Average Error: 15.2 → 0.4
Time: 7.9s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\left(8 \cdot \frac{\sin \left(x \cdot 0.5\right)}{3}\right) \cdot \log \left(e^{\frac{\sin \left(x \cdot 0.5\right)}{\sin x}}\right)\]
\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(8 \cdot \frac{\sin \left(x \cdot 0.5\right)}{3}\right) \cdot \log \left(e^{\frac{\sin \left(x \cdot 0.5\right)}{\sin x}}\right)
double f(double x) {
        double r524907 = 8.0;
        double r524908 = 3.0;
        double r524909 = r524907 / r524908;
        double r524910 = x;
        double r524911 = 0.5;
        double r524912 = r524910 * r524911;
        double r524913 = sin(r524912);
        double r524914 = r524909 * r524913;
        double r524915 = r524914 * r524913;
        double r524916 = sin(r524910);
        double r524917 = r524915 / r524916;
        return r524917;
}
↓
double f(double x) {
        double r524918 = 8.0;
        double r524919 = x;
        double r524920 = 0.5;
        double r524921 = r524919 * r524920;
        double r524922 = sin(r524921);
        double r524923 = 3.0;
        double r524924 = r524922 / r524923;
        double r524925 = r524918 * r524924;
        double r524926 = sin(r524919);
        double r524927 = r524922 / r524926;
        double r524928 = exp(r524927);
        double r524929 = log(r524928);
        double r524930 = r524925 * r524929;
        return r524930;
}

Error

Bits error versus x

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Results

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Target

Original15.2
Target0.3
Herbie0.4
\[\frac{\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}\]

Derivation

  1. Initial program 15.2

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

  3. Applied *-un-lft-identity15.2

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

  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 div-inv0.5

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

  8. Applied associate-*l*0.5

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

  9. Simplified0.3

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

  11. Applied add-log-exp0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2020045 
(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)))