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

Average Error: 34.3 → 0.5

Time: 13.2s

Precision: 64

Math

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

↓

\[\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \sqrt{\left|\frac{z}{t}\right|} \cdot \left(\left|{\left(\left|\frac{z}{t}\right|\right)}^{\frac{3}{4}}\right| \cdot \left|{\left(\left|\frac{z}{t}\right|\right)}^{\frac{3}{4}}\right|\right)\right)\]

C

double f(double x, double y, double z, double t) {
        double r415220 = x;
        double r415221 = r415220 * r415220;
        double r415222 = y;
        double r415223 = r415222 * r415222;
        double r415224 = r415221 / r415223;
        double r415225 = z;
        double r415226 = r415225 * r415225;
        double r415227 = t;
        double r415228 = r415227 * r415227;
        double r415229 = r415226 / r415228;
        double r415230 = r415224 + r415229;
        return r415230;
}

↓

double f(double x, double y, double z, double t) {
        double r415231 = x;
        double r415232 = y;
        double r415233 = r415231 / r415232;
        double r415234 = z;
        double r415235 = t;
        double r415236 = r415234 / r415235;
        double r415237 = fabs(r415236);
        double r415238 = sqrt(r415237);
        double r415239 = 0.75;
        double r415240 = pow(r415237, r415239);
        double r415241 = fabs(r415240);
        double r415242 = r415241 * r415241;
        double r415243 = r415238 * r415242;
        double r415244 = fma(r415233, r415233, r415243);
        return r415244;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	34.3
Target	0.4
Herbie	0.5

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

Derivation

1. Initial program 34.3

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

2. Simplified 19.8

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

4. Applied add-sqr-sqrt 19.8

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

5. Simplified 19.8

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

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

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

17. Simplified 0.5

    \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \sqrt{\left|\frac{z}{t}\right|} \cdot \left(\color{blue}{\left|{\left(\left|\frac{z}{t}\right|\right)}^{\frac{3}{4}}\right|} \cdot \sqrt{{\left(\left|\frac{z}{t}\right|\right)}^{\frac{3}{2}}}\right)\right)\]

18. Simplified 0.5

    \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \sqrt{\left|\frac{z}{t}\right|} \cdot \left(\left|{\left(\left|\frac{z}{t}\right|\right)}^{\frac{3}{4}}\right| \cdot \color{blue}{\left|{\left(\left|\frac{z}{t}\right|\right)}^{\frac{3}{4}}\right|}\right)\right)\]

19. Final simplification 0.5

    \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \sqrt{\left|\frac{z}{t}\right|} \cdot \left(\left|{\left(\left|\frac{z}{t}\right|\right)}^{\frac{3}{4}}\right| \cdot \left|{\left(\left|\frac{z}{t}\right|\right)}^{\frac{3}{4}}\right|\right)\right)\]

Reproduce

herbie shell --seed 2019235 +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))))