Average Error: 0.2 → 0.2
Time: 13.7s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, \frac{1}{21}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{5}, \frac{1}{5}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{3}, \frac{2}{3}, 2 \cdot \left|x\right|\right)\right)\right)\right|\]
\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|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, \frac{1}{21}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{5}, \frac{1}{5}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{3}, \frac{2}{3}, 2 \cdot \left|x\right|\right)\right)\right)\right|
double f(double x) {
        double r127413 = 1.0;
        double r127414 = atan2(1.0, 0.0);
        double r127415 = sqrt(r127414);
        double r127416 = r127413 / r127415;
        double r127417 = 2.0;
        double r127418 = x;
        double r127419 = fabs(r127418);
        double r127420 = r127417 * r127419;
        double r127421 = 3.0;
        double r127422 = r127417 / r127421;
        double r127423 = r127419 * r127419;
        double r127424 = r127423 * r127419;
        double r127425 = r127422 * r127424;
        double r127426 = r127420 + r127425;
        double r127427 = 5.0;
        double r127428 = r127413 / r127427;
        double r127429 = r127424 * r127419;
        double r127430 = r127429 * r127419;
        double r127431 = r127428 * r127430;
        double r127432 = r127426 + r127431;
        double r127433 = 21.0;
        double r127434 = r127413 / r127433;
        double r127435 = r127430 * r127419;
        double r127436 = r127435 * r127419;
        double r127437 = r127434 * r127436;
        double r127438 = r127432 + r127437;
        double r127439 = r127416 * r127438;
        double r127440 = fabs(r127439);
        return r127440;
}
↓
double f(double x) {
        double r127441 = 1.0;
        double r127442 = atan2(1.0, 0.0);
        double r127443 = sqrt(r127442);
        double r127444 = r127441 / r127443;
        double r127445 = x;
        double r127446 = fabs(r127445);
        double r127447 = 7.0;
        double r127448 = pow(r127446, r127447);
        double r127449 = 21.0;
        double r127450 = r127441 / r127449;
        double r127451 = 5.0;
        double r127452 = pow(r127446, r127451);
        double r127453 = 5.0;
        double r127454 = r127441 / r127453;
        double r127455 = 3.0;
        double r127456 = pow(r127446, r127455);
        double r127457 = 2.0;
        double r127458 = 3.0;
        double r127459 = r127457 / r127458;
        double r127460 = r127457 * r127446;
        double r127461 = fma(r127456, r127459, r127460);
        double r127462 = fma(r127452, r127454, r127461);
        double r127463 = fma(r127448, r127450, r127462);
        double r127464 = r127444 * r127463;
        double r127465 = fabs(r127464);
        return r127465;
}

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. Simplified0.6

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

  4. Applied associate-/r/0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(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)))))))