1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
|
/***************************************************************************
* Copyright (c) Johan Mabille, Sylvain Corlay, Wolf Vollprecht and *
* Martin Renou *
* Copyright (c) QuantStack *
* Copyright (c) Serge Guelton *
* *
* Distributed under the terms of the BSD 3-Clause License. *
* *
* The full license is in the file LICENSE, distributed with this software. *
****************************************************************************/
#include <cmath>
#include <cstdint>
#include <cstring>
namespace xsimd
{
namespace detail
{
/* origin: boost/simd/arch/common/scalar/function/rem_pio2.hpp */
/*
* ====================================================
* copyright 2016 NumScale SAS
*
* Distributed under the Boost Software License, Version 1.0.
* (See copy at http://boost.org/LICENSE_1_0.txt)
* ====================================================
*/
#if defined(_MSC_VER)
#define ONCE0 \
__pragma(warning(push)) \
__pragma(warning(disable : 4127)) while (0) \
__pragma(warning(pop)) /**/
#else
#define ONCE0 while (0)
#endif
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
#if defined(__GNUC__) && defined(__BYTE_ORDER__)
#if __BYTE_ORDER__ == __ORDER_LITTLE_ENDIAN__
#define XSIMD_LITTLE_ENDIAN
#endif
#elif defined(_WIN32)
// We can safely assume that Windows is always little endian
#define XSIMD_LITTLE_ENDIAN
#elif defined(i386) || defined(i486) || defined(intel) || defined(x86) || defined(i86pc) || defined(__alpha) || defined(__osf__)
#define XSIMD_LITTLE_ENDIAN
#endif
#ifdef XSIMD_LITTLE_ENDIAN
#define LOW_WORD_IDX 0
#define HIGH_WORD_IDX sizeof(std::uint32_t)
#else
#define LOW_WORD_IDX sizeof(std::uint32_t)
#define HIGH_WORD_IDX 0
#endif
#define GET_HIGH_WORD(i, d) \
do \
{ \
double f = (d); \
std::memcpy(&(i), reinterpret_cast<char*>(&f) + HIGH_WORD_IDX, \
sizeof(std::uint32_t)); \
} \
ONCE0 \
/**/
#define GET_LOW_WORD(i, d) \
do \
{ \
double f = (d); \
std::memcpy(&(i), reinterpret_cast<char*>(&f) + LOW_WORD_IDX, \
sizeof(std::uint32_t)); \
} \
ONCE0 \
/**/
#define SET_HIGH_WORD(d, v) \
do \
{ \
double f = (d); \
std::uint32_t value = (v); \
std::memcpy(reinterpret_cast<char*>(&f) + HIGH_WORD_IDX, \
&value, sizeof(std::uint32_t)); \
(d) = f; \
} \
ONCE0 \
/**/
#define SET_LOW_WORD(d, v) \
do \
{ \
double f = (d); \
std::uint32_t value = (v); \
std::memcpy(reinterpret_cast<char*>(&f) + LOW_WORD_IDX, \
&value, sizeof(std::uint32_t)); \
(d) = f; \
} \
ONCE0 \
/**/
/*
* __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
* double x[],y[]; int e0,nx,prec; int ipio2[];
*
* __kernel_rem_pio2 return the last three digits of N with
* y = x - N*pi/2
* so that |y| < pi/2.
*
* The method is to compute the integer (mod 8) and fraction parts of
* (2/pi)*x without doing the full multiplication. In general we
* skip the part of the product that are known to be a huge integer (
* more accurately, = 0 mod 8 ). Thus the number of operations are
* independent of the exponent of the input.
*
* (2/pi) is represented by an array of 24-bit integers in ipio2[].
*
* Input parameters:
* x[] The input value (must be positive) is broken into nx
* pieces of 24-bit integers in double precision format.
* x[i] will be the i-th 24 bit of x. The scaled exponent
* of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
* match x's up to 24 bits.
*
* Example of breaking a double positive z into x[0]+x[1]+x[2]:
* e0 = ilogb(z)-23
* z = scalbn(z,-e0)
* for i = 0,1,2
* x[i] = floor(z)
* z = (z-x[i])*2**24
*
*
* y[] ouput result in an array of double precision numbers.
* The dimension of y[] is:
* 24-bit precision 1
* 53-bit precision 2
* 64-bit precision 2
* 113-bit precision 3
* The actual value is the sum of them. Thus for 113-bit
* precison, one may have to do something like:
*
* long double t,w,r_head, r_tail;
* t = (long double)y[2] + (long double)y[1];
* w = (long double)y[0];
* r_head = t+w;
* r_tail = w - (r_head - t);
*
* e0 The exponent of x[0]
*
* nx dimension of x[]
*
* prec an integer indicating the precision:
* 0 24 bits (single)
* 1 53 bits (double)
* 2 64 bits (extended)
* 3 113 bits (quad)
*
* ipio2[]
* integer array, contains the (24*i)-th to (24*i+23)-th
* bit of 2/pi after binary point. The corresponding
* floating value is
*
* ipio2[i] * 2^(-24(i+1)).
*
* External function:
* double scalbn(), floor();
*
*
* Here is the description of some local variables:
*
* jk jk+1 is the initial number of terms of ipio2[] needed
* in the computation. The recommended value is 2,3,4,
* 6 for single, double, extended,and quad.
*
* jz local integer variable indicating the number of
* terms of ipio2[] used.
*
* jx nx - 1
*
* jv index for pointing to the suitable ipio2[] for the
* computation. In general, we want
* ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
* is an integer. Thus
* e0-3-24*jv >= 0 or (e0-3)/24 >= jv
* Hence jv = max(0,(e0-3)/24).
*
* jp jp+1 is the number of terms in PIo2[] needed, jp = jk.
*
* q[] double array with integral value, representing the
* 24-bits chunk of the product of x and 2/pi.
*
* q0 the corresponding exponent of q[0]. Note that the
* exponent for q[i] would be q0-24*i.
*
* PIo2[] double precision array, obtained by cutting pi/2
* into 24 bits chunks.
*
* f[] ipio2[] in floating point
*
* iq[] integer array by breaking up q[] in 24-bits chunk.
*
* fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
*
* ih integer. If >0 it indicates q[] is >= 0.5, hence
* it also indicates the *sign* of the result.
*
*/
inline int32_t __kernel_rem_pio2(double* x, double* y, int32_t e0, int32_t nx, int32_t prec, const int32_t* ipio2) noexcept
{
static const int32_t init_jk[] = { 2, 3, 4, 6 }; /* initial value for jk */
static const double PIo2[] = {
1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
};
static const double
zero
= 0.0,
one = 1.0,
two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
int32_t jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih;
double z, fw, f[20], fq[20], q[20];
/* initialize jk*/
jk = init_jk[prec];
jp = jk;
/* determine jx,jv,q0, note that 3>q0 */
jx = nx - 1;
jv = (e0 - 3) / 24;
if (jv < 0)
jv = 0;
q0 = e0 - 24 * (jv + 1);
/* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
j = jv - jx;
m = jx + jk;
for (i = 0; i <= m; i++, j++)
f[i] = (j < 0) ? zero : (double)ipio2[j];
/* compute q[0],q[1],...q[jk] */
for (i = 0; i <= jk; i++)
{
for (j = 0, fw = 0.0; j <= jx; j++)
fw += x[j] * f[jx + i - j];
q[i] = fw;
}
jz = jk;
recompute:
/* distill q[] into iq[] reversingly */
for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--)
{
fw = (double)((int32_t)(twon24 * z));
iq[i] = (int)(z - two24 * fw);
z = q[j - 1] + fw;
}
/* compute n */
z = std::scalbn(z, q0); /* actual value of z */
z -= 8.0 * std::floor(z * 0.125); /* trim off integer >= 8 */
n = (int32_t)z;
z -= (double)n;
ih = 0;
if (q0 > 0)
{ /* need iq[jz-1] to determine n */
i = (iq[jz - 1] >> (24 - q0));
n += i;
iq[jz - 1] -= i << (24 - q0);
ih = iq[jz - 1] >> (23 - q0);
}
else if (q0 == 0)
ih = iq[jz - 1] >> 23;
else if (z >= 0.5)
ih = 2;
if (ih > 0)
{ /* q > 0.5 */
n += 1;
carry = 0;
for (i = 0; i < jz; i++)
{ /* compute 1-q */
j = iq[i];
if (carry == 0)
{
if (j != 0)
{
carry = 1;
iq[i] = 0x1000000 - j;
}
}
else
iq[i] = 0xffffff - j;
}
if (q0 > 0)
{ /* rare case: chance is 1 in 12 */
switch (q0)
{
case 1:
iq[jz - 1] &= 0x7fffff;
break;
case 2:
iq[jz - 1] &= 0x3fffff;
break;
}
}
if (ih == 2)
{
z = one - z;
if (carry != 0)
z -= std::scalbn(one, q0);
}
}
/* check if recomputation is needed */
if (z == zero)
{
j = 0;
for (i = jz - 1; i >= jk; i--)
j |= iq[i];
if (j == 0)
{ /* need recomputation */
for (k = 1; iq[jk - k] == 0; k++)
; /* k = no. of terms needed */
for (i = jz + 1; i <= jz + k; i++)
{ /* add q[jz+1] to q[jz+k] */
f[jx + i] = (double)ipio2[jv + i];
for (j = 0, fw = 0.0; j <= jx; j++)
fw += x[j] * f[jx + i - j];
q[i] = fw;
}
jz += k;
goto recompute;
}
}
/* chop off zero terms */
if (z == 0.0)
{
jz -= 1;
q0 -= 24;
while (iq[jz] == 0)
{
jz--;
q0 -= 24;
}
}
else
{ /* break z into 24-bit if necessary */
z = std::scalbn(z, -q0);
if (z >= two24)
{
fw = (double)((int32_t)(twon24 * z));
iq[jz] = (int32_t)(z - two24 * fw);
jz += 1;
q0 += 24;
iq[jz] = (int32_t)fw;
}
else
iq[jz] = (int32_t)z;
}
/* convert integer "bit" chunk to floating-point value */
fw = scalbn(one, q0);
for (i = jz; i >= 0; i--)
{
q[i] = fw * (double)iq[i];
fw *= twon24;
}
/* compute PIo2[0,...,jp]*q[jz,...,0] */
for (i = jz; i >= 0; i--)
{
for (fw = 0.0, k = 0; k <= jp && k <= jz - i; k++)
fw += PIo2[k] * q[i + k];
fq[jz - i] = fw;
}
/* compress fq[] into y[] */
switch (prec)
{
case 0:
fw = 0.0;
for (i = jz; i >= 0; i--)
fw += fq[i];
y[0] = (ih == 0) ? fw : -fw;
break;
case 1:
case 2:
fw = 0.0;
for (i = jz; i >= 0; i--)
fw += fq[i];
y[0] = (ih == 0) ? fw : -fw;
fw = fq[0] - fw;
for (i = 1; i <= jz; i++)
fw += fq[i];
y[1] = (ih == 0) ? fw : -fw;
break;
case 3: /* painful */
for (i = jz; i > 0; i--)
{
fw = fq[i - 1] + fq[i];
fq[i] += fq[i - 1] - fw;
fq[i - 1] = fw;
}
for (i = jz; i > 1; i--)
{
fw = fq[i - 1] + fq[i];
fq[i] += fq[i - 1] - fw;
fq[i - 1] = fw;
}
for (fw = 0.0, i = jz; i >= 2; i--)
fw += fq[i];
if (ih == 0)
{
y[0] = fq[0];
y[1] = fq[1];
y[2] = fw;
}
else
{
y[0] = -fq[0];
y[1] = -fq[1];
y[2] = -fw;
}
}
return n & 7;
}
inline std::int32_t __ieee754_rem_pio2(double x, double* y) noexcept
{
static const std::int32_t two_over_pi[] = {
0xA2F983,
0x6E4E44,
0x1529FC,
0x2757D1,
0xF534DD,
0xC0DB62,
0x95993C,
0x439041,
0xFE5163,
0xABDEBB,
0xC561B7,
0x246E3A,
0x424DD2,
0xE00649,
0x2EEA09,
0xD1921C,
0xFE1DEB,
0x1CB129,
0xA73EE8,
0x8235F5,
0x2EBB44,
0x84E99C,
0x7026B4,
0x5F7E41,
0x3991D6,
0x398353,
0x39F49C,
0x845F8B,
0xBDF928,
0x3B1FF8,
0x97FFDE,
0x05980F,
0xEF2F11,
0x8B5A0A,
0x6D1F6D,
0x367ECF,
0x27CB09,
0xB74F46,
0x3F669E,
0x5FEA2D,
0x7527BA,
0xC7EBE5,
0xF17B3D,
0x0739F7,
0x8A5292,
0xEA6BFB,
0x5FB11F,
0x8D5D08,
0x560330,
0x46FC7B,
0x6BABF0,
0xCFBC20,
0x9AF436,
0x1DA9E3,
0x91615E,
0xE61B08,
0x659985,
0x5F14A0,
0x68408D,
0xFFD880,
0x4D7327,
0x310606,
0x1556CA,
0x73A8C9,
0x60E27B,
0xC08C6B,
};
static const std::int32_t npio2_hw[] = {
0x3FF921FB,
0x400921FB,
0x4012D97C,
0x401921FB,
0x401F6A7A,
0x4022D97C,
0x4025FDBB,
0x402921FB,
0x402C463A,
0x402F6A7A,
0x4031475C,
0x4032D97C,
0x40346B9C,
0x4035FDBB,
0x40378FDB,
0x403921FB,
0x403AB41B,
0x403C463A,
0x403DD85A,
0x403F6A7A,
0x40407E4C,
0x4041475C,
0x4042106C,
0x4042D97C,
0x4043A28C,
0x40446B9C,
0x404534AC,
0x4045FDBB,
0x4046C6CB,
0x40478FDB,
0x404858EB,
0x404921FB,
};
/*
* invpio2: 53 bits of 2/pi
* pio2_1: first 33 bit of pi/2
* pio2_1t: pi/2 - pio2_1
* pio2_2: second 33 bit of pi/2
* pio2_2t: pi/2 - (pio2_1+pio2_2)
* pio2_3: third 33 bit of pi/2
* pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
*/
static const double
zero
= 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
double z = 0., w, t, r, fn;
double tx[3];
std::int32_t e0, i, j, nx, n, ix, hx;
std::uint32_t low;
GET_HIGH_WORD(hx, x); /* high word of x */
ix = hx & 0x7fffffff;
if (ix <= 0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */
{
y[0] = x;
y[1] = 0;
return 0;
}
if (ix < 0x4002d97c)
{ /* |x| < 3pi/4, special case with n=+-1 */
if (hx > 0)
{
z = x - pio2_1;
if (ix != 0x3ff921fb)
{ /* 33+53 bit pi is good enough */
y[0] = z - pio2_1t;
y[1] = (z - y[0]) - pio2_1t;
}
else
{ /* near pi/2, use 33+33+53 bit pi */
z -= pio2_2;
y[0] = z - pio2_2t;
y[1] = (z - y[0]) - pio2_2t;
}
return 1;
}
else
{ /* negative x */
z = x + pio2_1;
if (ix != 0x3ff921fb)
{ /* 33+53 bit pi is good enough */
y[0] = z + pio2_1t;
y[1] = (z - y[0]) + pio2_1t;
}
else
{ /* near pi/2, use 33+33+53 bit pi */
z += pio2_2;
y[0] = z + pio2_2t;
y[1] = (z - y[0]) + pio2_2t;
}
return -1;
}
}
if (ix <= 0x413921fb)
{ /* |x| ~<= 2^19*(pi/2), medium_ size */
t = std::fabs(x);
n = (std::int32_t)(t * invpio2 + half);
fn = (double)n;
r = t - fn * pio2_1;
w = fn * pio2_1t; /* 1st round good to 85 bit */
if ((n < 32) && (n > 0) && (ix != npio2_hw[n - 1]))
{
y[0] = r - w; /* quick check no cancellation */
}
else
{
std::uint32_t high;
j = ix >> 20;
y[0] = r - w;
GET_HIGH_WORD(high, y[0]);
i = j - static_cast<int32_t>((high >> 20) & 0x7ff);
if (i > 16)
{ /* 2nd iteration needed, good to 118 */
t = r;
w = fn * pio2_2;
r = t - w;
w = fn * pio2_2t - ((t - r) - w);
y[0] = r - w;
GET_HIGH_WORD(high, y[0]);
i = j - static_cast<int32_t>((high >> 20) & 0x7ff);
if (i > 49)
{ /* 3rd iteration need, 151 bits acc */
t = r; /* will cover all possible cases */
w = fn * pio2_3;
r = t - w;
w = fn * pio2_3t - ((t - r) - w);
y[0] = r - w;
}
}
}
y[1] = (r - y[0]) - w;
if (hx < 0)
{
y[0] = -y[0];
y[1] = -y[1];
return -n;
}
else
return n;
}
/*
* all other (large) arguments
*/
if (ix >= 0x7ff00000)
{ /* x is inf or NaN */
y[0] = y[1] = x - x;
return 0;
}
/* set z = scalbn(|x|,ilogb(x)-23) */
GET_LOW_WORD(low, x);
SET_LOW_WORD(z, low);
e0 = (ix >> 20) - 1046; /* e0 = ilogb(z)-23; */
SET_HIGH_WORD(z, static_cast<uint32_t>(ix - (e0 << 20)));
for (i = 0; i < 2; i++)
{
tx[i] = (double)((std::int32_t)(z));
z = (z - tx[i]) * two24;
}
tx[2] = z;
nx = 3;
while (tx[nx - 1] == zero)
nx--; /* skip zero term */
n = __kernel_rem_pio2(tx, y, e0, nx, 2, two_over_pi);
if (hx < 0)
{
y[0] = -y[0];
y[1] = -y[1];
return -n;
}
return n;
}
}
#undef XSIMD_LITTLE_ENDIAN
#undef SET_LOW_WORD
#undef SET_HIGH_WORD
#undef GET_LOW_WORD
#undef GET_HIGH_WORD
#undef HIGH_WORD_IDX
#undef LOW_WORD_IDX
#undef ONCE0
}
|
