Average Error: 28.6 → 0.1
Time: 18.7s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\frac{y + \frac{x + z}{y} \cdot \left(x - z\right)}{2}\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\frac{y + \frac{x + z}{y} \cdot \left(x - z\right)}{2}
double f(double x, double y, double z) {
        double r495119 = x;
        double r495120 = r495119 * r495119;
        double r495121 = y;
        double r495122 = r495121 * r495121;
        double r495123 = r495120 + r495122;
        double r495124 = z;
        double r495125 = r495124 * r495124;
        double r495126 = r495123 - r495125;
        double r495127 = 2.0;
        double r495128 = r495121 * r495127;
        double r495129 = r495126 / r495128;
        return r495129;
}
↓
double f(double x, double y, double z) {
        double r495130 = y;
        double r495131 = x;
        double r495132 = z;
        double r495133 = r495131 + r495132;
        double r495134 = r495133 / r495130;
        double r495135 = r495131 - r495132;
        double r495136 = r495134 * r495135;
        double r495137 = r495130 + r495136;
        double r495138 = 2.0;
        double r495139 = r495137 / r495138;
        return r495139;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Simplified12.7

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

  4. Applied difference-of-squares12.7

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

  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 associate-/r/0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"

  :herbie-target
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))