Average Error: 30.2 → 0.5
Time: 18.3s
Precision: 64
Internal Precision: 1344
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0159564017437261:\\ \;\;\;\;(\left(\frac{\sqrt[3]{x}}{x \cdot x}\right) \cdot \left(\frac{\frac{5}{81}}{x} + \frac{-1}{9}\right) + \left(\frac{\sqrt[3]{x}}{x} \cdot \frac{1}{3}\right))_*\\ \mathbf{elif}\;x \le 5296.848279415423:\\ \;\;\;\;(\left(\sqrt{\sqrt[3]{1 + x}}\right) \cdot \left(\sqrt{\sqrt[3]{1 + x}}\right) + \left(-\sqrt[3]{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\sqrt[3]{x}}{x \cdot x}\right) \cdot \left(\frac{\frac{5}{81}}{x} + \frac{-1}{9}\right) + \left(\frac{1}{\frac{\frac{x}{\frac{1}{3}}}{\sqrt[3]{x}}}\right))_*\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0159564017437261

    1. Initial program 59.2

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

    2. Taylor expanded around -inf 62.4

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

    3. Simplified1.2

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

    5. Applied associate-/r/1.1

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

    if -1.0159564017437261 < x < 5296.848279415423

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied fma-neg0.1

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

    if 5296.848279415423 < x

    1. Initial program 60.2

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

    2. Taylor expanded around -inf 62.4

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

    3. Simplified0.7

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

    5. Applied clear-num0.8

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0159564017437261:\\ \;\;\;\;(\left(\frac{\sqrt[3]{x}}{x \cdot x}\right) \cdot \left(\frac{\frac{5}{81}}{x} + \frac{-1}{9}\right) + \left(\frac{\sqrt[3]{x}}{x} \cdot \frac{1}{3}\right))_*\\ \mathbf{elif}\;x \le 5296.848279415423:\\ \;\;\;\;(\left(\sqrt{\sqrt[3]{1 + x}}\right) \cdot \left(\sqrt{\sqrt[3]{1 + x}}\right) + \left(-\sqrt[3]{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\sqrt[3]{x}}{x \cdot x}\right) \cdot \left(\frac{\frac{5}{81}}{x} + \frac{-1}{9}\right) + \left(\frac{1}{\frac{\frac{x}{\frac{1}{3}}}{\sqrt[3]{x}}}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 18.3s)Debug logProfile

herbie shell --seed 2018254 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))