Average Error: 14.3 → 0.6
Time: 6.3s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\left(8 \cdot \frac{\frac{\sin \left(0.5 \cdot x\right)}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{3}}\right) \cdot \frac{\sin \left(0.5 \cdot x\right)}{\sin x}\]
\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(8 \cdot \frac{\frac{\sin \left(0.5 \cdot x\right)}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{3}}\right) \cdot \frac{\sin \left(0.5 \cdot x\right)}{\sin x}
double f(double x) {
        double r674840 = 8.0;
        double r674841 = 3.0;
        double r674842 = r674840 / r674841;
        double r674843 = x;
        double r674844 = 0.5;
        double r674845 = r674843 * r674844;
        double r674846 = sin(r674845);
        double r674847 = r674842 * r674846;
        double r674848 = r674847 * r674846;
        double r674849 = sin(r674843);
        double r674850 = r674848 / r674849;
        return r674850;
}
↓
double f(double x) {
        double r674851 = 8.0;
        double r674852 = 0.5;
        double r674853 = x;
        double r674854 = r674852 * r674853;
        double r674855 = sin(r674854);
        double r674856 = 3.0;
        double r674857 = cbrt(r674856);
        double r674858 = r674857 * r674857;
        double r674859 = r674855 / r674858;
        double r674860 = r674859 / r674857;
        double r674861 = r674851 * r674860;
        double r674862 = sin(r674853);
        double r674863 = r674855 / r674862;
        double r674864 = r674861 * r674863;
        return r674864;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.3
Target0.3
Herbie0.6
\[\frac{\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}\]

Derivation

  1. Initial program 14.3

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

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

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

  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 add-cube-cbrt0.3

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

  13. Applied associate-/r*0.6

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

  14. Final simplification0.6

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

Reproduce

herbie shell --seed 2019353 +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)))