Average Error: 14.6 → 0.3
Time: 16.1s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\frac{\sin \left(0.5 \cdot x\right)}{\sin x} \cdot \frac{\sin \left(0.5 \cdot x\right) \cdot 8}{3}\]
\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{\sin \left(0.5 \cdot x\right)}{\sin x} \cdot \frac{\sin \left(0.5 \cdot x\right) \cdot 8}{3}
double f(double x) {
        double r404728 = 8.0;
        double r404729 = 3.0;
        double r404730 = r404728 / r404729;
        double r404731 = x;
        double r404732 = 0.5;
        double r404733 = r404731 * r404732;
        double r404734 = sin(r404733);
        double r404735 = r404730 * r404734;
        double r404736 = r404735 * r404734;
        double r404737 = sin(r404731);
        double r404738 = r404736 / r404737;
        return r404738;
}
↓
double f(double x) {
        double r404739 = 0.5;
        double r404740 = x;
        double r404741 = r404739 * r404740;
        double r404742 = sin(r404741);
        double r404743 = sin(r404740);
        double r404744 = r404742 / r404743;
        double r404745 = 8.0;
        double r404746 = r404742 * r404745;
        double r404747 = 3.0;
        double r404748 = r404746 / r404747;
        double r404749 = r404744 * r404748;
        return r404749;
}

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. Using strategy rm

  7. Applied pow10.5

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

  8. Applied pow10.5

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

  9. Applied pow-prod-down0.5

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

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019179 
(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)))