§4.45 Methods of Computation

The function $\ln x$ can always be computed from its ascending power series after preliminary scaling. Suppose first $1/10\le x\le 10$. Then we take square roots repeatedly until $|y|$ is sufficiently small, where

After computing $\ln \left(1+y\right)$ from (4.6.1)

For other values of $x$ set $x={10}^m\xi$, where $1/10\le \xi \le 10$ and $m\in \mathbb{Z}$. Then

Let $x$ have any real value. First, rescale via

Then

and since $|y|\le \frac{1}{2}\ln 10=1.15\mathrm{\dots }$, ${\mathrm{e}}^y$ can be computed straightforwardly from (4.2.19).

Let $x$ have any real value. We first compute $\xi =x/\pi$, followed by

Then

and since $|\theta |\le \frac{1}{2}\pi =1.57\mathrm{\dots }$, $\sin \theta$ and $\cos \theta$ can be computed straightforwardly from (4.19.1) and (4.19.2).

The other trigonometric functions can be found from the definitions (4.14.4)–(4.14.7).

The function $\arctan x$ can always be computed from its ascending power series after preliminary transformations to reduce the size of $x$. From (4.24.15) with $u=v=({(1+x^2)}^{1/2}-1)/x$, we have

Beginning with $x_0=x$, generate the sequence

until $x_n$ is sufficiently small. We then compute $\arctan x_n$ from (4.24.3), followed by

Another method, when $x$ is large, is to sum

compare (4.24.4).

As an example, take $x=9.47376$. Then

From (4.24.3) $\arctan x_4=0.09160\mathrm{\dots }$. From (4.45.10)

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, $\arcsin x=\arctan \left(x{(1-x^2)}^{-1/2}\right)$.

The hyperbolic functions can be computed directly from the definitions (4.28.1)–(4.28.7). The inverses $\mathrm{arcsinh}$, $\mathrm{arccosh}$, and $\mathrm{arctanh}$ can be computed from the logarithmic forms given in §4.37(iv), with real arguments. For $\mathrm{arccsch}$, $\mathrm{arcsech}$, and $\mathrm{arccoth}$ we have (4.37.7)–(4.37.9).

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).