Average Error: 28.5 → 0.1
Time: 17.0s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\frac{y + \frac{x - z}{\frac{y}{z + x}}}{2}\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\frac{y + \frac{x - z}{\frac{y}{z + x}}}{2}
double f(double x, double y, double z) {
        double r587568 = x;
        double r587569 = r587568 * r587568;
        double r587570 = y;
        double r587571 = r587570 * r587570;
        double r587572 = r587569 + r587571;
        double r587573 = z;
        double r587574 = r587573 * r587573;
        double r587575 = r587572 - r587574;
        double r587576 = 2.0;
        double r587577 = r587570 * r587576;
        double r587578 = r587575 / r587577;
        return r587578;
}
↓
double f(double x, double y, double z) {
        double r587579 = y;
        double r587580 = x;
        double r587581 = z;
        double r587582 = r587580 - r587581;
        double r587583 = r587581 + r587580;
        double r587584 = r587579 / r587583;
        double r587585 = r587582 / r587584;
        double r587586 = r587579 + r587585;
        double r587587 = 2.0;
        double r587588 = r587586 / r587587;
        return r587588;
}

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

Original28.5
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 28.5

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

  2. Simplified0.1

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

  4. Applied clear-num0.2

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

  5. Simplified0.2

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

  7. Applied *-un-lft-identity0.2

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

  8. Applied *-un-lft-identity0.2

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

  9. Applied times-frac0.2

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

  10. Applied *-un-lft-identity0.2

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

  11. Applied times-frac0.2

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

  12. Applied associate-*l*0.2

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

  13. Simplified0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2019194 +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)))