Average Error: 14.9 → 0.3
Time: 43.4s
Precision: 64
\[\frac{\left(\frac{8.0}{3.0} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\frac{\sin \left(x \cdot 0.5\right) \cdot 8.0}{\frac{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}{\frac{1}{3.0}}}\]
\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}
↓
\frac{\sin \left(x \cdot 0.5\right) \cdot 8.0}{\frac{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}{\frac{1}{3.0}}}
double f(double x) {
        double r24744779 = 8.0;
        double r24744780 = 3.0;
        double r24744781 = r24744779 / r24744780;
        double r24744782 = x;
        double r24744783 = 0.5;
        double r24744784 = r24744782 * r24744783;
        double r24744785 = sin(r24744784);
        double r24744786 = r24744781 * r24744785;
        double r24744787 = r24744786 * r24744785;
        double r24744788 = sin(r24744782);
        double r24744789 = r24744787 / r24744788;
        return r24744789;
}
↓
double f(double x) {
        double r24744790 = x;
        double r24744791 = 0.5;
        double r24744792 = r24744790 * r24744791;
        double r24744793 = sin(r24744792);
        double r24744794 = 8.0;
        double r24744795 = r24744793 * r24744794;
        double r24744796 = sin(r24744790);
        double r24744797 = r24744796 / r24744793;
        double r24744798 = 1.0;
        double r24744799 = 3.0;
        double r24744800 = r24744798 / r24744799;
        double r24744801 = r24744797 / r24744800;
        double r24744802 = r24744795 / r24744801;
        return r24744802;
}

Error

Bits error versus x

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Results

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Target

Original14.9
Target0.3
Herbie0.3
\[\frac{\frac{8.0 \cdot \sin \left(x \cdot 0.5\right)}{3.0}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}\]

Derivation

  1. Initial program 14.9

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

  3. Applied *-un-lft-identity14.9

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

  4. Applied *-commutative14.9

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

  5. Applied associate-*l*14.9

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

  6. Applied times-frac0.5

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

  7. Simplified0.5

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

  9. Applied associate-/l*0.5

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

  11. Applied div-inv0.5

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

  12. Applied associate-/l*0.5

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

  13. Applied associate-*r/0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2019158 
(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)))