Average Error: 58.7 → 3.8
Time: 59.8s
Precision: 64
Internal Precision: 2368
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l}
\mathbf{if}\;\sqrt[3]{\frac{1}{b} + \frac{1}{a}} \le -2.965304485368676 \cdot 10^{-35}:\\
\;\;\;\;\frac{1}{b} + \frac{1}{a}\\
\mathbf{if}\;\sqrt[3]{\frac{1}{b} + \frac{1}{a}} \le 4.3005371419799126 \cdot 10^{-36}:\\
\;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\
\mathbf{else}:\\
\;\;\;\;\frac{1}{b} + \frac{1}{a}\\
\end{array}\]
Target
| Original | 58.7 |
|---|
| Target | 14.4 |
|---|
| Herbie | 3.8 |
|---|
\[\frac{a + b}{a \cdot b}\]
Derivation
- Split input into 2 regimes
if (cbrt (+ (/ 1 b) (/ 1 a))) < -2.965304485368676e-35 or 4.3005371419799126e-36 < (cbrt (+ (/ 1 b) (/ 1 a)))
Initial program 60.9
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
Taylor expanded around 0 1.2
\[\leadsto \color{blue}{\frac{1}{b} + \frac{1}{a}}\]
if -2.965304485368676e-35 < (cbrt (+ (/ 1 b) (/ 1 a))) < 4.3005371419799126e-36
Initial program 33.2
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
- Recombined 2 regimes into one program.
Runtime
herbie shell --seed '#(1071821486 549052472 3784827256 1559736200 3548510075 881134285)'
(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))))
