Average Error: 15.1 → 0.3
Time: 16.1s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\frac{\frac{\sin \left(0.5 \cdot x\right)}{\sin x} \cdot \left(\sin \left(0.5 \cdot x\right) \cdot 8\right)}{3}\]
\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(0.5 \cdot x\right)}{\sin x} \cdot \left(\sin \left(0.5 \cdot x\right) \cdot 8\right)}{3}
double f(double x) {
        double r29326424 = 8.0;
        double r29326425 = 3.0;
        double r29326426 = r29326424 / r29326425;
        double r29326427 = x;
        double r29326428 = 0.5;
        double r29326429 = r29326427 * r29326428;
        double r29326430 = sin(r29326429);
        double r29326431 = r29326426 * r29326430;
        double r29326432 = r29326431 * r29326430;
        double r29326433 = sin(r29326427);
        double r29326434 = r29326432 / r29326433;
        return r29326434;
}
↓
double f(double x) {
        double r29326435 = 0.5;
        double r29326436 = x;
        double r29326437 = r29326435 * r29326436;
        double r29326438 = sin(r29326437);
        double r29326439 = sin(r29326436);
        double r29326440 = r29326438 / r29326439;
        double r29326441 = 8.0;
        double r29326442 = r29326438 * r29326441;
        double r29326443 = r29326440 * r29326442;
        double r29326444 = 3.0;
        double r29326445 = r29326443 / r29326444;
        return r29326445;
}

Error

Bits error versus x

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Results

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Target

Original15.1
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 15.1

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

  3. Applied *-un-lft-identity15.1

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

  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. Applied associate-*l/0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019192 
(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)))