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

Average Error: 27.9 → 0.2

Time: 3.2s

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 f(double x, double y, double z) {
        double r869595 = x;
        double r869596 = r869595 * r869595;
        double r869597 = y;
        double r869598 = r869597 * r869597;
        double r869599 = r869596 + r869598;
        double r869600 = z;
        double r869601 = r869600 * r869600;
        double r869602 = r869599 - r869601;
        double r869603 = 2.0;
        double r869604 = r869597 * r869603;
        double r869605 = r869602 / r869604;
        return r869605;
}

↓

double f(double x, double y, double z) {
        double r869606 = 0.5;
        double r869607 = y;
        double r869608 = x;
        double r869609 = r869608 / r869607;
        double r869610 = r869608 * r869609;
        double r869611 = r869607 + r869610;
        double r869612 = z;
        double r869613 = r869612 / r869607;
        double r869614 = r869612 * r869613;
        double r869615 = r869611 - r869614;
        double r869616 = r869606 * r869615;
        return r869616;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	27.9
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 27.9

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

2. Taylor expanded around 0 12.0

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

3. Simplified 12.0

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

5. Applied *-un-lft-identity 12.0

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

6. Applied add-sqr-sqrt 37.6

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

7. Applied unpow-prod-down 37.6

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

8. Applied times-frac 34.9

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

9. Simplified 34.9

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

10. Simplified 6.8

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

12. Applied *-un-lft-identity 6.8

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

13. Applied add-sqr-sqrt 34.9

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

14. Applied unpow-prod-down 34.9

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

15. Applied times-frac 31.6

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

16. Simplified 31.5

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

17. Simplified 0.2

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

18. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019362 
(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)))