Average Error: 14.4 → 0.4
Time: 15.1s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\frac{\sin \left(0.5 \cdot x\right) \cdot 8}{3} \cdot \frac{1}{\sqrt[3]{{\left(\frac{\sin x}{\sin \left(x \cdot 0.5\right)}\right)}^{3}}}\]
\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(0.5 \cdot x\right) \cdot 8}{3} \cdot \frac{1}{\sqrt[3]{{\left(\frac{\sin x}{\sin \left(x \cdot 0.5\right)}\right)}^{3}}}
double f(double x) {
        double r390557 = 8.0;
        double r390558 = 3.0;
        double r390559 = r390557 / r390558;
        double r390560 = x;
        double r390561 = 0.5;
        double r390562 = r390560 * r390561;
        double r390563 = sin(r390562);
        double r390564 = r390559 * r390563;
        double r390565 = r390564 * r390563;
        double r390566 = sin(r390560);
        double r390567 = r390565 / r390566;
        return r390567;
}
↓
double f(double x) {
        double r390568 = 0.5;
        double r390569 = x;
        double r390570 = r390568 * r390569;
        double r390571 = sin(r390570);
        double r390572 = 8.0;
        double r390573 = r390571 * r390572;
        double r390574 = 3.0;
        double r390575 = r390573 / r390574;
        double r390576 = 1.0;
        double r390577 = sin(r390569);
        double r390578 = r390569 * r390568;
        double r390579 = sin(r390578);
        double r390580 = r390577 / r390579;
        double r390581 = 3.0;
        double r390582 = pow(r390580, r390581);
        double r390583 = cbrt(r390582);
        double r390584 = r390576 / r390583;
        double r390585 = r390575 * r390584;
        return r390585;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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Target

Original14.4
Target0.3
Herbie0.4
\[\frac{\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}\]

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 associate-*l/0.3

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

  8. Simplified0.3

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

  10. Applied clear-num0.3

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

  12. Applied add-cbrt-cube19.9

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

  13. Applied add-cbrt-cube21.1

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

  14. Applied cbrt-undiv21.1

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

  15. Simplified0.4

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

  16. Final simplification0.4

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

Reproduce

herbie shell --seed 2019303 
(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)))