Average Error: 15.2 → 0.3
Time: 10.5s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\]
\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}
↓
\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}
double f(double x) {
        double r644492 = 8.0;
        double r644493 = 3.0;
        double r644494 = r644492 / r644493;
        double r644495 = x;
        double r644496 = 0.5;
        double r644497 = r644495 * r644496;
        double r644498 = sin(r644497);
        double r644499 = r644494 * r644498;
        double r644500 = r644499 * r644498;
        double r644501 = sin(r644495);
        double r644502 = r644500 / r644501;
        return r644502;
}
↓
double f(double x) {
        double r644503 = 8.0;
        double r644504 = x;
        double r644505 = 0.5;
        double r644506 = r644504 * r644505;
        double r644507 = sin(r644506);
        double r644508 = r644503 * r644507;
        double r644509 = 3.0;
        double r644510 = r644508 / r644509;
        double r644511 = sin(r644504);
        double r644512 = r644507 / r644511;
        double r644513 = r644510 * r644512;
        return r644513;
}

Error

Bits error versus x

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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. Final simplification0.3

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

Reproduce

herbie shell --seed 2020045 +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)))