Average Error: 28.5 → 0.2
Time: 21.0s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\frac{\mathsf{fma}\left(\left(x + z\right) \cdot \frac{1}{y}, x - z, y\right)}{2}\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\frac{\mathsf{fma}\left(\left(x + z\right) \cdot \frac{1}{y}, x - z, y\right)}{2}
double f(double x, double y, double z) {
        double r420001 = x;
        double r420002 = r420001 * r420001;
        double r420003 = y;
        double r420004 = r420003 * r420003;
        double r420005 = r420002 + r420004;
        double r420006 = z;
        double r420007 = r420006 * r420006;
        double r420008 = r420005 - r420007;
        double r420009 = 2.0;
        double r420010 = r420003 * r420009;
        double r420011 = r420008 / r420010;
        return r420011;
}
↓
double f(double x, double y, double z) {
        double r420012 = x;
        double r420013 = z;
        double r420014 = r420012 + r420013;
        double r420015 = 1.0;
        double r420016 = y;
        double r420017 = r420015 / r420016;
        double r420018 = r420014 * r420017;
        double r420019 = r420012 - r420013;
        double r420020 = fma(r420018, r420019, r420016);
        double r420021 = 2.0;
        double r420022 = r420020 / r420021;
        return r420022;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original28.5
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 28.5

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

  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 div-inv0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019322 +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)))