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Author Topic: Fast Sinus() and Cosinus() algos  (Read 16956 times)

jj2007

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Fast Sinus() and Cosinus() algos
« on: September 10, 2015, 12:08:13 AM »
Hi folks,
Just uploaded a new version featuring faster Sinus() and Cosinus() algos. The absolute error is rather small, here compared to the FPU fsin and fcos functions:
Code: [Select]
x       sin x                   error
-90     -1.000000000000000000   0.0
-45     -0.7071067811865475244  -9.38e-18
0       0.0     0.0
45      0.7071067811865475244   9.38e-18
90      1.000000000000000000    0.0
135     0.7071067811865475244   -2.82e-17
180     0.0     -5.31e-17
225     -0.7071067811865475244  -4.69e-17
270     -1.000000000000000000   0.0
315     -0.7071067811865475244  6.57e-17
360     0.0     1.06e-16
405     0.7071067811865475244   8.44e-17
450     1.000000000000000000    0.0

x       cos x                   error
-90     0.0     0.0
-45     0.7071067811865475244   -9.38e-18
0       1.000000000000000000    0.0
45      0.7071067811865475244   -9.38e-18
90      0.0     -2.65e-17
135     -0.7071067811865475244  -2.81e-17
180     -1.000000000000000000   0.0
225     -0.7071067811865475244  4.69e-17
270     0.0     7.96e-17
315     0.7071067811865475244   6.56e-17
360     1.000000000000000000    0.0
405     0.7071067811865475244   -8.45e-17
450     0.0     -1.33e-16
On my machine, they are almost five times as fast. Can I have some timings please? Thanks.

Code: [Select]
Intel(R) Core(TM) i5-2450M CPU @ 2.50GHz
293 ms fpu fsin         71 ms MasmBasic
295 ms fpu fsin         61 ms MasmBasic
294 ms fpu fsin         61 ms MasmBasic
295 ms fpu fsin         66 ms MasmBasic
290 ms fpu fsin         60 ms MasmBasic
290 ms fpu fsin         61 ms MasmBasic
299 ms fpu fsin         61 ms MasmBasic
288 ms fpu fsin         67 ms MasmBasic
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FORTRANS

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Re: Fast Sinus() and Cosinus() algos
« Reply #1 on: September 10, 2015, 01:50:30 AM »
Intel(R) Pentium(R) M processor 1.70GHz
459 ms fpu fsin         167 ms MasmBasic
433 ms fpu fsin         167 ms MasmBasic
442 ms fpu fsin         166 ms MasmBasic
451 ms fpu fsin         167 ms MasmBasic
456 ms fpu fsin         167 ms MasmBasic
432 ms fpu fsin         167 ms MasmBasic
443 ms fpu fsin         166 ms MasmBasic
449 ms fpu fsin         167 ms MasmBasic
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TouEnMasm

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Re: Fast Sinus() and Cosinus() algos
« Reply #2 on: September 10, 2015, 02:33:32 AM »
C:\Users\luce\Downloads>fastsincos
Intel(R) Core(TM) i3-4150 CPU @ 3.50GHz
236 ms fpu fsin         46 ms MasmBasic
258 ms fpu fsin         51 ms MasmBasic
238 ms fpu fsin         56 ms MasmBasic
239 ms fpu fsin         46 ms MasmBasic
241 ms fpu fsin         46 ms MasmBasic
240 ms fpu fsin         46 ms MasmBasic
241 ms fpu fsin         68 ms MasmBasic
238 ms fpu fsin         46 ms MasmBasic
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Fa is a musical note to play with CL

sinsi

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Re: Fast Sinus() and Cosinus() algos
« Reply #3 on: September 10, 2015, 04:47:59 AM »
AMD A10-7850K APU with Radeon(TM) R7 Graphics
252 ms fpu fsin         76 ms MasmBasic
250 ms fpu fsin         73 ms MasmBasic
249 ms fpu fsin         75 ms MasmBasic
250 ms fpu fsin         76 ms MasmBasic
250 ms fpu fsin         76 ms MasmBasic
248 ms fpu fsin         74 ms MasmBasic
249 ms fpu fsin         76 ms MasmBasic
249 ms fpu fsin         76 ms MasmBasic
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jj2007

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Re: Fast Sinus() and Cosinus() algos
« Reply #4 on: September 10, 2015, 04:59:29 AM »
Thanxalot :icon14:

One more, with CRT sin added:
Code: [Select]
Intel(R) Celeron(R) CPU  N2840  @ 2.16GHz
411 ms fpu fsin         1074 ms CRT sin         175 ms MasmBasic
409 ms fpu fsin         1096 ms CRT sin         180 ms MasmBasic
401 ms fpu fsin         1070 ms CRT sin         174 ms MasmBasic
397 ms fpu fsin         1098 ms CRT sin         181 ms MasmBasic
407 ms fpu fsin         1072 ms CRT sin         173 ms MasmBasic
395 ms fpu fsin         1054 ms CRT sin         172 ms MasmBasic
392 ms fpu fsin         1055 ms CRT sin         172 ms MasmBasic
394 ms fpu fsin         1054 ms CRT sin         172 ms MasmBasic
Precision of Sinus() is marginally better than the CRT algo.
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zedd151

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Re: Fast Sinus() and Cosinus() algos
« Reply #5 on: September 10, 2015, 05:01:16 AM »
Code: [Select]
Genuine Intel(R) CPU           T2060  @ 1.60GHz
524 ms fpu fsin         179 ms MasmBasic
472 ms fpu fsin         211 ms MasmBasic
463 ms fpu fsin         175 ms MasmBasic
460 ms fpu fsin         175 ms MasmBasic
449 ms fpu fsin         175 ms MasmBasic
459 ms fpu fsin         175 ms MasmBasic
462 ms fpu fsin         175 ms MasmBasic
458 ms fpu fsin         175 ms MasmBasic

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Regards, zedd.
:tongue:

dedndave

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Re: Fast Sinus() and Cosinus() algos
« Reply #6 on: September 10, 2015, 06:29:56 AM »
prescott w/htt
Code: [Select]
Intel(R) Pentium(R) 4 CPU 3.00GHz
438 ms fpu fsin         156 ms MasmBasic
445 ms fpu fsin         149 ms MasmBasic
431 ms fpu fsin         167 ms MasmBasic
441 ms fpu fsin         147 ms MasmBasic
433 ms fpu fsin         147 ms MasmBasic
432 ms fpu fsin         150 ms MasmBasic
434 ms fpu fsin         146 ms MasmBasic
431 ms fpu fsin         149 ms MasmBasic
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zedd151

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Re: Fast Sinus() and Cosinus() algos
« Reply #7 on: September 10, 2015, 08:30:22 AM »
Same computer as my previous posting, but this time I started up with 'Minimal' services running,

just to see if there is much difference.

Code: [Select]
Genuine Intel(R) CPU           T2060  @ 1.60GHz
451 ms fpu fsin         175 ms MasmBasic
457 ms fpu fsin         175 ms MasmBasic
460 ms fpu fsin         175 ms MasmBasic
455 ms fpu fsin         175 ms MasmBasic
447 ms fpu fsin         175 ms MasmBasic
462 ms fpu fsin         178 ms MasmBasic
460 ms fpu fsin         175 ms MasmBasic
455 ms fpu fsin         175 ms MasmBasic

11 year old Gateway laptop.
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Regards, zedd.
:tongue:

Siekmanski

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Re: Fast Sinus() and Cosinus() algos
« Reply #8 on: September 10, 2015, 06:48:23 PM »
Code: [Select]
Intel(R) Core(TM) i7-4930K CPU @ 3.40GHz
254 ms fpu fsin         50 ms MasmBasic
245 ms fpu fsin         50 ms MasmBasic
245 ms fpu fsin         50 ms MasmBasic
245 ms fpu fsin         50 ms MasmBasic
244 ms fpu fsin         50 ms MasmBasic
245 ms fpu fsin         50 ms MasmBasic
245 ms fpu fsin         50 ms MasmBasic
245 ms fpu fsin         50 ms MasmBasic
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jj2007

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Re: Fast Sinus() and Cosinus() algos
« Reply #9 on: September 10, 2015, 06:58:24 PM »
Quote from: zedd151 on September 10, 2015, 08:30:22 AM
Same computer as my previous posting, but this time I started up with 'Minimal' services running,

just to see if there is much difference.
..
11 year old Gateway laptop.

No difference for the lowest values. The algo grabs its share and doesn't care for services :P

@Marinus: This was inspired by your earlier lolengine/remez post, thanks :icon14:
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Siekmanski

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Re: Fast Sinus() and Cosinus() algos
« Reply #10 on: September 10, 2015, 09:30:39 PM »
Hi Jochen,
Did you use his c++ example source code to get the Chebyshev coefficients ?
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jj2007

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Re: Fast Sinus() and Cosinus() algos
« Reply #11 on: September 11, 2015, 03:50:49 AM »
Quote from: Siekmanski on September 10, 2015, 09:30:39 PM
Did you use his c++ example source code to get the Chebyshev coefficients ?

I tried these coefficients for fastsin2:
Quote from: Siekmanski on April 15, 2015, 04:36:41 PM
And this one is our goldmine i think,
http://lolengine.net/wiki/doc/maths/remez
...
There is even a faster one with 4 coeffs,

Code: [Select]
double fastsin2(double x)
{
    const double a3 = -1.666665709650470145824129400050267289858e-1;
    const double a5 = 8.333017291562218127986291618761571373087e-3;
    const double a7 = -1.980661520135080504411629636078917643846e-4;
    const double a9 = 2.600054767890361277123254766503271638682e-6;

    return x + x*x*x * (a3 + x*x * (a5 + x*x * (a7 + x*x * a9))));
}

Problem is they are reasonably precise only for the 0...90 degrees range. Of course, with known symmetries you can cover the whole range, but that makes it a bit slower.

In the end, I redesigned my SetPoly3 for REAL10 precision and voilà, fast and precise :biggrin:
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qWord

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Re: Fast Sinus() and Cosinus() algos
« Reply #12 on: September 12, 2015, 02:55:41 AM »
Quote from: jj2007 on September 11, 2015, 03:50:49 AM
Problem is they are reasonably precise only for the 0...90 degrees range. Of course, with known symmetries you can cover the whole range, but that makes it a bit slower.
Just to note that the range reduction is the complicated part here and critical for the precision.
It might be also interesting to see plots of the relative error compared to the FPU (or any high-precision library with correct rounding)...
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jj2007

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Re: Fast Sinus() and Cosinus() algos
« Reply #13 on: September 12, 2015, 07:41:43 PM »
Quote from: qWord on September 12, 2015, 02:55:41 AM
It might be also interesting to see plots of the relative error compared to the FPU (or any high-precision library with correct rounding)...
Can't offer plots right now, but here is a table version:
Code: [Select]
x       MasmBasic Sinus(x)
        -0.707106781186547524400844362104849039284...(Wolfram Alpha)
                                 error fpu-CRT  error fpu-MB
-45     -0.7071067811865475244   -5.33e-17      -9.38e-18
-30     -0.5000000000000000001   -4.78e-17      -7.72e-18
-15     -0.2588190451025207624   -1.86e-17      -4.31e-18
0       0.0                      0.0            0.0
15      0.2588190451025207624    1.86e-17       4.31e-18
30      0.5000000000000000001    4.78e-17       7.72e-18
45      0.7071067811865475244    5.33e-17       9.38e-18
60      0.8660254037844386468    4.13e-17       8.89e-18
75      0.9659258262890682867    -3.12e-17      5.64e-18
90      1.000000000000000000     0.0            0.0
105     0.9659258262890682867    -1.75e-17      -8.08e-18
120     0.8660254037844386468    -4.32e-17      -1.77e-17
135     0.7071067811865475244    -2.02e-17      -2.82e-17
150     0.5000000000000000001    9.38e-17       -3.83e-17
165     0.2588190451025207624    -2.08e-16      -4.69e-17
180     0.0                      -6.94e-17      -5.31e-17
195     -0.2588190451025207624   8.82e-17       -5.56e-17
210     -0.5000000000000000001   1.65e-16       -5.37e-17
225     -0.7071067811865475244   -1.58e-17      -4.69e-17
240     -0.8660254037844386468   -2.37e-16      -3.54e-17
255     -0.9659258262890682867   4.49e-17       -1.94e-17
270     -1.000000000000000000    0.0            0.0
285     -0.9659258262890682867   1.15e-16       2.18e-17
300     -0.8660254037844386468   -9.44e-17      4.42e-17
315     -0.7071067811865475244   9.37e-17       6.57e-17
330     -0.5000000000000000001   3.60e-16       8.42e-17
345     -0.2588190451025207624   -1.77e-16      9.82e-17
360     0.0                      1.39e-16       1.06e-16
375     0.2588190451025207624    4.16e-16       1.07e-16
390     0.5000000000000000001    -9.96e-17      9.96e-17
405     0.7071067811865475244    8.93e-17       8.44e-17

x       MasmBasic Cosinus(x)
        0.707106781186547524400844362104849039284...(Wolfram Alpha)
                                 error fpu-CRT  error fpu-MB
-45     0.7071067811865475244    -3.90e-17      -9.38e-18
-30     0.8660254037844386468    -5.64e-17      -4.39e-18
-15     0.9659258262890682867    -2.43e-17      -1.19e-18
0       1.000000000000000000     0.0            0.0
15      0.9659258262890682867    -2.43e-17      -1.19e-18
30      0.8660254037844386468    -5.64e-17      -4.39e-18
45      0.7071067811865475244    -3.90e-17      -9.38e-18
60      0.5000000000000000001    -9.57e-17      -1.53e-17
75      0.2588190451025207624    4.42e-17       -2.13e-17
90      0.0                      -3.47e-17      -2.65e-17
105     -0.2588190451025207624   1.18e-16       -2.99e-17
120     -0.5000000000000000001   -1.91e-16      -3.07e-17
135     -0.7071067811865475244   -3.45e-17      -2.81e-17
150     -0.8660254037844386468   8.29e-17       -2.21e-17
165     -0.9659258262890682867   -7.30e-17      -1.25e-17
180     -1.000000000000000000    0.0            0.0
195     -0.9659258262890682867   1.06e-17       1.50e-17
210     -0.8660254037844386468   -8.12e-17      3.10e-17
225     -0.7071067811865475244   1.12e-16       4.69e-17
240     -0.5000000000000000001   3.83e-16       6.12e-17
255     -0.2588190451025207624   -2.07e-16      7.26e-17
270     0.0                      1.04e-16       7.96e-17
285     0.2588190451025207624    3.86e-16       8.12e-17
300     0.5000000000000000001    -1.88e-16      7.67e-17
315     0.7071067811865475244    1.08e-16       6.56e-17
330     0.8660254037844386468    2.24e-16       4.87e-17
345     0.9659258262890682867    -5.18e-17      2.62e-17
360     1.000000000000000000     0.0            0.0
375     0.9659258262890682867    -1.08e-16      -2.87e-17
390     0.8660254037844386468    1.08e-16       -5.75e-17
405     0.7071067811865475244    -7.49e-17      -8.45e-17

Intel(R) Core(TM) i5-2450M CPU @ 2.50GHz
287 ms fpu fsin         610 ms CRT sin          64 ms MasmBasic
292 ms fpu fsin         604 ms CRT sin          62 ms MasmBasic
294 ms fpu fsin         604 ms CRT sin          62 ms MasmBasic
289 ms fpu fsin         604 ms CRT sin          62 ms MasmBasic
288 ms fpu fsin         601 ms CRT sin          62 ms MasmBasic
289 ms fpu fsin         601 ms CRT sin          63 ms MasmBasic
300 ms fpu fsin         600 ms CRT sin          64 ms MasmBasic
286 ms fpu fsin         612 ms CRT sin          62 ms MasmBasic
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zedd151

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Re: Fast Sinus() and Cosinus() algos
« Reply #14 on: September 12, 2015, 10:45:25 PM »
From my slow assed laptop
Quote
Genuine Intel(R) CPU           T2060  @ 1.60GHz (SSE3)

Code: [Select]
x       MasmBasic Sinus(x)
        -0.707106781186547524400844362104849039284...(Wolfram Alpha)
                                 error fpu-CRT  error fpu-MB
-45     -0.7071067811865475244   -5.33e-17      -9.38e-18
-30     -0.5000000000000000001   -4.78e-17      -7.72e-18
-15     -0.2588190451025207624   -1.86e-17      -4.31e-18
0       0.0                      0.0            0.0
15      0.2588190451025207624    1.86e-17       4.31e-18
30      0.5000000000000000001    4.78e-17       7.72e-18
45      0.7071067811865475244    5.33e-17       9.38e-18
60      0.8660254037844386468    4.13e-17       8.89e-18
75      0.9659258262890682867    -3.12e-17      5.64e-18
90      1.000000000000000000     0.0            0.0
105     0.9659258262890682867    -1.75e-17      -8.08e-18
120     0.8660254037844386468    -4.32e-17      -1.77e-17
135     0.7071067811865475244    -2.02e-17      -2.82e-17
150     0.5000000000000000001    9.38e-17       -3.83e-17
165     0.2588190451025207624    -2.08e-16      -4.69e-17
180     0.0                      -6.94e-17      -5.31e-17
195     -0.2588190451025207624   8.82e-17       -5.56e-17
210     -0.5000000000000000001   1.65e-16       -5.37e-17
225     -0.7071067811865475244   -1.58e-17      -4.69e-17
240     -0.8660254037844386468   -2.37e-16      -3.54e-17
255     -0.9659258262890682867   4.49e-17       -1.94e-17
270     -1.000000000000000000    0.0            0.0
285     -0.9659258262890682867   1.15e-16       2.18e-17
300     -0.8660254037844386468   -9.44e-17      4.42e-17
315     -0.7071067811865475244   9.37e-17       6.57e-17
330     -0.5000000000000000001   3.60e-16       8.42e-17
345     -0.2588190451025207624   -1.77e-16      9.82e-17
360     0.0                      1.39e-16       1.06e-16
375     0.2588190451025207624    4.16e-16       1.07e-16
390     0.5000000000000000001    -9.96e-17      9.96e-17
405     0.7071067811865475244    8.93e-17       8.44e-17

x       MasmBasic Cosinus(x)
        0.707106781186547524400844362104849039284...(Wolfram Alpha)
                                 error fpu-CRT  error fpu-MB
-45     0.7071067811865475244    -3.90e-17      -9.38e-18
-30     0.8660254037844386468    -5.64e-17      -4.39e-18
-15     0.9659258262890682867    -2.43e-17      -1.19e-18
0       1.000000000000000000     0.0            0.0
15      0.9659258262890682867    -2.43e-17      -1.19e-18
30      0.8660254037844386468    -5.64e-17      -4.39e-18
45      0.7071067811865475244    -3.90e-17      -9.38e-18
60      0.5000000000000000001    -9.57e-17      -1.53e-17
75      0.2588190451025207624    4.42e-17       -2.13e-17
90      0.0                      -3.47e-17      -2.65e-17
105     -0.2588190451025207624   1.18e-16       -2.99e-17
120     -0.5000000000000000001   -1.91e-16      -3.07e-17
135     -0.7071067811865475244   -3.45e-17      -2.81e-17
150     -0.8660254037844386468   8.29e-17       -2.21e-17
165     -0.9659258262890682867   -7.30e-17      -1.25e-17
180     -1.000000000000000000    0.0            0.0
195     -0.9659258262890682867   1.06e-17       1.50e-17
210     -0.8660254037844386468   -8.12e-17      3.10e-17
225     -0.7071067811865475244   1.12e-16       4.69e-17
240     -0.5000000000000000001   3.83e-16       6.12e-17
255     -0.2588190451025207624   -2.07e-16      7.26e-17
270     0.0                      1.04e-16       7.96e-17
285     0.2588190451025207624    3.86e-16       8.12e-17
300     0.5000000000000000001    -1.88e-16      7.67e-17
315     0.7071067811865475244    1.08e-16       6.56e-17
330     0.8660254037844386468    2.24e-16       4.87e-17
345     0.9659258262890682867    -5.18e-17      2.62e-17
360     1.000000000000000000     0.0            0.0
375     0.9659258262890682867    -1.08e-16      -2.87e-17
390     0.8660254037844386468    1.08e-16       -5.75e-17
405     0.7071067811865475244    -7.49e-17      -8.45e-17

Genuine Intel(R) CPU           T2060  @ 1.60GHz
531 ms fpu fsin         1264 ms CRT sin         198 ms MasmBasic
461 ms fpu fsin         1323 ms CRT sin         196 ms MasmBasic
462 ms fpu fsin         1257 ms CRT sin         195 ms MasmBasic
462 ms fpu fsin         1256 ms CRT sin         196 ms MasmBasic
461 ms fpu fsin         1265 ms CRT sin         196 ms MasmBasic
461 ms fpu fsin         1268 ms CRT sin         196 ms MasmBasic
487 ms fpu fsin         1333 ms CRT sin         201 ms MasmBasic
462 ms fpu fsin         1267 ms CRT sin         197 ms MasmBasic
Logged
Regards, zedd.
:tongue:
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