Average Error: 0.2 → 0.2
Time: 26.6s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left({\left(\left|x\right|\right)}^{5} \cdot \frac{1}{5} + \left(\left|x\right| \cdot \left(\left|x\right| \cdot \frac{2}{3}\right) + 2\right) \cdot \left|x\right|\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right|\]
\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({\left(\left|x\right|\right)}^{5} \cdot \frac{1}{5} + \left(\left|x\right| \cdot \left(\left|x\right| \cdot \frac{2}{3}\right) + 2\right) \cdot \left|x\right|\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right|
double f(double x) {
        double r5923008 = 1.0;
        double r5923009 = atan2(1.0, 0.0);
        double r5923010 = sqrt(r5923009);
        double r5923011 = r5923008 / r5923010;
        double r5923012 = 2.0;
        double r5923013 = x;
        double r5923014 = fabs(r5923013);
        double r5923015 = r5923012 * r5923014;
        double r5923016 = 3.0;
        double r5923017 = r5923012 / r5923016;
        double r5923018 = r5923014 * r5923014;
        double r5923019 = r5923018 * r5923014;
        double r5923020 = r5923017 * r5923019;
        double r5923021 = r5923015 + r5923020;
        double r5923022 = 5.0;
        double r5923023 = r5923008 / r5923022;
        double r5923024 = r5923019 * r5923014;
        double r5923025 = r5923024 * r5923014;
        double r5923026 = r5923023 * r5923025;
        double r5923027 = r5923021 + r5923026;
        double r5923028 = 21.0;
        double r5923029 = r5923008 / r5923028;
        double r5923030 = r5923025 * r5923014;
        double r5923031 = r5923030 * r5923014;
        double r5923032 = r5923029 * r5923031;
        double r5923033 = r5923027 + r5923032;
        double r5923034 = r5923011 * r5923033;
        double r5923035 = fabs(r5923034);
        return r5923035;
}
↓
double f(double x) {
        double r5923036 = 0.047619047619047616;
        double r5923037 = x;
        double r5923038 = fabs(r5923037);
        double r5923039 = 7.0;
        double r5923040 = pow(r5923038, r5923039);
        double r5923041 = r5923036 * r5923040;
        double r5923042 = 5.0;
        double r5923043 = pow(r5923038, r5923042);
        double r5923044 = 0.2;
        double r5923045 = r5923043 * r5923044;
        double r5923046 = 0.6666666666666666;
        double r5923047 = r5923038 * r5923046;
        double r5923048 = r5923038 * r5923047;
        double r5923049 = 2.0;
        double r5923050 = r5923048 + r5923049;
        double r5923051 = r5923050 * r5923038;
        double r5923052 = r5923045 + r5923051;
        double r5923053 = r5923041 + r5923052;
        double r5923054 = 1.0;
        double r5923055 = atan2(1.0, 0.0);
        double r5923056 = r5923054 / r5923055;
        double r5923057 = sqrt(r5923056);
        double r5923058 = r5923053 * r5923057;
        double r5923059 = fabs(r5923058);
        return r5923059;
}

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 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

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019163 
(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)))))))