Average Error: 15.3 → 0.4
Time: 7.9s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3}\right)\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\]
\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3}\right)\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}
double f(double x) {
        double r625033 = 8.0;
        double r625034 = 3.0;
        double r625035 = r625033 / r625034;
        double r625036 = x;
        double r625037 = 0.5;
        double r625038 = r625036 * r625037;
        double r625039 = sin(r625038);
        double r625040 = r625035 * r625039;
        double r625041 = r625040 * r625039;
        double r625042 = sin(r625036);
        double r625043 = r625041 / r625042;
        return r625043;
}
↓
double f(double x) {
        double r625044 = 8.0;
        double r625045 = x;
        double r625046 = 0.5;
        double r625047 = r625045 * r625046;
        double r625048 = sin(r625047);
        double r625049 = r625044 * r625048;
        double r625050 = 3.0;
        double r625051 = r625049 / r625050;
        double r625052 = expm1(r625051);
        double r625053 = log1p(r625052);
        double r625054 = sin(r625045);
        double r625055 = r625048 / r625054;
        double r625056 = r625053 * r625055;
        return r625056;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.3
Target0.3
Herbie0.4
\[\frac{\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}\]

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 *-un-lft-identity15.3

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

  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. Simplified0.3

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

  7. Applied log1p-expm1-u0.4

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

  8. Final simplification0.4

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

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)))