Average Error: 40.6 → 1.0
Time: 30.9s
Precision: 64
Internal Precision: 1344
\[1 - {\left(1 - x\right)}^{\left(\frac{1}{3}\right)}\]
\[\begin{array}{l} \mathbf{if}\;1 - {\left(1 - x\right)}^{\left(\frac{1}{3}\right)} \le -8.669224468689961 \cdot 10^{-05}:\\ \;\;\;\;\frac{\sqrt[3]{-x}}{x} \cdot \left(\frac{1}{3} + \frac{\frac{1}{9}}{x}\right) - \left(\sqrt[3]{-x} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{1}{9} + \frac{1}{3}\right) \cdot x\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes
  2. if (- 1 (pow (- 1 x) (/ 1 3))) < -8.669224468689961e-05

    1. Initial program 5.1

      \[1 - {\left(1 - x\right)}^{\left(\frac{1}{3}\right)}\]
    2. Taylor expanded around inf 63.7

      \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{1}{x}\right)\right)}}{x} + \left(1 + \frac{1}{9} \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{1}{x}\right)\right)}}{{x}^{2}}\right)\right) - e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{1}{x}\right)\right)}}\]
    3. Applied simplify1.6

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

    if -8.669224468689961e-05 < (- 1 (pow (- 1 x) (/ 1 3)))

    1. Initial program 58.8

      \[1 - {\left(1 - x\right)}^{\left(\frac{1}{3}\right)}\]
    2. Taylor expanded around 0 59.1

      \[\leadsto 1 - \color{blue}{\left(1 - \left(\frac{1}{9} \cdot {x}^{2} + \frac{1}{3} \cdot x\right)\right)}\]
    3. Applied simplify0.8

      \[\leadsto \color{blue}{\left(\frac{1}{3} + x \cdot \frac{1}{9}\right) \cdot x}\]
  3. Recombined 2 regimes into one program.
  4. Applied simplify1.0

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;1 - {\left(1 - x\right)}^{\left(\frac{1}{3}\right)} \le -8.669224468689961 \cdot 10^{-05}:\\ \;\;\;\;\frac{\sqrt[3]{-x}}{x} \cdot \left(\frac{1}{3} + \frac{\frac{1}{9}}{x}\right) - \left(\sqrt[3]{-x} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{1}{9} + \frac{1}{3}\right) \cdot x\\ \end{array}}\]

Runtime

Time bar (total: 30.9s)Debug log

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