Average Error: 28.4 → 0.2
Time: 13.5s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\frac{\frac{x - z}{y} \cdot \left(x + z\right) + y}{2}\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\frac{\frac{x - z}{y} \cdot \left(x + z\right) + y}{2}
double f(double x, double y, double z) {
        double r882078 = x;
        double r882079 = r882078 * r882078;
        double r882080 = y;
        double r882081 = r882080 * r882080;
        double r882082 = r882079 + r882081;
        double r882083 = z;
        double r882084 = r882083 * r882083;
        double r882085 = r882082 - r882084;
        double r882086 = 2.0;
        double r882087 = r882080 * r882086;
        double r882088 = r882085 / r882087;
        return r882088;
}
↓
double f(double x, double y, double z) {
        double r882089 = x;
        double r882090 = z;
        double r882091 = r882089 - r882090;
        double r882092 = y;
        double r882093 = r882091 / r882092;
        double r882094 = r882089 + r882090;
        double r882095 = r882093 * r882094;
        double r882096 = r882095 + r882092;
        double r882097 = 2.0;
        double r882098 = r882096 / r882097;
        return r882098;
}

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

Original28.4
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 28.4

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

  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

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019351 +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)))