Average Error: 15.0 → 0.3
Time: 20.6s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\left(8 \cdot \frac{\sin \left(0.5 \cdot x\right)}{3}\right) \cdot \log \left(e^{\frac{\sin \left(x \cdot 0.5\right)}{\sin x}}\right)\]
\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}
↓
\left(8 \cdot \frac{\sin \left(0.5 \cdot x\right)}{3}\right) \cdot \log \left(e^{\frac{\sin \left(x \cdot 0.5\right)}{\sin x}}\right)
double f(double x) {
        double r464448 = 8.0;
        double r464449 = 3.0;
        double r464450 = r464448 / r464449;
        double r464451 = x;
        double r464452 = 0.5;
        double r464453 = r464451 * r464452;
        double r464454 = sin(r464453);
        double r464455 = r464450 * r464454;
        double r464456 = r464455 * r464454;
        double r464457 = sin(r464451);
        double r464458 = r464456 / r464457;
        return r464458;
}
↓
double f(double x) {
        double r464459 = 8.0;
        double r464460 = 0.5;
        double r464461 = x;
        double r464462 = r464460 * r464461;
        double r464463 = sin(r464462);
        double r464464 = 3.0;
        double r464465 = r464463 / r464464;
        double r464466 = r464459 * r464465;
        double r464467 = r464461 * r464460;
        double r464468 = sin(r464467);
        double r464469 = sin(r464461);
        double r464470 = r464468 / r464469;
        double r464471 = exp(r464470);
        double r464472 = log(r464471);
        double r464473 = r464466 * r464472;
        return r464473;
}

Error

Bits error versus x

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Results

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Target

Original15.0
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 15.0

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

  3. Applied *-un-lft-identity15.0

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

  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 div-inv0.5

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

  8. Applied associate-*l*0.5

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

  9. Simplified0.3

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

  11. Applied add-log-exp0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019198 +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)))