Average Error: 0.2 → 0.2
Time: 21.0s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\left(1 \cdot \left(\frac{{\left(\left|x\right|\right)}^{7}}{21} + \frac{{\left(\left|x\right|\right)}^{5}}{5}\right) + 2 \cdot \left(\frac{{\left(\left|x\right|\right)}^{3}}{3} + \left|x\right|\right)\right) \cdot \frac{1}{\frac{\sqrt{\pi}}{1}}\right|\]
\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(1 \cdot \left(\frac{{\left(\left|x\right|\right)}^{7}}{21} + \frac{{\left(\left|x\right|\right)}^{5}}{5}\right) + 2 \cdot \left(\frac{{\left(\left|x\right|\right)}^{3}}{3} + \left|x\right|\right)\right) \cdot \frac{1}{\frac{\sqrt{\pi}}{1}}\right|
double f(double x) {
        double r99543 = 1.0;
        double r99544 = atan2(1.0, 0.0);
        double r99545 = sqrt(r99544);
        double r99546 = r99543 / r99545;
        double r99547 = 2.0;
        double r99548 = x;
        double r99549 = fabs(r99548);
        double r99550 = r99547 * r99549;
        double r99551 = 3.0;
        double r99552 = r99547 / r99551;
        double r99553 = r99549 * r99549;
        double r99554 = r99553 * r99549;
        double r99555 = r99552 * r99554;
        double r99556 = r99550 + r99555;
        double r99557 = 5.0;
        double r99558 = r99543 / r99557;
        double r99559 = r99554 * r99549;
        double r99560 = r99559 * r99549;
        double r99561 = r99558 * r99560;
        double r99562 = r99556 + r99561;
        double r99563 = 21.0;
        double r99564 = r99543 / r99563;
        double r99565 = r99560 * r99549;
        double r99566 = r99565 * r99549;
        double r99567 = r99564 * r99566;
        double r99568 = r99562 + r99567;
        double r99569 = r99546 * r99568;
        double r99570 = fabs(r99569);
        return r99570;
}
↓
double f(double x) {
        double r99571 = 1.0;
        double r99572 = x;
        double r99573 = fabs(r99572);
        double r99574 = 7.0;
        double r99575 = pow(r99573, r99574);
        double r99576 = 21.0;
        double r99577 = r99575 / r99576;
        double r99578 = 5.0;
        double r99579 = pow(r99573, r99578);
        double r99580 = 5.0;
        double r99581 = r99579 / r99580;
        double r99582 = r99577 + r99581;
        double r99583 = r99571 * r99582;
        double r99584 = 2.0;
        double r99585 = 3.0;
        double r99586 = pow(r99573, r99585);
        double r99587 = 3.0;
        double r99588 = r99586 / r99587;
        double r99589 = r99588 + r99573;
        double r99590 = r99584 * r99589;
        double r99591 = r99583 + r99590;
        double r99592 = 1.0;
        double r99593 = atan2(1.0, 0.0);
        double r99594 = sqrt(r99593);
        double r99595 = r99594 / r99571;
        double r99596 = r99592 / r99595;
        double r99597 = r99591 * r99596;
        double r99598 = fabs(r99597);
        return r99598;
}

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. Simplified0.6

    \[\leadsto \color{blue}{\left|\frac{1 \cdot \left(\frac{{\left(\left|x\right|\right)}^{7}}{21} + \frac{{\left(\left|x\right|\right)}^{5}}{5}\right) + 2 \cdot \left(\frac{{\left(\left|x\right|\right)}^{3}}{3} + \left|x\right|\right)}{\frac{\sqrt{\pi}}{1}}\right|}\]
  3. Using strategy rm

  4. Applied div-inv0.2

    \[\leadsto \left|\color{blue}{\left(1 \cdot \left(\frac{{\left(\left|x\right|\right)}^{7}}{21} + \frac{{\left(\left|x\right|\right)}^{5}}{5}\right) + 2 \cdot \left(\frac{{\left(\left|x\right|\right)}^{3}}{3} + \left|x\right|\right)\right) \cdot \frac{1}{\frac{\sqrt{\pi}}{1}}}\right|\]

  5. Final simplification0.2

    \[\leadsto \left|\left(1 \cdot \left(\frac{{\left(\left|x\right|\right)}^{7}}{21} + \frac{{\left(\left|x\right|\right)}^{5}}{5}\right) + 2 \cdot \left(\frac{{\left(\left|x\right|\right)}^{3}}{3} + \left|x\right|\right)\right) \cdot \frac{1}{\frac{\sqrt{\pi}}{1}}\right|\]

Reproduce

herbie shell --seed 2019323 
(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)))))))