Average Error: 28.2 → 0.2
Time: 4.4s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[0.5 \cdot \left(\left(y + \frac{\frac{x}{y}}{\frac{1}{x}}\right) - \left|z\right| \cdot \frac{\left|z\right|}{y}\right)\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
0.5 \cdot \left(\left(y + \frac{\frac{x}{y}}{\frac{1}{x}}\right) - \left|z\right| \cdot \frac{\left|z\right|}{y}\right)
double f(double x, double y, double z) {
        double r689909 = x;
        double r689910 = r689909 * r689909;
        double r689911 = y;
        double r689912 = r689911 * r689911;
        double r689913 = r689910 + r689912;
        double r689914 = z;
        double r689915 = r689914 * r689914;
        double r689916 = r689913 - r689915;
        double r689917 = 2.0;
        double r689918 = r689911 * r689917;
        double r689919 = r689916 / r689918;
        return r689919;
}
↓
double f(double x, double y, double z) {
        double r689920 = 0.5;
        double r689921 = y;
        double r689922 = x;
        double r689923 = r689922 / r689921;
        double r689924 = 1.0;
        double r689925 = r689924 / r689922;
        double r689926 = r689923 / r689925;
        double r689927 = r689921 + r689926;
        double r689928 = z;
        double r689929 = fabs(r689928);
        double r689930 = r689929 / r689921;
        double r689931 = r689929 * r689930;
        double r689932 = r689927 - r689931;
        double r689933 = r689920 * r689932;
        return r689933;
}

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

Original28.2
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 28.2

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

  2. Taylor expanded around 0 12.3

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

  3. Simplified12.3

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

  5. Applied unpow212.3

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

  6. Applied associate-/l*6.9

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

  8. Applied *-un-lft-identity6.9

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - \frac{{z}^{2}}{\color{blue}{1 \cdot y}}\right)\]

  9. Applied add-sqr-sqrt6.9

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - \frac{\color{blue}{\sqrt{{z}^{2}} \cdot \sqrt{{z}^{2}}}}{1 \cdot y}\right)\]

  10. Applied times-frac6.9

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - \color{blue}{\frac{\sqrt{{z}^{2}}}{1} \cdot \frac{\sqrt{{z}^{2}}}{y}}\right)\]

  11. Simplified6.9

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

  12. Simplified0.1

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

  14. Applied div-inv0.2

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

  15. Applied associate-/r*0.2

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

  16. Final simplification0.2

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

Reproduce

herbie shell --seed 2020027 +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)))