Embedded Software and Motor Control Libraries for PXR40xx

Figure 4-19. Course of the function GFLIB_Atan
The GFLIB_Atan_F32 function is implemented with consideration to fixed point
fractional arithmetic. As can be further seen from Figure 4-19, the arctangent values are
identical for the input ranges [-1, 0) and [0, 1).
Moreover, it can be observed from Figure 4-19 that the course of the arctan(x) function
output for a ratio in interval 1. is identical, but with the opposite sign, to the output for a
ratio in interval 2. Therefore, the approximation of the arctangent function over the entire
defined range of input ratios can be simplified to the approximation for a ratio in range
[0, 1), and then, depending on the input ratio, the result will be negated. In order to
increase the accuracy of approximation without the need for a higher order polynomial,
the interval [0, 1) is further divided into eight equally spaced sub intervals, and minimax
polynomial approximation is done for each interval respectively. Such a division results
in eight sets of polynomial coefficients. Moreover, it allows using a polynomial of only
the 3rd order to achieve an accuracy of less than 0.5LSB (on the upper 16 bits of 32-bit
results) across the entire range of input ratios.
The GFLIB_Atan_F32 function uses fixed point fractional arithmetic, so to cast the
fractional value of the output angle [-0.25, 0.25) into the correct range [- π/4, π/4), the
fixed point output angle can be multiplied by π for an angle in radians. Then, the fixed
point fractional implementation of the minimax approximation polynomial, used for
calculation of each sub sector, is defined as follows:
Equation GFLIB_Atan_Eq1
Embedded Software and Motor Control Libraries for PXR40xx, Rev. 1.0
Chapter 4 API References
Freescale Semiconductor, Inc. 235