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

Average Error: 28.5 → 0.2

Time: 3.2s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r714999 = x;
        double r715000 = r714999 * r714999;
        double r715001 = y;
        double r715002 = r715001 * r715001;
        double r715003 = r715000 + r715002;
        double r715004 = z;
        double r715005 = r715004 * r715004;
        double r715006 = r715003 - r715005;
        double r715007 = 2.0;
        double r715008 = r715001 * r715007;
        double r715009 = r715006 / r715008;
        return r715009;
}

↓

double f(double x, double y, double z) {
        double r715010 = 0.5;
        double r715011 = y;
        double r715012 = x;
        double r715013 = r715012 / r715011;
        double r715014 = r715012 * r715013;
        double r715015 = r715011 + r715014;
        double r715016 = z;
        double r715017 = 1.0;
        double r715018 = pow(r715016, r715017);
        double r715019 = r715018 / r715011;
        double r715020 = r715018 * r715019;
        double r715021 = r715015 - r715020;
        double r715022 = r715010 * r715021;
        return r715022;
}

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 sqr-pow 12.8

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

6. Applied associate-/l* 6.9

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

7. Simplified 6.9

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

9. Applied *-un-lft-identity 6.9

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

10. Applied add-sqr-sqrt 35.7

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

11. Applied unpow-prod-down 35.7

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

12. Applied times-frac 32.3

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

13. Simplified 32.3

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

14. Simplified 0.2

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

16. Applied div-inv 0.2

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

17. Simplified 0.2

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

18. Final simplification 0.2

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

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)))