Average Error: 15.3 → 0.3
Time: 4.4s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\frac{8 \cdot \sin \left(0.5 \cdot x\right)}{\frac{\sin x}{\sin \left(0.5 \cdot x\right)} \cdot 3}\]
\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{8 \cdot \sin \left(0.5 \cdot x\right)}{\frac{\sin x}{\sin \left(0.5 \cdot x\right)} \cdot 3}
double f(double x) {
        double r641113 = 8.0;
        double r641114 = 3.0;
        double r641115 = r641113 / r641114;
        double r641116 = x;
        double r641117 = 0.5;
        double r641118 = r641116 * r641117;
        double r641119 = sin(r641118);
        double r641120 = r641115 * r641119;
        double r641121 = r641120 * r641119;
        double r641122 = sin(r641116);
        double r641123 = r641121 / r641122;
        return r641123;
}
↓
double f(double x) {
        double r641124 = 8.0;
        double r641125 = 0.5;
        double r641126 = x;
        double r641127 = r641125 * r641126;
        double r641128 = sin(r641127);
        double r641129 = r641124 * r641128;
        double r641130 = sin(r641126);
        double r641131 = r641130 / r641128;
        double r641132 = 3.0;
        double r641133 = r641131 * r641132;
        double r641134 = r641129 / r641133;
        return r641134;
}

Error

Bits error versus x

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Results

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Target

Original15.3
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 15.3

    \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
  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/0.3

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

  11. Applied associate-/l/0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(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)))