Average Error: 14.8 → 0.1
Time: 44.2s
Precision: 64
Internal Precision: 576
$\tan^{-1} \left(\sqrt{\left(\left(-a\right) \cdot a\right) \cdot \log x}\right)$
$\begin{array}{l} \mathbf{if}\;a \le -2.0127090758672 \cdot 10^{-311}:\\ \;\;\;\;\tan^{-1} \left(\sqrt{-\log x} \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(a \cdot \sqrt{-\log x}\right)\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 2 regimes
2. ## if a < -2.0127090758672e-311

1. Initial program 15.3

$\tan^{-1} \left(\sqrt{\left(\left(-a\right) \cdot a\right) \cdot \log x}\right)$
2. Taylor expanded around -inf 62.3

$\leadsto \tan^{-1} \color{blue}{\left(-1 \cdot \left(a \cdot \sqrt{-1 \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}\right)\right)}$
3. Applied simplify0.1

$\leadsto \color{blue}{\tan^{-1} \left(\sqrt{-\log x} \cdot \left(-a\right)\right)}$

## if -2.0127090758672e-311 < a

1. Initial program 14.3

$\tan^{-1} \left(\sqrt{\left(\left(-a\right) \cdot a\right) \cdot \log x}\right)$
2. Taylor expanded around 0 0.1

$\leadsto \tan^{-1} \color{blue}{\left(a \cdot \sqrt{-\log x}\right)}$
3. Recombined 2 regimes into one program.

# Runtime

Time bar (total: 44.2s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'
(FPCore (a x)
:name "atan(sqrt(-a*a*log(x)))"
(atan (sqrt (* (* (- a) a) (log x)))))