Average Error: 0.2 → 0.2
Time: 27.4s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\mathsf{fma}\left(\frac{1}{5}, {\left(\left|x\right|\right)}^{5}, \mathsf{fma}\left(\left|x\right|, 2, \mathsf{fma}\left(\frac{1}{21}, {\left(\left|x\right|\right)}^{7}, \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \frac{2}{3}\right)\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right|\]
\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|\mathsf{fma}\left(\frac{1}{5}, {\left(\left|x\right|\right)}^{5}, \mathsf{fma}\left(\left|x\right|, 2, \mathsf{fma}\left(\frac{1}{21}, {\left(\left|x\right|\right)}^{7}, \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \frac{2}{3}\right)\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right|
double f(double x) {
        double r6258502 = 1.0;
        double r6258503 = atan2(1.0, 0.0);
        double r6258504 = sqrt(r6258503);
        double r6258505 = r6258502 / r6258504;
        double r6258506 = 2.0;
        double r6258507 = x;
        double r6258508 = fabs(r6258507);
        double r6258509 = r6258506 * r6258508;
        double r6258510 = 3.0;
        double r6258511 = r6258506 / r6258510;
        double r6258512 = r6258508 * r6258508;
        double r6258513 = r6258512 * r6258508;
        double r6258514 = r6258511 * r6258513;
        double r6258515 = r6258509 + r6258514;
        double r6258516 = 5.0;
        double r6258517 = r6258502 / r6258516;
        double r6258518 = r6258513 * r6258508;
        double r6258519 = r6258518 * r6258508;
        double r6258520 = r6258517 * r6258519;
        double r6258521 = r6258515 + r6258520;
        double r6258522 = 21.0;
        double r6258523 = r6258502 / r6258522;
        double r6258524 = r6258519 * r6258508;
        double r6258525 = r6258524 * r6258508;
        double r6258526 = r6258523 * r6258525;
        double r6258527 = r6258521 + r6258526;
        double r6258528 = r6258505 * r6258527;
        double r6258529 = fabs(r6258528);
        return r6258529;
}
↓
double f(double x) {
        double r6258530 = 0.2;
        double r6258531 = x;
        double r6258532 = fabs(r6258531);
        double r6258533 = 5.0;
        double r6258534 = pow(r6258532, r6258533);
        double r6258535 = 2.0;
        double r6258536 = 0.047619047619047616;
        double r6258537 = 7.0;
        double r6258538 = pow(r6258532, r6258537);
        double r6258539 = r6258532 * r6258532;
        double r6258540 = r6258532 * r6258539;
        double r6258541 = 0.6666666666666666;
        double r6258542 = r6258540 * r6258541;
        double r6258543 = fma(r6258536, r6258538, r6258542);
        double r6258544 = fma(r6258532, r6258535, r6258543);
        double r6258545 = fma(r6258530, r6258534, r6258544);
        double r6258546 = 1.0;
        double r6258547 = atan2(1.0, 0.0);
        double r6258548 = r6258546 / r6258547;
        double r6258549 = sqrt(r6258548);
        double r6258550 = r6258545 * r6258549;
        double r6258551 = fabs(r6258550);
        return r6258551;
}

Error

Bits error versus x

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

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

  3. Simplified0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019163 +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)))))))