Average Error: 39.2 → 0.1
Time: 16.2s
Precision: 64
Internal Precision: 1344
$\log \left(\frac{x}{1 + x}\right)$
$\begin{array}{l} \mathbf{if}\;x \le -7181.902587127047 \lor \neg \left(x \le 7113.091247550719\right):\\ \;\;\;\;\frac{\left(\frac{\frac{1}{2}}{x} - 1\right) - \frac{\frac{1}{3}}{x \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{\frac{x}{x + 1}}\right) + \log \left(\sqrt{\frac{x}{x + 1}}\right)\\ \end{array}$

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Results

 In Out
Enter valid numbers for all inputs

Derivation

1. Split input into 2 regimes
2. if x < -7181.902587127047 or 7113.091247550719 < x

1. Initial program 59.3

$\log \left(\frac{x}{1 + x}\right)$
2. Taylor expanded around inf 0.0

$\leadsto \color{blue}{\frac{1}{2} \cdot \frac{1}{{x}^{2}} - \left(\frac{1}{x} + \frac{1}{3} \cdot \frac{1}{{x}^{3}}\right)}$
3. Simplified0.0

$\leadsto \color{blue}{\frac{\left(\frac{\frac{1}{2}}{x} - 1\right) - \frac{\frac{1}{3}}{x \cdot x}}{x}}$

if -7181.902587127047 < x < 7113.091247550719

1. Initial program 0.1

$\log \left(\frac{x}{1 + x}\right)$
2. Using strategy rm

$\leadsto \log \color{blue}{\left(\sqrt{\frac{x}{1 + x}} \cdot \sqrt{\frac{x}{1 + x}}\right)}$
4. Applied log-prod0.2

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

$\leadsto \begin{array}{l} \mathbf{if}\;x \le -7181.902587127047 \lor \neg \left(x \le 7113.091247550719\right):\\ \;\;\;\;\frac{\left(\frac{\frac{1}{2}}{x} - 1\right) - \frac{\frac{1}{3}}{x \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{\frac{x}{x + 1}}\right) + \log \left(\sqrt{\frac{x}{x + 1}}\right)\\ \end{array}$

Runtime

Time bar (total: 16.2s)Debug log

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