Error

Derivation

1. Split input into 2 regimes

2. if x < 2.024352565067993e+17

   1. Initial program 1.8

      \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
   2. Using strategy rm

   3. Applied pow1/31.9

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{x}^{\frac{1}{3}}}\]
   4. Using strategy rm

   5. Applied flip3--2.0

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x + 1}\right)}^{3} - {\left({x}^{\frac{1}{3}}\right)}^{3}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left({x}^{\frac{1}{3}} \cdot {x}^{\frac{1}{3}} + \sqrt[3]{x + 1} \cdot {x}^{\frac{1}{3}}\right)}}\]

   6. Taylor expanded around inf 0.2

      \[\leadsto \frac{\color{blue}{1}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left({x}^{\frac{1}{3}} \cdot {x}^{\frac{1}{3}} + \sqrt[3]{x + 1} \cdot {x}^{\frac{1}{3}}\right)}\]
   7. Using strategy rm

   8. Applied add-cbrt-cube0.2

      \[\leadsto \frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left({x}^{\frac{1}{3}} \cdot \color{blue}{\sqrt[3]{\left({x}^{\frac{1}{3}} \cdot {x}^{\frac{1}{3}}\right) \cdot {x}^{\frac{1}{3}}}} + \sqrt[3]{x + 1} \cdot {x}^{\frac{1}{3}}\right)}\]

   9. Applied unpow1/30.2

      \[\leadsto \frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\color{blue}{\sqrt[3]{x}} \cdot \sqrt[3]{\left({x}^{\frac{1}{3}} \cdot {x}^{\frac{1}{3}}\right) \cdot {x}^{\frac{1}{3}}} + \sqrt[3]{x + 1} \cdot {x}^{\frac{1}{3}}\right)}\]

   10. Applied cbrt-unprod0.2

       \[\leadsto \frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\color{blue}{\sqrt[3]{x \cdot \left(\left({x}^{\frac{1}{3}} \cdot {x}^{\frac{1}{3}}\right) \cdot {x}^{\frac{1}{3}}\right)}} + \sqrt[3]{x + 1} \cdot {x}^{\frac{1}{3}}\right)}\]

   11. Simplified0.2

       \[\leadsto \frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{\color{blue}{x \cdot x}} + \sqrt[3]{x + 1} \cdot {x}^{\frac{1}{3}}\right)}\]

   if 2.024352565067993e+17 < x

   1. Initial program 61.3

      \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
   2. Using strategy rm

   3. Applied pow1/360.5

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{x}^{\frac{1}{3}}}\]

   4. Taylor expanded around -inf 62.4

      \[\leadsto \color{blue}{\left(\frac{5}{81} \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{{x}^{3}} + \frac{1}{3} \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{x}\right) - \frac{1}{9} \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{{x}^{2}}}\]

   5. Simplified0.7

      \[\leadsto \color{blue}{(\left(\frac{\sqrt[3]{x}}{x \cdot x}\right) \cdot \left(\frac{\frac{5}{81}}{x} + \frac{-1}{9}\right) + \left(\frac{\sqrt[3]{x}}{\frac{x}{\frac{1}{3}}}\right))_*}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 2.024352565067993 \cdot 10^{+17}:\\ \;\;\;\;\frac{1}{\left(\sqrt[3]{x \cdot x} + \sqrt[3]{1 + x} \cdot {x}^{\frac{1}{3}}\right) + \sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\sqrt[3]{x}}{x \cdot x}\right) \cdot \left(\frac{-1}{9} + \frac{\frac{5}{81}}{x}\right) + \left(\frac{\sqrt[3]{x}}{\frac{x}{\frac{1}{3}}}\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019022 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))

Details

Time bar (total: 45.8s)Debug log