Average Error: 28.9 → 0.2
Time: 16.5s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\frac{\frac{x - z}{\frac{y}{x + z}} + y}{2}\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\frac{\frac{x - z}{\frac{y}{x + z}} + y}{2}
double f(double x, double y, double z) {
        double r732377 = x;
        double r732378 = r732377 * r732377;
        double r732379 = y;
        double r732380 = r732379 * r732379;
        double r732381 = r732378 + r732380;
        double r732382 = z;
        double r732383 = r732382 * r732382;
        double r732384 = r732381 - r732383;
        double r732385 = 2.0;
        double r732386 = r732379 * r732385;
        double r732387 = r732384 / r732386;
        return r732387;
}
↓
double f(double x, double y, double z) {
        double r732388 = x;
        double r732389 = z;
        double r732390 = r732388 - r732389;
        double r732391 = y;
        double r732392 = r732388 + r732389;
        double r732393 = r732391 / r732392;
        double r732394 = r732390 / r732393;
        double r732395 = r732394 + r732391;
        double r732396 = 2.0;
        double r732397 = r732395 / r732396;
        return r732397;
}

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

Original28.9
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 28.9

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

  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 clear-num0.2

    \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{\frac{y}{x + z}}}, x - z, y\right)}{2}\]
  5. Using strategy rm

  6. Applied fma-udef0.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

    \[\leadsto \frac{\frac{x - z}{\frac{y}{x + z}} + y}{2}\]

Reproduce

herbie shell --seed 2020042 +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)))