\[\frac{1}{x + 1} - \frac{1}{x}\]

⬇

\[\begin{array}{l} \mathbf{if}\;x \le -8.658073314327725 \cdot 10^{-05}:\\ \;\;\;\;\frac{1}{{x}^{3}} - \left(\frac{1}{{x}^{4}} + {x}^{\left(-2\right)}\right)\\ \mathbf{if}\;x \le 2486138704986031.0:\\ \;\;\;\;\frac{x - \left(1 + x\right)}{\left(x + 1\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{3}} - \left(\frac{1}{{x}^{4}} + {x}^{\left(-2\right)}\right)\\ \end{array}\]

Derivation

Split input into 2 regimes.
if x < -8.658073314327725e-05 or 2486138704986031.0 < x
1. Initial program 29.4
  \[\frac{1}{x + 1} - \frac{1}{x}\]
2. Applied taylor 0.7
  \[\leadsto \frac{1}{{x}^{3}} - \left(\frac{1}{{x}^{4}} + \frac{1}{{x}^2}\right)\]
3. Taylor expanded around inf 0.7
  
  \[\leadsto \color{blue}{\frac{1}{{x}^{3}} - \left(\frac{1}{{x}^{4}} + \frac{1}{{x}^2}\right)}\]
4. Using strategy rm
5. Applied pow2 0.7
  \[\leadsto \frac{1}{{x}^{3}} - \left(\frac{1}{{x}^{4}} + \frac{1}{\color{blue}{{x}^{2}}}\right)\]
6. Applied pow-flip 0.0
  \[\leadsto \frac{1}{{x}^{3}} - \left(\frac{1}{{x}^{4}} + \color{blue}{{x}^{\left(-2\right)}}\right)\]
if -8.658073314327725e-05 < x < 2486138704986031.0
1. Initial program 0.7
  \[\frac{1}{x + 1} - \frac{1}{x}\]
2. Using strategy rm
3. Applied frac-sub 0.0
  \[\leadsto \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 1}{\left(x + 1\right) \cdot x}}\]
4. Applied simplify 0.0
  \[\leadsto \frac{\color{blue}{x - \left(1 + x\right)}}{\left(x + 1\right) \cdot x}\]
Recombined 2 regimes into one program.
Removed slow pow expressions

Runtime

Please include this information when filing a bug report:

herbie --seed '#(3361196376 1818039471 2480134366 3803070597 1215164070 3576802717)'
(FPCore (x)
  :name "NMSE problem 3.3.1"
  (- (/ 1 (+ x 1)) (/ 1 x)))

Error

Derivation

`if x < -8.658073314327725e-05 or 2486138704986031.0 < x`

`if -8.658073314327725e-05 < x < 2486138704986031.0`

Runtime