aboutsummaryrefslogtreecommitdiffstats
path: root/doc/transforms.md
blob: 78f3f68d65204bb058be1b7f6ed7b16c43b81510 (plain) (blame)
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
The basis transforms used for FFT and various other derived functions are based
on the following unrollings.
The functions can be easily adapted to double precision floats as well.

# Parity permutation
The basis transforms described here all use the following permutation:

``` C
void ff_tx_gen_split_radix_parity_revtab(int *revtab, int len, int inv,
                                         int basis, int dual_stride);
```
Parity means even and odd complex numbers will be split, e.g. the even
coefficients will come first, after which the odd coefficients will be
placed. For example, a 4-point transform's coefficients after reordering:
`z[0].re, z[0].im, z[2].re, z[2].im, z[1].re, z[1].im, z[3].re, z[3].im`

The basis argument is the length of the largest non-composite transform
supported, and also implies that the basis/2 transform is supported as well,
as the split-radix algorithm requires it to be.

The dual_stride argument indicates that both the basis, as well as the
basis/2 transforms support doing two transforms at once, and the coefficients
will be interleaved between each pair in a split-radix like so (stride == 2):
`tx1[0], tx1[2], tx2[0], tx2[2], tx1[1], tx1[3], tx2[1], tx2[3]`
A non-zero number switches this on, with the value indicating the stride
(how many values of 1 transform to put first before switching to the other).
Must be a power of two or 0. Must be less than the basis.
Value will be clipped to the transform size, so for a basis of 16 and a
dual_stride of 8, dual 8-point transforms will be laid out as if dual_stride
was set to 4.
Usually you'll set this to half the complex numbers that fit in a single
register or 0. This allows to reuse SSE functions as dual-transform
functions in AVX mode.
If length is smaller than basis/2 this function will not do anything.

# 4-point FFT transform
The only permutation this transform needs is to swap the `z[1]` and `z[2]`
elements when performing an inverse transform, which in the assembly code is
hardcoded with the function itself being templated and duplicated for each
direction.

``` C
static void fft4(FFTComplex *z)
{
    FFTSample r1 = z[0].re - z[2].re;
    FFTSample r2 = z[0].im - z[2].im;
    FFTSample r3 = z[1].re - z[3].re;
    FFTSample r4 = z[1].im - z[3].im;
    /* r5-r8 second transform */

    FFTSample t1 = z[0].re + z[2].re;
    FFTSample t2 = z[0].im + z[2].im;
    FFTSample t3 = z[1].re + z[3].re;
    FFTSample t4 = z[1].im + z[3].im;
    /* t5-t8 second transform */

    /* 1sub + 1add = 2 instructions */

    /* 2 shufs */
    FFTSample a3 = t1 - t3;
    FFTSample a4 = t2 - t4;
    FFTSample b3 = r1 - r4;
    FFTSample b2 = r2 - r3;

    FFTSample a1 = t1 + t3;
    FFTSample a2 = t2 + t4;
    FFTSample b1 = r1 + r4;
    FFTSample b4 = r2 + r3;
    /* 1 add 1 sub 3 shufs */

    z[0].re = a1;
    z[0].im = a2;
    z[2].re = a3;
    z[2].im = a4;

    z[1].re = b1;
    z[1].im = b2;
    z[3].re = b3;
    z[3].im = b4;
}
```

# 8-point AVX FFT transform
Input must be pre-permuted using the parity lookup table, generated via
`ff_tx_gen_split_radix_parity_revtab`.

``` C
static void fft8(FFTComplex *z)
{
    FFTSample r1 = z[0].re - z[4].re;
    FFTSample r2 = z[0].im - z[4].im;
    FFTSample r3 = z[1].re - z[5].re;
    FFTSample r4 = z[1].im - z[5].im;

    FFTSample r5 = z[2].re - z[6].re;
    FFTSample r6 = z[2].im - z[6].im;
    FFTSample r7 = z[3].re - z[7].re;
    FFTSample r8 = z[3].im - z[7].im;

    FFTSample q1 = z[0].re + z[4].re;
    FFTSample q2 = z[0].im + z[4].im;
    FFTSample q3 = z[1].re + z[5].re;
    FFTSample q4 = z[1].im + z[5].im;

    FFTSample q5 = z[2].re + z[6].re;
    FFTSample q6 = z[2].im + z[6].im;
    FFTSample q7 = z[3].re + z[7].re;
    FFTSample q8 = z[3].im + z[7].im;

    FFTSample s3 = q1 - q3;
    FFTSample s1 = q1 + q3;
    FFTSample s4 = q2 - q4;
    FFTSample s2 = q2 + q4;

    FFTSample s7 = q5 - q7;
    FFTSample s5 = q5 + q7;
    FFTSample s8 = q6 - q8;
    FFTSample s6 = q6 + q8;

    FFTSample e1 = s1 * -1;
    FFTSample e2 = s2 * -1;
    FFTSample e3 = s3 * -1;
    FFTSample e4 = s4 * -1;

    FFTSample e5 = s5 *  1;
    FFTSample e6 = s6 *  1;
    FFTSample e7 = s7 * -1;
    FFTSample e8 = s8 *  1;

    FFTSample w1 =  e5 - e1;
    FFTSample w2 =  e6 - e2;
    FFTSample w3 =  e8 - e3;
    FFTSample w4 =  e7 - e4;

    FFTSample w5 =  s1 - e5;
    FFTSample w6 =  s2 - e6;
    FFTSample w7 =  s3 - e8;
    FFTSample w8 =  s4 - e7;

    z[0].re = w1;
    z[0].im = w2;
    z[2].re = w3;
    z[2].im = w4;
    z[4].re = w5;
    z[4].im = w6;
    z[6].re = w7;
    z[6].im = w8;

    FFTSample z1 = r1 - r4;
    FFTSample z2 = r1 + r4;
    FFTSample z3 = r3 - r2;
    FFTSample z4 = r3 + r2;

    FFTSample z5 = r5 - r6;
    FFTSample z6 = r5 + r6;
    FFTSample z7 = r7 - r8;
    FFTSample z8 = r7 + r8;

    z3 *= -1;
    z5 *= -M_SQRT1_2;
    z6 *= -M_SQRT1_2;
    z7 *=  M_SQRT1_2;
    z8 *=  M_SQRT1_2;

    FFTSample t5 = z7 - z6;
    FFTSample t6 = z8 + z5;
    FFTSample t7 = z8 - z5;
    FFTSample t8 = z7 + z6;

    FFTSample u1 =  z2 + t5;
    FFTSample u2 =  z3 + t6;
    FFTSample u3 =  z1 - t7;
    FFTSample u4 =  z4 + t8;

    FFTSample u5 =  z2 - t5;
    FFTSample u6 =  z3 - t6;
    FFTSample u7 =  z1 + t7;
    FFTSample u8 =  z4 - t8;

    z[1].re = u1;
    z[1].im = u2;
    z[3].re = u3;
    z[3].im = u4;
    z[5].re = u5;
    z[5].im = u6;
    z[7].re = u7;
    z[7].im = u8;
}
```

As you can see, there are 2 independent paths, one for even and one for odd coefficients.
This theme continues throughout the document. Note that in the actual assembly code,
the paths are interleaved to improve unit saturation and CPU dependency tracking, so
to more clearly see them, you'll need to deinterleave the instructions.

# 8-point SSE/ARM64 FFT transform
Input must be pre-permuted using the parity lookup table, generated via
`ff_tx_gen_split_radix_parity_revtab`.

``` C
static void fft8(FFTComplex *z)
{
    FFTSample r1 = z[0].re - z[4].re;
    FFTSample r2 = z[0].im - z[4].im;
    FFTSample r3 = z[1].re - z[5].re;
    FFTSample r4 = z[1].im - z[5].im;

    FFTSample j1 = z[2].re - z[6].re;
    FFTSample j2 = z[2].im - z[6].im;
    FFTSample j3 = z[3].re - z[7].re;
    FFTSample j4 = z[3].im - z[7].im;

    FFTSample q1 = z[0].re + z[4].re;
    FFTSample q2 = z[0].im + z[4].im;
    FFTSample q3 = z[1].re + z[5].re;
    FFTSample q4 = z[1].im + z[5].im;

    FFTSample k1 = z[2].re + z[6].re;
    FFTSample k2 = z[2].im + z[6].im;
    FFTSample k3 = z[3].re + z[7].re;
    FFTSample k4 = z[3].im + z[7].im;
    /* 2 add 2 sub = 4 */

    /* 2 shufs, 1 add 1 sub = 4 */
    FFTSample s1 = q1 + q3;
    FFTSample s2 = q2 + q4;
    FFTSample g1 = k3 + k1;
    FFTSample g2 = k2 + k4;

    FFTSample s3 = q1 - q3;
    FFTSample s4 = q2 - q4;
    FFTSample g4 = k3 - k1;
    FFTSample g3 = k2 - k4;

    /* 1 unpack + 1 shuffle = 2 */

    /* 1 add */
    FFTSample w1 =  s1 + g1;
    FFTSample w2 =  s2 + g2;
    FFTSample w3 =  s3 + g3;
    FFTSample w4 =  s4 + g4;

    /* 1 sub */
    FFTSample h1 =  s1 - g1;
    FFTSample h2 =  s2 - g2;
    FFTSample h3 =  s3 - g3;
    FFTSample h4 =  s4 - g4;

    z[0].re = w1;
    z[0].im = w2;
    z[2].re = w3;
    z[2].im = w4;
    z[4].re = h1;
    z[4].im = h2;
    z[6].re = h3;
    z[6].im = h4;

    /* 1 shuf + 1 shuf + 1 xor + 1 addsub */
    FFTSample z1 = r1 + r4;
    FFTSample z2 = r2 - r3;
    FFTSample z3 = r1 - r4;
    FFTSample z4 = r2 + r3;

    /* 1 mult */
    j1 *=  M_SQRT1_2;
    j2 *= -M_SQRT1_2;
    j3 *= -M_SQRT1_2;
    j4 *=  M_SQRT1_2;

    /* 1 shuf + 1 addsub */
    FFTSample l2 = j1 - j2;
    FFTSample l1 = j2 + j1;
    FFTSample l4 = j3 - j4;
    FFTSample l3 = j4 + j3;

    /* 1 shuf + 1 addsub */
    FFTSample t1 = l3 - l2;
    FFTSample t2 = l4 + l1;
    FFTSample t3 = l1 - l4;
    FFTSample t4 = l2 + l3;

    /* 1 add */
    FFTSample u1 =  z1 - t1;
    FFTSample u2 =  z2 - t2;
    FFTSample u3 =  z3 - t3;
    FFTSample u4 =  z4 - t4;

    /* 1 sub */
    FFTSample o1 =  z1 + t1;
    FFTSample o2 =  z2 + t2;
    FFTSample o3 =  z3 + t3;
    FFTSample o4 =  z4 + t4;

    z[1].re = u1;
    z[1].im = u2;
    z[3].re = u3;
    z[3].im = u4;
    z[5].re = o1;
    z[5].im = o2;
    z[7].re = o3;
    z[7].im = o4;
}
```

Most functions here are highly tuned to use x86's addsub instruction to save on
external sign mask loading.

# 16-point AVX FFT transform
This version expects the output of the 8 and 4-point transforms to follow the
even/odd convention established above.

``` C
static void fft16(FFTComplex *z)
{
    FFTSample cos_16_1 = 0.92387950420379638671875f;
    FFTSample cos_16_3 = 0.3826834261417388916015625f;

    fft8(z);
    fft4(z+8);
    fft4(z+10);

    FFTSample s[32];

    /*
        xorps m1, m1 - free
        mulps m0
        shufps m1, m1, m0
        xorps
        addsub
        shufps
        mulps
        mulps
        addps
        or (fma3)
        shufps
        shufps
        mulps
        mulps
        fma
        fma
     */

    s[0]  =  z[8].re*( 1) - z[8].im*( 0);
    s[1]  =  z[8].im*( 1) + z[8].re*( 0);
    s[2]  =  z[9].re*( 1) - z[9].im*(-1);
    s[3]  =  z[9].im*( 1) + z[9].re*(-1);

    s[4]  = z[10].re*( 1) - z[10].im*( 0);
    s[5]  = z[10].im*( 1) + z[10].re*( 0);
    s[6]  = z[11].re*( 1) - z[11].im*( 1);
    s[7]  = z[11].im*( 1) + z[11].re*( 1);

    s[8]  = z[12].re*(  cos_16_1) - z[12].im*( -cos_16_3);
    s[9]  = z[12].im*(  cos_16_1) + z[12].re*( -cos_16_3);
    s[10] = z[13].re*(  cos_16_3) - z[13].im*( -cos_16_1);
    s[11] = z[13].im*(  cos_16_3) + z[13].re*( -cos_16_1);

    s[12] = z[14].re*(  cos_16_1) - z[14].im*(  cos_16_3);
    s[13] = z[14].im*( -cos_16_1) + z[14].re*( -cos_16_3);
    s[14] = z[15].re*(  cos_16_3) - z[15].im*(  cos_16_1);
    s[15] = z[15].im*( -cos_16_3) + z[15].re*( -cos_16_1);

    s[2] *=  M_SQRT1_2;
    s[3] *=  M_SQRT1_2;
    s[5] *= -1;
    s[6] *=  M_SQRT1_2;
    s[7] *= -M_SQRT1_2;

    FFTSample w5 =  s[0] + s[4];
    FFTSample w6 =  s[1] - s[5];
    FFTSample x5 =  s[2] + s[6];
    FFTSample x6 =  s[3] - s[7];

    FFTSample w3 =  s[4] - s[0];
    FFTSample w4 =  s[5] + s[1];
    FFTSample x3 =  s[6] - s[2];
    FFTSample x4 =  s[7] + s[3];

    FFTSample y5 =  s[8] + s[12];
    FFTSample y6 =  s[9] - s[13];
    FFTSample u5 = s[10] + s[14];
    FFTSample u6 = s[11] - s[15];

    FFTSample y3 = s[12] - s[8];
    FFTSample y4 = s[13] + s[9];
    FFTSample u3 = s[14] - s[10];
    FFTSample u4 = s[15] + s[11];

    /* 2xorps, 2vperm2fs, 2 adds, 2 vpermilps = 8 */

    FFTSample o1  = z[0].re + w5;
    FFTSample o2  = z[0].im + w6;
    FFTSample o5  = z[1].re + x5;
    FFTSample o6  = z[1].im + x6;
    FFTSample o9  = z[2].re + w4; //h
    FFTSample o10 = z[2].im + w3;
    FFTSample o13 = z[3].re + x4;
    FFTSample o14 = z[3].im + x3;

    FFTSample o17 = z[0].re - w5;
    FFTSample o18 = z[0].im - w6;
    FFTSample o21 = z[1].re - x5;
    FFTSample o22 = z[1].im - x6;
    FFTSample o25 = z[2].re - w4; //h
    FFTSample o26 = z[2].im - w3;
    FFTSample o29 = z[3].re - x4;
    FFTSample o30 = z[3].im - x3;

    FFTSample o3  = z[4].re + y5;
    FFTSample o4  = z[4].im + y6;
    FFTSample o7  = z[5].re + u5;
    FFTSample o8  = z[5].im + u6;
    FFTSample o11 = z[6].re + y4; //h
    FFTSample o12 = z[6].im + y3;
    FFTSample o15 = z[7].re + u4;
    FFTSample o16 = z[7].im + u3;

    FFTSample o19 = z[4].re - y5;
    FFTSample o20 = z[4].im - y6;
    FFTSample o23 = z[5].re - u5;
    FFTSample o24 = z[5].im - u6;
    FFTSample o27 = z[6].re - y4; //h
    FFTSample o28 = z[6].im - y3;
    FFTSample o31 = z[7].re - u4;
    FFTSample o32 = z[7].im - u3;

    /* This is just deinterleaving, happens separately */
    z[0]  = (FFTComplex){  o1,  o2 };
    z[1]  = (FFTComplex){  o3,  o4 };
    z[2]  = (FFTComplex){  o5,  o6 };
    z[3]  = (FFTComplex){  o7,  o8 };
    z[4]  = (FFTComplex){  o9, o10 };
    z[5]  = (FFTComplex){ o11, o12 };
    z[6]  = (FFTComplex){ o13, o14 };
    z[7]  = (FFTComplex){ o15, o16 };

    z[8]  = (FFTComplex){ o17, o18 };
    z[9]  = (FFTComplex){ o19, o20 };
    z[10] = (FFTComplex){ o21, o22 };
    z[11] = (FFTComplex){ o23, o24 };
    z[12] = (FFTComplex){ o25, o26 };
    z[13] = (FFTComplex){ o27, o28 };
    z[14] = (FFTComplex){ o29, o30 };
    z[15] = (FFTComplex){ o31, o32 };
}
```

# AVX split-radix synthesis
To create larger transforms, the following unrolling of the C split-radix
function is used.

``` C
#define BF(x, y, a, b)                           \
    do {                                         \
        x = (a) - (b);                           \
        y = (a) + (b);                           \
    } while (0)

#define BUTTERFLIES(a0,a1,a2,a3)               \
    do {                                       \
        r0=a0.re;                              \
        i0=a0.im;                              \
        r1=a1.re;                              \
        i1=a1.im;                              \
        BF(q3, q5, q5, q1);                    \
        BF(a2.re, a0.re, r0, q5);              \
        BF(a3.im, a1.im, i1, q3);              \
        BF(q4, q6, q2, q6);                    \
        BF(a3.re, a1.re, r1, q4);              \
        BF(a2.im, a0.im, i0, q6);              \
    } while (0)

#undef TRANSFORM
#define TRANSFORM(a0,a1,a2,a3,wre,wim)         \
    do {                                       \
        CMUL(q1, q2, a2.re, a2.im, wre, -wim); \
        CMUL(q5, q6, a3.re, a3.im, wre,  wim); \
        BUTTERFLIES(a0, a1, a2, a3);           \
    } while (0)

#define CMUL(dre, dim, are, aim, bre, bim)       \
    do {                                         \
        (dre) = (are) * (bre) - (aim) * (bim);   \
        (dim) = (are) * (bim) + (aim) * (bre);   \
    } while (0)

static void recombine(FFTComplex *z, const FFTSample *cos,
                      unsigned int n)
{
    const int o1 = 2*n;
    const int o2 = 4*n;
    const int o3 = 6*n;
    const FFTSample *wim = cos + o1 - 7;
    FFTSample q1, q2, q3, q4, q5, q6, r0, i0, r1, i1;

#if 0
    for (int i = 0; i < n; i += 4) {
#endif

#if 0
        TRANSFORM(z[ 0 + 0], z[ 0 + 4], z[o2 + 0], z[o2 + 2], cos[0], wim[7]);
        TRANSFORM(z[ 0 + 1], z[ 0 + 5], z[o2 + 1], z[o2 + 3], cos[2], wim[5]);
        TRANSFORM(z[ 0 + 2], z[ 0 + 6], z[o2 + 4], z[o2 + 6], cos[4], wim[3]);
        TRANSFORM(z[ 0 + 3], z[ 0 + 7], z[o2 + 5], z[o2 + 7], cos[6], wim[1]);

        TRANSFORM(z[o1 + 0], z[o1 + 4], z[o3 + 0], z[o3 + 2], cos[1], wim[6]);
        TRANSFORM(z[o1 + 1], z[o1 + 5], z[o3 + 1], z[o3 + 3], cos[3], wim[4]);
        TRANSFORM(z[o1 + 2], z[o1 + 6], z[o3 + 4], z[o3 + 6], cos[5], wim[2]);
        TRANSFORM(z[o1 + 3], z[o1 + 7], z[o3 + 5], z[o3 + 7], cos[7], wim[0]);
#else
        FFTSample h[8], j[8], r[8], w[8];
        FFTSample t[8];
        FFTComplex *m0 = &z[0];
        FFTComplex *m1 = &z[4];
        FFTComplex *m2 = &z[o2 + 0];
        FFTComplex *m3 = &z[o2 + 4];

        const FFTSample *t1  = &cos[0];
        const FFTSample *t2  = &wim[0];

        /* 2 loads (tabs) */

        /* 2 vperm2fs, 2 shufs (im), 2 shufs (tabs) */
        /* 1 xor, 1 add, 1 sub, 4 mults OR 2 mults, 2 fmas */
        /* 13 OR 10ish (-2 each for second passovers!) */

        w[0] = m2[0].im*t1[0] - m2[0].re*t2[7];
        w[1] = m2[0].re*t1[0] + m2[0].im*t2[7];
        w[2] = m2[1].im*t1[2] - m2[1].re*t2[5];
        w[3] = m2[1].re*t1[2] + m2[1].im*t2[5];
        w[4] = m3[0].im*t1[4] - m3[0].re*t2[3];
        w[5] = m3[0].re*t1[4] + m3[0].im*t2[3];
        w[6] = m3[1].im*t1[6] - m3[1].re*t2[1];
        w[7] = m3[1].re*t1[6] + m3[1].im*t2[1];

        j[0] = m2[2].im*t1[0] + m2[2].re*t2[7];
        j[1] = m2[2].re*t1[0] - m2[2].im*t2[7];
        j[2] = m2[3].im*t1[2] + m2[3].re*t2[5];
        j[3] = m2[3].re*t1[2] - m2[3].im*t2[5];
        j[4] = m3[2].im*t1[4] + m3[2].re*t2[3];
        j[5] = m3[2].re*t1[4] - m3[2].im*t2[3];
        j[6] = m3[3].im*t1[6] + m3[3].re*t2[1];
        j[7] = m3[3].re*t1[6] - m3[3].im*t2[1];

        /* 1 add + 1 shuf */
        t[1] = j[0] + w[0];
        t[0] = j[1] + w[1];
        t[3] = j[2] + w[2];
        t[2] = j[3] + w[3];
        t[5] = j[4] + w[4];
        t[4] = j[5] + w[5];
        t[7] = j[6] + w[6];
        t[6] = j[7] + w[7];

        /* 1 sub + 1 xor */
        r[0] =  (w[0] - j[0]);
        r[1] = -(w[1] - j[1]);
        r[2] =  (w[2] - j[2]);
        r[3] = -(w[3] - j[3]);
        r[4] =  (w[4] - j[4]);
        r[5] = -(w[5] - j[5]);
        r[6] =  (w[6] - j[6]);
        r[7] = -(w[7] - j[7]);

        /* Min: 2 subs, 2 adds, 2 vperm2fs (OPTIONAL) */
        m2[0].re = m0[0].re - t[0];
        m2[0].im = m0[0].im - t[1];
        m2[1].re = m0[1].re - t[2];
        m2[1].im = m0[1].im - t[3];
        m3[0].re = m0[2].re - t[4];
        m3[0].im = m0[2].im - t[5];
        m3[1].re = m0[3].re - t[6];
        m3[1].im = m0[3].im - t[7];

        m2[2].re = m1[0].re - r[0];
        m2[2].im = m1[0].im - r[1];
        m2[3].re = m1[1].re - r[2];
        m2[3].im = m1[1].im - r[3];
        m3[2].re = m1[2].re - r[4];
        m3[2].im = m1[2].im - r[5];
        m3[3].re = m1[3].re - r[6];
        m3[3].im = m1[3].im - r[7];

        m0[0].re = m0[0].re + t[0];
        m0[0].im = m0[0].im + t[1];
        m0[1].re = m0[1].re + t[2];
        m0[1].im = m0[1].im + t[3];
        m0[2].re = m0[2].re + t[4];
        m0[2].im = m0[2].im + t[5];
        m0[3].re = m0[3].re + t[6];
        m0[3].im = m0[3].im + t[7];

        m1[0].re = m1[0].re + r[0];
        m1[0].im = m1[0].im + r[1];
        m1[1].re = m1[1].re + r[2];
        m1[1].im = m1[1].im + r[3];
        m1[2].re = m1[2].re + r[4];
        m1[2].im = m1[2].im + r[5];
        m1[3].re = m1[3].re + r[6];
        m1[3].im = m1[3].im + r[7];

        /* Identical for below, but with the following parameters */
        m0 = &z[o1];
        m1 = &z[o1 + 4];
        m2 = &z[o3 + 0];
        m3 = &z[o3 + 4];
        t1  = &cos[1];
        t2  = &wim[-1];

        w[0] = m2[0].im*t1[0] - m2[0].re*t2[7];
        w[1] = m2[0].re*t1[0] + m2[0].im*t2[7];
        w[2] = m2[1].im*t1[2] - m2[1].re*t2[5];
        w[3] = m2[1].re*t1[2] + m2[1].im*t2[5];
        w[4] = m3[0].im*t1[4] - m3[0].re*t2[3];
        w[5] = m3[0].re*t1[4] + m3[0].im*t2[3];
        w[6] = m3[1].im*t1[6] - m3[1].re*t2[1];
        w[7] = m3[1].re*t1[6] + m3[1].im*t2[1];

        j[0] = m2[2].im*t1[0] + m2[2].re*t2[7];
        j[1] = m2[2].re*t1[0] - m2[2].im*t2[7];
        j[2] = m2[3].im*t1[2] + m2[3].re*t2[5];
        j[3] = m2[3].re*t1[2] - m2[3].im*t2[5];
        j[4] = m3[2].im*t1[4] + m3[2].re*t2[3];
        j[5] = m3[2].re*t1[4] - m3[2].im*t2[3];
        j[6] = m3[3].im*t1[6] + m3[3].re*t2[1];
        j[7] = m3[3].re*t1[6] - m3[3].im*t2[1];

        /* 1 add + 1 shuf */
        t[1] = j[0] + w[0];
        t[0] = j[1] + w[1];
        t[3] = j[2] + w[2];
        t[2] = j[3] + w[3];
        t[5] = j[4] + w[4];
        t[4] = j[5] + w[5];
        t[7] = j[6] + w[6];
        t[6] = j[7] + w[7];

        /* 1 sub + 1 xor */
        r[0] =  (w[0] - j[0]);
        r[1] = -(w[1] - j[1]);
        r[2] =  (w[2] - j[2]);
        r[3] = -(w[3] - j[3]);
        r[4] =  (w[4] - j[4]);
        r[5] = -(w[5] - j[5]);
        r[6] =  (w[6] - j[6]);
        r[7] = -(w[7] - j[7]);

        /* Min: 2 subs, 2 adds, 2 vperm2fs (OPTIONAL) */
        m2[0].re = m0[0].re - t[0];
        m2[0].im = m0[0].im - t[1];
        m2[1].re = m0[1].re - t[2];
        m2[1].im = m0[1].im - t[3];
        m3[0].re = m0[2].re - t[4];
        m3[0].im = m0[2].im - t[5];
        m3[1].re = m0[3].re - t[6];
        m3[1].im = m0[3].im - t[7];

        m2[2].re = m1[0].re - r[0];
        m2[2].im = m1[0].im - r[1];
        m2[3].re = m1[1].re - r[2];
        m2[3].im = m1[1].im - r[3];
        m3[2].re = m1[2].re - r[4];
        m3[2].im = m1[2].im - r[5];
        m3[3].re = m1[3].re - r[6];
        m3[3].im = m1[3].im - r[7];

        m0[0].re = m0[0].re + t[0];
        m0[0].im = m0[0].im + t[1];
        m0[1].re = m0[1].re + t[2];
        m0[1].im = m0[1].im + t[3];
        m0[2].re = m0[2].re + t[4];
        m0[2].im = m0[2].im + t[5];
        m0[3].re = m0[3].re + t[6];
        m0[3].im = m0[3].im + t[7];

        m1[0].re = m1[0].re + r[0];
        m1[0].im = m1[0].im + r[1];
        m1[1].re = m1[1].re + r[2];
        m1[1].im = m1[1].im + r[3];
        m1[2].re = m1[2].re + r[4];
        m1[2].im = m1[2].im + r[5];
        m1[3].re = m1[3].re + r[6];
        m1[3].im = m1[3].im + r[7];
#endif

#if 0
        z   +=   4; // !!!
        cos += 2*4;
        wim -= 2*4;
    }
#endif
}
```

The macros used are identical to those in the generic C version, only with all
variable declarations exported to the function body.
An important point here is that the high frequency registers (m2 and m3) have
their high and low halves swapped in the output. This is intentional, as the
inputs must also have the same layout, and therefore, the input swapping is only
performed once for the bottom-most basis transform, with all subsequent combinations
using the already swapped halves.

Also note that this function requires a special iteration way, due to coefficients
beginning to overlap, particularly `[o1]` with `[0]` after the second iteration.
To iterate further, set `z = &z[16]` via `z += 8` for the second iteration. After
the 4th iteration, the layout resets, so repeat the same.