Average Error: 28.5 → 0.2
Time: 3.5s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - \frac{{z}^{1}}{\frac{y}{z}}\right)\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - \frac{{z}^{1}}{\frac{y}{z}}\right)
double f(double x, double y, double z) {
        double r739783 = x;
        double r739784 = r739783 * r739783;
        double r739785 = y;
        double r739786 = r739785 * r739785;
        double r739787 = r739784 + r739786;
        double r739788 = z;
        double r739789 = r739788 * r739788;
        double r739790 = r739787 - r739789;
        double r739791 = 2.0;
        double r739792 = r739785 * r739791;
        double r739793 = r739790 / r739792;
        return r739793;
}
↓
double f(double x, double y, double z) {
        double r739794 = 0.5;
        double r739795 = y;
        double r739796 = x;
        double r739797 = r739795 / r739796;
        double r739798 = r739796 / r739797;
        double r739799 = r739795 + r739798;
        double r739800 = z;
        double r739801 = 1.0;
        double r739802 = pow(r739800, r739801);
        double r739803 = r739795 / r739800;
        double r739804 = r739802 / r739803;
        double r739805 = r739799 - r739804;
        double r739806 = r739794 * r739805;
        return r739806;
}

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
Herbie0.2
\[y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right)\]

Derivation

  1. Initial program 28.5

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

  2. Taylor expanded around 0 12.5

    \[\leadsto \color{blue}{\left(0.5 \cdot y + 0.5 \cdot \frac{{x}^{2}}{y}\right) - 0.5 \cdot \frac{{z}^{2}}{y}}\]

  3. Simplified12.5

    \[\leadsto \color{blue}{0.5 \cdot \left(\left(y + \frac{{x}^{2}}{y}\right) - \frac{{z}^{2}}{y}\right)}\]
  4. Using strategy rm

  5. Applied unpow212.5

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{\color{blue}{x \cdot x}}{y}\right) - \frac{{z}^{2}}{y}\right)\]

  6. Applied associate-/l*6.7

    \[\leadsto 0.5 \cdot \left(\left(y + \color{blue}{\frac{x}{\frac{y}{x}}}\right) - \frac{{z}^{2}}{y}\right)\]
  7. Using strategy rm

  8. Applied sqr-pow6.7

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - \frac{\color{blue}{{z}^{\left(\frac{2}{2}\right)} \cdot {z}^{\left(\frac{2}{2}\right)}}}{y}\right)\]

  9. Applied associate-/l*0.2

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - \color{blue}{\frac{{z}^{\left(\frac{2}{2}\right)}}{\frac{y}{{z}^{\left(\frac{2}{2}\right)}}}}\right)\]

  10. Simplified0.2

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - \frac{{z}^{\left(\frac{2}{2}\right)}}{\color{blue}{\frac{y}{z}}}\right)\]

  11. Final simplification0.2

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - \frac{{z}^{1}}{\frac{y}{z}}\right)\]

Reproduce

herbie shell --seed 2020062 
(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)))