Average Error: 30.1 → 0.4
Time: 3.1m
Precision: 64
Internal Precision: 1344
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -128718.44875201154 \lor \neg \left(x \le 65248.20123858831\right):\\ \;\;\;\;(\left(\frac{\sqrt[3]{x}}{x \cdot x}\right) \cdot \left(\frac{\frac{5}{81}}{x} + \frac{-1}{9}\right) + \left(\frac{\sqrt[3]{x}}{\frac{x}{\frac{1}{3}}}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\left(1 + x\right) + x}}{(\left((\left(\sqrt[3]{1 + x}\right) \cdot \left(\sqrt[3]{1 + x}\right) + \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right))_*\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\sqrt[3]{1 + x} \cdot \left(1 + x\right)\right))_*} \cdot \left(\sqrt[3]{1 + \left(x + x\right)} \cdot \sqrt[3]{1 + \left(x + x\right)}\right)}{\sqrt[3]{1 + x} + \sqrt[3]{x}}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -128718.44875201154 or 65248.20123858831 < x

    1. Initial program 60.3

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

    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. Simplified0.6

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

    if -128718.44875201154 < x < 65248.20123858831

    1. Initial program 0.2

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

    3. Applied flip--0.3

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} - \sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + 1} + \sqrt[3]{x}}}\]
    4. Using strategy rm

    5. Applied flip3--0.3

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

    7. Applied *-un-lft-identity0.3

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

    8. Applied add-cube-cbrt0.3

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

    9. Applied times-frac0.3

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

    10. Simplified0.3

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

    11. Simplified0.1

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -128718.44875201154 \lor \neg \left(x \le 65248.20123858831\right):\\ \;\;\;\;(\left(\frac{\sqrt[3]{x}}{x \cdot x}\right) \cdot \left(\frac{\frac{5}{81}}{x} + \frac{-1}{9}\right) + \left(\frac{\sqrt[3]{x}}{\frac{x}{\frac{1}{3}}}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\left(1 + x\right) + x}}{(\left((\left(\sqrt[3]{1 + x}\right) \cdot \left(\sqrt[3]{1 + x}\right) + \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right))_*\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\sqrt[3]{1 + x} \cdot \left(1 + x\right)\right))_*} \cdot \left(\sqrt[3]{1 + \left(x + x\right)} \cdot \sqrt[3]{1 + \left(x + x\right)}\right)}{\sqrt[3]{1 + x} + \sqrt[3]{x}}\\ \end{array}\]

Runtime

Time bar (total: 3.1m)Debug logProfile

herbie shell --seed 2018248 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))