Average Error: 53.0 → 2.4
Time: 30.6s
Precision: 64
Internal Precision: 2368
\[\frac{{2}^{\left(\frac{1}{3}\right)}}{{\left(\left(-3\right) \cdot x + \sqrt{4 + 9 \cdot {x}^{2}}\right)}^{\left(\frac{1}{3}\right)}} - \frac{{\left(\left(-3\right) \cdot x + \sqrt{4 + 9 \cdot {x}^{2}}\right)}^{\left(\frac{1}{3}\right)}}{{2}^{\left(\frac{1}{3}\right)}}\]
\[\begin{array}{l} \mathbf{if}\;x \le -0.6805997101349137:\\ \;\;\;\;\left(\left(\frac{\sqrt[3]{2}}{{\left(e^{\frac{1}{3}}\right)}^{\left(\log 6 - \log \left(\frac{-1}{x}\right)\right)}} + \frac{\frac{\frac{5}{729} \cdot \sqrt[3]{2}}{{x}^{4}}}{{\left(e^{\frac{1}{3}}\right)}^{\left(\log 6 - \log \left(\frac{-1}{x}\right)\right)}}\right) + \frac{4}{729} \cdot \frac{{\left(e^{\frac{1}{3}}\right)}^{\left(\log 6 - \log \left(\frac{-1}{x}\right)\right)}}{\frac{{x}^{4}}{\sqrt[3]{\frac{1}{2}}}}\right) - \left(\left(\left(\frac{\sqrt[3]{2}}{x \cdot x} \cdot \frac{1}{27}\right) \cdot {\left(e^{-\frac{1}{3}}\right)}^{\left(\log 6 - \log \left(\frac{-1}{x}\right)\right)} + {\left(e^{\frac{1}{3}}\right)}^{\left(\log 6 - \log \left(\frac{-1}{x}\right)\right)} \cdot \sqrt[3]{\frac{1}{2}}\right) + \frac{1}{27} \cdot \frac{{\left(e^{\frac{1}{3}}\right)}^{\left(\log 6 - \log \left(\frac{-1}{x}\right)\right)}}{\frac{x \cdot x}{\sqrt[3]{\frac{1}{2}}}}\right)\\ \mathbf{elif}\;x \le 0.6839042917823821:\\ \;\;\;\;\left(x + \frac{1}{3} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\frac{\frac{1}{27} \cdot \sqrt[3]{2}}{x \cdot x}}{{\left(e^{\frac{1}{3}}\right)}^{\left(\log \frac{2}{3} - \log x\right)}} + \left(\frac{\sqrt[3]{\frac{1}{2}}}{x} \cdot \frac{1}{27}\right) \cdot \frac{{\left(\frac{1}{x}\right)}^{\frac{1}{3}}}{\frac{x}{{\frac{2}{3}}^{\frac{1}{3}}}}\right) - \left(\frac{\frac{\sqrt[3]{2} \cdot \frac{4}{729}}{{x}^{4}}}{{\left(e^{\frac{1}{3}}\right)}^{\left(\log \frac{2}{3} - \log x\right)}} + {\left(\frac{1}{x}\right)}^{\frac{1}{3}} \cdot \left(\sqrt[3]{\frac{1}{2}} \cdot {\frac{2}{3}}^{\frac{1}{3}}\right)\right)\right) - \left(\frac{{\left(\frac{1}{x}\right)}^{\frac{1}{3}}}{\frac{{x}^{4}}{{\frac{2}{3}}^{\frac{1}{3}}}} \cdot \left(\frac{5}{729} \cdot \sqrt[3]{\frac{1}{2}}\right) - \frac{\sqrt[3]{2}}{{\left(e^{\frac{1}{3}}\right)}^{\left(\log \frac{2}{3} - \log x\right)}}\right)\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes
  2. if x < -0.6805997101349137

    1. Initial program 33.1

      \[\frac{{2}^{\left(\frac{1}{3}\right)}}{{\left(\left(-3\right) \cdot x + \sqrt{4 + 9 \cdot {x}^{2}}\right)}^{\left(\frac{1}{3}\right)}} - \frac{{\left(\left(-3\right) \cdot x + \sqrt{4 + 9 \cdot {x}^{2}}\right)}^{\left(\frac{1}{3}\right)}}{{2}^{\left(\frac{1}{3}\right)}}\]
    2. Initial simplification33.1

      \[\leadsto \frac{{2}^{\left(\frac{1}{3}\right)}}{{\left(\sqrt{9 \cdot \left(x \cdot x\right) + 4} - 3 \cdot x\right)}^{\left(\frac{1}{3}\right)}} - \frac{{\left(\sqrt{9 \cdot \left(x \cdot x\right) + 4} - 3 \cdot x\right)}^{\left(\frac{1}{3}\right)}}{{2}^{\left(\frac{1}{3}\right)}}\]
    3. Taylor expanded around -inf 5.0

      \[\leadsto \color{blue}{\left({2}^{\frac{1}{3}} \cdot \frac{1}{e^{\frac{1}{3} \cdot \left(\log 6 - \log \left(\frac{-1}{x}\right)\right)}} + \left(\frac{4}{729} \cdot \left(\frac{e^{\frac{1}{3} \cdot \left(\log 6 - \log \left(\frac{-1}{x}\right)\right)}}{{x}^{4}} \cdot {\frac{1}{2}}^{\frac{1}{3}}\right) + \frac{5}{729} \cdot \left(\frac{1}{e^{\frac{1}{3} \cdot \left(\log 6 - \log \left(\frac{-1}{x}\right)\right)} \cdot {x}^{4}} \cdot {2}^{\frac{1}{3}}\right)\right)\right) - \left(\frac{1}{27} \cdot \left(\frac{e^{\frac{1}{3} \cdot \left(\log 6 - \log \left(\frac{-1}{x}\right)\right)}}{{x}^{2}} \cdot {\frac{1}{2}}^{\frac{1}{3}}\right) + \left(e^{\frac{1}{3} \cdot \left(\log 6 - \log \left(\frac{-1}{x}\right)\right)} \cdot {\frac{1}{2}}^{\frac{1}{3}} + \frac{1}{27} \cdot \left({2}^{\frac{1}{3}} \cdot \frac{1}{e^{\frac{1}{3} \cdot \left(\log 6 - \log \left(\frac{-1}{x}\right)\right)} \cdot {x}^{2}}\right)\right)\right)}\]
    4. Simplified4.5

      \[\leadsto \color{blue}{\left(\frac{{\left(e^{\frac{1}{3}}\right)}^{\left(\log 6 - \log \left(\frac{-1}{x}\right)\right)}}{\frac{{x}^{4}}{\sqrt[3]{\frac{1}{2}}}} \cdot \frac{4}{729} + \left(\frac{\sqrt[3]{2}}{{\left(e^{\frac{1}{3}}\right)}^{\left(\log 6 - \log \left(\frac{-1}{x}\right)\right)}} + \frac{\frac{\sqrt[3]{2} \cdot \frac{5}{729}}{{x}^{4}}}{{\left(e^{\frac{1}{3}}\right)}^{\left(\log 6 - \log \left(\frac{-1}{x}\right)\right)}}\right)\right) - \left(\left(\left(\frac{1}{27} \cdot \frac{\sqrt[3]{2}}{x \cdot x}\right) \cdot {\left(e^{-\frac{1}{3}}\right)}^{\left(\log 6 - \log \left(\frac{-1}{x}\right)\right)} + {\left(e^{\frac{1}{3}}\right)}^{\left(\log 6 - \log \left(\frac{-1}{x}\right)\right)} \cdot \sqrt[3]{\frac{1}{2}}\right) + \frac{1}{27} \cdot \frac{{\left(e^{\frac{1}{3}}\right)}^{\left(\log 6 - \log \left(\frac{-1}{x}\right)\right)}}{\frac{x \cdot x}{\sqrt[3]{\frac{1}{2}}}}\right)}\]

    if -0.6805997101349137 < x < 0.6839042917823821

    1. Initial program 58.6

      \[\frac{{2}^{\left(\frac{1}{3}\right)}}{{\left(\left(-3\right) \cdot x + \sqrt{4 + 9 \cdot {x}^{2}}\right)}^{\left(\frac{1}{3}\right)}} - \frac{{\left(\left(-3\right) \cdot x + \sqrt{4 + 9 \cdot {x}^{2}}\right)}^{\left(\frac{1}{3}\right)}}{{2}^{\left(\frac{1}{3}\right)}}\]
    2. Initial simplification58.6

      \[\leadsto \frac{{2}^{\left(\frac{1}{3}\right)}}{{\left(\sqrt{9 \cdot \left(x \cdot x\right) + 4} - 3 \cdot x\right)}^{\left(\frac{1}{3}\right)}} - \frac{{\left(\sqrt{9 \cdot \left(x \cdot x\right) + 4} - 3 \cdot x\right)}^{\left(\frac{1}{3}\right)}}{{2}^{\left(\frac{1}{3}\right)}}\]
    3. Taylor expanded around 0 0.2

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

    if 0.6839042917823821 < x

    1. Initial program 61.6

      \[\frac{{2}^{\left(\frac{1}{3}\right)}}{{\left(\left(-3\right) \cdot x + \sqrt{4 + 9 \cdot {x}^{2}}\right)}^{\left(\frac{1}{3}\right)}} - \frac{{\left(\left(-3\right) \cdot x + \sqrt{4 + 9 \cdot {x}^{2}}\right)}^{\left(\frac{1}{3}\right)}}{{2}^{\left(\frac{1}{3}\right)}}\]
    2. Initial simplification61.6

      \[\leadsto \frac{{2}^{\left(\frac{1}{3}\right)}}{{\left(\sqrt{9 \cdot \left(x \cdot x\right) + 4} - 3 \cdot x\right)}^{\left(\frac{1}{3}\right)}} - \frac{{\left(\sqrt{9 \cdot \left(x \cdot x\right) + 4} - 3 \cdot x\right)}^{\left(\frac{1}{3}\right)}}{{2}^{\left(\frac{1}{3}\right)}}\]
    3. Taylor expanded around inf 5.2

      \[\leadsto \color{blue}{\left({2}^{\frac{1}{3}} \cdot \frac{1}{e^{\frac{1}{3} \cdot \left(\log \left(\frac{1}{x}\right) + \log \frac{2}{3}\right)}} + \left(\frac{1}{27} \cdot \left({2}^{\frac{1}{3}} \cdot \frac{1}{{x}^{2} \cdot e^{\frac{1}{3} \cdot \left(\log \left(\frac{1}{x}\right) + \log \frac{2}{3}\right)}}\right) + \frac{1}{27} \cdot \left(\frac{e^{\frac{1}{3} \cdot \left(\log \left(\frac{1}{x}\right) + \log \frac{2}{3}\right)}}{{x}^{2}} \cdot {\frac{1}{2}}^{\frac{1}{3}}\right)\right)\right) - \left({\frac{1}{2}}^{\frac{1}{3}} \cdot e^{\frac{1}{3} \cdot \left(\log \left(\frac{1}{x}\right) + \log \frac{2}{3}\right)} + \left(\frac{4}{729} \cdot \left({2}^{\frac{1}{3}} \cdot \frac{1}{{x}^{4} \cdot e^{\frac{1}{3} \cdot \left(\log \left(\frac{1}{x}\right) + \log \frac{2}{3}\right)}}\right) + \frac{5}{729} \cdot \left(\frac{e^{\frac{1}{3} \cdot \left(\log \left(\frac{1}{x}\right) + \log \frac{2}{3}\right)}}{{x}^{4}} \cdot {\frac{1}{2}}^{\frac{1}{3}}\right)\right)\right)}\]
    4. Simplified4.7

      \[\leadsto \color{blue}{\left(\left(\frac{\frac{\frac{1}{27} \cdot \sqrt[3]{2}}{x \cdot x}}{{\left(e^{\frac{1}{3}}\right)}^{\left(\log \frac{2}{3} - \log x\right)}} + \left(\frac{1}{27} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{x}\right) \cdot \frac{{\left(\frac{1}{x}\right)}^{\frac{1}{3}}}{\frac{x}{{\frac{2}{3}}^{\frac{1}{3}}}}\right) - \left({\left(\frac{1}{x}\right)}^{\frac{1}{3}} \cdot \left({\frac{2}{3}}^{\frac{1}{3}} \cdot \sqrt[3]{\frac{1}{2}}\right) + \frac{\frac{\frac{4}{729} \cdot \sqrt[3]{2}}{{x}^{4}}}{{\left(e^{\frac{1}{3}}\right)}^{\left(\log \frac{2}{3} - \log x\right)}}\right)\right) - \left(\left(\sqrt[3]{\frac{1}{2}} \cdot \frac{5}{729}\right) \cdot \frac{{\left(\frac{1}{x}\right)}^{\frac{1}{3}}}{\frac{{x}^{4}}{{\frac{2}{3}}^{\frac{1}{3}}}} - \frac{\sqrt[3]{2}}{{\left(e^{\frac{1}{3}}\right)}^{\left(\log \frac{2}{3} - \log x\right)}}\right)}\]
  3. Recombined 3 regimes into one program.
  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.6805997101349137:\\ \;\;\;\;\left(\left(\frac{\sqrt[3]{2}}{{\left(e^{\frac{1}{3}}\right)}^{\left(\log 6 - \log \left(\frac{-1}{x}\right)\right)}} + \frac{\frac{\frac{5}{729} \cdot \sqrt[3]{2}}{{x}^{4}}}{{\left(e^{\frac{1}{3}}\right)}^{\left(\log 6 - \log \left(\frac{-1}{x}\right)\right)}}\right) + \frac{4}{729} \cdot \frac{{\left(e^{\frac{1}{3}}\right)}^{\left(\log 6 - \log \left(\frac{-1}{x}\right)\right)}}{\frac{{x}^{4}}{\sqrt[3]{\frac{1}{2}}}}\right) - \left(\left(\left(\frac{\sqrt[3]{2}}{x \cdot x} \cdot \frac{1}{27}\right) \cdot {\left(e^{-\frac{1}{3}}\right)}^{\left(\log 6 - \log \left(\frac{-1}{x}\right)\right)} + {\left(e^{\frac{1}{3}}\right)}^{\left(\log 6 - \log \left(\frac{-1}{x}\right)\right)} \cdot \sqrt[3]{\frac{1}{2}}\right) + \frac{1}{27} \cdot \frac{{\left(e^{\frac{1}{3}}\right)}^{\left(\log 6 - \log \left(\frac{-1}{x}\right)\right)}}{\frac{x \cdot x}{\sqrt[3]{\frac{1}{2}}}}\right)\\ \mathbf{elif}\;x \le 0.6839042917823821:\\ \;\;\;\;\left(x + \frac{1}{3} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\frac{\frac{1}{27} \cdot \sqrt[3]{2}}{x \cdot x}}{{\left(e^{\frac{1}{3}}\right)}^{\left(\log \frac{2}{3} - \log x\right)}} + \left(\frac{\sqrt[3]{\frac{1}{2}}}{x} \cdot \frac{1}{27}\right) \cdot \frac{{\left(\frac{1}{x}\right)}^{\frac{1}{3}}}{\frac{x}{{\frac{2}{3}}^{\frac{1}{3}}}}\right) - \left(\frac{\frac{\sqrt[3]{2} \cdot \frac{4}{729}}{{x}^{4}}}{{\left(e^{\frac{1}{3}}\right)}^{\left(\log \frac{2}{3} - \log x\right)}} + {\left(\frac{1}{x}\right)}^{\frac{1}{3}} \cdot \left(\sqrt[3]{\frac{1}{2}} \cdot {\frac{2}{3}}^{\frac{1}{3}}\right)\right)\right) - \left(\frac{{\left(\frac{1}{x}\right)}^{\frac{1}{3}}}{\frac{{x}^{4}}{{\frac{2}{3}}^{\frac{1}{3}}}} \cdot \left(\frac{5}{729} \cdot \sqrt[3]{\frac{1}{2}}\right) - \frac{\sqrt[3]{2}}{{\left(e^{\frac{1}{3}}\right)}^{\left(\log \frac{2}{3} - \log x\right)}}\right)\\ \end{array}\]

Runtime

Time bar (total: 30.6s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (x)
  :name "2^(1/3)/(-3 x + sqrt(4 + 9 x^2))^(1/3) - (-3 x + sqrt(4 + 9 x^2))^(1/3)/2^(1/3)"
  (- (/ (pow 2 (/ 1 3)) (pow (+ (* (- 3) x) (sqrt (+ 4 (* 9 (pow x 2))))) (/ 1 3))) (/ (pow (+ (* (- 3) x) (sqrt (+ 4 (* 9 (pow x 2))))) (/ 1 3)) (pow 2 (/ 1 3)))))