Average Error: 39.0 → 0.2
Time: 10.3s
Precision: 64
Internal Precision: 1344
$\log \left(1 + x\right)$
$\begin{array}{l} \mathbf{if}\;x \le 0.0001756469204950462:\\ \;\;\;\;x + \left(x \cdot \frac{1}{3} - \frac{1}{2}\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{x + 1}\right) + \log \left(\sqrt{x + 1}\right)\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 2 regimes
2. ## if x < 0.0001756469204950462

1. Initial program 58.7

$\log \left(1 + x\right)$
2. Initial simplification58.7

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

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

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

## if 0.0001756469204950462 < x

1. Initial program 0.0

$\log \left(1 + x\right)$
2. Initial simplification0.0

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

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

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

$\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.0001756469204950462:\\ \;\;\;\;x + \left(x \cdot \frac{1}{3} - \frac{1}{2}\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{x + 1}\right) + \log \left(\sqrt{x + 1}\right)\\ \end{array}$

# Runtime

Time bar (total: 10.3s)Debug log

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