Average Error: 30.0 → 0.4
Time: 26.4s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -89867.88213118578:\\ \;\;\;\;\left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{-1}\right)\right) + \frac{\sqrt[3]{x}}{x} \cdot \left(\frac{1}{3} + \frac{\frac{-1}{9}}{x}\right)\\ \mathbf{elif}\;x \le 69526.91935895119:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{-1}\right)\right) + \frac{\left(\frac{1}{9} - \frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x}\right) \cdot \sqrt[3]{x}}{x \cdot \left(\frac{1}{3} - \frac{\frac{-1}{9}}{x}\right)}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -89867.88213118578

    1. Initial program 60.4

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

    2. Taylor expanded around -inf 62.4

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

    3. Simplified0.7

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

    4. Taylor expanded around 0 62.4

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

    5. Simplified0.7

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

    if -89867.88213118578 < x < 69526.91935895119

    1. Initial program 0.1

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

    2. Taylor expanded around 0 30.4

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

    3. Simplified0.1

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

    if 69526.91935895119 < x

    1. Initial program 60.5

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

    2. Taylor expanded around -inf 62.4

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

    3. Simplified0.7

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

    4. Taylor expanded around 0 59.8

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

    5. Simplified0.7

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

    7. Applied flip-+0.7

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

    8. Applied frac-times0.7

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

    9. Simplified0.7

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -89867.88213118578:\\ \;\;\;\;\left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{-1}\right)\right) + \frac{\sqrt[3]{x}}{x} \cdot \left(\frac{1}{3} + \frac{\frac{-1}{9}}{x}\right)\\ \mathbf{elif}\;x \le 69526.91935895119:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{-1}\right)\right) + \frac{\left(\frac{1}{9} - \frac{\frac{-1}{9}}{x} \cdot \frac{\frac{-1}{9}}{x}\right) \cdot \sqrt[3]{x}}{x \cdot \left(\frac{1}{3} - \frac{\frac{-1}{9}}{x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019100 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))