Average Error: 15.3 → 0.5
Time: 6.7s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{\frac{8}{3}} \cdot \left(\sqrt{\frac{8}{3}} \cdot \sin \left(x \cdot 0.5\right)\right)\right)\right)}{\frac{\sin x}{\sin \left(0.5 \cdot x\right)}}\]
\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{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{\frac{8}{3}} \cdot \left(\sqrt{\frac{8}{3}} \cdot \sin \left(x \cdot 0.5\right)\right)\right)\right)}{\frac{\sin x}{\sin \left(0.5 \cdot x\right)}}
double f(double x) {
        double r616512 = 8.0;
        double r616513 = 3.0;
        double r616514 = r616512 / r616513;
        double r616515 = x;
        double r616516 = 0.5;
        double r616517 = r616515 * r616516;
        double r616518 = sin(r616517);
        double r616519 = r616514 * r616518;
        double r616520 = r616519 * r616518;
        double r616521 = sin(r616515);
        double r616522 = r616520 / r616521;
        return r616522;
}
↓
double f(double x) {
        double r616523 = 8.0;
        double r616524 = 3.0;
        double r616525 = r616523 / r616524;
        double r616526 = sqrt(r616525);
        double r616527 = x;
        double r616528 = 0.5;
        double r616529 = r616527 * r616528;
        double r616530 = sin(r616529);
        double r616531 = r616526 * r616530;
        double r616532 = r616526 * r616531;
        double r616533 = expm1(r616532);
        double r616534 = log1p(r616533);
        double r616535 = sin(r616527);
        double r616536 = r616528 * r616527;
        double r616537 = sin(r616536);
        double r616538 = r616535 / r616537;
        double r616539 = r616534 / r616538;
        return r616539;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.3
Target0.3
Herbie0.5
\[\frac{\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}\]

Derivation

  1. Initial program 15.3

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

  3. Applied associate-/l*0.5

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

  4. Simplified0.5

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

  6. Applied add-sqr-sqrt0.5

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

  7. Applied associate-*l*0.4

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

  9. Applied log1p-expm1-u0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2020046 +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)))