Average Error: 28.6 → 0.2
Time: 19.0s
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 r558696 = x;
        double r558697 = r558696 * r558696;
        double r558698 = y;
        double r558699 = r558698 * r558698;
        double r558700 = r558697 + r558699;
        double r558701 = z;
        double r558702 = r558701 * r558701;
        double r558703 = r558700 - r558702;
        double r558704 = 2.0;
        double r558705 = r558698 * r558704;
        double r558706 = r558703 / r558705;
        return r558706;
}
↓
double f(double x, double y, double z) {
        double r558707 = x;
        double r558708 = z;
        double r558709 = r558707 - r558708;
        double r558710 = y;
        double r558711 = r558707 + r558708;
        double r558712 = r558710 / r558711;
        double r558713 = r558709 / r558712;
        double r558714 = r558713 + r558710;
        double r558715 = 2.0;
        double r558716 = r558714 / r558715;
        return r558716;
}

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
Target0.1
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 28.6

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

  2. Simplified0.1

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{x + z}{y}, x - z, y\right)}{2}}\]
  3. 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 2019323 +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)))