Average Error: 14.4 → 0.3
Time: 16.3s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\left(8 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin \left(0.5 \cdot x\right)}{3}\right)\right)\right) \cdot \frac{1}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}\]
\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 \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin \left(0.5 \cdot x\right)}{3}\right)\right)\right) \cdot \frac{1}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}
double f(double x) {
        double r462127 = 8.0;
        double r462128 = 3.0;
        double r462129 = r462127 / r462128;
        double r462130 = x;
        double r462131 = 0.5;
        double r462132 = r462130 * r462131;
        double r462133 = sin(r462132);
        double r462134 = r462129 * r462133;
        double r462135 = r462134 * r462133;
        double r462136 = sin(r462130);
        double r462137 = r462135 / r462136;
        return r462137;
}
↓
double f(double x) {
        double r462138 = 8.0;
        double r462139 = 0.5;
        double r462140 = x;
        double r462141 = r462139 * r462140;
        double r462142 = sin(r462141);
        double r462143 = 3.0;
        double r462144 = r462142 / r462143;
        double r462145 = log1p(r462144);
        double r462146 = expm1(r462145);
        double r462147 = r462138 * r462146;
        double r462148 = 1.0;
        double r462149 = sin(r462140);
        double r462150 = r462140 * r462139;
        double r462151 = sin(r462150);
        double r462152 = r462149 / r462151;
        double r462153 = r462148 / r462152;
        double r462154 = r462147 * r462153;
        return r462154;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

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 div-inv0.5

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

  8. Applied associate-*l*0.5

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

  9. Simplified0.3

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

  11. Applied expm1-log1p-u0.3

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

  13. Applied clear-num0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2019303 +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)))