- Notifications
You must be signed in to change notification settings - Fork 1
Expand file tree
/
Copy pathslides.html
More file actions
Latest commit
587 lines (562 loc) · 155 KB
/
Copy pathslides.html
File metadata and controls
587 lines (562 loc) · 155 KB
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
<!DOCTYPE html><htmllang="en-US"><head><metacharset="UTF-8"><metaname="viewport" content="width=device-width,height=device-height,initial-scale=1.0"><metaname="apple-mobile-web-app-capable" content="yes"><metahttp-equiv="X-UA-Compatible" content="ie=edge"><metaproperty="og:type" content="website"><metaname="twitter:card" content="summary"><style>@media screen{body,html{background:#000;height:100%;margin:0;overflow:hidden}[data-bespoke-marp-fragment=inactive]{visibility:hidden}.bespoke-marp-osc{display:none;opacity:0}.bespoke-marp-parent{bottom:0;left:0;position:absolute;right:0;top:0}.bespoke-marp-parent>.bespoke-marp-osc{background:rgba(0,0,0,.65);border-radius:7px;bottom:50px;color:#fff;display:block;font-family:Helvetica,Arial,sans-serif;font-size:16px;left:50%;line-height:0;opacity:1;padding:12px;position:absolute;touch-action:manipulation;transform:translateX(-50%);transition:opacity .2s linear;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;white-space:nowrap;z-index:1}.bespoke-marp-parent>.bespoke-marp-osc>*{margin-left:6px}.bespoke-marp-parent>.bespoke-marp-osc>:first-child{margin-left:0}.bespoke-marp-parent>.bespoke-marp-osc>span{opacity:.8}.bespoke-marp-parent>.bespoke-marp-osc>span[data-bespoke-marp-osc=page]{display:inline-block;min-width:140px;text-align:center}.bespoke-marp-parent>.bespoke-marp-osc>button{-webkit-appearance:none;-moz-appearance:none;appearance:none;background-color:transparent;border:0;color:inherit;cursor:pointer;font-size:inherit;opacity:.8;outline:none;padding:0;transition:opacity .2s linear;-webkit-tap-highlight-color:transparent}.bespoke-marp-parent>.bespoke-marp-osc>button:disabled{cursor:not-allowed;opacity:.15!important}.bespoke-marp-parent>.bespoke-marp-osc>button:hover{opacity:1}.bespoke-marp-parent>.bespoke-marp-osc>button:hover:active{opacity:.6}.bespoke-marp-parent>.bespoke-marp-osc>button:hover:not(:disabled){transition:none}.bespoke-marp-parent>.bespoke-marp-osc>button[data-bespoke-marp-osc=prev]{background:transparent url("data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHZpZXdCb3g9IjAgMCAxMDAgMTAwIj48cGF0aCBmaWxsPSJub25lIiBzdHJva2U9IiNmZmYiIHN0cm9rZS1saW5lY2FwPSJyb3VuZCIgc3Ryb2tlLWxpbmVqb2luPSJyb3VuZCIgc3Ryb2tlLXdpZHRoPSI1IiBkPSJNNjggOTBMMjggNTBsNDAtNDAiLz48L3N2Zz4=") no-repeat 50%;background-size:contain;height:32px;overflow:hidden;text-indent:100%;white-space:nowrap;width:32px}.bespoke-marp-parent>.bespoke-marp-osc>button[data-bespoke-marp-osc=next]{background:transparent url("data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHZpZXdCb3g9IjAgMCAxMDAgMTAwIj48cGF0aCBmaWxsPSJub25lIiBzdHJva2U9IiNmZmYiIHN0cm9rZS1saW5lY2FwPSJyb3VuZCIgc3Ryb2tlLWxpbmVqb2luPSJyb3VuZCIgc3Ryb2tlLXdpZHRoPSI1IiBkPSJNMzIgOTBsNDAtNDAtNDAtNDAiLz48L3N2Zz4=") no-repeat 50%;background-size:contain;height:32px;overflow:hidden;text-indent:100%;white-space:nowrap;width:32px}.bespoke-marp-parent>.bespoke-marp-osc>button[data-bespoke-marp-osc=fullscreen]{background:transparent url("data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHZpZXdCb3g9IjAgMCAxMDAgMTAwIj48ZGVmcz48c3R5bGU+LmF7ZmlsbDpub25lO3N0cm9rZTojZmZmO3N0cm9rZS1saW5lY2FwOnJvdW5kO3N0cm9rZS1saW5lam9pbjpyb3VuZDtzdHJva2Utd2lkdGg6NXB4fTwvc3R5bGU+PC9kZWZzPjxyZWN0IGNsYXNzPSJhIiB4PSIxMCIgeT0iMjAiIHdpZHRoPSI4MCIgaGVpZ2h0PSI2MCIgcng9IjUuNjciLz48cGF0aCBjbGFzcz0iYSIgZD0iTTQwIDcwSDIwVjUwbTIwIDBMMjAgNzBtNDAtNDBoMjB2MjBtLTIwIDBsMjAtMjAiLz48L3N2Zz4=") no-repeat 50%;background-size:contain;height:32px;overflow:hidden;text-indent:100%;white-space:nowrap;width:32px}.bespoke-marp-parent>.bespoke-marp-osc>button[data-bespoke-marp-osc=fullscreen].exit{background-image:url("data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHZpZXdCb3g9IjAgMCAxMDAgMTAwIj48ZGVmcz48c3R5bGU+LmF7ZmlsbDpub25lO3N0cm9rZTojZmZmO3N0cm9rZS1saW5lY2FwOnJvdW5kO3N0cm9rZS1saW5lam9pbjpyb3VuZDtzdHJva2Utd2lkdGg6NXB4fTwvc3R5bGU+PC9kZWZzPjxyZWN0IGNsYXNzPSJhIiB4PSIxMCIgeT0iMjAiIHdpZHRoPSI4MCIgaGVpZ2h0PSI2MCIgcng9IjUuNjciLz48cGF0aCBjbGFzcz0iYSIgZD0iTTIwIDUwaDIwdjIwbS0yMCAwbDIwLTIwbTQwIDBINjBWMzBtMjAgMEw2MCA1MCIvPjwvc3ZnPg==")}.bespoke-marp-parent.bespoke-marp-inactive{cursor:none}.bespoke-marp-parent.bespoke-marp-inactive>.bespoke-marp-osc{opacity:0;pointer-events:none}svg.bespoke-marp-slide{height:100%;left:0;pointer-events:none;position:absolute;top:0;width:100%;z-index:-1}svg.bespoke-marp-slide.bespoke-marp-active{pointer-events:auto;z-index:0}.bespoke-progress-parent{background:#222;display:flex;height:5px;width:100%}.bespoke-progress-parent+.bespoke-marp-parent{top:5px}.bespoke-progress-parent .bespoke-progress-bar{flex:000;background:#0288d1;transition:flex-basis .2scubic-bezier(0,1,1,1)}}@media print{.bespoke-marp-osc,.bespoke-progress-parent{display:none!important;transition:none!important}.bespoke-marp-parent{top:0}}</style><style>@charset"UTF-8";@importurl("https://fonts.googleapis.com/css?family=Lato:400,900|Roboto+Mono:400,700");article#presentation>svg>foreignObject>section {
width:1280px;
height:720px;
box-sizing: border-box;
overflow: hidden;
position: relative;
scroll-snap-align: center center;
}article#presentation>svg>foreignObject>section::after {
bottom:0;
content:attr(data-marpit-pagination);
padding: inherit;
pointer-events: none;
position: absolute;
right:0;
}article#presentation>svg>foreignObject>section:not([data-marpit-pagination])::after {
display: none;
}/* Normalization */article#presentation>svg>foreignObject>sectionh1 {
font-size:2em;
margin:0.67em0;
}@page {
size:1280px720px;
margin:0;
}@media print {html,body {
background-color:#fff;
margin:0;
page-break-inside: avoid;
break-inside: avoid-page;
}
article#presentation>svg>foreignObject>section {
page-break-before: always;
break-before: page;
}
article#presentation>svg>foreignObject>section,article#presentation>svg>foreignObject>section* {
-webkit-print-color-adjust: exact !important;
color-adjust: exact !important;
}
article#presentation>svg[data-marpit-svg] {
display: block;
height:100vh;
width:100vw;
}
}@font-face{font-family:KaTeX_AMS;src:url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_AMS-Regular.woff2') format("woff2"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_AMS-Regular.woff') format("woff"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_AMS-Regular.ttf') format("truetype");font-weight:400;font-style:normal}@font-face{font-family:KaTeX_Caligraphic;src:url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Caligraphic-Bold.woff2') format("woff2"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Caligraphic-Bold.woff') format("woff"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Caligraphic-Bold.ttf') format("truetype");font-weight:700;font-style:normal}@font-face{font-family:KaTeX_Caligraphic;src:url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Caligraphic-Regular.woff2') format("woff2"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Caligraphic-Regular.woff') format("woff"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Caligraphic-Regular.ttf') format("truetype");font-weight:400;font-style:normal}@font-face{font-family:KaTeX_Fraktur;src:url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Fraktur-Bold.woff2') format("woff2"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Fraktur-Bold.woff') format("woff"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Fraktur-Bold.ttf') format("truetype");font-weight:700;font-style:normal}@font-face{font-family:KaTeX_Fraktur;src:url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Fraktur-Regular.woff2') format("woff2"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Fraktur-Regular.woff') format("woff"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Fraktur-Regular.ttf') format("truetype");font-weight:400;font-style:normal}@font-face{font-family:KaTeX_Main;src:url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Main-Bold.woff2') format("woff2"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Main-Bold.woff') format("woff"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Main-Bold.ttf') format("truetype");font-weight:700;font-style:normal}@font-face{font-family:KaTeX_Main;src:url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Main-BoldItalic.woff2') format("woff2"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Main-BoldItalic.woff') format("woff"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Main-BoldItalic.ttf') format("truetype");font-weight:700;font-style:italic}@font-face{font-family:KaTeX_Main;src:url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Main-Italic.woff2') format("woff2"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Main-Italic.woff') format("woff"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Main-Italic.ttf') format("truetype");font-weight:400;font-style:italic}@font-face{font-family:KaTeX_Main;src:url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Main-Regular.woff2') format("woff2"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Main-Regular.woff') format("woff"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Main-Regular.ttf') format("truetype");font-weight:400;font-style:normal}@font-face{font-family:KaTeX_Math;src:url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Math-BoldItalic.woff2') format("woff2"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Math-BoldItalic.woff') format("woff"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Math-BoldItalic.ttf') format("truetype");font-weight:700;font-style:italic}@font-face{font-family:KaTeX_Math;src:url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Math-Italic.woff2') format("woff2"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Math-Italic.woff') format("woff"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Math-Italic.ttf') format("truetype");font-weight:400;font-style:italic}@font-face{font-family:"KaTeX_SansSerif";src:url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_SansSerif-Bold.woff2') format("woff2"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_SansSerif-Bold.woff') format("woff"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_SansSerif-Bold.ttf') format("truetype");font-weight:700;font-style:normal}@font-face{font-family:"KaTeX_SansSerif";src:url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_SansSerif-Italic.woff2') format("woff2"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_SansSerif-Italic.woff') format("woff"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_SansSerif-Italic.ttf') format("truetype");font-weight:400;font-style:italic}@font-face{font-family:"KaTeX_SansSerif";src:url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_SansSerif-Regular.woff2') format("woff2"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_SansSerif-Regular.woff') format("woff"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_SansSerif-Regular.ttf') format("truetype");font-weight:400;font-style:normal}@font-face{font-family:KaTeX_Script;src:url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Script-Regular.woff2') format("woff2"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Script-Regular.woff') format("woff"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Script-Regular.ttf') format("truetype");font-weight:400;font-style:normal}@font-face{font-family:KaTeX_Size1;src:url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Size1-Regular.woff2') format("woff2"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Size1-Regular.woff') format("woff"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Size1-Regular.ttf') format("truetype");font-weight:400;font-style:normal}@font-face{font-family:KaTeX_Size2;src:url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Size2-Regular.woff2') format("woff2"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Size2-Regular.woff') format("woff"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Size2-Regular.ttf') format("truetype");font-weight:400;font-style:normal}@font-face{font-family:KaTeX_Size3;src:url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Size3-Regular.woff2') format("woff2"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Size3-Regular.woff') format("woff"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Size3-Regular.ttf') format("truetype");font-weight:400;font-style:normal}@font-face{font-family:KaTeX_Size4;src:url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Size4-Regular.woff2') format("woff2"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Size4-Regular.woff') format("woff"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Size4-Regular.ttf') format("truetype");font-weight:400;font-style:normal}@font-face{font-family:KaTeX_Typewriter;src:url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Typewriter-Regular.woff2') format("woff2"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Typewriter-Regular.woff') format("woff"),url('https://cdn.jsdelivr.net/npm/katex@0.10.2/dist/fonts/KaTeX_Typewriter-Regular.ttf') format("truetype");font-weight:400;font-style:normal}article#presentation>svg>foreignObject>section .katex{font:normal 1.21em KaTeX_Main,Times New Roman,serif;line-height:1.2;text-indent:0;text-rendering:auto}article#presentation>svg>foreignObject>section .katex*{-ms-high-contrast-adjust:none!important}article#presentation>svg>foreignObject>section .katex .katex-version:after{content:"0.10.2"}article#presentation>svg>foreignObject>section .katex .katex-mathml{position:absolute;clip:rect(1px,1px,1px,1px);padding:0;border:0;height:1px;width:1px;overflow:hidden}article#presentation>svg>foreignObject>section .katex .katex-html>.newline{display:block}article#presentation>svg>foreignObject>section .katex .base{position:relative;white-space:nowrap;width:-webkit-min-content;width:-moz-min-content;width:min-content}article#presentation>svg>foreignObject>section .katex .base,article#presentation>svg>foreignObject>section .katex .strut{display:inline-block}article#presentation>svg>foreignObject>section .katex .textbf{font-weight:700}article#presentation>svg>foreignObject>section .katex .textit{font-style:italic}article#presentation>svg>foreignObject>section .katex .textrm{font-family:KaTeX_Main}article#presentation>svg>foreignObject>section .katex .textsf{font-family:KaTeX_SansSerif}article#presentation>svg>foreignObject>section .katex .texttt{font-family:KaTeX_Typewriter}article#presentation>svg>foreignObject>section .katex .mathdefault{font-family:KaTeX_Math;font-style:italic}article#presentation>svg>foreignObject>section .katex .mathit{font-family:KaTeX_Main;font-style:italic}article#presentation>svg>foreignObject>section .katex .mathrm{font-style:normal}article#presentation>svg>foreignObject>section .katex .mathbf{font-family:KaTeX_Main;font-weight:700}article#presentation>svg>foreignObject>section .katex .boldsymbol{font-family:KaTeX_Math;font-weight:700;font-style:italic}article#presentation>svg>foreignObject>section .katex .amsrm,article#presentation>svg>foreignObject>section .katex .mathbb,article#presentation>svg>foreignObject>section .katex .textbb{font-family:KaTeX_AMS}article#presentation>svg>foreignObject>section .katex .mathcal{font-family:KaTeX_Caligraphic}article#presentation>svg>foreignObject>section .katex .mathfrak,article#presentation>svg>foreignObject>section .katex .textfrak{font-family:KaTeX_Fraktur}article#presentation>svg>foreignObject>section .katex .mathtt{font-family:KaTeX_Typewriter}article#presentation>svg>foreignObject>section .katex .mathscr,article#presentation>svg>foreignObject>section .katex .textscr{font-family:KaTeX_Script}article#presentation>svg>foreignObject>section .katex .mathsf,article#presentation>svg>foreignObject>section .katex .textsf{font-family:KaTeX_SansSerif}article#presentation>svg>foreignObject>section .katex .mathboldsf,article#presentation>svg>foreignObject>section .katex .textboldsf{font-family:KaTeX_SansSerif;font-weight:700}article#presentation>svg>foreignObject>section .katex .mathitsf,article#presentation>svg>foreignObject>section .katex .textitsf{font-family:KaTeX_SansSerif;font-style:italic}article#presentation>svg>foreignObject>section .katex .mainrm{font-family:KaTeX_Main;font-style:normal}article#presentation>svg>foreignObject > section .katex .vlist-t{display:inline-table;table-layout:fixed}article#presentation > svg > foreignObject > section .katex .vlist-r{display:table-row}article#presentation > svg > foreignObject > section .katex .vlist{display:table-cell;vertical-align:bottom;position:relative}article#presentation > svg > foreignObject > section .katex .vlist>span{display:block;height:0;position:relative}article#presentation > svg > foreignObject > section .katex .vlist>span>span{display:inline-block}article#presentation > svg > foreignObject > section .katex .vlist>span>.pstrut{overflow:hidden;width:0}article#presentation > svg > foreignObject > section .katex .vlist-t2{margin-right:-2px}article#presentation > svg > foreignObject > section .katex .vlist-s{display:table-cell;vertical-align:bottom;font-size:1px;width:2px;min-width:2px}article#presentation > svg > foreignObject > section .katex .msupsub{text-align:left}article#presentation > svg > foreignObject > section .katex .mfrac>span>span{text-align:center}article#presentation > svg > foreignObject > section .katex .mfrac .frac-line{display:inline-block;width:100%;border-bottom-style:solid}article#presentation > svg > foreignObject > section .katex .hdashline,article#presentation > svg > foreignObject > section .katex .hline,article#presentation > svg > foreignObject > section .katex .mfrac .frac-line,article#presentation > svg > foreignObject > section .katex .overline .overline-line,article#presentation > svg > foreignObject > section .katex .rule,article#presentation > svg > foreignObject > section .katex .underline .underline-line{min-height:1px}article#presentation > svg > foreignObject > section .katex .mspace{display:inline-block}article#presentation > svg > foreignObject > section .katex .clap,article#presentation > svg > foreignObject > section .katex .llap,article#presentation > svg > foreignObject > section .katex .rlap{width:0;position:relative}article#presentation > svg > foreignObject > section .katex .clap>.inner,article#presentation > svg > foreignObject > section .katex .llap>.inner,article#presentation > svg > foreignObject > section .katex .rlap>.inner{position:absolute}article#presentation > svg > foreignObject > section .katex .clap>.fix,article#presentation > svg > foreignObject > section .katex .llap>.fix,article#presentation > svg > foreignObject > section .katex .rlap>.fix{display:inline-block}article#presentation > svg > foreignObject > section .katex .llap>.inner{right:0}article#presentation > svg > foreignObject > section .katex .clap>.inner,article#presentation > svg > foreignObject > section .katex .rlap>.inner{left:0}article#presentation > svg > foreignObject > section .katex .clap>.inner>span{margin-left:-50%;margin-right:50%}article#presentation > svg > foreignObject > section .katex .rule{display:inline-block;border:0 solid;position:relative}article#presentation > svg > foreignObject > section .katex .hline,article#presentation > svg > foreignObject > section .katex .overline .overline-line,article#presentation > svg > foreignObject > section .katex .underline .underline-line{display:inline-block;width:100%;border-bottom-style:solid}article#presentation > svg > foreignObject > section .katex .hdashline{display:inline-block;width:100%;border-bottom-style:dashed}article#presentation > svg > foreignObject > section .katex .sqrt>.root{margin-left:.27777778em;margin-right:-.55555556em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer,article#presentation > svg > foreignObject > section .katex .sizing{display:inline-block}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size1.size1,article#presentation > svg > foreignObject > section .katex .sizing.reset-size1.size1{font-size:1em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size1.size2,article#presentation > svg > foreignObject > section .katex .sizing.reset-size1.size2{font-size:1.2em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size1.size3,article#presentation > svg > foreignObject > section .katex .sizing.reset-size1.size3{font-size:1.4em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size1.size4,article#presentation > svg > foreignObject > section .katex .sizing.reset-size1.size4{font-size:1.6em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size1.size5,article#presentation > svg > foreignObject > section .katex .sizing.reset-size1.size5{font-size:1.8em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size1.size6,article#presentation > svg > foreignObject > section .katex .sizing.reset-size1.size6{font-size:2em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size1.size7,article#presentation > svg > foreignObject > section .katex .sizing.reset-size1.size7{font-size:2.4em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size1.size8,article#presentation > svg > foreignObject > section .katex .sizing.reset-size1.size8{font-size:2.88em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size1.size9,article#presentation > svg > foreignObject > section .katex .sizing.reset-size1.size9{font-size:3.456em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size1.size10,article#presentation > svg > foreignObject > section .katex .sizing.reset-size1.size10{font-size:4.148em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size1.size11,article#presentation > svg > foreignObject > section .katex .sizing.reset-size1.size11{font-size:4.976em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size2.size1,article#presentation > svg > foreignObject > section .katex .sizing.reset-size2.size1{font-size:.83333333em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size2.size2,article#presentation > svg > foreignObject > section .katex .sizing.reset-size2.size2{font-size:1em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size2.size3,article#presentation > svg > foreignObject > section .katex .sizing.reset-size2.size3{font-size:1.16666667em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size2.size4,article#presentation > svg > foreignObject > section .katex .sizing.reset-size2.size4{font-size:1.33333333em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size2.size5,article#presentation > svg > foreignObject > section .katex .sizing.reset-size2.size5{font-size:1.5em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size2.size6,article#presentation > svg > foreignObject > section .katex .sizing.reset-size2.size6{font-size:1.66666667em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size2.size7,article#presentation > svg > foreignObject > section .katex .sizing.reset-size2.size7{font-size:2em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size2.size8,article#presentation > svg > foreignObject > section .katex .sizing.reset-size2.size8{font-size:2.4em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size2.size9,article#presentation > svg > foreignObject > section .katex .sizing.reset-size2.size9{font-size:2.88em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size2.size10,article#presentation > svg > foreignObject > section .katex .sizing.reset-size2.size10{font-size:3.45666667em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size2.size11,article#presentation > svg > foreignObject > section .katex .sizing.reset-size2.size11{font-size:4.14666667em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size3.size1,article#presentation > svg > foreignObject > section .katex .sizing.reset-size3.size1{font-size:.71428571em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size3.size2,article#presentation > svg > foreignObject > section .katex .sizing.reset-size3.size2{font-size:.85714286em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size3.size3,article#presentation > svg > foreignObject > section .katex .sizing.reset-size3.size3{font-size:1em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size3.size4,article#presentation > svg > foreignObject > section .katex .sizing.reset-size3.size4{font-size:1.14285714em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size3.size5,article#presentation > svg > foreignObject > section .katex .sizing.reset-size3.size5{font-size:1.28571429em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size3.size6,article#presentation > svg > foreignObject > section .katex .sizing.reset-size3.size6{font-size:1.42857143em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size3.size7,article#presentation > svg > foreignObject > section .katex .sizing.reset-size3.size7{font-size:1.71428571em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size3.size8,article#presentation > svg > foreignObject > section .katex .sizing.reset-size3.size8{font-size:2.05714286em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size3.size9,article#presentation > svg > foreignObject > section .katex .sizing.reset-size3.size9{font-size:2.46857143em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size3.size10,article#presentation > svg > foreignObject > section .katex .sizing.reset-size3.size10{font-size:2.96285714em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size3.size11,article#presentation > svg > foreignObject > section .katex .sizing.reset-size3.size11{font-size:3.55428571em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size4.size1,article#presentation > svg > foreignObject > section .katex .sizing.reset-size4.size1{font-size:.625em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size4.size2,article#presentation > svg > foreignObject > section .katex .sizing.reset-size4.size2{font-size:.75em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size4.size3,article#presentation > svg > foreignObject > section .katex .sizing.reset-size4.size3{font-size:.875em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size4.size4,article#presentation > svg > foreignObject > section .katex .sizing.reset-size4.size4{font-size:1em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size4.size5,article#presentation > svg > foreignObject > section .katex .sizing.reset-size4.size5{font-size:1.125em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size4.size6,article#presentation > svg > foreignObject > section .katex .sizing.reset-size4.size6{font-size:1.25em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size4.size7,article#presentation > svg > foreignObject > section .katex .sizing.reset-size4.size7{font-size:1.5em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size4.size8,article#presentation > svg > foreignObject > section .katex .sizing.reset-size4.size8{font-size:1.8em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size4.size9,article#presentation > svg > foreignObject > section .katex .sizing.reset-size4.size9{font-size:2.16em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size4.size10,article#presentation > svg > foreignObject > section .katex .sizing.reset-size4.size10{font-size:2.5925em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size4.size11,article#presentation > svg > foreignObject > section .katex .sizing.reset-size4.size11{font-size:3.11em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size5.size1,article#presentation > svg > foreignObject > section .katex .sizing.reset-size5.size1{font-size:.55555556em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size5.size2,article#presentation > svg > foreignObject > section .katex .sizing.reset-size5.size2{font-size:.66666667em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size5.size3,article#presentation > svg > foreignObject > section .katex .sizing.reset-size5.size3{font-size:.77777778em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size5.size4,article#presentation > svg > foreignObject > section .katex .sizing.reset-size5.size4{font-size:.88888889em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size5.size5,article#presentation > svg > foreignObject > section .katex .sizing.reset-size5.size5{font-size:1em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size5.size6,article#presentation > svg > foreignObject > section .katex .sizing.reset-size5.size6{font-size:1.11111111em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size5.size7,article#presentation > svg > foreignObject > section .katex .sizing.reset-size5.size7{font-size:1.33333333em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size5.size8,article#presentation > svg > foreignObject > section .katex .sizing.reset-size5.size8{font-size:1.6em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size5.size9,article#presentation > svg > foreignObject > section .katex .sizing.reset-size5.size9{font-size:1.92em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size5.size10,article#presentation > svg > foreignObject > section .katex .sizing.reset-size5.size10{font-size:2.30444444em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size5.size11,article#presentation > svg > foreignObject > section .katex .sizing.reset-size5.size11{font-size:2.76444444em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size6.size1,article#presentation > svg > foreignObject > section .katex .sizing.reset-size6.size1{font-size:.5em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size6.size2,article#presentation > svg > foreignObject > section .katex .sizing.reset-size6.size2{font-size:.6em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size6.size3,article#presentation > svg > foreignObject > section .katex .sizing.reset-size6.size3{font-size:.7em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size6.size4,article#presentation > svg > foreignObject > section .katex .sizing.reset-size6.size4{font-size:.8em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size6.size5,article#presentation > svg > foreignObject > section .katex .sizing.reset-size6.size5{font-size:.9em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size6.size6,article#presentation > svg > foreignObject > section .katex .sizing.reset-size6.size6{font-size:1em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size6.size7,article#presentation > svg > foreignObject > section .katex .sizing.reset-size6.size7{font-size:1.2em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size6.size8,article#presentation > svg > foreignObject > section .katex .sizing.reset-size6.size8{font-size:1.44em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size6.size9,article#presentation > svg > foreignObject > section .katex .sizing.reset-size6.size9{font-size:1.728em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size6.size10,article#presentation > svg > foreignObject > section .katex .sizing.reset-size6.size10{font-size:2.074em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size6.size11,article#presentation > svg > foreignObject > section .katex .sizing.reset-size6.size11{font-size:2.488em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size7.size1,article#presentation > svg > foreignObject > section .katex .sizing.reset-size7.size1{font-size:.41666667em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size7.size2,article#presentation > svg > foreignObject > section .katex .sizing.reset-size7.size2{font-size:.5em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size7.size3,article#presentation > svg > foreignObject > section .katex .sizing.reset-size7.size3{font-size:.58333333em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size7.size4,article#presentation > svg > foreignObject > section .katex .sizing.reset-size7.size4{font-size:.66666667em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size7.size5,article#presentation > svg > foreignObject > section .katex .sizing.reset-size7.size5{font-size:.75em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size7.size6,article#presentation > svg > foreignObject > section .katex .sizing.reset-size7.size6{font-size:.83333333em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size7.size7,article#presentation > svg > foreignObject > section .katex .sizing.reset-size7.size7{font-size:1em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size7.size8,article#presentation > svg > foreignObject > section .katex .sizing.reset-size7.size8{font-size:1.2em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size7.size9,article#presentation > svg > foreignObject > section .katex .sizing.reset-size7.size9{font-size:1.44em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size7.size10,article#presentation > svg > foreignObject > section .katex .sizing.reset-size7.size10{font-size:1.72833333em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size7.size11,article#presentation > svg > foreignObject > section .katex .sizing.reset-size7.size11{font-size:2.07333333em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size8.size1,article#presentation > svg > foreignObject > section .katex .sizing.reset-size8.size1{font-size:.34722222em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size8.size2,article#presentation > svg > foreignObject > section .katex .sizing.reset-size8.size2{font-size:.41666667em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size8.size3,article#presentation > svg > foreignObject > section .katex .sizing.reset-size8.size3{font-size:.48611111em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size8.size4,article#presentation > svg > foreignObject > section .katex .sizing.reset-size8.size4{font-size:.55555556em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size8.size5,article#presentation > svg > foreignObject > section .katex .sizing.reset-size8.size5{font-size:.625em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size8.size6,article#presentation > svg > foreignObject > section .katex .sizing.reset-size8.size6{font-size:.69444444em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size8.size7,article#presentation > svg > foreignObject > section .katex .sizing.reset-size8.size7{font-size:.83333333em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size8.size8,article#presentation > svg > foreignObject > section .katex .sizing.reset-size8.size8{font-size:1em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size8.size9,article#presentation > svg > foreignObject > section .katex .sizing.reset-size8.size9{font-size:1.2em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size8.size10,article#presentation > svg > foreignObject > section .katex .sizing.reset-size8.size10{font-size:1.44027778em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size8.size11,article#presentation > svg > foreignObject > section .katex .sizing.reset-size8.size11{font-size:1.72777778em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size9.size1,article#presentation > svg > foreignObject > section .katex .sizing.reset-size9.size1{font-size:.28935185em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size9.size2,article#presentation > svg > foreignObject > section .katex .sizing.reset-size9.size2{font-size:.34722222em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size9.size3,article#presentation > svg > foreignObject > section .katex .sizing.reset-size9.size3{font-size:.40509259em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size9.size4,article#presentation > svg > foreignObject > section .katex .sizing.reset-size9.size4{font-size:.46296296em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size9.size5,article#presentation > svg > foreignObject > section .katex .sizing.reset-size9.size5{font-size:.52083333em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size9.size6,article#presentation > svg > foreignObject > section .katex .sizing.reset-size9.size6{font-size:.5787037em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size9.size7,article#presentation > svg > foreignObject > section .katex .sizing.reset-size9.size7{font-size:.69444444em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size9.size8,article#presentation > svg > foreignObject > section .katex .sizing.reset-size9.size8{font-size:.83333333em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size9.size9,article#presentation > svg > foreignObject > section .katex .sizing.reset-size9.size9{font-size:1em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size9.size10,article#presentation > svg > foreignObject > section .katex .sizing.reset-size9.size10{font-size:1.20023148em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size9.size11,article#presentation > svg > foreignObject > section .katex .sizing.reset-size9.size11{font-size:1.43981481em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size10.size1,article#presentation > svg > foreignObject > section .katex .sizing.reset-size10.size1{font-size:.24108004em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size10.size2,article#presentation > svg > foreignObject > section .katex .sizing.reset-size10.size2{font-size:.28929605em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size10.size3,article#presentation > svg > foreignObject > section .katex .sizing.reset-size10.size3{font-size:.33751205em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size10.size4,article#presentation > svg > foreignObject > section .katex .sizing.reset-size10.size4{font-size:.38572806em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size10.size5,article#presentation > svg > foreignObject > section .katex .sizing.reset-size10.size5{font-size:.43394407em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size10.size6,article#presentation > svg > foreignObject > section .katex .sizing.reset-size10.size6{font-size:.48216008em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size10.size7,article#presentation > svg > foreignObject > section .katex .sizing.reset-size10.size7{font-size:.57859209em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size10.size8,article#presentation > svg > foreignObject > section .katex .sizing.reset-size10.size8{font-size:.69431051em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size10.size9,article#presentation > svg > foreignObject > section .katex .sizing.reset-size10.size9{font-size:.83317261em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size10.size10,article#presentation > svg > foreignObject > section .katex .sizing.reset-size10.size10{font-size:1em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size10.size11,article#presentation > svg > foreignObject > section .katex .sizing.reset-size10.size11{font-size:1.19961427em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size11.size1,article#presentation > svg > foreignObject > section .katex .sizing.reset-size11.size1{font-size:.20096463em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size11.size2,article#presentation > svg > foreignObject > section .katex .sizing.reset-size11.size2{font-size:.24115756em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size11.size3,article#presentation > svg > foreignObject > section .katex .sizing.reset-size11.size3{font-size:.28135048em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size11.size4,article#presentation > svg > foreignObject > section .katex .sizing.reset-size11.size4{font-size:.32154341em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size11.size5,article#presentation > svg > foreignObject > section .katex .sizing.reset-size11.size5{font-size:.36173633em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size11.size6,article#presentation > svg > foreignObject > section .katex .sizing.reset-size11.size6{font-size:.40192926em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size11.size7,article#presentation > svg > foreignObject > section .katex .sizing.reset-size11.size7{font-size:.48231511em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size11.size8,article#presentation > svg > foreignObject > section .katex .sizing.reset-size11.size8{font-size:.57877814em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size11.size9,article#presentation > svg > foreignObject > section .katex .sizing.reset-size11.size9{font-size:.69453376em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size11.size10,article#presentation > svg > foreignObject > section .katex .sizing.reset-size11.size10{font-size:.83360129em}article#presentation > svg > foreignObject > section .katex .fontsize-ensurer.reset-size11.size11,article#presentation > svg > foreignObject > section .katex .sizing.reset-size11.size11{font-size:1em}article#presentation > svg > foreignObject > section .katex .delimsizing.size1{font-family:KaTeX_Size1}article#presentation > svg > foreignObject > section .katex .delimsizing.size2{font-family:KaTeX_Size2}article#presentation > svg > foreignObject > section .katex .delimsizing.size3{font-family:KaTeX_Size3}article#presentation > svg > foreignObject > section .katex .delimsizing.size4{font-family:KaTeX_Size4}article#presentation > svg > foreignObject > section .katex .delimsizing.mult .delim-size1>span{font-family:KaTeX_Size1}article#presentation > svg > foreignObject > section .katex .delimsizing.mult .delim-size4>span{font-family:KaTeX_Size4}article#presentation > svg > foreignObject > section .katex .nulldelimiter{display:inline-block;width:.12em}article#presentation > svg > foreignObject > section .katex .delimcenter,article#presentation > svg > foreignObject > section .katex .op-symbol{position:relative}article#presentation > svg > foreignObject > section .katex .op-symbol.small-op{font-family:KaTeX_Size1}article#presentation > svg > foreignObject > section .katex .op-symbol.large-op{font-family:KaTeX_Size2}article#presentation > svg > foreignObject > section .katex .accent>.vlist-t,article#presentation > svg > foreignObject > section .katex .op-limits>.vlist-t{text-align:center}article#presentation > svg > foreignObject > section .katex .accent .accent-body{position:relative}article#presentation > svg > foreignObject > section .katex .accent .accent-body:not(.accent-full){width:0}article#presentation > svg > foreignObject > section .katex .overlay{display:block}article#presentation > svg > foreignObject > section .katex .mtable .vertical-separator{display:inline-block;margin:0 -.025em;border-right:.05em solid;min-width:1px}article#presentation > svg > foreignObject > section .katex .mtable .vs-dashed{border-right:.05em dashed}article#presentation > svg > foreignObject > section .katex .mtable .arraycolsep{display:inline-block}article#presentation > svg > foreignObject > section .katex .mtable .col-align-c>.vlist-t{text-align:center}article#presentation > svg > foreignObject > section .katex .mtable .col-align-l>.vlist-t{text-align:left}article#presentation > svg > foreignObject > section .katex .mtable .col-align-r>.vlist-t{text-align:right}article#presentation > svg > foreignObject > section .katex .svg-align{text-align:left}article#presentation > svg > foreignObject > section .katex svg{display:block;position:absolute;width:100%;height:inherit;fill:currentColor;stroke:currentColor;fill-rule:nonzero;fill-opacity:1;stroke-width:1;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1}article#presentation > svg > foreignObject > section .katex svg path{stroke:none}article#presentation > svg > foreignObject > section .katex img{border-style:none;min-width:0;min-height:0;max-width:none;max-height:none}article#presentation > svg > foreignObject > section .katex .stretchy{width:100%;display:block;position:relative;overflow:hidden}article#presentation > svg > foreignObject > section .katex .stretchy:after,article#presentation > svg > foreignObject > section .katex .stretchy:before{content:""}article#presentation > svg > foreignObject > section .katex .hide-tail{width:100%;position:relative;overflow:hidden}article#presentation > svg > foreignObject > section .katex .halfarrow-left{position:absolute;left:0;width:50.2%;overflow:hidden}article#presentation > svg > foreignObject > section .katex .halfarrow-right{position:absolute;right:0;width:50.2%;overflow:hidden}article#presentation > svg > foreignObject > section .katex .brace-left{position:absolute;left:0;width:25.1%;overflow:hidden}article#presentation > svg > foreignObject > section .katex .brace-center{position:absolute;left:25%;width:50%;overflow:hidden}article#presentation > svg > foreignObject > section .katex .brace-right{position:absolute;right:0;width:25.1%;overflow:hidden}article#presentation > svg > foreignObject > section .katex .x-arrow-pad{padding:0 .5em}article#presentation > svg > foreignObject > section .katex .mover,article#presentation > svg > foreignObject > section .katex .munder,article#presentation > svg > foreignObject > section .katex .x-arrow{text-align:center}article#presentation > svg > foreignObject > section .katex .boxpad{padding:0 .3em}article#presentation > svg > foreignObject > section .katex .fbox,article#presentation > svg > foreignObject > section .katex .fcolorbox{box-sizing:border-box;border:.04em solid}article#presentation > svg > foreignObject > section .katex .cancel-pad{padding:0 .2em}article#presentation > svg > foreignObject > section .katex .cancel-lap{margin-left:-.2em;margin-right:-.2em}article#presentation > svg > foreignObject > section .katex .sout{border-bottom-style:solid;border-bottom-width:.08em}article#presentation > svg > foreignObject > section .katex-display{display:block;margin:1em 0;text-align:center}article#presentation > svg > foreignObject > section .katex-display>.katex{display:block;text-align:center;white-space:nowrap}article#presentation > svg > foreignObject > section .katex-display>.katex>.katex-html{display:block;position:relative}article#presentation > svg > foreignObject > section .katex-display>.katex>.katex-html>.tag{position:absolute;right:0}article#presentation > svg > foreignObject > section .katex-display.leqno>.katex>.katex-html>.tag{left:0;right:auto}article#presentation > svg > foreignObject > section .katex-display.fleqn>.katex{text-align:left}article#presentation > svg > foreignObject > section .katex-display{margin:0}article#presentation > svg > foreignObject > section svg[data-marp-fitting-math]{--preserve-aspect-ratio:xMidYMid meet}article#presentation > svg > foreignObject > section svg[data-marp-fitting-math] [data-marp-fitting-svg-content]{margin:0 auto}article#presentation > svg > foreignObject > section svg[data-marp-fitting=svg]{display:block;height:auto;width:100%}@supports (-ms-ime-align:auto){article#presentation > svg > foreignObject > section svg[data-marp-fitting=svg]{position:static}}article#presentation > svg > foreignObject > section svg[data-marp-fitting=svg].__reflow__{content:""}@supports (-ms-ime-align:auto){article#presentation > svg > foreignObject > section svg[data-marp-fitting=svg].__reflow__{position:relative}}article#presentation > svg > foreignObject > section [data-marp-fitting-svg-content]{display:table;white-space:nowrap}article#presentation > svg > foreignObject > section [data-marp-fitting-svg-content-wrap]{white-space:pre}article#presentation > svg > foreignObject > section img[data-marp-twemoji]{background:transparent;height:1em;margin:0 .05em 0 .1em;vertical-align:-.1em;width:1em}
/*!
* Marp / Marpit Gaia theme.
*
* @theme gaia
* @author Yuki Hattori
*
* @auto-scaling true
* @size 4:3 960px 720px
*/article#presentation>svg>foreignObject>section .hljs{display:block;overflow-x:auto;padding:.5em;background:#000;color:#f8f8f8}article#presentation>svg>foreignObject>section .hljs-comment,article#presentation>svg>foreignObject>section .hljs-quote{color:#aeaeae;font-style:italic}article#presentation>svg>foreignObject>section .hljs-keyword,article#presentation>svg>foreignObject>section .hljs-selector-tag,article#presentation>svg>foreignObject>section .hljs-type{color:#e28964}article#presentation>svg>foreignObject>section .hljs-string{color:#65b042}article#presentation>svg>foreignObject>section .hljs-subst{color:#daefa3}article#presentation>svg>foreignObject>section .hljs-link,article#presentation>svg>foreignObject>section .hljs-regexp{color:#e9c062}article#presentation>svg>foreignObject>section .hljs-name,article#presentation>svg>foreignObject>section .hljs-section,article#presentation>svg>foreignObject>section .hljs-tag,article#presentation>svg>foreignObject>section .hljs-title{color:#89bdff}article#presentation>svg>foreignObject>section .hljs-class .hljs-title,article#presentation>svg>foreignObject>section .hljs-doctag{text-decoration:underline}article#presentation>svg>foreignObject>section .hljs-bullet,article#presentation>svg>foreignObject>section .hljs-number,article#presentation>svg>foreignObject>section .hljs-symbol{color:#3387cc}article#presentation>svg>foreignObject>section .hljs-params,article#presentation>svg>foreignObject>section .hljs-template-variable,article#presentation>svg>foreignObject>section .hljs-variable{color:#3e87e3}article#presentation>svg>foreignObject>section .hljs-attribute{color:#cda869}article#presentation>svg>foreignObject>section .hljs-meta{color:#8996a8}article#presentation>svg>foreignObject>section .hljs-formula{background-color:#0e2231;color:#f8f8f8;font-style:italic}article#presentation>svg>foreignObject>section .hljs-addition{background-color:#253b22;color:#f8f8f8}article#presentation>svg>foreignObject>section .hljs-deletion{background-color:#420e09;color:#f8f8f8}article#presentation>svg>foreignObject>section .hljs-selector-class{color:#9b703f}article#presentation>svg>foreignObject>section .hljs-selector-id{color:#8b98ab}article#presentation>svg>foreignObject>section .hljs-emphasis{font-style:italic}article#presentation>svg>foreignObject>section .hljs-strong{font-weight:700}article#presentation>svg>foreignObject>sectionsvg[data-marp-fitting=svg]{max-height:580px}article#presentation>svg>foreignObject>sectionh1,article#presentation>svg>foreignObject>sectionh2,article#presentation>svg>foreignObject>sectionh3,article#presentation>svg>foreignObject>sectionh4,article#presentation>svg>foreignObject>sectionh5,article#presentation>svg>foreignObject>sectionh6{margin:.5em00}article#presentation>svg>foreignObject>sectionh1strong,article#presentation>svg>foreignObject>sectionh2strong,article#presentation>svg>foreignObject>sectionh3strong,article#presentation>svg>foreignObject>sectionh4strong,article#presentation>svg>foreignObject>sectionh5strong,article#presentation>svg>foreignObject>sectionh6strong{font-weight:inherit}article#presentation>svg>foreignObject>sectionh1{font-size:1.8em}article#presentation>svg>foreignObject>sectionh2{font-size:1.5em}article#presentation>svg>foreignObject>sectionh3{font-size:1.3em}article#presentation>svg>foreignObject>sectionh4{font-size:1.1em}article#presentation>svg>foreignObject>sectionh5{font-size:1em}article#presentation>svg>foreignObject>sectionh6{font-size:.9em}article#presentation>svg>foreignObject>sectionblockquote,article#presentation>svg>foreignObject>sectionp{margin:1em00}article#presentation>svg>foreignObject>sectionol>li,article#presentation>svg>foreignObject>sectionul>li{margin:.3em00}article#presentation>svg>foreignObject>sectionol>li>p,article#presentation>svg>foreignObject>sectionul>li>p{margin:.6em00}article#presentation>svg>foreignObject>sectioncode{display:inline-block;font-family:Roboto Mono,monospace;font-size:.8em;letter-spacing:0;margin:-.1em.15em;padding:.1em.2em;vertical-align:baseline}article#presentation>svg>foreignObject>sectionpre{display:block;margin:1em00;min-height:1em;overflow:visible}article#presentation>svg>foreignObject>sectionprecode{box-sizing:border-box;margin:0;min-width:100%;padding:.5em;font-size:.7em}article#presentation>svg>foreignObject>sectionprecodesvg[data-marp-fitting=svg]{max-height:calc(580px-1em)}article#presentation>svg>foreignObject>sectionblockquote{margin:1em00;padding:01em;position:relative}article#presentation>svg>foreignObject>sectionblockquote:after,article#presentation>svg>foreignObject>sectionblockquote:before{content:"“";display:block;font-family:Times New Roman,serif;font-weight:700;position:absolute}article#presentation>svg>foreignObject>sectionblockquote:before{top:0;left:0}article#presentation>svg>foreignObject>sectionblockquote:after{right:0;bottom:0;transform:rotate(180deg)}article#presentation>svg>foreignObject>sectionblockquote>:first-child{margin-top:0}article#presentation>svg>foreignObject>sectionmark{background:transparent}article#presentation > svg > foreignObject > section table{border-spacing:0;border-collapse:collapse;margin:1em 0 0}article#presentation > svg > foreignObject > section table td,article#presentation > svg > foreignObject > section table th{padding:.2em .4em;border-width:1px;border-style:solid}article#presentation > svg > foreignObject > section{background-image:linear-gradient(135deg,hsla(0,0%,53.3%,0),hsla(0,0%,53.3%,.02) 50%,hsla(0,0%,100%,0) 0,hsla(0,0%,100%,.05));font-size:35px;font-family:Lato,Avenir Next,Avenir,Trebuchet MS,Segoe UI,sans-serif;height:720px;line-height:1.35;letter-spacing:1.25px;padding:70px;width:1280px;color:#455a64;background-color:#fff8e1}article#presentation > svg > foreignObject > section>:first-child,article#presentation > svg > foreignObject > section>header:first-child+*{margin-top:0}article#presentation > svg > foreignObject > section a,article#presentation > svg > foreignObject > section mark{color:#0288d1}article#presentation > svg > foreignObject > section code{background:#6a7a7d;color:#fff8e1}article#presentation > svg > foreignObject > section h1 strong,article#presentation > svg > foreignObject > section h2 strong,article#presentation > svg > foreignObject > section h3 strong,article#presentation > svg > foreignObject > section h4 strong,article#presentation > svg > foreignObject > section h5 strong,article#presentation > svg > foreignObject > section h6 strong{color:#0288d1}article#presentation > svg > foreignObject > section pre>code{background:#455a64}article#presentation > svg > foreignObject > section blockquote:after,article#presentation > svg > foreignObject > section blockquote:before,article#presentation > svg > foreignObject > section footer,article#presentation > svg > foreignObject > section header,article#presentation > svg > foreignObject > section section:after{color:#6a7a7d}article#presentation > svg > foreignObject > section table td,article#presentation > svg > foreignObject > section table th{border-color:#455a64}article#presentation > svg > foreignObject > section table thead th{background:#455a64;color:#fff8e1}article#presentation > svg > foreignObject > section table tbody>tr:nth-child(odd) td,article#presentation > svg > foreignObject > section table tbody>tr:nth-child(odd) th{background:rgba(69,90,100,.1)}article#presentation > svg > foreignObject > section.invert{color:#fff8e1;background-color:#455a64}article#presentation > svg > foreignObject > section.invert a,article#presentation > svg > foreignObject > section.invert mark{color:#81d4fa}article#presentation > svg > foreignObject > section.invert code{background:#dad8c8;color:#455a64}article#presentation > svg > foreignObject > section.invert h1 strong,article#presentation > svg > foreignObject > section.invert h2 strong,article#presentation > svg > foreignObject > section.invert h3 strong,article#presentation > svg > foreignObject > section.invert h4 strong,article#presentation > svg > foreignObject > section.invert h5 strong,article#presentation > svg > foreignObject > section.invert h6 strong{color:#81d4fa}article#presentation > svg > foreignObject > section.invert pre>code{background:#fff8e1}article#presentation > svg > foreignObject > section.invert blockquote:after,article#presentation > svg > foreignObject > section.invert blockquote:before,article#presentation > svg > foreignObject > section.invert footer,article#presentation > svg > foreignObject > section.invert header,article#presentation > svg > foreignObject > section.invert section:after{color:#dad8c8}article#presentation > svg > foreignObject > section.invert table td,article#presentation > svg > foreignObject > section.invert table th{border-color:#fff8e1}article#presentation > svg > foreignObject > section.invert table thead th{background:#fff8e1;color:#455a64}article#presentation > svg > foreignObject > section.invert table tbody>tr:nth-child(odd) td,article#presentation > svg > foreignObject > section.invert table tbody>tr:nth-child(odd) th{background:rgba(255,248,225,.1)}article#presentation > svg > foreignObject > section.gaia{color:#fff8e1;background-color:#0288d1}article#presentation > svg > foreignObject > section.gaia a,article#presentation > svg > foreignObject > section.gaia mark{color:#81d4fa}article#presentation > svg > foreignObject > section.gaia code{background:#cce2de;color:#0288d1}article#presentation > svg > foreignObject > section.gaia h1 strong,article#presentation > svg > foreignObject > section.gaia h2 strong,article#presentation > svg > foreignObject > section.gaia h3 strong,article#presentation > svg > foreignObject > section.gaia h4 strong,article#presentation > svg > foreignObject > section.gaia h5 strong,article#presentation > svg > foreignObject > section.gaia h6 strong{color:#81d4fa}article#presentation > svg > foreignObject > section.gaia pre>code{background:#fff8e1}article#presentation > svg > foreignObject > section.gaia blockquote:after,article#presentation > svg > foreignObject > section.gaia blockquote:before,article#presentation > svg > foreignObject > section.gaia footer,article#presentation > svg > foreignObject > section.gaia header,article#presentation > svg > foreignObject > section.gaia section:after{color:#cce2de}article#presentation > svg > foreignObject > section.gaia table td,article#presentation > svg > foreignObject > section.gaia table th{border-color:#fff8e1}article#presentation > svg > foreignObject > section.gaia table thead th{background:#fff8e1;color:#0288d1}article#presentation > svg > foreignObject > section.gaia table tbody>tr:nth-child(odd) td,article#presentation > svg > foreignObject > section.gaia table tbody>tr:nth-child(odd) th{background:rgba(255,248,225,.1)}article#presentation > svg > foreignObject > section.lead{display:flex;flex-direction:column;flex-wrap:nowrap;justify-content:center}article#presentation > svg > foreignObject > section.lead h1,article#presentation > svg > foreignObject > section.lead h2,article#presentation > svg > foreignObject > section.lead h3,article#presentation > svg > foreignObject > section.lead h4,article#presentation > svg > foreignObject > section.lead h5,article#presentation > svg > foreignObject > section.lead h6{text-align:center}article#presentation > svg > foreignObject > section.lead h1 svg[data-marp-fitting=svg],article#presentation > svg > foreignObject > section.lead h2 svg[data-marp-fitting=svg],article#presentation > svg > foreignObject > section.lead h3 svg[data-marp-fitting=svg],article#presentation > svg > foreignObject > section.lead h4 svg[data-marp-fitting=svg],article#presentation > svg > foreignObject > section.lead h5 svg[data-marp-fitting=svg],article#presentation > svg > foreignObject > section.lead h6 svg[data-marp-fitting=svg]{--preserve-aspect-ratio:xMidYMid meet}article#presentation > svg > foreignObject > section.lead p{text-align:center}article#presentation > svg > foreignObject > section.lead blockquote>h1,article#presentation > svg > foreignObject > section.lead blockquote>h2,article#presentation > svg > foreignObject > section.lead blockquote>h3,article#presentation > svg > foreignObject > section.lead blockquote>h4,article#presentation > svg > foreignObject > section.lead blockquote>h5,article#presentation > svg > foreignObject > section.lead blockquote>h6,article#presentation > svg > foreignObject > section.lead blockquote>p{text-align:left}article#presentation > svg > foreignObject > section.lead blockquote svg[data-marp-fitting=svg]:not([data-marp-fitting-math]){--preserve-aspect-ratio:xMinYMin meet}article#presentation > svg > foreignObject > section.lead ol>li>p,article#presentation > svg > foreignObject > section.lead ul>li>p{text-align:left}article#presentation > svg > foreignObject > section.lead table{margin-left:auto;margin-right:auto}article#presentation > svg > foreignObject > section footer,article#presentation > svg > foreignObject > section header,article#presentation > svg > foreignObject > section:after{box-sizing:border-box;font-size:66%;height:70px;line-height:50px;overflow:hidden;padding:10px 25px;position:absolute}article#presentation > svg > foreignObject > section header{top:0}article#presentation > svg > foreignObject > section footer,article#presentation > svg > foreignObject > section header{left:0;right:0}article#presentation > svg > foreignObject > section footer{bottom:0}article#presentation > svg > foreignObject > section:after{right:0;bottom:0;font-size:80%}article#presentation > svg > foreignObject > section[data-marpit-advanced-background="background"] {
padding:0!important;
}article#presentation>svg>foreignObject>section[data-marpit-advanced-background="background"]::before,
article#presentation>svg>foreignObject>section[data-marpit-advanced-background="background"]::after,
article#presentation>svg>foreignObject>section[data-marpit-advanced-background="content"]::before,
article#presentation>svg>foreignObject>section[data-marpit-advanced-background="content"]::after {
display: none !important;
}article#presentation>svg>foreignObject>section[data-marpit-advanced-background="background"] >div[data-marpit-advanced-background-container] {
all: initial;
display: flex;
flex-direction: row;
height:100%;
overflow: hidden;
width:100%;
}article#presentation>svg>foreignObject>section[data-marpit-advanced-background="background"] >div[data-marpit-advanced-background-container][data-marpit-advanced-background-direction="vertical"] {
flex-direction: column;
}article#presentation>svg>foreignObject>section[data-marpit-advanced-background="background"][data-marpit-advanced-background-split] >div[data-marpit-advanced-background-container] {
width:var(--marpit-advanced-background-split,50%);
}article#presentation>svg>foreignObject>section[data-marpit-advanced-background="background"][data-marpit-advanced-background-split="right"] >div[data-marpit-advanced-background-container] {
margin-left:calc(100%-var(--marpit-advanced-background-split,50%));
}article#presentation>svg>foreignObject>section[data-marpit-advanced-background="background"] >div[data-marpit-advanced-background-container] >figure {
all: initial;
background-position: center;
background-repeat: no-repeat;
background-size: cover;
flex: auto;
margin:0;
}article#presentation>svg>foreignObject>section[data-marpit-advanced-background="content"],
article#presentation>svg>foreignObject>section[data-marpit-advanced-background="pseudo"] {
background: transparent !important;
}article#presentation>svg>foreignObject>section[data-marpit-advanced-background="pseudo"],
article#presentation>svg[data-marpit-svg] >foreignObject[data-marpit-advanced-background="pseudo"] {
pointer-events: none !important;
}article#presentation>svg>foreignObject>section[data-marpit-advanced-background-split] {
width:100%;
height:100%;
}</style></head><body><divclass="bespoke-marp-osc"><buttondata-bespoke-marp-osc="prev" tabindex="-1" title="Previous slide">Previous slide</button><spandata-bespoke-marp-osc="page"></span><buttondata-bespoke-marp-osc="next" tabindex="-1" title="Next slide">Next slide</button><buttondata-bespoke-marp-osc="fullscreen" tabindex="-1" title="Toggle fullscreen (f)">Toggle fullscreen</button></div><articleid="presentation"><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="1" data-class="lead invert" data-theme="gaia" class="lead invert" style="--class:lead invert;--theme:gaia;">
<h1>A simple project mixing parallel Python and C++ codes</h1>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="2" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>We will show...</h1>
<ul>
<li>how to configure a platform-independent project mixing Python and C++ with <ahref="https://cmake.org">cmake</a> and <ahref="https://github.com/pybind/pybind11">pybind11</a></li>
<li>how to call Python from C++ and vice versa</li>
<li>how to write efficient parallel code mixing C++ and Python by circumventing Python's GIL</li>
<li>how to unit-test C++ code</li>
<li>how to produce meaningful stack traces in case of segfaults (like stack trace exceptions in Java or Python)</li>
<li>how to setup a cmake-based mixed C++/Python library with setuptools (e.g. for installation with pip)</li>
</ul>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="3" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Requirements</h1>
<ul>
<li>good C++ compiler with C++-17 standard support: e.g. <ahref="https://gcc.gnu.org">GCC</a> or <ahref="https://clang.llvm.org">Clang</a></li>
<li><ahref="https://cmake.org">CMake</a>: platform-independent configuration tool for C++</li>
<li><ahref="https://github.com/pybind/pybind11">pybind11</a>: high-level C++ directives to call Python from C++ and vice versa; built upon cython</li>
<li><ahref="https://github.com/bombela/backward-cpp">backward</a>: magically adds stack trace capabilities for C++ without additional code (only if project compiled in debug mode)</li>
<li><ahref="https://github.com/catchorg/Catch2">Catch2</a>: beautiful unit tests in C++</li>
</ul>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="4" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Project architecture</h1>
<pre><codeclass="language-bash"><svgdata-marp-fitting="svg" data-marp-fitting-code><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap>simple-cpp-python
|-- CMakeLists.txt <spanclass="hljs-comment"># project configuration</span>
|-- code.hh <spanclass="hljs-comment"># C++ functions code</span>
|-- package.cc <spanclass="hljs-comment"># export C++ functions to python</span>
|-- tests.cc <spanclass="hljs-comment"># C++ unit tests</span>
|-- script.py <spanclass="hljs-comment"># python code calling C++ code</span>
|-- setup.py <spanclass="hljs-comment"># setuptools installation script</span>
|-- __init__.py <spanclass="hljs-comment"># init script loading the compiled library</span>
|-- build <spanclass="hljs-comment"># created after project configuration (see next slide)</span>
|-- __simple_cpp_python.cpython-37m-darwin.so <spanclass="hljs-comment"># C++ python extension</span>
</span></span></foreignObject></svg></code></pre>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="5" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Project configuration: CMakeLists.txt</h1>
<pre><codeclass="language-cmake"><svgdata-marp-fitting="svg" data-marp-fitting-code><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap><spanclass="hljs-keyword">CMAKE_MINIMUM_REQUIRED</span>(VERSION <spanclass="hljs-number">3.10</span>.<spanclass="hljs-number">2</span>)
<spanclass="hljs-keyword">PROJECT</span>(simple_cpp_python)
<spanclass="hljs-keyword">ENABLE_LANGUAGE</span>(CXX)
<spanclass="hljs-keyword">SET</span>(CMAKE_CXX_STANDARD <spanclass="hljs-number">17</span>) <spanclass="hljs-comment"># C++-17 required for high-level code parallelization</span>
<spanclass="hljs-keyword">SET</span>(CMAKE_CXX_STANDARD_REQUIRED <spanclass="hljs-keyword">ON</span>)
<spanclass="hljs-keyword">FIND_PACKAGE</span>(TBB REQUIRED) <spanclass="hljs-comment"># TBB required by GCC's implementation of C++-17</span>
<spanclass="hljs-keyword">FIND_PACKAGE</span>(pybind11 REQUIRED)
<spanclass="hljs-keyword">FIND_PACKAGE</span>(Catch2 REQUIRED)
<spanclass="hljs-keyword">FIND_PACKAGE</span>(Backward) <spanclass="hljs-comment"># optional package</span>
PYBIND11_ADD_MODULE(__simple_cpp_python package.cc)
<spanclass="hljs-keyword">TARGET_LINK_LIBRARIES</span>(__simple_cpp_python PRIVATE <spanclass="hljs-variable">${TBB_IMPORTED_TARGETS}</span>)
</span></span></foreignObject></svg></code></pre>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="6" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Configuring the project</h1>
<ol>
<li>Go to the directory where you installed <code>simple-cpp-python</code></li>
<li>Create a <code>build</code> directory there</li>
<li>Run the following commands (or use an IDE like visual studio code):<pre><codeclass="language-bash"><svgdata-marp-fitting="svg" data-marp-fitting-code><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap><spanclass="hljs-built_in">cd</span> build
cmake .. <spanclass="hljs-comment"># configure the project for your platform (creates a Makefile)</span>
make <spanclass="hljs-comment"># compile the python extension library</span>
</span></span></foreignObject></svg></code></pre>
</li>
</ol>
<p>You should now have a file under <code>build</code> called<br/>
<code>__simple_cpp_python.cpython-37m-darwin.so</code> that you can import in Python.</p>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="7" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Selecting different python interpreters</h1>
<p>In the previous slide, cmake via the pybind11 package (line<br/>
<code>FIND_PACKAGE(pybind11 REQUIRED)</code> in <em>CMakeLists.txt</em>) automatically selects the default python interpreter of my shell.</p>
<p>To select a different python interpreter you need to configure with the following command:<br/>
<code>cmake -DPYTHON_EXECUTABLE="/path/to/my/python/interpreter" ..</code></p>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="8" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Computing Lagrange's trigonometric identities</h1>
<p>See the <ahref="https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Lagrange's_trigonometric_identities">Wikipedia</a> page. We gonna compute:</p>
<p><svgdata-marp-fitting="svg" data-marp-fitting-math><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap><spanclass="katex-display"><spanclass="katex"><spanclass="katex-mathml"><math><semantics><mtablerowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstylescriptlevel="0" displaystyle="true"><mrow><mi>f</mi><mostretchy="false">(</mo><mi>θ</mi><moseparator="true">,</mo><mi>N</mi><mostretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstylescriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mi>sin</mi><mo></mo><mostretchy="false">(</mo><mi>n</mi><mi>θ</mi><mostretchy="false">)</mo><mo>+</mo><mi>cos</mi><mo></mo><mostretchy="false">(</mo><mi>n</mi><mi>θ</mi><mostretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstylescriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstylescriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mofence="true">(</mo><mi>cot</mi><mo></mo><mrow><mofence="true">(</mo><mfrac><mi>θ</mi><mn>2</mn></mfrac><mofence="true">)</mo></mrow><mo>−</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mi>sin</mi><mo></mo><mrow><mofence="true">(</mo><mrow><mofence="true">(</mo><mi>N</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mofence="true">)</mo></mrow><mi>θ</mi><mofence="true">)</mo></mrow><mo>−</mo><mi>cos</mi><mo></mo><mrow><mofence="true">(</mo><mrow><mofence="true">(</mo><mi>N</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mofence="true">)</mo></mrow><mi>θ</mi><mofence="true">)</mo></mrow></mrow><mrow><mi>sin</mi><mo></mo><mrow><mofence="true">(</mo><mfrac><mi>θ</mi><mn>2</mn></mfrac><mofence="true">)</mo></mrow></mrow></mfrac><mofence="true">)</mo></mrow></mrow></mstyle></mtd></mtr></mtable><annotationencoding="application/x-tex"> \begin{aligned}
f(\theta, N) &= \sum_{n=1}^N \sin(n \theta) + \cos(n \theta) \\
& = \frac{1}{2} \left( \cot \left(\frac{\theta}{2}\right) - 1 + \frac{\sin \left( \left(N+\frac{1}{2}\right) \theta \right) - \cos \left( \left(N+\frac{1}{2}\right) \theta \right)}{\sin \left(\frac{\theta}{2}\right)} \right)
\end{aligned} </annotation></semantics></math></span><spanclass="katex-html" aria-hidden="true"><spanclass="base"><spanclass="strut" style="height:6.695479000000001em;vertical-align:-3.0977395000000003em;"></span><spanclass="mord"><spanclass="mtable"><spanclass="col-align-r"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist" style="height:3.5977395000000003em;"><spanstyle="top:-5.5977395em;"><spanclass="pstrut" style="height:3.828336em;"></span><spanclass="mord"><spanclass="mord mathdefault" style="margin-right:0.10764em;">f</span><spanclass="mopen">(</span><spanclass="mord mathdefault" style="margin-right:0.02778em;">θ</span><spanclass="mpunct">,</span><spanclass="mspace" style="margin-right:0.16666666666666666em;"></span><spanclass="mord mathdefault" style="margin-right:0.10903em;">N</span><spanclass="mclose">)</span></span></span><spanstyle="top:-2.2806265em;"><spanclass="pstrut" style="height:3.828336em;"></span><spanclass="mord"></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist" style="height:3.0977395000000003em;"><span></span></span></span></span></span><spanclass="col-align-l"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist" style="height:3.5977395000000003em;"><spanstyle="top:-5.5977395em;"><spanclass="pstrut" style="height:3.828336em;"></span><spanclass="mord"><spanclass="mord"></span><spanclass="mspace" style="margin-right:0.2777777777777778em;"></span><spanclass="mrel">=</span><spanclass="mspace" style="margin-right:0.2777777777777778em;"></span><spanclass="mop op-limits"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist" style="height:1.8283360000000002em;"><spanstyle="top:-1.882887em;margin-left:0em;"><spanclass="pstrut" style="height:3.05em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mathdefault mtight">n</span><spanclass="mrel mtight">=</span><spanclass="mord mtight">1</span></span></span></span><spanstyle="top:-3.050005em;"><spanclass="pstrut" style="height:3.05em;"></span><span><spanclass="mop op-symbol large-op">∑</span></span></span><spanstyle="top:-4.3000050000000005em;margin-left:0em;"><spanclass="pstrut" style="height:3.05em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathdefault mtight" style="margin-right:0.10903em;">N</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist" style="height:1.267113em;"><span></span></span></span></span></span><spanclass="mspace" style="margin-right:0.16666666666666666em;"></span><spanclass="mop">sin</span><spanclass="mopen">(</span><spanclass="mord mathdefault">n</span><spanclass="mord mathdefault" style="margin-right:0.02778em;">θ</span><spanclass="mclose">)</span><spanclass="mspace" style="margin-right:0.2222222222222222em;"></span><spanclass="mbin">+</span><spanclass="mspace" style="margin-right:0.2222222222222222em;"></span><spanclass="mop">cos</span><spanclass="mopen">(</span><spanclass="mord mathdefault">n</span><spanclass="mord mathdefault" style="margin-right:0.02778em;">θ</span><spanclass="mclose">)</span></span></span><spanstyle="top:-2.2806265em;"><spanclass="pstrut" style="height:3.828336em;"></span><spanclass="mord"><spanclass="mord"></span><spanclass="mspace" style="margin-right:0.2777777777777778em;"></span><spanclass="mrel">=</span><spanclass="mspace" style="margin-right:0.2777777777777778em;"></span><spanclass="mord"><spanclass="mopen nulldelimiter"></span><spanclass="mfrac"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist" style="height:1.32144em;"><spanstyle="top:-2.314em;"><spanclass="pstrut" style="height:3em;"></span><spanclass="mord"><spanclass="mord">2</span></span></span><spanstyle="top:-3.23em;"><spanclass="pstrut" style="height:3em;"></span><spanclass="frac-line" style="border-bottom-width:0.04em;"></span></span><spanstyle="top:-3.677em;"><spanclass="pstrut" style="height:3em;"></span><spanclass="mord"><spanclass="mord">1</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist" style="height:0.686em;"><span></span></span></span></span></span><spanclass="mclose nulldelimiter"></span></span><spanclass="mspace" style="margin-right:0.16666666666666666em;"></span><spanclass="minner"><spanclass="mopen delimcenter" style="top:0em;"><spanclass="delimsizing size4">(</span></span><spanclass="mop">cot</span><spanclass="mspace" style="margin-right:0.16666666666666666em;"></span><spanclass="minner"><spanclass="mopen delimcenter" style="top:0em;"><spanclass="delimsizing size3">(</span></span><spanclass="mord"><spanclass="mopen nulldelimiter"></span><spanclass="mfrac"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist" style="height:1.37144em;"><spanstyle="top:-2.314em;"><spanclass="pstrut" style="height:3em;"></span><spanclass="mord"><spanclass="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.59001em;"><span style="top:-2.229892em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">sin</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801079999999999em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.7400100000000003em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">sin</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.120118em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size4">)</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.0977395000000003em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></foreignObject></svg></p><p>with <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>θ</mi><mo>=</mo><mfrac><mi>π</mi><mn>8</mn></mfrac><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>4</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>4</mn><mi>k</mi><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\theta = \frac{\pi}{8} = \sum_{k=0}^{+\infty} \frac{1}{(4k+1)(4k+3)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.040392em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.695392em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">π</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.431231em;vertical-align:-0.52em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.911231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">4</span><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span><span class="mopen mtight">(</span><span class="mord mtight">4</span><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">3</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.52em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></p>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="9" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Why such a dummy useless nested loops computation?</h1>
<br/>
<spanstyle="font-size:95%">
Only to show:
(1) the benefit of parallezing code mixing Python and C++
(2) how to call python (inner loop) from C++ (outer loop)
<p>Outer loop (i.e. sum from <spanclass="katex"><spanclass="katex-mathml"><math><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotationencoding="application/x-tex">n=1</annotation></semantics></math></span><spanclass="katex-html" aria-hidden="true"><spanclass="base"><spanclass="strut" style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathdefault">n</span><spanclass="mspace" style="margin-right:0.2777777777777778em;"></span><spanclass="mrel">=</span><spanclass="mspace" style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut" style="height:0.64444em;vertical-align:0em;"></span><spanclass="mord">1</span></span></span></span> to <spanclass="katex"><spanclass="katex-mathml"><math><semantics><mrow><mi>N</mi></mrow><annotationencoding="application/x-tex">N</annotation></semantics></math></span><spanclass="katex-html" aria-hidden="true"><spanclass="base"><spanclass="strut" style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathdefault" style="margin-right:0.10903em;">N</span></span></span></span>) used to demonstrate parallelization (compute each element of the sum in parallel)</p>
<p>Overhead of parallel for loops mitigated if computation of each element of the loop takes a lot of CPU cycles --> compute <spanclass="katex"><spanclass="katex-mathml"><math><semantics><mrow><mi>sin</mi><mo></mo><mostretchy="false">(</mo><mi>n</mi><mi>θ</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x-tex">\sin(n\theta)</annotation></semantics></math></span><spanclass="katex-html" aria-hidden="true"><spanclass="base"><spanclass="strut" style="height:1em;vertical-align:-0.25em;"></span><spanclass="mop">sin</span><spanclass="mopen">(</span><spanclass="mord mathdefault">n</span><spanclass="mord mathdefault" style="margin-right:0.02778em;">θ</span><spanclass="mclose">)</span></span></span></span> and <spanclass="katex"><spanclass="katex-mathml"><math><semantics><mrow><mi>cos</mi><mo></mo><mostretchy="false">(</mo><mi>n</mi><mi>θ</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x-tex">\cos(n\theta)</annotation></semantics></math></span><spanclass="katex-html" aria-hidden="true"><spanclass="base"><spanclass="strut" style="height:1em;vertical-align:-0.25em;"></span><spanclass="mop">cos</span><spanclass="mopen">(</span><spanclass="mord mathdefault">n</span><spanclass="mord mathdefault" style="margin-right:0.02778em;">θ</span><spanclass="mclose">)</span></span></span></span> with <spanclass="katex"><spanclass="katex-mathml"><math><semantics><mrow><mi>θ</mi><mo>=</mo><mfrac><mi>π</mi><mn>8</mn></mfrac></mrow><annotationencoding="application/x-tex">\theta=\frac{\pi}{8}</annotation></semantics></math></span><spanclass="katex-html" aria-hidden="true"><spanclass="base"><spanclass="strut" style="height:0.69444em;vertical-align:0em;"></span><spanclass="mord mathdefault" style="margin-right:0.02778em;">θ</span><spanclass="mspace" style="margin-right:0.2777777777777778em;"></span><spanclass="mrel">=</span><spanclass="mspace" style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut" style="height:1.040392em;vertical-align:-0.345em;"></span><spanclass="mord"><spanclass="mopen nulldelimiter"></span><spanclass="mfrac"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist" style="height:0.695392em;"><spanstyle="top:-2.6550000000000002em;"><spanclass="pstrut" style="height:3em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mtight">8</span></span></span></span><spanstyle="top:-3.23em;"><spanclass="pstrut" style="height:3em;"></span><spanclass="frac-line" style="border-bottom-width:0.04em;"></span></span><spanstyle="top:-3.394em;"><spanclass="pstrut" style="height:3em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mathdefault mtight" style="margin-right:0.03588em;">π</span></span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist" style="height:0.345em;"><span></span></span></span></span></span><spanclass="mclose nulldelimiter"></span></span></span></span></span> as a series (inner loop).<br/>
</span></p>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="10" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Full python code</h1>
<pre><codeclass="language-python"><svgdata-marp-fitting="svg" data-marp-fitting-code><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap><spanclass="hljs-keyword">import</span> math, operator, time, sys
<spanclass="hljs-keyword">from</span> pathos <spanclass="hljs-keyword">import</span> multiprocessing
<spanclass="hljs-keyword">from</span> functools <spanclass="hljs-keyword">import</span> reduce
<spanclass="hljs-function"><spanclass="hljs-keyword">def</span><spanclass="hljs-title">python_inner_loop</span><spanclass="hljs-params">(M)</span>:</span><spanclass="hljs-comment"># computes pi/8 as a sequential series</span>
<spanclass="hljs-keyword">return</span> reduce(operator.add, [<spanclass="hljs-number">1</span> / ((<spanclass="hljs-number">4</span>*k + <spanclass="hljs-number">1</span>) * (<spanclass="hljs-number">4</span>*k + <spanclass="hljs-number">3</span>)) <spanclass="hljs-keyword">for</span> k <spanclass="hljs-keyword">in</span> range(M)], <spanclass="hljs-number">0</span>)
<spanclass="hljs-function"><spanclass="hljs-keyword">def</span><spanclass="hljs-title">python_outer_loop</span><spanclass="hljs-params">(xf, N, M)</span>:</span><spanclass="hljs-comment"># computes the parallel series of cos(n * theta) + sin(n * theta)</span>
pool = multiprocessing.Pool()
v = pool.map(<spanclass="hljs-keyword">lambda</span> n: math.cos((n+<spanclass="hljs-number">1</span>) * xf(M)) + math.sin((n+<spanclass="hljs-number">1</span>) * xf(M)), [n <spanclass="hljs-keyword">for</span> n <spanclass="hljs-keyword">in</span> range(N)])
<spanclass="hljs-keyword">return</span> reduce(operator.add, v, <spanclass="hljs-number">0</span>)
<spanclass="hljs-function"><spanclass="hljs-keyword">def</span><spanclass="hljs-title">full_python</span><spanclass="hljs-params">(N, M)</span>:</span>
<spanclass="hljs-keyword">return</span> python_outer_loop(python_inner_loop, N, M)
<spanclass="hljs-keyword">if</span> __name__ == <spanclass="hljs-string">"__main__"</span>:
start = time.time()
r = full_python(<spanclass="hljs-number">1000</span>, <spanclass="hljs-number">10000</span>)
end = time.time()
print(<spanclass="hljs-string">'full python result: '</span> + str(r) + <spanclass="hljs-string">' in '</span> + str(end-start) + <spanclass="hljs-string">' seconds'</span>)
x = math.pi / <spanclass="hljs-number">8</span>
r = <spanclass="hljs-number">0.5</span> * ((<spanclass="hljs-number">1.0</span> / math.tan(x / <spanclass="hljs-number">2</span>)) - <spanclass="hljs-number">1</span> + (math.sin(x * (N + <spanclass="hljs-number">0.5</span>)) - math.cos(x * (N + <spanclass="hljs-number">0.5</span>))) / math.sin(x / <spanclass="hljs-number">2</span>))
print(<spanclass="hljs-string">'exact result: '</span> + str(r))
</span></span></foreignObject></svg></code></pre>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="11" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Full python code (cont.)</h1>
<p>Result:</p>
<pre><codeclass="language-bash"><svgdata-marp-fitting="svg" data-marp-fitting-code><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap>$ python script.py
full python result: 4.046217839811811 <spanclass="hljs-keyword">in</span> 1.1689767837524414 seconds
exact result: 4.027339492125908
</span></span></foreignObject></svg></code></pre>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="12" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Full C++ version</h1>
<pre><codeclass="language-cpp"><svgdata-marp-fitting="svg" data-marp-fitting-code><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap><spanclass="hljs-function"><spanclass="hljs-keyword">double</span><spanclass="hljs-title">cpp_inner_loop</span><spanclass="hljs-params">(<spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">unsigned</span><spanclass="hljs-keyword">int</span>& M)</span></span>{
<spanclass="hljs-built_in">std</span>::<spanclass="hljs-built_in">vector</span><<spanclass="hljs-keyword">double</span>> v(M);
<spanclass="hljs-built_in">std</span>::iota(v.begin(), v.end(), <spanclass="hljs-number">0</span>); <spanclass="hljs-comment">// fills the vector v with [0, 1, 2, ..., M-1]</span>
<spanclass="hljs-built_in">std</span>::transform(<spanclass="hljs-built_in">std</span>::execution::seq, v.begin(), v.end(), v.begin(), [](<spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">double</span>& k){ <spanclass="hljs-comment">// sequential loop</span>
<spanclass="hljs-keyword">return</span><spanclass="hljs-number">1.0</span> / ((<spanclass="hljs-number">4</span>*k + <spanclass="hljs-number">1</span>) * (<spanclass="hljs-number">4</span>*k + <spanclass="hljs-number">3</span>));
});
<spanclass="hljs-keyword">return</span><spanclass="hljs-built_in">std</span>::reduce(<spanclass="hljs-built_in">std</span>::execution::seq, v.begin(), v.end(), <spanclass="hljs-number">0.0</span>, <spanclass="hljs-built_in">std</span>::plus<>());
}
<spanclass="hljs-function"><spanclass="hljs-keyword">double</span><spanclass="hljs-title">cpp_outer_loop</span><spanclass="hljs-params">(<spanclass="hljs-keyword">const</span><spanclass="hljs-built_in">std</span>::function<<spanclass="hljs-keyword">double</span> (<spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">unsigned</span><spanclass="hljs-keyword">int</span>&)>& xf, <spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">unsigned</span><spanclass="hljs-keyword">int</span>& N, <spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">unsigned</span><spanclass="hljs-keyword">int</span>& M)</span></span>{
<spanclass="hljs-built_in">std</span>::<spanclass="hljs-built_in">vector</span><<spanclass="hljs-keyword">double</span>> v(N);
<spanclass="hljs-built_in">std</span>::iota(v.begin(), v.end(), <spanclass="hljs-number">0</span>); <spanclass="hljs-comment">// fills the vector v with [0, 1, 2, ..., N-1]</span>
<spanclass="hljs-built_in">std</span>::transform(<spanclass="hljs-built_in">std</span>::execution::par, v.begin(), v.end(), v.begin(), [&xf, &M](<spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">auto</span>& n){ <spanclass="hljs-comment">// parallel loop</span>
<spanclass="hljs-keyword">return</span><spanclass="hljs-built_in">std</span>::<spanclass="hljs-built_in">cos</span>((n+<spanclass="hljs-number">1</span>) * xf(M)) + <spanclass="hljs-built_in">std</span>::<spanclass="hljs-built_in">sin</span>((n+<spanclass="hljs-number">1</span>) * xf(M));
});
<spanclass="hljs-keyword">return</span><spanclass="hljs-built_in">std</span>::reduce(<spanclass="hljs-built_in">std</span>::execution::par, v.begin(), v.end(), <spanclass="hljs-number">0.0</span>, <spanclass="hljs-built_in">std</span>::plus<>());
}
<spanclass="hljs-function"><spanclass="hljs-keyword">double</span><spanclass="hljs-title">full_cpp</span><spanclass="hljs-params">(<spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">unsigned</span><spanclass="hljs-keyword">int</span>& N, <spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">unsigned</span><spanclass="hljs-keyword">int</span>& M)</span></span>{
<spanclass="hljs-keyword">return</span> cpp_outer_loop(cpp_inner_loop, N, M);
}
</span></span></foreignObject></svg></code></pre>
<spanstyle="font-size:75%">
Note how std::transform() is called with std::execution::seq or std::execution::par.
</span>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="13" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Exporting the C++ code to python</h1>
<p>In <em><ahref="http://package.cc">package.cc</a></em>:</p>
<pre><codeclass="language-cpp"><svgdata-marp-fitting="svg" data-marp-fitting-code><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap><spanclass="hljs-meta">#<spanclass="hljs-meta-keyword">include</span><spanclass="hljs-meta-string">"code.hh"</span></span>
<spanclass="hljs-meta">#<spanclass="hljs-meta-keyword">include</span><spanclass="hljs-meta-string"><pybind11/pybind11.h></span></span>
<spanclass="hljs-keyword">namespace</span> py = pybind11;
PYBIND11_MODULE(__simple_cpp_python, m) {
m.doc() = <spanclass="hljs-string">"pybind11 example plugin"</span>; <spanclass="hljs-comment">// optional module docstring</span>
m.def(<spanclass="hljs-string">"full_cpp"</span>, &full_cpp, <spanclass="hljs-string">"A function which computes the sum of
Lagrangian's trigonometric identities - full c++"</span>);
}
</span></span></foreignObject></svg></code></pre>
<spanstyle="font-size:75%">
It will compile a python extension library called __simple_cpp_python.cpython-37m-darwin.so (on MacOS) containing the full_cpp() python function.
</span>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="14" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Calling the C++ code from python</h1>
<p>In <em><ahref="http://script.py">script.py</a></em>:</p>
<pre><codeclass="language-python"><svgdata-marp-fitting="svg" data-marp-fitting-code><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap><spanclass="hljs-keyword">import</span> build.__simple_cpp_python <spanclass="hljs-keyword">as</span> wow
<spanclass="hljs-keyword">if</span> __name__ == <spanclass="hljs-string">"__main__"</span>:
start = time.time()
r = wow.full_cpp(<spanclass="hljs-number">1000</span>, <spanclass="hljs-number">10000</span>)
end = time.time()
print(<spanclass="hljs-string">'full c++ result: '</span> + str(r) + <spanclass="hljs-string">' in '</span> + str(end-start) + <spanclass="hljs-string">' seconds'</span>)
</span></span></foreignObject></svg></code></pre>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="15" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Running the script and comparing full Python vs full C++</h1>
<p>Result:</p>
<pre><codeclass="language-bash"><svgdata-marp-fitting="svg" data-marp-fitting-code><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap>$ python script.py
full python result: 4.046217839811811 <spanclass="hljs-keyword">in</span> 1.1679670810699463 seconds
full c++ result: 4.046217839811809 <spanclass="hljs-keyword">in</span> 0.008682012557983398 seconds
exact result: 4.027339492125908
</span></span></foreignObject></svg></code></pre>
<p>On this example <strong>pure C++ is 134.53x faster than pure Python</strong>; both using a sequential inner loop and a parallel outer loop.</p>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="16" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Mixing C++ (outer loop) and Python (inner loop)</h1>
<p>In <em>code.hh</em> (<code>xf</code> will be a Python lambda function when calling <code>cpp_outer_loop_gil</code> from Python):</p>
<pre><code><svgdata-marp-fitting="svg" data-marp-fitting-code><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap><spanclass="hljs-function"><spanclass="hljs-keyword">double</span><spanclass="hljs-title">cpp_outer_loop_gil</span><spanclass="hljs-params">(<spanclass="hljs-keyword">const</span><spanclass="hljs-built_in">std</span>::function<<spanclass="hljs-keyword">double</span> (<spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">unsigned</span><spanclass="hljs-keyword">int</span>&)>& xf,
<spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">unsigned</span><spanclass="hljs-keyword">int</span>& N, <spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">unsigned</span><spanclass="hljs-keyword">int</span>& M)</span></span>{
py::gil_scoped_release release; <spanclass="hljs-comment">// Release Python's GIL to enable OS multithreading</span>
<spanclass="hljs-built_in">std</span>::<spanclass="hljs-built_in">vector</span><<spanclass="hljs-keyword">double</span>> v(N);
<spanclass="hljs-built_in">std</span>::iota(v.begin(), v.end(), <spanclass="hljs-number">0</span>);
<spanclass="hljs-built_in">std</span>::transform(<spanclass="hljs-built_in">std</span>::execution::par, v.begin(), v.end(), v.begin(), [&xf, &M](<spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">auto</span>& n){
py::gil_scoped_acquire acquire; <spanclass="hljs-comment">// Acquire the GIL to prevent Python race conditions</span>
<spanclass="hljs-keyword">return</span><spanclass="hljs-built_in">std</span>::<spanclass="hljs-built_in">cos</span>((n+<spanclass="hljs-number">1</span>) * xf(M)) + <spanclass="hljs-built_in">std</span>::<spanclass="hljs-built_in">sin</span>((n+<spanclass="hljs-number">1</span>) * xf(M)); <spanclass="hljs-comment">// xf() calls a Python lambda</span>
});
<spanclass="hljs-keyword">return</span><spanclass="hljs-built_in">std</span>::reduce(<spanclass="hljs-built_in">std</span>::execution::par, v.begin(), v.end(), <spanclass="hljs-number">0.0</span>, <spanclass="hljs-built_in">std</span>::plus<>());
}
</span></span></foreignObject></svg></code></pre>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="17" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Calling the C++ function from Python</h1>
<p>In <em><ahref="http://script.py">script.py</a></em>:</p>
<pre><codeclass="language-python"><svgdata-marp-fitting="svg" data-marp-fitting-code><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap><spanclass="hljs-keyword">import</span> build.__simple_cpp_python <spanclass="hljs-keyword">as</span> wow
<spanclass="hljs-keyword">if</span> __name__ == <spanclass="hljs-string">"__main__"</span>:
start = time.time()
r = wow.cpp_outer_loop_gil(<spanclass="hljs-keyword">lambda</span> M: python_inner_loop(M), <spanclass="hljs-number">1000</span>, <spanclass="hljs-number">10000</span>)
end = time.time()
print(<spanclass="hljs-string">'mixed result: '</span> + str(r) + <spanclass="hljs-string">' in '</span> + str(end-start) + <spanclass="hljs-string">' seconds'</span>)
</span></span></foreignObject></svg></code></pre>
<p>In short: Python calls a C++ function that calls a long-running Python function in a parallel loop.</p>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="18" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Running the script</h1>
<p>Result:</p>
<pre><codeclass="language-bash"><svgdata-marp-fitting="svg" data-marp-fitting-code><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap>$ python script.py
full python result: 4.046217839811811 <spanclass="hljs-keyword">in</span> 1.239090919494629 seconds
full c++ result: 4.046217839811811 <spanclass="hljs-keyword">in</span> 0.009485006332397461 seconds
mixed result: 4.046217839811811 <spanclass="hljs-keyword">in</span> 4.09197211265564 seconds
exact result: 4.027339492125908
</span></span></foreignObject></svg></code></pre>
<p>The mixed C++/Python code is <strong>3.3x slower</strong> than the pure Python code! (and 431x slower than the pure C++ code)<br/>
That's because of <strong>Python's GIL</strong> that acts like a mutex that forces to sequentialize the C++ parallel loop...</p>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="19" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Analysis</h1>
<ul>
<li>Python's GIL forces the outer C++ parallel loop to call each inner element in sequence</li>
<li>Not happening in pure Python because full python code parallelism is implemented using <em>multiprocessing</em> instead of <em>multithreading</em> (thus circumventing the GIL)</li>
<li>Solution: <strong>call each inner Python function from a different Python process!</strong> (but each Python inner loop running in sequence as previously)</li>
</ul>
<p>Idea: <strong>1 C++ thread <=> 1 Python process</strong></p>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="20" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>C++ code calling concurrent python processes in parallel</h1>
<pre><codeclass="language-cpp"><svgdata-marp-fitting="svg" data-marp-fitting-code><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap><spanclass="hljs-function"><spanclass="hljs-keyword">double</span><spanclass="hljs-title">cpp_outer_loop_process</span><spanclass="hljs-params">(<spanclass="hljs-keyword">const</span><spanclass="hljs-built_in">std</span>::function<<spanclass="hljs-keyword">void</span> (<spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">unsigned</span><spanclass="hljs-keyword">int</span>&, <spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">unsigned</span><spanclass="hljs-keyword">int</span>&)>& xg,
<spanclass="hljs-keyword">const</span><spanclass="hljs-built_in">std</span>::function<<spanclass="hljs-keyword">double</span> (<spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">unsigned</span><spanclass="hljs-keyword">int</span>&)>& xf,
<spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">unsigned</span><spanclass="hljs-keyword">int</span>& N, <spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">unsigned</span><spanclass="hljs-keyword">int</span>& M)</span></span>{
py::gil_scoped_release release;
<spanclass="hljs-built_in">std</span>::<spanclass="hljs-built_in">vector</span><<spanclass="hljs-keyword">double</span>> v(N);
<spanclass="hljs-built_in">std</span>::iota(v.begin(), v.end(), <spanclass="hljs-number">0</span>);
<spanclass="hljs-built_in">std</span>::for_each(<spanclass="hljs-built_in">std</span>::execution::par, v.begin(), v.end(), [&xg, &M](<spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">auto</span>& n){
py::gil_scoped_acquire acquire;
xg(n, M); <spanclass="hljs-comment">// Call to Python function that launches a detached process computing the inner loop</span>
});
<spanclass="hljs-built_in">std</span>::transform(<spanclass="hljs-built_in">std</span>::execution::par, v.begin(), v.end(), v.begin(), [&xf](<spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">auto</span>& n){
py::gil_scoped_acquire acquire;
<spanclass="hljs-keyword">return</span><spanclass="hljs-built_in">std</span>::<spanclass="hljs-built_in">cos</span>((n+<spanclass="hljs-number">1</span>) * xf(n)) + <spanclass="hljs-built_in">std</span>::<spanclass="hljs-built_in">sin</span>((n+<spanclass="hljs-number">1</span>) * xf(n)); <spanclass="hljs-comment">// xf(n) gets the result of xg(n, M)</span>
});
<spanclass="hljs-keyword">return</span><spanclass="hljs-built_in">std</span>::reduce(<spanclass="hljs-built_in">std</span>::execution::par, v.begin(), v.end(), <spanclass="hljs-number">0.0</span>, <spanclass="hljs-built_in">std</span>::plus<>());
}
</span></span></foreignObject></svg></code></pre>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="21" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Calling the C++ function from Python</h1>
<pre><codeclass="language-python"><svgdata-marp-fitting="svg" data-marp-fitting-code><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap><spanclass="hljs-function"><spanclass="hljs-keyword">def</span><spanclass="hljs-title">mixed_op</span><spanclass="hljs-params">(pool, lr, n, M)</span>:</span><spanclass="hljs-comment"># this our xg() viewed from C++</span>
lr[n] = pool.apply_async(<spanclass="hljs-keyword">lambda</span> : python_inner_loop(M)) <spanclass="hljs-comment"># GIL freed after launch</span>
<spanclass="hljs-function"><spanclass="hljs-keyword">def</span><spanclass="hljs-title">mixed_get</span><spanclass="hljs-params">(lr, n)</span>:</span><spanclass="hljs-comment"># this our xf() viewed from C++</span>
<spanclass="hljs-keyword">return</span> lr[n].get() <spanclass="hljs-comment"># blocks the process until the result is ready</span>
<spanclass="hljs-keyword">if</span> __name__ == <spanclass="hljs-string">"__main__"</span>:
start = time.time()
pool = multiprocessing.Pool()
lr = [<spanclass="hljs-literal">None</span>]*N <spanclass="hljs-comment"># lr[] saves the result of each detached process</span>
r = wow.cpp_outer_loop_process(<spanclass="hljs-keyword">lambda</span> n, M : mixed_op(pool, lr, n, M),
<spanclass="hljs-keyword">lambda</span> n : mixed_get(lr, n), <spanclass="hljs-number">1000</span>, <spanclass="hljs-number">10000</span>)
end = time.time()
print(<spanclass="hljs-string">'mixed result (process): '</span> + str(r) + <spanclass="hljs-string">' in '</span> + str(end-start) + <spanclass="hljs-string">' seconds'</span>)
</span></span></foreignObject></svg></code></pre>
<spanstyle="font-size:70%">
The GIL blocks the (only) main Python thread just to launch processes and to get their results.
</span>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="22" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Running the script</h1>
<pre><codeclass="language-bash"><svgdata-marp-fitting="svg" data-marp-fitting-code><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap>$ python script.py
full python result: 4.046217839811811 <spanclass="hljs-keyword">in</span> 1.1558029651641846 seconds
full c++ result: 4.046217839811814 <spanclass="hljs-keyword">in</span> 0.00848531723022461 seconds
mixed result (GIL): 4.046217839811811 <spanclass="hljs-keyword">in</span> 4.034562110900879 seconds
mixed result (process): 4.046217839811811 <spanclass="hljs-keyword">in</span> 0.6684191226959229 seconds
exact result: 4.027339492125908
</span></span></foreignObject></svg></code></pre>
<p>The mixed C++/Python process-based implementation is now <strong>1.73x faster than the full parallel python</strong> one (but still 78.77x slower than the full parallel C++ one...). Improvements expected to increase as inner Python code takes more CPU time.</p>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="23" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Performance analysis (with 4 cores)</h1>
<p>Speedup vs full (parallel) Python: the higher the better (log scale)</p>
<table>
<thead>
<tr>
<th><imgsrc="comparison_M.png" alt="width:550px" style="width:550px;" /></th>
<th><imgsrc="comparison_N.png" alt="width:550px" style="width:550px;" /></th>
</tr>
</thead>
<tbody></tbody>
</table>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="24" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Using native python objects</h1>
<pre><codeclass="language-cpp"><svgdata-marp-fitting="svg" data-marp-fitting-code><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap><spanclass="hljs-function"><spanclass="hljs-keyword">double</span><spanclass="hljs-title">cpp_outer_loop_process</span><spanclass="hljs-params">(<spanclass="hljs-keyword">const</span><spanclass="hljs-built_in">std</span>::function<<spanclass="hljs-keyword">void</span> (<spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">unsigned</span><spanclass="hljs-keyword">int</span>&, <spanclass="hljs-keyword">const</span> py::dict&)>& xg,
<spanclass="hljs-keyword">const</span><spanclass="hljs-built_in">std</span>::function<<spanclass="hljs-keyword">double</span> (<spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">unsigned</span><spanclass="hljs-keyword">int</span>&)>& xf,
<spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">unsigned</span><spanclass="hljs-keyword">int</span>& N, <spanclass="hljs-keyword">const</span> py::dict& d)</span></span>{
py::gil_scoped_release release;
<spanclass="hljs-built_in">std</span>::<spanclass="hljs-built_in">vector</span><<spanclass="hljs-keyword">double</span>> v(N);
<spanclass="hljs-built_in">std</span>::iota(v.begin(), v.end(), <spanclass="hljs-number">0</span>);
<spanclass="hljs-built_in">std</span>::for_each(<spanclass="hljs-built_in">std</span>::execution::par, v.begin(), v.end(), [&xg, &M](<spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">auto</span>& n){
py::gil_scoped_acquire acquire;
xg(n, d); <spanclass="hljs-comment">// Work with a native python dictionary d (no copy)</span>
});
<spanclass="hljs-built_in">std</span>::transform(<spanclass="hljs-built_in">std</span>::execution::par, v.begin(), v.end(), v.begin(), [&xf](<spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">auto</span>& n){
py::gil_scoped_acquire acquire;
<spanclass="hljs-keyword">return</span><spanclass="hljs-built_in">std</span>::<spanclass="hljs-built_in">cos</span>((n+<spanclass="hljs-number">1</span>) * xf(n)) + <spanclass="hljs-built_in">std</span>::<spanclass="hljs-built_in">sin</span>((n+<spanclass="hljs-number">1</span>) * xf(n)); <spanclass="hljs-comment">// xf(n) gets the result of xg(n, M)</span>
});
<spanclass="hljs-keyword">return</span><spanclass="hljs-built_in">std</span>::reduce(<spanclass="hljs-built_in">std</span>::execution::par, v.begin(), v.end(), <spanclass="hljs-number">0.0</span>, <spanclass="hljs-built_in">std</span>::plus<>());
}
</span></span></foreignObject></svg></code></pre>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="25" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>C++ templates: one generic outer loop for either Python or C++ inner loop</h1>
<pre><codeclass="language-cpp"><svgdata-marp-fitting="svg" data-marp-fitting-code><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap><spanclass="hljs-keyword">template</span> <<spanclass="hljs-keyword">typename</span> ExecutionModel, <spanclass="hljs-keyword">typename</span> InnerType> <spanclass="hljs-comment">// declare generic types</span>
<spanclass="hljs-function"><spanclass="hljs-keyword">double</span><spanclass="hljs-title">cpp_outer_loop_process</span><spanclass="hljs-params">(<spanclass="hljs-keyword">const</span><spanclass="hljs-built_in">std</span>::function<<spanclass="hljs-keyword">void</span> (<spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">unsigned</span><spanclass="hljs-keyword">int</span>&, <spanclass="hljs-keyword">const</span> InnerType&)>& xg,
<spanclass="hljs-keyword">const</span><spanclass="hljs-built_in">std</span>::function<<spanclass="hljs-keyword">double</span> (<spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">unsigned</span><spanclass="hljs-keyword">int</span>&)>& xf,
<spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">unsigned</span><spanclass="hljs-keyword">int</span>& N, <spanclass="hljs-keyword">const</span> InnerType& d, <spanclass="hljs-keyword">const</span> ExecutionModel& em)</span></span>{
<spanclass="hljs-keyword">typename</span> ExecutionModel::Release release; <spanclass="hljs-comment">// py::gil_scoped_release if xg() is a Python lambda otherwise empty class</span>
<spanclass="hljs-built_in">std</span>::<spanclass="hljs-built_in">vector</span><<spanclass="hljs-keyword">double</span>> v(N);
<spanclass="hljs-built_in">std</span>::iota(v.begin(), v.end(), <spanclass="hljs-number">0</span>);
<spanclass="hljs-built_in">std</span>::for_each(<spanclass="hljs-built_in">std</span>::execution::par, v.begin(), v.end(), [&xg, &M](<spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">auto</span>& n){
<spanclass="hljs-keyword">typename</span> ExecutionModel::Acquire acquire; <spanclass="hljs-comment">// py::gil_scoped_acquire if xg() is a Python lambda otherwise empty class</span>
xg(n, d); <spanclass="hljs-comment">// d is a generic type (C++ type or e.g. python native type via pybind11)</span>
});
<spanclass="hljs-built_in">std</span>::transform(<spanclass="hljs-built_in">std</span>::execution::par, v.begin(), v.end(), v.begin(), [&xf](<spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">auto</span>& n){
<spanclass="hljs-keyword">typename</span> ExecutionModel::Acquire acquire; <spanclass="hljs-comment">// py::gil_scoped_acquire if xg() is a Python lambda otherwise empty class</span>
<spanclass="hljs-keyword">return</span><spanclass="hljs-built_in">std</span>::<spanclass="hljs-built_in">cos</span>((n+<spanclass="hljs-number">1</span>) * xf(n)) + <spanclass="hljs-built_in">std</span>::<spanclass="hljs-built_in">sin</span>((n+<spanclass="hljs-number">1</span>) * xf(n)); <spanclass="hljs-comment">// xf(n) gets the result of xg(n, M)</span>
});
<spanclass="hljs-keyword">return</span><spanclass="hljs-built_in">std</span>::reduce(<spanclass="hljs-built_in">std</span>::execution::par, v.begin(), v.end(), <spanclass="hljs-number">0.0</span>, <spanclass="hljs-built_in">std</span>::plus<>());
}
</span></span></foreignObject></svg></code></pre>
<spanstyle="font-size:60%">
ExecutionModel is a generic type that must define Release and Acquire classes (empty in case of pure C++).
</span>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="26" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Using the generic C++ outer loop with C++ inner function (from C++)</h1>
<pre><codeclass="language-cpp"><svgdata-marp-fitting="svg" data-marp-fitting-code><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap><spanclass="hljs-class"><spanclass="hljs-keyword">struct</span><spanclass="hljs-title">CPPExecutionModel</span> {</span>
<spanclass="hljs-class"><spanclass="hljs-keyword">struct</span><spanclass="hljs-title">Acquire</span> {</span>};
<spanclass="hljs-class"><spanclass="hljs-keyword">struct</span><spanclass="hljs-title">Release</span> {</span>};
};
<spanclass="hljs-function"><spanclass="hljs-keyword">void</span><spanclass="hljs-title">cpp_xg</span><spanclass="hljs-params">(<spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">unsigned</span><spanclass="hljs-keyword">int</span>& n, <spanclass="hljs-keyword">const</span><spanclass="hljs-built_in">std</span>::<spanclass="hljs-built_in">map</span><<spanclass="hljs-built_in">std</span>::<spanclass="hljs-built_in">string</span>, <spanclass="hljs-keyword">double</span>>& m)</span></span>{...};
<spanclass="hljs-function"><spanclass="hljs-keyword">double</span><spanclass="hljs-title">cpp_xf</span><spanclass="hljs-params">(<spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">unsigned</span><spanclass="hljs-keyword">int</span>& n)</span></span>{...};
<spanclass="hljs-keyword">unsigned</span><spanclass="hljs-keyword">int</span> N = ...;
<spanclass="hljs-built_in">std</span>::<spanclass="hljs-built_in">map</span><<spanclass="hljs-built_in">std</span>::<spanclass="hljs-built_in">string</span>, <spanclass="hljs-keyword">double</span>> m = ...;
<spanclass="hljs-keyword">double</span> r = cpp_outer_loop_process(cpp_xg, cpp_xf, N, m, CPPExecutionModel());
</span></span></foreignObject></svg></code></pre>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="27" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Using the generic C++ outer loop with Python inner function (from Python)</h1>
<pre><codeclass="language-cpp"><svgdata-marp-fitting="svg" data-marp-fitting-code><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap><spanclass="hljs-class"><spanclass="hljs-keyword">struct</span><spanclass="hljs-title">PYExecutionModel</span> {</span>
<spanclass="hljs-keyword">typedef</span> py::gil_scoped_acquire Acquire;
<spanclass="hljs-keyword">typedef</span> py::gil_scoped_release Release;
};
<spanclass="hljs-function"><spanclass="hljs-keyword">double</span><spanclass="hljs-title">export_cpp_outer_loop_process</span><spanclass="hljs-params">(<spanclass="hljs-keyword">const</span><spanclass="hljs-built_in">std</span>::function<<spanclass="hljs-keyword">void</span> (<spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">unsigned</span><spanclass="hljs-keyword">int</span>&, <spanclass="hljs-keyword">const</span> py::dict&)>& xg,
<spanclass="hljs-keyword">const</span><spanclass="hljs-built_in">std</span>::function<<spanclass="hljs-keyword">double</span> (<spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">unsigned</span><spanclass="hljs-keyword">int</span>&)>& xf,
<spanclass="hljs-keyword">const</span><spanclass="hljs-keyword">unsigned</span><spanclass="hljs-keyword">int</span>& N, <spanclass="hljs-keyword">const</span> py::dict& d)</span></span>{
<spanclass="hljs-keyword">return</span> cpp_outer_loop_process(xg, xf, N, d, PYExecutionModel()); <spanclass="hljs-comment">// instantiate generic C++ outer loop function for Python</span>
}
PYBIND11_MODULE(__simple_cpp_python, m) {
m.def(<spanclass="hljs-string">"cpp_outer_loop_process"</span>, &export_cpp_outer_loop_process); <spanclass="hljs-comment">// Export the specialized function to Python</span>
}
</span></span></foreignObject></svg></code></pre>
<pre><codeclass="language-python"><svgdata-marp-fitting="svg" data-marp-fitting-code><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap>r = export_cpp_outer_loop_process(<spanclass="hljs-keyword">lambda</span> n, d: ..., <spanclass="hljs-keyword">lambda</span> n: ..., N, my_dict) <spanclass="hljs-comment"># use with native python dict</span>
</span></span></foreignObject></svg></code></pre>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="28" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Unit-testing the C++ code</h1>
<p>Append the following lines to your <em>CMakeLists.txt</em>:</p>
<pre><codeclass="language-cmake"><svgdata-marp-fitting="svg" data-marp-fitting-code><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap><spanclass="hljs-keyword">INCLUDE</span>(CTest)
<spanclass="hljs-keyword">INCLUDE</span>(Catch)
<spanclass="hljs-keyword">ADD_EXECUTABLE</span>(<spanclass="hljs-keyword">test</span>-cpp-inner-loop tests.cc)
<spanclass="hljs-keyword">TARGET_INCLUDE_DIRECTORIES</span>(<spanclass="hljs-keyword">test</span>-cpp-inner-loop PRIVATE <spanclass="hljs-variable">${PYTHON_INCLUDE_DIRS}</span>)
<spanclass="hljs-keyword">TARGET_LINK_LIBRARIES</span>(<spanclass="hljs-keyword">test</span>-cpp-inner-loop PRIVATE
Catch2::Catch2 <spanclass="hljs-variable">${TBB_IMPORTED_TARGETS}</span><spanclass="hljs-variable">${PYTHON_LIBRARIES}</span>)
CATCH_DISCOVER_TESTS(<spanclass="hljs-keyword">test</span>-cpp-inner-loop)
</span></span></foreignObject></svg></code></pre>
<p>It will add a unit test program called test-cpp-inner-loop.<br/>
Catch2 takes care of automatically finding your unit tests in <em><ahref="http://tests.cc">tests.cc</a></em></p>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="29" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Unit-testing the C++ code (cont.)</h1>
<p>Contents of <em><ahref="http://tests.cc">tests.cc</a></em>:</p>
<pre><codeclass="language-cpp"><svgdata-marp-fitting="svg" data-marp-fitting-code><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap><spanclass="hljs-meta">#<spanclass="hljs-meta-keyword">define</span> CATCH_CONFIG_MAIN</span>
<spanclass="hljs-meta">#<spanclass="hljs-meta-keyword">include</span><spanclass="hljs-meta-string"><catch2/catch.hpp></span></span>
<spanclass="hljs-meta">#<spanclass="hljs-meta-keyword">include</span><spanclass="hljs-meta-string"><cmath></span></span>
<spanclass="hljs-meta">#<spanclass="hljs-meta-keyword">include</span><spanclass="hljs-meta-string">"code.hh"</span></span>
TEST_CASE(<spanclass="hljs-string">"Test the approximate computation of PI/8.0"</span>, <spanclass="hljs-string">"[PI computation]"</span>) { <spanclass="hljs-comment">// declare a new unit test</span>
<spanclass="hljs-keyword">double</span> r1 = cpp_inner_loop(<spanclass="hljs-number">10</span>);
<spanclass="hljs-keyword">double</span> r2 = cpp_inner_loop(<spanclass="hljs-number">100</span>);
<spanclass="hljs-keyword">double</span> r3 = cpp_inner_loop(<spanclass="hljs-number">1000</span>);
<spanclass="hljs-keyword">double</span> v = <spanclass="hljs-built_in">std</span>::<spanclass="hljs-built_in">acos</span>(<spanclass="hljs-number">-1.0</span>) / <spanclass="hljs-number">8.0</span>;
REQUIRE( <spanclass="hljs-built_in">std</span>::<spanclass="hljs-built_in">fabs</span>(v - r1) < <spanclass="hljs-number">1e-1</span> );
REQUIRE( <spanclass="hljs-built_in">std</span>::<spanclass="hljs-built_in">fabs</span>(v - r2) < <spanclass="hljs-number">1e-3</span> );
REQUIRE( <spanclass="hljs-built_in">std</span>::<spanclass="hljs-built_in">fabs</span>(v - r3) < <spanclass="hljs-number">1e-5</span> );
}
</span></span></foreignObject></svg></code></pre>
<spanstyle="font-size:65%">
You can write scenario tests, check values or exceptions, write conditonal tests, and much more...
</span>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="30" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Unit test result</h1>
<pre><codeclass="language-bash"><svgdata-marp-fitting="svg" data-marp-fitting-code><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap>$ build/<spanclass="hljs-built_in">test</span>-cpp-inner-loop
<spanclass="hljs-built_in">test</span>-cpp-inner-loop is a Catch v2.9.2 host application.
Run with -? <spanclass="hljs-keyword">for</span> options
-------------------------------------------------------------------------------
Test the approximate computation of PI/8.0
-------------------------------------------------------------------------------
/Users/teichteil_fl/Projects/simple-cpp-python/tests.cc:6
...............................................................................
/Users/teichteil_fl/Projects/simple-cpp-python/tests.cc:13: FAILED:
REQUIRE( std::fabs(v - r3) < 1e-5 )
with expansion:
0.0000625 < 0.00001
===============================================================================
<spanclass="hljs-built_in">test</span> cases: 1 | 1 failed
assertions: 3 | 2 passed | 1 failed
</span></span></foreignObject></svg></code></pre>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="31" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Finally: replace segfault message by useful stack trace message</h1>
<p>Link your target against Backward libraries in <em>CMakeLists.txt</em>, e.g.:</p>
<pre><codeclass="language-cmake"><svgdata-marp-fitting="svg" data-marp-fitting-code><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap><spanclass="hljs-keyword">TARGET_LINK_LIBRARIES</span>(<spanclass="hljs-keyword">test</span>-cpp-inner-loop PRIVATE Backward::Backward
Catch2::Catch2 <spanclass="hljs-variable">${TBB_IMPORTED_TARGETS}</span><spanclass="hljs-variable">${PYTHON_LIBRARIES}</span>)
</span></span></foreignObject></svg></code></pre>
<p>Add the following line at the top of your source file, e.g. in <em><ahref="http://tests.cc">tests.cc</a></em>:</p>
<pre><codeclass="language-cpp"><svgdata-marp-fitting="svg" data-marp-fitting-code><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap>include <spanclass="hljs-string">"backward.hpp"</span>
<spanclass="hljs-keyword">namespace</span> backward { backward::SignalHandling sh; }
</span></span></foreignObject></svg></code></pre>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="32" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Replace segfault message by useful stack trace message</h1>
<p>Add a silly test that should produce a segfault (<imgclass="emoji" draggable="false" alt="⚠️" src="https://twemoji.maxcdn.com/2/svg/26a0.svg" data-marp-twemoji=""/> bad old school C++ coding style from the 90s on top of a silly bug!), e.g.:</p>
<pre><codeclass="language-cpp"><svgdata-marp-fitting="svg" data-marp-fitting-code><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap>TEST_CASE(<spanclass="hljs-string">"Stupid test to demonstrate Backward"</span>, <spanclass="hljs-string">"[backward]"</span>) {
<spanclass="hljs-keyword">int</span>* v = <spanclass="hljs-literal">nullptr</span>; <spanclass="hljs-comment">// v is a null pointer</span>
REQUIRE( v[<spanclass="hljs-number">0</span>] == <spanclass="hljs-number">0</span> ); <spanclass="hljs-comment">// crash: Backward provides a stack trace </span>
}
</span></span></foreignObject></svg></code></pre>
<p><em>Note: we put this intentional crash test in a unit test for simplicity but Backward can trace any C++ code of your project</em></p>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="33" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Backward stack trace result</h1>
<p>Instead of an ungentle <em>sefgault</em> message we get a more informative trace (well, depend on the context) that helps debug your program:</p>
<pre><codeclass="language-bash"><svgdata-marp-fitting="svg" data-marp-fitting-code><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap>Stack trace (most recent call last):
<spanclass="hljs-comment">#2 Object "libsystem_platform.dylib", at 0x7fff5932fb3c, in _sigtramp + 28</span>
<spanclass="hljs-comment">#1 Object "test-cpp-inner-loop", at 0x10ae52478, in backward::SignalHandling::sig_handler(int, __siginfo*, void*) + 8</span>
<spanclass="hljs-comment">#0 Object "test-cpp-inner-loop", at 0x10ae5207d, in backward::SignalHandling::handleSignal(int, __siginfo*, void*) + 77</span>
</span></span></foreignObject></svg></code></pre>
<p>Without Backward we would simply get a message like:</p>
<pre><codeclass="language-bash"><svgdata-marp-fitting="svg" data-marp-fitting-code><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap>[1] 46461 segmentation fault build/<spanclass="hljs-built_in">test</span>-cpp-inner-loop
</span></span></foreignObject></svg></code></pre>
<p>Note that Backward is nearly non-intrusive in your code.</p>
</section>
</foreignObject></svg><svgdata-marpit-svg="" viewBox="0 0 1280 720"><foreignObjectwidth="1280" height="720"><sectionid="34" data-class="invert" data-theme="gaia" class="invert" style="--class:invert;--theme:gaia;">
<h1>Setting-up an installation script with cmake and setuptools</h1>
<p>Look at <em><ahref="http://setup.py">setup.py</a></em> (too large to be copied here) -> provides <em>CMakeExtension</em> and <em>CMakeBuild</em> classes<br/>
In <code>__init__.py</code>:</p>
<pre><codeclass="language-python"><svgdata-marp-fitting="svg" data-marp-fitting-code><foreignObject><spandata-marp-fitting-svg-content><spandata-marp-fitting-svg-content-wrap><spanclass="hljs-keyword">import</span> sys
<spanclass="hljs-keyword">import</span> os
sys.path.append(os.path.dirname(os.path.realpath(__file__)))
<spanclass="hljs-keyword">import</span> __simple_cpp_python <spanclass="hljs-keyword">as</span> simple_cpp_python
</span></span></foreignObject></svg></code></pre>
<p>That's it! Now you can <code>pip install -e .</code> and load <code>simple_cpp_python</code></p>
</section>
</foreignObject></svg></article><script>!function(){"use strict";constt=()=>"Apple Computer, Inc."===navigator.vendor?[e]:[];functione(t){Array.from(document.getElementsByTagName("svg"),e=>{if(e.hasAttribute("data-marpit-svg")){const{clientHeight:r,clientWidth:a}=e;e.style.transform||(e.style.transform="translateZ(0)");consti=t||e.currentScale||1,n=e.viewBox.baseVal.width/i,s=e.viewBox.baseVal.height/i,o=Math.min(r/s,a/n);Array.from(e.querySelectorAll(":scope > foreignObject"),t=>{conste=t.x.baseVal.value,i=t.y.baseVal.value;Array.from(t.querySelectorAll(":scope > section"),t=>{t.style.transformOrigin||(t.style.transformOrigin="0 0");constl=(a-o*n)/2-e,c=(r-o*s)/2-i;t.style.transform=`translate3d(${l}px,${c}px,0) scale(${o}) translate(${e}px,${i}px)`})})}})}constr="data-marp-fitting",a="data-marp-fitting-code",i="data-marp-fitting-math",n=(t,e,r)=>{if(t.getAttribute(e)!==r)returnt.setAttribute(e,r),!0};!function(e=!0){consts=()=>{for(consteoft())e();Array.from(document.querySelectorAll(`svg[${r}="svg"]`),t=>{conste=t.firstChild,r=e.firstChild,{scrollWidth:s,scrollHeight:o}=r;letl,c=1;if(t.hasAttribute(a)&&(l=t.parentElement.parentElement),t.hasAttribute(i)&&(l=t.parentElement),l){constt=getComputedStyle(l),e=Math.ceil(l.clientWidth-parseFloat(t.paddingLeft)-parseFloat(t.paddingRight));e&&(c=e)}constm=Math.max(s,c),p=Math.max(o,1),g=`0 0 ${m}${p}`;n(e,"width",`${m}`),n(e,"height",`${p}`),n(t,"preserveAspectRatio",getComputedStyle(t).getPropertyValue("--preserve-aspect-ratio")||"xMinYMin meet"),n(t,"viewBox",g)&&t.classList.toggle("__reflow__")}),e&&window.requestAnimationFrame(s)};s()}()}();
</script><script>!function(){"use strict";vare={from:function(e,t){varn=1===(e.parent||e).nodeType?e.parent||e:document.querySelector(e.parent||e),r=[].filter.call("string"==typeofe.slides?n.querySelectorAll(e.slides):e.slides||n.children,function(e){return"SCRIPT"!==e.nodeName}),i=r[0],l={},o=function(e,t){r[e]&&(c("deactivate",u(i,t)),i=r[e],c("activate",u(i,t)))},s=function(e,t){varn=r.indexOf(i)+e;c(e>0?"next":"prev",u(i,t))&&o(n,t)},a=function(e,t){l[e]=(l[e]||[]).filter(function(e){returne!==t})},c=function(e,t){return(l[e]||[]).reduce(function(e,n){returne&&!1!==n(t)},!0)},u=function(e,t){return(t=t||{}).index=r.indexOf(e),t.slide=e,t},d={on:function(e,t){return(l[e]||(l[e]=[])).push(t),a.bind(null,e,t)},off:a,fire:c,slide:function(e,t){if(!arguments.length)returnr.indexOf(i);c("slide",u(r[e],t))&&o(e,t)},next:s.bind(null,1),prev:s.bind(null,-1),parent:n,slides:r};return(t||[]).forEach(function(e){e(d)}),o(0),d}},t=function(){returnfunction(e){e.slides.forEach(function(e){e.addEventListener("keydown",function(e){(/INPUT|TEXTAREA|SELECT/.test(e.target.nodeName)||"true"===e.target.contentEditable)&&e.stopPropagation()})})}};functionn(e){e.parent.classList.add("bespoke-marp-parent"),e.slides.map(e=>e.classList.add("bespoke-marp-slide")),e.on("activate",t=>{e.slides.map(e=>e.classList.remove("bespoke-marp-active")),t.slide.classList.add("bespoke-marp-active")})}functionr(e=2e3){returnt=>{letn;functionr(){n&&clearTimeout(n),n=setTimeout(()=>{t.parent.classList.add("bespoke-marp-inactive")},e),t.parent.classList.remove("bespoke-marp-inactive")}document.addEventListener("mousedown",r),document.addEventListener("mousemove",r),document.addEventListener("touchend",r),setTimeout(r,0)}}functioni(e){lett=0,n=0;constr=e.slides.map(e=>[null,...e.querySelectorAll("[data-marpit-fragment]")]),i=e=>void0!==r[t][n+e],l=(i,l)=>{t=i,n=l,r.forEach((e,t)=>{e.forEach((e,n)=>{if(null==e)return;constr=t<i||t===i&&n<=l;e.setAttribute("data-bespoke-marp-fragment",r?"active":"inactive"),t===i&&n===l?e.setAttribute("data-bespoke-marp-current-fragment","current"):e.removeAttribute("data-bespoke-marp-current-fragment")})}),e.fire("fragment",{slide:e.slides[i],index:i,fragments:r[i],fragmentIndex:l})};e.on("next",()=>{if(i(1))returnl(t,n+1),!1;conste=t+1;r[e]&&l(e,0)}),e.on("prev",()=>{if(i(-1))returnl(t,n-1),!1;const e=t-1;r[e]&&l(e,r[e].length-1)}),e.on("slide",({index:e,fragment:t})=>{let n=0;if(void 0!==t){const i=r[e];if(i){const{length:e}=i;n=-1===t?e-1:Math.min(Math.max(t,0),e-1)}}l(e,n)}),l(0,0)}var l,o=function(e,t){return e(t={exports:{}},t.exports),t.exports}(function(e){!function(){var t="undefined"!=typeof window&&void 0!==window.document?window.document:{},n=e.exports,r="undefined"!=typeof Element&&"ALLOW_KEYBOARD_INPUT"in Element,i=function(){for(var e,n=[["requestFullscreen","exitFullscreen","fullscreenElement","fullscreenEnabled","fullscreenchange","fullscreenerror"],["webkitRequestFullscreen","webkitExitFullscreen","webkitFullscreenElement","webkitFullscreenEnabled","webkitfullscreenchange","webkitfullscreenerror"],["webkitRequestFullScreen","webkitCancelFullScreen","webkitCurrentFullScreenElement","webkitCancelFullScreen","webkitfullscreenchange","webkitfullscreenerror"],["mozRequestFullScreen","mozCancelFullScreen","mozFullScreenElement","mozFullScreenEnabled","mozfullscreenchange","mozfullscreenerror"],["msRequestFullscreen","msExitFullscreen","msFullscreenElement","msFullscreenEnabled","MSFullscreenChange","MSFullscreenError"]],r=0,i=n.length,l={};r<i;r++)if((e=n[r])&&e[1]in t){for(r=0;r<e.length;r++)l[n[0][r]]=e[r];return l}return!1}(),l={change:i.fullscreenchange,error:i.fullscreenerror},o={request:function(e){return new Promise(function(n){var l=i.requestFullscreen,o=function(){this.off("change",o),n()}.bind(this);e=e||t.documentElement,/ Version\/5\.1(?:\.\d+)? Safari\//.test(navigator.userAgent)?e[l]():e[l](r?Element.ALLOW_KEYBOARD_INPUT:{}),this.on("change",o)}.bind(this))},exit:function(){return new Promise(function(e){if(this.isFullscreen){var n=function(){this.off("change",n),e()}.bind(this);t[i.exitFullscreen](),this.on("change",n)}else e()}.bind(this))},toggle:function(e){return this.isFullscreen?this.exit():this.request(e)},onchange:function(e){this.on("change",e)},onerror:function(e){this.on("error",e)},on:function(e,n){var r=l[e];r&&t.addEventListener(r,n,!1)},off:function(e,n){var r=l[e];r&&t.removeEventListener(r,n,!1)},raw:i};i?(Object.defineProperties(o,{isFullscreen:{get:function(){return Boolean(t[i.fullscreenElement])}},element:{enumerable:!0,get:function(){return t[i.fullscreenElement]}},enabled:{enumerable:!0,get:function(){return Boolean(t[i.fullscreenEnabled])}}}),n?(e.exports=o,e.exports.default=o):window.screenfull=o):n?e.exports=!1:window.screenfull=!1}()});function s(e){e.fullscreen=()=>o&&o.toggle(document.body),document.addEventListener("keydown",t=>{70!==t.which&&122!==t.which||t.altKey||t.ctrlKey||t.metaKey||!o||!o.enabled||(e.fullscreen(),t.preventDefault())})}function a(e={}){const t=Object.assign({history:!0},e);return e=>{function n(){const t=parseInt(window.location.hash.slice(1),10);Number.isNaN(t)||function(t){const n=Math.max(0,Math.min(t,e.slides.length-1));n!==e.slide()&&e.slide(n)}(t-1)}setTimeout(()=>{n(),e.on("activate",e=>{t.history?window.location.hash=e.index+1:window.location.replace(`#${e.index+1}`)}),window.addEventListener("hashchange",n)},0)}}function c(e={}){const t=Object.assign({interval:200},e);return e=>{document.addEventListener("keydown",t=>{(32===t.which&&t.shiftKey||33===t.which||37===t.which||38===t.which)&&e.prev(),(32===t.which&&!t.shiftKey||34===t.which||39===t.which||40===t.which)&&e.next(),35===t.which&&e.slide(e.slides.length-1,{fragment:-1}),36===t.which&&e.slide(0)});let n,r,i=0;e.parent.addEventListener("wheel",o=>{let s=!1;const a=(e,t)=>{e&&(s=s||function(e,t){return function(e,t){const n=t===l.X?"Width":"Height";return e[`client${n}`]<e[`scroll${n}`]}(e,t)&&function(e,t){const{overflow:n}=e,r=e[`overflow${t}`];return"auto"===n||"scroll"===n||"auto"===r||"scroll"===r}(getComputedStyle(e),t)}(e,t)),e&&e.parentElement&&a(e.parentElement,t)};if(0!==o.deltaX&&a(o.target,l.X),0!==o.deltaY&&a(o.target,l.Y),s)return;o.preventDefault(),r&&clearTimeout(r),r=setTimeout(()=>{n=0},t.interval);const c=Date.now()-i<t.interval,u=Math.sqrt(Math.pow(o.deltaX,2)+Math.pow(o.deltaY,2)),d=u<=n;if(n=u,c||d)return;let f;(o.deltaX>0||o.deltaY>0)&&(f="next"),(o.deltaX<0||o.deltaY<0)&&(f="prev"),f&&(e[f](),i=Date.now())})}}function u(e=".bespoke-marp-osc"){const t=document.querySelector(e);if(!t)return()=>{};const n=(e,n)=>{t.querySelectorAll(`[data-bespoke-marp-osc=${JSON.stringify(e)}]`).forEach(n)};return o&&!o.enabled&&n("fullscreen",e=>e.style.display="none"),e=>{t.addEventListener("click",t=>{if(t.target instanceof HTMLElement)switch(t.target.dataset.bespokeMarpOsc){case"next":e.next();break;case"prev":e.prev();break;case"fullscreen":"function"==typeof e.fullscreen&&o&&o.enabled&&e.fullscreen()}}),e.parent.appendChild(t),e.on("activate",({index:t})=>{n("page",n=>n.textContent=`Page ${t+1} of ${e.slides.length}`)}),e.on("fragment",({index:t,fragments:r,fragmentIndex:i})=>{n("prev",e=>e.disabled=0===t&&0===i),n("next",n=>n.disabled=t===e.slides.length-1&&i===r.length-1)}),o&&o.onchange(()=>n("fullscreen",e=>e.classList.toggle("exit",o&&o.isFullscreen)))}}!function(e){e.X="X",e.Y="Y"}(l||(l={}));const d=".bespoke-progress-bar";function f(e){e.on("activate",t=>{document.querySelectorAll(d).forEach(n=>{n.style.flexBasis=`${100*t.index/(e.slides.length-1)}%`})})}function h(e={}){const t=Object.assign({slope:Math.tan(-35*Math.PI/180),swipeThreshold:30},e);return e=>{let n;const r=e.parent,i=e=>{const t=r.getBoundingClientRect();return{x:e.pageX-(t.left+t.right)/2,y:e.pageY-(t.top+t.bottom)/2}};r.addEventListener("touchstart",e=>{n=1===e.touches.length?i(e.touches[0]):void 0}),r.addEventListener("touchmove",e=>{if(n)if(1===e.touches.length){e.preventDefault();const t=i(e.touches[0]),r=t.x-n.x,l=t.y-n.y;n.delta=Math.sqrt(Math.pow(Math.abs(r),2)+Math.pow(Math.abs(l),2)),n.radian=Math.atan2(r,l)}else n=void 0}),r.addEventListener("touchend",r=>{if(n){if(n.delta&&n.delta>=t.swipeThreshold){let i=n.radian-t.slope;i=(i+Math.PI)%(2*Math.PI)-Math.PI,e[i<0?"next":"prev"](),r.stopPropagation()}n=void 0}})}}e.from(document.getElementById("presentation"),[t(),n,r(),a({history:!1}),c(),s,f,h(),u(),i])}();
</script></body></html>