Average Error: 14.9 → 0.4
Time: 5.4s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3}\right)\right) \cdot \frac{\sin \left(0.5 \cdot x\right)}{\sin x}\]
\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(0.5 \cdot x\right)}{\sin x}
double f(double x) {
        double r563714 = 8.0;
        double r563715 = 3.0;
        double r563716 = r563714 / r563715;
        double r563717 = x;
        double r563718 = 0.5;
        double r563719 = r563717 * r563718;
        double r563720 = sin(r563719);
        double r563721 = r563716 * r563720;
        double r563722 = r563721 * r563720;
        double r563723 = sin(r563717);
        double r563724 = r563722 / r563723;
        return r563724;
}
↓
double f(double x) {
        double r563725 = 8.0;
        double r563726 = x;
        double r563727 = 0.5;
        double r563728 = r563726 * r563727;
        double r563729 = sin(r563728);
        double r563730 = r563725 * r563729;
        double r563731 = 3.0;
        double r563732 = r563730 / r563731;
        double r563733 = expm1(r563732);
        double r563734 = log1p(r563733);
        double r563735 = r563727 * r563726;
        double r563736 = sin(r563735);
        double r563737 = sin(r563726);
        double r563738 = r563736 / r563737;
        double r563739 = r563734 * r563738;
        return r563739;
}

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

Derivation

  1. Initial program 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 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-u0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2020100 +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)))