Average Error: 14.8 → 0.3
Time: 8.6s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\left(8 \cdot \frac{\sin \left(0.5 \cdot x\right)}{3}\right) \cdot \frac{\sin \left(0.5 \cdot x\right)}{\sin x}\]
\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(0.5 \cdot x\right)}{3}\right) \cdot \frac{\sin \left(0.5 \cdot x\right)}{\sin x}
double f(double x) {
        double r447816 = 8.0;
        double r447817 = 3.0;
        double r447818 = r447816 / r447817;
        double r447819 = x;
        double r447820 = 0.5;
        double r447821 = r447819 * r447820;
        double r447822 = sin(r447821);
        double r447823 = r447818 * r447822;
        double r447824 = r447823 * r447822;
        double r447825 = sin(r447819);
        double r447826 = r447824 / r447825;
        return r447826;
}
↓
double f(double x) {
        double r447827 = 8.0;
        double r447828 = 0.5;
        double r447829 = x;
        double r447830 = r447828 * r447829;
        double r447831 = sin(r447830);
        double r447832 = 3.0;
        double r447833 = r447831 / r447832;
        double r447834 = r447827 * r447833;
        double r447835 = sin(r447829);
        double r447836 = r447831 / r447835;
        double r447837 = r447834 * r447836;
        return r447837;
}

Error

Bits error versus x

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Results

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Target

Original14.8
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 14.8

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

  3. Applied *-un-lft-identity14.8

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

  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 div-inv0.5

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

  9. Applied associate-*l*0.5

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

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 1978988140 
(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)))