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

Average Error: 34.1 → 0.5

Time: 14.1s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r466927 = x;
        double r466928 = r466927 * r466927;
        double r466929 = y;
        double r466930 = r466929 * r466929;
        double r466931 = r466928 / r466930;
        double r466932 = z;
        double r466933 = r466932 * r466932;
        double r466934 = t;
        double r466935 = r466934 * r466934;
        double r466936 = r466933 / r466935;
        double r466937 = r466931 + r466936;
        return r466937;
}

↓

double f(double x, double y, double z, double t) {
        double r466938 = x;
        double r466939 = y;
        double r466940 = r466938 / r466939;
        double r466941 = z;
        double r466942 = t;
        double r466943 = r466941 / r466942;
        double r466944 = fabs(r466943);
        double r466945 = sqrt(r466944);
        double r466946 = 1.5;
        double r466947 = pow(r466944, r466946);
        double r466948 = r466945 * r466947;
        double r466949 = fma(r466940, r466940, r466948);
        return r466949;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	34.1
Target	0.4
Herbie	0.5

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

Derivation

1. Initial program 34.1

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

2. Simplified 19.3

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z \cdot z}{t \cdot t}\right)}\]
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 19.3

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

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. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1"

  :herbie-target
  (+ (pow (/ x y) 2.0) (pow (/ z t) 2.0))

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