Average Error: 28.2 → 0.2
Time: 18.5s
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 r478713 = x;
        double r478714 = r478713 * r478713;
        double r478715 = y;
        double r478716 = r478715 * r478715;
        double r478717 = r478714 + r478716;
        double r478718 = z;
        double r478719 = r478718 * r478718;
        double r478720 = r478717 - r478719;
        double r478721 = 2.0;
        double r478722 = r478715 * r478721;
        double r478723 = r478720 / r478722;
        return r478723;
}
↓
double f(double x, double y, double z) {
        double r478724 = x;
        double r478725 = z;
        double r478726 = r478724 - r478725;
        double r478727 = y;
        double r478728 = r478724 + r478725;
        double r478729 = r478727 / r478728;
        double r478730 = r478726 / r478729;
        double r478731 = r478730 + r478727;
        double r478732 = 2.0;
        double r478733 = r478731 / r478732;
        return r478733;
}

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)))