Average Error: 28.6 → 0.1
Time: 12.9s
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 r748728 = x;
        double r748729 = r748728 * r748728;
        double r748730 = y;
        double r748731 = r748730 * r748730;
        double r748732 = r748729 + r748731;
        double r748733 = z;
        double r748734 = r748733 * r748733;
        double r748735 = r748732 - r748734;
        double r748736 = 2.0;
        double r748737 = r748730 * r748736;
        double r748738 = r748735 / r748737;
        return r748738;
}
↓
double f(double x, double y, double z) {
        double r748739 = x;
        double r748740 = z;
        double r748741 = r748739 + r748740;
        double r748742 = y;
        double r748743 = r748741 / r748742;
        double r748744 = r748739 - r748740;
        double r748745 = fma(r748743, r748744, r748742);
        double r748746 = 2.0;
        double r748747 = r748745 / r748746;
        return r748747;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original28.6
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 28.6

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

  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 2020045 +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)))