Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A

Average Error: 28.8 → 0.2

Time: 4.0s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r739846 = x;
        double r739847 = r739846 * r739846;
        double r739848 = y;
        double r739849 = r739848 * r739848;
        double r739850 = r739847 + r739849;
        double r739851 = z;
        double r739852 = r739851 * r739851;
        double r739853 = r739850 - r739852;
        double r739854 = 2.0;
        double r739855 = r739848 * r739854;
        double r739856 = r739853 / r739855;
        return r739856;
}

↓

double f(double x, double y, double z) {
        double r739857 = 0.5;
        double r739858 = y;
        double r739859 = x;
        double r739860 = r739859 / r739858;
        double r739861 = r739859 * r739860;
        double r739862 = r739858 + r739861;
        double r739863 = z;
        double r739864 = r739863 / r739858;
        double r739865 = r739863 * r739864;
        double r739866 = r739862 - r739865;
        double r739867 = r739857 * r739866;
        return r739867;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	28.8
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 28.8

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

2. Simplified 28.8

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

3. Taylor expanded around 0 12.8

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

4. Simplified 12.8

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

6. Applied *-un-lft-identity 12.8

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

7. Applied add-sqr-sqrt 38.0

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

8. Applied unpow-prod-down 38.0

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

9. Applied times-frac 35.2

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

10. Simplified 35.2

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

11. Simplified 7.2

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

13. Applied *-un-lft-identity 7.2

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

14. Applied add-sqr-sqrt 36.2

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

15. Applied unpow-prod-down 36.2

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

16. Applied times-frac 32.6

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

17. Simplified 32.6

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

18. Simplified 0.2

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

19. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020025 +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)))