Average Error: 0.2 → 0.2
Time: 25.9s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3}\right) + \left|x\right| \cdot \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{4} + \frac{1}{21} \cdot {\left(\left|x\right|\right)}^{6}\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right|\]
\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|\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3}\right) + \left|x\right| \cdot \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{4} + \frac{1}{21} \cdot {\left(\left|x\right|\right)}^{6}\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right|
double f(double x) {
        double r123546 = 1.0;
        double r123547 = atan2(1.0, 0.0);
        double r123548 = sqrt(r123547);
        double r123549 = r123546 / r123548;
        double r123550 = 2.0;
        double r123551 = x;
        double r123552 = fabs(r123551);
        double r123553 = r123550 * r123552;
        double r123554 = 3.0;
        double r123555 = r123550 / r123554;
        double r123556 = r123552 * r123552;
        double r123557 = r123556 * r123552;
        double r123558 = r123555 * r123557;
        double r123559 = r123553 + r123558;
        double r123560 = 5.0;
        double r123561 = r123546 / r123560;
        double r123562 = r123557 * r123552;
        double r123563 = r123562 * r123552;
        double r123564 = r123561 * r123563;
        double r123565 = r123559 + r123564;
        double r123566 = 21.0;
        double r123567 = r123546 / r123566;
        double r123568 = r123563 * r123552;
        double r123569 = r123568 * r123552;
        double r123570 = r123567 * r123569;
        double r123571 = r123565 + r123570;
        double r123572 = r123549 * r123571;
        double r123573 = fabs(r123572);
        return r123573;
}
↓
double f(double x) {
        double r123574 = 2.0;
        double r123575 = x;
        double r123576 = fabs(r123575);
        double r123577 = r123574 * r123576;
        double r123578 = 3.0;
        double r123579 = r123574 / r123578;
        double r123580 = 3.0;
        double r123581 = pow(r123576, r123580);
        double r123582 = r123579 * r123581;
        double r123583 = r123577 + r123582;
        double r123584 = 1.0;
        double r123585 = 5.0;
        double r123586 = r123584 / r123585;
        double r123587 = 4.0;
        double r123588 = pow(r123576, r123587);
        double r123589 = r123586 * r123588;
        double r123590 = 21.0;
        double r123591 = r123584 / r123590;
        double r123592 = 6.0;
        double r123593 = pow(r123576, r123592);
        double r123594 = r123591 * r123593;
        double r123595 = r123589 + r123594;
        double r123596 = r123576 * r123595;
        double r123597 = r123583 + r123596;
        double r123598 = atan2(1.0, 0.0);
        double r123599 = sqrt(r123598);
        double r123600 = r123584 / r123599;
        double r123601 = r123597 * r123600;
        double r123602 = fabs(r123601);
        return r123602;
}

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. Simplified0.2

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

  3. Final simplification0.2

    \[\leadsto \left|\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3}\right) + \left|x\right| \cdot \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{4} + \frac{1}{21} \cdot {\left(\left|x\right|\right)}^{6}\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right|\]

Reproduce

herbie shell --seed 2019303 
(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)))))))