Average Error: 28.2 → 0.1
Time: 20.0s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\frac{y + \left(x + z\right) \cdot \frac{x - z}{y}}{2}\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\frac{y + \left(x + z\right) \cdot \frac{x - z}{y}}{2}
double f(double x, double y, double z) {
        double r503098 = x;
        double r503099 = r503098 * r503098;
        double r503100 = y;
        double r503101 = r503100 * r503100;
        double r503102 = r503099 + r503101;
        double r503103 = z;
        double r503104 = r503103 * r503103;
        double r503105 = r503102 - r503104;
        double r503106 = 2.0;
        double r503107 = r503100 * r503106;
        double r503108 = r503105 / r503107;
        return r503108;
}
↓
double f(double x, double y, double z) {
        double r503109 = y;
        double r503110 = x;
        double r503111 = z;
        double r503112 = r503110 + r503111;
        double r503113 = r503110 - r503111;
        double r503114 = r503113 / r503109;
        double r503115 = r503112 * r503114;
        double r503116 = r503109 + r503115;
        double r503117 = 2.0;
        double r503118 = r503116 / r503117;
        return r503118;
}

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
Herbie0.1
\[y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right)\]

Derivation

  1. Initial program 28.2

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

  2. Simplified12.2

    \[\leadsto \color{blue}{\frac{y + \frac{x \cdot x - z \cdot z}{y}}{2}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity12.2

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

  5. Applied difference-of-squares12.2

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

  6. Applied times-frac0.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019212 +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)))