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

Average Error: 28.6 → 0.4

Time: 3.1s

Precision: 64

Math

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

↓

\[0.5 \cdot \left(\left(y + x \cdot \frac{x}{y}\right) - \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \left(\sqrt[3]{z} \cdot \frac{z}{y}\right)\right)\]

C

double f(double x, double y, double z) {
        double r660788 = x;
        double r660789 = r660788 * r660788;
        double r660790 = y;
        double r660791 = r660790 * r660790;
        double r660792 = r660789 + r660791;
        double r660793 = z;
        double r660794 = r660793 * r660793;
        double r660795 = r660792 - r660794;
        double r660796 = 2.0;
        double r660797 = r660790 * r660796;
        double r660798 = r660795 / r660797;
        return r660798;
}

↓

double f(double x, double y, double z) {
        double r660799 = 0.5;
        double r660800 = y;
        double r660801 = x;
        double r660802 = r660801 / r660800;
        double r660803 = r660801 * r660802;
        double r660804 = r660800 + r660803;
        double r660805 = z;
        double r660806 = cbrt(r660805);
        double r660807 = r660806 * r660806;
        double r660808 = r660805 / r660800;
        double r660809 = r660806 * r660808;
        double r660810 = r660807 * r660809;
        double r660811 = r660804 - r660810;
        double r660812 = r660799 * r660811;
        return r660812;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	28.6
Target	0.2
Herbie	0.4

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

Derivation

1. Initial program 28.6

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

2. Taylor expanded around 0 12.2

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

3. Simplified 12.2

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

5. Applied *-un-lft-identity 12.2

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

6. Applied add-sqr-sqrt 38.5

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

7. Applied unpow-prod-down 38.5

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

8. Applied times-frac 35.9

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

9. Simplified 35.9

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

10. Simplified 6.7

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

12. Applied *-un-lft-identity 6.7

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

13. Applied add-sqr-sqrt 35.8

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

14. Applied unpow-prod-down 35.8

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

15. Applied times-frac 32.5

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

16. Simplified 32.4

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

17. Simplified 0.1

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

19. Applied add-cube-cbrt 0.4

    \[\leadsto 0.5 \cdot \left(\left(y + x \cdot \frac{x}{y}\right) - \color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)} \cdot \frac{z}{y}\right)\]

20. Applied associate-*l* 0.4

    \[\leadsto 0.5 \cdot \left(\left(y + x \cdot \frac{x}{y}\right) - \color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \left(\sqrt[3]{z} \cdot \frac{z}{y}\right)}\right)\]

21. Final simplification 0.4

    \[\leadsto 0.5 \cdot \left(\left(y + x \cdot \frac{x}{y}\right) - \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \left(\sqrt[3]{z} \cdot \frac{z}{y}\right)\right)\]

Reproduce

herbie shell --seed 2020021 
(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)))