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

Average Error: 28.5 → 0.2

Time: 3.0s

Precision: 64

Math

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

↓

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

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 + (fabs(x) * (fabs(x) / y))) - (z * (z / y))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	28.5
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 28.5

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

2. Taylor expanded around 0 12.8

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

3. Simplified 12.8

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

5. Applied unpow2 12.8

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

6. Applied associate-/l* 6.9

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

8. Applied *-un-lft-identity 6.9

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

9. Applied add-sqr-sqrt 6.9

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

10. Applied times-frac 6.9

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

11. Simplified 6.9

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

12. Simplified 0.2

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

14. Applied add-sqr-sqrt 31.6

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

15. Applied *-un-lft-identity 31.6

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

16. Applied times-frac 31.6

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

17. Applied add-sqr-sqrt 31.6

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

18. Applied times-frac 31.6

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

19. Simplified 31.6

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

20. Simplified 0.2

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

21. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020060 
(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)))