Average Error: 0.2 → 0.2
Time: 42.9s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{6}\right) \cdot \left|x\right|\right)\right|\]
\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|\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) + \left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{6}\right) \cdot \left|x\right|\right)\right|
double f(double x) {
        double r94662 = 1.0;
        double r94663 = atan2(1.0, 0.0);
        double r94664 = sqrt(r94663);
        double r94665 = r94662 / r94664;
        double r94666 = 2.0;
        double r94667 = x;
        double r94668 = fabs(r94667);
        double r94669 = r94666 * r94668;
        double r94670 = 3.0;
        double r94671 = r94666 / r94670;
        double r94672 = r94668 * r94668;
        double r94673 = r94672 * r94668;
        double r94674 = r94671 * r94673;
        double r94675 = r94669 + r94674;
        double r94676 = 5.0;
        double r94677 = r94662 / r94676;
        double r94678 = r94673 * r94668;
        double r94679 = r94678 * r94668;
        double r94680 = r94677 * r94679;
        double r94681 = r94675 + r94680;
        double r94682 = 21.0;
        double r94683 = r94662 / r94682;
        double r94684 = r94679 * r94668;
        double r94685 = r94684 * r94668;
        double r94686 = r94683 * r94685;
        double r94687 = r94681 + r94686;
        double r94688 = r94665 * r94687;
        double r94689 = fabs(r94688);
        return r94689;
}
↓
double f(double x) {
        double r94690 = 1.0;
        double r94691 = atan2(1.0, 0.0);
        double r94692 = sqrt(r94691);
        double r94693 = r94690 / r94692;
        double r94694 = 2.0;
        double r94695 = x;
        double r94696 = fabs(r94695);
        double r94697 = r94694 * r94696;
        double r94698 = 3.0;
        double r94699 = r94694 / r94698;
        double r94700 = r94696 * r94696;
        double r94701 = r94700 * r94696;
        double r94702 = r94699 * r94701;
        double r94703 = r94697 + r94702;
        double r94704 = 5.0;
        double r94705 = r94690 / r94704;
        double r94706 = r94701 * r94696;
        double r94707 = r94706 * r94696;
        double r94708 = r94705 * r94707;
        double r94709 = r94703 + r94708;
        double r94710 = 21.0;
        double r94711 = r94690 / r94710;
        double r94712 = 6.0;
        double r94713 = pow(r94696, r94712);
        double r94714 = r94711 * r94713;
        double r94715 = r94714 * r94696;
        double r94716 = r94709 + r94715;
        double r94717 = r94693 * r94716;
        double r94718 = fabs(r94717);
        return r94718;
}

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 associate-*r*0.2

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

  4. Simplified0.2

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \color{blue}{\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{6}\right)} \cdot \left|x\right|\right)\right|\]

  5. Final simplification0.2

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{6}\right) \cdot \left|x\right|\right)\right|\]

Reproduce

herbie shell --seed 2019306 +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)))))))