Average Error: 0.2 → 0.2
Time: 1.3m
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\left(\frac{2}{3} \cdot \left|x\right|\right) \cdot \left|x\right| + 2\right) \cdot \left|x\right|\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right|\]
\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|\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\left(\frac{2}{3} \cdot \left|x\right|\right) \cdot \left|x\right| + 2\right) \cdot \left|x\right|\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right|
double f(double x) {
        double r18776969 = 1.0;
        double r18776970 = atan2(1.0, 0.0);
        double r18776971 = sqrt(r18776970);
        double r18776972 = r18776969 / r18776971;
        double r18776973 = 2.0;
        double r18776974 = x;
        double r18776975 = fabs(r18776974);
        double r18776976 = r18776973 * r18776975;
        double r18776977 = 3.0;
        double r18776978 = r18776973 / r18776977;
        double r18776979 = r18776975 * r18776975;
        double r18776980 = r18776979 * r18776975;
        double r18776981 = r18776978 * r18776980;
        double r18776982 = r18776976 + r18776981;
        double r18776983 = 5.0;
        double r18776984 = r18776969 / r18776983;
        double r18776985 = r18776980 * r18776975;
        double r18776986 = r18776985 * r18776975;
        double r18776987 = r18776984 * r18776986;
        double r18776988 = r18776982 + r18776987;
        double r18776989 = 21.0;
        double r18776990 = r18776969 / r18776989;
        double r18776991 = r18776986 * r18776975;
        double r18776992 = r18776991 * r18776975;
        double r18776993 = r18776990 * r18776992;
        double r18776994 = r18776988 + r18776993;
        double r18776995 = r18776972 * r18776994;
        double r18776996 = fabs(r18776995);
        return r18776996;
}
↓
double f(double x) {
        double r18776997 = 0.047619047619047616;
        double r18776998 = x;
        double r18776999 = fabs(r18776998);
        double r18777000 = 7.0;
        double r18777001 = pow(r18776999, r18777000);
        double r18777002 = r18776997 * r18777001;
        double r18777003 = 0.2;
        double r18777004 = 5.0;
        double r18777005 = pow(r18776999, r18777004);
        double r18777006 = r18777003 * r18777005;
        double r18777007 = 0.6666666666666666;
        double r18777008 = r18777007 * r18776999;
        double r18777009 = r18777008 * r18776999;
        double r18777010 = 2.0;
        double r18777011 = r18777009 + r18777010;
        double r18777012 = r18777011 * r18776999;
        double r18777013 = r18777006 + r18777012;
        double r18777014 = r18777002 + r18777013;
        double r18777015 = 1.0;
        double r18777016 = atan2(1.0, 0.0);
        double r18777017 = r18777015 / r18777016;
        double r18777018 = sqrt(r18777017);
        double r18777019 = r18777014 * r18777018;
        double r18777020 = fabs(r18777019);
        return r18777020;
}

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. Taylor expanded around -inf 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

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019104 
(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)))))))