Average Error: 0.2 → 0.2
Time: 27.7s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\frac{\frac{1}{21} \cdot \left(\left(\left|x\right| \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right)\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right)}{\sqrt{\pi}} + \left(\left(2 \cdot \left|x\right| + \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \frac{2}{3}\right) + \frac{1}{5} \cdot \left(\left|x\right| \cdot \left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right|\]
\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|\frac{\frac{1}{21} \cdot \left(\left(\left|x\right| \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right)\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right)}{\sqrt{\pi}} + \left(\left(2 \cdot \left|x\right| + \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \frac{2}{3}\right) + \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)\right) \cdot \frac{1}{\sqrt{\pi}}\right|
double f(double x) {
        double r2863630 = 1.0;
        double r2863631 = atan2(1.0, 0.0);
        double r2863632 = sqrt(r2863631);
        double r2863633 = r2863630 / r2863632;
        double r2863634 = 2.0;
        double r2863635 = x;
        double r2863636 = fabs(r2863635);
        double r2863637 = r2863634 * r2863636;
        double r2863638 = 3.0;
        double r2863639 = r2863634 / r2863638;
        double r2863640 = r2863636 * r2863636;
        double r2863641 = r2863640 * r2863636;
        double r2863642 = r2863639 * r2863641;
        double r2863643 = r2863637 + r2863642;
        double r2863644 = 5.0;
        double r2863645 = r2863630 / r2863644;
        double r2863646 = r2863641 * r2863636;
        double r2863647 = r2863646 * r2863636;
        double r2863648 = r2863645 * r2863647;
        double r2863649 = r2863643 + r2863648;
        double r2863650 = 21.0;
        double r2863651 = r2863630 / r2863650;
        double r2863652 = r2863647 * r2863636;
        double r2863653 = r2863652 * r2863636;
        double r2863654 = r2863651 * r2863653;
        double r2863655 = r2863649 + r2863654;
        double r2863656 = r2863633 * r2863655;
        double r2863657 = fabs(r2863656);
        return r2863657;
}
↓
double f(double x) {
        double r2863658 = 0.047619047619047616;
        double r2863659 = x;
        double r2863660 = fabs(r2863659);
        double r2863661 = r2863660 * r2863660;
        double r2863662 = r2863661 * r2863661;
        double r2863663 = r2863660 * r2863662;
        double r2863664 = r2863663 * r2863661;
        double r2863665 = r2863658 * r2863664;
        double r2863666 = atan2(1.0, 0.0);
        double r2863667 = sqrt(r2863666);
        double r2863668 = r2863665 / r2863667;
        double r2863669 = 2.0;
        double r2863670 = r2863669 * r2863660;
        double r2863671 = r2863661 * r2863660;
        double r2863672 = 0.6666666666666666;
        double r2863673 = r2863671 * r2863672;
        double r2863674 = r2863670 + r2863673;
        double r2863675 = 0.2;
        double r2863676 = r2863671 * r2863660;
        double r2863677 = r2863660 * r2863676;
        double r2863678 = r2863675 * r2863677;
        double r2863679 = r2863674 + r2863678;
        double r2863680 = 1.0;
        double r2863681 = r2863680 / r2863667;
        double r2863682 = r2863679 * r2863681;
        double r2863683 = r2863668 + r2863682;
        double r2863684 = fabs(r2863683);
        return r2863684;
}

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

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

  4. Applied associate-*r*0.2

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

  6. Applied distribute-rgt-in0.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019153 
(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)))))))