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

Average Error: 33.8 → 0.5

Time: 12.6s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \sqrt{\sqrt{\left|\frac{z}{t}\right|}} \cdot \left(\sqrt{\sqrt{\left|\frac{z}{t}\right|}} \cdot {\left(\left|\frac{z}{t}\right|\right)}^{\frac{3}{2}}\right)\right)\]

C

double f(double x, double y, double z, double t) {
        double r639801 = x;
        double r639802 = r639801 * r639801;
        double r639803 = y;
        double r639804 = r639803 * r639803;
        double r639805 = r639802 / r639804;
        double r639806 = z;
        double r639807 = r639806 * r639806;
        double r639808 = t;
        double r639809 = r639808 * r639808;
        double r639810 = r639807 / r639809;
        double r639811 = r639805 + r639810;
        return r639811;
}

↓

double f(double x, double y, double z, double t) {
        double r639812 = x;
        double r639813 = y;
        double r639814 = r639812 / r639813;
        double r639815 = z;
        double r639816 = t;
        double r639817 = r639815 / r639816;
        double r639818 = fabs(r639817);
        double r639819 = sqrt(r639818);
        double r639820 = sqrt(r639819);
        double r639821 = 1.5;
        double r639822 = pow(r639818, r639821);
        double r639823 = r639820 * r639822;
        double r639824 = r639820 * r639823;
        double r639825 = fma(r639814, r639814, r639824);
        return r639825;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	33.8
Target	0.4
Herbie	0.5

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

Derivation

1. Initial program 33.8

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

2. Simplified 19.2

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z \cdot z}{t \cdot t}\right)}\]
3. Using strategy rm

4. Applied add-sqr-sqrt 19.3

   \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \color{blue}{\sqrt{\frac{z \cdot z}{t \cdot t}} \cdot \sqrt{\frac{z \cdot z}{t \cdot t}}}\right)\]

5. Simplified 19.2

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

6. Simplified 0.4

   \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \left|\frac{z}{t}\right| \cdot \color{blue}{\left|\frac{z}{t}\right|}\right)\]
7. Using strategy rm

8. Applied add-sqr-sqrt 0.5

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

9. Applied associate-*l* 0.5

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

10. Simplified 0.6

    \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \sqrt{\left|\frac{z}{t}\right|} \cdot \color{blue}{{\left(\sqrt{\left|\frac{z}{t}\right|}\right)}^{3}}\right)\]
11. Using strategy rm

12. Applied pow1/2 0.6

    \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \sqrt{\left|\frac{z}{t}\right|} \cdot {\color{blue}{\left({\left(\left|\frac{z}{t}\right|\right)}^{\frac{1}{2}}\right)}}^{3}\right)\]

13. Applied pow-pow 0.5

    \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \sqrt{\left|\frac{z}{t}\right|} \cdot \color{blue}{{\left(\left|\frac{z}{t}\right|\right)}^{\left(\frac{1}{2} \cdot 3\right)}}\right)\]

14. Simplified 0.5

    \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \sqrt{\left|\frac{z}{t}\right|} \cdot {\left(\left|\frac{z}{t}\right|\right)}^{\color{blue}{\frac{3}{2}}}\right)\]
15. Using strategy rm

16. Applied add-sqr-sqrt 0.5

    \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \sqrt{\color{blue}{\sqrt{\left|\frac{z}{t}\right|} \cdot \sqrt{\left|\frac{z}{t}\right|}}} \cdot {\left(\left|\frac{z}{t}\right|\right)}^{\frac{3}{2}}\right)\]

17. Applied sqrt-prod 0.5

    \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \color{blue}{\left(\sqrt{\sqrt{\left|\frac{z}{t}\right|}} \cdot \sqrt{\sqrt{\left|\frac{z}{t}\right|}}\right)} \cdot {\left(\left|\frac{z}{t}\right|\right)}^{\frac{3}{2}}\right)\]

18. Applied associate-*l* 0.5

    \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \color{blue}{\sqrt{\sqrt{\left|\frac{z}{t}\right|}} \cdot \left(\sqrt{\sqrt{\left|\frac{z}{t}\right|}} \cdot {\left(\left|\frac{z}{t}\right|\right)}^{\frac{3}{2}}\right)}\right)\]

19. Final simplification 0.5

    \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \sqrt{\sqrt{\left|\frac{z}{t}\right|}} \cdot \left(\sqrt{\sqrt{\left|\frac{z}{t}\right|}} \cdot {\left(\left|\frac{z}{t}\right|\right)}^{\frac{3}{2}}\right)\right)\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(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))))