- Published on
2 的幂次方 ——《C/C++ 位运算黑科技 02》
- Authors
- Name
- Homing So
原理
现在我们使用的二进制码表示都很简单：1、2、4、8、16 ······
仔细观察就可以发现：在一串二进制数中, 如果只出现一个 1, 它就是 2 的幂次方
代码
power2_1.cc
template<typename T, class = std::enable_if_t<std::is_integral_v<T>>>
inline bool power2_1(T v) {
return v && !(v & (v - 1));
}
或者
power2_2.cc
template<typename T, class = std::enable_if_t<std::is_integral_v<T>>>
inline bool power2_2(T v) {
return v && ((v & -v) == v);
}
原理剖析
方法一
因为 2 的幂次方只有一个 1, 我们只需要去掉最后一个 1 后判断是否等于 0 即可。
v & (v - 1);
上面的代码就能够去掉最低位的 1, 原理也很简单：减 1 会使最低位 1 变为 0, 并在更低位产生 1, 其他位不变。而与上自身之后, 这些 1 和 之前的最低位 1 就会被清除掉。
但是 0 是一个特例, 因此我们要把它排除掉：
v && !(v & (v - 1));
方法二
法二和法一类似, 首先我们需要知道 v & -v 有什么用, v & -v 其实就是获取一个二进制数的从低位到高位的第一个 1 的位索引。以 111 为例, 111 的补码为 001, 111 & 001 = 001；以 110 为例, 110 的补码为 010, 110 & 010 = 010；
显而易见, 如果一个数的位索引等于它本身, 那么它就是 2 的幂次方。
Benchmark
#include "benchmark/benchmark.h"
template<typename T, class = std::enable_if_t<std::is_integral_v<T>>>
inline bool power2_1(T v) {
return v && !(v & (v - 1));
}
template<typename T, class = std::enable_if_t<std::is_integral_v<T>>>
inline bool power2_2(T v) {
return v && ((v & -v) == v);
}
static void BM_power2_1(benchmark::State &state) {
for (auto _: state) {
benchmark::DoNotOptimize(power2_1(state.range(0)));
}
}
static void BM_power2_2(benchmark::State &state) {
for (auto _: state) {
benchmark::DoNotOptimize(power2_2(state.range(0)));
}
}
BENCHMARK(BM_power2_1)->RangeMultiplier(32)->Range(INT64_MIN, INT64_MAX);
BENCHMARK(BM_power2_2)->RangeMultiplier(32)->Range(INT64_MIN, INT64_MAX);
BENCHMARK_MAIN();
下面是使用 MacBook Air (M1, 2020) 和 Apple clang 13.1.6 得到的结果
/Users/hominsu/CLionProjects/bit-hacks-bench/cmake-build-release-appleclang/bench/power2
Unable to determine clock rate from sysctl: hw.cpufrequency: No such file or directory
2022-03-26T13:24:41+08:00
Running /Users/hominsu/CLionProjects/bit-hacks-bench/cmake-build-release-appleclang/bench/power2
Run on (8 X 24.0935 MHz CPU s)
CPU Caches:
L1 Data 64 KiB (x8)
L1 Instruction 128 KiB (x8)
L2 Unified 4096 KiB (x2)
Load Average: 1.38, 1.45, 1.71
---------------------------------------------------------------------------
Benchmark Time CPU Iterations
---------------------------------------------------------------------------
BM_power2_1/-9223372036854775808 0.443 ns 0.443 ns 1000000000
BM_power2_1/-1152921504606846976 0.443 ns 0.443 ns 1000000000
BM_power2_1/-36028797018963968 0.443 ns 0.443 ns 1000000000
BM_power2_1/-1125899906842624 0.443 ns 0.443 ns 1000000000
BM_power2_1/-35184372088832 0.443 ns 0.443 ns 1000000000
BM_power2_1/-1099511627776 0.444 ns 0.444 ns 1000000000
BM_power2_1/-34359738368 0.443 ns 0.443 ns 1000000000
BM_power2_1/-1073741824 0.444 ns 0.444 ns 1000000000
BM_power2_1/-33554432 0.444 ns 0.444 ns 1000000000
BM_power2_1/-1048576 0.444 ns 0.444 ns 1000000000
BM_power2_1/-32768 0.443 ns 0.443 ns 1000000000
BM_power2_1/-1024 0.443 ns 0.443 ns 1000000000
BM_power2_1/-32 0.444 ns 0.444 ns 1000000000
BM_power2_1/-1 0.444 ns 0.444 ns 1000000000
BM_power2_1/0 0.314 ns 0.314 ns 1000000000
BM_power2_1/1 0.444 ns 0.443 ns 1000000000
BM_power2_1/32 0.444 ns 0.444 ns 1000000000
BM_power2_1/1024 0.443 ns 0.443 ns 1000000000
BM_power2_1/32768 0.443 ns 0.443 ns 1000000000
BM_power2_1/1048576 0.443 ns 0.443 ns 1000000000
BM_power2_1/33554432 0.446 ns 0.446 ns 1000000000
BM_power2_1/1073741824 0.443 ns 0.443 ns 1000000000
BM_power2_1/34359738368 0.443 ns 0.443 ns 1000000000
BM_power2_1/1099511627776 0.444 ns 0.444 ns 1000000000
BM_power2_1/35184372088832 0.443 ns 0.443 ns 1000000000
BM_power2_1/1125899906842624 0.444 ns 0.444 ns 1000000000
BM_power2_1/36028797018963968 0.443 ns 0.443 ns 1000000000
BM_power2_1/1152921504606846976 0.443 ns 0.443 ns 1000000000
BM_power2_1/9223372036854775807 0.444 ns 0.444 ns 1000000000
BM_power2_2/-9223372036854775808 0.443 ns 0.443 ns 1000000000
BM_power2_2/-1152921504606846976 0.443 ns 0.443 ns 1000000000
BM_power2_2/-36028797018963968 0.444 ns 0.444 ns 1000000000
BM_power2_2/-1125899906842624 0.444 ns 0.444 ns 1000000000
BM_power2_2/-35184372088832 0.443 ns 0.443 ns 1000000000
BM_power2_2/-1099511627776 0.443 ns 0.443 ns 1000000000
BM_power2_2/-34359738368 0.444 ns 0.444 ns 1000000000
BM_power2_2/-1073741824 0.444 ns 0.444 ns 1000000000
BM_power2_2/-33554432 0.443 ns 0.443 ns 1000000000
BM_power2_2/-1048576 0.444 ns 0.444 ns 1000000000
BM_power2_2/-32768 0.444 ns 0.444 ns 1000000000
BM_power2_2/-1024 0.445 ns 0.445 ns 1000000000
BM_power2_2/-32 0.444 ns 0.444 ns 1000000000
BM_power2_2/-1 0.443 ns 0.443 ns 1000000000
BM_power2_2/0 0.313 ns 0.313 ns 1000000000
BM_power2_2/1 0.443 ns 0.443 ns 1000000000
BM_power2_2/32 0.444 ns 0.444 ns 1000000000
BM_power2_2/1024 0.444 ns 0.443 ns 1000000000
BM_power2_2/32768 0.443 ns 0.443 ns 1000000000
BM_power2_2/1048576 0.443 ns 0.443 ns 1000000000
BM_power2_2/33554432 0.444 ns 0.444 ns 1000000000
BM_power2_2/1073741824 0.443 ns 0.443 ns 1000000000
BM_power2_2/34359738368 0.443 ns 0.443 ns 1000000000
BM_power2_2/1099511627776 0.443 ns 0.443 ns 1000000000
BM_power2_2/35184372088832 0.443 ns 0.443 ns 1000000000
BM_power2_2/1125899906842624 0.444 ns 0.444 ns 1000000000
BM_power2_2/36028797018963968 0.445 ns 0.445 ns 1000000000
BM_power2_2/1152921504606846976 0.444 ns 0.444 ns 1000000000
BM_power2_2/9223372036854775807 0.450 ns 0.449 ns 1000000000
下面是使用 i5-9500 和 gcc 8.5.0 (Red Hat 8.5.0-10) 在 CentOS-8-Stream 下得到的结果
/tmp/tmp.CtmwmpTLjC/cmake-build-release-1104/bench/power2
2022-03-26T13:30:11+08:00
Running /tmp/tmp.CtmwmpTLjC/cmake-build-release-1104/bench/power2
Run on (6 X 4099.87 MHz CPU s)
CPU Caches:
L1 Data 32 KiB (x6)
L1 Instruction 32 KiB (x6)
L2 Unified 256 KiB (x6)
L3 Unified 9216 KiB (x1)
Load Average: 3.17, 1.60, 1.17
---------------------------------------------------------------------------
Benchmark Time CPU Iterations
---------------------------------------------------------------------------
BM_power2_1/-9223372036854775808 0.487 ns 0.487 ns 1000000000
BM_power2_1/-1152921504606846976 0.496 ns 0.495 ns 1000000000
BM_power2_1/-36028797018963968 0.490 ns 0.489 ns 1000000000
BM_power2_1/-1125899906842624 0.489 ns 0.489 ns 1000000000
BM_power2_1/-35184372088832 0.485 ns 0.484 ns 1000000000
BM_power2_1/-1099511627776 0.493 ns 0.492 ns 1000000000
BM_power2_1/-34359738368 0.488 ns 0.488 ns 1000000000
BM_power2_1/-1073741824 0.491 ns 0.490 ns 1000000000
BM_power2_1/-33554432 0.489 ns 0.488 ns 1000000000
BM_power2_1/-1048576 0.496 ns 0.495 ns 1000000000
BM_power2_1/-32768 0.491 ns 0.490 ns 1000000000
BM_power2_1/-1024 0.491 ns 0.490 ns 1000000000
BM_power2_1/-32 0.484 ns 0.484 ns 1000000000
BM_power2_1/-1 0.495 ns 0.494 ns 1000000000
BM_power2_1/0 0.886 ns 0.885 ns 788464796
BM_power2_1/1 0.486 ns 0.486 ns 1000000000
BM_power2_1/32 0.491 ns 0.490 ns 1000000000
BM_power2_1/1024 0.489 ns 0.489 ns 1000000000
BM_power2_1/32768 0.491 ns 0.491 ns 1000000000
BM_power2_1/1048576 0.491 ns 0.490 ns 1000000000
BM_power2_1/33554432 0.494 ns 0.493 ns 1000000000
BM_power2_1/1073741824 0.484 ns 0.484 ns 1000000000
BM_power2_1/34359738368 0.492 ns 0.491 ns 1000000000
BM_power2_1/1099511627776 0.491 ns 0.490 ns 1000000000
BM_power2_1/35184372088832 0.495 ns 0.495 ns 1000000000
BM_power2_1/1125899906842624 0.484 ns 0.483 ns 1000000000
BM_power2_1/36028797018963968 0.493 ns 0.492 ns 1000000000
BM_power2_1/1152921504606846976 0.491 ns 0.490 ns 1000000000
BM_power2_1/9223372036854775807 0.496 ns 0.495 ns 1000000000
BM_power2_2/-9223372036854775808 0.552 ns 0.551 ns 1000000000
BM_power2_2/-1152921504606846976 0.552 ns 0.552 ns 1000000000
BM_power2_2/-36028797018963968 0.561 ns 0.560 ns 1000000000
BM_power2_2/-1125899906842624 0.546 ns 0.546 ns 1000000000
BM_power2_2/-35184372088832 0.551 ns 0.550 ns 1000000000
BM_power2_2/-1099511627776 0.553 ns 0.553 ns 1000000000
BM_power2_2/-34359738368 0.552 ns 0.551 ns 1000000000
BM_power2_2/-1073741824 0.552 ns 0.552 ns 1000000000
BM_power2_2/-33554432 0.553 ns 0.552 ns 1000000000
BM_power2_2/-1048576 0.553 ns 0.552 ns 1000000000
BM_power2_2/-32768 0.545 ns 0.545 ns 1000000000
BM_power2_2/-1024 0.554 ns 0.553 ns 1000000000
BM_power2_2/-32 0.548 ns 0.547 ns 1000000000
BM_power2_2/-1 0.546 ns 0.546 ns 1000000000
BM_power2_2/0 0.493 ns 0.493 ns 1000000000
BM_power2_2/1 0.553 ns 0.553 ns 1000000000
BM_power2_2/32 0.554 ns 0.553 ns 1000000000
BM_power2_2/1024 0.545 ns 0.544 ns 1000000000
BM_power2_2/32768 0.555 ns 0.555 ns 1000000000
BM_power2_2/1048576 0.550 ns 0.549 ns 1000000000
BM_power2_2/33554432 0.550 ns 0.549 ns 1000000000
BM_power2_2/1073741824 0.555 ns 0.554 ns 1000000000
BM_power2_2/34359738368 0.551 ns 0.550 ns 1000000000
BM_power2_2/1099511627776 0.553 ns 0.553 ns 1000000000
BM_power2_2/35184372088832 0.553 ns 0.552 ns 1000000000
BM_power2_2/1125899906842624 0.552 ns 0.552 ns 1000000000
BM_power2_2/36028797018963968 0.552 ns 0.552 ns 1000000000
BM_power2_2/1152921504606846976 0.551 ns 0.551 ns 1000000000
BM_power2_2/9223372036854775807 0.554 ns 0.553 ns 1000000000