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

# Try it out

Results

 In Out
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))))