An Analytic Expression for the Inverse Involute

Table 1

Summary of the approximations to the inverse involute function.

Formula	Max error (rad); Range	FLOP CR = cube root
Existing in the literature
Dudley u = 1.441 · x^1/3 − 0.366 · x	1.2E − 3 rad (4.12529′); Range = 0°–45°	3 + CR
Laskin (2 iterations)	1.7E − 5 rad = 3.506″; Range = 0°–45°	Uses trigonometric functions
Cheng (9 terms): equation (3)	1.58E − 9 rad = 3.2589E − 4″; Range = 0°–45°	38 + CR
Cheng (2 terms) (economization):	1.308E − 3 rad = 4.4998′; Range = 0°–45°	3 + CR
Liu (6 terms)	2.75E − 5 rad = 5.68″; Range = 0°–45°	21 + CR
Liu (rational inverse trigonometric function):	1.09E − 9 rad = 2.261E − 4″; Range = 0°–45°	Uses inverse trigonometric functions
New formulae
	2.22E − 4 rad = 0.763′; Range = 0°–45°	6 + CR
	8.847E − 6 rad = 1.82″; Range = 0°–45°	5 + CR
	6.6E − 7 rad = 0.137″; Range = 0°–35°	3 + CR