Average Error: 15.1 → 0.3
Time: 8.2s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\frac{8 \cdot \frac{\sin \left(\frac{1}{2} \cdot x\right)}{3}}{\frac{\sin x}{\sin \left(\frac{1}{2} \cdot x\right)}}\]
\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 \frac{\sin \left(\frac{1}{2} \cdot x\right)}{3}}{\frac{\sin x}{\sin \left(\frac{1}{2} \cdot x\right)}}
double f(double x) {
        double r612151 = 8.0;
        double r612152 = 3.0;
        double r612153 = r612151 / r612152;
        double r612154 = x;
        double r612155 = 0.5;
        double r612156 = r612154 * r612155;
        double r612157 = sin(r612156);
        double r612158 = r612153 * r612157;
        double r612159 = r612158 * r612157;
        double r612160 = sin(r612154);
        double r612161 = r612159 / r612160;
        return r612161;
}
↓
double f(double x) {
        double r612162 = 8.0;
        double r612163 = 1.0;
        double r612164 = 2.0;
        double r612165 = r612163 / r612164;
        double r612166 = x;
        double r612167 = r612165 * r612166;
        double r612168 = sin(r612167);
        double r612169 = 3.0;
        double r612170 = r612168 / r612169;
        double r612171 = r612162 * r612170;
        double r612172 = sin(r612166);
        double r612173 = r612172 / r612168;
        double r612174 = r612171 / r612173;
        return r612174;
}

Error

Bits error versus x

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Results

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Target

Original15.1
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 15.1

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

    \[\leadsto \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\color{blue}{\frac{\sin x}{\sin \left(\frac{1}{2} \cdot x\right)}}}\]
  5. Using strategy rm

  6. Applied div-inv0.5

    \[\leadsto \frac{\color{blue}{\left(8 \cdot \frac{1}{3}\right)} \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(\frac{1}{2} \cdot x\right)}}\]

  7. Applied associate-*l*0.5

    \[\leadsto \frac{\color{blue}{8 \cdot \left(\frac{1}{3} \cdot \sin \left(x \cdot 0.5\right)\right)}}{\frac{\sin x}{\sin \left(\frac{1}{2} \cdot x\right)}}\]

  8. Simplified0.3

    \[\leadsto \frac{8 \cdot \color{blue}{\frac{\sin \left(\frac{1}{2} \cdot x\right)}{3}}}{\frac{\sin x}{\sin \left(\frac{1}{2} \cdot x\right)}}\]

  9. Final simplification0.3

    \[\leadsto \frac{8 \cdot \frac{\sin \left(\frac{1}{2} \cdot x\right)}{3}}{\frac{\sin x}{\sin \left(\frac{1}{2} \cdot x\right)}}\]

Reproduce

herbie shell --seed 2019304 
(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)))