Average Error: 0.2 → 0.2
Time: 8.7s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|1 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.66666666666666663 \cdot {\left(\left|x\right|\right)}^{3} + \left(0.20000000000000001 \cdot {\left(\left|x\right|\right)}^{5} + \left(2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)\right)\right|\]
\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|1 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.66666666666666663 \cdot {\left(\left|x\right|\right)}^{3} + \left(0.20000000000000001 \cdot {\left(\left|x\right|\right)}^{5} + \left(2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)\right)\right|
double f(double x) {
        double r138513 = 1.0;
        double r138514 = atan2(1.0, 0.0);
        double r138515 = sqrt(r138514);
        double r138516 = r138513 / r138515;
        double r138517 = 2.0;
        double r138518 = x;
        double r138519 = fabs(r138518);
        double r138520 = r138517 * r138519;
        double r138521 = 3.0;
        double r138522 = r138517 / r138521;
        double r138523 = r138519 * r138519;
        double r138524 = r138523 * r138519;
        double r138525 = r138522 * r138524;
        double r138526 = r138520 + r138525;
        double r138527 = 5.0;
        double r138528 = r138513 / r138527;
        double r138529 = r138524 * r138519;
        double r138530 = r138529 * r138519;
        double r138531 = r138528 * r138530;
        double r138532 = r138526 + r138531;
        double r138533 = 21.0;
        double r138534 = r138513 / r138533;
        double r138535 = r138530 * r138519;
        double r138536 = r138535 * r138519;
        double r138537 = r138534 * r138536;
        double r138538 = r138532 + r138537;
        double r138539 = r138516 * r138538;
        double r138540 = fabs(r138539);
        return r138540;
}
↓
double f(double x) {
        double r138541 = 1.0;
        double r138542 = 1.0;
        double r138543 = atan2(1.0, 0.0);
        double r138544 = r138542 / r138543;
        double r138545 = sqrt(r138544);
        double r138546 = 0.6666666666666666;
        double r138547 = x;
        double r138548 = fabs(r138547);
        double r138549 = 3.0;
        double r138550 = pow(r138548, r138549);
        double r138551 = r138546 * r138550;
        double r138552 = 0.2;
        double r138553 = 5.0;
        double r138554 = pow(r138548, r138553);
        double r138555 = r138552 * r138554;
        double r138556 = 2.0;
        double r138557 = r138556 * r138548;
        double r138558 = 0.047619047619047616;
        double r138559 = 7.0;
        double r138560 = pow(r138548, r138559);
        double r138561 = r138558 * r138560;
        double r138562 = r138557 + r138561;
        double r138563 = r138555 + r138562;
        double r138564 = r138551 + r138563;
        double r138565 = r138545 * r138564;
        double r138566 = r138541 * r138565;
        double r138567 = fabs(r138566);
        return r138567;
}

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. Taylor expanded around 0 0.2

    \[\leadsto \left|\color{blue}{1 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.66666666666666663 \cdot {\left(\left|x\right|\right)}^{3} + \left(0.20000000000000001 \cdot {\left(\left|x\right|\right)}^{5} + \left(2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)\right)}\right|\]

  3. Final simplification0.2

    \[\leadsto \left|1 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.66666666666666663 \cdot {\left(\left|x\right|\right)}^{3} + \left(0.20000000000000001 \cdot {\left(\left|x\right|\right)}^{5} + \left(2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)\right)\right|\]

Reproduce

herbie shell --seed 2020003 
(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)))))))