Average Error: 27.4 → 0.2
Time: 5.0m
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2.0}\]
↓
\[\frac{y + \left(\frac{x}{\frac{y}{x}} - \frac{\frac{z}{y}}{\frac{1}{z}}\right)}{2.0}\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2.0}
↓
\frac{y + \left(\frac{x}{\frac{y}{x}} - \frac{\frac{z}{y}}{\frac{1}{z}}\right)}{2.0}
double f(double x, double y, double z) {
        double r40281724 = x;
        double r40281725 = r40281724 * r40281724;
        double r40281726 = y;
        double r40281727 = r40281726 * r40281726;
        double r40281728 = r40281725 + r40281727;
        double r40281729 = z;
        double r40281730 = r40281729 * r40281729;
        double r40281731 = r40281728 - r40281730;
        double r40281732 = 2.0;
        double r40281733 = r40281726 * r40281732;
        double r40281734 = r40281731 / r40281733;
        return r40281734;
}
↓
double f(double x, double y, double z) {
        double r40281735 = y;
        double r40281736 = x;
        double r40281737 = r40281735 / r40281736;
        double r40281738 = r40281736 / r40281737;
        double r40281739 = z;
        double r40281740 = r40281739 / r40281735;
        double r40281741 = 1.0;
        double r40281742 = r40281741 / r40281739;
        double r40281743 = r40281740 / r40281742;
        double r40281744 = r40281738 - r40281743;
        double r40281745 = r40281735 + r40281744;
        double r40281746 = 2.0;
        double r40281747 = r40281745 / r40281746;
        return r40281747;
}

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

Original27.4
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 27.4

    \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2.0}\]

  2. Simplified12.1

    \[\leadsto \color{blue}{\frac{y + \frac{\left(x - z\right) \cdot \left(z + x\right)}{y}}{2.0}}\]

  3. Taylor expanded around 0 12.2

    \[\leadsto \frac{y + \color{blue}{\left(\frac{{x}^{2}}{y} - \frac{{z}^{2}}{y}\right)}}{2.0}\]

  4. Simplified0.1

    \[\leadsto \frac{y + \color{blue}{\left(\frac{x}{\frac{y}{x}} - \frac{z}{\frac{y}{z}}\right)}}{2.0}\]
  5. Using strategy rm

  6. Applied div-inv0.2

    \[\leadsto \frac{y + \left(\frac{x}{\frac{y}{x}} - \frac{z}{\color{blue}{y \cdot \frac{1}{z}}}\right)}{2.0}\]

  7. Applied associate-/r*0.2

    \[\leadsto \frac{y + \left(\frac{x}{\frac{y}{x}} - \color{blue}{\frac{\frac{z}{y}}{\frac{1}{z}}}\right)}{2.0}\]

  8. Final simplification0.2

    \[\leadsto \frac{y + \left(\frac{x}{\frac{y}{x}} - \frac{\frac{z}{y}}{\frac{1}{z}}\right)}{2.0}\]

Reproduce

herbie shell --seed 2019163 +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)))