Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3

Average Error: 32.2 → 12.8

Time: 9.8s

Precision: 64

Math

\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot x \le 2.757556978790064116743818529955834991566 \cdot 10^{-173}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \le 1.02761016817255949891811789914490538041 \cdot 10^{266}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \sqrt[3]{{\left(\frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

C

double f(double x, double y) {
        double r444781 = x;
        double r444782 = r444781 * r444781;
        double r444783 = y;
        double r444784 = 4.0;
        double r444785 = r444783 * r444784;
        double r444786 = r444785 * r444783;
        double r444787 = r444782 - r444786;
        double r444788 = r444782 + r444786;
        double r444789 = r444787 / r444788;
        return r444789;
}

↓

double f(double x, double y) {
        double r444790 = x;
        double r444791 = r444790 * r444790;
        double r444792 = 2.757556978790064e-173;
        bool r444793 = r444791 <= r444792;
        double r444794 = 1.0;
        double r444795 = -r444794;
        double r444796 = 1.0276101681725595e+266;
        bool r444797 = r444791 <= r444796;
        double r444798 = y;
        double r444799 = 4.0;
        double r444800 = r444798 * r444799;
        double r444801 = r444800 * r444798;
        double r444802 = r444791 + r444801;
        double r444803 = r444791 / r444802;
        double r444804 = r444801 / r444802;
        double r444805 = 3.0;
        double r444806 = pow(r444804, r444805);
        double r444807 = cbrt(r444806);
        double r444808 = r444803 - r444807;
        double r444809 = 1.0;
        double r444810 = r444797 ? r444808 : r444809;
        double r444811 = r444793 ? r444795 : r444810;
        return r444811;
}

Error

Bits error versus x

Bits error versus y

Target

Original	32.2
Target	31.9
Herbie	12.8

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \lt 0.9743233849626781184483093056769575923681:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (* x x) < 2.757556978790064e-173

   1. Initial program 26.7

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
   2. Using strategy rm

   3. Applied div-sub 26.7

      \[\leadsto \color{blue}{\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\]

   4. Taylor expanded around 0 12.0

      \[\leadsto \color{blue}{-1}\]

   if 2.757556978790064e-173 < (* x x) < 1.0276101681725595e+266

   1. Initial program 16.5

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
   2. Using strategy rm

   3. Applied div-sub 16.4

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

   5. Applied add-cbrt-cube 29.0

      \[\leadsto \frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{\color{blue}{\sqrt[3]{\left(\left(x \cdot x + \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)}}}\]

   6. Applied add-cbrt-cube 29.3

      \[\leadsto \frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot 4\right) \cdot \color{blue}{\sqrt[3]{\left(y \cdot y\right) \cdot y}}}{\sqrt[3]{\left(\left(x \cdot x + \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)}}\]

   7. Applied add-cbrt-cube 29.3

      \[\leadsto \frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(y \cdot \color{blue}{\sqrt[3]{\left(4 \cdot 4\right) \cdot 4}}\right) \cdot \sqrt[3]{\left(y \cdot y\right) \cdot y}}{\sqrt[3]{\left(\left(x \cdot x + \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)}}\]

   8. Applied add-cbrt-cube 29.3

      \[\leadsto \frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\left(\color{blue}{\sqrt[3]{\left(y \cdot y\right) \cdot y}} \cdot \sqrt[3]{\left(4 \cdot 4\right) \cdot 4}\right) \cdot \sqrt[3]{\left(y \cdot y\right) \cdot y}}{\sqrt[3]{\left(\left(x \cdot x + \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)}}\]

   9. Applied cbrt-unprod 29.3

      \[\leadsto \frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\color{blue}{\sqrt[3]{\left(\left(y \cdot y\right) \cdot y\right) \cdot \left(\left(4 \cdot 4\right) \cdot 4\right)}} \cdot \sqrt[3]{\left(y \cdot y\right) \cdot y}}{\sqrt[3]{\left(\left(x \cdot x + \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)}}\]

   10. Applied cbrt-unprod 30.5

       \[\leadsto \frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{\color{blue}{\sqrt[3]{\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot \left(\left(4 \cdot 4\right) \cdot 4\right)\right) \cdot \left(\left(y \cdot y\right) \cdot y\right)}}}{\sqrt[3]{\left(\left(x \cdot x + \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)}}\]

   11. Applied cbrt-undiv 30.4

       \[\leadsto \frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \color{blue}{\sqrt[3]{\frac{\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot \left(\left(4 \cdot 4\right) \cdot 4\right)\right) \cdot \left(\left(y \cdot y\right) \cdot y\right)}{\left(\left(x \cdot x + \left(y \cdot 4\right) \cdot y\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)\right) \cdot \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)}}}\]

   12. Simplified 16.4

       \[\leadsto \frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \sqrt[3]{\color{blue}{{\left(\frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)}^{3}}}\]

   if 1.0276101681725595e+266 < (* x x)

   1. Initial program 58.3

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]

   2. Taylor expanded around inf 9.2

      \[\leadsto \color{blue}{1}\]
3. Recombined 3 regimes into one program.

4. Final simplification 12.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \le 2.757556978790064116743818529955834991566 \cdot 10^{-173}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \le 1.02761016817255949891811789914490538041 \cdot 10^{266}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \sqrt[3]{{\left(\frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< (/ (- (* x x) (* (* y 4) y)) (+ (* x x) (* (* y 4) y))) 0.9743233849626781) (- (/ (* x x) (+ (* x x) (* (* y y) 4))) (/ (* (* y y) 4) (+ (* x x) (* (* y y) 4)))) (- (pow (/ x (sqrt (+ (* x x) (* (* y y) 4)))) 2) (/ (* (* y y) 4) (+ (* x x) (* (* y y) 4)))))

  (/ (- (* x x) (* (* y 4) y)) (+ (* x x) (* (* y 4) y))))