Average Error: 15.2 → 0.3
Time: 6.7s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\frac{\sin \left(x \cdot 0.5\right)}{\sin x} \cdot \left(8 \cdot \frac{\sin \left(x \cdot 0.5\right)}{3}\right)\]
\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{\sin \left(x \cdot 0.5\right)}{\sin x} \cdot \left(8 \cdot \frac{\sin \left(x \cdot 0.5\right)}{3}\right)
double f(double x) {
        double r631226 = 8.0;
        double r631227 = 3.0;
        double r631228 = r631226 / r631227;
        double r631229 = x;
        double r631230 = 0.5;
        double r631231 = r631229 * r631230;
        double r631232 = sin(r631231);
        double r631233 = r631228 * r631232;
        double r631234 = r631233 * r631232;
        double r631235 = sin(r631229);
        double r631236 = r631234 / r631235;
        return r631236;
}
↓
double f(double x) {
        double r631237 = x;
        double r631238 = 0.5;
        double r631239 = r631237 * r631238;
        double r631240 = sin(r631239);
        double r631241 = sin(r631237);
        double r631242 = r631240 / r631241;
        double r631243 = 8.0;
        double r631244 = 3.0;
        double r631245 = r631240 / r631244;
        double r631246 = r631243 * r631245;
        double r631247 = r631242 * r631246;
        return r631247;
}

Error

Bits error versus x

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Results

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Target

Original15.2
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 15.2

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

  3. Applied *-un-lft-identity15.2

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

  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 div-inv0.5

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

  8. Applied associate-*l*0.5

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

  9. Simplified0.3

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

  11. Applied *-un-lft-identity0.3

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

  12. Applied *-un-lft-identity0.3

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

  13. Applied times-frac0.3

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

  14. Simplified0.3

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

  15. Final simplification0.3

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

Reproduce

herbie shell --seed 2020045 
(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)))