Fast tangent and secant approximation

• Input range (-π/4, π/4) expressed as binary angle measurement (0xe000, 0xffff] U [0x0000, 0x2000)
• 512 byte lookup table (256b each for tangent and secant)
• Q18 Fixed-point result accurate within 2 bits

Table generation

gen_values.py

• Use lolremez Remez Method library to generate polynomial of degree 2 for 32 divisions of input range
• Incorporate fixed-point shift and base offset into constant term

gen_array.py

• Pack coefficients into 8 bytes per division
• Output C++ header with array and constants

Full-range arctan2 using CORDIC

• Output angle expressed as binary angle measurement [0x0, 0xffff]
• 10 iterations of CORDIC vectoring mode to find angle
• Angle accurate to within 5 bits (32 / 0x10000 approx. 0.0031 rad)
• Hypotenuse optional
  • Calculated by dividing result x by gain $\prod_{i=0}^{9} \sqrt{1 + 2^{-2 \cdot i}} \approx 1.64676$

Branchless ARMv4 implementation

Tangent

• 16 ALU operations
• 1 multiply and 1 multiply-accumulate operation
• 2 load operations
• Completes in 30 cycles including return (on system with 1 cycle word read)

Secant

• Same polynomial evaluation core as tangent
• 3 cycles to adjust input followed by tangent core, 33 cycles

Arctangent

• 89 cycles including return, all ALU

  • 12 cycles for input adjustment to range (-π/4, π/4)
  • 75 cycles for 10 CORDIC iterations
  • 2 cycles for negative angle adjustment

• 3 cycles for Q8 hypotenuse gain adjustment (if used)

  • See header for hypotenuse use and gain adjustment

ARMv4 source including table