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

Average Error: 27.7 → 0.4

Time: 3.0s

Precision: 64

Math

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

↓

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

C

double code(double x, double y, double z) {
	return ((((x * x) + (y * y)) - (z * z)) / (y * 2.0));
}

↓

double code(double x, double y, double z) {
	return (0.5 * ((y + ((cbrt((x * (x / y))) * cbrt((x * (x / y)))) * cbrt((x * (x / y))))) - (z * (z / y))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	27.7
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 27.7

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

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

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

6. Applied add-sqr-sqrt 38.5

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

7. Applied unpow-prod-down 38.5

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

8. Applied times-frac 35.5

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

9. Simplified 35.5

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

10. Simplified 6.4

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

12. Applied *-un-lft-identity 6.4

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

13. Applied add-sqr-sqrt 35.1

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

14. Applied unpow-prod-down 35.1

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

15. Applied times-frac 31.9

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

16. Simplified 31.9

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

17. Simplified 0.2

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

19. Applied add-cube-cbrt 0.4

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

20. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020100 
(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)))