Average Error: 28.4 → 0.1
Time: 21.5s
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 r629347 = x;
        double r629348 = r629347 * r629347;
        double r629349 = y;
        double r629350 = r629349 * r629349;
        double r629351 = r629348 + r629350;
        double r629352 = z;
        double r629353 = r629352 * r629352;
        double r629354 = r629351 - r629353;
        double r629355 = 2.0;
        double r629356 = r629349 * r629355;
        double r629357 = r629354 / r629356;
        return r629357;
}
↓
double f(double x, double y, double z) {
        double r629358 = x;
        double r629359 = z;
        double r629360 = r629358 + r629359;
        double r629361 = y;
        double r629362 = r629360 / r629361;
        double r629363 = r629358 - r629359;
        double r629364 = fma(r629362, r629363, r629361);
        double r629365 = 2.0;
        double r629366 = r629364 / r629365;
        return r629366;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original28.4
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 28.4

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

  2. Simplified0.1

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

  4. Applied *-un-lft-identity0.1

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

  5. Applied *-un-lft-identity0.1

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

  6. Applied times-frac0.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019304 +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)))