Average Error: 28.9 → 0.5
Time: 45.0s
Precision: 64
Internal Precision: 1344
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.9924714463196989 \lor \neg \left(x \le 4122.6992300157335\right):\\ \;\;\;\;(\left(\frac{\sqrt[3]{x}}{x \cdot x}\right) \cdot \left(\frac{\frac{5}{81}}{x} + \frac{-1}{9}\right) + \left(\frac{\sqrt[3]{x}}{\frac{x}{\frac{1}{3}}}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\sqrt{\sqrt[3]{1 + x}}\right) \cdot \left(\sqrt{\sqrt[3]{1 + x}}\right) + \left(-\sqrt[3]{x}\right))_*\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -0.9924714463196989 or 4122.6992300157335 < x

    1. Initial program 59.8

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

    2. Initial simplification59.8

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

    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. Simplified1.0

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

    if -0.9924714463196989 < x < 4122.6992300157335

    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-sqr-sqrt0.1

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

    5. Applied fma-neg0.1

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.9924714463196989 \lor \neg \left(x \le 4122.6992300157335\right):\\ \;\;\;\;(\left(\frac{\sqrt[3]{x}}{x \cdot x}\right) \cdot \left(\frac{\frac{5}{81}}{x} + \frac{-1}{9}\right) + \left(\frac{\sqrt[3]{x}}{\frac{x}{\frac{1}{3}}}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\sqrt{\sqrt[3]{1 + x}}\right) \cdot \left(\sqrt{\sqrt[3]{1 + x}}\right) + \left(-\sqrt[3]{x}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 45.0s)Debug logProfile

herbie shell --seed 2018252 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))