Embedded Software and Motor Control Libraries for PXR40xx

Equation GFLIB_Tan_Eq1
Because both sin(x) and cos(x) are defined on interval [- π, π), function tan(x) is equal to
zero when sin(x)=0 and is equal to infinity when cos(x)=0. Therefore, the tangent
function has asymptotes at n* π/2 for n= , , ... The graph of tan(x) is shown in Figure
4-40.
Figure 4-40. Course of the function GFLIB_Tan
The GFLIB_Tan_F32 function is implemented with consideration to fixed point
fractional arithmetic, hence all tangent values falling beyond [-1, 1) are truncated to -1
and 1 respectively. This truncation is applied for angles in the ranges [-3 π/4, - π/4) and
[ π/4, 3 π/4). As can be further seen from Figure 4-40, tangent values are identical for
angles in the ranges:
1. [- π/4, 0) and [3 π/4, π)
2. [- π, -3 π/4) and [0, π/4)
Chapter 4 API References
Moreover, it can be observed from Figure 4-40 that the course of the tan(x) function
output for angles in interval 1. is identical, but with the opposite sign, to output for angles
in interval 2. Therefore, the approximation of the tangent function over the entire defined
range of input angles can be simplified to an approximation for angles in the range [0, π/
4), and then, depending on the input angle, the result will be negated. In order to increase
the accuracy of approximation without the need for a higher order polynomial, the
interval [0, π/4) is further divided into eight equally spaced sub intervals, and 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 4th order
to achieve an accuracy of less than 0.5LSB (on the upper 16 bits of 32-bit results) across
the full range of input angles.
Embedded Software and Motor Control Libraries for PXR40xx, Rev. 1.0
Freescale Semiconductor, Inc. 421