Average Error: 15.0 → 0.5
Time: 21.1s
Precision: 64
\[\frac{\left(\frac{8.0}{3.0} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\left(\frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot 8.0\right) \cdot \frac{1}{3.0}\]
\frac{\left(\frac{8.0}{3.0} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}
↓
\left(\frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot 8.0\right) \cdot \frac{1}{3.0}
double f(double x) {
        double r20768736 = 8.0;
        double r20768737 = 3.0;
        double r20768738 = r20768736 / r20768737;
        double r20768739 = x;
        double r20768740 = 0.5;
        double r20768741 = r20768739 * r20768740;
        double r20768742 = sin(r20768741);
        double r20768743 = r20768738 * r20768742;
        double r20768744 = r20768743 * r20768742;
        double r20768745 = sin(r20768739);
        double r20768746 = r20768744 / r20768745;
        return r20768746;
}
↓
double f(double x) {
        double r20768747 = x;
        double r20768748 = 0.5;
        double r20768749 = r20768747 * r20768748;
        double r20768750 = sin(r20768749);
        double r20768751 = sin(r20768747);
        double r20768752 = r20768751 / r20768750;
        double r20768753 = r20768750 / r20768752;
        double r20768754 = 8.0;
        double r20768755 = r20768753 * r20768754;
        double r20768756 = 1.0;
        double r20768757 = 3.0;
        double r20768758 = r20768756 / r20768757;
        double r20768759 = r20768755 * r20768758;
        return r20768759;
}

Error

Bits error versus x

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Results

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Target

Original15.0
Target0.3
Herbie0.5
\[\frac{\frac{8.0 \cdot \sin \left(x \cdot 0.5\right)}{3.0}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}\]

Derivation

  1. Initial program 15.0

    \[\frac{\left(\frac{8.0}{3.0} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity15.0

    \[\leadsto \frac{\left(\frac{8.0}{3.0} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\color{blue}{1 \cdot \sin x}}\]

  4. Applied times-frac0.5

    \[\leadsto \color{blue}{\frac{\frac{8.0}{3.0} \cdot \sin \left(x \cdot 0.5\right)}{1} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}}\]

  5. Simplified0.3

    \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{3.0}{8.0}}} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\]
  6. Using strategy rm

  7. Applied div-inv0.3

    \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{3.0 \cdot \frac{1}{8.0}}} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\]

  8. Applied *-un-lft-identity0.3

    \[\leadsto \frac{\color{blue}{1 \cdot \sin \left(x \cdot 0.5\right)}}{3.0 \cdot \frac{1}{8.0}} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\]

  9. Applied times-frac0.5

    \[\leadsto \color{blue}{\left(\frac{1}{3.0} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\frac{1}{8.0}}\right)} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\]

  10. Applied associate-*l*0.5

    \[\leadsto \color{blue}{\frac{1}{3.0} \cdot \left(\frac{\sin \left(x \cdot 0.5\right)}{\frac{1}{8.0}} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\right)}\]

  11. Simplified0.5

    \[\leadsto \frac{1}{3.0} \cdot \color{blue}{\left(\frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot 8.0\right)}\]

  12. Final simplification0.5

    \[\leadsto \left(\frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot 8.0\right) \cdot \frac{1}{3.0}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A"

  :herbie-target
  (/ (/ (* 8.0 (sin (* x 0.5))) 3.0) (/ (sin x) (sin (* x 0.5))))

  (/ (* (* (/ 8.0 3.0) (sin (* x 0.5))) (sin (* x 0.5))) (sin x)))