Average Error: 14.6 → 0.3
Time: 5.9s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\left(8 \cdot \frac{\sin \left(0.5 \cdot x\right)}{3}\right) \cdot \frac{\sin \left(0.5 \cdot x\right)}{\sin x}\]
\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{\sin \left(0.5 \cdot x\right)}{3}\right) \cdot \frac{\sin \left(0.5 \cdot x\right)}{\sin x}
double f(double x) {
        double r697786 = 8.0;
        double r697787 = 3.0;
        double r697788 = r697786 / r697787;
        double r697789 = x;
        double r697790 = 0.5;
        double r697791 = r697789 * r697790;
        double r697792 = sin(r697791);
        double r697793 = r697788 * r697792;
        double r697794 = r697793 * r697792;
        double r697795 = sin(r697789);
        double r697796 = r697794 / r697795;
        return r697796;
}
↓
double f(double x) {
        double r697797 = 8.0;
        double r697798 = 0.5;
        double r697799 = x;
        double r697800 = r697798 * r697799;
        double r697801 = sin(r697800);
        double r697802 = 3.0;
        double r697803 = r697801 / r697802;
        double r697804 = r697797 * r697803;
        double r697805 = sin(r697799);
        double r697806 = r697801 / r697805;
        double r697807 = r697804 * r697806;
        return r697807;
}

Error

Bits error versus x

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Results

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Target

Original14.6
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 14.6

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

  3. Applied *-un-lft-identity14.6

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

  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. Final simplification0.3

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

Reproduce

herbie shell --seed 2020001 
(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)))