Average Error: 28.5 → 0.2
Time: 3.2s
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 r651938 = x;
        double r651939 = r651938 * r651938;
        double r651940 = y;
        double r651941 = r651940 * r651940;
        double r651942 = r651939 + r651941;
        double r651943 = z;
        double r651944 = r651943 * r651943;
        double r651945 = r651942 - r651944;
        double r651946 = 2.0;
        double r651947 = r651940 * r651946;
        double r651948 = r651945 / r651947;
        return r651948;
}
↓
double f(double x, double y, double z) {
        double r651949 = 0.5;
        double r651950 = y;
        double r651951 = x;
        double r651952 = r651950 / r651951;
        double r651953 = r651951 / r651952;
        double r651954 = r651950 + r651953;
        double r651955 = z;
        double r651956 = r651950 / r651955;
        double r651957 = r651955 / r651956;
        double r651958 = r651954 - r651957;
        double r651959 = r651949 * r651958;
        return r651959;
}

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 unpow26.7

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

  9. Applied associate-/l*0.2

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

  10. 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 
(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)))