Hello,
I found your webpage 'Fun with sinusoïds', particulary the approximation with
parabol segments. I made some tests with higer degrees, with terms in x^2, x^4
and x^6, I optain a maximal error smaller than 1.5*10^-5 for an amplitude of 1.
This is a maximal error of one LSB in 16bit audio. Adding a term in x^8 would
make it exact for 16bit audio with just two multiplications more...

Here is my test code in c I used in a Pure Data external. It compute a sine or a
cosine with a period of one, between 0 and 1. (Other constants would of course
make it possible also for a period of 2*pi ):

Regards
Basile Graf

//CODE:
//----------------------------------------------------

#define COEFF0 1                 //POSITIVE!!
#define COEFF2 19.73640067359408 //NEGATIVE!!
#define COEFF4 64.66421708408689 //POSITIVE!!
#define COEFF6 78.10890090530666 //NEGATIVE!!

float x2_poly_6, x4_poly_6;

static __inline t_float sinpoly6(t_float x)
{
if (x<=0.5)
{
x-=0.25;

x2_poly_6 = x*x;
x4_poly_6 = x2_poly_6*x2_poly_6;

return  COEFF0 - COEFF2*x2_poly_6 + COEFF4*x4_poly_6 -
COEFF6*x2_poly_6*x4_poly_6;
}
else
{
x-=.75;

x2_poly_6 = x*x;
x4_poly_6 = x2_poly_6*x2_poly_6;

return -COEFF0 + COEFF2*x2_poly_6 - COEFF4*x4_poly_6 +
COEFF6*x2_poly_6*x4_poly_6;
}

}

static __inline float cospoly6(float x)
{
if (x<=0.25)
{
x2_poly_6 = x*x;
x4_poly_6 = x2_poly_6*x2_poly_6;

return COEFF0 - COEFF2*x2_poly_6 + COEFF4*x4_poly_6 -
COEFF6*x2_poly_6*x4_poly_6;
}
else if (x>=.75)
{
x-=1;

x2_poly_6 = x*x;
x4_poly_6 = x2_poly_6*x2_poly_6;

return COEFF0 - COEFF2*x2_poly_6 + COEFF4*x4_poly_6 -
COEFF6*x2_poly_6*x4_poly_6;
}
else
{
x-=0.5;

x2_poly_6 = x*x;
x4_poly_6 = x2_poly_6*x2_poly_6;

return -COEFF0 + COEFF2*x2_poly_6 - COEFF4*x4_poly_6 +
COEFF6*x2_poly_6*x4_poly_6;
}

}