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

Average Error: 28.6 → 0.1

Time: 3.4s

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

↓

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	28.6
Target	0.2
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 28.6

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

2. Taylor expanded around 0 12.9

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

3. Simplified 12.9

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

5. Applied unpow2 12.9

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

6. Applied associate-/l* 7.2

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

8. Applied *-un-lft-identity 7.2

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

9. Applied add-sqr-sqrt 35.4

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

10. Applied unpow-prod-down 35.4

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

11. Applied times-frac 31.8

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

12. Simplified 31.8

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

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 32.0

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

16. Applied *-un-lft-identity 32.0

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

17. Applied times-frac 32.0

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

18. Applied add-sqr-sqrt 32.0

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

19. Applied times-frac 32.0

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

20. Simplified 32.0

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

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 2020140 
(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 2.0)))