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为什么我的一个函数运行得这么慢？

[英]Why is one of my functions running so much slower?

 原文  2015-11-05 23:37:31  9   2    c/ numerical-methods/ gmp/ natural-logarithm

问题描述

I'm writing an implementation of a couple of methods to find the natural logs of numbers using GMP in C. I have two functions, both of which work, but one of which runs a lot slower than the other. 我正在编写一些方法的实现，以便在C中使用GMP查找数字的自然日志。我有两个函数，两个函数都有效，但其中一个运行速度比另一个慢很多。 The issue is that it is the one that I expected to be faster that is the slowest. 问题是，我认为它是速度最慢的那个。

Below are the two relevant functions, while the full file with int main can be found here , while the required ln_2.txt file is here . 下面是两个相关的函数，而int main的完整文件可以在这里找到，而所需的ln_2.txt文件在这里 。 

void taylor_log(mpf_t R, 
                const mpf_t N, 
                const mpf_t T)
{
    mpf_t x, y, r, pr, tmp, d;
    int n = 1;
    mpf_init(x);
    mpf_init(y);
    mpf_init(tmp);
    mpf_init(d);

    mpf_sub_ui(x, N, 1);
    mpf_init_set(y, x);

    mpf_init_set(r, x);
    mpf_init_set_ui(pr, 0);    

    mpf_sub(d, r, pr);
    mpf_abs(d, d);
    while(mpf_cmp(d, T) > 0)
    {
        mpf_set(pr, r);

        mpf_mul(y, y, x);
        mpf_div_ui(tmp, y, ++n);
        mpf_sub(r, r, tmp);

        mpf_mul(y, y, x);
        mpf_div_ui(tmp, y, ++n);
        mpf_add(r, r, tmp);

        mpf_sub(d, r, pr);
        mpf_abs(d,d);
    }
    printf("%d\n", n);
    mpf_set(R, r);
 }



void hyperbolic_log(mpf_t R,
                    const mpf_t N,
                    const mpf_t T)
{
    mpf_t x, x2, r, pr, tmp, d;
    int n = 1;   
    mpf_init(x);
    mpf_init(x2);
    mpf_init(tmp);
    mpf_init(d);

    mpf_sub_ui(x, N, 1);
    mpf_add_ui(tmp, N, 1);
    mpf_div(x, x, tmp);

    mpf_init_set(r, x);
    mpf_init_set_ui(pr, 0);

    mpf_mul(x2, x, x);

    mpf_sub(d, r, pr);
    mpf_abs(d,d);
    while(mpf_cmp(d, T) > 0)
    {
        mpf_set(pr, r);
        ++n;

        mpf_mul(x, x, x2);
        mpf_div_ui(tmp, x, ++n);
        mpf_add(r, r, tmp);

        mpf_sub(d, r, pr);
        mpf_abs(d,d);
    }
    printf("%d\n", n);
    mpf_mul_ui(R, r, 2);
}

Now the second function should, in theory at least, run quicker as it has fewer instructions per loop and typically executes fewer loops due to a quicker convergence. 从理论上讲，第二个函数至少应该运行得更快，因为每个循环的指令更少，并且由于更快的收敛，通常执行更少的循环。 This is not what I see in practice, as when I calculated ln(2) to 10000 decimal places using at least 33296 bits in the calculations, both gave the correct result but the first method completed in about 0.150s seconds while the second takes about 1 second. 这不是我在实践中看到的，因为当我在计算中使用至少33296位计算ln（2）到10000个小数位时，两者都给出了正确的结果，但第一种方法在大约0.150秒内完成，而第二种方法在第二种方法中完成1秒。

I have no idea what is causing one function to run so much slower than the other, and any help would be appreciated. 我不知道是什么导致一个函数运行速度比另一个慢得多，任何帮助将不胜感激。

Edit: For clarity I forgot to mention that before being passed to the functions the values are normalized to the range [0.5, 1). 编辑：为了清楚起见，我忘了提及在传递给函数之前，值被归一化到范围[0.5,1]。 This is the same for both functions I've confirmed this works in the program, so technically my issue is when 0.5 is provided to the algorithms. 对于我已经确认在程序中有效的两个函数，这是相同的，所以从技术上讲，我的问题是当算法提供0.5时。

2 个解决方案

解决方案1
 1  2015-11-06 06:27:55

Running the program with added timing functions, the performance hit comes, for N = 2 , from the mpf_mul operations. 使用添加的计时功能运行程序，从mpf_mul操作中，对于N = 2 ，性能命中。

In the Taylor case, 在泰勒案中， 

mpf_mul(y, y, x);

has x being -0.5 from mpf_get_d_2exp((long *)&e, N)) - 1 , which in the gmp structure has a size of 1, whatever the precision requested. 如果x是-0.5来自mpf_get_d_2exp((long *)&e, N)) - 1 ，它在gmp结构中的大小为1，无论请求的精度如何。

In the Hyperbolic case, 在双曲线情况下， 

mpf_mul(x, x, x2);

has x2 being 0.111... from x2是0.111...来自 

T  = mpf_get_d_2exp((long *)&e, N))
x  = (T - 1)/(T + 1)
x2 = x * x

which in the gmp structure has the size of the requested precision. 在gmp结构中具有所请求精度的大小。 The multiplication in this case is much more expensive. 在这种情况下的乘法要贵得多。

edit 编辑

It seems actually the culprit is mpf_div(x, x, tmp) which forces gmp to use the whole precision. 实际上，罪魁祸首是mpf_div(x, x, tmp) ，它迫使gmp使用整个精度。 x2 being calculated from x gets the same precision, thus the mpf_mul are slower. 从x计算的x2具有相同的精度，因此mpf_mul较慢。

解决方案2
 0  2015-11-06 07:18:56

I've benchmarked your program for both cases. 我已针对这两种情况对您的程序进行了基准测试。

  • These are nanosecond resolution. 这些是纳秒分辨率。
  • They compensate for the overhead of the benchmark calls [the compensation is calculated by the bnctst at the beginning] 它们补偿了基准调用的开销[补偿由开头的bnctst计算]
  • Although the output is floating point fractional seconds, during the benchmark run they're using 64 bit integer times 虽然输出是浮点小数秒，但在基准测试期间，它们使用的是64位整数倍
  • Outer functions are benchmarked. 外部功能是基准测试。
  • Loops are benchmarked. 循环是基准测试。
  • All mpf_* have their own. 所有mpf_*都有自己的。
  • If a benchmark is invoked more than once (eg loops), it will show average, count, and min/max) in addition to total. 如果多次调用基准（例如循环），除了总数之外，它还将显示平均值，计数值和最小值/最大值。
  • The benchmarks are output in the order the calls appear in your source file. 基准测试按照调用在源文件中出现的顺序输出。

The gmp_printf calls have a wrapper around them that wraps the output at 78 chars and the final output for hyperbolic has been elided to fit in the SO 30,000 character limit. gmp_printf调用在它们周围有一个包装器，它将输出包装在78个字符，并且双曲线的最终输出已被省略，以符合SO 30,000字符限制。 

23:32:14.233952283 NEWDAY 11/05/15
23:32:14.233952283 ph: starting 13378 ...
23:32:14.234557628 ph: ARGV tayhypgo ...

--------------------------------------------------------------------------------
23:32:16 tayhypgo: ARGV 2 10000 33296 1
23:32:17 ovrgo: SDIR /home/cae/preserve/ovrbnc/tayhyp
23:32:18 ovrgo: /home/cae/preserve/ovrstk/gen/tayhyp/tayhyp 2 10000 33296 1
bnctst: BEST bncmin=21 bncmax=-1 skipcnt=1000
bnctst: AVG tot=0.000000000

bnctst: SKP tot=0.000023975 avg=0.000000023 cnt=1,000
bnctst: BNCDMP   min=0.000000000 max=0.000000000

2
PRINTF 0.50000000000000000000
Case 1
33207
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tayhyp: BNCDMP mpf_set_default_prec(p) tot=0.000003063
tayhyp: BNCDMP mpf_init(nlog2) tot=0.000002623
tayhyp: BNCDMP mpf_inp_str(nlog2,l_2_in,10) tot=0.003018814
tayhyp: BNCDMP mpf_init_set_str(N,argv[1],10) tot=0.000001509
tayhyp: BNCDMP mpf_init_set_ui(T,10) tot=0.000000533
tayhyp: BNCDMP mpf_pow_ui(T,T,n) tot=0.000086848
tayhyp: BNCDMP mpf_ui_div(T,1,T) tot=0.000380477
tayhyp: BNCDMP mpf_init(V) tot=0.000000271
tayhyp: BNCDMP mpf_set_d(V,mpf_get_d_2exp(&e,N)) tot=0.000001757
tayhyp: BNCDMP mpf_init(R) tot=0.000000037
tayhyp: BNCDMP mpf_init(tmp) tot=0.000000084 avg=0.000000042 cnt=2
tayhyp: BNCDMP   min=0.000000058 max=0.000000068
tayhyp: BNCDMP mpf_mul_ui(tmp,nlog2,e) tot=0.000002245
tayhyp: BNCDMP mpf_add(R,R,tmp) tot=0.000001428
tayhyp: BNCDMP mpf_add(R,R,tmp) tot=0.224560256
tayhyp: BNCDMP mpf_init(x) tot=0.000000362
tayhyp: BNCDMP mpf_init(y) tot=0.000000037
tayhyp: BNCDMP mpf_init(d) tot=0.000000037
tayhyp: BNCDMP mpf_sub_ui(x,N,1) tot=0.000001437
tayhyp: BNCDMP mpf_init_set(y,x) tot=0.000000344
tayhyp: BNCDMP mpf_init_set(r,x) tot=0.000000050
tayhyp: BNCDMP mpf_init_set_ui(pr,0) tot=0.000000056
tayhyp: BNCDMP mpf_sub(d,r,pr) tot=0.017311085 avg=0.000001042 cnt=16,604
tayhyp: BNCDMP   min=0.000000223 max=0.000063322
tayhyp: BNCDMP mpf_abs(d,d) tot=0.000164212 avg=0.000000009 cnt=16,604
tayhyp: BNCDMP   min=0.000000026 max=0.000003257
tayhyp: BNCDMP mpf_abs(d,d) tot=0.223701027 avg=0.000013472 cnt=16,604
tayhyp: BNCDMP   min=0.000000109 max=0.000129095
tayhyp: BNCDMP cmpflg = mpf_cmp(d,T) tot=0.000159926 avg=0.000000009 cnt=16,604
tayhyp: BNCDMP   min=0.000000021 max=0.000009950
tayhyp: BNCDMP mpf_set(pr,r) tot=0.004447240 avg=0.000000267 cnt=16,603
tayhyp: BNCDMP   min=0.000000210 max=0.000004657
tayhyp: BNCDMP mpf_mul(y,y,x) tot=0.030792804 avg=0.000000927 cnt=33,206
tayhyp: BNCDMP   min=0.000000080 max=0.000088428
tayhyp: BNCDMP mpf_div_ui(tmp,y,++n) tot=0.128409872 avg=0.000003867 cnt=33,206
tayhyp: BNCDMP   min=0.000003728 max=0.000109781
tayhyp: BNCDMP mpf_sub(r,r,tmp) tot=0.017006018 avg=0.000001024 cnt=16,603
tayhyp: BNCDMP   min=0.000000651 max=0.000090142
tayhyp: BNCDMP mpf_add(r,r,tmp) tot=0.016594638 avg=0.000000999 cnt=16,603
tayhyp: BNCDMP   min=0.000000635 max=0.000064980
tayhyp: BNCDMP mpf_set(R,r) tot=0.000000239
23:32:19 tayhypgo: 2 10000 33296 1 -- complete

--------------------------------------------------------------------------------
23:32:19 tayhypgo: ARGV 2 10000 33296 2
23:32:19 ovrgo: SDIR /home/cae/preserve/ovrbnc/tayhyp
23:32:21 ovrgo: /home/cae/preserve/ovrstk/gen/tayhyp/tayhyp 2 10000 33296 2
bnctst: BEST bncmin=19 bncmax=-1 skipcnt=1000
bnctst: AVG tot=0.000000000

bnctst: SKP tot=0.000024750 avg=0.000000024 cnt=1,000
bnctst: BNCDMP   min=0.000000000 max=0.000000000

2
PRINTF 0.50000000000000000000
Case 2
20951
PRINTF elided for space
tayhyp: BNCDMP mpf_set_default_prec(p) tot=0.000003154
tayhyp: BNCDMP mpf_init(nlog2) tot=0.000002571
tayhyp: BNCDMP mpf_inp_str(nlog2,l_2_in,10) tot=0.002921480
tayhyp: BNCDMP mpf_init_set_str(N,argv[1],10) tot=0.000001506
tayhyp: BNCDMP mpf_init_set_ui(T,10) tot=0.000000554
tayhyp: BNCDMP mpf_pow_ui(T,T,n) tot=0.000090689
tayhyp: BNCDMP mpf_ui_div(T,1,T) tot=0.000290122
tayhyp: BNCDMP mpf_init(V) tot=0.000000179
tayhyp: BNCDMP mpf_set_d(V,mpf_get_d_2exp(&e,N)) tot=0.000001484
tayhyp: BNCDMP mpf_init(R) tot=0.000000051
tayhyp: BNCDMP mpf_init(tmp) tot=0.000000066 avg=0.000000033 cnt=2
tayhyp: BNCDMP   min=0.000000052 max=0.000000052
tayhyp: BNCDMP mpf_mul_ui(tmp,nlog2,e) tot=0.000001359
tayhyp: BNCDMP mpf_add(R,R,tmp) tot=0.000001408
tayhyp: BNCDMP mpf_init(x) tot=0.000000339
tayhyp: BNCDMP mpf_init(d) tot=0.000000039
tayhyp: BNCDMP mpf_sub_ui(x,N,1) tot=0.000001328
tayhyp: BNCDMP mpf_init_set(r,x) tot=0.000000537
tayhyp: BNCDMP mpf_init_set_ui(pr,0) tot=0.000000054
tayhyp: BNCDMP mpf_sub(d,r,pr) tot=0.011097537 avg=0.000001059 cnt=10,476
tayhyp: BNCDMP   min=0.000000739 max=0.000004082
tayhyp: BNCDMP mpf_abs(d,d) tot=0.000123286 avg=0.000000011 cnt=10,476
tayhyp: BNCDMP   min=0.000000021 max=0.000000364
tayhyp: BNCDMP cmpflg = mpf_cmp(d,T) tot=0.000141611 avg=0.000000013 cnt=10,476
tayhyp: BNCDMP   min=0.000000022 max=0.000002173
tayhyp: BNCDMP mpf_set(pr,r) tot=0.002824968 avg=0.000000269 cnt=10,475
tayhyp: BNCDMP   min=0.000000265 max=0.000003199
tayhyp: BNCDMP mpf_add(r,r,tmp) tot=0.010823055 avg=0.000001033 cnt=10,475
tayhyp: BNCDMP   min=0.000000713 max=0.000004003
tayhyp: BNCDMP mpf_set(R,r) tot=1.366819061
tayhyp: BNCDMP mpf_init(x2) tot=0.000000041
tayhyp: BNCDMP mpf_add_ui(tmp,N,1) tot=0.000000385
tayhyp: BNCDMP mpf_div(x,x,tmp) tot=0.000011269
tayhyp: BNCDMP mpf_mul(x2,x,x) tot=0.000098311
tayhyp: BNCDMP mpf_abs(d,d) tot=1.366190674 avg=0.000130411 cnt=10,476
tayhyp: BNCDMP   min=0.000000117 max=0.000460681
tayhyp: BNCDMP mpf_mul(x,x,x2) tot=1.296999859 avg=0.000123818 cnt=10,475
tayhyp: BNCDMP   min=0.000121270 max=0.000453871
tayhyp: BNCDMP mpf_div_ui(tmp,x,++n) tot=0.040474597 avg=0.000003863 cnt=10,475
tayhyp: BNCDMP   min=0.000003748 max=0.000011291
tayhyp: BNCDMP mpf_mul_ui(R,r,2) tot=0.000002962
23:32:22 tayhypgo: 2 10000 33296 2 -- complete

23:32:22.758210659 ph: complete (ELAPSED: 00:00:08.522868633)

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