Average Error: 29.3 → 0.1
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 r644646 = x;
        double r644647 = r644646 * r644646;
        double r644648 = y;
        double r644649 = r644648 * r644648;
        double r644650 = r644647 + r644649;
        double r644651 = z;
        double r644652 = r644651 * r644651;
        double r644653 = r644650 - r644652;
        double r644654 = 2.0;
        double r644655 = r644648 * r644654;
        double r644656 = r644653 / r644655;
        return r644656;
}
↓
double f(double x, double y, double z) {
        double r644657 = 0.5;
        double r644658 = y;
        double r644659 = x;
        double r644660 = r644658 / r644659;
        double r644661 = r644659 / r644660;
        double r644662 = r644658 + r644661;
        double r644663 = z;
        double r644664 = r644658 / r644663;
        double r644665 = r644663 / r644664;
        double r644666 = r644662 - r644665;
        double r644667 = r644657 * r644666;
        return r644667;
}

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

Original29.3
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 29.3

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

  2. Taylor expanded around 0 12.8

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

  3. Simplified12.8

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

  5. Applied unpow212.8

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

  6. Applied associate-/l*6.9

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

  8. Applied unpow26.9

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

  9. Applied associate-/l*0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2020036 +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)))