Average Error: 38.6 → 0.5
Time: 9.6s
Precision: 64
Internal Precision: 1344
$e^{x} - 1$
$\begin{array}{l} \mathbf{if}\;e^{x} - 1 \le -1.4757203902199797 \cdot 10^{-11}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{3} - 1}{e^{x + x} + \left(1 + e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right)\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 2 regimes
2. ## if (- (exp x) 1) < -1.4757203902199797e-11

1. Initial program 0.6

$e^{x} - 1$
2. Using strategy rm
3. Applied flip3--0.6

$\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)}}$
4. Applied simplify0.6

$\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{3} - 1}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)}$
5. Applied simplify0.6

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

## if -1.4757203902199797e-11 < (- (exp x) 1)

1. Initial program 59.0

$e^{x} - 1$
2. Taylor expanded around 0 0.5

$\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2} + \left(x + \frac{1}{6} \cdot {x}^{3}\right)}$
3. Applied simplify0.5

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

# Runtime

Time bar (total: 9.6s)Debug log

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