Re: Minimax approximation for cosine
From: Richard Mathar (mathar_at_amer.strw.leidenuniv.nl)
Date: 11/18/04
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Date: Thu, 18 Nov 2004 21:01:19 +0000 (UTC)
In article <9215d7ac.0411141747.23763c56@posting.google.com>,
glenlow@pixelglow.com (Glen Low) writes:
>> It might be better to use identities to reduce x to the interval [0,
>> Pi/2], and then divide that interval into two subintervals, say [0,
>> beta*Pi] and [beta*Pi, Pi/2].
>
>The Maclaurin expansion of cos(x) around x = 0 is a polynomial with
>even powers. Therefore it's highly likely that a minimax polynomial
>with even powers only will be a better fit than one with all powers.
>Such a polynomial will have the additional benefit of P(x) = P(-x)
>because of the even powers, so that I can just use the interval [0,
>Pi/2] to get a result that works correctly with [-Pi/2, Pi/2].
>
>After some thought on the bus to the day job, I came up with this.
>
>Usually in minimax approximations, you divide out any zeros of the
>function because otherwise the algorithm chokes e.g. if f(x) = 0 for x
>= k, then you can't do a minimax approximation on f(x), but you have
>to do it on f(x)/(x-k).
>
>The obvious function for cos(x) in [0, Pi/2] is cos(x)/(x-Pi/2) since
>cos(Pi/2)=0. But the resulting polynomial has all powers, and then
>multiply by (x-Pi/2) still gives me all powers back.
>
>So I thought I'd try: cos(x)/{(x-Pi/2)(x+Pi/2)} = cos(x)/(x^2 -
>Pi^2/4)... at x = Pi/2, this tends to 2/Pi. Substituing x -> x^.5, we
>have cos(x^.5)/(x - Pi^2/4). Assuming that P'(x) is the minimax
>polynomial for this, then P(x) = P'(x^2)(x^2 - Pi^2/4), so nicely P(x)
>has all even powers too.
>
>> Use a minimax polynomial for cos(x) for
>> x in [0, beta*Pi] and a minimax polynomial for sin(Pi/2 - x) for x in
>> [beta*Pi, Pi/2], where beta is chosen to balance the accuracy and the
>> execution time of the two approximations.
>
>I did the analysis for sin(x) and found a polynomial of degree 9 with
>odd powers only was sufficient to get enough accuracy for IEEE 754
>float (24 bits). I tried a couple of splits of the range [0, Pi/2] but
>none actually reduced the polynomial degree while maintaining
>accuracy. Of course if there is such a split, the hardware pipeline
>for fma would allow me to do both intervals almost simultaneously, so
>there'd be no loss in speed.
>
>I'm assuming that a similar situation would apply with cosine, we'll
>see...
>
>> How much accuracy do you need? I'm sure that there are newer books on
>> the topic, but I might be able to suggest values of beta and the
>> coefficients from an ancient (1967) book on approximation of
>> functions.
>
>The relative error in sin(x) or cos(x) must be < 2^-24 or about
>5.96e-8.
>
>Cheers,
>Glen Low, Pixelglow Software
>www.pixelglow.com
One can extend the example 5.6 in my script in
http://arXiv.org/abs/math.CA/0403344
for the Chebyshev polynomial for cos(pi/2*x)/(1-x^2) by two higher orders
than actually quoted in the example in the script.
This yields the expansion coefficients for the Chebyshev polynomials
b[0] = .890365196792210693146129746935
b[2] = -.107274469488517592983676091401
b[4] = .233370052015861433579008625011e-2
b[6] = -.264479401843810274578975201202e-4
b[8] = .184330753236829605876190693440e-6
or
.999999999071824518322659698297-.233700504645726537695303066267*x^2+.\
199685982106370967412399614777e-1*x^4-.893522758728821257757025461367e-3*x^6+.\
235943364143141895521524087603e-4*x^8
which should be good for a relative error of about 1.05e-9 in this
combined function. If one wants to go for the Chebyshev approximation
(some sort of minimum *absolute* error) for cos(pi*x/2)/(1-x^2):
these are also given in terms of Bessel functions there.
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