\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le 3773.1647317338056:\\ \;\;\;\;{\left(1 + x\right)}^{\frac{1}{3}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;(\left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) \cdot \frac{-1}{9} + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81}\right))_*\right))_*\\ \end{array}\]

Derivation

Split input into 2 regimes
if x < 3773.1647317338056
1. Initial program 0.1
  \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
2. Using strategy rm
3. Applied pow1/30.1
  \[\leadsto \color{blue}{{\left(x + 1\right)}^{\frac{1}{3}}} - \sqrt[3]{x}\]
if 3773.1647317338056 < x
1. Initial program 60.1
  \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
2. Taylor expanded around inf 32.8
  \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + \frac{5}{81} \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - \frac{1}{9} \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}}\]
3. Simplified30.6
  \[\leadsto \color{blue}{(\left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) \cdot \frac{-1}{9} + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81}\right))_*\right))_*}\]
Recombined 2 regimes into one program.
Final simplification15.2
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le 3773.1647317338056:\\ \;\;\;\;{\left(1 + x\right)}^{\frac{1}{3}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;(\left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) \cdot \frac{-1}{9} + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81}\right))_*\right))_*\\ \end{array}\]

Runtime

herbie shell --seed 2018274 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))

Error

Derivation

`if x < 3773.1647317338056`

`if 3773.1647317338056 < x`

Runtime