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

⬇

\[\begin{array}{l} \mathbf{if}\;x \le -1399953536007.6062:\\ \;\;\;\;\frac{1}{{x}^{3}} - \left({x}^{\left(-2\right)} + \frac{1}{{x}^{4}}\right)\\ \mathbf{if}\;x \le 849008901.3057916:\\ \;\;\;\;\frac{x - \left(1 + x\right)}{\left(x + 1\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{3}} - \left({x}^{\left(-2\right)} + \frac{1}{{x}^{4}}\right)\\ \end{array}\]

Derivation

Split input into 2 regimes.
if x < -1399953536007.6062 or 849008901.3057916 < x
1. Initial program 29.7
  \[\frac{1}{x + 1} - \frac{1}{x}\]
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 -1399953536007.6062 < x < 849008901.3057916
1. Initial program 0.6
  \[\frac{1}{x + 1} - \frac{1}{x}\]
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 '#(1639775192 589002392 3355153433 1792592353 1460638544 2009226429)'
(FPCore (x)
  :name "NMSE problem 3.3.1"
  (- (/ 1 (+ x 1)) (/ 1 x)))

Error

Derivation

`if x < -1399953536007.6062 or 849008901.3057916 < x`

`if -1399953536007.6062 < x < 849008901.3057916`

Runtime