1 | // |
---|
2 | // Lol Engine |
---|
3 | // |
---|
4 | // Copyright: (c) 2010-2011 Sam Hocevar <sam@hocevar.net> |
---|
5 | // This program is free software; you can redistribute it and/or |
---|
6 | // modify it under the terms of the Do What The Fuck You Want To |
---|
7 | // Public License, Version 2, as published by Sam Hocevar. See |
---|
8 | // http://sam.zoy.org/projects/COPYING.WTFPL for more details. |
---|
9 | // |
---|
10 | |
---|
11 | #if defined HAVE_CONFIG_H |
---|
12 | # include "config.h" |
---|
13 | #endif |
---|
14 | |
---|
15 | #include <cstring> |
---|
16 | #include <cstdio> |
---|
17 | |
---|
18 | #include "core.h" |
---|
19 | |
---|
20 | using namespace std; |
---|
21 | |
---|
22 | namespace lol |
---|
23 | { |
---|
24 | |
---|
25 | real::real(float f) { *this = (double)f; } |
---|
26 | real::real(int i) { *this = (double)i; } |
---|
27 | real::real(unsigned int i) { *this = (double)i; } |
---|
28 | |
---|
29 | real::real(double d) |
---|
30 | { |
---|
31 | union { double d; uint64_t x; } u = { d }; |
---|
32 | |
---|
33 | uint32_t sign = (u.x >> 63) << 31; |
---|
34 | uint32_t exponent = (u.x << 1) >> 53; |
---|
35 | |
---|
36 | switch (exponent) |
---|
37 | { |
---|
38 | case 0x00: |
---|
39 | m_signexp = sign; |
---|
40 | break; |
---|
41 | case 0x7ff: |
---|
42 | m_signexp = sign | 0x7fffffffu; |
---|
43 | break; |
---|
44 | default: |
---|
45 | m_signexp = sign | (exponent + (1 << 30) - (1 << 10)); |
---|
46 | break; |
---|
47 | } |
---|
48 | |
---|
49 | m_mantissa[0] = u.x >> 36; |
---|
50 | m_mantissa[1] = u.x >> 20; |
---|
51 | m_mantissa[2] = u.x >> 4; |
---|
52 | m_mantissa[3] = u.x << 12; |
---|
53 | memset(m_mantissa + 4, 0, sizeof(m_mantissa) - 4 * sizeof(m_mantissa[0])); |
---|
54 | } |
---|
55 | |
---|
56 | real::operator float() const { return (float)(double)(*this); } |
---|
57 | real::operator int() const { return (int)(double)(*this); } |
---|
58 | real::operator unsigned int() const { return (unsigned int)(double)(*this); } |
---|
59 | |
---|
60 | real::operator double() const |
---|
61 | { |
---|
62 | union { double d; uint64_t x; } u; |
---|
63 | |
---|
64 | /* Get sign */ |
---|
65 | u.x = m_signexp >> 31; |
---|
66 | u.x <<= 11; |
---|
67 | |
---|
68 | /* Compute new exponent */ |
---|
69 | uint32_t exponent = (m_signexp << 1) >> 1; |
---|
70 | int e = (int)exponent - (1 << 30) + (1 << 10); |
---|
71 | |
---|
72 | if (e < 0) |
---|
73 | u.x <<= 52; |
---|
74 | else if (e >= 0x7ff) |
---|
75 | { |
---|
76 | u.x |= 0x7ff; |
---|
77 | u.x <<= 52; |
---|
78 | } |
---|
79 | else |
---|
80 | { |
---|
81 | u.x |= e; |
---|
82 | |
---|
83 | /* Store mantissa if necessary */ |
---|
84 | u.x <<= 16; |
---|
85 | u.x |= m_mantissa[0]; |
---|
86 | u.x <<= 16; |
---|
87 | u.x |= m_mantissa[1]; |
---|
88 | u.x <<= 16; |
---|
89 | u.x |= m_mantissa[2]; |
---|
90 | u.x <<= 4; |
---|
91 | u.x |= m_mantissa[3] >> 12; |
---|
92 | /* Rounding */ |
---|
93 | u.x += (m_mantissa[3] >> 11) & 1; |
---|
94 | } |
---|
95 | |
---|
96 | return u.d; |
---|
97 | } |
---|
98 | |
---|
99 | real real::operator -() const |
---|
100 | { |
---|
101 | real ret = *this; |
---|
102 | ret.m_signexp ^= 0x80000000u; |
---|
103 | return ret; |
---|
104 | } |
---|
105 | |
---|
106 | real real::operator +(real const &x) const |
---|
107 | { |
---|
108 | if (x.m_signexp << 1 == 0) |
---|
109 | return *this; |
---|
110 | |
---|
111 | /* Ensure both arguments are positive. Otherwise, switch signs, |
---|
112 | * or replace + with -. */ |
---|
113 | if (m_signexp >> 31) |
---|
114 | return -(-*this + -x); |
---|
115 | |
---|
116 | if (x.m_signexp >> 31) |
---|
117 | return *this - (-x); |
---|
118 | |
---|
119 | /* Ensure *this has the larger exponent (no need for the mantissa to |
---|
120 | * be larger, as in subtraction). Otherwise, switch. */ |
---|
121 | if ((m_signexp << 1) < (x.m_signexp << 1)) |
---|
122 | return x + *this; |
---|
123 | |
---|
124 | real ret; |
---|
125 | |
---|
126 | int e1 = m_signexp - (1 << 30) + 1; |
---|
127 | int e2 = x.m_signexp - (1 << 30) + 1; |
---|
128 | |
---|
129 | int bigoff = (e1 - e2) / (sizeof(uint16_t) * 8); |
---|
130 | int off = e1 - e2 - bigoff * (sizeof(uint16_t) * 8); |
---|
131 | |
---|
132 | if (bigoff > BIGITS) |
---|
133 | return *this; |
---|
134 | |
---|
135 | ret.m_signexp = m_signexp; |
---|
136 | |
---|
137 | uint32_t carry = 0; |
---|
138 | for (int i = BIGITS; i--; ) |
---|
139 | { |
---|
140 | carry += m_mantissa[i]; |
---|
141 | if (i - bigoff >= 0) |
---|
142 | carry += x.m_mantissa[i - bigoff] >> off; |
---|
143 | |
---|
144 | if (i - bigoff > 0) |
---|
145 | carry += (x.m_mantissa[i - bigoff - 1] << (16 - off)) & 0xffffu; |
---|
146 | else if (i - bigoff == 0) |
---|
147 | carry += 0x0001u << (16 - off); |
---|
148 | |
---|
149 | ret.m_mantissa[i] = carry; |
---|
150 | carry >>= 16; |
---|
151 | } |
---|
152 | |
---|
153 | /* Renormalise in case we overflowed the mantissa */ |
---|
154 | if (carry) |
---|
155 | { |
---|
156 | carry--; |
---|
157 | for (int i = 0; i < BIGITS; i++) |
---|
158 | { |
---|
159 | uint16_t tmp = ret.m_mantissa[i]; |
---|
160 | ret.m_mantissa[i] = (carry << 15) | (tmp >> 1); |
---|
161 | carry = tmp & 0x0001u; |
---|
162 | } |
---|
163 | ret.m_signexp++; |
---|
164 | } |
---|
165 | |
---|
166 | return ret; |
---|
167 | } |
---|
168 | |
---|
169 | real real::operator -(real const &x) const |
---|
170 | { |
---|
171 | if (x.m_signexp << 1 == 0) |
---|
172 | return *this; |
---|
173 | |
---|
174 | /* Ensure both arguments are positive. Otherwise, switch signs, |
---|
175 | * or replace - with +. */ |
---|
176 | if (m_signexp >> 31) |
---|
177 | return -(-*this + x); |
---|
178 | |
---|
179 | if (x.m_signexp >> 31) |
---|
180 | return (*this) + (-x); |
---|
181 | |
---|
182 | /* Ensure *this is larger than x */ |
---|
183 | if (*this < x) |
---|
184 | return -(x - *this); |
---|
185 | |
---|
186 | real ret; |
---|
187 | |
---|
188 | int e1 = m_signexp - (1 << 30) + 1; |
---|
189 | int e2 = x.m_signexp - (1 << 30) + 1; |
---|
190 | |
---|
191 | int bigoff = (e1 - e2) / (sizeof(uint16_t) * 8); |
---|
192 | int off = e1 - e2 - bigoff * (sizeof(uint16_t) * 8); |
---|
193 | |
---|
194 | if (bigoff > BIGITS) |
---|
195 | return *this; |
---|
196 | |
---|
197 | ret.m_signexp = m_signexp; |
---|
198 | |
---|
199 | int32_t carry = 0; |
---|
200 | for (int i = 0; i < bigoff; i++) |
---|
201 | { |
---|
202 | carry -= x.m_mantissa[BIGITS - i]; |
---|
203 | carry = (carry & 0xffff0000u) | (carry >> 16); |
---|
204 | } |
---|
205 | carry -= x.m_mantissa[BIGITS - 1 - bigoff] & ((1 << off) - 1); |
---|
206 | carry /= (1 << off); |
---|
207 | |
---|
208 | for (int i = BIGITS; i--; ) |
---|
209 | { |
---|
210 | carry += m_mantissa[i]; |
---|
211 | if (i - bigoff >= 0) |
---|
212 | carry -= x.m_mantissa[i - bigoff] >> off; |
---|
213 | |
---|
214 | if (i - bigoff > 0) |
---|
215 | carry -= (x.m_mantissa[i - bigoff - 1] << (16 - off)) & 0xffffu; |
---|
216 | else if (i - bigoff == 0) |
---|
217 | carry -= 0x0001u << (16 - off); |
---|
218 | |
---|
219 | ret.m_mantissa[i] = carry; |
---|
220 | carry = (carry & 0xffff0000u) | (carry >> 16); |
---|
221 | } |
---|
222 | |
---|
223 | carry += 1; |
---|
224 | |
---|
225 | /* Renormalise if we underflowed the mantissa */ |
---|
226 | if (carry == 0) |
---|
227 | { |
---|
228 | /* How much do we need to shift the mantissa? FIXME: this could |
---|
229 | * be computed above */ |
---|
230 | off = 0; |
---|
231 | for (int i = 0; i < BIGITS; i++) |
---|
232 | { |
---|
233 | if (!ret.m_mantissa[i]) |
---|
234 | { |
---|
235 | off += sizeof(uint16_t) * 8; |
---|
236 | continue; |
---|
237 | } |
---|
238 | |
---|
239 | for (uint16_t tmp = ret.m_mantissa[i]; tmp < 0x8000u; tmp <<= 1) |
---|
240 | off++; |
---|
241 | break; |
---|
242 | } |
---|
243 | if (off == BIGITS * sizeof(uint16_t) * 8) |
---|
244 | ret.m_signexp &= 0x80000000u; |
---|
245 | else |
---|
246 | { |
---|
247 | off++; /* Shift one more to get rid of the leading one */ |
---|
248 | ret.m_signexp -= off; |
---|
249 | |
---|
250 | bigoff = off / (sizeof(uint16_t) * 8); |
---|
251 | off -= bigoff * sizeof(uint16_t) * 8; |
---|
252 | |
---|
253 | for (int i = 0; i < BIGITS; i++) |
---|
254 | { |
---|
255 | uint16_t tmp = 0; |
---|
256 | if (i + bigoff < BIGITS) |
---|
257 | tmp |= ret.m_mantissa[i + bigoff] << off; |
---|
258 | if (i + bigoff + 1 < BIGITS) |
---|
259 | tmp |= ret.m_mantissa[i + bigoff + 1] >> (16 - off); |
---|
260 | ret.m_mantissa[i] = tmp; |
---|
261 | } |
---|
262 | } |
---|
263 | } |
---|
264 | |
---|
265 | return ret; |
---|
266 | } |
---|
267 | |
---|
268 | real real::operator *(real const &x) const |
---|
269 | { |
---|
270 | real ret; |
---|
271 | |
---|
272 | if (m_signexp << 1 == 0 || x.m_signexp << 1 == 0) |
---|
273 | { |
---|
274 | ret = (m_signexp << 1 == 0) ? *this : x; |
---|
275 | ret.m_signexp ^= x.m_signexp & 0x80000000u; |
---|
276 | return ret; |
---|
277 | } |
---|
278 | |
---|
279 | ret.m_signexp = (m_signexp ^ x.m_signexp) & 0x80000000u; |
---|
280 | int e = (m_signexp & 0x7fffffffu) - (1 << 30) + 1 |
---|
281 | + (x.m_signexp & 0x7fffffffu) - (1 << 30) + 1; |
---|
282 | |
---|
283 | /* Accumulate low order product; no need to store it, we just |
---|
284 | * want the carry value */ |
---|
285 | uint64_t carry = 0; |
---|
286 | for (int i = 0; i < BIGITS; i++) |
---|
287 | { |
---|
288 | for (int j = 0; j < i + 1; j++) |
---|
289 | carry += (uint32_t)m_mantissa[BIGITS - 1 - j] |
---|
290 | * (uint32_t)x.m_mantissa[BIGITS - 1 + j - i]; |
---|
291 | carry >>= 16; |
---|
292 | } |
---|
293 | |
---|
294 | for (int i = 0; i < BIGITS; i++) |
---|
295 | { |
---|
296 | for (int j = i + 1; j < BIGITS; j++) |
---|
297 | carry += (uint32_t)m_mantissa[BIGITS - 1 - j] |
---|
298 | * (uint32_t)x.m_mantissa[j - 1 - i]; |
---|
299 | |
---|
300 | carry += m_mantissa[BIGITS - 1 - i]; |
---|
301 | carry += x.m_mantissa[BIGITS - 1 - i]; |
---|
302 | ret.m_mantissa[BIGITS - 1 - i] = carry & 0xffffu; |
---|
303 | carry >>= 16; |
---|
304 | } |
---|
305 | |
---|
306 | /* Renormalise in case we overflowed the mantissa */ |
---|
307 | if (carry) |
---|
308 | { |
---|
309 | carry--; |
---|
310 | for (int i = 0; i < BIGITS; i++) |
---|
311 | { |
---|
312 | uint16_t tmp = ret.m_mantissa[i]; |
---|
313 | ret.m_mantissa[i] = (carry << 15) | (tmp >> 1); |
---|
314 | carry = tmp & 0x0001u; |
---|
315 | } |
---|
316 | e++; |
---|
317 | } |
---|
318 | |
---|
319 | ret.m_signexp |= e + (1 << 30) - 1; |
---|
320 | |
---|
321 | return ret; |
---|
322 | } |
---|
323 | |
---|
324 | real real::operator /(real const &x) const |
---|
325 | { |
---|
326 | return *this * re(x); |
---|
327 | } |
---|
328 | |
---|
329 | real &real::operator +=(real const &x) |
---|
330 | { |
---|
331 | real tmp = *this; |
---|
332 | return *this = tmp + x; |
---|
333 | } |
---|
334 | |
---|
335 | real &real::operator -=(real const &x) |
---|
336 | { |
---|
337 | real tmp = *this; |
---|
338 | return *this = tmp - x; |
---|
339 | } |
---|
340 | |
---|
341 | real &real::operator *=(real const &x) |
---|
342 | { |
---|
343 | real tmp = *this; |
---|
344 | return *this = tmp * x; |
---|
345 | } |
---|
346 | |
---|
347 | real &real::operator /=(real const &x) |
---|
348 | { |
---|
349 | real tmp = *this; |
---|
350 | return *this = tmp / x; |
---|
351 | } |
---|
352 | |
---|
353 | real real::operator <<(int x) const |
---|
354 | { |
---|
355 | real tmp = *this; |
---|
356 | return tmp <<= x; |
---|
357 | } |
---|
358 | |
---|
359 | real real::operator >>(int x) const |
---|
360 | { |
---|
361 | real tmp = *this; |
---|
362 | return tmp >>= x; |
---|
363 | } |
---|
364 | |
---|
365 | real &real::operator <<=(int x) |
---|
366 | { |
---|
367 | if (m_signexp << 1) |
---|
368 | m_signexp += x; |
---|
369 | return *this; |
---|
370 | } |
---|
371 | |
---|
372 | real &real::operator >>=(int x) |
---|
373 | { |
---|
374 | if (m_signexp << 1) |
---|
375 | m_signexp -= x; |
---|
376 | return *this; |
---|
377 | } |
---|
378 | |
---|
379 | bool real::operator ==(real const &x) const |
---|
380 | { |
---|
381 | if ((m_signexp << 1) == 0 && (x.m_signexp << 1) == 0) |
---|
382 | return true; |
---|
383 | |
---|
384 | if (m_signexp != x.m_signexp) |
---|
385 | return false; |
---|
386 | |
---|
387 | return memcmp(m_mantissa, x.m_mantissa, sizeof(m_mantissa)) == 0; |
---|
388 | } |
---|
389 | |
---|
390 | bool real::operator !=(real const &x) const |
---|
391 | { |
---|
392 | return !(*this == x); |
---|
393 | } |
---|
394 | |
---|
395 | bool real::operator <(real const &x) const |
---|
396 | { |
---|
397 | /* Ensure both numbers are positive */ |
---|
398 | if (m_signexp >> 31) |
---|
399 | return (x.m_signexp >> 31) ? -*this > -x : true; |
---|
400 | |
---|
401 | if (x.m_signexp >> 31) |
---|
402 | return false; |
---|
403 | |
---|
404 | /* Compare all relevant bits */ |
---|
405 | if (m_signexp != x.m_signexp) |
---|
406 | return m_signexp < x.m_signexp; |
---|
407 | |
---|
408 | for (int i = 0; i < BIGITS; i++) |
---|
409 | if (m_mantissa[i] != x.m_mantissa[i]) |
---|
410 | return m_mantissa[i] < x.m_mantissa[i]; |
---|
411 | |
---|
412 | return false; |
---|
413 | } |
---|
414 | |
---|
415 | bool real::operator <=(real const &x) const |
---|
416 | { |
---|
417 | return !(*this > x); |
---|
418 | } |
---|
419 | |
---|
420 | bool real::operator >(real const &x) const |
---|
421 | { |
---|
422 | /* Ensure both numbers are positive */ |
---|
423 | if (m_signexp >> 31) |
---|
424 | return (x.m_signexp >> 31) ? -*this < -x : false; |
---|
425 | |
---|
426 | if (x.m_signexp >> 31) |
---|
427 | return true; |
---|
428 | |
---|
429 | /* Compare all relevant bits */ |
---|
430 | if (m_signexp != x.m_signexp) |
---|
431 | return m_signexp > x.m_signexp; |
---|
432 | |
---|
433 | for (int i = 0; i < BIGITS; i++) |
---|
434 | if (m_mantissa[i] != x.m_mantissa[i]) |
---|
435 | return m_mantissa[i] > x.m_mantissa[i]; |
---|
436 | |
---|
437 | return false; |
---|
438 | } |
---|
439 | |
---|
440 | bool real::operator >=(real const &x) const |
---|
441 | { |
---|
442 | return !(*this < x); |
---|
443 | } |
---|
444 | |
---|
445 | bool real::operator !() const |
---|
446 | { |
---|
447 | return !(bool)*this; |
---|
448 | } |
---|
449 | |
---|
450 | real::operator bool() const |
---|
451 | { |
---|
452 | /* A real is "true" if it is non-zero (exponent is non-zero) AND |
---|
453 | * not NaN (exponent is not full bits OR higher order mantissa is zero) */ |
---|
454 | uint32_t exponent = m_signexp << 1; |
---|
455 | return exponent && (~exponent || m_mantissa[0] == 0); |
---|
456 | } |
---|
457 | |
---|
458 | real re(real const &x) |
---|
459 | { |
---|
460 | if (!(x.m_signexp << 1)) |
---|
461 | { |
---|
462 | real ret = x; |
---|
463 | ret.m_signexp = x.m_signexp | 0x7fffffffu; |
---|
464 | ret.m_mantissa[0] = 0; |
---|
465 | return ret; |
---|
466 | } |
---|
467 | |
---|
468 | /* Use the system's float inversion to approximate 1/x */ |
---|
469 | union { float f; uint32_t x; } u = { 1.0f }, v = { 1.0f }; |
---|
470 | v.x |= (uint32_t)x.m_mantissa[0] << 7; |
---|
471 | v.x |= (uint32_t)x.m_mantissa[1] >> 9; |
---|
472 | v.f = 1.0 / v.f; |
---|
473 | |
---|
474 | real ret; |
---|
475 | ret.m_mantissa[0] = (v.x >> 7) & 0xffffu; |
---|
476 | ret.m_mantissa[1] = (v.x << 9) & 0xffffu; |
---|
477 | |
---|
478 | uint32_t sign = x.m_signexp & 0x80000000u; |
---|
479 | ret.m_signexp = sign; |
---|
480 | |
---|
481 | int exponent = (x.m_signexp & 0x7fffffffu) + 1; |
---|
482 | exponent = -exponent + (v.x >> 23) - (u.x >> 23); |
---|
483 | ret.m_signexp |= (exponent - 1) & 0x7fffffffu; |
---|
484 | |
---|
485 | /* Five steps of Newton-Raphson seems enough for 32-bigit reals. */ |
---|
486 | real two = 2; |
---|
487 | ret = ret * (two - ret * x); |
---|
488 | ret = ret * (two - ret * x); |
---|
489 | ret = ret * (two - ret * x); |
---|
490 | ret = ret * (two - ret * x); |
---|
491 | ret = ret * (two - ret * x); |
---|
492 | |
---|
493 | return ret; |
---|
494 | } |
---|
495 | |
---|
496 | real sqrt(real const &x) |
---|
497 | { |
---|
498 | /* if zero, return x */ |
---|
499 | if (!(x.m_signexp << 1)) |
---|
500 | return x; |
---|
501 | |
---|
502 | /* if negative, return NaN */ |
---|
503 | if (x.m_signexp >> 31) |
---|
504 | { |
---|
505 | real ret; |
---|
506 | ret.m_signexp = 0x7fffffffu; |
---|
507 | ret.m_mantissa[0] = 0xffffu; |
---|
508 | return ret; |
---|
509 | } |
---|
510 | |
---|
511 | /* Use the system's float inversion to approximate 1/sqrt(x). First |
---|
512 | * we construct a float in the [1..4[ range that has roughly the same |
---|
513 | * mantissa as our real. Its exponent is 0 or 1, depending on the |
---|
514 | * partity of x. The final exponent is 0, -1 or -2. We use the final |
---|
515 | * exponent and final mantissa to pre-fill the result. */ |
---|
516 | union { float f; uint32_t x; } u = { 1.0f }, v = { 2.0f }; |
---|
517 | v.x -= ((x.m_signexp & 1) << 23); |
---|
518 | v.x |= (uint32_t)x.m_mantissa[0] << 7; |
---|
519 | v.x |= (uint32_t)x.m_mantissa[1] >> 9; |
---|
520 | v.f = 1.0 / sqrtf(v.f); |
---|
521 | |
---|
522 | real ret; |
---|
523 | ret.m_mantissa[0] = (v.x >> 7) & 0xffffu; |
---|
524 | ret.m_mantissa[1] = (v.x << 9) & 0xffffu; |
---|
525 | |
---|
526 | uint32_t sign = x.m_signexp & 0x80000000u; |
---|
527 | ret.m_signexp = sign; |
---|
528 | |
---|
529 | uint32_t exponent = (x.m_signexp & 0x7fffffffu); |
---|
530 | exponent = ((1 << 30) + (1 << 29) -1) - (exponent + 1) / 2; |
---|
531 | exponent = exponent + (v.x >> 23) - (u.x >> 23); |
---|
532 | ret.m_signexp |= exponent & 0x7fffffffu; |
---|
533 | |
---|
534 | /* Five steps of Newton-Raphson seems enough for 32-bigit reals. */ |
---|
535 | real three = 3; |
---|
536 | ret = ret * (three - ret * ret * x); |
---|
537 | ret.m_signexp--; |
---|
538 | ret = ret * (three - ret * ret * x); |
---|
539 | ret.m_signexp--; |
---|
540 | ret = ret * (three - ret * ret * x); |
---|
541 | ret.m_signexp--; |
---|
542 | ret = ret * (three - ret * ret * x); |
---|
543 | ret.m_signexp--; |
---|
544 | ret = ret * (three - ret * ret * x); |
---|
545 | ret.m_signexp--; |
---|
546 | |
---|
547 | return ret * x; |
---|
548 | } |
---|
549 | |
---|
550 | real fabs(real const &x) |
---|
551 | { |
---|
552 | real ret = x; |
---|
553 | ret.m_signexp &= 0x7fffffffu; |
---|
554 | return ret; |
---|
555 | } |
---|
556 | |
---|
557 | static real fastlog(real const &x) |
---|
558 | { |
---|
559 | /* This fast log method is tuned to work on the [1..2] range and |
---|
560 | * no effort whatsoever was made to improve convergence outside this |
---|
561 | * domain of validity. It can converge pretty fast, provided we use |
---|
562 | * the following variable substitutions: |
---|
563 | * y = sqrt(x) |
---|
564 | * z = (y - 1) / (y + 1) |
---|
565 | * |
---|
566 | * And the following identities: |
---|
567 | * ln(x) = 2 ln(y) |
---|
568 | * = 2 ln((1 + z) / (1 - z)) |
---|
569 | * = 4 z (1 + z^2 / 3 + z^4 / 5 + z^6 / 7...) |
---|
570 | * |
---|
571 | * Any additional sqrt() call would halve the convergence time, but |
---|
572 | * would also impact the final precision. For now we stick with one |
---|
573 | * sqrt() call. */ |
---|
574 | real y = sqrt(x); |
---|
575 | real z = (y - (real)1) / (y + (real)1), z2 = z * z, zn = z2; |
---|
576 | real sum = 1.0; |
---|
577 | |
---|
578 | for (int i = 3; i < 200; i += 2) |
---|
579 | { |
---|
580 | sum += zn / (real)i; |
---|
581 | zn *= z2; |
---|
582 | } |
---|
583 | |
---|
584 | return z * (sum << 2); |
---|
585 | } |
---|
586 | |
---|
587 | static real LOG_2 = fastlog((real)2); |
---|
588 | |
---|
589 | real log(real const &x) |
---|
590 | { |
---|
591 | /* Strategy for log(x): if x = 2^E*M then log(x) = E log(2) + log(M), |
---|
592 | * with the property that M is in [1..2[, so fastlog() applies here. */ |
---|
593 | real tmp = x; |
---|
594 | if (x.m_signexp >> 31 || x.m_signexp == 0) |
---|
595 | { |
---|
596 | tmp.m_signexp = 0xffffffffu; |
---|
597 | tmp.m_mantissa[0] = 0xffffu; |
---|
598 | return tmp; |
---|
599 | } |
---|
600 | tmp.m_signexp = (1 << 30) - 1; |
---|
601 | return (real)(x.m_signexp - (1 << 30) + 1) * LOG_2 + fastlog(tmp); |
---|
602 | } |
---|
603 | |
---|
604 | real exp(real const &x) |
---|
605 | { |
---|
606 | /* Strategy for exp(x): the Taylor series does not converge very fast |
---|
607 | * with large positive or negative values. |
---|
608 | * |
---|
609 | * However, we know that the result is going to be in the form M*2^E, |
---|
610 | * where M is the mantissa and E the exponent. We first try to predict |
---|
611 | * a value for E, which is approximately log2(exp(x)) = x / log(2). |
---|
612 | * |
---|
613 | * Let E0 be an integer close to x / log(2). We need to find a value x0 |
---|
614 | * such that exp(x) = 2^E0 * exp(x0). We get x0 = x - E0 log(2). |
---|
615 | * |
---|
616 | * Thus the final algorithm: |
---|
617 | * int E0 = x / log(2) |
---|
618 | * real x0 = x - E0 log(2) |
---|
619 | * real x1 = exp(x0) |
---|
620 | * return x1 * 2^E0 |
---|
621 | */ |
---|
622 | int e0 = x / LOG_2; |
---|
623 | real x0 = x - (real)e0 * LOG_2; |
---|
624 | real x1 = 1.0, fact = 1.0, xn = x0; |
---|
625 | |
---|
626 | for (int i = 1; i < 100; i++) |
---|
627 | { |
---|
628 | fact *= (real)i; |
---|
629 | x1 += xn / fact; |
---|
630 | xn *= x0; |
---|
631 | } |
---|
632 | |
---|
633 | x1.m_signexp += e0; |
---|
634 | return x1; |
---|
635 | } |
---|
636 | |
---|
637 | real sin(real const &x) |
---|
638 | { |
---|
639 | real ret = 0.0, fact = 1.0, xn = x, x2 = x * x; |
---|
640 | |
---|
641 | for (int i = 1; ; i += 2) |
---|
642 | { |
---|
643 | real newret = ret + xn / fact; |
---|
644 | if (ret == newret) |
---|
645 | break; |
---|
646 | ret = newret; |
---|
647 | xn *= x2; |
---|
648 | fact *= (real)(-(i + 1) * (i + 2)); |
---|
649 | } |
---|
650 | |
---|
651 | return ret; |
---|
652 | } |
---|
653 | |
---|
654 | real cos(real const &x) |
---|
655 | { |
---|
656 | real ret = 0.0, fact = 1.0, xn = 1.0, x2 = x * x; |
---|
657 | |
---|
658 | for (int i = 1; ; i += 2) |
---|
659 | { |
---|
660 | real newret = ret + xn / fact; |
---|
661 | if (ret == newret) |
---|
662 | break; |
---|
663 | ret = newret; |
---|
664 | xn *= x2; |
---|
665 | fact *= (real)(-i * (i + 1)); |
---|
666 | } |
---|
667 | |
---|
668 | return ret; |
---|
669 | } |
---|
670 | |
---|
671 | real atan(real const &x) |
---|
672 | { |
---|
673 | /* Computing atan(x): we choose a different Taylor series depending on |
---|
674 | * the value of x to help with convergence. |
---|
675 | * |
---|
676 | * If |x| < 0.5 we evaluate atan(y) near 0: |
---|
677 | * atan(y) = y - y^3/3 + y^5/5 - y^7/7 + y^9/9 ... |
---|
678 | * |
---|
679 | * If 0.5 <= |x| < 1.5 we evaluate atan(1+y) near 0: |
---|
680 | * atan(1+y) = π/4 + y/(1*2^1) - y^2/(2*2^1) + y^3/(3*2^2) |
---|
681 | * - y^5/(5*2^3) + y^6/(6*2^3) - y^7/(7*2^4) |
---|
682 | * + y^9/(9*2^5) - y^10/(10*2^5) + y^11/(11*2^6) ... |
---|
683 | * |
---|
684 | * If 1.5 <= |x| < 2 we evaluate atan(sqrt(3)+2y) near 0: |
---|
685 | * atan(sqrt(3)+2y) = π/3 + 1/2 y - sqrt(3)/2 y^2/2 |
---|
686 | * + y^3/3 - sqrt(3)/2 y^4/4 + 1/2 y^5/5 |
---|
687 | * - 1/2 y^7/7 + sqrt(3)/2 y^8/8 |
---|
688 | * - y^9/9 + sqrt(3)/2 y^10/10 - 1/2 y^11/11 |
---|
689 | * + 1/2 y^13/13 - sqrt(3)/2 y^14/14 |
---|
690 | * + y^15/15 - sqrt(3)/2 y^16/16 + 1/2 y^17/17 ... |
---|
691 | * |
---|
692 | * If |x| >= 2 we evaluate atan(y) near +∞: |
---|
693 | * atan(y) = π/2 - y^-1 + y^-3/3 - y^-5/5 + y^-7/7 - y^-9/9 ... |
---|
694 | */ |
---|
695 | real absx = fabs(x); |
---|
696 | |
---|
697 | if (absx < (real::R_1 >> 1)) |
---|
698 | { |
---|
699 | real ret = x, xn = x, mx2 = -x * x; |
---|
700 | for (int i = 3; i < 100; i += 2) |
---|
701 | { |
---|
702 | xn *= mx2; |
---|
703 | ret += xn / (real)i; |
---|
704 | } |
---|
705 | return ret; |
---|
706 | } |
---|
707 | |
---|
708 | real ret = 0; |
---|
709 | |
---|
710 | if (absx < (real::R_3 >> 1)) |
---|
711 | { |
---|
712 | real y = real::R_1 - absx; |
---|
713 | real yn = y, my2 = -y * y; |
---|
714 | for (int i = 0; i < 200; i += 2) |
---|
715 | { |
---|
716 | ret += (yn / (real)(2 * i + 1)) >> (i + 1); |
---|
717 | yn *= y; |
---|
718 | ret += (yn / (real)(2 * i + 2)) >> (i + 1); |
---|
719 | yn *= y; |
---|
720 | ret += (yn / (real)(2 * i + 3)) >> (i + 2); |
---|
721 | yn *= my2; |
---|
722 | } |
---|
723 | ret = real::R_PI_4 - ret; |
---|
724 | } |
---|
725 | else if (absx < real::R_2) |
---|
726 | { |
---|
727 | real y = (absx - real::R_SQRT3) >> 1; |
---|
728 | real yn = y, my2 = -y * y; |
---|
729 | for (int i = 1; i < 200; i += 6) |
---|
730 | { |
---|
731 | ret += (yn / (real)i) >> 1; |
---|
732 | yn *= y; |
---|
733 | ret -= (real::R_SQRT3 >> 1) * yn / (real)(i + 1); |
---|
734 | yn *= y; |
---|
735 | ret += yn / (real)(i + 2); |
---|
736 | yn *= y; |
---|
737 | ret -= (real::R_SQRT3 >> 1) * yn / (real)(i + 3); |
---|
738 | yn *= y; |
---|
739 | ret += (yn / (real)(i + 4)) >> 1; |
---|
740 | yn *= my2; |
---|
741 | } |
---|
742 | ret = real::R_PI_3 + ret; |
---|
743 | } |
---|
744 | else |
---|
745 | { |
---|
746 | real y = re(absx); |
---|
747 | real yn = y, my2 = -y * y; |
---|
748 | ret = y; |
---|
749 | for (int i = 3; i < 120; i += 2) |
---|
750 | { |
---|
751 | yn *= my2; |
---|
752 | ret += yn / (real)i; |
---|
753 | } |
---|
754 | ret = real::R_PI_2 - ret; |
---|
755 | } |
---|
756 | |
---|
757 | /* Propagate sign */ |
---|
758 | ret.m_signexp |= (x.m_signexp & 0x80000000u); |
---|
759 | return ret; |
---|
760 | } |
---|
761 | |
---|
762 | void real::print(int ndigits) const |
---|
763 | { |
---|
764 | real const r1 = 1, r10 = 10; |
---|
765 | real x = *this; |
---|
766 | |
---|
767 | if (x.m_signexp >> 31) |
---|
768 | { |
---|
769 | printf("-"); |
---|
770 | x = -x; |
---|
771 | } |
---|
772 | |
---|
773 | /* Normalise x so that mantissa is in [1..9.999] */ |
---|
774 | int exponent = 0; |
---|
775 | if (x.m_signexp) |
---|
776 | { |
---|
777 | for (real div = r1, newdiv; true; div = newdiv) |
---|
778 | { |
---|
779 | newdiv = div * r10; |
---|
780 | if (x < newdiv) |
---|
781 | { |
---|
782 | x /= div; |
---|
783 | break; |
---|
784 | } |
---|
785 | exponent++; |
---|
786 | } |
---|
787 | for (real mul = 1, newx; true; mul *= r10) |
---|
788 | { |
---|
789 | newx = x * mul; |
---|
790 | if (newx >= r1) |
---|
791 | { |
---|
792 | x = newx; |
---|
793 | break; |
---|
794 | } |
---|
795 | exponent--; |
---|
796 | } |
---|
797 | } |
---|
798 | |
---|
799 | /* Print digits */ |
---|
800 | for (int i = 0; i < ndigits; i++) |
---|
801 | { |
---|
802 | int digit = (int)x; |
---|
803 | printf("%i", digit); |
---|
804 | if (i == 0) |
---|
805 | printf("."); |
---|
806 | x -= real(digit); |
---|
807 | x *= r10; |
---|
808 | } |
---|
809 | |
---|
810 | /* Print exponent information */ |
---|
811 | if (exponent < 0) |
---|
812 | printf("e-%i", -exponent); |
---|
813 | else if (exponent > 0) |
---|
814 | printf("e+%i", exponent); |
---|
815 | |
---|
816 | printf("\n"); |
---|
817 | } |
---|
818 | |
---|
819 | static real fast_pi() |
---|
820 | { |
---|
821 | /* Approximate Pi using Machin's formula: 16*atan(1/5)-4*atan(1/239) */ |
---|
822 | real ret = 0.0, x0 = 5.0, x1 = 239.0; |
---|
823 | real const m0 = -x0 * x0, m1 = -x1 * x1, r16 = 16.0, r4 = 4.0; |
---|
824 | |
---|
825 | /* Degree 240 is required for 512-bit mantissa precision */ |
---|
826 | for (int i = 1; i < 240; i += 2) |
---|
827 | { |
---|
828 | ret += r16 / (x0 * (real)i) - r4 / (x1 * (real)i); |
---|
829 | x0 *= m0; |
---|
830 | x1 *= m1; |
---|
831 | } |
---|
832 | |
---|
833 | return ret; |
---|
834 | } |
---|
835 | |
---|
836 | real const real::R_0 = (real)0.0; |
---|
837 | real const real::R_1 = (real)1.0; |
---|
838 | real const real::R_2 = (real)2.0; |
---|
839 | real const real::R_3 = (real)3.0; |
---|
840 | real const real::R_10 = (real)10.0; |
---|
841 | |
---|
842 | real const real::R_E = exp(R_1); |
---|
843 | real const real::R_LN2 = log(R_2); |
---|
844 | real const real::R_LN10 = log(R_10); |
---|
845 | real const real::R_LOG2E = re(R_LN2); |
---|
846 | real const real::R_LOG10E = re(R_LN10); |
---|
847 | real const real::R_PI = fast_pi(); |
---|
848 | real const real::R_PI_2 = R_PI >> 1; |
---|
849 | real const real::R_PI_3 = R_PI / R_3; |
---|
850 | real const real::R_PI_4 = R_PI >> 2; |
---|
851 | real const real::R_1_PI = re(R_PI); |
---|
852 | real const real::R_2_PI = R_1_PI << 1; |
---|
853 | real const real::R_2_SQRTPI = re(sqrt(R_PI)) << 1; |
---|
854 | real const real::R_SQRT2 = sqrt(R_2); |
---|
855 | real const real::R_SQRT3 = sqrt(R_3); |
---|
856 | real const real::R_SQRT1_2 = R_SQRT2 >> 1; |
---|
857 | |
---|
858 | } /* namespace lol */ |
---|
859 | |
---|