Average Error: 14.5 → 0.6
Time: 9.0s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\frac{\sin x}{\sin \left(0.5 \cdot x\right)}}\right)\right)\]
\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{8 \cdot \sin \left(x \cdot 0.5\right)}{3} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\frac{\sin x}{\sin \left(0.5 \cdot x\right)}}\right)\right)
double f(double x) {
        double r3508 = 8.0;
        double r3509 = 3.0;
        double r3510 = r3508 / r3509;
        double r3511 = x;
        double r3512 = 0.5;
        double r3513 = r3511 * r3512;
        double r3514 = sin(r3513);
        double r3515 = r3510 * r3514;
        double r3516 = r3515 * r3514;
        double r3517 = sin(r3511);
        double r3518 = r3516 / r3517;
        return r3518;
}
↓
double f(double x) {
        double r3519 = 8.0;
        double r3520 = x;
        double r3521 = 0.5;
        double r3522 = r3520 * r3521;
        double r3523 = sin(r3522);
        double r3524 = r3519 * r3523;
        double r3525 = 3.0;
        double r3526 = r3524 / r3525;
        double r3527 = 1.0;
        double r3528 = sin(r3520);
        double r3529 = r3521 * r3520;
        double r3530 = sin(r3529);
        double r3531 = r3528 / r3530;
        double r3532 = r3527 / r3531;
        double r3533 = expm1(r3532);
        double r3534 = log1p(r3533);
        double r3535 = r3526 * r3534;
        return r3535;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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Target

Original14.5
Target0.3
Herbie0.6
\[\frac{\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}\]

Derivation

  1. Initial program 14.5

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

  3. Applied *-un-lft-identity14.5

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

  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. Simplified0.5

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

  8. Applied associate-*l/0.3

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

  10. Applied log1p-expm1-u0.6

    \[\leadsto \frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin \left(0.5 \cdot x\right)}{\sin x}\right)\right)}\]
  11. Using strategy rm

  12. Applied clear-num0.6

    \[\leadsto \frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{1}{\frac{\sin x}{\sin \left(0.5 \cdot x\right)}}}\right)\right)\]

  13. Final simplification0.6

    \[\leadsto \frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\frac{\sin x}{\sin \left(0.5 \cdot x\right)}}\right)\right)\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(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)))