Embedded Software and Motor Control Libraries for PXR40xx

Function GFLIB_Cos_F32
Because the GFLIB_Cos_F32 function is implemented with consideration to fixed point
fractional arithmetic, all variables are normalized to fit into the [-1, 1) range. Therefore,
in order to cast the fractional value of the input angle f32In [-1, 1) into the correct range
[- π, π), the input f32In must be multiplied by π. So, the fixed point fractional
implementation of the GFLIB_Sin_F32 function, using 9th order Taylor approximation,
is given as follows:
Equation GFLIB_Cos_Eq3
The 9th order polynomial approximation of the sine function has a very good accuracy in
the range [- π/2, π/2) of the argument, but in wider ranges the calculation error quickly
increases. To minimize the error without having to use a higher order polynomial, the
symmetry of the sine function sin(x) = sin( π - x) is utilized. Therefore, the input
argument is transferred to be always in the range [- π/2, π/2) and the Taylor polynomial
is calculated only in the range of the argument [- π/2, π/2).
To make calculations more precise, the given argument value f32In (that is to be
transferred into the range [-0.5, 0.5) due to the sine function symmetry) is shifted by 1 bit
to the left (multiplied by 2). Then, the value of f32In 2 , used in the calculations, is in the
range [-1, 1) instead of [-0.25, 0.25]. Shifting the input value by 1 bit to the left will
increase the accuracy of the calculated sin( π * f32In) function. Implementing such a
scale on the approximation function described by equation GFLIB_Cos_Eq2, results in
the following:
Equation GFLIB_Cos_Eq4
Equation GFLIB_Cos_Eq3 can be further rewritten into the following form:
Equation GFLIB_Cos_Eq5
Embedded Software and Motor Control Libraries for PXR40xx, Rev. 1.0
318 Freescale Semiconductor, Inc.