Average Error: 0.2 → 0.2
Time: 25.0s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\left(1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(0.6666666666666666296592325124947819858789, {\left(\left|x\right|\right)}^{3}, \mathsf{fma}\left(0.2000000000000000111022302462515654042363, {\left(\left|x\right|\right)}^{5}, \mathsf{fma}\left(2, \left|x\right|, 0.04761904761904761640423089374962728470564 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)\right|\]
\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|\left(1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(0.6666666666666666296592325124947819858789, {\left(\left|x\right|\right)}^{3}, \mathsf{fma}\left(0.2000000000000000111022302462515654042363, {\left(\left|x\right|\right)}^{5}, \mathsf{fma}\left(2, \left|x\right|, 0.04761904761904761640423089374962728470564 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)\right|
double f(double x) {
        double r89982 = 1.0;
        double r89983 = atan2(1.0, 0.0);
        double r89984 = sqrt(r89983);
        double r89985 = r89982 / r89984;
        double r89986 = 2.0;
        double r89987 = x;
        double r89988 = fabs(r89987);
        double r89989 = r89986 * r89988;
        double r89990 = 3.0;
        double r89991 = r89986 / r89990;
        double r89992 = r89988 * r89988;
        double r89993 = r89992 * r89988;
        double r89994 = r89991 * r89993;
        double r89995 = r89989 + r89994;
        double r89996 = 5.0;
        double r89997 = r89982 / r89996;
        double r89998 = r89993 * r89988;
        double r89999 = r89998 * r89988;
        double r90000 = r89997 * r89999;
        double r90001 = r89995 + r90000;
        double r90002 = 21.0;
        double r90003 = r89982 / r90002;
        double r90004 = r89999 * r89988;
        double r90005 = r90004 * r89988;
        double r90006 = r90003 * r90005;
        double r90007 = r90001 + r90006;
        double r90008 = r89985 * r90007;
        double r90009 = fabs(r90008);
        return r90009;
}
↓
double f(double x) {
        double r90010 = 1.0;
        double r90011 = 1.0;
        double r90012 = atan2(1.0, 0.0);
        double r90013 = r90011 / r90012;
        double r90014 = sqrt(r90013);
        double r90015 = r90010 * r90014;
        double r90016 = 0.6666666666666666;
        double r90017 = x;
        double r90018 = fabs(r90017);
        double r90019 = 3.0;
        double r90020 = pow(r90018, r90019);
        double r90021 = 0.2;
        double r90022 = 5.0;
        double r90023 = pow(r90018, r90022);
        double r90024 = 2.0;
        double r90025 = 0.047619047619047616;
        double r90026 = 7.0;
        double r90027 = pow(r90018, r90026);
        double r90028 = r90025 * r90027;
        double r90029 = fma(r90024, r90018, r90028);
        double r90030 = fma(r90021, r90023, r90029);
        double r90031 = fma(r90016, r90020, r90030);
        double r90032 = r90015 * r90031;
        double r90033 = fabs(r90032);
        return r90033;
}

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

    \[\leadsto \left|\color{blue}{1 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666296592325124947819858789 \cdot {\left(\left|x\right|\right)}^{3} + \left(0.2000000000000000111022302462515654042363 \cdot {\left(\left|x\right|\right)}^{5} + \left(2 \cdot \left|x\right| + 0.04761904761904761640423089374962728470564 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)\right)}\right|\]

  3. Simplified0.2

    \[\leadsto \left|\color{blue}{\left(1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(0.6666666666666666296592325124947819858789, {\left(\left|x\right|\right)}^{3}, \mathsf{fma}\left(0.2000000000000000111022302462515654042363, {\left(\left|x\right|\right)}^{5}, \mathsf{fma}\left(2, \left|x\right|, 0.04761904761904761640423089374962728470564 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)}\right|\]

  4. Final simplification0.2

    \[\leadsto \left|\left(1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(0.6666666666666666296592325124947819858789, {\left(\left|x\right|\right)}^{3}, \mathsf{fma}\left(0.2000000000000000111022302462515654042363, {\left(\left|x\right|\right)}^{5}, \mathsf{fma}\left(2, \left|x\right|, 0.04761904761904761640423089374962728470564 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)\right|\]

Reproduce

herbie shell --seed 2019325 +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)))))))