Average Error: 14.4 → 0.3
Time: 4.3s
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 r610147 = 8.0;
        double r610148 = 3.0;
        double r610149 = r610147 / r610148;
        double r610150 = x;
        double r610151 = 0.5;
        double r610152 = r610150 * r610151;
        double r610153 = sin(r610152);
        double r610154 = r610149 * r610153;
        double r610155 = r610154 * r610153;
        double r610156 = sin(r610150);
        double r610157 = r610155 / r610156;
        return r610157;
}
↓
double f(double x) {
        double r610158 = 8.0;
        double r610159 = 0.5;
        double r610160 = x;
        double r610161 = r610159 * r610160;
        double r610162 = sin(r610161);
        double r610163 = 3.0;
        double r610164 = r610162 / r610163;
        double r610165 = r610158 * r610164;
        double r610166 = sin(r610160);
        double r610167 = r610162 / r610166;
        double r610168 = r610165 * r610167;
        return r610168;
}

Error

Bits error versus x

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Results

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Target

Original14.4
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 14.4

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

  3. Applied associate-/l*0.5

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

  4. Simplified0.5

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

  6. Applied div-inv0.5

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

  7. Applied associate-*l*0.5

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

  8. Simplified0.3

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

  10. Applied associate-/r/0.4

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

  12. Applied div-inv0.5

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

  13. Applied associate-*l*0.4

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

  14. Simplified0.3

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

  15. 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 2020049 
(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)))