Average Error: 15.3 → 0.3
Time: 8.0s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\]
\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}
↓
\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}
double f(double x) {
        double r734879 = 8.0;
        double r734880 = 3.0;
        double r734881 = r734879 / r734880;
        double r734882 = x;
        double r734883 = 0.5;
        double r734884 = r734882 * r734883;
        double r734885 = sin(r734884);
        double r734886 = r734881 * r734885;
        double r734887 = r734886 * r734885;
        double r734888 = sin(r734882);
        double r734889 = r734887 / r734888;
        return r734889;
}
↓
double f(double x) {
        double r734890 = 8.0;
        double r734891 = x;
        double r734892 = 0.5;
        double r734893 = r734891 * r734892;
        double r734894 = sin(r734893);
        double r734895 = r734890 * r734894;
        double r734896 = 3.0;
        double r734897 = r734895 / r734896;
        double r734898 = sin(r734891);
        double r734899 = r734894 / r734898;
        double r734900 = r734897 * r734899;
        return r734900;
}

Error

Bits error versus x

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Results

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Target

Original15.3
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 15.3

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

  3. Applied 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 div-inv0.5

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

  6. Simplified0.5

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

  8. Applied *-un-lft-identity0.5

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

  9. Applied *-un-lft-identity0.5

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

  10. Applied times-frac0.5

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

  11. Applied associate-*l*0.5

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

  12. Simplified0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2020046 
(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)))