Average Error: 0.2 → 0.2
Time: 43.8s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\frac{1}{5} \cdot \left(\left|x\right| \cdot \left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(\left|x\right| \cdot 2 + \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \frac{2}{3}\right)\right) + \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right)\right) \cdot \frac{1}{21}\right)\right|\]
\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|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\frac{1}{5} \cdot \left(\left|x\right| \cdot \left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(\left|x\right| \cdot 2 + \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \frac{2}{3}\right)\right) + \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right)\right) \cdot \frac{1}{21}\right)\right|
double f(double x) {
        double r2253713 = 1.0;
        double r2253714 = atan2(1.0, 0.0);
        double r2253715 = sqrt(r2253714);
        double r2253716 = r2253713 / r2253715;
        double r2253717 = 2.0;
        double r2253718 = x;
        double r2253719 = fabs(r2253718);
        double r2253720 = r2253717 * r2253719;
        double r2253721 = 3.0;
        double r2253722 = r2253717 / r2253721;
        double r2253723 = r2253719 * r2253719;
        double r2253724 = r2253723 * r2253719;
        double r2253725 = r2253722 * r2253724;
        double r2253726 = r2253720 + r2253725;
        double r2253727 = 5.0;
        double r2253728 = r2253713 / r2253727;
        double r2253729 = r2253724 * r2253719;
        double r2253730 = r2253729 * r2253719;
        double r2253731 = r2253728 * r2253730;
        double r2253732 = r2253726 + r2253731;
        double r2253733 = 21.0;
        double r2253734 = r2253713 / r2253733;
        double r2253735 = r2253730 * r2253719;
        double r2253736 = r2253735 * r2253719;
        double r2253737 = r2253734 * r2253736;
        double r2253738 = r2253732 + r2253737;
        double r2253739 = r2253716 * r2253738;
        double r2253740 = fabs(r2253739);
        return r2253740;
}
↓
double f(double x) {
        double r2253741 = 1.0;
        double r2253742 = atan2(1.0, 0.0);
        double r2253743 = sqrt(r2253742);
        double r2253744 = r2253741 / r2253743;
        double r2253745 = 0.2;
        double r2253746 = x;
        double r2253747 = fabs(r2253746);
        double r2253748 = r2253747 * r2253747;
        double r2253749 = r2253748 * r2253747;
        double r2253750 = r2253749 * r2253747;
        double r2253751 = r2253747 * r2253750;
        double r2253752 = r2253745 * r2253751;
        double r2253753 = 2.0;
        double r2253754 = r2253747 * r2253753;
        double r2253755 = 0.6666666666666666;
        double r2253756 = r2253749 * r2253755;
        double r2253757 = r2253754 + r2253756;
        double r2253758 = r2253752 + r2253757;
        double r2253759 = r2253747 * r2253751;
        double r2253760 = r2253747 * r2253759;
        double r2253761 = 0.047619047619047616;
        double r2253762 = r2253760 * r2253761;
        double r2253763 = r2253758 + r2253762;
        double r2253764 = r2253744 * r2253763;
        double r2253765 = fabs(r2253764);
        return r2253765;
}

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. Final simplification0.2

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

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(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)))))))