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

Average Error: 28.7 → 0.2

Time: 4.8s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r804863 = x;
        double r804864 = r804863 * r804863;
        double r804865 = y;
        double r804866 = r804865 * r804865;
        double r804867 = r804864 + r804866;
        double r804868 = z;
        double r804869 = r804868 * r804868;
        double r804870 = r804867 - r804869;
        double r804871 = 2.0;
        double r804872 = r804865 * r804871;
        double r804873 = r804870 / r804872;
        return r804873;
}

↓

double f(double x, double y, double z) {
        double r804874 = 0.5;
        double r804875 = y;
        double r804876 = x;
        double r804877 = 1.0;
        double r804878 = pow(r804876, r804877);
        double r804879 = r804878 / r804875;
        double r804880 = r804876 * r804879;
        double r804881 = r804875 + r804880;
        double r804882 = z;
        double r804883 = pow(r804882, r804877);
        double r804884 = r804875 / r804882;
        double r804885 = r804883 / r804884;
        double r804886 = r804881 - r804885;
        double r804887 = r804874 * r804886;
        return r804887;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	28.7
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 28.7

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

2. Taylor expanded around 0 12.3

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

3. Simplified 12.3

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

5. Applied sqr-pow 12.3

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

6. Applied associate-/l* 6.7

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

7. Simplified 6.7

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

9. Applied sqr-pow 6.7

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

10. Applied associate-/l* 0.2

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

11. Simplified 0.2

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

13. Applied add-sqr-sqrt 32.5

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

14. Applied *-un-lft-identity 32.5

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

15. Applied times-frac 32.5

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

16. Applied add-sqr-sqrt 32.5

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

17. Applied unpow-prod-down 32.5

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

18. Applied times-frac 32.5

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

19. Simplified 32.5

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

20. Simplified 0.2

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

21. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020089 
(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)))