Average Error: 28.6 → 0.2
Time: 21.3s
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 r408634 = x;
        double r408635 = r408634 * r408634;
        double r408636 = y;
        double r408637 = r408636 * r408636;
        double r408638 = r408635 + r408637;
        double r408639 = z;
        double r408640 = r408639 * r408639;
        double r408641 = r408638 - r408640;
        double r408642 = 2.0;
        double r408643 = r408636 * r408642;
        double r408644 = r408641 / r408643;
        return r408644;
}
↓
double f(double x, double y, double z) {
        double r408645 = x;
        double r408646 = z;
        double r408647 = r408645 - r408646;
        double r408648 = y;
        double r408649 = r408645 + r408646;
        double r408650 = r408648 / r408649;
        double r408651 = r408647 / r408650;
        double r408652 = r408651 + r408648;
        double r408653 = 2.0;
        double r408654 = r408652 / r408653;
        return r408654;
}

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

Original28.6
Target0.1
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 28.6

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

  2. Simplified0.1

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{x + z}{y}, x - z, y\right)}{2}}\]
  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 2019323 +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)))