Average Error: 0.2 → 0.2
Time: 4.4s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left|{\left(\left|x\right|\right)}^{\frac{3}{2}}\right| \cdot \left|{\left(\left|x\right|\right)}^{\frac{3}{2}}\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\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(\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(\left|x\right|\right)}^{\frac{3}{2}}\right| \cdot \left|{\left(\left|x\right|\right)}^{\frac{3}{2}}\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|
double f(double x) {
        double r151745 = 1.0;
        double r151746 = atan2(1.0, 0.0);
        double r151747 = sqrt(r151746);
        double r151748 = r151745 / r151747;
        double r151749 = 2.0;
        double r151750 = x;
        double r151751 = fabs(r151750);
        double r151752 = r151749 * r151751;
        double r151753 = 3.0;
        double r151754 = r151749 / r151753;
        double r151755 = r151751 * r151751;
        double r151756 = r151755 * r151751;
        double r151757 = r151754 * r151756;
        double r151758 = r151752 + r151757;
        double r151759 = 5.0;
        double r151760 = r151745 / r151759;
        double r151761 = r151756 * r151751;
        double r151762 = r151761 * r151751;
        double r151763 = r151760 * r151762;
        double r151764 = r151758 + r151763;
        double r151765 = 21.0;
        double r151766 = r151745 / r151765;
        double r151767 = r151762 * r151751;
        double r151768 = r151767 * r151751;
        double r151769 = r151766 * r151768;
        double r151770 = r151764 + r151769;
        double r151771 = r151748 * r151770;
        double r151772 = fabs(r151771);
        return r151772;
}
↓
double f(double x) {
        double r151773 = 1.0;
        double r151774 = atan2(1.0, 0.0);
        double r151775 = sqrt(r151774);
        double r151776 = r151773 / r151775;
        double r151777 = 2.0;
        double r151778 = x;
        double r151779 = fabs(r151778);
        double r151780 = r151777 * r151779;
        double r151781 = 3.0;
        double r151782 = r151777 / r151781;
        double r151783 = r151779 * r151779;
        double r151784 = r151783 * r151779;
        double r151785 = r151782 * r151784;
        double r151786 = r151780 + r151785;
        double r151787 = 5.0;
        double r151788 = r151773 / r151787;
        double r151789 = r151784 * r151779;
        double r151790 = r151789 * r151779;
        double r151791 = r151788 * r151790;
        double r151792 = r151786 + r151791;
        double r151793 = 21.0;
        double r151794 = r151773 / r151793;
        double r151795 = 1.5;
        double r151796 = pow(r151779, r151795);
        double r151797 = fabs(r151796);
        double r151798 = r151797 * r151797;
        double r151799 = r151798 * r151779;
        double r151800 = r151799 * r151779;
        double r151801 = r151800 * r151779;
        double r151802 = r151801 * r151779;
        double r151803 = r151794 * r151802;
        double r151804 = r151792 + r151803;
        double r151805 = r151776 * r151804;
        double r151806 = fabs(r151805);
        return r151806;
}

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. Using strategy rm

  3. Applied add-sqr-sqrt0.2

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

  4. Simplified0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  :precision binary64
  (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)))))))