Average Error: 38.6 → 0.4
Time: 5.3s
Precision: 64
Internal Precision: 1344
$e^{x} - 1$
$\begin{array}{l} \mathbf{if}\;x \le -0.00016688961595810102:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{e^{x} \cdot e^{x} + \left(e^{x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{6} \cdot x + \frac{1}{2}\right) \cdot \left(x \cdot x\right) + x\\ \end{array}$

# Try it out

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# Derivation

1. Split input into 2 regimes
2. ## if x < -0.00016688961595810102

1. Initial program 0.1

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

$\leadsto e^{x} - 1$
3. Using strategy rm
4. Applied flip3--0.1

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

## if -0.00016688961595810102 < x

1. Initial program 58.5

$e^{x} - 1$
2. Initial simplification58.5

$\leadsto e^{x} - 1$
3. Taylor expanded around 0 0.5

$\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{3} + \left(\frac{1}{2} \cdot {x}^{2} + x\right)}$
4. Simplified0.5

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

$\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00016688961595810102:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{e^{x} \cdot e^{x} + \left(e^{x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{6} \cdot x + \frac{1}{2}\right) \cdot \left(x \cdot x\right) + x\\ \end{array}$

# Runtime

Time bar (total: 5.3s)Debug log

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