Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1

Average Error: 33.5 → 0.6

Time: 4.4s

Precision: 64

Math

\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]

↓

\[\sqrt{\left|\frac{x}{y}\right| \cdot \left|\frac{x}{y}\right| + \frac{z}{t} \cdot \frac{z}{t}} \cdot \sqrt{\left|\frac{x}{y}\right| \cdot \left|\frac{x}{y}\right| + \left(\sqrt[3]{\frac{z}{t} \cdot \frac{z}{t}} \cdot \sqrt[3]{\frac{z}{t} \cdot \frac{z}{t}}\right) \cdot \sqrt[3]{\frac{z}{t} \cdot \frac{z}{t}}}\]

C

double f(double x, double y, double z, double t) {
        double r716763 = x;
        double r716764 = r716763 * r716763;
        double r716765 = y;
        double r716766 = r716765 * r716765;
        double r716767 = r716764 / r716766;
        double r716768 = z;
        double r716769 = r716768 * r716768;
        double r716770 = t;
        double r716771 = r716770 * r716770;
        double r716772 = r716769 / r716771;
        double r716773 = r716767 + r716772;
        return r716773;
}

↓

double f(double x, double y, double z, double t) {
        double r716774 = x;
        double r716775 = y;
        double r716776 = r716774 / r716775;
        double r716777 = fabs(r716776);
        double r716778 = r716777 * r716777;
        double r716779 = z;
        double r716780 = t;
        double r716781 = r716779 / r716780;
        double r716782 = r716781 * r716781;
        double r716783 = r716778 + r716782;
        double r716784 = sqrt(r716783);
        double r716785 = cbrt(r716782);
        double r716786 = r716785 * r716785;
        double r716787 = r716786 * r716785;
        double r716788 = r716778 + r716787;
        double r716789 = sqrt(r716788);
        double r716790 = r716784 * r716789;
        return r716790;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	33.5
Target	0.4
Herbie	0.6

\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

1. Initial program 33.5

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
2. Using strategy rm

3. Applied times-frac 18.5

   \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}}\]
4. Using strategy rm

5. Applied add-sqr-sqrt 18.5

   \[\leadsto \color{blue}{\sqrt{\frac{x \cdot x}{y \cdot y}} \cdot \sqrt{\frac{x \cdot x}{y \cdot y}}} + \frac{z}{t} \cdot \frac{z}{t}\]

6. Simplified 18.5

   \[\leadsto \color{blue}{\left|\frac{x}{y}\right|} \cdot \sqrt{\frac{x \cdot x}{y \cdot y}} + \frac{z}{t} \cdot \frac{z}{t}\]

7. Simplified 0.4

   \[\leadsto \left|\frac{x}{y}\right| \cdot \color{blue}{\left|\frac{x}{y}\right|} + \frac{z}{t} \cdot \frac{z}{t}\]
8. Using strategy rm

9. Applied add-sqr-sqrt 0.4

   \[\leadsto \color{blue}{\sqrt{\left|\frac{x}{y}\right| \cdot \left|\frac{x}{y}\right| + \frac{z}{t} \cdot \frac{z}{t}} \cdot \sqrt{\left|\frac{x}{y}\right| \cdot \left|\frac{x}{y}\right| + \frac{z}{t} \cdot \frac{z}{t}}}\]
10. Using strategy rm

11. Applied add-cube-cbrt 0.6

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

12. Final simplification 0.6

    \[\leadsto \sqrt{\left|\frac{x}{y}\right| \cdot \left|\frac{x}{y}\right| + \frac{z}{t} \cdot \frac{z}{t}} \cdot \sqrt{\left|\frac{x}{y}\right| \cdot \left|\frac{x}{y}\right| + \left(\sqrt[3]{\frac{z}{t} \cdot \frac{z}{t}} \cdot \sqrt[3]{\frac{z}{t} \cdot \frac{z}{t}}\right) \cdot \sqrt[3]{\frac{z}{t} \cdot \frac{z}{t}}}\]

Reproduce

herbie shell --seed 2019353 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1"
  :precision binary64

  :herbie-target
  (+ (pow (/ x y) 2) (pow (/ z t) 2))

  (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))