Average Error: 14.8 → 0.5
Time: 18.0s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\left(\frac{8}{3} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\right) \cdot \sin \left(x \cdot 0.5\right)\]
\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}
↓
\left(\frac{8}{3} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\right) \cdot \sin \left(x \cdot 0.5\right)
double f(double x) {
        double r477716 = 8.0;
        double r477717 = 3.0;
        double r477718 = r477716 / r477717;
        double r477719 = x;
        double r477720 = 0.5;
        double r477721 = r477719 * r477720;
        double r477722 = sin(r477721);
        double r477723 = r477718 * r477722;
        double r477724 = r477723 * r477722;
        double r477725 = sin(r477719);
        double r477726 = r477724 / r477725;
        return r477726;
}
↓
double f(double x) {
        double r477727 = 8.0;
        double r477728 = 3.0;
        double r477729 = r477727 / r477728;
        double r477730 = x;
        double r477731 = 0.5;
        double r477732 = r477730 * r477731;
        double r477733 = sin(r477732);
        double r477734 = sin(r477730);
        double r477735 = r477733 / r477734;
        double r477736 = r477729 * r477735;
        double r477737 = r477736 * r477733;
        return r477737;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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Target

Original14.8
Target0.3
Herbie0.5
\[\frac{\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}\]

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. 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 *-un-lft-identity0.3

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

  11. Applied times-frac0.5

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

  12. Applied associate-*l*0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2019325 +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)))