Average Error: 22.4 → 18.6
Time: 2.2s
Precision: binary64
\[\frac{\left(1 - alphaD\right) + \sqrt{\left(\left(1 - 2 \cdot alphaD\right) + \left(4 \cdot Kp\right) \cdot alphaD\right) + alphaD \cdot alphaD}}{2 \cdot \sqrt{\left(\left(1 - 2 \cdot alphaD\right) + \left(4 \cdot Kp\right) \cdot alphaD\right) + alphaD \cdot alphaD}}\]
↓
\[\frac{1 - \frac{alphaD - 1}{\sqrt{\left(\left(1 - 2 \cdot alphaD\right) + \left(4 \cdot Kp\right) \cdot alphaD\right) + alphaD \cdot alphaD}}}{2}\]
\frac{\left(1 - alphaD\right) + \sqrt{\left(\left(1 - 2 \cdot alphaD\right) + \left(4 \cdot Kp\right) \cdot alphaD\right) + alphaD \cdot alphaD}}{2 \cdot \sqrt{\left(\left(1 - 2 \cdot alphaD\right) + \left(4 \cdot Kp\right) \cdot alphaD\right) + alphaD \cdot alphaD}}
↓
\frac{1 - \frac{alphaD - 1}{\sqrt{\left(\left(1 - 2 \cdot alphaD\right) + \left(4 \cdot Kp\right) \cdot alphaD\right) + alphaD \cdot alphaD}}}{2}
double code(double alphaD, double Kp) {
	return ((double) (((double) (((double) (1.0 - alphaD)) + ((double) sqrt(((double) (((double) (((double) (1.0 - ((double) (2.0 * alphaD)))) + ((double) (((double) (4.0 * Kp)) * alphaD)))) + ((double) (alphaD * alphaD)))))))) / ((double) (2.0 * ((double) sqrt(((double) (((double) (((double) (1.0 - ((double) (2.0 * alphaD)))) + ((double) (((double) (4.0 * Kp)) * alphaD)))) + ((double) (alphaD * alphaD))))))))));
}
↓
double code(double alphaD, double Kp) {
	return ((double) (((double) (1.0 - ((double) (((double) (alphaD - 1.0)) / ((double) sqrt(((double) (((double) (((double) (1.0 - ((double) (2.0 * alphaD)))) + ((double) (((double) (4.0 * Kp)) * alphaD)))) + ((double) (alphaD * alphaD)))))))))) / 2.0));
}

Error

Bits error versus alphaD

Bits error versus Kp

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 22.4

    \[\frac{\left(1 - alphaD\right) + \sqrt{\left(\left(1 - 2 \cdot alphaD\right) + \left(4 \cdot Kp\right) \cdot alphaD\right) + alphaD \cdot alphaD}}{2 \cdot \sqrt{\left(\left(1 - 2 \cdot alphaD\right) + \left(4 \cdot Kp\right) \cdot alphaD\right) + alphaD \cdot alphaD}}\]

  2. Simplified18.6

    \[\leadsto \color{blue}{\frac{1 - \frac{alphaD - 1}{\sqrt{\left(\left(1 - 2 \cdot alphaD\right) + \left(4 \cdot Kp\right) \cdot alphaD\right) + alphaD \cdot alphaD}}}{2}}\]

  3. Final simplification18.6

    \[\leadsto \frac{1 - \frac{alphaD - 1}{\sqrt{\left(\left(1 - 2 \cdot alphaD\right) + \left(4 \cdot Kp\right) \cdot alphaD\right) + alphaD \cdot alphaD}}}{2}\]

Reproduce

herbie shell --seed 2020152 
(FPCore (alphaD Kp)
  :name "(/ (+ (- 1 alphaD) (sqrt (+ (+ (- 1.0 (* 2.0 alphaD)) (* (* 4.0 Kp) alphaD)) (* alphaD alphaD)))) (* 2 (sqrt (+ (+ (- 1.0 (* 2.0 alphaD)) (* (* 4.0 Kp) alphaD)) (* alphaD alphaD)))))"
  :precision binary64
  (/ (+ (- 1.0 alphaD) (sqrt (+ (+ (- 1.0 (* 2.0 alphaD)) (* (* 4.0 Kp) alphaD)) (* alphaD alphaD)))) (* 2.0 (sqrt (+ (+ (- 1.0 (* 2.0 alphaD)) (* (* 4.0 Kp) alphaD)) (* alphaD alphaD))))))