forked from pytorch/pytorch.github.io
-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathnumerical_accuracy.html
907 lines (713 loc) · 46 KB
/
numerical_accuracy.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
<!DOCTYPE html>
<!--[if IE 8]><html class="no-js lt-ie9" lang="en" > <![endif]-->
<!--[if gt IE 8]><!--> <html class="no-js" lang="en" > <!--<![endif]-->
<head>
<meta charset="utf-8">
<meta name="generator" content="Docutils 0.18.1: http://docutils.sourceforge.net/" />
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>Numerical accuracy — PyTorch 1.13 documentation</title>
<link rel="canonical" href="https://pytorch.org/docs/stable/notes/numerical_accuracy.html"/>
<link rel="stylesheet" href="../_static/css/theme.css" type="text/css" />
<!-- <link rel="stylesheet" href="../_static/pygments.css" type="text/css" /> -->
<link rel="stylesheet" href="../_static/pygments.css" type="text/css" />
<link rel="stylesheet" href="../_static/css/theme.css" type="text/css" />
<link rel="stylesheet" href="../_static/copybutton.css" type="text/css" />
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/[email protected]/dist/katex.min.css" type="text/css" />
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/[email protected]/dist/katex.min.css" type="text/css" />
<link rel="stylesheet" href="../_static/katex-math.css" type="text/css" />
<link rel="stylesheet" href="../_static/sphinx-dropdown.css" type="text/css" />
<link rel="stylesheet" href="../_static/panels-bootstrap.min.css" type="text/css" />
<link rel="stylesheet" href="../_static/css/jit.css" type="text/css" />
<link rel="index" title="Index" href="../genindex.html" />
<link rel="search" title="Search" href="../search.html" />
<link rel="next" title="Reproducibility" href="randomness.html" />
<link rel="prev" title="Multiprocessing best practices" href="multiprocessing.html" />
<!-- Google Analytics -->
<script async src="https://www.googletagmanager.com/gtag/js?id=UA-117752657-2"></script>
<script>
window.dataLayer = window.dataLayer || [];
function gtag(){dataLayer.push(arguments);}
gtag('js', new Date());
gtag('config', 'UA-117752657-2');
</script>
<!-- End Google Analytics -->
<script src="../_static/js/modernizr.min.js"></script>
<!-- Preload the theme fonts -->
<link rel="preload" href="../_static/fonts/FreightSans/freight-sans-book.woff2" as="font" type="font/woff2" crossorigin="anonymous">
<link rel="preload" href="../_static/fonts/FreightSans/freight-sans-medium.woff2" as="font" type="font/woff2" crossorigin="anonymous">
<link rel="preload" href="../_static/fonts/IBMPlexMono/IBMPlexMono-Medium.woff2" as="font" type="font/woff2" crossorigin="anonymous">
<link rel="preload" href="../_static/fonts/FreightSans/freight-sans-bold.woff2" as="font" type="font/woff2" crossorigin="anonymous">
<link rel="preload" href="../_static/fonts/FreightSans/freight-sans-medium-italic.woff2" as="font" type="font/woff2" crossorigin="anonymous">
<link rel="preload" href="../_static/fonts/IBMPlexMono/IBMPlexMono-SemiBold.woff2" as="font" type="font/woff2" crossorigin="anonymous">
<!-- Preload the katex fonts -->
<link rel="preload" href="https://cdn.jsdelivr.net/npm/[email protected]/dist/fonts/KaTeX_Math-Italic.woff2" as="font" type="font/woff2" crossorigin="anonymous">
<link rel="preload" href="https://cdn.jsdelivr.net/npm/[email protected]/dist/fonts/KaTeX_Main-Regular.woff2" as="font" type="font/woff2" crossorigin="anonymous">
<link rel="preload" href="https://cdn.jsdelivr.net/npm/[email protected]/dist/fonts/KaTeX_Main-Bold.woff2" as="font" type="font/woff2" crossorigin="anonymous">
<link rel="preload" href="https://cdn.jsdelivr.net/npm/[email protected]/dist/fonts/KaTeX_Size1-Regular.woff2" as="font" type="font/woff2" crossorigin="anonymous">
<link rel="preload" href="https://cdn.jsdelivr.net/npm/[email protected]/dist/fonts/KaTeX_Size4-Regular.woff2" as="font" type="font/woff2" crossorigin="anonymous">
<link rel="preload" href="https://cdn.jsdelivr.net/npm/[email protected]/dist/fonts/KaTeX_Size2-Regular.woff2" as="font" type="font/woff2" crossorigin="anonymous">
<link rel="preload" href="https://cdn.jsdelivr.net/npm/[email protected]/dist/fonts/KaTeX_Size3-Regular.woff2" as="font" type="font/woff2" crossorigin="anonymous">
<link rel="preload" href="https://cdn.jsdelivr.net/npm/[email protected]/dist/fonts/KaTeX_Caligraphic-Regular.woff2" as="font" type="font/woff2" crossorigin="anonymous">
<link rel="stylesheet" href="https://use.fontawesome.com/releases/v5.15.2/css/all.css" integrity="sha384-vSIIfh2YWi9wW0r9iZe7RJPrKwp6bG+s9QZMoITbCckVJqGCCRhc+ccxNcdpHuYu" crossorigin="anonymous">
</head>
<div class="container-fluid header-holder tutorials-header" id="header-holder">
<div class="container">
<div class="header-container">
<a class="header-logo" href="https://pytorch.org/" aria-label="PyTorch"></a>
<div class="main-menu">
<ul>
<li>
<a href="https://pytorch.org/get-started">Get Started</a>
</li>
<li>
<a href="https://pytorch.org/ecosystem">Ecosystem</a>
</li>
<li>
<a href="https://pytorch.org/mobile">Mobile</a>
</li>
<li>
<a href="https://pytorch.org/blog/">Blog</a>
</li>
<li>
<a href="https://pytorch.org/tutorials">Tutorials</a>
</li>
<li class="active docs-active">
<div id="resourcesDropdownButton" data-toggle="resources-dropdown" class="resources-dropdown">
<a class="resource-option with-down-orange-arrow">
Docs
</a>
<div class="resources-dropdown-menu">
<a class="doc-dropdown-option nav-dropdown-item" href="https://pytorch.org/docs/stable/index.html">
<span class="dropdown-title">PyTorch</span>
<p></p>
</a>
<a class="doc-dropdown-option nav-dropdown-item" href="https://pytorch.org/audio/stable/index.html">
<span class="dropdown-title">torchaudio</span>
<p></p>
</a>
<a class="doc-dropdown-option nav-dropdown-item" href="https://pytorch.org/text/stable/index.html">
<span class="dropdown-title">torchtext</span>
<p></p>
</a>
<a class="doc-dropdown-option nav-dropdown-item" href="https://pytorch.org/vision/stable/index.html">
<span class="dropdown-title">torchvision</span>
<p></p>
</a>
<a class="doc-dropdown-option nav-dropdown-item" href="https://pytorch.org/torcharrow">
<span class="dropdown-title">torcharrow</span>
<p></p>
</a>
<a class="doc-dropdown-option nav-dropdown-item" href="https://pytorch.org/data">
<span class="dropdown-title">TorchData</span>
<p></p>
</a>
<a class="doc-dropdown-option nav-dropdown-item" href="https://pytorch.org/torchrec">
<span class="dropdown-title">TorchRec</span>
<p></p>
</a>
<a class="doc-dropdown-option nav-dropdown-item" href="https://pytorch.org/serve/">
<span class="dropdown-title">TorchServe</span>
<p></p>
</a>
<a class="doc-dropdown-option nav-dropdown-item" href="https://pytorch.org/torchx/">
<span class="dropdown-title">TorchX</span>
<p></p>
</a>
<a class="doc-dropdown-option nav-dropdown-item" href="https://pytorch.org/xla">
<span class="dropdown-title">PyTorch on XLA Devices</span>
<p></p>
</a>
</div>
</li>
<li>
<div id="resourcesDropdownButton" data-toggle="resources-dropdown" class="resources-dropdown">
<a class="resource-option with-down-arrow">
Resources
</a>
<div class="resources-dropdown-menu">
<a class="nav-dropdown-item" href="https://pytorch.org/features">
<span class="dropdown-title">About</span>
<p>Learn about PyTorch’s features and capabilities</p>
</a>
<a class="nav-dropdown-item" href="https://pytorch.org/foundation">
<span class="dropdown-title">PyTorch Foundation</span>
<p>Learn about the PyTorch foundation</p>
</a>
<a class="nav-dropdown-item" href="https://pytorch.org/#community-module">
<span class="dropdown-title">Community</span>
<p>Join the PyTorch developer community to contribute, learn, and get your questions answered.</p>
</a>
<a class="nav-dropdown-item" href="https://pytorch.org/community-stories">
<span class="dropdown-title">Community Stories</span>
<p>Learn how our community solves real, everyday machine learning problems with PyTorch.</p>
</a>
<a class="nav-dropdown-item" href="https://pytorch.org/resources">
<span class="dropdown-title">Developer Resources</span>
<p>Find resources and get questions answered</p>
</a>
<a class="nav-dropdown-item" href="https://pytorch.org/events">
<span class="dropdown-title">Events</span>
<p>Find events, webinars, and podcasts</p>
</a>
<a class="nav-dropdown-item" href="https://discuss.pytorch.org/" target="_blank">
<span class="dropdown-title">Forums</span>
<p>A place to discuss PyTorch code, issues, install, research</p>
</a>
<a class="nav-dropdown-item" href="https://pytorch.org/hub">
<span class="dropdown-title">Models (Beta)</span>
<p>Discover, publish, and reuse pre-trained models</p>
</a>
</div>
</div>
</li>
<li>
<a href="https://github.com/pytorch/pytorch">GitHub</a>
</li>
</ul>
</div>
<a class="main-menu-open-button" href="#" data-behavior="open-mobile-menu"></a>
</div>
</div>
</div>
<body class="pytorch-body">
<div class="table-of-contents-link-wrapper">
<span>Table of Contents</span>
<a href="#" class="toggle-table-of-contents" data-behavior="toggle-table-of-contents"></a>
</div>
<nav data-toggle="wy-nav-shift" class="pytorch-left-menu" id="pytorch-left-menu">
<div class="pytorch-side-scroll">
<div class="pytorch-menu pytorch-menu-vertical" data-spy="affix" role="navigation" aria-label="main navigation">
<div class="pytorch-left-menu-search">
<div class="version">
<a href='https://pytorch.org/docs/versions.html'>1.13 ▼</a>
</div>
<div role="search">
<form id="rtd-search-form" class="wy-form" action="../search.html" method="get">
<input type="text" name="q" placeholder="Search Docs" />
<input type="hidden" name="check_keywords" value="yes" />
<input type="hidden" name="area" value="default" />
</form>
</div>
</div>
<p class="caption" role="heading"><span class="caption-text">Community</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../community/build_ci_governance.html">PyTorch Governance | Build + CI</a></li>
<li class="toctree-l1"><a class="reference internal" href="../community/contribution_guide.html">PyTorch Contribution Guide</a></li>
<li class="toctree-l1"><a class="reference internal" href="../community/design.html">PyTorch Design Philosophy</a></li>
<li class="toctree-l1"><a class="reference internal" href="../community/governance.html">PyTorch Governance | Mechanics</a></li>
<li class="toctree-l1"><a class="reference internal" href="../community/persons_of_interest.html">PyTorch Governance | Maintainers</a></li>
</ul>
<p class="caption" role="heading"><span class="caption-text">Developer Notes</span></p>
<ul class="current">
<li class="toctree-l1"><a class="reference internal" href="amp_examples.html">CUDA Automatic Mixed Precision examples</a></li>
<li class="toctree-l1"><a class="reference internal" href="autograd.html">Autograd mechanics</a></li>
<li class="toctree-l1"><a class="reference internal" href="broadcasting.html">Broadcasting semantics</a></li>
<li class="toctree-l1"><a class="reference internal" href="cpu_threading_torchscript_inference.html">CPU threading and TorchScript inference</a></li>
<li class="toctree-l1"><a class="reference internal" href="cuda.html">CUDA semantics</a></li>
<li class="toctree-l1"><a class="reference internal" href="ddp.html">Distributed Data Parallel</a></li>
<li class="toctree-l1"><a class="reference internal" href="extending.html">Extending PyTorch</a></li>
<li class="toctree-l1"><a class="reference internal" href="faq.html">Frequently Asked Questions</a></li>
<li class="toctree-l1"><a class="reference internal" href="gradcheck.html">Gradcheck mechanics</a></li>
<li class="toctree-l1"><a class="reference internal" href="hip.html">HIP (ROCm) semantics</a></li>
<li class="toctree-l1"><a class="reference internal" href="large_scale_deployments.html">Features for large-scale deployments</a></li>
<li class="toctree-l1"><a class="reference internal" href="modules.html">Modules</a></li>
<li class="toctree-l1"><a class="reference internal" href="mps.html">MPS backend</a></li>
<li class="toctree-l1"><a class="reference internal" href="multiprocessing.html">Multiprocessing best practices</a></li>
<li class="toctree-l1 current"><a class="current reference internal" href="#">Numerical accuracy</a></li>
<li class="toctree-l1"><a class="reference internal" href="randomness.html">Reproducibility</a></li>
<li class="toctree-l1"><a class="reference internal" href="serialization.html">Serialization semantics</a></li>
<li class="toctree-l1"><a class="reference internal" href="windows.html">Windows FAQ</a></li>
</ul>
<p class="caption" role="heading"><span class="caption-text">Language Bindings</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../cpp_index.html">C++</a></li>
<li class="toctree-l1"><a class="reference external" href="https://pytorch.org/javadoc/">Javadoc</a></li>
<li class="toctree-l1"><a class="reference internal" href="../deploy.html">torch::deploy</a></li>
</ul>
<p class="caption" role="heading"><span class="caption-text">Python API</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../torch.html">torch</a></li>
<li class="toctree-l1"><a class="reference internal" href="../nn.html">torch.nn</a></li>
<li class="toctree-l1"><a class="reference internal" href="../nn.functional.html">torch.nn.functional</a></li>
<li class="toctree-l1"><a class="reference internal" href="../tensors.html">torch.Tensor</a></li>
<li class="toctree-l1"><a class="reference internal" href="../tensor_attributes.html">Tensor Attributes</a></li>
<li class="toctree-l1"><a class="reference internal" href="../tensor_view.html">Tensor Views</a></li>
<li class="toctree-l1"><a class="reference internal" href="../amp.html">torch.amp</a></li>
<li class="toctree-l1"><a class="reference internal" href="../autograd.html">torch.autograd</a></li>
<li class="toctree-l1"><a class="reference internal" href="../library.html">torch.library</a></li>
<li class="toctree-l1"><a class="reference internal" href="../cuda.html">torch.cuda</a></li>
<li class="toctree-l1"><a class="reference internal" href="../backends.html">torch.backends</a></li>
<li class="toctree-l1"><a class="reference internal" href="../distributed.html">torch.distributed</a></li>
<li class="toctree-l1"><a class="reference internal" href="../distributed.algorithms.join.html">torch.distributed.algorithms.join</a></li>
<li class="toctree-l1"><a class="reference internal" href="../distributed.elastic.html">torch.distributed.elastic</a></li>
<li class="toctree-l1"><a class="reference internal" href="../fsdp.html">torch.distributed.fsdp</a></li>
<li class="toctree-l1"><a class="reference internal" href="../distributed.optim.html">torch.distributed.optim</a></li>
<li class="toctree-l1"><a class="reference internal" href="../distributions.html">torch.distributions</a></li>
<li class="toctree-l1"><a class="reference internal" href="../fft.html">torch.fft</a></li>
<li class="toctree-l1"><a class="reference internal" href="../futures.html">torch.futures</a></li>
<li class="toctree-l1"><a class="reference internal" href="../fx.html">torch.fx</a></li>
<li class="toctree-l1"><a class="reference internal" href="../hub.html">torch.hub</a></li>
<li class="toctree-l1"><a class="reference internal" href="../jit.html">torch.jit</a></li>
<li class="toctree-l1"><a class="reference internal" href="../linalg.html">torch.linalg</a></li>
<li class="toctree-l1"><a class="reference internal" href="../monitor.html">torch.monitor</a></li>
<li class="toctree-l1"><a class="reference internal" href="../special.html">torch.special</a></li>
<li class="toctree-l1"><a class="reference internal" href="../torch.overrides.html">torch.overrides</a></li>
<li class="toctree-l1"><a class="reference internal" href="../package.html">torch.package</a></li>
<li class="toctree-l1"><a class="reference internal" href="../profiler.html">torch.profiler</a></li>
<li class="toctree-l1"><a class="reference internal" href="../nn.init.html">torch.nn.init</a></li>
<li class="toctree-l1"><a class="reference internal" href="../onnx.html">torch.onnx</a></li>
<li class="toctree-l1"><a class="reference internal" href="../optim.html">torch.optim</a></li>
<li class="toctree-l1"><a class="reference internal" href="../complex_numbers.html">Complex Numbers</a></li>
<li class="toctree-l1"><a class="reference internal" href="../ddp_comm_hooks.html">DDP Communication Hooks</a></li>
<li class="toctree-l1"><a class="reference internal" href="../pipeline.html">Pipeline Parallelism</a></li>
<li class="toctree-l1"><a class="reference internal" href="../quantization.html">Quantization</a></li>
<li class="toctree-l1"><a class="reference internal" href="../rpc.html">Distributed RPC Framework</a></li>
<li class="toctree-l1"><a class="reference internal" href="../random.html">torch.random</a></li>
<li class="toctree-l1"><a class="reference internal" href="../masked.html">torch.masked</a></li>
<li class="toctree-l1"><a class="reference internal" href="../nested.html">torch.nested</a></li>
<li class="toctree-l1"><a class="reference internal" href="../sparse.html">torch.sparse</a></li>
<li class="toctree-l1"><a class="reference internal" href="../storage.html">torch.Storage</a></li>
<li class="toctree-l1"><a class="reference internal" href="../testing.html">torch.testing</a></li>
<li class="toctree-l1"><a class="reference internal" href="../benchmark_utils.html">torch.utils.benchmark</a></li>
<li class="toctree-l1"><a class="reference internal" href="../bottleneck.html">torch.utils.bottleneck</a></li>
<li class="toctree-l1"><a class="reference internal" href="../checkpoint.html">torch.utils.checkpoint</a></li>
<li class="toctree-l1"><a class="reference internal" href="../cpp_extension.html">torch.utils.cpp_extension</a></li>
<li class="toctree-l1"><a class="reference internal" href="../data.html">torch.utils.data</a></li>
<li class="toctree-l1"><a class="reference internal" href="../jit_utils.html">torch.utils.jit</a></li>
<li class="toctree-l1"><a class="reference internal" href="../dlpack.html">torch.utils.dlpack</a></li>
<li class="toctree-l1"><a class="reference internal" href="../mobile_optimizer.html">torch.utils.mobile_optimizer</a></li>
<li class="toctree-l1"><a class="reference internal" href="../model_zoo.html">torch.utils.model_zoo</a></li>
<li class="toctree-l1"><a class="reference internal" href="../tensorboard.html">torch.utils.tensorboard</a></li>
<li class="toctree-l1"><a class="reference internal" href="../type_info.html">Type Info</a></li>
<li class="toctree-l1"><a class="reference internal" href="../named_tensor.html">Named Tensors</a></li>
<li class="toctree-l1"><a class="reference internal" href="../name_inference.html">Named Tensors operator coverage</a></li>
<li class="toctree-l1"><a class="reference internal" href="../config_mod.html">torch.__config__</a></li>
</ul>
<p class="caption" role="heading"><span class="caption-text">Libraries</span></p>
<ul>
<li class="toctree-l1"><a class="reference external" href="https://pytorch.org/audio/stable">torchaudio</a></li>
<li class="toctree-l1"><a class="reference external" href="https://pytorch.org/data">TorchData</a></li>
<li class="toctree-l1"><a class="reference external" href="https://pytorch.org/torchrec">TorchRec</a></li>
<li class="toctree-l1"><a class="reference external" href="https://pytorch.org/serve">TorchServe</a></li>
<li class="toctree-l1"><a class="reference external" href="https://pytorch.org/text/stable">torchtext</a></li>
<li class="toctree-l1"><a class="reference external" href="https://pytorch.org/vision/stable">torchvision</a></li>
<li class="toctree-l1"><a class="reference external" href="http://pytorch.org/xla/">PyTorch on XLA Devices</a></li>
</ul>
</div>
</div>
</nav>
<div class="pytorch-container">
<div class="pytorch-page-level-bar" id="pytorch-page-level-bar">
<div class="pytorch-breadcrumbs-wrapper">
<div role="navigation" aria-label="breadcrumbs navigation">
<ul class="pytorch-breadcrumbs">
<li>
<a href="../index.html">
Docs
</a> >
</li>
<li>Numerical accuracy</li>
<li class="pytorch-breadcrumbs-aside">
<a href="../_sources/notes/numerical_accuracy.rst.txt" rel="nofollow"><img src="../_static/images/view-page-source-icon.svg"></a>
</li>
</ul>
</div>
</div>
<div class="pytorch-shortcuts-wrapper" id="pytorch-shortcuts-wrapper">
Shortcuts
</div>
</div>
<section data-toggle="wy-nav-shift" id="pytorch-content-wrap" class="pytorch-content-wrap">
<div class="pytorch-content-left">
<div class="rst-content">
<div role="main" class="main-content" itemscope="itemscope" itemtype="http://schema.org/Article">
<article itemprop="articleBody" id="pytorch-article" class="pytorch-article">
<section id="numerical-accuracy">
<span id="id1"></span><h1>Numerical accuracy<a class="headerlink" href="#numerical-accuracy" title="Permalink to this heading">¶</a></h1>
<p>In modern computers, floating point numbers are represented using IEEE 754 standard.
For more details on floating point arithmetics and IEEE 754 standard, please see
<a class="reference external" href="https://en.wikipedia.org/wiki/Floating-point_arithmetic">Floating point arithmetic</a>
In particular, note that floating point provides limited accuracy (about 7 decimal digits
for single precision floating point numbers, about 16 decimal digits for double precision
floating point numbers) and that floating point addition and multiplication are not
associative, so the order of the operations affects the results.
Because of this, PyTorch is not guaranteed
to produce bitwise identical results for floating point computations that are
mathematically identical. Similarly, bitwise identical results are not guaranteed across
PyTorch releases, individual commits, or different platforms. In particular, CPU and GPU
results can be different even for bitwise-identical inputs and even after controlling for
the sources of randomness.</p>
<section id="batched-computations-or-slice-computations">
<h2>Batched computations or slice computations<a class="headerlink" href="#batched-computations-or-slice-computations" title="Permalink to this heading">¶</a></h2>
<p>Many operations in PyTorch support batched computation, where the same operation is performed
for the elements of the batches of inputs. An example of this is <a class="reference internal" href="../generated/torch.mm.html#torch.mm" title="torch.mm"><code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.mm()</span></code></a> and
<a class="reference internal" href="../generated/torch.bmm.html#torch.bmm" title="torch.bmm"><code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.bmm()</span></code></a>. It is possible to implement batched computation as a loop over batch elements,
and apply the necessary math operations to the individual batch elements, for efficiency reasons
we are not doing that, and typically perform computation for the whole batch. The mathematical
libraries that we are calling, and PyTorch internal implementations of operations can produces
slightly different results in this case, compared to non-batched computations. In particular,
let <code class="docutils literal notranslate"><span class="pre">A</span></code> and <code class="docutils literal notranslate"><span class="pre">B</span></code> be 3D tensors with the dimensions suitable for batched matrix multiplication.
Then <code class="docutils literal notranslate"><span class="pre">(A@B)[0]</span></code> (the first element of the batched result) is not guaranteed to be bitwise
identical to <code class="docutils literal notranslate"><span class="pre">A[0]@B[0]</span></code> (the matrix product of the first elements of the input batches)
even though mathematically it’s an identical computation.</p>
<p>Similarly, an operation applied to a tensor slice is not guaranteed to produce results that are
identical to the slice of the result of the same operation applied to the full tensor. E.g. let
<code class="docutils literal notranslate"><span class="pre">A</span></code> be a 2-dimentional tensor. <code class="docutils literal notranslate"><span class="pre">A.sum(-1)[0]</span></code> is not guaranteed to be bitwise equal to
<code class="docutils literal notranslate"><span class="pre">A[:,0].sum()</span></code>.</p>
</section>
<section id="extremal-values">
<h2>Extremal values<a class="headerlink" href="#extremal-values" title="Permalink to this heading">¶</a></h2>
<p>When inputs contain large values such that intermediate results may overflow the range of the
used datatype, the end result may overflow too, even though it is representable in the original
datatype. E.g.:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">torch</span>
<span class="n">a</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mf">1e20</span><span class="p">,</span> <span class="mf">1e20</span><span class="p">])</span> <span class="c1"># fp32 type by default</span>
<span class="n">a</span><span class="o">.</span><span class="n">norm</span><span class="p">()</span> <span class="c1"># produces tensor(inf)</span>
<span class="n">a</span><span class="o">.</span><span class="n">double</span><span class="p">()</span><span class="o">.</span><span class="n">norm</span><span class="p">()</span> <span class="c1"># produces tensor(1.4142e+20, dtype=torch.float64), representable in fp32</span>
</pre></div>
</div>
</section>
<section id="tensorfloat-32-tf32-on-nvidia-ampere-devices">
<h2>TensorFloat-32(TF32) on Nvidia Ampere devices<a class="headerlink" href="#tensorfloat-32-tf32-on-nvidia-ampere-devices" title="Permalink to this heading">¶</a></h2>
<p>On Ampere Nvidia GPUs, PyTorch can use TensorFloat32 (TF32) to speed up mathematically intensive operations, in particular matrix multiplications and convolutions.
When an operation is performed using TF32 tensor cores, only the first 10 bits of the input mantissa are read.
This may reduce accuracy and produce surprising results (e.g., multiplying a matrix by the identity matrix may produce results that are different from the input).
By default, TF32 tensor cores are disabled for matrix multiplications and enabled for convolutions, although most neural network workloads have the same convergence behavior when using TF32 as they have with fp32.
We recommend enabling TF32 tensor cores for matrix multiplications with <code class="docutils literal notranslate"><span class="pre">torch.backends.cuda.matmul.allow_tf32</span> <span class="pre">=</span> <span class="pre">True</span></code> if your network does not need full float32 precision.
If your network needs full float32 precision for both matrix multiplications and convolutions, then TF32 tensor cores can also be disabled for convolutions with <code class="docutils literal notranslate"><span class="pre">torch.backends.cudnn.allow_tf32</span> <span class="pre">=</span> <span class="pre">False</span></code>.</p>
<p>For more information see <a class="reference internal" href="cuda.html#tf32-on-ampere"><span class="std std-ref">TensorFloat32</span></a>.</p>
</section>
<section id="reduced-precision-reduction-for-fp16-gemms">
<h2>Reduced Precision Reduction for FP16 GEMMs<a class="headerlink" href="#reduced-precision-reduction-for-fp16-gemms" title="Permalink to this heading">¶</a></h2>
<p>Half-precision GEMM operations are typically done with intermediate accumulations (reduction) in single-precision for numerical accuracy and improved resilience to overflow. For performance, certain GPU architectures, especially more recent ones, allow a few truncations of the intermediate accumulation results to the reduced precision (e.g., half-precision). This change is often benign from the perspective of model convergence, though it may lead to unexpected results (e.g., <code class="docutils literal notranslate"><span class="pre">inf</span></code> values when the final result should be be representable in half-precision).
If reduced-precision reductions are problematic, they can be turned off with
<code class="docutils literal notranslate"><span class="pre">torch.backends.cuda.matmul.allow_fp16_reduced_precision_reduction</span> <span class="pre">=</span> <span class="pre">False</span></code></p>
<p>For more information see <a class="reference internal" href="cuda.html#fp16reducedprecision"><span class="std std-ref">allow_fp16_reduced_precision_reduction</span></a></p>
</section>
<section id="reduced-precision-fp16-and-bf16-gemms-and-convolutions-on-amd-instinct-mi200-devices">
<span id="fp16-on-mi200"></span><h2>Reduced Precision FP16 and BF16 GEMMs and Convolutions on AMD Instinct MI200 devices<a class="headerlink" href="#reduced-precision-fp16-and-bf16-gemms-and-convolutions-on-amd-instinct-mi200-devices" title="Permalink to this heading">¶</a></h2>
<p>On AMD Instinct MI200 GPUs, the FP16 and BF16 V_DOT2 and MFMA matrix instructions flush input and output denormal values to zero. FP32 and FP64 MFMA matrix instructions do not flush input and output denormal values to zero. The affected instructions are only used by rocBLAS (GEMM) and MIOpen (convolution) kernels; all other PyTorch operations will not encounter this behavior. All other supported AMD GPUs will not encounter this behavior.</p>
<p>rocBLAS and MIOpen provide alternate implementations for affected FP16 operations. Alternate implementations for BF16 operations are not provided; BF16 numbers have a larger dynamic range than FP16 numbers and are less likely to encounter denormal values. For the FP16 alternate implementations, FP16 input values are cast to an intermediate BF16 value and then cast back to FP16 output after the accumulate FP32 operations. In this way, the input and output types are unchanged.</p>
<p>When training using FP16 precision, some models may fail to converge with FP16 denorms flushed to zero. Denormal values more frequently occur in the backward pass of training during gradient calculation. PyTorch by default will use the rocBLAS and MIOpen alternate implementations during the backward pass. The default behavior can be overridden using environment variables, ROCBLAS_INTERNAL_FP16_ALT_IMPL and MIOPEN_DEBUG_CONVOLUTION_ATTRIB_FP16_ALT_IMPL. The behavior of these environment variables is as follows:</p>
<table class="docutils align-default">
<thead>
<tr class="row-odd"><th class="head"></th>
<th class="head"><p>forward</p></th>
<th class="head"><p>backward</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p>Env unset</p></td>
<td><p>original</p></td>
<td><p>alternate</p></td>
</tr>
<tr class="row-odd"><td><p>Env set to 1</p></td>
<td><p>alternate</p></td>
<td><p>alternate</p></td>
</tr>
<tr class="row-even"><td><p>Env set to 0</p></td>
<td><p>original</p></td>
<td><p>original</p></td>
</tr>
</tbody>
</table>
<p>The following is the list of operations where rocBLAS may be used:</p>
<ul class="simple">
<li><p>torch.addbmm</p></li>
<li><p>torch.addmm</p></li>
<li><p>torch.baddbmm</p></li>
<li><p>torch.bmm</p></li>
<li><p>torch.mm</p></li>
<li><p>torch.nn.GRUCell</p></li>
<li><p>torch.nn.LSTMCell</p></li>
<li><p>torch.nn.Linear</p></li>
<li><p>torch.sparse.addmm</p></li>
<li><p>the following torch._C._ConvBackend implementations:</p>
<ul>
<li><p>slowNd</p></li>
<li><p>slowNd_transposed</p></li>
<li><p>slowNd_dilated</p></li>
<li><p>slowNd_dilated_transposed</p></li>
</ul>
</li>
</ul>
<p>The following is the list of operations where MIOpen may be used:</p>
<ul class="simple">
<li><p>torch.nn.Conv[Transpose]Nd</p></li>
<li><p>the following torch._C._ConvBackend implementations:</p>
<ul>
<li><p>ConvBackend::Miopen</p></li>
<li><p>ConvBackend::MiopenDepthwise</p></li>
<li><p>ConvBackend::MiopenTranspose</p></li>
</ul>
</li>
</ul>
</section>
</section>
</article>
</div>
<footer>
<div class="rst-footer-buttons" role="navigation" aria-label="footer navigation">
<a href="randomness.html" class="btn btn-neutral float-right" title="Reproducibility" accesskey="n" rel="next">Next <img src="../_static/images/chevron-right-orange.svg" class="next-page"></a>
<a href="multiprocessing.html" class="btn btn-neutral" title="Multiprocessing best practices" accesskey="p" rel="prev"><img src="../_static/images/chevron-right-orange.svg" class="previous-page"> Previous</a>
</div>
<hr>
<div role="contentinfo">
<p>
© Copyright 2022, PyTorch Contributors.
</p>
</div>
<div>
Built with <a href="http://sphinx-doc.org/">Sphinx</a> using a <a href="https://github.com/rtfd/sphinx_rtd_theme">theme</a> provided by <a href="https://readthedocs.org">Read the Docs</a>.
</div>
</footer>
</div>
</div>
<div class="pytorch-content-right" id="pytorch-content-right">
<div class="pytorch-right-menu" id="pytorch-right-menu">
<div class="pytorch-side-scroll" id="pytorch-side-scroll-right">
<ul>
<li><a class="reference internal" href="#">Numerical accuracy</a><ul>
<li><a class="reference internal" href="#batched-computations-or-slice-computations">Batched computations or slice computations</a></li>
<li><a class="reference internal" href="#extremal-values">Extremal values</a></li>
<li><a class="reference internal" href="#tensorfloat-32-tf32-on-nvidia-ampere-devices">TensorFloat-32(TF32) on Nvidia Ampere devices</a></li>
<li><a class="reference internal" href="#reduced-precision-reduction-for-fp16-gemms">Reduced Precision Reduction for FP16 GEMMs</a></li>
<li><a class="reference internal" href="#reduced-precision-fp16-and-bf16-gemms-and-convolutions-on-amd-instinct-mi200-devices">Reduced Precision FP16 and BF16 GEMMs and Convolutions on AMD Instinct MI200 devices</a></li>
</ul>
</li>
</ul>
</div>
</div>
</div>
</section>
</div>
<script type="text/javascript" id="documentation_options" data-url_root="../" src="../_static/documentation_options.js"></script>
<script data-url_root="../" id="documentation_options" src="../_static/documentation_options.js"></script>
<script src="../_static/jquery.js"></script>
<script src="../_static/underscore.js"></script>
<script src="../_static/_sphinx_javascript_frameworks_compat.js"></script>
<script src="../_static/doctools.js"></script>
<script src="../_static/clipboard.min.js"></script>
<script src="../_static/copybutton.js"></script>
<script type="text/javascript" src="../_static/js/vendor/popper.min.js"></script>
<script type="text/javascript" src="../_static/js/vendor/bootstrap.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/list.js/1.5.0/list.min.js"></script>
<script type="text/javascript" src="../_static/js/theme.js"></script>
<script type="text/javascript">
jQuery(function () {
SphinxRtdTheme.Navigation.enable(true);
});
</script>
<script script type="text/javascript">
var collapsedSections = ['Developer Notes', 'Language Bindings', 'Libraries', 'Community'];
</script>
<img height="1" width="1" style="border-style:none;" alt="" src="https://www.googleadservices.com/pagead/conversion/795629140/?label=txkmCPmdtosBENSssfsC&guid=ON&script=0"/>
<!-- Begin Footer -->
<div class="container-fluid docs-tutorials-resources" id="docs-tutorials-resources">
<div class="container">
<div class="row">
<div class="col-md-4 text-center">
<h2>Docs</h2>
<p>Access comprehensive developer documentation for PyTorch</p>
<a class="with-right-arrow" href="https://pytorch.org/docs/stable/index.html">View Docs</a>
</div>
<div class="col-md-4 text-center">
<h2>Tutorials</h2>
<p>Get in-depth tutorials for beginners and advanced developers</p>
<a class="with-right-arrow" href="https://pytorch.org/tutorials">View Tutorials</a>
</div>
<div class="col-md-4 text-center">
<h2>Resources</h2>
<p>Find development resources and get your questions answered</p>
<a class="with-right-arrow" href="https://pytorch.org/resources">View Resources</a>
</div>
</div>
</div>
</div>
<footer class="site-footer">
<div class="container footer-container">
<div class="footer-logo-wrapper">
<a href="https://pytorch.org/" class="footer-logo"></a>
</div>
<div class="footer-links-wrapper">
<div class="footer-links-col">
<ul>
<li class="list-title"><a href="https://pytorch.org/">PyTorch</a></li>
<li><a href="https://pytorch.org/get-started">Get Started</a></li>
<li><a href="https://pytorch.org/features">Features</a></li>
<li><a href="https://pytorch.org/ecosystem">Ecosystem</a></li>
<li><a href="https://pytorch.org/blog/">Blog</a></li>
<li><a href="https://github.com/pytorch/pytorch/blob/master/CONTRIBUTING.md">Contributing</a></li>
</ul>
</div>
<div class="footer-links-col">
<ul>
<li class="list-title"><a href="https://pytorch.org/resources">Resources</a></li>
<li><a href="https://pytorch.org/tutorials">Tutorials</a></li>
<li><a href="https://pytorch.org/docs/stable/index.html">Docs</a></li>
<li><a href="https://discuss.pytorch.org" target="_blank">Discuss</a></li>
<li><a href="https://github.com/pytorch/pytorch/issues" target="_blank">Github Issues</a></li>
<li><a href="https://pytorch.org/assets/brand-guidelines/PyTorch-Brand-Guidelines.pdf" target="_blank">Brand Guidelines</a></li>
</ul>
</div>
<div class="footer-links-col">
<ul>
<li class="list-title">Stay up to date</li>
<li><a href="https://www.facebook.com/pytorch" target="_blank">Facebook</a></li>
<li><a href="https://twitter.com/pytorch" target="_blank">Twitter</a></li>
<li><a href="https://www.youtube.com/pytorch" target="_blank">YouTube</a></li>
<li><a href="https://www.linkedin.com/company/pytorch" target="_blank">LinkedIn</a></li>
</ul>
</div>
<div class="footer-links-col">
<ul>
<li class="list-title">PyTorch Podcasts</li>
<li><a href="https://open.spotify.com/show/6UzHKeiy368jKfQMKKvJY5" target="_blank">Spotify</a></li>
<li><a href="https://podcasts.apple.com/us/podcast/pytorch-developer-podcast/id1566080008" target="_blank">Apple</a></li>
<li><a href="https://www.google.com/podcasts?feed=aHR0cHM6Ly9mZWVkcy5zaW1wbGVjYXN0LmNvbS9PQjVGa0lsOA%3D%3D" target="_blank">Google</a></li>
<li><a href="https://music.amazon.com/podcasts/7a4e6f0e-26c2-49e9-a478-41bd244197d0/PyTorch-Developer-Podcast?" target="_blank">Amazon</a></li>
</ul>
</div>
</div>
<div class="privacy-policy">
<ul>
<li class="privacy-policy-links"><a href="https://www.linuxfoundation.org/terms/" target="_blank">Terms</a></li>
<li class="privacy-policy-links">|</li>
<li class="privacy-policy-links"><a href="https://www.linuxfoundation.org/privacy-policy/" target="_blank">Privacy</a></li>
</ul>
</div>
<div class="copyright">
<p>© Copyright The Linux Foundation. The PyTorch Foundation is a project of The Linux Foundation.
For web site terms of use, trademark policy and other policies applicable to The PyTorch Foundation please see
<a href="www.linuxfoundation.org/policies/">www.linuxfoundation.org/policies/</a>. The PyTorch Foundation supports the PyTorch open source
project, which has been established as PyTorch Project a Series of LF Projects, LLC. For policies applicable to the PyTorch Project a Series of LF Projects, LLC,
please see <a href="www.lfprojects.org/policies/">www.lfprojects.org/policies/</a>.</p>
</div>
</div>
</footer>
<div class="cookie-banner-wrapper">
<div class="container">
<p class="gdpr-notice">To analyze traffic and optimize your experience, we serve cookies on this site. By clicking or navigating, you agree to allow our usage of cookies. As the current maintainers of this site, Facebook’s Cookies Policy applies. Learn more, including about available controls: <a href="https://www.facebook.com/policies/cookies/">Cookies Policy</a>.</p>
<img class="close-button" src="../_static/images/pytorch-x.svg">
</div>
</div>
<!-- End Footer -->
<!-- Begin Mobile Menu -->
<div class="mobile-main-menu">
<div class="container-fluid">
<div class="container">
<div class="mobile-main-menu-header-container">
<a class="header-logo" href="https://pytorch.org/" aria-label="PyTorch"></a>
<a class="main-menu-close-button" href="#" data-behavior="close-mobile-menu"></a>
</div>
</div>
</div>
<div class="mobile-main-menu-links-container">
<div class="main-menu">
<ul>
<li>
<a href="https://pytorch.org/get-started">Get Started</a>
</li>
<li>
<a href="https://pytorch.org/ecosystem">Ecosystem</a>
</li>
<li>
<a href="https://pytorch.org/mobile">Mobile</a>
</li>
<li>
<a href="https://pytorch.org/blog/">Blog</a>
</li>
<li>
<a href="https://pytorch.org/tutorials">Tutorials</a>
</li>
<li class="resources-mobile-menu-title" class="active">
Docs
</li>
<ul class="resources-mobile-menu-items">
<li>
<a href="https://pytorch.org/docs/stable/index.html">PyTorch</a>
</li>
<li>
<a href="https://pytorch.org/audio/stable/index.html">torchaudio</a>
</li>
<li>
<a href="https://pytorch.org/text/stable/index.html">torchtext</a>
</li>
<li>
<a href="https://pytorch.org/vision/stable/index.html">torchvision</a>
</li>
<li>
<a href="https://pytorch.org/torcharrow">torcharrow</a>
</li>
<li>
<a href="https://pytorch.org/data">TorchData</a>
</li>
<li>
<a href="https://pytorch.org/torchrec">TorchRec</a>
</li>
<li>
<a href="https://pytorch.org/serve/">TorchServe</a>
</li>
<li>
<a href="https://pytorch.org/torchx/">TorchX</a>
</li>
<li>
<a href="https://pytorch.org/xla">PyTorch on XLA Devices</a>
</li>
</ul>
<li class="resources-mobile-menu-title">
Resources
</li>
<ul class="resources-mobile-menu-items">
<li>
<a href="https://pytorch.org/features">About</a>
</li>
<li>
<a href="https://pytorch.org/foundation">PyTorch Foundation</a>
</li>
<li>
<a href="https://pytorch.org/#community-module">Community</a>
</li>
<li>
<a href="https://pytorch.org/community-stories">Community Stories</a>
</li>
<li>
<a href="https://pytorch.org/resources">Developer Resources</a>
</li>
<li>
<a href="https://pytorch.org/events">Events</a>
</li>
<li>
<a href="https://discuss.pytorch.org/">Forums</a>
</li>
<li>
<a href="https://pytorch.org/hub">Models (Beta)</a>
</li>
</ul>
<li>
<a href="https://github.com/pytorch/pytorch">Github</a>
</li>
</ul>
</div>
</div>
</div>
<!-- End Mobile Menu -->
<script type="text/javascript" src="../_static/js/vendor/anchor.min.js"></script>
<script type="text/javascript">
$(document).ready(function() {
mobileMenu.bind();
mobileTOC.bind();
pytorchAnchors.bind();
sideMenus.bind();
scrollToAnchor.bind();
highlightNavigation.bind();
mainMenuDropdown.bind();
filterTags.bind();
// Add class to links that have code blocks, since we cannot create links in code blocks
$("article.pytorch-article a span.pre").each(function(e) {
$(this).closest("a").addClass("has-code");
});
})
</script>
</body>
</html>