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

Average Error: 33.8 → 0.5

Time: 15.7s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r413309 = x;
        double r413310 = r413309 * r413309;
        double r413311 = y;
        double r413312 = r413311 * r413311;
        double r413313 = r413310 / r413312;
        double r413314 = z;
        double r413315 = r413314 * r413314;
        double r413316 = t;
        double r413317 = r413316 * r413316;
        double r413318 = r413315 / r413317;
        double r413319 = r413313 + r413318;
        return r413319;
}

↓

double f(double x, double y, double z, double t) {
        double r413320 = x;
        double r413321 = y;
        double r413322 = r413320 / r413321;
        double r413323 = z;
        double r413324 = t;
        double r413325 = r413323 / r413324;
        double r413326 = fabs(r413325);
        double r413327 = sqrt(r413326);
        double r413328 = r413326 * r413327;
        double r413329 = r413328 * r413327;
        double r413330 = fma(r413322, r413322, r413329);
        return r413330;
}

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 18.9

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

4. Applied add-sqr-sqrt 18.9

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

5. Simplified 18.9

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

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

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

9. Applied associate-*r* 0.5

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

10. Simplified 0.5

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

12. Applied add-cube-cbrt 0.8

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

13. Simplified 0.5

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

14. Simplified 0.5

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

15. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019304 +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))))