Average Error: 15.1 → 0.3
Time: 15.8s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\frac{\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(0.5 \cdot x\right)}}}{3}\]
\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{\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(0.5 \cdot x\right)}}}{3}
double f(double x) {
        double r366765 = 8.0;
        double r366766 = 3.0;
        double r366767 = r366765 / r366766;
        double r366768 = x;
        double r366769 = 0.5;
        double r366770 = r366768 * r366769;
        double r366771 = sin(r366770);
        double r366772 = r366767 * r366771;
        double r366773 = r366772 * r366771;
        double r366774 = sin(r366768);
        double r366775 = r366773 / r366774;
        return r366775;
}
↓
double f(double x) {
        double r366776 = 8.0;
        double r366777 = x;
        double r366778 = 0.5;
        double r366779 = r366777 * r366778;
        double r366780 = sin(r366779);
        double r366781 = r366776 * r366780;
        double r366782 = sin(r366777);
        double r366783 = r366778 * r366777;
        double r366784 = sin(r366783);
        double r366785 = r366782 / r366784;
        double r366786 = r366781 / r366785;
        double r366787 = 3.0;
        double r366788 = r366786 / r366787;
        return r366788;
}

Error

Bits error versus x

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Results

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Target

Original15.1
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 15.1

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

  3. Applied associate-/l*0.5

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

  5. Applied associate-*l/0.3

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

  6. Applied associate-/l/0.3

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

  8. Applied add-cbrt-cube0.4

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

  9. Simplified0.4

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

  11. Applied associate-/r*0.4

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

  12. Simplified0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(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)))