Average Error: 28.7 → 0.1
Time: 25.2s
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 r384803 = x;
        double r384804 = r384803 * r384803;
        double r384805 = y;
        double r384806 = r384805 * r384805;
        double r384807 = r384804 + r384806;
        double r384808 = z;
        double r384809 = r384808 * r384808;
        double r384810 = r384807 - r384809;
        double r384811 = 2.0;
        double r384812 = r384805 * r384811;
        double r384813 = r384810 / r384812;
        return r384813;
}
↓
double f(double x, double y, double z) {
        double r384814 = x;
        double r384815 = z;
        double r384816 = r384814 + r384815;
        double r384817 = y;
        double r384818 = r384816 / r384817;
        double r384819 = r384814 - r384815;
        double r384820 = fma(r384818, r384819, r384817);
        double r384821 = 2.0;
        double r384822 = r384820 / r384821;
        return r384822;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original28.7
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 28.7

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

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019326 +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)))