Average Error: 29.3 → 15.8
Time: 21.0s
Precision: 64
Internal Precision: 128
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 0.8940235824808939:\\ \;\;\;\;1 - (x \cdot \left((\frac{1}{9} \cdot x + \frac{-1}{3})_*\right) + \left(\sqrt[3]{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) \cdot \frac{-1}{9} + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}}\right))_*\right))_*\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < 0.8940235824808939

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt0.1

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

    4. Applied cbrt-prod0.1

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

    5. Applied fma-neg0.1

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

    7. Applied add-cube-cbrt0.1

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

    8. Applied cbrt-prod0.1

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

    9. Applied cbrt-prod0.1

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

    10. Taylor expanded around 0 0.4

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

    11. Simplified0.4

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

    if 0.8940235824808939 < x

    1. Initial program 59.5

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

    2. Taylor expanded around inf 33.7

      \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + \frac{5}{81} \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - \frac{1}{9} \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}}\]

    3. Simplified31.6

      \[\leadsto \color{blue}{(\left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) \cdot \frac{-1}{9} + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81}\right))_*\right))_*}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification15.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.8940235824808939:\\ \;\;\;\;1 - (x \cdot \left((\frac{1}{9} \cdot x + \frac{-1}{3})_*\right) + \left(\sqrt[3]{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) \cdot \frac{-1}{9} + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}}\right))_*\right))_*\\ \end{array}\]

Runtime

Time bar (total: 21.0s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes29.415.815.413.997.6%
herbie shell --seed 2018354 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))