从a previous question中考虑这些定义:
type Algebra f a = f a -> a
cata :: Functor f => Algebra f b -> Fix f -> b
cata alg = alg . fmap (cata alg) . unFix
fixcata :: Functor f => Algebra f b -> Fix f -> b
fixcata alg = fix $ \f -> alg . fmap f . unFix
type CoAlgebra f a = a -> f a
ana :: Functor f => CoAlgebra f a -> a -> Fix f
ana coalg = Fix . fmap (ana coalg) . coalg
fixana :: Functor f => CoAlgebra f a -> a -> Fix f
fixana coalg = fix $ \f -> Fix . fmap f . coalg
我运行了一些基准测试,结果令我感到惊讶。 criterion
报告的内容类似于十倍的加速,特别是在启用O2
时。我想知道是什么导致了这么大的改进,并开始严重怀疑我的基准测试能力。
这是我使用的确切criterion
代码:
smallWord, largeWord :: Word
smallWord = 2^10
largeWord = 2^20
shortEnv, longEnv :: Fix Maybe
shortEnv = ana coAlg smallWord
longEnv = ana coAlg largeWord
benchCata = nf (cata alg)
benchFixcata = nf (fixcata alg)
benchAna = nf (ana coAlg)
benchFixana = nf (fixana coAlg)
main = defaultMain
[ bgroup "cata"
[ bgroup "short input"
[ env (return shortEnv) $ \x -> bench "cata" (benchCata x)
, env (return shortEnv) $ \x -> bench "fixcata" (benchFixcata x)
]
, bgroup "long input"
[ env (return longEnv) $ \x -> bench "cata" (benchCata x)
, env (return longEnv) $ \x -> bench "fixcata" (benchFixcata x)
]
]
, bgroup "ana"
[ bgroup "small word"
[ bench "ana" $ benchAna smallWord
, bench "fixana" $ benchFixana smallWord
]
, bgroup "large word"
[ bench "ana" $ benchAna largeWord
, bench "fixana" $ benchFixana largeWord
]
]
]
还有一些辅助代码:
alg :: Algebra Maybe Word
alg Nothing = 0
alg (Just x) = succ x
coAlg :: CoAlgebra Maybe Word
coAlg 0 = Nothing
coAlg x = Just (pred x)
使用O0
编译,数字非常均匀。使用O2
,fix~
函数似乎优于普通函数:
benchmarking cata/short input/cata
time 31.67 μs (31.10 μs .. 32.26 μs)
0.999 R² (0.998 R² .. 1.000 R²)
mean 31.20 μs (31.05 μs .. 31.46 μs)
std dev 633.9 ns (385.3 ns .. 1.029 μs)
variance introduced by outliers: 18% (moderately inflated)
benchmarking cata/short input/fixcata
time 2.422 μs (2.407 μs .. 2.440 μs)
1.000 R² (1.000 R² .. 1.000 R²)
mean 2.399 μs (2.388 μs .. 2.410 μs)
std dev 37.12 ns (31.44 ns .. 47.06 ns)
variance introduced by outliers: 14% (moderately inflated)
如果有人可以确认或发现缺陷,我将不胜感激。
*我在这个场合用ghc 8.2.2
编译了东西。)
postscriptum
This post from back in 2012详细阐述了fix
的表现。 (感谢@chi
获取链接。)
答案 0 :(得分:5)
这是由于how the fixed point is computed by fix
。
上面的@duplode指出了这一点(我自己在related question中)。无论如何,我们可以总结如下问题。
我们有那个
fix f = f (fix f)
有效,但在每次递归时都会进行fix f
次新调用。取而代之的是,
fix f = go
where go = f go
计算避免该调用的相同定点。在库中fix
以更有效的方式实现。
回到这个问题,考虑cata
的以下三个实现:
cata :: Functor f => Algebra f b -> Fix f -> b
cata alg' = alg' . fmap (cata alg') . unFix
cata2 :: Functor f => Algebra f b -> Fix f -> b
cata2 alg' = go
where
go = alg' . fmap go . unFix
fixcata :: Functor f => Algebra f b -> Fix f -> b
fixcata alg' = fix $ \f -> alg' . fmap f . unFix
第一个在每次递归时调用cata alg'
。第二个没有。第三个也没有,因为库fix
是有效的。
事实上,即使使用OP使用的相同测试,我们也可以使用Criterion来确认这一点:
benchmarking cata/short input/cata
time 16.58 us (16.54 us .. 16.62 us)
1.000 R² (1.000 R² .. 1.000 R²)
mean 16.62 us (16.58 us .. 16.65 us)
std dev 111.6 ns (89.76 ns .. 144.0 ns)
benchmarking cata/short input/cata2
time 1.746 us (1.742 us .. 1.749 us)
1.000 R² (1.000 R² .. 1.000 R²)
mean 1.741 us (1.736 us .. 1.744 us)
std dev 12.69 ns (10.50 ns .. 17.31 ns)
benchmarking cata/short input/fixcata
time 2.010 us (2.003 us .. 2.016 us)
1.000 R² (1.000 R² .. 1.000 R²)
mean 2.006 us (2.001 us .. 2.011 us)
std dev 16.40 ns (14.05 ns .. 19.27 ns)
长期投入也表明了改善。
benchmarking cata/long input/cata
time 119.3 ms (113.4 ms .. 125.8 ms)
0.996 R² (0.992 R² .. 1.000 R²)
mean 119.8 ms (117.7 ms .. 121.7 ms)
std dev 2.924 ms (2.073 ms .. 4.064 ms)
variance introduced by outliers: 11% (moderately inflated)
benchmarking cata/long input/cata2
time 17.89 ms (17.43 ms .. 18.36 ms)
0.996 R² (0.992 R² .. 0.999 R²)
mean 18.02 ms (17.49 ms .. 18.62 ms)
std dev 1.362 ms (853.9 us .. 2.022 ms)
variance introduced by outliers: 33% (moderately inflated)
benchmarking cata/long input/fixcata
time 18.03 ms (17.56 ms .. 18.50 ms)
0.996 R² (0.992 R² .. 0.999 R²)
mean 18.17 ms (17.57 ms .. 18.72 ms)
std dev 1.365 ms (852.1 us .. 2.045 ms)
variance introduced by outliers: 33% (moderately inflated)
我还尝试了ana
,观察到类似改进的ana2
的效果与fixana
一致。也没有惊喜。