Average Error: 27.7 → 0.2
Time: 14.3s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\frac{\mathsf{fma}\left(\frac{1}{\frac{y}{x + z}}, x - z, y\right)}{2}\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\frac{\mathsf{fma}\left(\frac{1}{\frac{y}{x + z}}, x - z, y\right)}{2}
double f(double x, double y, double z) {
        double r730713 = x;
        double r730714 = r730713 * r730713;
        double r730715 = y;
        double r730716 = r730715 * r730715;
        double r730717 = r730714 + r730716;
        double r730718 = z;
        double r730719 = r730718 * r730718;
        double r730720 = r730717 - r730719;
        double r730721 = 2.0;
        double r730722 = r730715 * r730721;
        double r730723 = r730720 / r730722;
        return r730723;
}
↓
double f(double x, double y, double z) {
        double r730724 = 1.0;
        double r730725 = y;
        double r730726 = x;
        double r730727 = z;
        double r730728 = r730726 + r730727;
        double r730729 = r730725 / r730728;
        double r730730 = r730724 / r730729;
        double r730731 = r730726 - r730727;
        double r730732 = fma(r730730, r730731, r730725);
        double r730733 = 2.0;
        double r730734 = r730732 / r730733;
        return r730734;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original27.7
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 27.7

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

  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. Final simplification0.2

    \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{\frac{y}{x + z}}, x - z, y\right)}{2}\]

Reproduce

herbie shell --seed 2019350 +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)))