Average Error: 14.3 → 0.3
Time: 15.8s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\frac{8 \cdot \sin \left(0.5 \cdot x\right)}{3} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\]
\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}
↓
\frac{8 \cdot \sin \left(0.5 \cdot x\right)}{3} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}
double f(double x) {
        double r428043 = 8.0;
        double r428044 = 3.0;
        double r428045 = r428043 / r428044;
        double r428046 = x;
        double r428047 = 0.5;
        double r428048 = r428046 * r428047;
        double r428049 = sin(r428048);
        double r428050 = r428045 * r428049;
        double r428051 = r428050 * r428049;
        double r428052 = sin(r428046);
        double r428053 = r428051 / r428052;
        return r428053;
}
↓
double f(double x) {
        double r428054 = 8.0;
        double r428055 = 0.5;
        double r428056 = x;
        double r428057 = r428055 * r428056;
        double r428058 = sin(r428057);
        double r428059 = r428054 * r428058;
        double r428060 = 3.0;
        double r428061 = r428059 / r428060;
        double r428062 = r428056 * r428055;
        double r428063 = sin(r428062);
        double r428064 = sin(r428056);
        double r428065 = r428063 / r428064;
        double r428066 = r428061 * r428065;
        return r428066;
}

Error

Bits error versus x

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Results

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Target

Original14.3
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 14.3

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

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

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

  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

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

  8. Applied associate-*l*0.5

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

  9. Simplified0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019209 
(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)))