // represents w. However we have to pay attention to low, high and w's
381 // imprecision.
382 static bool DigitGen(DiyFp low,
383 DiyFp w,
384 DiyFp high,
385 Vector<char> buffer,
386 int* length,
387 int* kappa) {
388 ASSERT(low.e() == w.e() && w.e() == high.e());
389 ASSERT(low.f() + 1 <= high.f() - 1);
390 ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent);
391 // low, w and high are imprecise, but by less than one ulp (unit in the last
392 // place).
393 // If we remove (resp. add) 1 ulp from low (resp. high) we are certain that
394 // the new numbers are outside of the interval we want the final
395 // representation to lie in.
396 // Inversely adding (resp. removing) 1 ulp from low (resp. high) would yield
397 // numbers that are certain to lie in the interval. We will use this fact
398 // later on.
399 // We will now start by generating the digits within the uncertain
400 // interval. Later we will weed out representations that lie outside the safe
401 // interval and thus _might_ lie outside the correct interval.
402 uint64_t unit = 1;
403 DiyFp too_low = DiyFp(low.f() - unit, low.e());
404 DiyFp too_high = DiyFp(high.f() + unit, high.e());
405 // too_low and too_high are guaranteed to lie outside the interval we want the
406 // generated number in.
407 DiyFp unsafe_interval = DiyFp::Minus(too_high, too_low);
408 // We now cut the input number into two parts: the integral digits and the
409 // fractionals. We will not write any decimal separator though, but adapt
410 // kappa instead.
411 // Reminder: we are currently computing the digits (stored inside the buffer)
412 // such that: too_low < buffer * 10^kappa < too_high
413 // We use too_high for the digit_generation and stop as soon as possible.
414 // If we stop early we effectively round down.
415 DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());
416 // Division by one is a shift.
417 uint32_t integrals = static_cast<uint32_t>(too_high.f() >> -one.e());
418 // Modulo by one is an and.
419 uint64_t fractionals = too_high.f() & (one.f() - 1);
420 uint32_t divisor;
421 int divisor_exponent;
422 BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()), &divisor,
423 &divisor_exponent);
424 *kappa = divisor_exponent + 1;
425 *length = 0;
426 // Loop invariant: buffer = too_high / 10^kappa (integer division)
427 // The invariant holds for the first iteration: kappa has been initialized
428 // with the divisor exponent + 1. And the divisor is the biggest power of ten
429 // that is smaller than integrals.
430 while (*kappa > 0) {
431 char digit = static_cast<char>(integrals / divisor);
432 buffer[*length] = '0' + digit;
433 (*length)++;
434 integrals %= divisor;
435 (*kappa)--;
436 // Note that kappa now equals the exponent of the divisor and that the
437 // invariant thus holds again.
438 uint64_t rest =
439 (static_cast<uint64_t>(integrals) << -one.e()) + fractionals;
440 // Invariant: too_high = buffer * 10^kappa + DiyFp(rest, one.e())
441 // Reminder: unsafe_interval.e() == one.e()
442 if (rest < unsafe_interval.f()) {
443 // Rounding down (by not emitting the remaining digits) yields a number
444 // that lies within the unsafe interval.
445 return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f(),
446 unsafe_interval.f(), rest,
447 static_cast<uint64_t>(divisor) << -one.e(), unit);
339 } 448 }
340 449 divisor /= 10;
341 450 }
342 // Generates the digits of input number w. 451
343 // w is a floating-point number (DiyFp), consisting of a significand and an 452 // The integrals have been generated. We are at the point of the decimal
344 // exponent. Its exponent is bounded by kMinimalTargetExponent and 453 // separator. In the following loop we simply multiply the remaining digits by
345 // kMaximalTargetExponent. 454 // 10 and divide by one. We just need to pay attention to multiply associated
346 // Hence -60 <= w.e() <= -32. 455 // data (like the interval or 'unit'), too.
347 // 456 // Note that the multiplication by 10 does not overflow, because w.e >= -60
348 // Returns false if it fails, in which case the generated digits in the buff er 457 // and thus one.e >= -60.
349 // should not be used. 458 ASSERT(one.e() >= -60);
350 // Preconditions: 459 ASSERT(fractionals < one.f());
351 // * low, w and high are correct up to 1 ulp (unit in the last place). That 460 ASSERT(UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f());
352 // is, their error must be less than a unit of their last digits. 461 while (true) {
353 // * low.e() == w.e() == high.e() 462 fractionals *= 10;
354 // * low < w < high, and taking into account their error: low~ <= high~ 463 unit *= 10;
355 // * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent 464 unsafe_interval.set_f(unsafe_interval.f() * 10);
356 // Postconditions: returns false if procedure fails. 465 // Integer division by one.
357 // otherwise: 466 char digit = static_cast<char>(fractionals >> -one.e());
358 // * buffer is not null-terminated, but len contains the number of digit s. 467 buffer[*length] = '0' + digit;
359 // * buffer contains the shortest possible decimal digit-sequence 468 (*length)++;
360 // such that LOW < buffer * 10^kappa < HIGH, where LOW and HIGH are th e 469 fractionals &= one.f() - 1; // Modulo by one.
361 // correct values of low and high (without their error). 470 (*kappa)--;
362 // * if more than one decimal representation gives the minimal number of 471 if (fractionals < unsafe_interval.f()) {
363 // decimal digits then the one closest to W (where W is the correct va lue 472 return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f() * unit,
364 // of w) is chosen. 473 unsafe_interval.f(), fractionals, one.f(), unit);
365 // Remark: this procedure takes into account the imprecision of its input
366 // numbers. If the precision is not enough to guarantee all the postcondit ions
367 // then false is returned. This usually happens rarely (~0.5%).
368 //
369 // Say, for the sake of example, that
370 // w.e() == -48, and w.f() == 0x1234567890abcdef
371 // w's value can be computed by w.f() * 2^w.e()
372 // We can obtain w's integral digits by simply shifting w.f() by -w.e().
373 // -> w's integral part is 0x1234
374 // w's fractional part is therefore 0x567890abcdef.
375 // Printing w's integral part is easy (simply print 0x1234 in decimal).
376 // In order to print its fraction we repeatedly multiply the fraction by 10 and
377 // get each digit. Example the first digit after the point would be computed by
378 // (0x567890abcdef * 10) >> 48. -> 3
379 // The whole thing becomes slightly more complicated because we want to stop
380 // once we have enough digits. That is, once the digits inside the buffer
381 // represent 'w' we can stop. Everything inside the interval low - high
382 // represents w. However we have to pay attention to low, high and w's
383 // imprecision.
384 static bool DigitGen(DiyFp low,
385 DiyFp w,
386 DiyFp high,
387 Vector<char> buffer,
388 int* length,
389 int* kappa) {
390 ASSERT(low.e() == w.e() && w.e() == high.e());
391 ASSERT(low.f() + 1 <= high.f() - 1);
392 ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponen t);
393 // low, w and high are imprecise, but by less than one ulp (unit in the last
394 // place).
395 // If we remove (resp. add) 1 ulp from low (resp. high) we are certain t hat
396 // the new numbers are outside of the interval we want the final
397 // representation to lie in.
398 // Inversely adding (resp. removing) 1 ulp from low (resp. high) would y ield
399 // numbers that are certain to lie in the interval. We will use this fac t
400 // later on.
401 // We will now start by generating the digits within the uncertain
402 // interval. Later we will weed out representations that lie outside the safe
403 // interval and thus _might_ lie outside the correct interval.
404 uint64_t unit = 1;
405 DiyFp too_low = DiyFp(low.f() - unit, low.e());
406 DiyFp too_high = DiyFp(high.f() + unit, high.e());
407 // too_low and too_high are guaranteed to lie outside the interval we wa nt the
408 // generated number in.
409 DiyFp unsafe_interval = DiyFp::Minus(too_high, too_low);
410 // We now cut the input number into two parts: the integral digits and t he
411 // fractionals. We will not write any decimal separator though, but adap t
412 // kappa instead.
413 // Reminder: we are currently computing the digits (stored inside the bu ffer)
414 // such that: too_low < buffer * 10^kappa < too_high
415 // We use too_high for the digit_generation and stop as soon as possible .
416 // If we stop early we effectively round down.
417 DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());
418 // Division by one is a shift.
419 uint32_t integrals = static_cast<uint32_t>(too_high.f() >> -one.e());
420 // Modulo by one is an and.
421 uint64_t fractionals = too_high.f() & (one.f() - 1);
422 uint32_t divisor;
423 int divisor_exponent;
424 BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),
425 &divisor, &divisor_exponent);
426 *kappa = divisor_exponent + 1;
427 *length = 0;
428 // Loop invariant: buffer = too_high / 10^kappa (integer division)
429 // The invariant holds for the first iteration: kappa has been initializ ed
430 // with the divisor exponent + 1. And the divisor is the biggest power o f ten
431 // that is smaller than integrals.
432 while (*kappa > 0) {
433 char digit = static_cast<char>(integrals / divisor);
434 buffer[*length] = '0' + digit;
435 (*length)++;
436 integrals %= divisor;
437 (*kappa)--;
438 // Note that kappa now equals the exponent of the divisor and that t he
439 // invariant thus holds again.
440 uint64_t rest =
441 (static_cast<uint64_t>(integrals) << -one.e()) + fractionals;
442 // Invariant: too_high = buffer * 10^kappa + DiyFp(rest, one.e())
443 // Reminder: unsafe_interval.e() == one.e()
444 if (rest < unsafe_interval.f()) {
445 // Rounding down (by not emitting the remaining digits) yields a number
446 // that lies within the unsafe interval.
447 return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f(),
448 unsafe_interval.f(), rest,
449 static_cast<uint64_t>(divisor) << -one.e(), uni t);
450 }
451 divisor /= 10;
452 }
453
454 // The integrals have been generated. We are at the point of the decimal
455 // separator. In the following loop we simply multiply the remaining dig its by
456 // 10 and divide by one. We just need to pay attention to multiply assoc iated
457 // data (like the interval or 'unit'), too.
458 // Note that the multiplication by 10 does not overflow, because w.e >= -60
459 // and thus one.e >= -60.
460 ASSERT(one.e() >= -60);
461 ASSERT(fractionals < one.f());
462 ASSERT(UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f());
463 while (true) {
464 fractionals *= 10;
465 unit *= 10;
466 unsafe_interval.set_f(unsafe_interval.f() * 10);
467 // Integer division by one.
468 char digit = static_cast<char>(fractionals >> -one.e());
469 buffer[*length] = '0' + digit;
470 (*length)++;
471 fractionals &= one.f() - 1; // Modulo by one.
472 (*kappa)--;
473 if (fractionals < unsafe_interval.f()) {
474 return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f() * unit,
475 unsafe_interval.f(), fractionals, one.f(), unit );
476 }
477 }
478 } 474 }
479 475 }
480 476 }
481 477
482 // Generates (at most) requested_digits digits of input number w. 478 // Generates (at most) requested_digits digits of input number w.
483 // w is a floating-point number (DiyFp), consisting of a significand and an 479 // w is a floating-point number (DiyFp), consisting of a significand and an
484 // exponent. Its exponent is bounded by kMinimalTargetExponent and 480 // exponent. Its exponent is bounded by kMinimalTargetExponent and
485 // kMaximalTargetExponent. 481 // kMaximalTargetExponent.
486 // Hence -60 <= w.e() <= -32. 482 // Hence -60 <= w.e() <= -32.
487 // 483 //
488 // Returns false if it fails, in which case the generated digits in the buff er 484 // Returns false if it fails, in which case the generated digits in the buffer
489 // should not be used. 485 // should not be used.
490 // Preconditions: 486 // Preconditions:
491 // * w is correct up to 1 ulp (unit in the last place). That 487 // * w is correct up to 1 ulp (unit in the last place). That
492 // is, its error must be strictly less than a unit of its last digit. 488 // is, its error must be strictly less than a unit of its last digit.
493 // * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent 489 // * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent
494 // 490 //
495 // Postconditions: returns false if procedure fails. 491 // Postconditions: returns false if procedure fails.
496 // otherwise: 492 // otherwise:
497 // * buffer is not null-terminated, but length contains the number of 493 // * buffer is not null-terminated, but length contains the number of
498 // digits. 494 // digits.
499 // * the representation in buffer is the most precise representation of 495 // * the representation in buffer is the most precise representation of
500 // requested_digits digits. 496 // requested_digits digits.
501 // * buffer contains at most requested_digits digits of w. If there are less 497 // * buffer contains at most requested_digits digits of w. If there are less
502 // than requested_digits digits then some trailing '0's have been remo ved. 498 // than requested_digits digits then some trailing '0's have been removed.
503 // * kappa is such that 499 // * kappa is such that
504 // w = buffer * 10^kappa + eps with |eps| < 10^kappa / 2. 500 // w = buffer * 10^kappa + eps with |eps| < 10^kappa / 2.
505 // 501 //
506 // Remark: This procedure takes into account the imprecision of its input 502 // Remark: This procedure takes into account the imprecision of its input
507 // numbers. If the precision is not enough to guarantee all the postcondit ions 503 // numbers. If the precision is not enough to guarantee all the postconditions
508 // then false is returned. This usually happens rarely, but the failure-ra te 504 // then false is returned. This usually happens rarely, but the failure-rate
509 // increases with higher requested_digits. 505 // increases with higher requested_digits.
510 static bool DigitGenCounted(DiyFp w, 506 static bool DigitGenCounted(DiyFp w,
511 int requested_digits, 507 int requested_digits,
512 Vector<char> buffer, 508 Vector<char> buffer,
513 int* length, 509 int* length,
514 int* kappa) { 510 int* kappa) {
515 ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponen t); 511 ASSERT(kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent);
516 ASSERT(kMinimalTargetExponent >= -60); 512 ASSERT(kMinimalTargetExponent >= -60);
517 ASSERT(kMaximalTargetExponent <= -32); 513 ASSERT(kMaximalTargetExponent <= -32);
518 // w is assumed to have an error less than 1 unit. Whenever w is scaled we 514 // w is assumed to have an error less than 1 unit. Whenever w is scaled we
519 // also scale its error. 515 // also scale its error.
520 uint64_t w_error = 1; 516 uint64_t w_error = 1;
521 // We cut the input number into two parts: the integral digits and the 517 // We cut the input number into two parts: the integral digits and the
522 // fractional digits. We don't emit any decimal separator, but adapt kap pa 518 // fractional digits. We don't emit any decimal separator, but adapt kappa
523 // instead. Example: instead of writing "1.2" we put "12" into the buffe r and 519 // instead. Example: instead of writing "1.2" we put "12" into the buffer and
524 // increase kappa by 1. 520 // increase kappa by 1.
525 DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e()); 521 DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());
526 // Division by one is a shift. 522 // Division by one is a shift.
527 uint32_t integrals = static_cast<uint32_t>(w.f() >> -one.e()); 523 uint32_t integrals = static_cast<uint32_t>(w.f() >> -one.e());
528 // Modulo by one is an and. 524 // Modulo by one is an and.
529 uint64_t fractionals = w.f() & (one.f() - 1); 525 uint64_t fractionals = w.f() & (one.f() - 1);
530 uint32_t divisor; 526 uint32_t divisor;
531 int divisor_exponent; 527 int divisor_exponent;
532 BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()), 528 BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()), &divisor,
533 &divisor, &divisor_exponent); 529 &divisor_exponent);
534 *kappa = divisor_exponent + 1; 530 *kappa = divisor_exponent + 1;
535 *length = 0; 531 *length = 0;
536 532
537 // Loop invariant: buffer = w / 10^kappa (integer division) 533 // Loop invariant: buffer = w / 10^kappa (integer division)
538 // The invariant holds for the first iteration: kappa has been initializ ed 534 // The invariant holds for the first iteration: kappa has been initialized
539 // with the divisor exponent + 1. And the divisor is the biggest power o f ten 535 // with the divisor exponent + 1. And the divisor is the biggest power of ten
540 // that is smaller than 'integrals'. 536 // that is smaller than 'integrals'.
541 while (*kappa > 0) { 537 while (*kappa > 0) {
542 char digit = static_cast<char>(integrals / divisor); 538 char digit = static_cast<char>(integrals / divisor);
543 buffer[*length] = '0' + digit; 539 buffer[*length] = '0' + digit;
544 (*length)++; 540 (*length)++;
545 requested_digits--; 541 requested_digits--;
546 integrals %= divisor; 542 integrals %= divisor;
547 (*kappa)--; 543 (*kappa)--;
548 // Note that kappa now equals the exponent of the divisor and that t he 544 // Note that kappa now equals the exponent of the divisor and that the
549 // invariant thus holds again. 545 // invariant thus holds again.
550 if (requested_digits == 0) break; 546 if (requested_digits == 0)
551 divisor /= 10; 547 break;
552 } 548 divisor /= 10;
553 549 }
554 if (requested_digits == 0) { 550
555 uint64_t rest = 551 if (requested_digits == 0) {
556 (static_cast<uint64_t>(integrals) << -one.e()) + fractionals; 552 uint64_t rest =
557 return RoundWeedCounted(buffer, *length, rest, 553 (static_cast<uint64_t>(integrals) << -one.e()) + fractionals;
558 static_cast<uint64_t>(divisor) << -one.e(), w_error, 554 return RoundWeedCounted(buffer, *length, rest,
559 kappa); 555 static_cast<uint64_t>(divisor) << -one.e(), w_error,
560 } 556 kappa);
561 557 }
562 // The integrals have been generated. We are at the point of the decimal 558
563 // separator. In the following loop we simply multiply the remaining dig its by 559 // The integrals have been generated. We are at the point of the decimal
564 // 10 and divide by one. We just need to pay attention to multiply assoc iated 560 // separator. In the following loop we simply multiply the remaining digits by
565 // data (the 'unit'), too. 561 // 10 and divide by one. We just need to pay attention to multiply associated
566 // Note that the multiplication by 10 does not overflow, because w.e >= -60 562 // data (the 'unit'), too.
567 // and thus one.e >= -60. 563 // Note that the multiplication by 10 does not overflow, because w.e >= -60
568 ASSERT(one.e() >= -60); 564 // and thus one.e >= -60.
569 ASSERT(fractionals < one.f()); 565 ASSERT(one.e() >= -60);
570 ASSERT(UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f()); 566 ASSERT(fractionals < one.f());
571 while (requested_digits > 0 && fractionals > w_error) { 567 ASSERT(UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f());
572 fractionals *= 10; 568 while (requested_digits > 0 && fractionals > w_error) {
573 w_error *= 10; 569 fractionals *= 10;
574 // Integer division by one. 570 w_error *= 10;
575 char digit = static_cast<char>(fractionals >> -one.e()); 571 // Integer division by one.
576 buffer[*length] = '0' + digit; 572 char digit = static_cast<char>(fractionals >> -one.e());
577 (*length)++; 573 buffer[*length] = '0' + digit;
578 requested_digits--; 574 (*length)++;
579 fractionals &= one.f() - 1; // Modulo by one. 575 requested_digits--;
580 (*kappa)--; 576 fractionals &= one.f() - 1; // Modulo by one.
581 } 577 (*kappa)--;
582 if (requested_digits != 0) return false; 578 }
583 return RoundWeedCounted(buffer, *length, fractionals, one.f(), w_error, 579 if (requested_digits != 0)
584 kappa); 580 return false;
585 } 581 return RoundWeedCounted(buffer, *length, fractionals, one.f(), w_error,
586 582 kappa);
587 583 }
588 // Provides a decimal representation of v. 584
589 // Returns true if it succeeds, otherwise the result cannot be trusted. 585 // Provides a decimal representation of v.
590 // There will be *length digits inside the buffer (not null-terminated). 586 // Returns true if it succeeds, otherwise the result cannot be trusted.
591 // If the function returns true then 587 // There will be *length digits inside the buffer (not null-terminated).
592 // v == (double) (buffer * 10^decimal_exponent). 588 // If the function returns true then
593 // The digits in the buffer are the shortest representation possible: no 589 // v == (double) (buffer * 10^decimal_exponent).
594 // 0.09999999999999999 instead of 0.1. The shorter representation will even be 590 // The digits in the buffer are the shortest representation possible: no
595 // chosen even if the longer one would be closer to v. 591 // 0.09999999999999999 instead of 0.1. The shorter representation will even be
596 // The last digit will be closest to the actual v. That is, even if several 592 // chosen even if the longer one would be closer to v.
597 // digits might correctly yield 'v' when read again, the closest will be 593 // The last digit will be closest to the actual v. That is, even if several
598 // computed. 594 // digits might correctly yield 'v' when read again, the closest will be
599 static bool Grisu3(double v, 595 // computed.
600 Vector<char> buffer, 596 static bool Grisu3(double v,
601 int* length, 597 Vector<char> buffer,
602 int* decimal_exponent) { 598 int* length,
603 DiyFp w = Double(v).AsNormalizedDiyFp(); 599 int* decimal_exponent) {
604 // boundary_minus and boundary_plus are the boundaries between v and its 600 DiyFp w = Double(v).AsNormalizedDiyFp();
605 // closest floating-point neighbors. Any number strictly between 601 // boundary_minus and boundary_plus are the boundaries between v and its
606 // boundary_minus and boundary_plus will round to v when convert to a do uble. 602 // closest floating-point neighbors. Any number strictly between
607 // Grisu3 will never output representations that lie exactly on a bounda ry. 603 // boundary_minus and boundary_plus will round to v when convert to a double.
608 DiyFp boundary_minus, boundary_plus; 604 // Grisu3 will never output representations that lie exactly on a boundary.
609 Double(v).NormalizedBoundaries(&boundary_minus, &boundary_plus); 605 DiyFp boundary_minus, boundary_plus;
610 ASSERT(boundary_plus.e() == w.e()); 606 Double(v).NormalizedBoundaries(&boundary_minus, &boundary_plus);
611 DiyFp ten_mk; // Cached power of ten: 10^-k 607 ASSERT(boundary_plus.e() == w.e());
612 int mk; // -k 608 DiyFp ten_mk; // Cached power of ten: 10^-k
613 int ten_mk_minimal_binary_exponent = 609 int mk; // -k
614 kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize); 610 int ten_mk_minimal_binary_exponent =
615 int ten_mk_maximal_binary_exponent = 611 kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize);
616 kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize); 612 int ten_mk_maximal_binary_exponent =
617 PowersOfTenCache::GetCachedPowerForBinaryExponentRange( 613 kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize);
618 ten_mk_minimal_bi nary_exponent, 614 PowersOfTenCache::GetCachedPowerForBinaryExponentRange(
619 ten_mk_maximal_bi nary_exponent, 615 ten_mk_minimal_binary_exponent, ten_mk_maximal_binary_exponent, &ten_mk,
620 &ten_mk, &mk); 616 &mk);
621 ASSERT((kMinimalTargetExponent <= w.e() + ten_mk.e() + 617 ASSERT(
622 DiyFp::kSignificandSize) && 618 (kMinimalTargetExponent <=
623 (kMaximalTargetExponent >= w.e() + ten_mk.e() + 619 w.e() + ten_mk.e() + DiyFp::kSignificandSize) &&
624 DiyFp::kSignificandSize)); 620 (kMaximalTargetExponent >= w.e() + ten_mk.e() + DiyFp::kSignificandSize));
625 // Note that ten_mk is only an approximation of 10^-k. A DiyFp only cont ains a 621 // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a
626 // 64 bit significand and ten_mk is thus only precise up to 64 bits. 622 // 64 bit significand and ten_mk is thus only precise up to 64 bits.
627 623
628 // The DiyFp::Times procedure rounds its result, and ten_mk is approxima ted 624 // The DiyFp::Times procedure rounds its result, and ten_mk is approximated
629 // too. The variable scaled_w (as well as scaled_boundary_minus/plus) ar e now 625 // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now
630 // off by a small amount. 626 // off by a small amount.
631 // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_ w. 627 // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w.
632 // In other words: let f = scaled_w.f() and e = scaled_w.e(), then 628 // In other words: let f = scaled_w.f() and e = scaled_w.e(), then
633 // (f-1) * 2^e < w*10^k < (f+1) * 2^e 629 // (f-1) * 2^e < w*10^k < (f+1) * 2^e
634 DiyFp scaled_w = DiyFp::Times(w, ten_mk); 630 DiyFp scaled_w = DiyFp::Times(w, ten_mk);
635 ASSERT(scaled_w.e() == 631 ASSERT(scaled_w.e() ==
636 boundary_plus.e() + ten_mk.e() + DiyFp::kSignificandSize); 632 boundary_plus.e() + ten_mk.e() + DiyFp::kSignificandSize);
637 // In theory it would be possible to avoid some recomputations by comput ing 633 // In theory it would be possible to avoid some recomputations by computing
638 // the difference between w and boundary_minus/plus (a power of 2) and t o 634 // the difference between w and boundary_minus/plus (a power of 2) and to
639 // compute scaled_boundary_minus/plus by subtracting/adding from 635 // compute scaled_boundary_minus/plus by subtracting/adding from
640 // scaled_w. However the code becomes much less readable and the speed 636 // scaled_w. However the code becomes much less readable and the speed
641 // enhancements are not terriffic. 637 // enhancements are not terriffic.
642 DiyFp scaled_boundary_minus = DiyFp::Times(boundary_minus, ten_mk); 638 DiyFp scaled_boundary_minus = DiyFp::Times(boundary_minus, ten_mk);
643 DiyFp scaled_boundary_plus = DiyFp::Times(boundary_plus, ten_mk); 639 DiyFp scaled_boundary_plus = DiyFp::Times(boundary_plus, ten_mk);
644 640
645 // DigitGen will generate the digits of scaled_w. Therefore we have 641 // DigitGen will generate the digits of scaled_w. Therefore we have
646 // v == (double) (scaled_w * 10^-mk). 642 // v == (double) (scaled_w * 10^-mk).
647 // Set decimal_exponent == -mk and pass it to DigitGen. If scaled_w is n ot an 643 // Set decimal_exponent == -mk and pass it to DigitGen. If scaled_w is not an
648 // integer than it will be updated. For instance if scaled_w == 1.23 the n 644 // integer than it will be updated. For instance if scaled_w == 1.23 then
649 // the buffer will be filled with "123" und the decimal_exponent will be 645 // the buffer will be filled with "123" und the decimal_exponent will be
650 // decreased by 2. 646 // decreased by 2.
651 int kappa; 647 int kappa;
652 bool result = DigitGen(scaled_boundary_minus, scaled_w, scaled_boundary_ plus, 648 bool result = DigitGen(scaled_boundary_minus, scaled_w, scaled_boundary_plus,
653 buffer, length, &kappa); 649 buffer, length, &kappa);
654 *decimal_exponent = -mk + kappa; 650 *decimal_exponent = -mk + kappa;
655 return result; 651 return result;
656 } 652 }
657 653
658 654 // The "counted" version of grisu3 (see above) only generates requested_digits
659 // The "counted" version of grisu3 (see above) only generates requested_digi ts 655 // number of digits. This version does not generate the shortest representation,
660 // number of digits. This version does not generate the shortest representat ion, 656 // and with enough requested digits 0.1 will at some point print as 0.9999999...
661 // and with enough requested digits 0.1 will at some point print as 0.999999 9... 657 // Grisu3 is too imprecise for real halfway cases (1.5 will not work) and
662 // Grisu3 is too imprecise for real halfway cases (1.5 will not work) and 658 // therefore the rounding strategy for halfway cases is irrelevant.
663 // therefore the rounding strategy for halfway cases is irrelevant. 659 static bool Grisu3Counted(double v,
664 static bool Grisu3Counted(double v, 660 int requested_digits,
665 int requested_digits, 661 Vector<char> buffer,
666 Vector<char> buffer, 662 int* length,
667 int* length, 663 int* decimal_exponent) {
668 int* decimal_exponent) { 664 DiyFp w = Double(v).AsNormalizedDiyFp();
669 DiyFp w = Double(v).AsNormalizedDiyFp(); 665 DiyFp ten_mk; // Cached power of ten: 10^-k
670 DiyFp ten_mk; // Cached power of ten: 10^-k 666 int mk; // -k
671 int mk; // -k 667 int ten_mk_minimal_binary_exponent =
672 int ten_mk_minimal_binary_exponent = 668 kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize);
673 kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize); 669 int ten_mk_maximal_binary_exponent =
674 int ten_mk_maximal_binary_exponent = 670 kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize);
675 kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize); 671 PowersOfTenCache::GetCachedPowerForBinaryExponentRange(
676 PowersOfTenCache::GetCachedPowerForBinaryExponentRange( 672 ten_mk_minimal_binary_exponent, ten_mk_maximal_binary_exponent, &ten_mk,
677 ten_mk_minimal_bi nary_exponent, 673 &mk);
678 ten_mk_maximal_bi nary_exponent, 674 ASSERT(
679 &ten_mk, &mk); 675 (kMinimalTargetExponent <=
680 ASSERT((kMinimalTargetExponent <= w.e() + ten_mk.e() + 676 w.e() + ten_mk.e() + DiyFp::kSignificandSize) &&
681 DiyFp::kSignificandSize) && 677 (kMaximalTargetExponent >= w.e() + ten_mk.e() + DiyFp::kSignificandSize));
682 (kMaximalTargetExponent >= w.e() + ten_mk.e() + 678 // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a
683 DiyFp::kSignificandSize)); 679 // 64 bit significand and ten_mk is thus only precise up to 64 bits.
684 // Note that ten_mk is only an approximation of 10^-k. A DiyFp only cont ains a 680
685 // 64 bit significand and ten_mk is thus only precise up to 64 bits. 681 // The DiyFp::Times procedure rounds its result, and ten_mk is approximated
686 682 // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now
687 // The DiyFp::Times procedure rounds its result, and ten_mk is approxima ted 683 // off by a small amount.
688 // too. The variable scaled_w (as well as scaled_boundary_minus/plus) ar e now 684 // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w.
689 // off by a small amount. 685 // In other words: let f = scaled_w.f() and e = scaled_w.e(), then
690 // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_ w. 686 // (f-1) * 2^e < w*10^k < (f+1) * 2^e
691 // In other words: let f = scaled_w.f() and e = scaled_w.e(), then 687 DiyFp scaled_w = DiyFp::Times(w, ten_mk);
692 // (f-1) * 2^e < w*10^k < (f+1) * 2^e 688
693 DiyFp scaled_w = DiyFp::Times(w, ten_mk); 689 // We now have (double) (scaled_w * 10^-mk).
694 690 // DigitGen will generate the first requested_digits digits of scaled_w and
695 // We now have (double) (scaled_w * 10^-mk). 691 // return together with a kappa such that scaled_w ~= buffer * 10^kappa. (It
696 // DigitGen will generate the first requested_digits digits of scaled_w and 692 // will not always be exactly the same since DigitGenCounted only produces a
697 // return together with a kappa such that scaled_w ~= buffer * 10^kappa. (It 693 // limited number of digits.)
698 // will not always be exactly the same since DigitGenCounted only produc es a 694 int kappa;
699 // limited number of digits.) 695 bool result =
700 int kappa; 696 DigitGenCounted(scaled_w, requested_digits, buffer, length, &kappa);
701 bool result = DigitGenCounted(scaled_w, requested_digits, 697 *decimal_exponent = -mk + kappa;
702 buffer, length, &kappa); 698 return result;
703 *decimal_exponent = -mk + kappa; 699 }
704 return result; 700
705 } 701 bool FastDtoa(double v,
706 702 FastDtoaMode mode,
707 703 int requested_digits,
708 bool FastDtoa(double v, 704 Vector<char> buffer,
709 FastDtoaMode mode, 705 int* length,
710 int requested_digits, 706 int* decimal_point) {
711 Vector<char> buffer, 707 ASSERT(v > 0);
712 int* length, 708 ASSERT(!Double(v).IsSpecial());
713 int* decimal_point) { 709
714 ASSERT(v > 0); 710 bool result = false;
715 ASSERT(!Double(v).IsSpecial()); 711 int decimal_exponent = 0;
716 712 switch (mode) {
717 bool result = false; 713 case FAST_DTOA_SHORTEST:
718 int decimal_exponent = 0; 714 result = Grisu3(v, buffer, length, &decimal_exponent);
719 switch (mode) { 715 break;
720 case FAST_DTOA_SHORTEST: 716 case FAST_DTOA_PRECISION:
721 result = Grisu3(v, buffer, length, &decimal_exponent); 717 result =
722 break; 718 Grisu3Counted(v, requested_digits, buffer, length, &decimal_exponent);
723 case FAST_DTOA_PRECISION: 719 break;
724 result = Grisu3Counted(v, requested_digits, 720 default:
725 buffer, length, &decimal_exponent); 721 UNREACHABLE();
726 break; 722 }
727 default: 723 if (result) {
728 UNREACHABLE(); 724 *decimal_point = *length + decimal_exponent;
729 } 725 buffer[*length] = '\0';
730 if (result) { 726 }
731 *decimal_point = *length + decimal_exponent; 727 return result;
732 buffer[*length] = '\0'; 728 }
733 }
734 return result;
735 }
736 729
737 } // namespace double_conversion 730 } // namespace double_conversion
738 731
739 } // namespace WTF 732 } // namespace WTF
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