Average Error: 14.6 → 0.3
Time: 17.2s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\frac{\sin \left(0.5 \cdot x\right)}{\sin x} \cdot \frac{\sin \left(0.5 \cdot x\right) \cdot 8}{3}\]
\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}
↓
\frac{\sin \left(0.5 \cdot x\right)}{\sin x} \cdot \frac{\sin \left(0.5 \cdot x\right) \cdot 8}{3}
double f(double x) {
        double r32059805 = 8.0;
        double r32059806 = 3.0;
        double r32059807 = r32059805 / r32059806;
        double r32059808 = x;
        double r32059809 = 0.5;
        double r32059810 = r32059808 * r32059809;
        double r32059811 = sin(r32059810);
        double r32059812 = r32059807 * r32059811;
        double r32059813 = r32059812 * r32059811;
        double r32059814 = sin(r32059808);
        double r32059815 = r32059813 / r32059814;
        return r32059815;
}
↓
double f(double x) {
        double r32059816 = 0.5;
        double r32059817 = x;
        double r32059818 = r32059816 * r32059817;
        double r32059819 = sin(r32059818);
        double r32059820 = sin(r32059817);
        double r32059821 = r32059819 / r32059820;
        double r32059822 = 8.0;
        double r32059823 = r32059819 * r32059822;
        double r32059824 = 3.0;
        double r32059825 = r32059823 / r32059824;
        double r32059826 = r32059821 * r32059825;
        return r32059826;
}

Error

Bits error versus x

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Results

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Target

Original14.6
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 14.6

    \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
  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. Final simplification0.3

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

Reproduce

herbie shell --seed 2019179 
(FPCore (x)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A"

  :herbie-target
  (/ (/ (* 8.0 (sin (* x 0.5))) 3.0) (/ (sin x) (sin (* x 0.5))))

  (/ (* (* (/ 8.0 3.0) (sin (* x 0.5))) (sin (* x 0.5))) (sin x)))