Average Error: 14.8 → 0.1
Time: 49.2s
Precision: 64
Internal Precision: 576
$\sin \left(\tan^{-1} \left(\sqrt{\left(\left(-a\right) \cdot a\right) \cdot \log x}\right)\right)$
$\begin{array}{l} \mathbf{if}\;a \le -2.0127090758672 \cdot 10^{-311}:\\ \;\;\;\;\sin \left(\tan^{-1} \left(\left(-a\right) \cdot \sqrt{-\log x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\tan^{-1} \left(a \cdot \sqrt{-\log x}\right)\right)\\ \end{array}$

# 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

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

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

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

## if -2.0127090758672e-311 < a

1. Initial program 14.3

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

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

# Runtime

Time bar (total: 49.2s)Debug log

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