Average Error: 0.2 → 0.2
Time: 11.7s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\left(\left|x\right| \cdot \left(2 + \left({\left(\left|x\right|\right)}^{4} \cdot \frac{1}{5} + {\left(\left|x\right|\right)}^{6} \cdot \frac{1}{21}\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}} + \frac{1}{\sqrt{\pi}} \cdot \frac{2}{\frac{3}{{\left(\left|x\right|\right)}^{3}}}\right|\]
\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(\left|x\right| \cdot \left(2 + \left({\left(\left|x\right|\right)}^{4} \cdot \frac{1}{5} + {\left(\left|x\right|\right)}^{6} \cdot \frac{1}{21}\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}} + \frac{1}{\sqrt{\pi}} \cdot \frac{2}{\frac{3}{{\left(\left|x\right|\right)}^{3}}}\right|
double f(double x) {
        double r491034 = 1.0;
        double r491035 = atan2(1.0, 0.0);
        double r491036 = sqrt(r491035);
        double r491037 = r491034 / r491036;
        double r491038 = 2.0;
        double r491039 = x;
        double r491040 = fabs(r491039);
        double r491041 = r491038 * r491040;
        double r491042 = 3.0;
        double r491043 = r491038 / r491042;
        double r491044 = r491040 * r491040;
        double r491045 = r491044 * r491040;
        double r491046 = r491043 * r491045;
        double r491047 = r491041 + r491046;
        double r491048 = 5.0;
        double r491049 = r491034 / r491048;
        double r491050 = r491045 * r491040;
        double r491051 = r491050 * r491040;
        double r491052 = r491049 * r491051;
        double r491053 = r491047 + r491052;
        double r491054 = 21.0;
        double r491055 = r491034 / r491054;
        double r491056 = r491051 * r491040;
        double r491057 = r491056 * r491040;
        double r491058 = r491055 * r491057;
        double r491059 = r491053 + r491058;
        double r491060 = r491037 * r491059;
        double r491061 = fabs(r491060);
        return r491061;
}
↓
double f(double x) {
        double r491062 = x;
        double r491063 = fabs(r491062);
        double r491064 = 2.0;
        double r491065 = 4.0;
        double r491066 = pow(r491063, r491065);
        double r491067 = 1.0;
        double r491068 = 5.0;
        double r491069 = r491067 / r491068;
        double r491070 = r491066 * r491069;
        double r491071 = 6.0;
        double r491072 = pow(r491063, r491071);
        double r491073 = 21.0;
        double r491074 = r491067 / r491073;
        double r491075 = r491072 * r491074;
        double r491076 = r491070 + r491075;
        double r491077 = r491064 + r491076;
        double r491078 = r491063 * r491077;
        double r491079 = atan2(1.0, 0.0);
        double r491080 = sqrt(r491079);
        double r491081 = r491067 / r491080;
        double r491082 = r491078 * r491081;
        double r491083 = 3.0;
        double r491084 = 3.0;
        double r491085 = pow(r491063, r491084);
        double r491086 = r491083 / r491085;
        double r491087 = r491064 / r491086;
        double r491088 = r491081 * r491087;
        double r491089 = r491082 + r491088;
        double r491090 = fabs(r491089);
        return r491090;
}

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. Simplified0.2

    \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + \left({\left(\left|x\right|\right)}^{4} \cdot \frac{1}{5} + {\left(\left|x\right|\right)}^{6} \cdot \frac{1}{21}\right)\right) + \frac{2}{\frac{3}{{\left(\left|x\right|\right)}^{3}}}\right)\right|}\]
  3. Using strategy rm

  4. Applied distribute-lft-in0.2

    \[\leadsto \left|\color{blue}{\frac{1}{\sqrt{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + \left({\left(\left|x\right|\right)}^{4} \cdot \frac{1}{5} + {\left(\left|x\right|\right)}^{6} \cdot \frac{1}{21}\right)\right)\right) + \frac{1}{\sqrt{\pi}} \cdot \frac{2}{\frac{3}{{\left(\left|x\right|\right)}^{3}}}}\right|\]

  5. Simplified0.2

    \[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot \left(2 + \left({\left(\left|x\right|\right)}^{4} \cdot \frac{1}{5} + {\left(\left|x\right|\right)}^{6} \cdot \frac{1}{21}\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}} + \frac{1}{\sqrt{\pi}} \cdot \frac{2}{\frac{3}{{\left(\left|x\right|\right)}^{3}}}\right|\]

  6. Final simplification0.2

    \[\leadsto \left|\left(\left|x\right| \cdot \left(2 + \left({\left(\left|x\right|\right)}^{4} \cdot \frac{1}{5} + {\left(\left|x\right|\right)}^{6} \cdot \frac{1}{21}\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}} + \frac{1}{\sqrt{\pi}} \cdot \frac{2}{\frac{3}{{\left(\left|x\right|\right)}^{3}}}\right|\]

Reproduce

herbie shell --seed 2020042 
(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)))))))