Stoilen from CUDA file
/* approximate sine on -pi/4...+pi/4 */
static __forceinline__ float __internal_sin_kernel(float x)
{
  float x2, z;

  x2 = x * x;
  z  =                         -1.95152959e-4f;
  z  = __internal_fmad (z, x2,  8.33216087e-3f);
  z  = __internal_fmad (z, x2, -1.66666546e-1f);
  z  = z * x2;
  z  = __internal_fmad (z, x, x);
  return z;
}

CUDASin[x_]:= Module[
{x2,z},
x2 =x*x;
z=-1.95152959*(10^(-4));
z=z*x2+8.33216087*(10^(-3));
z=z*x2-1.66666546*(10^(-1));
z=z*x2;
z=z*x+x
]
Expand[CUDASin[x]]
TPoly[x_]=N[Normal[Series[Sin[x],{x,0,7}]]]
LogPlot[{
Abs[CUDASin[x]-Sin[x]],
Abs[TPoly[x]-Sin[x]]
},{x,-π/4,π/4},
PlotRange->All]