Average Error: 14.4 → 0.3
Time: 15.5s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\frac{\sin \left(x \cdot 0.5\right)}{\sin x} \cdot \frac{8 \cdot \sin \left(0.5 \cdot x\right)}{3}\]
\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{\sin \left(x \cdot 0.5\right)}{\sin x} \cdot \frac{8 \cdot \sin \left(0.5 \cdot x\right)}{3}
double f(double x) {
        double r548091 = 8.0;
        double r548092 = 3.0;
        double r548093 = r548091 / r548092;
        double r548094 = x;
        double r548095 = 0.5;
        double r548096 = r548094 * r548095;
        double r548097 = sin(r548096);
        double r548098 = r548093 * r548097;
        double r548099 = r548098 * r548097;
        double r548100 = sin(r548094);
        double r548101 = r548099 / r548100;
        return r548101;
}
↓
double f(double x) {
        double r548102 = x;
        double r548103 = 0.5;
        double r548104 = r548102 * r548103;
        double r548105 = sin(r548104);
        double r548106 = sin(r548102);
        double r548107 = r548105 / r548106;
        double r548108 = 8.0;
        double r548109 = r548103 * r548102;
        double r548110 = sin(r548109);
        double r548111 = r548108 * r548110;
        double r548112 = 3.0;
        double r548113 = r548111 / r548112;
        double r548114 = r548107 * r548113;
        return r548114;
}

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 *-un-lft-identity14.4

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

  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. Using strategy rm

  7. Applied *-un-lft-identity0.5

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

  8. Applied *-un-lft-identity0.5

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

  9. Applied times-frac0.5

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

  10. Applied associate-*l*0.5

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019212 
(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)))