Average Error: 15.2 → 0.3
Time: 16.6s
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 r389528 = 8.0;
        double r389529 = 3.0;
        double r389530 = r389528 / r389529;
        double r389531 = x;
        double r389532 = 0.5;
        double r389533 = r389531 * r389532;
        double r389534 = sin(r389533);
        double r389535 = r389530 * r389534;
        double r389536 = r389535 * r389534;
        double r389537 = sin(r389531);
        double r389538 = r389536 / r389537;
        return r389538;
}
↓
double f(double x) {
        double r389539 = 8.0;
        double r389540 = 0.5;
        double r389541 = x;
        double r389542 = r389540 * r389541;
        double r389543 = sin(r389542);
        double r389544 = r389539 * r389543;
        double r389545 = 3.0;
        double r389546 = r389544 / r389545;
        double r389547 = r389541 * r389540;
        double r389548 = sin(r389547);
        double r389549 = sin(r389541);
        double r389550 = r389548 / r389549;
        double r389551 = r389546 * r389550;
        return r389551;
}

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 *-un-lft-identity0.5

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

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

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

  9. Applied times-frac0.5

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

  10. Applied associate-*r*0.5

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

  11. Simplified0.3

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

  12. 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 2019208 
(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)))