Average Error: 28.5 → 0.2
Time: 4.5s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - \frac{z}{\frac{y}{z}}\right)\]
\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}{\frac{y}{z}}\right)
double f(double x, double y, double z) {
        double r684952 = x;
        double r684953 = r684952 * r684952;
        double r684954 = y;
        double r684955 = r684954 * r684954;
        double r684956 = r684953 + r684955;
        double r684957 = z;
        double r684958 = r684957 * r684957;
        double r684959 = r684956 - r684958;
        double r684960 = 2.0;
        double r684961 = r684954 * r684960;
        double r684962 = r684959 / r684961;
        return r684962;
}
↓
double f(double x, double y, double z) {
        double r684963 = 0.5;
        double r684964 = y;
        double r684965 = x;
        double r684966 = r684964 / r684965;
        double r684967 = r684965 / r684966;
        double r684968 = r684964 + r684967;
        double r684969 = z;
        double r684970 = r684964 / r684969;
        double r684971 = r684969 / r684970;
        double r684972 = r684968 - r684971;
        double r684973 = r684963 * r684972;
        return r684973;
}

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. Simplified28.5

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

  3. Taylor expanded around 0 12.5

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

  4. Simplified12.5

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

  6. Applied unpow212.5

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

  7. Applied associate-/l*6.7

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

  9. Applied unpow26.7

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

  10. Applied associate-/l*0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2020062 +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)))