Average Error: 38.8 → 0.2
Time: 12.8s
Precision: 64
Internal Precision: 1344
$\log \left(\frac{1}{1 - x}\right)$
$\begin{array}{l} \mathbf{if}\;x \le -0.00012149033212153346:\\ \;\;\;\;\left(-\log \left(\sqrt{1 - x}\right)\right) + \left(-\log \left(\sqrt{1 - x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(\left(-x\right) - \left(x \cdot x\right) \cdot \left(x \cdot \frac{1}{3} + \frac{1}{2}\right)\right)\\ \end{array}$

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Results

 In Out
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Derivation

1. Split input into 2 regimes
2. if x < -0.00012149033212153346

1. Initial program 0.1

$\log \left(\frac{1}{1 - x}\right)$
2. Initial simplification0.1

$\leadsto -\log \left(1 - x\right)$
3. Using strategy rm

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

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

if -0.00012149033212153346 < x

1. Initial program 58.8

$\log \left(\frac{1}{1 - x}\right)$
2. Initial simplification58.8

$\leadsto -\log \left(1 - x\right)$
3. Taylor expanded around 0 0.2

$\leadsto -\color{blue}{\left(-\left(\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{2} \cdot {x}^{2} + x\right)\right)\right)}$
4. Simplified0.2

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

$\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00012149033212153346:\\ \;\;\;\;\left(-\log \left(\sqrt{1 - x}\right)\right) + \left(-\log \left(\sqrt{1 - x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(\left(-x\right) - \left(x \cdot x\right) \cdot \left(x \cdot \frac{1}{3} + \frac{1}{2}\right)\right)\\ \end{array}$

Runtime

Time bar (total: 12.8s)Debug log

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