Fixed point math routines

 XCore Addict
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 Joined: Wed May 11, 2016 3:50 pm
Fixed point math routines
I hate to ask, but does anybody have a fixed point math acos or asin functions written they could share? I'm trying to get these calls down some, they dont take forever at 42uS but when I'm performing multiple calls of them at the same time along with i2c transactions the timing can start to get tight. Alternatively if anyone could recommend some books, papers, or other libraries in whatever language with source, that describe how this would be done I'd appreciate it.

 XCore Expert
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 Joined: Thu Nov 26, 2015 11:47 pm
The quickest method, if you can use it, is a lookup table. acos(x) can be computed from asin(x) and asin(x) is just asin(x) so it seems to me you could have a pretty small asin lookup table (+ve input +ve output) if you don't need huge resolution. You can interpolate between points in your table to improve resolution a bit to trade off computation time with memory usage. I have done this for sin but not asin.
I have seen asin computed using NR or from atan (which is itself computed from continued fraction expansion) if you despise lookup tables. And then the fixed point math gets messy compared to floating point in my opinion. But I don't think for realworld resolutions it would save a tonne of memory and would use a lot more compute power.
Interested to hear if anyone knows better, like I said I've not actually implemented arcsin myself in fixed point.
I have seen asin computed using NR or from atan (which is itself computed from continued fraction expansion) if you despise lookup tables. And then the fixed point math gets messy compared to floating point in my opinion. But I don't think for realworld resolutions it would save a tonne of memory and would use a lot more compute power.
Interested to hear if anyone knows better, like I said I've not actually implemented arcsin myself in fixed point.

 XCore Expert
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Of course the lookup table + interpolation will get pretty inaccurate very close to input of 1 or +1, but NR will take a long time to converge there, too. You might want to convert some of the functions here (http://http.developer.nvidia.com/Cg/index_stdlib.html) from floating point to fixed point for acos and asin and see if that gives you better results. Won't be as fast a lookup table but will use less memory. I haven't done it myself either.

 XCore Addict
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Thanks, I'll check it out and see what works for me.

 Respected Member
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Hi,
Have a look at https://github.com/xmos/lib_dsp
It contains many of the functions, and they are all reasonably well optimised.
Cheers,
Henk
Have a look at https://github.com/xmos/lib_dsp
It contains many of the functions, and they are all reasonably well optimised.
Cheers,
Henk

 Respected Member
 Posts: 347
 Joined: Wed Jan 27, 2016 5:21 pm
Sorry  that appears to be closed at the moment. Whilst that is being sorted out, you can download the source here
<https://www.xmos.com/support/libraries?subcategory=DSP>
<https://www.xmos.com/support/libraries?subcategory=DSP>

 Respected Member
 Posts: 347
 Joined: Wed Jan 27, 2016 5:21 pm
Even more sorry  just realised you want the arc sin and arc cos. Cody and Waite will have an algorithm for them, I will try and dig it out.

 XCore Addict
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If you can find those that'd be great. As it is I use the xcore200 dsp math library everywhere it makes sense and it works really well. I looked for a copy of the book since it was cited but couldn't find one near.

 Experienced Member
 Posts: 96
 Joined: Mon Mar 22, 2010 8:55 pm
You can calculate arcsin from a reasonably simple infinite series. Look at:
https://en.wikipedia.org/wiki/Inverse_t ... ite_series
and choose a suitable cutoff point.
Arccos can be derived from that as arccos(z) = pi/2  arcsin(z).
Someone on stackexchange provided this as a pretty accurate floatingpoint implementation:
y = x*(1+x*x*(1/6+ x*x*(3/(2*4*5) + x*x*((1*3*5)/(2*4*6*7)))))
Looks like you could get a fixedpoint implementation out of that.
https://en.wikipedia.org/wiki/Inverse_t ... ite_series
and choose a suitable cutoff point.
Arccos can be derived from that as arccos(z) = pi/2  arcsin(z).
Someone on stackexchange provided this as a pretty accurate floatingpoint implementation:
y = x*(1+x*x*(1/6+ x*x*(3/(2*4*5) + x*x*((1*3*5)/(2*4*6*7)))))
Looks like you could get a fixedpoint implementation out of that.

 Experienced Member
 Posts: 96
 Joined: Mon Mar 22, 2010 8:55 pm
I think you'll need to do a range reduction to use the polynomial approach, though. This may not be suitable. See this:
http://stackoverflow.com/questions/7378 ... cfuncions
http://stackoverflow.com/questions/7378 ... cfuncions