Average Error: 30.2 → 0.4
Time: 21.6s
Precision: 64
Internal Precision: 1344
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3400.294910717929 \lor \neg \left(x \le 3581.54093585158\right):\\ \;\;\;\;(\left(\frac{\sqrt[3]{x}}{x \cdot x}\right) \cdot \left(\frac{\frac{5}{81}}{x} + \frac{-1}{9}\right) + \left(\frac{\sqrt[3]{x}}{\frac{x}{\frac{1}{3}}}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{1 + x}}\right) + \left(-\sqrt[3]{x}\right))_*\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -3400.294910717929 or 3581.54093585158 < x

    1. Initial program 60.1

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

    2. Initial simplification60.1

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

    4. Applied expm1-log1p-u60.1

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

    5. Taylor expanded around -inf 62.4

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

    6. Simplified0.7

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

    if -3400.294910717929 < x < 3581.54093585158

    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-cube-cbrt0.1

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

    5. Applied cbrt-prod0.2

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

    6. Applied fma-neg0.2

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3400.294910717929 \lor \neg \left(x \le 3581.54093585158\right):\\ \;\;\;\;(\left(\frac{\sqrt[3]{x}}{x \cdot x}\right) \cdot \left(\frac{\frac{5}{81}}{x} + \frac{-1}{9}\right) + \left(\frac{\sqrt[3]{x}}{\frac{x}{\frac{1}{3}}}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{1 + x}}\right) + \left(-\sqrt[3]{x}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 21.6s)Debug logProfile

herbie shell --seed 2018235 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))