Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A

Average Error: 14.7 → 0.5

Time: 18.0s

Precision: 64

Math

\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]

↓

\[\left(\frac{1}{3} \cdot \frac{\sin \left(0.5 \cdot x\right)}{\sin x}\right) \cdot \left(8 \cdot \sin \left(0.5 \cdot x\right)\right)\]

C

double f(double x) {
        double r402982 = 8.0;
        double r402983 = 3.0;
        double r402984 = r402982 / r402983;
        double r402985 = x;
        double r402986 = 0.5;
        double r402987 = r402985 * r402986;
        double r402988 = sin(r402987);
        double r402989 = r402984 * r402988;
        double r402990 = r402989 * r402988;
        double r402991 = sin(r402985);
        double r402992 = r402990 / r402991;
        return r402992;
}

↓

double f(double x) {
        double r402993 = 1.0;
        double r402994 = 3.0;
        double r402995 = r402993 / r402994;
        double r402996 = 0.5;
        double r402997 = x;
        double r402998 = r402996 * r402997;
        double r402999 = sin(r402998);
        double r403000 = sin(r402997);
        double r403001 = r402999 / r403000;
        double r403002 = r402995 * r403001;
        double r403003 = 8.0;
        double r403004 = r403003 * r402999;
        double r403005 = r403002 * r403004;
        return r403005;
}

Error

Bits error versus x

Target

Original	14.7
Target	0.3
Herbie	0.5

\[\frac{\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}\]

Derivation

1. Initial program 14.7

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
2. Using strategy rm

3. Applied *-un-lft-identity 14.7

   \[\leadsto \frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\color{blue}{1 \cdot \sin x}}\]

4. Applied times-frac 0.5

   \[\leadsto \color{blue}{\frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{1} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}}\]

5. Simplified 0.5

   \[\leadsto \color{blue}{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right)} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\]
6. Using strategy rm

7. Applied pow1 0.5

   \[\leadsto \left(\frac{8}{3} \cdot \color{blue}{{\left(\sin \left(x \cdot 0.5\right)\right)}^{1}}\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\]

8. Applied pow1 0.5

   \[\leadsto \left(\color{blue}{{\left(\frac{8}{3}\right)}^{1}} \cdot {\left(\sin \left(x \cdot 0.5\right)\right)}^{1}\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\]

9. Applied pow-prod-down 0.5

   \[\leadsto \color{blue}{{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right)}^{1}} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\]

10. Simplified 0.3

    \[\leadsto {\color{blue}{\left(\frac{8 \cdot \sin \left(0.5 \cdot x\right)}{3}\right)}}^{1} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\]
11. Using strategy rm

12. Applied div-inv 0.5

    \[\leadsto {\color{blue}{\left(\left(8 \cdot \sin \left(0.5 \cdot x\right)\right) \cdot \frac{1}{3}\right)}}^{1} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\]

13. Applied unpow-prod-down 0.5

    \[\leadsto \color{blue}{\left({\left(8 \cdot \sin \left(0.5 \cdot x\right)\right)}^{1} \cdot {\left(\frac{1}{3}\right)}^{1}\right)} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\]

14. Applied associate-*l* 0.5

    \[\leadsto \color{blue}{{\left(8 \cdot \sin \left(0.5 \cdot x\right)\right)}^{1} \cdot \left({\left(\frac{1}{3}\right)}^{1} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\right)}\]

15. Simplified 0.5

    \[\leadsto {\left(8 \cdot \sin \left(0.5 \cdot x\right)\right)}^{1} \cdot \color{blue}{\left(\frac{1}{3} \cdot \frac{\sin \left(0.5 \cdot x\right)}{\sin x}\right)}\]

16. Final simplification 0.5

    \[\leadsto \left(\frac{1}{3} \cdot \frac{\sin \left(0.5 \cdot x\right)}{\sin x}\right) \cdot \left(8 \cdot \sin \left(0.5 \cdot x\right)\right)\]

Reproduce

herbie shell --seed 2019305 
(FPCore (x)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A"
  :precision binary64

  :herbie-target
  (/ (/ (* 8 (sin (* x 0.5))) 3) (/ (sin x) (sin (* x 0.5))))

  (/ (* (* (/ 8 3) (sin (* x 0.5))) (sin (* x 0.5))) (sin x)))