Average Error: 14.8 → 0.3
Time: 4.7s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\frac{8 \cdot \sin \left(0.5 \cdot x\right)}{3 \cdot \frac{\sin x}{\sin \left(0.5 \cdot x\right)}}\]
\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}
↓
\frac{8 \cdot \sin \left(0.5 \cdot x\right)}{3 \cdot \frac{\sin x}{\sin \left(0.5 \cdot x\right)}}
double f(double x) {
        double r641608 = 8.0;
        double r641609 = 3.0;
        double r641610 = r641608 / r641609;
        double r641611 = x;
        double r641612 = 0.5;
        double r641613 = r641611 * r641612;
        double r641614 = sin(r641613);
        double r641615 = r641610 * r641614;
        double r641616 = r641615 * r641614;
        double r641617 = sin(r641611);
        double r641618 = r641616 / r641617;
        return r641618;
}
↓
double f(double x) {
        double r641619 = 8.0;
        double r641620 = 0.5;
        double r641621 = x;
        double r641622 = r641620 * r641621;
        double r641623 = sin(r641622);
        double r641624 = r641619 * r641623;
        double r641625 = 3.0;
        double r641626 = sin(r641621);
        double r641627 = r641626 / r641623;
        double r641628 = r641625 * r641627;
        double r641629 = r641624 / r641628;
        return r641629;
}

Error

Bits error versus x

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Results

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Target

Original14.8
Target0.3
Herbie0.3
\[\frac{\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}\]

Derivation

  1. Initial program 14.8

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

  3. Applied *-un-lft-identity14.8

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

  4. Applied times-frac0.5

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

  5. Simplified0.5

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

  6. Simplified0.5

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

  8. Applied div-inv0.5

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

  9. Applied associate-*l*0.5

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

  10. Simplified0.3

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

  12. Applied clear-num0.3

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

  14. Applied associate-*r/0.3

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

  15. Applied frac-times0.3

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

  16. Simplified0.3

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

  17. Final simplification0.3

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

Reproduce

herbie shell --seed 2020060 
(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)))