-
Notifications
You must be signed in to change notification settings - Fork 4
/
tutorial.cxx
484 lines (398 loc) · 16.9 KB
/
tutorial.cxx
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
#include <algorithm> // `std::sort`
#include <cmath> // `std::pow`
#include <cstdint> // `int32_t`
#include <cstdlib> // `std::rand`
#include <execution> // `std::execution::par_unseq`
#include <new> // `std::launder`
#include <random> // `std::mt19937`
#include <vector> // `std::algorithm`
#include <benchmark/benchmark.h>
namespace bm = benchmark;
static void i32_addition(bm::State &state) {
int32_t a = 0, b = 0, c = 0;
for (auto _ : state)
c = a + b;
// Silence "variable ‘c’ set but not used" warning
(void)c;
}
// The compiler will just optimize everything out.
// After the first run, the value of `c` won't change.
// The benchmark will show 0ns per iteration.
BENCHMARK(i32_addition);
static void i32_addition_random(bm::State &state) {
int32_t c = 0;
for (auto _ : state)
c = std::rand() + std::rand();
// Silence "variable ‘c’ set but not used" warning
(void)c;
}
// This run in 25ns, or about 100 CPU cycles.
// Is integer addition really that expensive?
BENCHMARK(i32_addition_random);
static void i32_addition_semi_random(bm::State &state) {
int32_t a = std::rand(), b = std::rand(), c = 0;
for (auto _ : state)
bm::DoNotOptimize(c = (++a) + (++b));
}
// We trigger the two `inc` instructions and the `add` on x86.
// This shouldn't take more then 0.7 ns on a modern CPU.
// So all the time spent - was in the `rand()`!
BENCHMARK(i32_addition_semi_random);
// Our `rand()` is 100 cycles on a single core, but it involves
// global state management, so it can be as slow 12'000 ns with
// just 8 threads.
BENCHMARK(i32_addition_random)->Threads(8);
BENCHMARK(i32_addition_semi_random)->Threads(8);
// ------------------------------------
// ## Let's do some basic math
// ### Maclaurin series
// ------------------------------------
static void f64_sin(bm::State &state) {
double argument = std::rand(), result = 0;
for (auto _ : state)
bm::DoNotOptimize(result = std::sin(argument += 1.0));
}
static void f64_sin_maclaurin(bm::State &state) {
double argument = std::rand(), result = 0;
for (auto _ : state) {
argument += 1.0;
result = argument - std::pow(argument, 3) / 6 + std::pow(argument, 5) / 120;
bm::DoNotOptimize(result);
}
}
// Lets compute the `sin(x)` via Maclaurin series.
// It will involve a fair share of floating point operations.
// We will only take the first 3 parts of the expansion:
// sin(x) ~ x - (x^3) / 3! + (x^5) / 5!
// https://en.wikipedia.org/wiki/Taylor_series
BENCHMARK(f64_sin);
BENCHMARK(f64_sin_maclaurin);
static void f64_sin_maclaurin_powless(bm::State &state) {
double argument = std::rand(), result = 0;
for (auto _ : state) {
argument += 1.0;
result = argument - (argument * argument * argument) / 6.0 +
(argument * argument * argument * argument * argument) / 120.0;
bm::DoNotOptimize(result);
}
}
// Help the compiler Help you!
// Instead of using the heavy generic operation - describe your special case to the compiler!
BENCHMARK(f64_sin_maclaurin_powless);
// We want to recommend them to avoid all IEEE-754 compliance checks at a single-function level.
// For that, "fast math" attributes can be used: https://simonbyrne.github.io/notes/fastmath/
// The problem is, compilers define function attributes in different ways.
// -ffast-math (and included by -Ofast) in GCC and Clang
// -fp-model=fast (the default) in ICC
// /fp:fast in MSVC
#if defined(__GNUC__) && !defined(__clang__)
// The old syntax in GCC is: __attribute__((optimize("-ffast-math")))
#define FAST_MATH [[gnu::optimize("-ffast-math")]]
#elif defined(__clang__)
#define FAST_MATH __attribute__((target("-ffast-math")))
#else
#define FAST_MATH
#endif
FAST_MATH static void f64_sin_maclaurin_with_fast_math(bm::State &state) {
double argument = std::rand(), result = 0;
for (auto _ : state) {
argument += 1.0;
result = (argument) - (argument * argument * argument) / 6.0 +
(argument * argument * argument * argument * argument) / 120.0;
bm::DoNotOptimize(result);
}
}
// Floating point math is not associative!
// So it's not reorderable! And it requires extra annotation!
// Use only when you work with low-mid precision numbers and values of similar magnitude.
// As always with IEEE-754, you have same number of elements in [-inf,-1], [-1,0], [0,1], [1,+inf].
// https://en.wikipedia.org/wiki/Double-precision_floating-point_format
BENCHMARK(f64_sin_maclaurin_with_fast_math);
// ------------------------------------
// ## Lets look at Integer Division
// ### If floating point arithmetic can be fast, what about integer division?
// ------------------------------------
static void i64_division(bm::State &state) {
int64_t a = std::rand(), b = std::rand(), c = 0;
for (auto _ : state)
bm::DoNotOptimize(c = (++a) / (++b));
}
// If we take 32-bit integers - their division can be performed via `double`
// without loss of accuracy. Result: 7ns, or 15x more expensive then addition.
BENCHMARK(i64_division);
static void i64_division_by_const(bm::State &state) {
int64_t b = 2147483647;
int64_t a = std::rand(), c = 0;
for (auto _ : state)
bm::DoNotOptimize(c = (++a) / *std::launder(&b));
}
// Let's fix a constant, but `std::launder` it a bit.
// So it looks like a generic pointer and not explicitly
// a constant as a developer might have seen.
// Result: more or less the same as before.
BENCHMARK(i64_division_by_const);
static void i64_division_by_constexpr(bm::State &state) {
constexpr int64_t b = 2147483647;
int64_t a = std::rand(), c = 0;
for (auto _ : state)
bm::DoNotOptimize(c = (++a) / b);
}
// But once we mark it as a `constexpr`, the compiler will replace
// heavy divisions with a combination of simpler shifts and multiplications.
// https://www.sciencedirect.com/science/article/pii/S2405844021015450
BENCHMARK(i64_division_by_constexpr);
// ------------------------------------
// ## Where else those tricks are needed
// ------------------------------------
#if defined(__GNUC__) && !defined(__clang__)
[[gnu::target("default")]] static void u64_population_count(bm::State &state) {
auto a = static_cast<uint64_t>(std::rand());
for (auto _ : state)
bm::DoNotOptimize(__builtin_popcount(++a));
}
BENCHMARK(u64_population_count);
[[gnu::target("popcnt")]] static void u64_population_count_x86(bm::State &state) {
auto a = static_cast<uint64_t>(std::rand());
for (auto _ : state)
bm::DoNotOptimize(__builtin_popcount(++a));
}
BENCHMARK(u64_population_count_x86);
#endif
// ------------------------------------
// ## Data Alignment
// ------------------------------------
constexpr size_t f32s_in_cache_line_k = 64 / sizeof(float);
constexpr size_t f32s_in_cache_line_half_k = f32s_in_cache_line_k / 2;
struct alignas(64) f32_array_t {
float raw[f32s_in_cache_line_k * 2];
};
static void f32_pairwise_accumulation(bm::State &state) {
f32_array_t a, b, c;
for (auto _ : state)
for (size_t i = f32s_in_cache_line_half_k; i != f32s_in_cache_line_half_k * 3; ++i)
bm::DoNotOptimize(c.raw[i] = a.raw[i] + b.raw[i]);
}
static void f32_pairwise_accumulation_aligned(bm::State &state) {
f32_array_t a, b, c;
for (auto _ : state)
for (size_t i = 0; i != f32s_in_cache_line_half_k; ++i)
bm::DoNotOptimize(c.raw[i] = a.raw[i] + b.raw[i]);
}
// Split load occurs in the first case and doesn't in the second.
// We do the same number of arithmetical operations, but:
// - first takes 8 ns
// - second takes 4 ns
BENCHMARK(f32_pairwise_accumulation)->MinTime(10);
BENCHMARK(f32_pairwise_accumulation_aligned)->MinTime(10);
// ------------------------------------
// ## Cost of Control Flow
// ------------------------------------
// The `if` statement and seemingly innocent ternary operator (condition ? a : b)
// can have a high for performance. It's especially noticeable, when conditional
// execution is happening at the scale of single bytes, like in text processing,
// parsing, search, compression, encoding, and so on.
//
// The CPU has a branch-predictor which is one of the most complex parts of the silicon.
// It memorizes the most common `if` statements, to allow "speculative execution".
// In other words, start processing the task (i + 1), before finishing the task (i).
//
// Those branch predictors are very powerful, and if you have a single `if` statement
// on your hot-path, it's not a big deal. But most programs are almost entirely built
// on `if` statements. On most modern CPUs up to 4096 branches will be memorized, but
// anything that goes beyond that, would work slower - 2.9 ns vs 0.7 ns for the following snippet.
static void cost_of_branching_for_different_depth(bm::State &state) {
auto count = static_cast<size_t>(state.range(0));
std::vector<int32_t> random_values(count);
std::generate_n(random_values.begin(), random_values.size(), &std::rand);
int32_t variable = 0;
size_t iteration = 0;
for (auto _ : state) {
int32_t random = random_values[(++iteration) & (count - 1)];
bm::DoNotOptimize(variable = (random & 1) ? (variable + random) : (variable * random));
}
}
BENCHMARK(cost_of_branching_for_different_depth)->RangeMultiplier(4)->Range(256, 32 * 1024);
// We don't have to generate a large array of random numbers to showcase the cost of branching.
// Simple one-line statement can be enough to cause the same 2.2 ns slowdown.
static void cost_of_branching_without_random_arrays(bm::State &state) {
int32_t a = std::rand(), b = std::rand(), c = 0;
for (auto _ : state)
bm::DoNotOptimize(c = (c & 1) ? ((a--) + (b)) : ((++b) - (a)));
}
BENCHMARK(cost_of_branching_without_random_arrays);
// Google Benchmark also provides it's own Control Flow primitives, to control timing.
// Those `PauseTiming` and `ResumeTiming` functions, however, are not free.
// In current implementation, they can easily take ~127 ns, or around 300 CPU cycles.
static void cost_of_pausing(bm::State &state) {
int32_t a = std::rand(), c = 0;
for (auto _ : state) {
state.PauseTiming();
++a;
state.ResumeTiming();
bm::DoNotOptimize(c += a);
}
}
BENCHMARK(cost_of_pausing);
// ------------------------------------
// ## Bulk Operations
// ------------------------------------
static void sorting(bm::State &state) {
auto count = static_cast<size_t>(state.range(0));
auto include_preprocessing = static_cast<bool>(state.range(1));
std::vector<int32_t> array(count);
std::iota(array.begin(), array.end(), 1);
for (auto _ : state) {
if (!include_preprocessing)
state.PauseTiming();
// Reverse order is the most classical worst case, but not the only one.
std::reverse(array.begin(), array.end());
if (!include_preprocessing)
state.ResumeTiming();
std::sort(array.begin(), array.end());
bm::DoNotOptimize(array.size());
}
}
// `std::sort` will invoke a modification of Quick-Sort.
// It's worst case complexity is ~O(N^2), but what the hell are those numbers??
BENCHMARK(sorting)->Args({3, false})->Args({3, true});
BENCHMARK(sorting)->Args({4, false})->Args({4, true});
template <bool include_preprocessing_k> static void sorting_template(bm::State &state) {
auto count = static_cast<size_t>(state.range(0));
std::vector<int32_t> array(count);
std::iota(array.begin(), array.end(), 1);
for (auto _ : state) {
if constexpr (!include_preprocessing_k)
state.PauseTiming();
std::reverse(array.begin(), array.end());
if constexpr (!include_preprocessing_k)
state.ResumeTiming();
std::sort(array.begin(), array.end());
bm::DoNotOptimize(array.size());
}
}
// Now, our control-flow will not affect the measurements!
// "Don't pay what you don't use" becomes: "Don't pay for what you can avoid!"
BENCHMARK_TEMPLATE(sorting_template, false)->Arg(3);
BENCHMARK_TEMPLATE(sorting_template, true)->Arg(3);
BENCHMARK_TEMPLATE(sorting_template, false)->Arg(4);
BENCHMARK_TEMPLATE(sorting_template, true)->Arg(4);
template <typename element_at> //
struct quick_sort_partition_gt {
using element_t = element_at;
std::int32_t operator()(element_t *arr, std::int32_t low, std::int32_t high) {
element_t pivot = arr[high];
std::int32_t i = low - 1;
for (std::int32_t j = low; j <= high - 1; j++) {
if (arr[j] >= pivot)
continue;
i++;
std::swap(arr[i], arr[j]);
}
std::swap(arr[i + 1], arr[high]);
return i + 1;
}
};
template <typename element_at> //
struct quick_sort_recursive_gt {
using element_t = element_at;
using quick_sort_partition_t = quick_sort_partition_gt<element_t>;
using quick_sort_recursive_t = quick_sort_recursive_gt<element_t>;
void operator()(element_t *arr, std::int32_t low, std::int32_t high) {
if (low >= high)
return;
auto pivot = quick_sort_partition_t{}(arr, low, high);
quick_sort_recursive_t{}(arr, low, pivot - 1);
quick_sort_recursive_t{}(arr, pivot + 1, high);
}
};
template <typename element_at> //
struct quick_sort_iterative_gt {
using element_t = element_at;
using quick_sort_partition_t = quick_sort_partition_gt<element_t>;
std::vector<std::int32_t> stack;
void operator()(element_t *arr, std::int32_t low, std::int32_t high) {
stack.resize((high - low + 1) * 2);
std::int32_t top = -1;
stack[++top] = low;
stack[++top] = high;
while (top >= 0) {
high = stack[top--];
low = stack[top--];
auto pivot = quick_sort_partition_t{}(arr, low, high);
// If there are elements on left side of pivot,
// then push left side to stack
if (low < pivot - 1) {
stack[++top] = low;
stack[++top] = pivot - 1;
}
// If there are elements on right side of pivot,
// then push right side to stack
if (pivot + 1 < high) {
stack[++top] = pivot + 1;
stack[++top] = high;
}
}
}
};
template <typename sorter_at, std::int32_t length_ak> //
static void cost_of_recursion(bm::State &state) {
using element_t = typename sorter_at::element_t;
sorter_at sorter;
std::vector<element_t> arr(static_cast<std::size_t>(length_ak));
for (auto _ : state) {
for (std::int32_t i = 0; i != length_ak; ++i)
arr[i] = length_ak - i;
sorter(arr.data(), 0, length_ak - 1);
}
}
BENCHMARK_TEMPLATE(cost_of_recursion, quick_sort_recursive_gt<std::int32_t>, 1024);
BENCHMARK_TEMPLATE(cost_of_recursion, quick_sort_iterative_gt<std::int32_t>, 1024);
BENCHMARK_TEMPLATE(cost_of_recursion, quick_sort_recursive_gt<std::int32_t>, 1024 * 1024);
BENCHMARK_TEMPLATE(cost_of_recursion, quick_sort_iterative_gt<std::int32_t>, 1024 * 1024);
BENCHMARK_TEMPLATE(cost_of_recursion, quick_sort_recursive_gt<std::int32_t>, 1024 * 1024 * 1024);
BENCHMARK_TEMPLATE(cost_of_recursion, quick_sort_iterative_gt<std::int32_t>, 1024 * 1024 * 1024);
// ------------------------------------
// ## Now that we know how fast algorithm works - lets scale it!
// ### And learn the rest of relevant functionality in the process
// ------------------------------------
template <typename execution_policy_t> static void super_sort(bm::State &state, execution_policy_t &&policy) {
auto count = static_cast<size_t>(state.range(0));
std::vector<int32_t> array(count);
std::iota(array.begin(), array.end(), 1);
for (auto _ : state) {
std::reverse(policy, array.begin(), array.end());
std::sort(policy, array.begin(), array.end());
bm::DoNotOptimize(array.size());
}
state.SetComplexityN(count);
state.SetItemsProcessed(count * state.iterations());
state.SetBytesProcessed(count * state.iterations() * sizeof(int32_t));
// Feel free to report something else:
// state.counters["temperature_on_mars"] = bm::Counter(-95.4);
}
#ifdef __cpp_lib_parallel_algorithm
// Let's try running on 1M to 4B entries.
// This means input sizes between 4 MB and 16 GB respectively.
BENCHMARK_CAPTURE(super_sort, seq, std::execution::seq)
->RangeMultiplier(8)
->Range(1l << 20, 1l << 32)
->MinTime(10)
->Complexity(bm::oNLogN);
BENCHMARK_CAPTURE(super_sort, par_unseq, std::execution::par_unseq)
->RangeMultiplier(8)
->Range(1l << 20, 1l << 32)
->MinTime(10)
->Complexity(bm::oNLogN);
// Without `UseRealTime()`, CPU time is used by default.
// Difference example: when you sleep your process it is no longer accumulating CPU time.
BENCHMARK_CAPTURE(super_sort, par_unseq, std::execution::par_unseq)
->RangeMultiplier(8)
->Range(1l << 20, 1l << 32)
->MinTime(10)
->Complexity(bm::oNLogN)
->UseRealTime();
#endif
// ------------------------------------
// ## Practical Investigation Example
// ------------------------------------
BENCHMARK_MAIN();