Average Error: 0.2 → 0.2
Time: 2.8m
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 r35116557 = 1.0;
        double r35116558 = atan2(1.0, 0.0);
        double r35116559 = sqrt(r35116558);
        double r35116560 = r35116557 / r35116559;
        double r35116561 = 2.0;
        double r35116562 = x;
        double r35116563 = fabs(r35116562);
        double r35116564 = r35116561 * r35116563;
        double r35116565 = 3.0;
        double r35116566 = r35116561 / r35116565;
        double r35116567 = r35116563 * r35116563;
        double r35116568 = r35116567 * r35116563;
        double r35116569 = r35116566 * r35116568;
        double r35116570 = r35116564 + r35116569;
        double r35116571 = 5.0;
        double r35116572 = r35116557 / r35116571;
        double r35116573 = r35116568 * r35116563;
        double r35116574 = r35116573 * r35116563;
        double r35116575 = r35116572 * r35116574;
        double r35116576 = r35116570 + r35116575;
        double r35116577 = 21.0;
        double r35116578 = r35116557 / r35116577;
        double r35116579 = r35116574 * r35116563;
        double r35116580 = r35116579 * r35116563;
        double r35116581 = r35116578 * r35116580;
        double r35116582 = r35116576 + r35116581;
        double r35116583 = r35116560 * r35116582;
        double r35116584 = fabs(r35116583);
        return r35116584;
}
↓
double f(double x) {
        double r35116585 = 1.0;
        double r35116586 = atan2(1.0, 0.0);
        double r35116587 = sqrt(r35116586);
        double r35116588 = r35116585 / r35116587;
        double r35116589 = 0.2;
        double r35116590 = x;
        double r35116591 = fabs(r35116590);
        double r35116592 = r35116591 * r35116591;
        double r35116593 = r35116592 * r35116591;
        double r35116594 = r35116593 * r35116591;
        double r35116595 = r35116591 * r35116594;
        double r35116596 = r35116589 * r35116595;
        double r35116597 = 2.0;
        double r35116598 = r35116591 * r35116597;
        double r35116599 = 0.6666666666666666;
        double r35116600 = r35116593 * r35116599;
        double r35116601 = r35116598 + r35116600;
        double r35116602 = r35116596 + r35116601;
        double r35116603 = r35116591 * r35116595;
        double r35116604 = r35116591 * r35116603;
        double r35116605 = 0.047619047619047616;
        double r35116606 = r35116604 * r35116605;
        double r35116607 = r35116602 + r35116606;
        double r35116608 = r35116588 * r35116607;
        double r35116609 = fabs(r35116608);
        return r35116609;
}

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 2019128 +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)))))))