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

⬇

\[\begin{array}{l} \mathbf{if}\;x \le -8005246051280362.0:\\ \;\;\;\;\frac{1}{{x}^{3}} - \left({x}^{\left(-2\right)} + \frac{1}{{x}^{4}}\right)\\ \mathbf{if}\;x \le 9271942351367.576:\\ \;\;\;\;\frac{x - \left(1 + x\right)}{\left(x + 1\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{3}} - \left({x}^{\left(-2\right)} + \frac{1}{{x}^{4}}\right)\\ \end{array}\]

Derivation

Split input into 2 regimes.
if x < -8005246051280362.0 or 9271942351367.576 < x
1. Initial program 28.5
  \[\frac{1}{x + 1} - \frac{1}{x}\]
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 -8005246051280362.0 < x < 9271942351367.576
1. Initial program 1.0
  \[\frac{1}{x + 1} - \frac{1}{x}\]
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 '#(3283856077 3183919125 3399458751 396155847 3688194147 1862413033)'
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))

Error

Derivation

`if x < -8005246051280362.0 or 9271942351367.576 < x`

`if -8005246051280362.0 < x < 9271942351367.576`

Runtime