Average Error: 7.5 → 0.1
Time: 15.6s
Precision: 64
Internal Precision: 576
$a + \log \left(\frac{x}{1 + a}\right)$
$\begin{array}{l} \mathbf{if}\;x \le -3.61649337106027 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(\frac{\frac{\frac{1}{2}}{a}}{a} - \log \left(\frac{-1}{x}\right)\right) - \left(\frac{1}{a} - \log \left(\frac{-1}{a}\right)\right)\right) + a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log x - \log \left(\sqrt{a + 1}\right)\right) - \log \left(\sqrt{a + 1}\right)\right) + a\\ \end{array}$

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# Derivation

1. Split input into 2 regimes
2. ## if x < -3.61649337106027e-310

1. Initial program 14.3

$a + \log \left(\frac{x}{1 + a}\right)$
2. Using strategy rm
3. Applied log-div63.8

$\leadsto a + \color{blue}{\left(\log x - \log \left(1 + a\right)\right)}$
4. Taylor expanded around -inf 0.2

$\leadsto a + \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{{a}^{2}} + \log \left(\frac{-1}{a}\right)\right) - \left(\log \left(\frac{-1}{x}\right) + \frac{1}{a}\right)\right)}$
5. Simplified0.2

$\leadsto a + \color{blue}{\left(\left(\frac{\frac{\frac{1}{2}}{a}}{a} - \log \left(\frac{-1}{x}\right)\right) - \left(\frac{1}{a} - \log \left(\frac{-1}{a}\right)\right)\right)}$

## if -3.61649337106027e-310 < x

1. Initial program 5.1

$a + \log \left(\frac{x}{1 + a}\right)$
2. Using strategy rm
3. Applied log-div0.0

$\leadsto a + \color{blue}{\left(\log x - \log \left(1 + a\right)\right)}$
4. Using strategy rm
5. Applied add-sqr-sqrt0.0

$\leadsto a + \left(\log x - \log \color{blue}{\left(\sqrt{1 + a} \cdot \sqrt{1 + a}\right)}\right)$
6. Applied log-prod0.0

$\leadsto a + \left(\log x - \color{blue}{\left(\log \left(\sqrt{1 + a}\right) + \log \left(\sqrt{1 + a}\right)\right)}\right)$
7. Applied associate--r+0.0

$\leadsto a + \color{blue}{\left(\left(\log x - \log \left(\sqrt{1 + a}\right)\right) - \log \left(\sqrt{1 + a}\right)\right)}$
3. Recombined 2 regimes into one program.
4. Final simplification0.1

$\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.61649337106027 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(\frac{\frac{\frac{1}{2}}{a}}{a} - \log \left(\frac{-1}{x}\right)\right) - \left(\frac{1}{a} - \log \left(\frac{-1}{a}\right)\right)\right) + a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log x - \log \left(\sqrt{a + 1}\right)\right) - \log \left(\sqrt{a + 1}\right)\right) + a\\ \end{array}$

# Runtime

Time bar (total: 15.6s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'
(FPCore (a x)
:name "a+log(x/(1+a))"
(+ a (log (/ x (+ 1 a)))))