Average Error: 28.2 → 0.2
Time: 17.7s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\frac{\frac{x - z}{\frac{y}{x + z}} + y}{2}\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\frac{\frac{x - z}{\frac{y}{x + z}} + y}{2}
double f(double x, double y, double z) {
        double r433658 = x;
        double r433659 = r433658 * r433658;
        double r433660 = y;
        double r433661 = r433660 * r433660;
        double r433662 = r433659 + r433661;
        double r433663 = z;
        double r433664 = r433663 * r433663;
        double r433665 = r433662 - r433664;
        double r433666 = 2.0;
        double r433667 = r433660 * r433666;
        double r433668 = r433665 / r433667;
        return r433668;
}
↓
double f(double x, double y, double z) {
        double r433669 = x;
        double r433670 = z;
        double r433671 = r433669 - r433670;
        double r433672 = y;
        double r433673 = r433669 + r433670;
        double r433674 = r433672 / r433673;
        double r433675 = r433671 / r433674;
        double r433676 = r433675 + r433672;
        double r433677 = 2.0;
        double r433678 = r433676 / r433677;
        return r433678;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original28.2
Target0.2
Herbie0.2
\[y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right)\]

Derivation

  1. Initial program 28.2

    \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]

  2. Simplified0.2

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{x + z}{y}, x - z, y\right)}{2}}\]
  3. Using strategy rm

  4. Applied clear-num0.2

    \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{\frac{y}{x + z}}}, x - z, y\right)}{2}\]
  5. Using strategy rm

  6. Applied fma-udef0.2

    \[\leadsto \frac{\color{blue}{\frac{1}{\frac{y}{x + z}} \cdot \left(x - z\right) + y}}{2}\]

  7. Simplified0.2

    \[\leadsto \frac{\color{blue}{\frac{x - z}{\frac{y}{x + z}}} + y}{2}\]

  8. Final simplification0.2

    \[\leadsto \frac{\frac{x - z}{\frac{y}{x + z}} + y}{2}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :herbie-target
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2)))