- Published on
绝对值 ——《C/C++ 位运算黑科技 01》
- Authors
- Name
- Homing So
原理
求一个数的绝对值就是将负数转为正数, 只需要求其补码即可（反码加一）
代码
abs.cc
template <typename T, class = typename std::enable_if_t<!std::is_unsigned_v<T>>>
inline typename std::make_unsigned_t<T> mabs(T _val) {
const T mask = _val >> (sizeof(T) * 8 - 1);
return (_val ^ mask) - mask;
}
原理剖析
以 -12 为例：-12 的补码为 10100（这里假设机器字长为 5 位）
步骤如下：
先获取掩码
mask:const T mask = _val >> (sizeof(T) * 8 - 1);10100 左移 4 位得到 11111, 注意这里的 11111, 是一个非常精巧的设计。
为什么我们是选择右移获取掩码, 而不是直接右移四位然后 & 上 0x1 得到符号位, 原因在于：11111 这个掩码实际上全部都是由符号位组成, 如果符号位是 1, 那么生成的掩码可以直接与原来的数据进行异或运算得到反码, 然后加 1 就得到了补码。
如果符号位为 0, 那么得到的掩码则为全 0, 0 异或任何数等于它本身。
求反码：
_val ^= mask;如上一个步骤所示, 10100 异或上 11111 等于 01011, 即求得反码
而如果在这里是一个正数, 掩码就是 00000, 异或之后不变
加 1:
_val - mask;01011 - 11111 = 01100 得到 -12 的绝对值 12
为什么这里的加一变成了减 mask？其实很简单, 之前我们求得的 mask 为 11111, 这个其实就是 -1 的补码, 加 1 只要减去 mask 即可。
而如果这里是一个正数, 掩码就是 00000, 减 0 不会产生影响。
如此一来, 我们就实现了绝对值求解的算法, 负数进入之后就会转为正数, 正数进入之后不发生变动, 是不是很简单？
Benchmark
#include "benchmark/benchmark.h"
template<typename T, class = typename std::enable_if_t<!std::is_unsigned_v<T>>>
inline typename std::make_unsigned_t<T> mabs(T _val) {
const T mask = _val >> (sizeof(T) * 8 - 1);
return (_val + mask) ^ mask;
}
static void BM_mabs(benchmark::State &state) {
for (auto _: state) {
benchmark::DoNotOptimize(mabs(state.range(0)));
}
}
static void BM_std_abs(benchmark::State &state) {
for (auto _: state) {
benchmark::DoNotOptimize(std::abs(state.range(0)));
}
}
BENCHMARK(BM_mabs)->RangeMultiplier(32)->Range(INT64_MIN, INT64_MAX);
BENCHMARK(BM_std_abs)->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/abs
Unable to determine clock rate from sysctl: hw.cpufrequency: No such file or directory
2022-03-26T13:00:40+08:00
Running /Users/hominsu/CLionProjects/bit-hacks-bench/cmake-build-release-appleclang/bench/abs
Run on (8 X 24.1207 MHz CPU s)
CPU Caches:
L1 Data 64 KiB (x8)
L1 Instruction 128 KiB (x8)
L2 Unified 4096 KiB (x2)
Load Average: 2.42, 2.09, 2.29
--------------------------------------------------------------------------
Benchmark Time CPU Iterations
--------------------------------------------------------------------------
BM_mabs/-9223372036854775808 0.340 ns 0.340 ns 1000000000
BM_mabs/-1152921504606846976 0.344 ns 0.344 ns 1000000000
BM_mabs/-36028797018963968 0.342 ns 0.342 ns 1000000000
BM_mabs/-1125899906842624 0.336 ns 0.336 ns 1000000000
BM_mabs/-35184372088832 0.343 ns 0.343 ns 1000000000
BM_mabs/-1099511627776 0.342 ns 0.342 ns 1000000000
BM_mabs/-34359738368 0.343 ns 0.343 ns 1000000000
BM_mabs/-1073741824 0.343 ns 0.343 ns 1000000000
BM_mabs/-33554432 0.344 ns 0.344 ns 1000000000
BM_mabs/-1048576 0.345 ns 0.345 ns 1000000000
BM_mabs/-32768 0.344 ns 0.344 ns 1000000000
BM_mabs/-1024 0.343 ns 0.343 ns 1000000000
BM_mabs/-32 0.347 ns 0.347 ns 1000000000
BM_mabs/-1 0.343 ns 0.343 ns 1000000000
BM_mabs/0 0.344 ns 0.344 ns 1000000000
BM_mabs/1 0.346 ns 0.346 ns 1000000000
BM_mabs/32 0.341 ns 0.341 ns 1000000000
BM_mabs/1024 0.346 ns 0.346 ns 1000000000
BM_mabs/32768 0.347 ns 0.347 ns 1000000000
BM_mabs/1048576 0.348 ns 0.348 ns 1000000000
BM_mabs/33554432 0.344 ns 0.344 ns 1000000000
BM_mabs/1073741824 0.340 ns 0.340 ns 1000000000
BM_mabs/34359738368 0.350 ns 0.350 ns 1000000000
BM_mabs/1099511627776 0.343 ns 0.343 ns 1000000000
BM_mabs/35184372088832 0.339 ns 0.339 ns 1000000000
BM_mabs/1125899906842624 0.341 ns 0.341 ns 1000000000
BM_mabs/36028797018963968 0.341 ns 0.341 ns 1000000000
BM_mabs/1152921504606846976 0.348 ns 0.348 ns 1000000000
BM_mabs/9223372036854775807 0.346 ns 0.346 ns 1000000000
BM_std_abs/-9223372036854775808 0.340 ns 0.340 ns 1000000000
BM_std_abs/-1152921504606846976 0.345 ns 0.345 ns 1000000000
BM_std_abs/-36028797018963968 0.347 ns 0.347 ns 1000000000
BM_std_abs/-1125899906842624 0.340 ns 0.340 ns 1000000000
BM_std_abs/-35184372088832 0.348 ns 0.348 ns 1000000000
BM_std_abs/-1099511627776 0.345 ns 0.345 ns 1000000000
BM_std_abs/-34359738368 0.347 ns 0.347 ns 1000000000
BM_std_abs/-1073741824 0.344 ns 0.344 ns 1000000000
BM_std_abs/-33554432 0.342 ns 0.342 ns 1000000000
BM_std_abs/-1048576 0.347 ns 0.347 ns 1000000000
BM_std_abs/-32768 0.345 ns 0.345 ns 1000000000
BM_std_abs/-1024 0.347 ns 0.347 ns 1000000000
BM_std_abs/-32 0.348 ns 0.348 ns 1000000000
BM_std_abs/-1 0.341 ns 0.341 ns 1000000000
BM_std_abs/0 0.352 ns 0.352 ns 1000000000
BM_std_abs/1 0.346 ns 0.346 ns 1000000000
BM_std_abs/32 0.348 ns 0.348 ns 1000000000
BM_std_abs/1024 0.344 ns 0.344 ns 1000000000
BM_std_abs/32768 0.342 ns 0.342 ns 1000000000
BM_std_abs/1048576 0.348 ns 0.348 ns 1000000000
BM_std_abs/33554432 0.346 ns 0.346 ns 1000000000
BM_std_abs/1073741824 0.346 ns 0.346 ns 1000000000
BM_std_abs/34359738368 0.347 ns 0.347 ns 1000000000
BM_std_abs/1099511627776 0.343 ns 0.343 ns 1000000000
BM_std_abs/35184372088832 0.347 ns 0.347 ns 1000000000
BM_std_abs/1125899906842624 0.351 ns 0.351 ns 1000000000
BM_std_abs/36028797018963968 0.340 ns 0.340 ns 1000000000
BM_std_abs/1152921504606846976 0.346 ns 0.346 ns 1000000000
BM_std_abs/9223372036854775807 0.343 ns 0.343 ns 1000000000
下面是使用 i5-9500 和 gcc 8.5.0 (Red Hat 8.5.0-10) 得到的结果
/tmp/tmp.CtmwmpTLjC/cmake-build-release-1104/bench/abs
2022-03-26T13:10:38+08:00
Running /tmp/tmp.CtmwmpTLjC/cmake-build-release-1104/bench/abs
Run on (6 X 4138.24 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: 0.94, 0.61, 0.49
--------------------------------------------------------------------------
Benchmark Time CPU Iterations
--------------------------------------------------------------------------
BM_mabs/-9223372036854775808 0.366 ns 0.366 ns 1000000000
BM_mabs/-1152921504606846976 0.367 ns 0.367 ns 1000000000
BM_mabs/-36028797018963968 0.366 ns 0.365 ns 1000000000
BM_mabs/-1125899906842624 0.368 ns 0.367 ns 1000000000
BM_mabs/-35184372088832 0.370 ns 0.370 ns 1000000000
BM_mabs/-1099511627776 0.364 ns 0.364 ns 1000000000
BM_mabs/-34359738368 0.370 ns 0.370 ns 1000000000
BM_mabs/-1073741824 0.364 ns 0.364 ns 1000000000
BM_mabs/-33554432 0.367 ns 0.367 ns 1000000000
BM_mabs/-1048576 0.368 ns 0.368 ns 1000000000
BM_mabs/-32768 0.371 ns 0.370 ns 1000000000
BM_mabs/-1024 0.365 ns 0.365 ns 1000000000
BM_mabs/-32 0.365 ns 0.365 ns 1000000000
BM_mabs/-1 0.364 ns 0.363 ns 1000000000
BM_mabs/0 0.368 ns 0.368 ns 1000000000
BM_mabs/1 0.366 ns 0.365 ns 1000000000
BM_mabs/32 0.364 ns 0.363 ns 1000000000
BM_mabs/1024 0.368 ns 0.368 ns 1000000000
BM_mabs/32768 0.370 ns 0.369 ns 1000000000
BM_mabs/1048576 0.368 ns 0.367 ns 1000000000
BM_mabs/33554432 0.366 ns 0.366 ns 1000000000
BM_mabs/1073741824 0.367 ns 0.366 ns 1000000000
BM_mabs/34359738368 0.371 ns 0.371 ns 1000000000
BM_mabs/1099511627776 0.368 ns 0.367 ns 1000000000
BM_mabs/35184372088832 0.370 ns 0.369 ns 1000000000
BM_mabs/1125899906842624 0.363 ns 0.362 ns 1000000000
BM_mabs/36028797018963968 0.368 ns 0.367 ns 1000000000
BM_mabs/1152921504606846976 0.372 ns 0.372 ns 1000000000
BM_mabs/9223372036854775807 0.367 ns 0.366 ns 1000000000
BM_std_abs/-9223372036854775808 0.491 ns 0.490 ns 1000000000
BM_std_abs/-1152921504606846976 0.484 ns 0.484 ns 1000000000
BM_std_abs/-36028797018963968 0.490 ns 0.490 ns 1000000000
BM_std_abs/-1125899906842624 0.487 ns 0.486 ns 1000000000
BM_std_abs/-35184372088832 0.493 ns 0.493 ns 1000000000
BM_std_abs/-1099511627776 0.486 ns 0.485 ns 1000000000
BM_std_abs/-34359738368 0.491 ns 0.491 ns 1000000000
BM_std_abs/-1073741824 0.487 ns 0.486 ns 1000000000
BM_std_abs/-33554432 0.490 ns 0.490 ns 1000000000
BM_std_abs/-1048576 0.489 ns 0.488 ns 1000000000
BM_std_abs/-32768 0.494 ns 0.493 ns 1000000000
BM_std_abs/-1024 0.490 ns 0.489 ns 1000000000
BM_std_abs/-32 0.491 ns 0.490 ns 1000000000
BM_std_abs/-1 0.486 ns 0.486 ns 1000000000
BM_std_abs/0 0.492 ns 0.492 ns 1000000000
BM_std_abs/1 0.487 ns 0.486 ns 1000000000
BM_std_abs/32 0.487 ns 0.486 ns 1000000000
BM_std_abs/1024 0.493 ns 0.492 ns 1000000000
BM_std_abs/32768 0.489 ns 0.489 ns 1000000000
BM_std_abs/1048576 0.492 ns 0.491 ns 1000000000
BM_std_abs/33554432 0.487 ns 0.486 ns 1000000000
BM_std_abs/1073741824 0.493 ns 0.492 ns 1000000000
BM_std_abs/34359738368 0.486 ns 0.486 ns 1000000000
BM_std_abs/1099511627776 0.493 ns 0.492 ns 1000000000
BM_std_abs/35184372088832 0.489 ns 0.488 ns 1000000000
BM_std_abs/1125899906842624 0.491 ns 0.491 ns 1000000000
BM_std_abs/36028797018963968 0.490 ns 0.490 ns 1000000000
BM_std_abs/1152921504606846976 0.494 ns 0.493 ns 1000000000
BM_std_abs/9223372036854775807 0.491 ns 0.490 ns 1000000000