Average Error: 28.4 → 0.2
Time: 22.3s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\frac{\frac{x + z}{y} \cdot \left(x - z\right) + y}{2}\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\frac{\frac{x + z}{y} \cdot \left(x - z\right) + y}{2}
double f(double x, double y, double z) {
        double r482223 = x;
        double r482224 = r482223 * r482223;
        double r482225 = y;
        double r482226 = r482225 * r482225;
        double r482227 = r482224 + r482226;
        double r482228 = z;
        double r482229 = r482228 * r482228;
        double r482230 = r482227 - r482229;
        double r482231 = 2.0;
        double r482232 = r482225 * r482231;
        double r482233 = r482230 / r482232;
        return r482233;
}
↓
double f(double x, double y, double z) {
        double r482234 = x;
        double r482235 = z;
        double r482236 = r482234 + r482235;
        double r482237 = y;
        double r482238 = r482236 / r482237;
        double r482239 = r482234 - r482235;
        double r482240 = r482238 * r482239;
        double r482241 = r482240 + r482237;
        double r482242 = 2.0;
        double r482243 = r482241 / r482242;
        return r482243;
}

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

Original28.4
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 28.4

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

  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 fma-udef0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019303 +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)))