Average Error: 40.0 → 1.5
Time: 34.4s
Precision: 64
Internal Precision: 1344
$1 - {\left(1 - x\right)}^{\left(\frac{1}{12}\right)}$
$\begin{array}{l} \mathbf{if}\;\left(\frac{11}{288} + \frac{253}{10368} \cdot x\right) \cdot \left(x \cdot x\right) + \sqrt[3]{{\left(\frac{1}{12} \cdot x\right)}^{3}} \le -0.4315739689459135:\\ \;\;\;\;1 - e^{\frac{\log \left(1 - x\right)}{12}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{11}{288} + \frac{253}{10368} \cdot x\right) \cdot \left(x \cdot x\right) + \frac{1}{12} \cdot x\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 2 regimes
2. ## if (+ (* (+ 11/288 (* 253/10368 x)) (* x x)) (cbrt (pow (* 1/12 x) 3))) < -0.4315739689459135

1. Initial program 3.2

$1 - {\left(1 - x\right)}^{\left(\frac{1}{12}\right)}$
2. Using strategy rm

$\leadsto 1 - {\color{blue}{\left(e^{\log \left(1 - x\right)}\right)}}^{\left(\frac{1}{12}\right)}$
4. Applied pow-exp3.2

$\leadsto 1 - \color{blue}{e^{\log \left(1 - x\right) \cdot \frac{1}{12}}}$
5. Applied simplify2.9

$\leadsto 1 - e^{\color{blue}{\frac{\log \left(1 - x\right)}{12}}}$

## if -0.4315739689459135 < (+ (* (+ 11/288 (* 253/10368 x)) (* x x)) (cbrt (pow (* 1/12 x) 3)))

1. Initial program 58.6

$1 - {\left(1 - x\right)}^{\left(\frac{1}{12}\right)}$
2. Taylor expanded around 0 0.8

$\leadsto \color{blue}{\frac{253}{10368} \cdot {x}^{3} + \left(\frac{11}{288} \cdot {x}^{2} + \frac{1}{12} \cdot x\right)}$
3. Applied simplify0.8

$\leadsto \color{blue}{\left(\frac{11}{288} + \frac{253}{10368} \cdot x\right) \cdot \left(x \cdot x\right) + \frac{1}{12} \cdot x}$
3. Recombined 2 regimes into one program.

# Runtime

Time bar (total: 34.4s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'
(FPCore (x)
:name "1-pow(1 - x, 1/12) "
(- 1 (pow (- 1 x) (/ 1 12))))