Average Error: 0.2 → 0.2
Time: 44.2s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\frac{1}{5} \cdot \left(\left|x\right| \cdot \left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(\left|x\right| \cdot 2 + \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \frac{2}{3}\right)\right) + \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right)\right) \cdot \frac{1}{21}\right)\right|\]
\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 \left(\left(\frac{1}{5} \cdot \left(\left|x\right| \cdot \left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(\left|x\right| \cdot 2 + \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \frac{2}{3}\right)\right) + \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right)\right) \cdot \frac{1}{21}\right)\right|
double f(double x) {
        double r5970034 = 1.0;
        double r5970035 = atan2(1.0, 0.0);
        double r5970036 = sqrt(r5970035);
        double r5970037 = r5970034 / r5970036;
        double r5970038 = 2.0;
        double r5970039 = x;
        double r5970040 = fabs(r5970039);
        double r5970041 = r5970038 * r5970040;
        double r5970042 = 3.0;
        double r5970043 = r5970038 / r5970042;
        double r5970044 = r5970040 * r5970040;
        double r5970045 = r5970044 * r5970040;
        double r5970046 = r5970043 * r5970045;
        double r5970047 = r5970041 + r5970046;
        double r5970048 = 5.0;
        double r5970049 = r5970034 / r5970048;
        double r5970050 = r5970045 * r5970040;
        double r5970051 = r5970050 * r5970040;
        double r5970052 = r5970049 * r5970051;
        double r5970053 = r5970047 + r5970052;
        double r5970054 = 21.0;
        double r5970055 = r5970034 / r5970054;
        double r5970056 = r5970051 * r5970040;
        double r5970057 = r5970056 * r5970040;
        double r5970058 = r5970055 * r5970057;
        double r5970059 = r5970053 + r5970058;
        double r5970060 = r5970037 * r5970059;
        double r5970061 = fabs(r5970060);
        return r5970061;
}
↓
double f(double x) {
        double r5970062 = 1.0;
        double r5970063 = atan2(1.0, 0.0);
        double r5970064 = sqrt(r5970063);
        double r5970065 = r5970062 / r5970064;
        double r5970066 = 0.2;
        double r5970067 = x;
        double r5970068 = fabs(r5970067);
        double r5970069 = r5970068 * r5970068;
        double r5970070 = r5970069 * r5970068;
        double r5970071 = r5970070 * r5970068;
        double r5970072 = r5970068 * r5970071;
        double r5970073 = r5970066 * r5970072;
        double r5970074 = 2.0;
        double r5970075 = r5970068 * r5970074;
        double r5970076 = 0.6666666666666666;
        double r5970077 = r5970070 * r5970076;
        double r5970078 = r5970075 + r5970077;
        double r5970079 = r5970073 + r5970078;
        double r5970080 = r5970068 * r5970072;
        double r5970081 = r5970068 * r5970080;
        double r5970082 = 0.047619047619047616;
        double r5970083 = r5970081 * r5970082;
        double r5970084 = r5970079 + r5970083;
        double r5970085 = r5970065 * r5970084;
        double r5970086 = fabs(r5970085);
        return r5970086;
}

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. Final simplification0.2

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

Reproduce

herbie shell --seed 2019151 
(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)))))))