Average Error: 14.9 → 0.4
Time: 5.3s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3}\right)\right) \cdot \frac{\sin \left(0.5 \cdot x\right)}{\sin x}\]
\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 r645110 = 8.0;
        double r645111 = 3.0;
        double r645112 = r645110 / r645111;
        double r645113 = x;
        double r645114 = 0.5;
        double r645115 = r645113 * r645114;
        double r645116 = sin(r645115);
        double r645117 = r645112 * r645116;
        double r645118 = r645117 * r645116;
        double r645119 = sin(r645113);
        double r645120 = r645118 / r645119;
        return r645120;
}
↓
double f(double x) {
        double r645121 = 8.0;
        double r645122 = x;
        double r645123 = 0.5;
        double r645124 = r645122 * r645123;
        double r645125 = sin(r645124);
        double r645126 = r645121 * r645125;
        double r645127 = 3.0;
        double r645128 = r645126 / r645127;
        double r645129 = expm1(r645128);
        double r645130 = log1p(r645129);
        double r645131 = r645123 * r645122;
        double r645132 = sin(r645131);
        double r645133 = sin(r645122);
        double r645134 = r645132 / r645133;
        double r645135 = r645130 * r645134;
        return r645135;
}

Error

Bits error versus x

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Results

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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 2020033 +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)))