Average Error: 29.5 → 0.2
Time: 11.8s
Precision: 64
Internal Precision: 1344
${e}^{x} - 1$
$\begin{array}{l} \mathbf{if}\;x \le -7.06650157794332 \cdot 10^{-07}:\\ \;\;\;\;\log \left(e^{{e}^{x} - 1}\right)\\ \mathbf{elif}\;x \le 1.2248036861576476 \cdot 10^{-06}:\\ \;\;\;\;x \cdot \log e + \left(\frac{1}{2} + \log e \cdot \left(x \cdot \frac{1}{6}\right)\right) \cdot \left(\left(x \cdot \log e\right) \cdot \left(x \cdot \log e\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\log \left(e^{{e}^{x} - 1}\right)} \cdot \sqrt[3]{\log \left(e^{{e}^{x} - 1}\right)}\right) \cdot \sqrt[3]{\log \left(e^{{e}^{x} - 1}\right)}\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 3 regimes
2. ## if x < -7.06650157794332e-07

1. Initial program 0.1

${e}^{x} - 1$
2. Initial simplification0.1

$\leadsto {e}^{x} - 1$
3. Using strategy rm

$\leadsto \color{blue}{\log \left(e^{{e}^{x} - 1}\right)}$

## if -7.06650157794332e-07 < x < 1.2248036861576476e-06

1. Initial program 59.2

${e}^{x} - 1$
2. Initial simplification59.2

$\leadsto {e}^{x} - 1$
3. Taylor expanded around 0 0.3

$\leadsto \color{blue}{\log e \cdot x + \left(\frac{1}{2} \cdot \left({\left(\log e\right)}^{2} \cdot {x}^{2}\right) + \frac{1}{6} \cdot \left({\left(\log e\right)}^{3} \cdot {x}^{3}\right)\right)}$
4. Simplified0.3

$\leadsto \color{blue}{\left(\left(\log e \cdot x\right) \cdot \left(\log e \cdot x\right)\right) \cdot \left(\left(\frac{1}{6} \cdot x\right) \cdot \log e + \frac{1}{2}\right) + \log e \cdot x}$

## if 1.2248036861576476e-06 < x

1. Initial program 0.1

${e}^{x} - 1$
2. Initial simplification0.1

$\leadsto {e}^{x} - 1$
3. Using strategy rm

$\leadsto \color{blue}{\log \left(e^{{e}^{x} - 1}\right)}$
5. Using strategy rm

$\leadsto \color{blue}{\left(\sqrt[3]{\log \left(e^{{e}^{x} - 1}\right)} \cdot \sqrt[3]{\log \left(e^{{e}^{x} - 1}\right)}\right) \cdot \sqrt[3]{\log \left(e^{{e}^{x} - 1}\right)}}$
3. Recombined 3 regimes into one program.
4. Final simplification0.2

$\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.06650157794332 \cdot 10^{-07}:\\ \;\;\;\;\log \left(e^{{e}^{x} - 1}\right)\\ \mathbf{elif}\;x \le 1.2248036861576476 \cdot 10^{-06}:\\ \;\;\;\;x \cdot \log e + \left(\frac{1}{2} + \log e \cdot \left(x \cdot \frac{1}{6}\right)\right) \cdot \left(\left(x \cdot \log e\right) \cdot \left(x \cdot \log e\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\log \left(e^{{e}^{x} - 1}\right)} \cdot \sqrt[3]{\log \left(e^{{e}^{x} - 1}\right)}\right) \cdot \sqrt[3]{\log \left(e^{{e}^{x} - 1}\right)}\\ \end{array}$

# Runtime

Time bar (total: 11.8s)Debug log

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