Average Error: 28.7 → 0.1
Time: 17.3s
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 r432523 = x;
        double r432524 = r432523 * r432523;
        double r432525 = y;
        double r432526 = r432525 * r432525;
        double r432527 = r432524 + r432526;
        double r432528 = z;
        double r432529 = r432528 * r432528;
        double r432530 = r432527 - r432529;
        double r432531 = 2.0;
        double r432532 = r432525 * r432531;
        double r432533 = r432530 / r432532;
        return r432533;
}
↓
double f(double x, double y, double z) {
        double r432534 = x;
        double r432535 = z;
        double r432536 = r432534 + r432535;
        double r432537 = y;
        double r432538 = r432536 / r432537;
        double r432539 = r432534 - r432535;
        double r432540 = fma(r432538, r432539, r432537);
        double r432541 = 2.0;
        double r432542 = r432540 / r432541;
        return r432542;
}

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. Simplified12.6

    \[\leadsto \color{blue}{\frac{y + \frac{x \cdot x - z \cdot z}{y}}{2}}\]
  3. Using strategy rm

  4. Applied difference-of-squares12.6

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

  5. Applied associate-/l*0.1

    \[\leadsto \frac{y + \color{blue}{\frac{x + z}{\frac{y}{x - z}}}}{2}\]
  6. Using strategy rm

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

    \[\leadsto \frac{y + \frac{x + z}{\frac{y}{x - z}}}{\color{blue}{1 \cdot 2}}\]

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

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

  9. Applied times-frac0.1

    \[\leadsto \color{blue}{\frac{1}{1} \cdot \frac{y + \frac{x + z}{\frac{y}{x - z}}}{2}}\]

  10. Simplified0.1

    \[\leadsto \color{blue}{1} \cdot \frac{y + \frac{x + z}{\frac{y}{x - z}}}{2}\]

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019305 +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)))