Average Error: 14.9 → 0.6
Time: 5.1s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\left(8 \cdot \frac{\sin \left(0.5 \cdot x\right)}{3}\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin \left(0.5 \cdot x\right)}{\sin x}\right)\right)\]
\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}
↓
\left(8 \cdot \frac{\sin \left(0.5 \cdot x\right)}{3}\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin \left(0.5 \cdot x\right)}{\sin x}\right)\right)
double f(double x) {
        double r592977 = 8.0;
        double r592978 = 3.0;
        double r592979 = r592977 / r592978;
        double r592980 = x;
        double r592981 = 0.5;
        double r592982 = r592980 * r592981;
        double r592983 = sin(r592982);
        double r592984 = r592979 * r592983;
        double r592985 = r592984 * r592983;
        double r592986 = sin(r592980);
        double r592987 = r592985 / r592986;
        return r592987;
}
↓
double f(double x) {
        double r592988 = 8.0;
        double r592989 = 0.5;
        double r592990 = x;
        double r592991 = r592989 * r592990;
        double r592992 = sin(r592991);
        double r592993 = 3.0;
        double r592994 = r592992 / r592993;
        double r592995 = r592988 * r592994;
        double r592996 = sin(r592990);
        double r592997 = r592992 / r592996;
        double r592998 = expm1(r592997);
        double r592999 = log1p(r592998);
        double r593000 = r592995 * r592999;
        return r593000;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.9
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 14.9

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

  3. Applied *-un-lft-identity14.9

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

  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 div-inv0.5

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

  9. Applied associate-*l*0.5

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

  10. Simplified0.3

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

  12. Applied log1p-expm1-u0.6

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

  13. Final simplification0.6

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

Reproduce

herbie shell --seed 2020036 +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)))