Average Error: 15.2 → 0.4
Time: 19.7s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{8 \cdot \sin \left(0.5 \cdot x\right)}{3}\right)\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\]
\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{8 \cdot \sin \left(0.5 \cdot x\right)}{3}\right)\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}
double f(double x) {
        double r488386 = 8.0;
        double r488387 = 3.0;
        double r488388 = r488386 / r488387;
        double r488389 = x;
        double r488390 = 0.5;
        double r488391 = r488389 * r488390;
        double r488392 = sin(r488391);
        double r488393 = r488388 * r488392;
        double r488394 = r488393 * r488392;
        double r488395 = sin(r488389);
        double r488396 = r488394 / r488395;
        return r488396;
}
↓
double f(double x) {
        double r488397 = 8.0;
        double r488398 = 0.5;
        double r488399 = x;
        double r488400 = r488398 * r488399;
        double r488401 = sin(r488400);
        double r488402 = r488397 * r488401;
        double r488403 = 3.0;
        double r488404 = r488402 / r488403;
        double r488405 = expm1(r488404);
        double r488406 = log1p(r488405);
        double r488407 = r488399 * r488398;
        double r488408 = sin(r488407);
        double r488409 = sin(r488399);
        double r488410 = r488408 / r488409;
        double r488411 = r488406 * r488410;
        return r488411;
}

Error

Bits error versus x

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Results

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Target

Original15.2
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 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. Simplified0.5

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

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

  10. Applied log1p-expm1-u0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019208 +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)))