Average Error: 30.1 → 0.4
Time: 15.5s
Precision: 64
Internal Precision: 128
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 4797.19566391008:\\ \;\;\;\;{\left(1 + x\right)}^{\frac{1}{3}} - {x}^{\frac{1}{3}}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\sqrt[3]{x}}{x \cdot x}\right) \cdot \left(\frac{\frac{5}{81}}{x} + \frac{-1}{9}\right) + \left(\frac{\frac{\sqrt[3]{x}}{x}}{3}\right))_*\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < 4797.19566391008

    1. Initial program 0.1

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

    3. Applied pow1/30.1

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

    5. Applied pow1/30.1

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

    if 4797.19566391008 < x

    1. Initial program 60.2

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

    3. Applied pow1/359.6

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

    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

      \[\leadsto \color{blue}{(\left(\frac{\sqrt[3]{x}}{x \cdot x}\right) \cdot \left(\frac{\frac{5}{81}}{x} + \frac{-1}{9}\right) + \left(\frac{\sqrt[3]{x}}{\frac{x}{\frac{1}{3}}}\right))_*}\]
    6. Using strategy rm

    7. Applied div-inv0.7

      \[\leadsto (\left(\frac{\sqrt[3]{x}}{x \cdot x}\right) \cdot \left(\frac{\frac{5}{81}}{x} + \frac{-1}{9}\right) + \left(\frac{\sqrt[3]{x}}{\color{blue}{x \cdot \frac{1}{\frac{1}{3}}}}\right))_*\]

    8. Applied associate-/r*0.6

      \[\leadsto (\left(\frac{\sqrt[3]{x}}{x \cdot x}\right) \cdot \left(\frac{\frac{5}{81}}{x} + \frac{-1}{9}\right) + \color{blue}{\left(\frac{\frac{\sqrt[3]{x}}{x}}{\frac{1}{\frac{1}{3}}}\right)})_*\]

    9. Simplified0.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 4797.19566391008:\\ \;\;\;\;{\left(1 + x\right)}^{\frac{1}{3}} - {x}^{\frac{1}{3}}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\sqrt[3]{x}}{x \cdot x}\right) \cdot \left(\frac{\frac{5}{81}}{x} + \frac{-1}{9}\right) + \left(\frac{\frac{\sqrt[3]{x}}{x}}{3}\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019026 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))