Average Error: 40.3 → 0.3
Time: 55.8s
Precision: 64
Internal Precision: 1344
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l}
\mathbf{if}\;\sqrt[3]{{\left(\left(x \cdot \frac{1}{6}\right) \cdot x + \left(1 + \frac{1}{2} \cdot x\right)\right)}^{3}} \le 1.0568894441533945:\\
\;\;\;\;\sqrt[3]{{\left(\left(x \cdot \frac{1}{6}\right) \cdot x + \left(1 + \frac{1}{2} \cdot x\right)\right)}^{3}}\\
\mathbf{else}:\\
\;\;\;\;\frac{e^{x}}{x} - \frac{1}{x}\\
\end{array}\]
Try it out
Enter valid numbers for all inputs
Target
| Original | 40.3 |
|---|
| Target | 39.5 |
|---|
| 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 (cbrt (pow (+ (* (* x 1/6) x) (+ 1 (* 1/2 x))) 3)) < 1.0568894441533945
Initial program 60.1
\[\frac{e^{x} - 1}{x}\]
Taylor expanded around 0 0.4
\[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2} + \left(1 + \frac{1}{2} \cdot x\right)}\]
- Using strategy
rm Applied add-cbrt-cube0.4
\[\leadsto \color{blue}{\sqrt[3]{\left(\left(\frac{1}{6} \cdot {x}^{2} + \left(1 + \frac{1}{2} \cdot x\right)\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + \left(1 + \frac{1}{2} \cdot x\right)\right)\right) \cdot \left(\frac{1}{6} \cdot {x}^{2} + \left(1 + \frac{1}{2} \cdot x\right)\right)}}\]
Applied simplify0.4
\[\leadsto \sqrt[3]{\color{blue}{{\left(\left(x \cdot \frac{1}{6}\right) \cdot x + \left(1 + \frac{1}{2} \cdot x\right)\right)}^{3}}}\]
if 1.0568894441533945 < (cbrt (pow (+ (* (* x 1/6) x) (+ 1 (* 1/2 x))) 3))
Initial program 0.0
\[\frac{e^{x} - 1}{x}\]
- Using strategy
rm Applied div-sub0.0
\[\leadsto \color{blue}{\frac{e^{x}}{x} - \frac{1}{x}}\]
- Recombined 2 regimes into one program.
Runtime
herbie shell --seed '#(1072743783 989954326 4239155542 3782239461 3602631542 1719177920)'
(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))
