Average Error: 29.8 → 0.6
Time: 18.8s
Precision: 64
Internal Precision: 128
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \le 0.00027039825459951317:\\ \;\;\;\;\frac{(\frac{-1}{9} \cdot \left(\sqrt[3]{\frac{1}{{x}^{4}}}\right) + \left((\left(\sqrt[3]{\frac{1}{{x}^{7}}}\right) \cdot \frac{4}{81} + \left(\sqrt[3]{\frac{1}{x}} \cdot \frac{2}{3}\right))_*\right))_*}{\sqrt[3]{1 + x} + \sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x} - \sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x} - \sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{1 + x} + \sqrt[3]{x}}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if (- (cbrt (+ x 1)) (cbrt x)) < 0.00027039825459951317

    1. Initial program 60.2

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

    3. Applied flip--60.2

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

    4. Taylor expanded around inf 5.0

      \[\leadsto \frac{\color{blue}{\left(\frac{4}{81} \cdot {\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}} + \frac{2}{3} \cdot {\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right) - \frac{1}{9} \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}}}}{\sqrt[3]{x + 1} + \sqrt[3]{x}}\]

    5. Simplified1.0

      \[\leadsto \frac{\color{blue}{(\frac{-1}{9} \cdot \left(\sqrt[3]{\frac{1}{{x}^{4}}}\right) + \left((\left(\sqrt[3]{\frac{1}{{x}^{7}}}\right) \cdot \frac{4}{81} + \left(\sqrt[3]{\frac{1}{x}} \cdot \frac{2}{3}\right))_*\right))_*}}{\sqrt[3]{x + 1} + \sqrt[3]{x}}\]

    if 0.00027039825459951317 < (- (cbrt (+ x 1)) (cbrt x))

    1. Initial program 0.2

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

    3. Applied flip--0.2

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

    5. Applied add-sqr-sqrt0.2

      \[\leadsto \frac{\color{blue}{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} - \sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} - \sqrt[3]{x} \cdot \sqrt[3]{x}}}}{\sqrt[3]{x + 1} + \sqrt[3]{x}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \le 0.00027039825459951317:\\ \;\;\;\;\frac{(\frac{-1}{9} \cdot \left(\sqrt[3]{\frac{1}{{x}^{4}}}\right) + \left((\left(\sqrt[3]{\frac{1}{{x}^{7}}}\right) \cdot \frac{4}{81} + \left(\sqrt[3]{\frac{1}{x}} \cdot \frac{2}{3}\right))_*\right))_*}{\sqrt[3]{1 + x} + \sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x} - \sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x} - \sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{1 + x} + \sqrt[3]{x}}\\ \end{array}\]

Runtime

Time bar (total: 18.8s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes29.80.60.429.499.4%
herbie shell --seed 2018351 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))