Average Error: 22.4 → 15.8
Time: 49.5s
Precision: 64
Internal Precision: 1344
$\frac{\left(1 - alphaD\right) + \sqrt{\left(\left(1.0 - 2.0 \cdot alphaD\right) + \left(4.0 \cdot Kp\right) \cdot alphaD\right) + alphaD \cdot alphaD}}{2 \cdot \sqrt{\left(\left(1.0 - 2.0 \cdot alphaD\right) + \left(4.0 \cdot Kp\right) \cdot alphaD\right) + alphaD \cdot alphaD}}$
$\begin{array}{l} \mathbf{if}\;alphaD \le 32524.336403369205:\\ \;\;\;\;\frac{\left(1 - alphaD\right) + \sqrt{1.0 - \left(\left(2.0 - alphaD\right) - Kp \cdot 4.0\right) \cdot alphaD}}{2 \cdot \sqrt{1.0 - \left(\left(2.0 - alphaD\right) - Kp \cdot 4.0\right) \cdot alphaD}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.0 \cdot \left(\frac{\frac{Kp}{alphaD}}{alphaD} + \left(\frac{Kp}{alphaD} + Kp\right)\right)}{2 \cdot \sqrt{1.0 - \left(\left(2.0 - alphaD\right) - Kp \cdot 4.0\right) \cdot alphaD}}\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 2 regimes
2. ## if alphaD < 32524.336403369205

1. Initial program 12.3

$\frac{\left(1 - alphaD\right) + \sqrt{\left(\left(1.0 - 2.0 \cdot alphaD\right) + \left(4.0 \cdot Kp\right) \cdot alphaD\right) + alphaD \cdot alphaD}}{2 \cdot \sqrt{\left(\left(1.0 - 2.0 \cdot alphaD\right) + \left(4.0 \cdot Kp\right) \cdot alphaD\right) + alphaD \cdot alphaD}}$
2. Initial simplification12.3

$\leadsto \frac{\sqrt{1.0 - \left(\left(2.0 - alphaD\right) - Kp \cdot 4.0\right) \cdot alphaD} + \left(1 - alphaD\right)}{\sqrt{1.0 - \left(\left(2.0 - alphaD\right) - Kp \cdot 4.0\right) \cdot alphaD} \cdot 2}$

1. Initial program 53.3

$\frac{\left(1 - alphaD\right) + \sqrt{\left(\left(1.0 - 2.0 \cdot alphaD\right) + \left(4.0 \cdot Kp\right) \cdot alphaD\right) + alphaD \cdot alphaD}}{2 \cdot \sqrt{\left(\left(1.0 - 2.0 \cdot alphaD\right) + \left(4.0 \cdot Kp\right) \cdot alphaD\right) + alphaD \cdot alphaD}}$
2. Initial simplification53.3

$\leadsto \frac{\sqrt{1.0 - \left(\left(2.0 - alphaD\right) - Kp \cdot 4.0\right) \cdot alphaD} + \left(1 - alphaD\right)}{\sqrt{1.0 - \left(\left(2.0 - alphaD\right) - Kp \cdot 4.0\right) \cdot alphaD} \cdot 2}$
3. Taylor expanded around inf 26.5

$\leadsto \frac{\color{blue}{2.0 \cdot \frac{Kp}{{alphaD}^{2}} + \left(2.0 \cdot Kp + 2.0 \cdot \frac{Kp}{alphaD}\right)}}{\sqrt{1.0 - \left(\left(2.0 - alphaD\right) - Kp \cdot 4.0\right) \cdot alphaD} \cdot 2}$
4. Simplified26.5

$\leadsto \frac{\color{blue}{\left(\frac{\frac{Kp}{alphaD}}{alphaD} + \left(Kp + \frac{Kp}{alphaD}\right)\right) \cdot 2.0}}{\sqrt{1.0 - \left(\left(2.0 - alphaD\right) - Kp \cdot 4.0\right) \cdot alphaD} \cdot 2}$
3. Recombined 2 regimes into one program.
4. Final simplification15.8

$\leadsto \begin{array}{l} \mathbf{if}\;alphaD \le 32524.336403369205:\\ \;\;\;\;\frac{\left(1 - alphaD\right) + \sqrt{1.0 - \left(\left(2.0 - alphaD\right) - Kp \cdot 4.0\right) \cdot alphaD}}{2 \cdot \sqrt{1.0 - \left(\left(2.0 - alphaD\right) - Kp \cdot 4.0\right) \cdot alphaD}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.0 \cdot \left(\frac{\frac{Kp}{alphaD}}{alphaD} + \left(\frac{Kp}{alphaD} + Kp\right)\right)}{2 \cdot \sqrt{1.0 - \left(\left(2.0 - alphaD\right) - Kp \cdot 4.0\right) \cdot alphaD}}\\ \end{array}$

# Runtime

Time bar (total: 49.5s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'