Average Error: 40.1 → 0.3
Time: 18.5s
Precision: 64
Internal Precision: 1344
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l}
\mathbf{if}\;x \le -0.00013592676717935897:\\
\;\;\;\;\frac{\left(\sqrt[3]{e^{x} - 1} \cdot \sqrt[3]{e^{x} - 1}\right) \cdot \left(\sqrt[3]{\sqrt[3]{e^{x} - 1} \cdot \sqrt[3]{e^{x} - 1}} \cdot \sqrt[3]{\sqrt[3]{e^{x} - 1}}\right)}{x}\\
\mathbf{else}:\\
\;\;\;\;\frac{1}{6} \cdot {x}^{2} + \left(1 + \frac{1}{2} \cdot x\right)\\
\end{array}\]
Target
| Original | 40.1 |
|---|
| Target | 39.3 |
|---|
| Herbie | 0.3 |
|---|
\[\begin{array}{l}
\mathbf{if}\;x \lt 1 \land x \gt -1:\\
\;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\
\mathbf{else}:\\
\;\;\;\;\frac{e^{x} - 1}{x}\\
\end{array}\]
Derivation
- Split input into 2 regimes
if x < -0.00013592676717935897
Initial program 0.1
\[\frac{e^{x} - 1}{x}\]
- Using strategy
rm Applied add-cube-cbrt0.1
\[\leadsto \frac{\color{blue}{\left(\sqrt[3]{e^{x} - 1} \cdot \sqrt[3]{e^{x} - 1}\right) \cdot \sqrt[3]{e^{x} - 1}}}{x}\]
- Using strategy
rm Applied add-cube-cbrt0.1
\[\leadsto \frac{\left(\sqrt[3]{e^{x} - 1} \cdot \sqrt[3]{e^{x} - 1}\right) \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{e^{x} - 1} \cdot \sqrt[3]{e^{x} - 1}\right) \cdot \sqrt[3]{e^{x} - 1}}}}{x}\]
Applied cbrt-prod0.1
\[\leadsto \frac{\left(\sqrt[3]{e^{x} - 1} \cdot \sqrt[3]{e^{x} - 1}\right) \cdot \color{blue}{\left(\sqrt[3]{\sqrt[3]{e^{x} - 1} \cdot \sqrt[3]{e^{x} - 1}} \cdot \sqrt[3]{\sqrt[3]{e^{x} - 1}}\right)}}{x}\]
if -0.00013592676717935897 < x
Initial program 60.3
\[\frac{e^{x} - 1}{x}\]
Taylor expanded around 0 0.5
\[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2} + \left(1 + \frac{1}{2} \cdot x\right)}\]
- Recombined 2 regimes into one program.
Runtime
herbie shell --seed '#(1071948828 1180510430 2986424009 997076509 406109801 420189285)'
(FPCore (x)
:name "Kahan's exp quotient"
:herbie-target
(if (and (< x 1) (> x -1)) (/ (- (exp x) 1) (log (exp x))) (/ (- (exp x) 1) x))
(/ (- (exp x) 1) x))
