Average Error: 14.5 → 0.6
Time: 5.0s
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 r864484 = 8.0;
        double r864485 = 3.0;
        double r864486 = r864484 / r864485;
        double r864487 = x;
        double r864488 = 0.5;
        double r864489 = r864487 * r864488;
        double r864490 = sin(r864489);
        double r864491 = r864486 * r864490;
        double r864492 = r864491 * r864490;
        double r864493 = sin(r864487);
        double r864494 = r864492 / r864493;
        return r864494;
}
↓
double f(double x) {
        double r864495 = 8.0;
        double r864496 = x;
        double r864497 = 0.5;
        double r864498 = r864496 * r864497;
        double r864499 = sin(r864498);
        double r864500 = r864495 * r864499;
        double r864501 = 3.0;
        double r864502 = r864500 / r864501;
        double r864503 = 1.0;
        double r864504 = sin(r864496);
        double r864505 = r864497 * r864496;
        double r864506 = sin(r864505);
        double r864507 = r864504 / r864506;
        double r864508 = r864503 / r864507;
        double r864509 = expm1(r864508);
        double r864510 = log1p(r864509);
        double r864511 = r864502 * r864510;
        return r864511;
}

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)))