Average Error: 33.1 → 0.3
Time: 13.2s
Precision: 64
Internal Precision: 1408
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l}
\mathbf{if}\;a \cdot x \le -9.120824479008042 \cdot 10^{-14}:\\
\;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\
\mathbf{else}:\\
\;\;\;\;x \cdot a + \left(x \cdot a\right) \cdot \left(\frac{1}{2} \cdot \left(x \cdot a\right)\right)\\
\end{array}\]
Target
| Original | 33.1 |
|---|
| Target | 7.8 |
|---|
| Herbie | 0.3 |
|---|
\[\begin{array}{l}
\mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\
\;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\
\mathbf{else}:\\
\;\;\;\;e^{a \cdot x} - 1\\
\end{array}\]
Derivation
- Split input into 2 regimes
if (* a x) < -9.120824479008042e-14
Initial program 0.8
\[e^{a \cdot x} - 1\]
- Using strategy
rm Applied add-log-exp0.8
\[\leadsto \color{blue}{\log \left(e^{e^{a \cdot x} - 1}\right)}\]
if -9.120824479008042e-14 < (* a x)
Initial program 47.5
\[e^{a \cdot x} - 1\]
Taylor expanded around 0 21.6
\[\leadsto \color{blue}{\frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right) + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x\right)}\]
Taylor expanded around inf 12.2
\[\leadsto \frac{1}{6} \cdot \color{blue}{0} + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x\right)\]
Applied simplify0.1
\[\leadsto \color{blue}{x \cdot a + \left(x \cdot a\right) \cdot \left(\frac{1}{2} \cdot \left(x \cdot a\right)\right)}\]
- Recombined 2 regimes into one program.
- Removed slow
pow expressions.
Runtime
herbie shell --seed '#(1062803647 245428163 493620569 3595423923 1908391097 2390014376)'
(FPCore (a x)
:name "expax (section 3.5)"
:herbie-target
(if (< (fabs (* a x)) 1/10) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))
(- (exp (* a x)) 1))
