Why is one of my functions running so much slower?

Question

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. The issue is that it is the one that I expected to be faster that is the slowest.

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.

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. 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.

I have no idea what is causing one function to run so much slower than the other, and any help would be appreciated.

Edit: For clarity I forgot to mention that before being passed to the functions the values are normalized to the range [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.

What are your argv values? I looked at your pastes and I would download and build as I have some performance analysis routines I could hook in. But, I'd have to guess at the argv values. And, a dumb/insulting question: can you prove that you've implemented both algs correctly? As a general rule, I'd add more comments (e.g. equations at top of file, and comments on a line by line basis that relate to the equation. In doing so, you might find you've made a trivial error with the gmp call (e.g. doing mul vs div, etc). Also, you mentioned timings but not loop counts for each. — Craig Estey, Nov 06 '15 at 00:02
Also, it appears you have only one test number. Would multiple ones help? Is the one you have guaranteed to be algorithm agnostic? When I do tests like this, I usually try several test numbers with edge cases, such as good for algA, bad for algB (or vice versa). You might just be getting "lucky". A few more numbers might help prove/disprove this — Craig Estey, Nov 06 '15 at 00:09
"fewer instructions per loop" . . . well, for arb-prec, instructions vary drastically in cost. Are the numbers in the slower one, for example, larger? — geometrian, Nov 06 '15 at 00:43
I tried a few other cases and they all give a similar result. The values the function takes in are only ever in the range of 0.5 (inclusive) to 1 (exclusive), so technically my test case is when the input is 0.5. Once compiled with `gcc` using `std=c99` to get `a.out` I run `./a.out 2 10000 33296 K` where K is 1 or 2, depending on the algorithm being tested. I got the timings from the user and real sections of linux's time command. The algorithms both are tested to work in the given ranges for any value in the range [0.5, 1) and the correct answer is found by calculations outside the functions — user163911, Nov 06 '15 at 00:45
@imallett I believe the values are in the same ranges as both functions use the same default precisions and perform calculations until they reach a matched precision. — user163911, Nov 06 '15 at 00:47
Have you actually verified that it executes fewer loops (say, actually counted number of loops)? I'm not up on the math, but I'm wondering whether it's actually converging faster or looping less frequently. Also, excuse my math dumb, but is there a reason you increment `n` twice per loop in the second case (should it really go up by 2 on each use; if so, seems like you could just increment once by 2, not twice in different places). — ShadowRanger, Nov 06 '15 at 00:47
Note that GMP's mpf is deprecated, you should use MPFR for any new code. — Marc Glisse, Nov 06 '15 at 08:05
@ShadowRanger Yeah, I confirmed fewer loops. For example to 100 decimal places the first function performs about 160 loops and the second about 100. The double increment is just part of the sum, it's a series whose denominator goes up by two each time. — user163911, Nov 06 '15 at 15:18

Déjà vu · Answer 1 · 2015-11-06T07:45:26.597

Running the program with added timing functions, the performance hit comes, for N = 2, from the mpf_mul operations.

In the Taylor case,

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.

In the Hyperbolic case,

mpf_mul(x, x, x2);

has x2 being 0.111... from

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. The multiplication in this case is much more expensive.

edit

It seems actually the culprit is mpf_div(x, x, tmp) which forces gmp to use the whole precision. x2 being calculated from x gets the same precision, thus the mpf_mul are slower.

Thanks, that makes sense. I thought it may have been something like that. — user163911, Nov 06 '15 at 15:25

Craig Estey · Answer 2 · 2015-11-06T08:27:55.747

I've benchmarked your program for both cases.

These are nanosecond resolution.
They compensate for the overhead of the benchmark calls [the compensation is calculated by the bnctst at the beginning]
Although the output is floating point fractional seconds, during the benchmark run they're using 64 bit integer times
Outer functions are benchmarked.
Loops are benchmarked.
All mpf_* have their own.
If a benchmark is invoked more than once (e.g. loops), it will show average, count, and min/max) in addition to total.
The benchmarks are output in the order the calls appear in your source file.

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.

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)

Why is one of my functions running so much slower?

2 Answers2