Average Error: 14.9 → 0.3
Time: 14.5s
Precision: 64
\[\frac{\left(\frac{8.0}{3.0} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\frac{\sin \left(0.5 \cdot x\right) \cdot 8.0}{3.0 \cdot \frac{\sin x}{\sin \left(0.5 \cdot x\right)}}\]
\frac{\left(\frac{8.0}{3.0} \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) \cdot 8.0}{3.0 \cdot \frac{\sin x}{\sin \left(0.5 \cdot x\right)}}
double f(double x) {
        double r11524125 = 8.0;
        double r11524126 = 3.0;
        double r11524127 = r11524125 / r11524126;
        double r11524128 = x;
        double r11524129 = 0.5;
        double r11524130 = r11524128 * r11524129;
        double r11524131 = sin(r11524130);
        double r11524132 = r11524127 * r11524131;
        double r11524133 = r11524132 * r11524131;
        double r11524134 = sin(r11524128);
        double r11524135 = r11524133 / r11524134;
        return r11524135;
}
↓
double f(double x) {
        double r11524136 = 0.5;
        double r11524137 = x;
        double r11524138 = r11524136 * r11524137;
        double r11524139 = sin(r11524138);
        double r11524140 = 8.0;
        double r11524141 = r11524139 * r11524140;
        double r11524142 = 3.0;
        double r11524143 = sin(r11524137);
        double r11524144 = r11524143 / r11524139;
        double r11524145 = r11524142 * r11524144;
        double r11524146 = r11524141 / r11524145;
        return r11524146;
}

Error

Bits error versus x

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Results

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Target

Original14.9
Target0.3
Herbie0.3
\[\frac{\frac{8.0 \cdot \sin \left(x \cdot 0.5\right)}{3.0}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}\]

Derivation

  1. Initial program 14.9

    \[\frac{\left(\frac{8.0}{3.0} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity14.9

    \[\leadsto \frac{\left(\frac{8.0}{3.0} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\color{blue}{1 \cdot \sin x}}\]

  4. Applied times-frac0.5

    \[\leadsto \color{blue}{\frac{\frac{8.0}{3.0} \cdot \sin \left(x \cdot 0.5\right)}{1} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}}\]

  5. Simplified0.3

    \[\leadsto \color{blue}{\frac{8.0 \cdot \sin \left(x \cdot 0.5\right)}{3.0}} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\]
  6. Using strategy rm

  7. Applied associate-*l/0.3

    \[\leadsto \color{blue}{\frac{\left(8.0 \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}}{3.0}}\]
  8. Using strategy rm

  9. Applied clear-num0.3

    \[\leadsto \frac{\left(8.0 \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \color{blue}{\frac{1}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}}}{3.0}\]
  10. Using strategy rm

  11. Applied un-div-inv0.3

    \[\leadsto \frac{\color{blue}{\frac{8.0 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}}}{3.0}\]

  12. Applied associate-/l/0.3

    \[\leadsto \color{blue}{\frac{8.0 \cdot \sin \left(x \cdot 0.5\right)}{3.0 \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}}}\]

  13. Final simplification0.3

    \[\leadsto \frac{\sin \left(0.5 \cdot x\right) \cdot 8.0}{3.0 \cdot \frac{\sin x}{\sin \left(0.5 \cdot x\right)}}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(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)))