Average Error: 28.9 → 0.2
Time: 14.7s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\frac{\mathsf{fma}\left(\frac{x + z}{y}, x - z, y\right)}{2}\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\frac{\mathsf{fma}\left(\frac{x + z}{y}, x - z, y\right)}{2}
double f(double x, double y, double z) {
        double r746903 = x;
        double r746904 = r746903 * r746903;
        double r746905 = y;
        double r746906 = r746905 * r746905;
        double r746907 = r746904 + r746906;
        double r746908 = z;
        double r746909 = r746908 * r746908;
        double r746910 = r746907 - r746909;
        double r746911 = 2.0;
        double r746912 = r746905 * r746911;
        double r746913 = r746910 / r746912;
        return r746913;
}
↓
double f(double x, double y, double z) {
        double r746914 = x;
        double r746915 = z;
        double r746916 = r746914 + r746915;
        double r746917 = y;
        double r746918 = r746916 / r746917;
        double r746919 = r746914 - r746915;
        double r746920 = fma(r746918, r746919, r746917);
        double r746921 = 2.0;
        double r746922 = r746920 / r746921;
        return r746922;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original28.9
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 28.9

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

  2. Simplified0.2

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2020042 +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)))