Average Error: 28.7 → 0.1
Time: 24.7s
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 r536476 = x;
        double r536477 = r536476 * r536476;
        double r536478 = y;
        double r536479 = r536478 * r536478;
        double r536480 = r536477 + r536479;
        double r536481 = z;
        double r536482 = r536481 * r536481;
        double r536483 = r536480 - r536482;
        double r536484 = 2.0;
        double r536485 = r536478 * r536484;
        double r536486 = r536483 / r536485;
        return r536486;
}
↓
double f(double x, double y, double z) {
        double r536487 = x;
        double r536488 = z;
        double r536489 = r536487 + r536488;
        double r536490 = y;
        double r536491 = r536489 / r536490;
        double r536492 = r536487 - r536488;
        double r536493 = fma(r536491, r536492, r536490);
        double r536494 = 2.0;
        double r536495 = r536493 / r536494;
        return r536495;
}

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)))