Average Error: 14.4 → 0.4
Time: 14.9s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\frac{\sin \left(0.5 \cdot x\right) \cdot 8}{3} \cdot \frac{1}{\sqrt[3]{{\left(\frac{\sin x}{\sin \left(x \cdot 0.5\right)}\right)}^{3}}}\]
\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{\sin \left(0.5 \cdot x\right) \cdot 8}{3} \cdot \frac{1}{\sqrt[3]{{\left(\frac{\sin x}{\sin \left(x \cdot 0.5\right)}\right)}^{3}}}
double f(double x) {
        double r1130763 = 8.0;
        double r1130764 = 3.0;
        double r1130765 = r1130763 / r1130764;
        double r1130766 = x;
        double r1130767 = 0.5;
        double r1130768 = r1130766 * r1130767;
        double r1130769 = sin(r1130768);
        double r1130770 = r1130765 * r1130769;
        double r1130771 = r1130770 * r1130769;
        double r1130772 = sin(r1130766);
        double r1130773 = r1130771 / r1130772;
        return r1130773;
}
↓
double f(double x) {
        double r1130774 = 0.5;
        double r1130775 = x;
        double r1130776 = r1130774 * r1130775;
        double r1130777 = sin(r1130776);
        double r1130778 = 8.0;
        double r1130779 = r1130777 * r1130778;
        double r1130780 = 3.0;
        double r1130781 = r1130779 / r1130780;
        double r1130782 = 1.0;
        double r1130783 = sin(r1130775);
        double r1130784 = r1130775 * r1130774;
        double r1130785 = sin(r1130784);
        double r1130786 = r1130783 / r1130785;
        double r1130787 = 3.0;
        double r1130788 = pow(r1130786, r1130787);
        double r1130789 = cbrt(r1130788);
        double r1130790 = r1130782 / r1130789;
        double r1130791 = r1130781 * r1130790;
        return r1130791;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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Target

Original14.4
Target0.3
Herbie0.4
\[\frac{\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}\]

Derivation

  1. Initial program 14.4

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

  3. Applied *-un-lft-identity14.4

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

  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. Using strategy rm

  7. Applied associate-*l/0.3

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

  8. Simplified0.3

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

  10. Applied clear-num0.3

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

  12. Applied add-cbrt-cube19.9

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

  13. Applied add-cbrt-cube21.1

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

  14. Applied cbrt-undiv21.1

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

  15. Simplified0.4

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

  16. Final simplification0.4

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

Reproduce

herbie shell --seed 2019303 
(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)))