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

Average Error: 29.2 → 0.4

Time: 3.9s

Precision: 64

Math

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

↓

\[0.5 \cdot \left(\left(y + \frac{1}{\sqrt[3]{\frac{y}{x}} \cdot \sqrt[3]{\frac{y}{x}}} \cdot \frac{{x}^{1}}{\sqrt[3]{\frac{y}{x}}}\right) - \left|z\right| \cdot \frac{\left|z\right|}{y}\right)\]

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 + ((1.0 / (cbrt((y / x)) * cbrt((y / x)))) * (pow(x, 1.0) / cbrt((y / x))))) - (fabs(z) * (fabs(z) / y))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	29.2
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 29.2

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

2. Simplified 29.2

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{2}}{y}}\]

3. Taylor expanded around 0 12.4

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

4. Simplified 12.4

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

6. Applied sqr-pow 12.4

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

7. Applied associate-/l* 6.8

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

8. Simplified 6.8

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

10. Applied *-un-lft-identity 6.8

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

11. Applied add-sqr-sqrt 6.8

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

12. Applied times-frac 6.8

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

13. Simplified 6.8

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

14. Simplified 0.2

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

16. Applied add-cube-cbrt 0.4

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{{x}^{\left(\frac{2}{2}\right)}}{\color{blue}{\left(\sqrt[3]{\frac{y}{x}} \cdot \sqrt[3]{\frac{y}{x}}\right) \cdot \sqrt[3]{\frac{y}{x}}}}\right) - \left|z\right| \cdot \frac{\left|z\right|}{y}\right)\]

17. Applied *-un-lft-identity 0.4

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{{\color{blue}{\left(1 \cdot x\right)}}^{\left(\frac{2}{2}\right)}}{\left(\sqrt[3]{\frac{y}{x}} \cdot \sqrt[3]{\frac{y}{x}}\right) \cdot \sqrt[3]{\frac{y}{x}}}\right) - \left|z\right| \cdot \frac{\left|z\right|}{y}\right)\]

18. Applied unpow-prod-down 0.4

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{\color{blue}{{1}^{\left(\frac{2}{2}\right)} \cdot {x}^{\left(\frac{2}{2}\right)}}}{\left(\sqrt[3]{\frac{y}{x}} \cdot \sqrt[3]{\frac{y}{x}}\right) \cdot \sqrt[3]{\frac{y}{x}}}\right) - \left|z\right| \cdot \frac{\left|z\right|}{y}\right)\]

19. Applied times-frac 0.4

    \[\leadsto 0.5 \cdot \left(\left(y + \color{blue}{\frac{{1}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\frac{y}{x}} \cdot \sqrt[3]{\frac{y}{x}}} \cdot \frac{{x}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\frac{y}{x}}}}\right) - \left|z\right| \cdot \frac{\left|z\right|}{y}\right)\]

20. Simplified 0.4

    \[\leadsto 0.5 \cdot \left(\left(y + \color{blue}{\frac{1}{\sqrt[3]{\frac{y}{x}} \cdot \sqrt[3]{\frac{y}{x}}}} \cdot \frac{{x}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\frac{y}{x}}}\right) - \left|z\right| \cdot \frac{\left|z\right|}{y}\right)\]

21. Simplified 0.4

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{1}{\sqrt[3]{\frac{y}{x}} \cdot \sqrt[3]{\frac{y}{x}}} \cdot \color{blue}{\frac{{x}^{1}}{\sqrt[3]{\frac{y}{x}}}}\right) - \left|z\right| \cdot \frac{\left|z\right|}{y}\right)\]

22. Final simplification 0.4

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{1}{\sqrt[3]{\frac{y}{x}} \cdot \sqrt[3]{\frac{y}{x}}} \cdot \frac{{x}^{1}}{\sqrt[3]{\frac{y}{x}}}\right) - \left|z\right| \cdot \frac{\left|z\right|}{y}\right)\]

Reproduce

herbie shell --seed 2020058 +o rules:numerics
(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)))