Average Error: 10.0 → 0.2
Time: 37.3s
Precision: 64
Internal Precision: 1152
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l}
\mathbf{if}\;x \le -128.92025171658022:\\
\;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{2}{{x}^{3}}\\
\mathbf{if}\;x \le 100.97752181982032:\\
\;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\
\mathbf{else}:\\
\;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{\frac{\frac{2}{x}}{x}}{x}\\
\end{array}\]
Target
| Original | 10.0 |
|---|
| Target | 0.2 |
|---|
| Herbie | 0.2 |
|---|
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]
Derivation
- Split input into 3 regimes
if x < -128.92025171658022
Initial program 19.5
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
Taylor expanded around inf 0.5
\[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{3}} + \left(2 \cdot \frac{1}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{7}}\right)}\]
Applied simplify0.1
\[\leadsto \color{blue}{\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{\frac{2}{x}}{x \cdot x}}\]
- Using strategy
rm Applied associate-/l/0.6
\[\leadsto \left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \color{blue}{\frac{2}{\left(x \cdot x\right) \cdot x}}\]
Applied simplify0.5
\[\leadsto \left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{2}{\color{blue}{{x}^{3}}}\]
if -128.92025171658022 < x < 100.97752181982032
Initial program 0.0
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
if 100.97752181982032 < x
Initial program 20.7
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
Taylor expanded around inf 0.4
\[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{3}} + \left(2 \cdot \frac{1}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{7}}\right)}\]
Applied simplify0.1
\[\leadsto \color{blue}{\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{\frac{2}{x}}{x \cdot x}}\]
- Using strategy
rm Applied associate-/r*0.1
\[\leadsto \left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \color{blue}{\frac{\frac{\frac{2}{x}}{x}}{x}}\]
- Recombined 3 regimes into one program.
Runtime
herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' +o rules:numerics
(FPCore (x)
:name "3frac (problem 3.3.3)"
:herbie-target
(/ 2 (* x (- (* x x) 1)))
(+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))
