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

Average Error: 28.8 → 0.1

Time: 3.6s

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 r678410 = x;
        double r678411 = r678410 * r678410;
        double r678412 = y;
        double r678413 = r678412 * r678412;
        double r678414 = r678411 + r678413;
        double r678415 = z;
        double r678416 = r678415 * r678415;
        double r678417 = r678414 - r678416;
        double r678418 = 2.0;
        double r678419 = r678412 * r678418;
        double r678420 = r678417 / r678419;
        return r678420;
}

↓

double f(double x, double y, double z) {
        double r678421 = 0.5;
        double r678422 = y;
        double r678423 = x;
        double r678424 = r678423 / r678422;
        double r678425 = r678423 * r678424;
        double r678426 = r678422 + r678425;
        double r678427 = z;
        double r678428 = r678427 / r678422;
        double r678429 = r678427 * r678428;
        double r678430 = r678426 - r678429;
        double r678431 = r678421 * r678430;
        return r678431;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	28.8
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 28.8

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

2. Taylor expanded around 0 12.4

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

3. Simplified 12.4

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

5. Applied unpow2 12.4

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

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-identity 6.9

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

9. Applied add-sqr-sqrt 35.9

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

10. Applied unpow-prod-down 35.9

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

11. Applied times-frac 32.5

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

12. Simplified 32.5

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

13. Simplified 0.1

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

15. Applied add-sqr-sqrt 32.3

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

16. Applied *-un-lft-identity 32.3

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

17. Applied times-frac 32.3

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

18. Applied add-sqr-sqrt 32.4

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

19. Applied times-frac 32.4

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

20. Simplified 32.3

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

21. Simplified 0.1

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

22. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020018 
(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)))