Average Error: 39.8 → 1.2
Time: 38.3s
Precision: 64
Internal Precision: 1344
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l}
\mathbf{if}\;\frac{e^{x} - 1}{x} \le -0.0:\\
\;\;\;\;\frac{1}{6} \cdot {x}^{2} + \left(1 + \frac{1}{2} \cdot x\right)\\
\mathbf{if}\;\frac{e^{x} - 1}{x} \le 2.3619944064204065 \cdot 10^{-12}:\\
\;\;\;\;\frac{\frac{{\left(e^{x}\right)}^{3} - 1}{e^{x + x} + \left(e^{x} + 1\right)}}{x}\\
\mathbf{else}:\\
\;\;\;\;\frac{1}{6} \cdot {x}^{2} + \left(1 + \frac{1}{2} \cdot x\right)\\
\end{array}\]
Target
| Original | 39.8 |
|---|
| Target | 39.0 |
|---|
| Herbie | 1.2 |
|---|
\[\begin{array}{l}
\mathbf{if}\;x \lt 1 \land x \gt -1:\\
\;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\
\mathbf{else}:\\
\;\;\;\;\frac{e^{x} - 1}{x}\\
\end{array}\]
Derivation
- Split input into 2 regimes
if (/ (- (exp x) 1) x) < -0.0 or 2.3619944064204065e-12 < (/ (- (exp x) 1) x)
Initial program 58.8
\[\frac{e^{x} - 1}{x}\]
Taylor expanded around 0 1.7
\[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2} + \left(1 + \frac{1}{2} \cdot x\right)}\]
if -0.0 < (/ (- (exp x) 1) x) < 2.3619944064204065e-12
Initial program 0
\[\frac{e^{x} - 1}{x}\]
- Using strategy
rm Applied flip3--0
\[\leadsto \frac{\color{blue}{\frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)}}}{x}\]
Applied simplify0
\[\leadsto \frac{\frac{\color{blue}{{\left(e^{x}\right)}^{3} - 1}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)}}{x}\]
Applied simplify0
\[\leadsto \frac{\frac{{\left(e^{x}\right)}^{3} - 1}{\color{blue}{e^{x + x} + \left(e^{x} + 1\right)}}}{x}\]
- Recombined 2 regimes into one program.
Runtime
herbie shell --seed '#(1071979731 1496239409 439705970 2863295848 982327776 189749553)'
(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))
