When evaluating polynomials although Horner's method is shorter to code, and has the fewest actual operations used, on modern architectures with deep pipelines I would recommend giving Estrin's scheme a go and let the profiler / accurate cpu meter logging tell you which one is best:
https://en.wikipedia.org/wiki/Horner%27s_method vs https://en.wikipedia.org/wiki/Estrin%27s_scheme Andy On Sat, 23 Feb 2019 at 07:31, robert bristow-johnson < r...@audioimagination.com> wrote: > > > this is just a guess at what the C code might look like to calculate one > sample. this uses Olli's 7th-order with continuous derivative. f0 is the > oscillator frequency which can vary but it must be non-negative. 1/T is > the sample rate. > > > > float sine_osc(float f0, float T) > { > static float phase = 1.0; // this is the only state. -2.0 <= phase > < +2.0 > > phase += 4.0*f0*T; > if (phase >= 2.0) > { > phase -= 4.0; > } > > if (phase < 0.0) > { > triangle = -phase - 1.0; > } > else > { > triangle = phase - 1.0; > } > > x2 = triangle*triangle; > return triangle*(1.570781972 - x2*(0.6458482979 - x2*(0.07935067784 - > x2*0.004284352588))); > } > > i haven't run this code nor checked it for syntax. but it's conceptually > so simple that i'll bet it works. > > r b-j > > ---------------------------- Original Message ---------------------------- > Subject: Re: [music-dsp] Time-variant 2nd-order sinusoidal resonator > From: "Olli Niemitalo" <o...@iki.fi> > Date: Thu, February 21, 2019 11:58 pm > To: "A discussion list for music-related DSP" < > music-dsp@music.columbia.edu> > -------------------------------------------------------------------------- > > > On Fri, Feb 22, 2019 at 9:08 AM robert bristow-johnson < > > r...@audioimagination.com> wrote: > > > >> i just got in touch with Olli, and this "triangle wave to sine wave" > >> shaper polynomial is discussed at this Stack Exchange: > >> > >> > >> > >> > https://dsp.stackexchange.com/questions/46629/finding-polynomial-approximations-of-a-sine-wave/46761#46761 > >> > > I'll just summarize the results here. The polynomials f(x) approximate > > sin(pi/2*x) for x=-1..1 and are solutions with minimum peak harmonic > > distortion compared to the fundamental frequency. Both solutions with > > continuous and discontinuous derivative are given. In summary: > > > > Shared polynomial approximation properties and constraints: > > x = -1..1, f(-1) = -1, f(0) = 0, f(1) = 1, and f(-x) = -f(x). > > > > If continuous derivative: > > f'(-1) = 0 and f'(1) = 0 for the anti-periodic extension f(x + 2) = > -f(x). > > > > 5th order, continuous derivative, -78.99 dB peak harmonic distortion: > > f(x) = 1.569778813*x - 0.6395576276*x^3 + 0.06977881382*x^5 > > > > 5th order, discontinuous derivative, -91.52 dB peak harmonic distortion: > > f(x) = 1.570034357*x - 0.6425216143*x^3 + 0.07248725712*x^5 > > > > 7th order, continuous derivative, -123.8368 dB peak harmonic distortion: > > f(x) = 1.570781972*x - 0.6458482979*x^3 + 0.07935067784*x^5 > > - 0.004284352588*x^7 > > > > 7th order, discontinuous derivative, -133.627 dB peak harmonic > distortion: > > f(x) = 1.5707953785726114835*x - > > 0.64590724797262922190*x^3 + 0.079473610232926783079*x^5 > > - 0.0043617408329090447344*x^7 > > > > Also the exact coefficients that are rational functions of pi are given > in > > the answer, in case anyone needs more precision. > > > > -olli > > _______________________________________________ > > dupswapdrop: music-dsp mailing list > > music-dsp@music.columbia.edu > > https://lists.columbia.edu/mailman/listinfo/music-dsp > > > > > > > > > -- > > r b-j r...@audioimagination.com > > "Imagination is more important than knowledge." > > > > > > > > _______________________________________________ > dupswapdrop: music-dsp mailing list > music-dsp@music.columbia.edu > https://lists.columbia.edu/mailman/listinfo/music-dsp
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