Average Error: 9.9 → 0.5
Time: 10.2s
Precision: 64
Internal precision: 1152
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
⬇
\[\begin{array}{l}
\mathbf{if}\;x \le -0.10283760294103053:\\
\;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{2}{x} \cdot \frac{1}{x \cdot x}\\
\mathbf{if}\;x \le 1850694803779224.5:\\
\;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\
\mathbf{else}:\\
\;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{\frac{2}{x}}{x \cdot x}\\
\end{array}\]
Target
| Original | 9.9 |
|---|
| Comparison | 0.3 |
|---|
| Herbie | 0.5 |
|---|
\[ \frac{2}{x \cdot \left({x}^2 - 1\right)} \]
Derivation
- Split input into 3 regimes.
-
if x < -0.10283760294103053
Initial program 19.4
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
Applied taylor 0.5
\[\leadsto 2 \cdot \frac{1}{{x}^{3}} + \left(2 \cdot \frac{1}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{7}}\right)\]
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 simplify 0.1
\[\leadsto \color{blue}{\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{\frac{2}{x}}{x \cdot x}}\]
- Using strategy
rm
Applied div-inv 0.1
\[\leadsto \left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \color{blue}{\frac{2}{x} \cdot \frac{1}{x \cdot x}}\]
if -0.10283760294103053 < x < 1850694803779224.5
Initial program 0.9
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
if 1850694803779224.5 < x
Initial program 19.3
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
Applied taylor 0.5
\[\leadsto 2 \cdot \frac{1}{{x}^{3}} + \left(2 \cdot \frac{1}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{7}}\right)\]
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 simplify 0.1
\[\leadsto \color{blue}{\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{\frac{2}{x}}{x \cdot x}}\]
- Recombined 3 regimes into one program.
- Removed slow pow expressions
Runtime
Please include this information when filing a bug report:
herbie shell --seed '#(1067901057 3396600083 3715501224 3126139233 3908045574 1593683916)'
(FPCore (x)
:name "3frac (problem 3.3.3)"
:target
(/ 2 (* x (- (sqr x) 1)))
(+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))
