Average Error: 14.7 → 0.4
Time: 19.0s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\frac{\sin \left(x \cdot 0.5\right)}{\sin x} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin \left(x \cdot 0.5\right)}{3} \cdot 8\right)\right)\]
\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(x \cdot 0.5\right)}{\sin x} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin \left(x \cdot 0.5\right)}{3} \cdot 8\right)\right)
double f(double x) {
        double r522295 = 8.0;
        double r522296 = 3.0;
        double r522297 = r522295 / r522296;
        double r522298 = x;
        double r522299 = 0.5;
        double r522300 = r522298 * r522299;
        double r522301 = sin(r522300);
        double r522302 = r522297 * r522301;
        double r522303 = r522302 * r522301;
        double r522304 = sin(r522298);
        double r522305 = r522303 / r522304;
        return r522305;
}
↓
double f(double x) {
        double r522306 = x;
        double r522307 = 0.5;
        double r522308 = r522306 * r522307;
        double r522309 = sin(r522308);
        double r522310 = sin(r522306);
        double r522311 = r522309 / r522310;
        double r522312 = 3.0;
        double r522313 = r522309 / r522312;
        double r522314 = 8.0;
        double r522315 = r522313 * r522314;
        double r522316 = expm1(r522315);
        double r522317 = log1p(r522316);
        double r522318 = r522311 * r522317;
        return r522318;
}

Error

Bits error versus x

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Results

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Target

Original14.7
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 14.7

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

  2. Simplified14.7

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

  4. Applied div-inv14.7

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

  5. Applied times-frac0.3

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

  6. Simplified0.3

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

  8. Applied log1p-expm1-u0.4

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019196 +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)))