Average Error: 14.5 → 0.3
Time: 9.0s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\frac{\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(0.5 \cdot x\right)}}}{3}\]
\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}
↓
\frac{\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(0.5 \cdot x\right)}}}{3}
double f(double x) {
        double r3655 = 8.0;
        double r3656 = 3.0;
        double r3657 = r3655 / r3656;
        double r3658 = x;
        double r3659 = 0.5;
        double r3660 = r3658 * r3659;
        double r3661 = sin(r3660);
        double r3662 = r3657 * r3661;
        double r3663 = r3662 * r3661;
        double r3664 = sin(r3658);
        double r3665 = r3663 / r3664;
        return r3665;
}
↓
double f(double x) {
        double r3666 = 8.0;
        double r3667 = x;
        double r3668 = 0.5;
        double r3669 = r3667 * r3668;
        double r3670 = sin(r3669);
        double r3671 = r3666 * r3670;
        double r3672 = sin(r3667);
        double r3673 = r3668 * r3667;
        double r3674 = sin(r3673);
        double r3675 = r3672 / r3674;
        double r3676 = r3671 / r3675;
        double r3677 = 3.0;
        double r3678 = r3676 / r3677;
        return r3678;
}

Error

Bits error versus x

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Results

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Target

Original14.5
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 14.5

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

  3. Applied *-un-lft-identity14.5

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

  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 associate-*l/0.3

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

  10. Applied clear-num0.3

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

  12. Applied associate-*l/0.3

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

  13. Simplified0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2020025 
(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)))