Average Error: 30.4 → 0.4
Time: 23.5s
Precision: 64
Internal Precision: 1344
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.003492093787603:\\ \;\;\;\;(\left(\frac{\frac{5}{81}}{x} - \frac{1}{9}\right) \cdot \left(\frac{\sqrt[3]{x}}{x \cdot x}\right) + \left(\left(\sqrt[3]{x} \cdot \frac{1}{3}\right) \cdot \frac{1}{x}\right))_*\\ \mathbf{elif}\;x \le 3953.5570327954874:\\ \;\;\;\;(\left(\sqrt{\sqrt[3]{1 + x}}\right) \cdot \left(\sqrt{\sqrt[3]{1 + x}}\right) + \left(-\sqrt[3]{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\frac{5}{81}}{x} - \frac{1}{9}\right) \cdot \left(\frac{\sqrt[3]{x}}{x \cdot x}\right) + \left(\left(\frac{1}{3} \cdot \frac{1}{x}\right) \cdot \sqrt[3]{x}\right))_*\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -1.003492093787603

    1. Initial program 59.7

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

    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. Simplified0.9

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

    5. Applied div-inv0.9

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

    6. Applied associate-*r*0.9

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

    if -1.003492093787603 < x < 3953.5570327954874

    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 3953.5570327954874 < 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. Simplified0.6

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

    5. Applied div-inv0.7

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

    6. Applied associate-*r*0.7

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

    8. Applied associate-*l*0.7

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.003492093787603:\\ \;\;\;\;(\left(\frac{\frac{5}{81}}{x} - \frac{1}{9}\right) \cdot \left(\frac{\sqrt[3]{x}}{x \cdot x}\right) + \left(\left(\sqrt[3]{x} \cdot \frac{1}{3}\right) \cdot \frac{1}{x}\right))_*\\ \mathbf{elif}\;x \le 3953.5570327954874:\\ \;\;\;\;(\left(\sqrt{\sqrt[3]{1 + x}}\right) \cdot \left(\sqrt{\sqrt[3]{1 + x}}\right) + \left(-\sqrt[3]{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\frac{5}{81}}{x} - \frac{1}{9}\right) \cdot \left(\frac{\sqrt[3]{x}}{x \cdot x}\right) + \left(\left(\frac{1}{3} \cdot \frac{1}{x}\right) \cdot \sqrt[3]{x}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 23.5s)Debug logProfile

herbie shell --seed 2018227 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))