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

⬇

\[\begin{array}{l} \mathbf{if}\;x \le -209135036385.79047:\\ \;\;\;\;\frac{1}{{x}^{3}} - \left({x}^{\left(-2\right)} + \frac{1}{{x}^{4}}\right)\\ \mathbf{if}\;x \le 290639.63932394225:\\ \;\;\;\;\frac{x - \left(1 + x\right)}{\left(x + 1\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{3}} - \left({x}^{\left(-2\right)} + \frac{1}{{x}^{4}}\right)\\ \end{array}\]

Derivation

Split input into 2 regimes.
if x < -209135036385.79047 or 290639.63932394225 < x
1. Initial program 28.4
  \[\frac{1}{x + 1} - \frac{1}{x}\]
2. Applied taylor 0.8
  \[\leadsto \frac{1}{{x}^{3}} - \left(\frac{1}{{x}^2} + \frac{1}{{x}^{4}}\right)\]
3. Taylor expanded around inf 0.8
  
  \[\leadsto \color{blue}{\frac{1}{{x}^{3}} - \left(\frac{1}{{x}^2} + \frac{1}{{x}^{4}}\right)}\]
4. Using strategy rm
5. Applied pow2 0.8
  \[\leadsto \frac{1}{{x}^{3}} - \left(\frac{1}{\color{blue}{{x}^{2}}} + \frac{1}{{x}^{4}}\right)\]
6. Applied pow-flip 0.0
  \[\leadsto \frac{1}{{x}^{3}} - \left(\color{blue}{{x}^{\left(-2\right)}} + \frac{1}{{x}^{4}}\right)\]
if -209135036385.79047 < x < 290639.63932394225
1. Initial program 0.4
  \[\frac{1}{x + 1} - \frac{1}{x}\]
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 shell --seed '#(772101555 1905824529 294602591 2478279198 2123125427 4197813737)'
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))

Error

Derivation

`if x < -209135036385.79047 or 290639.63932394225 < x`

`if -209135036385.79047 < x < 290639.63932394225`

Runtime