The function can always be computed from its ascending power series after preliminary scaling. Suppose first . Then we take square roots repeatedly until is sufficiently small, where
After computing from (4.6.1)
For other values of set , where and . Then
Let have any real value. First, rescale via
and since , can be computed straightforwardly from (4.2.19).
Let have any real value. We first compute , followed by
The function can always be computed from its ascending power series after preliminary transformations to reduce the size of . From (4.24.15) with , we have
Beginning with , generate the sequence
until is sufficiently small. We then compute from (4.24.3), followed by
Another method, when is large, is to sum
As an example, take . Then
As a check, from (4.45.11)
For the remaining inverse trigonometric functions, we may use the identities provided by the fourth row of Table 4.16.3. For example, .
See Luther (1995), Ziv (1991), Cody and Waite (1980), Rosenberg and McNamee (1976), Carlson (1972a). For interval-arithmetic algorithms, see Markov (1981). For Shift-and-Add and CORDIC algorithms, see Muller (1997), Merrheim (1994), Schelin (1983). For multiprecision methods, see Smith (1989), Brent (1976).
See §1.9(i) for the precise relationship of to the arctangent function.
The trigonometric functions may be computed from the definitions (4.14.1)–(4.14.7), and their inverses from the logarithmic forms in §4.23(iv), followed by (4.23.7)–(4.23.9). Similarly for the hyperbolic and inverse hyperbolic functions; compare (4.28.1)–(4.28.7), §4.37(iv), and (4.37.7)–(4.37.9).
For other methods see Miel (1981).
For the principal branch can be computed by solving the defining equation numerically, for example, by Newton’s rule (§3.8(ii)). Initial approximations are obtainable, for example, from the power series (4.13.6) (with ) when is close to , from the asymptotic expansion (4.13.10) when is large, and by numerical integration of the differential equation (4.13.4) (§3.7) for other values of .
Similarly for in the interval .