Average Error: 58.3 → 3.6
Time: 1.5m
Precision: 64
Internal Precision: 2368
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l}
\mathbf{if}\;\frac{1}{b} + \frac{1}{a} \le -4.635212546899196 \cdot 10^{-100}:\\
\;\;\;\;\frac{1}{b} + \frac{1}{a}\\
\mathbf{if}\;\frac{1}{b} + \frac{1}{a} \le 1.7540289770605972 \cdot 10^{-197}:\\
\;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\
\mathbf{else}:\\
\;\;\;\;\frac{1}{b} + \frac{1}{a}\\
\end{array}\]
Target
| Original | 58.3 |
|---|
| Target | 14.4 |
|---|
| Herbie | 3.6 |
|---|
\[\frac{a + b}{a \cdot b}\]
Derivation
- Split input into 2 regimes
if (+ (/ 1 b) (/ 1 a)) < -4.635212546899196e-100 or 1.7540289770605972e-197 < (+ (/ 1 b) (/ 1 a))
Initial program 60.1
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
Taylor expanded around 0 1.9
\[\leadsto \color{blue}{\frac{1}{b} + \frac{1}{a}}\]
if -4.635212546899196e-100 < (+ (/ 1 b) (/ 1 a)) < 1.7540289770605972e-197
Initial program 29.9
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
- Recombined 2 regimes into one program.
Runtime
herbie shell --seed '#(1071852389 864846987 1238109217 3425890003 4124793586 650694553)'
(FPCore (a b eps)
:name "expq3 (problem 3.4.2)"
:pre (and (< -1 eps) (< eps 1))
:herbie-target
(/ (+ a b) (* a b))
(/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))))
