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

↓

\[\begin{array}{l} \mathbf{if}\;\sqrt[3]{(x \cdot x + x)_*} \le 8773097.605766743:\\ \;\;\;\;\frac{\frac{{x}^{3} - {\left(x + 1\right)}^{3}}{(\left(x + 1\right) \cdot \left((2 \cdot x + 1)_*\right) + \left(x \cdot x\right))_*}}{(x \cdot x + x)_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{3}} - \left(\frac{1}{{x}^{4}} + \frac{1}{{x}^{2}}\right)\\ \end{array}\]

Derivation

Split input into 2 regimes
if (cbrt (fma x x x)) < 8773097.605766743
1. Initial program 0.4
  \[\frac{1}{x + 1} - \frac{1}{x}\]
2. Using strategy rm
3. Applied frac-sub0.0
  \[\leadsto \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 1}{\left(x + 1\right) \cdot x}}\]
4. Applied simplify0.0
  \[\leadsto \frac{\color{blue}{x - \left(x + 1\right)}}{\left(x + 1\right) \cdot x}\]
5. Applied simplify0.0
  \[\leadsto \frac{x - \left(x + 1\right)}{\color{blue}{(x \cdot x + x)_*}}\]
6. Using strategy rm
7. Applied flip3--0.4
  \[\leadsto \frac{\color{blue}{\frac{{x}^{3} - {\left(x + 1\right)}^{3}}{x \cdot x + \left(\left(x + 1\right) \cdot \left(x + 1\right) + x \cdot \left(x + 1\right)\right)}}}{(x \cdot x + x)_*}\]
8. Applied simplify0.4
  \[\leadsto \frac{\frac{{x}^{3} - {\left(x + 1\right)}^{3}}{\color{blue}{(\left(x + 1\right) \cdot \left((2 \cdot x + 1)_*\right) + \left(x \cdot x\right))_*}}}{(x \cdot x + x)_*}\]
if 8773097.605766743 < (cbrt (fma x x x))
1. Initial program 29.2
  \[\frac{1}{x + 1} - \frac{1}{x}\]
2. Taylor expanded around inf 0.8
  \[\leadsto \color{blue}{\frac{1}{{x}^{3}} - \left(\frac{1}{{x}^{4}} + \frac{1}{{x}^{2}}\right)}\]
Recombined 2 regimes into one program.

Runtime

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))

Error

Derivation

`if (cbrt (fma x x x)) < 8773097.605766743`

`if 8773097.605766743 < (cbrt (fma x x x))`

Runtime