[Chortle]
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I just got my compiler to compute a sine table using a taylor series expansion.
Real programmers use butterflies
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I just got my compiler to compute a sine table using a taylor series expansion.
Real programmers use butterflies
The biggest problem with accurate computation of trig. functions is not computation of the result from the reduced argument, but reducing the argument to the range [-Pi, Pi). For example, 10^22 is accurately representable as a double. sin(10^22) requires finding the exact remainder of division by 2*Pi, which is non-trivial. See the following article for details: [https://www.csee.umbc.edu/~phatak/645/supl/Ng-ArgReduction.pdf\](https://www.csee.umbc.edu/~phatak/645/supl/Ng-ArgReduction.pdf)
Freedom is the freedom to say that two plus two make four. If that is granted, all else follows. -- 6079 Smith W.
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You can do better: Fixed-Point Sine (and Cosine) for Embedded Systems | Null Hardware[^].
"In testa che avete, Signor di Ceprano?" -- Rigoletto
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... In Q15.
GCS d--(d-) s-/++ a C++++ U+++ P- L+@ E-- W++ N+ o+ K- w+++ O? M-- V? PS+ PE- Y+ PGP t+ 5? X R+++ tv-- b+(+++) DI+++ D++ G e++ h--- r+++ y+++* Weapons extension: ma- k++ F+2 X
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And M3, and M4 :D depensing on the system on chip RAM can be expensive real estate.
GCS d--(d-) s-/++ a C++++ U+++ P- L+@ E-- W++ N+ o+ K- w+++ O? M-- V? PS+ PE- Y+ PGP t+ 5? X R+++ tv-- b+(+++) DI+++ D++ G e++ h--- r+++ y+++* Weapons extension: ma- k++ F+2 X
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The biggest problem with accurate computation of trig. functions is not computation of the result from the reduced argument, but reducing the argument to the range [-Pi, Pi). For example, 10^22 is accurately representable as a double. sin(10^22) requires finding the exact remainder of division by 2*Pi, which is non-trivial. See the following article for details: [https://www.csee.umbc.edu/~phatak/645/supl/Ng-ArgReduction.pdf\](https://www.csee.umbc.edu/~phatak/645/supl/Ng-ArgReduction.pdf)
Freedom is the freedom to say that two plus two make four. If that is granted, all else follows. -- 6079 Smith W.
Restrict to [0 - Pi/2] and then flip the result depending on the argument. Either you get 1/4 of the memory usage or 4 times the interpolation points.
GCS d--(d-) s-/++ a C++++ U+++ P- L+@ E-- W++ N+ o+ K- w+++ O? M-- V? PS+ PE- Y+ PGP t+ 5? X R+++ tv-- b+(+++) DI+++ D++ G e++ h--- r+++ y+++* Weapons extension: ma- k++ F+2 X
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And M3, and M4 :D depensing on the system on chip RAM can be expensive real estate.
GCS d--(d-) s-/++ a C++++ U+++ P- L+@ E-- W++ N+ o+ K- w+++ O? M-- V? PS+ PE- Y+ PGP t+ 5? X R+++ tv-- b+(+++) DI+++ D++ G e++ h--- r+++ y+++* Weapons extension: ma- k++ F+2 X
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Quote:
and M4
What a damned luxury thingy you are messing with! :-D
"In testa che avete, Signor di Ceprano?" -- Rigoletto
The one we're probably about to use is awesome, it has independent UARTs with internal self managed FIFOs for reading and writing, a Timer Capture interface that almost makes coffee... With respect to my experience with M0 and the current M3 it's truly luxury.
GCS d--(d-) s-/++ a C++++ U+++ P- L+@ E-- W++ N+ o+ K- w+++ O? M-- V? PS+ PE- Y+ PGP t+ 5? X R+++ tv-- b+(+++) DI+++ D++ G e++ h--- r+++ y+++* Weapons extension: ma- k++ F+2 X
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Restrict to [0 - Pi/2] and then flip the result depending on the argument. Either you get 1/4 of the memory usage or 4 times the interpolation points.
GCS d--(d-) s-/++ a C++++ U+++ P- L+@ E-- W++ N+ o+ K- w+++ O? M-- V? PS+ PE- Y+ PGP t+ 5? X R+++ tv-- b+(+++) DI+++ D++ G e++ h--- r+++ y+++* Weapons extension: ma- k++ F+2 X
You still need to reduce the argument to that range, which is a major source of inaccuracy. That is what I pointed out.
Freedom is the freedom to say that two plus two make four. If that is granted, all else follows. -- 6079 Smith W.
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You still need to reduce the argument to that range, which is a major source of inaccuracy. That is what I pointed out.
Freedom is the freedom to say that two plus two make four. If that is granted, all else follows. -- 6079 Smith W.
Yeah I actually thought in fixed point, where accuracy is lost at the very beginning (but it's easier to calculate).
GCS d--(d-) s-/++ a C++++ U+++ P- L+@ E-- W++ N+ o+ K- w+++ O? M-- V? PS+ PE- Y+ PGP t+ 5? X R+++ tv-- b+(+++) DI+++ D++ G e++ h--- r+++ y+++* Weapons extension: ma- k++ F+2 X
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You still need to reduce the argument to that range, which is a major source of inaccuracy. That is what I pointed out.
Freedom is the freedom to say that two plus two make four. If that is granted, all else follows. -- 6079 Smith W.
For a theoretical math library function, true. But for practical applications, who uses angles beyond maybe a few tens of radians?
Software rusts. Simon Stephenson, ca 1994. So does this signature. me, 2012
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For a theoretical math library function, true. But for practical applications, who uses angles beyond maybe a few tens of radians?
Software rusts. Simon Stephenson, ca 1994. So does this signature. me, 2012
In order to calculate sin(x), you must first reduce the argument by calculating y = fmod(x, Pi/2). This entails subtraction of two numbers that may be very close to each other, i.e. you may get catastrophic cancellation and loss of accuracy. In order to avoid this (even for relatively small numbers), you must use a more accurate version of Pi than can be contained in a double. A high-quality floating-point library will perform the reduction at the highest accuracy for all finite values representable by double. This may be overkill for some embedded software (where, as you say, the angles may be small), but in that case there should be a way of indicating whether the result has any significance. Failure to do so can lead to some rather odd results... (The article I referred to in my earlier post ([https://www.csee.umbc.edu/~phatak/645/supl/Ng-ArgReduction.pdf\](https://www.csee.umbc.edu/~phatak/645/supl/Ng-ArgReduction.pdf)) has a table showing how different compilers miscalculated sin(10^22) and cos(10^22))
Freedom is the freedom to say that two plus two make four. If that is granted, all else follows. -- 6079 Smith W.
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In order to calculate sin(x), you must first reduce the argument by calculating y = fmod(x, Pi/2). This entails subtraction of two numbers that may be very close to each other, i.e. you may get catastrophic cancellation and loss of accuracy. In order to avoid this (even for relatively small numbers), you must use a more accurate version of Pi than can be contained in a double. A high-quality floating-point library will perform the reduction at the highest accuracy for all finite values representable by double. This may be overkill for some embedded software (where, as you say, the angles may be small), but in that case there should be a way of indicating whether the result has any significance. Failure to do so can lead to some rather odd results... (The article I referred to in my earlier post ([https://www.csee.umbc.edu/~phatak/645/supl/Ng-ArgReduction.pdf\](https://www.csee.umbc.edu/~phatak/645/supl/Ng-ArgReduction.pdf)) has a table showing how different compilers miscalculated sin(10^22) and cos(10^22))
Freedom is the freedom to say that two plus two make four. If that is granted, all else follows. -- 6079 Smith W.
Now y'all are talking python. I was over here talking C++. :laugh:
Real programmers use butterflies
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I just got my compiler to compute a sine table using a taylor series expansion.
Real programmers use butterflies
That was one of my first Fortran assignments back up in 82 or 83! Back then their approach to teaching software development was sort of like “Here’s a dictionary- now write a novel!” Don’t know that it has changed much (aside from the Fortran part!).
If you can't laugh at yourself - ask me and I will do it for you.
