Average Error: 15.2 → 0.3
Time: 19.5s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\frac{\frac{\sin \left(x \cdot 0.5\right)}{\sin x} \cdot \left(8 \cdot \sin \left(x \cdot 0.5\right)\right)}{3}\]
\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{\frac{\sin \left(x \cdot 0.5\right)}{\sin x} \cdot \left(8 \cdot \sin \left(x \cdot 0.5\right)\right)}{3}
double f(double x) {
        double r24705221 = 8.0;
        double r24705222 = 3.0;
        double r24705223 = r24705221 / r24705222;
        double r24705224 = x;
        double r24705225 = 0.5;
        double r24705226 = r24705224 * r24705225;
        double r24705227 = sin(r24705226);
        double r24705228 = r24705223 * r24705227;
        double r24705229 = r24705228 * r24705227;
        double r24705230 = sin(r24705224);
        double r24705231 = r24705229 / r24705230;
        return r24705231;
}
↓
double f(double x) {
        double r24705232 = x;
        double r24705233 = 0.5;
        double r24705234 = r24705232 * r24705233;
        double r24705235 = sin(r24705234);
        double r24705236 = sin(r24705232);
        double r24705237 = r24705235 / r24705236;
        double r24705238 = 8.0;
        double r24705239 = r24705238 * r24705235;
        double r24705240 = r24705237 * r24705239;
        double r24705241 = 3.0;
        double r24705242 = r24705240 / r24705241;
        return r24705242;
}

Error

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Results

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Target

Original15.2
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 15.2

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

  3. Applied *-un-lft-identity15.2

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

  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. Using strategy rm

  9. Applied *-un-lft-identity0.4

    \[\leadsto \color{blue}{\left(1 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3}\right)\right)\right)} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\]

  10. Applied associate-*l*0.4

    \[\leadsto \color{blue}{1 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3}\right)\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\right)}\]

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(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)))