Average Error: 14.6 → 0.4
Time: 17.9s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin \left(0.5 \cdot x\right) \cdot 8}{3}\right)\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\]
\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{\sin \left(0.5 \cdot x\right) \cdot 8}{3}\right)\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}
double f(double x) {
        double r430650 = 8.0;
        double r430651 = 3.0;
        double r430652 = r430650 / r430651;
        double r430653 = x;
        double r430654 = 0.5;
        double r430655 = r430653 * r430654;
        double r430656 = sin(r430655);
        double r430657 = r430652 * r430656;
        double r430658 = r430657 * r430656;
        double r430659 = sin(r430653);
        double r430660 = r430658 / r430659;
        return r430660;
}
↓
double f(double x) {
        double r430661 = 0.5;
        double r430662 = x;
        double r430663 = r430661 * r430662;
        double r430664 = sin(r430663);
        double r430665 = 8.0;
        double r430666 = r430664 * r430665;
        double r430667 = 3.0;
        double r430668 = r430666 / r430667;
        double r430669 = expm1(r430668);
        double r430670 = log1p(r430669);
        double r430671 = r430662 * r430661;
        double r430672 = sin(r430671);
        double r430673 = sin(r430662);
        double r430674 = r430672 / r430673;
        double r430675 = r430670 * r430674;
        return r430675;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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Target

Original14.6
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 14.6

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

  3. Applied *-un-lft-identity14.6

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

  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 associate-*l/0.3

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

  8. Simplified0.3

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

  10. Applied log1p-expm1-u0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019306 +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)))