Average Error: 0.2 → 0.2
Time: 23.3s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left(\frac{2}{3} + \left(\left(\frac{1}{21} \cdot \left|x\right|\right) \cdot \left|x\right| + \frac{1}{5}\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) + 2\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)\right|\]
\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|
↓
\left|\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left(\frac{2}{3} + \left(\left(\frac{1}{21} \cdot \left|x\right|\right) \cdot \left|x\right| + \frac{1}{5}\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) + 2\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)\right|
double f(double x) {
        double r4566324 = 1.0;
        double r4566325 = atan2(1.0, 0.0);
        double r4566326 = sqrt(r4566325);
        double r4566327 = r4566324 / r4566326;
        double r4566328 = 2.0;
        double r4566329 = x;
        double r4566330 = fabs(r4566329);
        double r4566331 = r4566328 * r4566330;
        double r4566332 = 3.0;
        double r4566333 = r4566328 / r4566332;
        double r4566334 = r4566330 * r4566330;
        double r4566335 = r4566334 * r4566330;
        double r4566336 = r4566333 * r4566335;
        double r4566337 = r4566331 + r4566336;
        double r4566338 = 5.0;
        double r4566339 = r4566324 / r4566338;
        double r4566340 = r4566335 * r4566330;
        double r4566341 = r4566340 * r4566330;
        double r4566342 = r4566339 * r4566341;
        double r4566343 = r4566337 + r4566342;
        double r4566344 = 21.0;
        double r4566345 = r4566324 / r4566344;
        double r4566346 = r4566341 * r4566330;
        double r4566347 = r4566346 * r4566330;
        double r4566348 = r4566345 * r4566347;
        double r4566349 = r4566343 + r4566348;
        double r4566350 = r4566327 * r4566349;
        double r4566351 = fabs(r4566350);
        return r4566351;
}
↓
double f(double x) {
        double r4566352 = x;
        double r4566353 = fabs(r4566352);
        double r4566354 = r4566353 * r4566353;
        double r4566355 = 0.6666666666666666;
        double r4566356 = 0.047619047619047616;
        double r4566357 = r4566356 * r4566353;
        double r4566358 = r4566357 * r4566353;
        double r4566359 = 0.2;
        double r4566360 = r4566358 + r4566359;
        double r4566361 = r4566360 * r4566354;
        double r4566362 = r4566355 + r4566361;
        double r4566363 = r4566354 * r4566362;
        double r4566364 = 2.0;
        double r4566365 = r4566363 + r4566364;
        double r4566366 = 1.0;
        double r4566367 = atan2(1.0, 0.0);
        double r4566368 = r4566366 / r4566367;
        double r4566369 = sqrt(r4566368);
        double r4566370 = r4566369 * r4566353;
        double r4566371 = r4566365 * r4566370;
        double r4566372 = fabs(r4566371);
        return r4566372;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

    \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]

  2. Simplified0.6

    \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(2 + \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left(\frac{2}{3} + \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \frac{1}{21}\right) + \frac{1}{5}\right)\right)\right)\right|}\]

  3. Taylor expanded around 0 0.2

    \[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right)} \cdot \left(2 + \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left(\frac{2}{3} + \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \frac{1}{21}\right) + \frac{1}{5}\right)\right)\right)\right|\]

  4. Final simplification0.2

    \[\leadsto \left|\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left(\frac{2}{3} + \left(\left(\frac{1}{21} \cdot \left|x\right|\right) \cdot \left|x\right| + \frac{1}{5}\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) + 2\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)\right|\]

Reproduce

herbie shell --seed 2019135 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  (fabs (* (/ 1 (sqrt PI)) (+ (+ (+ (* 2 (fabs x)) (* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1 21) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))))