Average Error: 28.9 → 0.3
Time: 11.6s
Precision: 64
Internal Precision: 1344
$e^{a \cdot x} - 1$
$\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.0011015980719615973:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(a \cdot \left(x \cdot \frac{1}{6}\right) + \frac{1}{2}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\\ \end{array}$

# Try it out

Results

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Enter valid numbers for all inputs

# Derivation

1. Split input into 2 regimes
2. ## if (* a x) < -0.0011015980719615973

1. Initial program 0.0

$e^{a \cdot x} - 1$
2. Using strategy rm

$\leadsto \color{blue}{\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}}$

## if -0.0011015980719615973 < (* a x)

1. Initial program 44.1

$e^{a \cdot x} - 1$
2. Taylor expanded around 0 13.9

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

$\leadsto \color{blue}{a \cdot x + \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot \left(\frac{1}{6} \cdot x\right) + \frac{1}{2}\right)}$
3. Recombined 2 regimes into one program.
4. Final simplification0.3

$\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.0011015980719615973:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(a \cdot \left(x \cdot \frac{1}{6}\right) + \frac{1}{2}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\\ \end{array}$

# Runtime

Time bar (total: 11.6s)Debug log

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