Average Error: 0.2 → 0.2
Time: 49.6s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\left(\frac{1}{21} \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right) + \left(\left(\sqrt[3]{\frac{1}{5}} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) \cdot \left(\sqrt[3]{\frac{1}{5}} \cdot \sqrt[3]{\frac{1}{5}}\right) + \left(\left|x\right| \cdot 2 + \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \frac{2}{3}\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right|\]
\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(\frac{1}{21} \cdot \left(\left|x\right| \cdot \left(\left|x\right| \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)\right) + \left(\left(\sqrt[3]{\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 \left(\sqrt[3]{\frac{1}{5}} \cdot \sqrt[3]{\frac{1}{5}}\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)\right) \cdot \frac{1}{\sqrt{\pi}}\right|
double f(double x) {
        double r4309530 = 1.0;
        double r4309531 = atan2(1.0, 0.0);
        double r4309532 = sqrt(r4309531);
        double r4309533 = r4309530 / r4309532;
        double r4309534 = 2.0;
        double r4309535 = x;
        double r4309536 = fabs(r4309535);
        double r4309537 = r4309534 * r4309536;
        double r4309538 = 3.0;
        double r4309539 = r4309534 / r4309538;
        double r4309540 = r4309536 * r4309536;
        double r4309541 = r4309540 * r4309536;
        double r4309542 = r4309539 * r4309541;
        double r4309543 = r4309537 + r4309542;
        double r4309544 = 5.0;
        double r4309545 = r4309530 / r4309544;
        double r4309546 = r4309541 * r4309536;
        double r4309547 = r4309546 * r4309536;
        double r4309548 = r4309545 * r4309547;
        double r4309549 = r4309543 + r4309548;
        double r4309550 = 21.0;
        double r4309551 = r4309530 / r4309550;
        double r4309552 = r4309547 * r4309536;
        double r4309553 = r4309552 * r4309536;
        double r4309554 = r4309551 * r4309553;
        double r4309555 = r4309549 + r4309554;
        double r4309556 = r4309533 * r4309555;
        double r4309557 = fabs(r4309556);
        return r4309557;
}
↓
double f(double x) {
        double r4309558 = 0.047619047619047616;
        double r4309559 = x;
        double r4309560 = fabs(r4309559);
        double r4309561 = r4309560 * r4309560;
        double r4309562 = r4309561 * r4309560;
        double r4309563 = r4309562 * r4309560;
        double r4309564 = r4309563 * r4309560;
        double r4309565 = r4309560 * r4309564;
        double r4309566 = r4309560 * r4309565;
        double r4309567 = r4309558 * r4309566;
        double r4309568 = 0.2;
        double r4309569 = cbrt(r4309568);
        double r4309570 = r4309569 * r4309564;
        double r4309571 = r4309569 * r4309569;
        double r4309572 = r4309570 * r4309571;
        double r4309573 = 2.0;
        double r4309574 = r4309560 * r4309573;
        double r4309575 = 0.6666666666666666;
        double r4309576 = r4309562 * r4309575;
        double r4309577 = r4309574 + r4309576;
        double r4309578 = r4309572 + r4309577;
        double r4309579 = r4309567 + r4309578;
        double r4309580 = 1.0;
        double r4309581 = atan2(1.0, 0.0);
        double r4309582 = sqrt(r4309581);
        double r4309583 = r4309580 / r4309582;
        double r4309584 = r4309579 * r4309583;
        double r4309585 = fabs(r4309584);
        return r4309585;
}

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-cube-cbrt0.2

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

  4. Applied associate-*l*0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019142 
(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)))))))