Average Error: 0.2 → 0.2
Time: 1.3m
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\left(\frac{2}{3} \cdot \left|x\right|\right) \cdot \left|x\right| + 2\right) \cdot \left|x\right|\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right|\]
\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(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\left(\frac{2}{3} \cdot \left|x\right|\right) \cdot \left|x\right| + 2\right) \cdot \left|x\right|\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right|
double f(double x) {
        double r30330381 = 1.0;
        double r30330382 = atan2(1.0, 0.0);
        double r30330383 = sqrt(r30330382);
        double r30330384 = r30330381 / r30330383;
        double r30330385 = 2.0;
        double r30330386 = x;
        double r30330387 = fabs(r30330386);
        double r30330388 = r30330385 * r30330387;
        double r30330389 = 3.0;
        double r30330390 = r30330385 / r30330389;
        double r30330391 = r30330387 * r30330387;
        double r30330392 = r30330391 * r30330387;
        double r30330393 = r30330390 * r30330392;
        double r30330394 = r30330388 + r30330393;
        double r30330395 = 5.0;
        double r30330396 = r30330381 / r30330395;
        double r30330397 = r30330392 * r30330387;
        double r30330398 = r30330397 * r30330387;
        double r30330399 = r30330396 * r30330398;
        double r30330400 = r30330394 + r30330399;
        double r30330401 = 21.0;
        double r30330402 = r30330381 / r30330401;
        double r30330403 = r30330398 * r30330387;
        double r30330404 = r30330403 * r30330387;
        double r30330405 = r30330402 * r30330404;
        double r30330406 = r30330400 + r30330405;
        double r30330407 = r30330384 * r30330406;
        double r30330408 = fabs(r30330407);
        return r30330408;
}
↓
double f(double x) {
        double r30330409 = 0.047619047619047616;
        double r30330410 = x;
        double r30330411 = fabs(r30330410);
        double r30330412 = 7.0;
        double r30330413 = pow(r30330411, r30330412);
        double r30330414 = r30330409 * r30330413;
        double r30330415 = 0.2;
        double r30330416 = 5.0;
        double r30330417 = pow(r30330411, r30330416);
        double r30330418 = r30330415 * r30330417;
        double r30330419 = 0.6666666666666666;
        double r30330420 = r30330419 * r30330411;
        double r30330421 = r30330420 * r30330411;
        double r30330422 = 2.0;
        double r30330423 = r30330421 + r30330422;
        double r30330424 = r30330423 * r30330411;
        double r30330425 = r30330418 + r30330424;
        double r30330426 = r30330414 + r30330425;
        double r30330427 = 1.0;
        double r30330428 = atan2(1.0, 0.0);
        double r30330429 = r30330427 / r30330428;
        double r30330430 = sqrt(r30330429);
        double r30330431 = r30330426 * r30330430;
        double r30330432 = fabs(r30330431);
        return r30330432;
}

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. Taylor expanded around inf 0.2

    \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(2 \cdot \left|x\right| + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + \frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)}\right|\]

  3. Simplified0.2

    \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left|x\right| \cdot \left(2 + \left(\frac{2}{3} \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7}\right)}\right|\]

  4. Final simplification0.2

    \[\leadsto \left|\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\left(\frac{2}{3} \cdot \left|x\right|\right) \cdot \left|x\right| + 2\right) \cdot \left|x\right|\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right|\]

Reproduce

herbie shell --seed 2019121 
(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)))))))