Average Error: 46.6 → 0.5
Time: 52.2s
Precision: 64
Internal Precision: 1344
$\frac{1 - {\left(x - 1\right)}^{4}}{24}$
$\begin{array}{l} \mathbf{if}\;\sqrt[3]{\left({\left(\frac{x}{24}\right)}^{3} \cdot \left(\left(x \cdot x\right) \cdot 4 - \left(x \cdot 6 - 4\right)\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot 4 - \left(x \cdot 6 - 4\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 4 - \left(x \cdot 6 - 4\right)\right)\right)} \le -3.4292842047890375 \cdot 10^{-05}:\\ \;\;\;\;\left(1 - {\left(x - 1\right)}^{4}\right) \cdot \frac{1}{24}\\ \mathbf{if}\;\sqrt[3]{\left({\left(\frac{x}{24}\right)}^{3} \cdot \left(\left(x \cdot x\right) \cdot 4 - \left(x \cdot 6 - 4\right)\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot 4 - \left(x \cdot 6 - 4\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 4 - \left(x \cdot 6 - 4\right)\right)\right)} \le 26040.35734553348:\\ \;\;\;\;\frac{x}{24} \cdot \left(\left(x \cdot 4\right) \cdot x - \left(x \cdot 6 - 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \sqrt{{\left(x - 1\right)}^{4}} \cdot \sqrt{{\left(x - 1\right)}^{4}}}{24}\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 3 regimes
2. ## if (cbrt (* (* (pow (/ x 24) 3) (- (* (* x x) 4) (- (* x 6) 4))) (* (- (* (* x x) 4) (- (* x 6) 4)) (- (* (* x x) 4) (- (* x 6) 4))))) < -3.4292842047890375e-05

1. Initial program 0.7

$\frac{1 - {\left(x - 1\right)}^{4}}{24}$
2. Using strategy rm
3. Applied div-inv0.8

$\leadsto \color{blue}{\left(1 - {\left(x - 1\right)}^{4}\right) \cdot \frac{1}{24}}$

## if -3.4292842047890375e-05 < (cbrt (* (* (pow (/ x 24) 3) (- (* (* x x) 4) (- (* x 6) 4))) (* (- (* (* x x) 4) (- (* x 6) 4)) (- (* (* x x) 4) (- (* x 6) 4))))) < 26040.35734553348

1. Initial program 58.6

$\frac{1 - {\left(x - 1\right)}^{4}}{24}$
2. Taylor expanded around 0 0.4

$\leadsto \frac{\color{blue}{\left(4 \cdot x + 4 \cdot {x}^{3}\right) - 6 \cdot {x}^{2}}}{24}$
3. Applied simplify0.4

$\leadsto \color{blue}{\frac{x}{24} \cdot \left(\left(x \cdot 4\right) \cdot x - \left(x \cdot 6 - 4\right)\right)}$

## if 26040.35734553348 < (cbrt (* (* (pow (/ x 24) 3) (- (* (* x x) 4) (- (* x 6) 4))) (* (- (* (* x x) 4) (- (* x 6) 4)) (- (* (* x x) 4) (- (* x 6) 4)))))

1. Initial program 0.3

$\frac{1 - {\left(x - 1\right)}^{4}}{24}$
2. Using strategy rm
$\leadsto \frac{1 - \color{blue}{\sqrt{{\left(x - 1\right)}^{4}} \cdot \sqrt{{\left(x - 1\right)}^{4}}}}{24}$
herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'