Average Error: 58.8 → 2.8
Time: 1.5m
Precision: 64
Internal Precision: 2432
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l}
\mathbf{if}\;\frac{1}{b} + \frac{1}{a} \le -3.9176709166158323 \cdot 10^{-202}:\\
\;\;\;\;\frac{1}{b} + \frac{1}{a}\\
\mathbf{if}\;\frac{1}{b} + \frac{1}{a} \le 4.4473138216824803 \cdot 10^{-160}:\\
\;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\
\mathbf{else}:\\
\;\;\;\;\frac{1}{b} + \frac{1}{a}\\
\end{array}\]
Target
| Original | 58.8 |
|---|
| Target | 14.1 |
|---|
| Herbie | 2.8 |
|---|
\[\frac{a + b}{a \cdot b}\]
Derivation
- Split input into 2 regimes
if (+ (/ 1 b) (/ 1 a)) < -3.9176709166158323e-202 or 4.4473138216824803e-160 < (+ (/ 1 b) (/ 1 a))
Initial program 59.9
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
Taylor expanded around 0 2.2
\[\leadsto \color{blue}{\frac{1}{b} + \frac{1}{a}}\]
if -3.9176709166158323e-202 < (+ (/ 1 b) (/ 1 a)) < 4.4473138216824803e-160
Initial program 23.4
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
- Recombined 2 regimes into one program.
Runtime
herbie shell --seed '#(1070833653 108281690 3330367898 3632331308 3494323072 43156186)'
(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))))
