Average Error: 29.7 → 0.4
Time: 28.6s
Precision: 64
Internal Precision: 1600
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5147.074943161286 \lor \neg \left(x \le 4295.283920242833\right):\\ \;\;\;\;(\left(\frac{\sqrt[3]{x}}{x \cdot x}\right) \cdot \left(\frac{\frac{5}{81}}{x} + \frac{-1}{9}\right) + \left(\frac{\sqrt[3]{x}}{x} \cdot \frac{1}{3}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\sqrt[3]{1 + x} - \sqrt[3]{x}}\right)\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -5147.074943161286 or 4295.283920242833 < x

    1. Initial program 60.2

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

    2. Initial simplification60.2

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

    3. Taylor expanded around -inf 62.4

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

    4. Simplified0.8

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

    6. Applied associate-/r/0.6

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

    if -5147.074943161286 < x < 4295.283920242833

    1. Initial program 0.1

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

    2. Initial simplification0.1

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

    4. Applied add-log-exp0.2

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

    5. Applied add-log-exp0.2

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

    6. Applied diff-log0.2

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

    7. Simplified0.1

      \[\leadsto \log \color{blue}{\left(e^{\sqrt[3]{x + 1} - \sqrt[3]{x}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5147.074943161286 \lor \neg \left(x \le 4295.283920242833\right):\\ \;\;\;\;(\left(\frac{\sqrt[3]{x}}{x \cdot x}\right) \cdot \left(\frac{\frac{5}{81}}{x} + \frac{-1}{9}\right) + \left(\frac{\sqrt[3]{x}}{x} \cdot \frac{1}{3}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\sqrt[3]{1 + x} - \sqrt[3]{x}}\right)\\ \end{array}\]

Runtime

Time bar (total: 28.6s)Debug logProfile

herbie shell --seed 2018250 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))