Average Error: 0.2 → 0.2
Time: 35.2s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|1 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666296592325124947819858789 \cdot {\left(\left|x\right|\right)}^{3} + \left(0.2000000000000000111022302462515654042363 \cdot {\left(\left|x\right|\right)}^{5} + \left(2 \cdot \left|x\right| + 0.04761904761904761640423089374962728470564 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)\right)\right|\]
\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|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|
double f(double x) {
        double r124930 = 1.0;
        double r124931 = atan2(1.0, 0.0);
        double r124932 = sqrt(r124931);
        double r124933 = r124930 / r124932;
        double r124934 = 2.0;
        double r124935 = x;
        double r124936 = fabs(r124935);
        double r124937 = r124934 * r124936;
        double r124938 = 3.0;
        double r124939 = r124934 / r124938;
        double r124940 = r124936 * r124936;
        double r124941 = r124940 * r124936;
        double r124942 = r124939 * r124941;
        double r124943 = r124937 + r124942;
        double r124944 = 5.0;
        double r124945 = r124930 / r124944;
        double r124946 = r124941 * r124936;
        double r124947 = r124946 * r124936;
        double r124948 = r124945 * r124947;
        double r124949 = r124943 + r124948;
        double r124950 = 21.0;
        double r124951 = r124930 / r124950;
        double r124952 = r124947 * r124936;
        double r124953 = r124952 * r124936;
        double r124954 = r124951 * r124953;
        double r124955 = r124949 + r124954;
        double r124956 = r124933 * r124955;
        double r124957 = fabs(r124956);
        return r124957;
}
↓
double f(double x) {
        double r124958 = 1.0;
        double r124959 = 1.0;
        double r124960 = atan2(1.0, 0.0);
        double r124961 = r124959 / r124960;
        double r124962 = sqrt(r124961);
        double r124963 = 0.6666666666666666;
        double r124964 = x;
        double r124965 = fabs(r124964);
        double r124966 = 3.0;
        double r124967 = pow(r124965, r124966);
        double r124968 = r124963 * r124967;
        double r124969 = 0.2;
        double r124970 = 5.0;
        double r124971 = pow(r124965, r124970);
        double r124972 = r124969 * r124971;
        double r124973 = 2.0;
        double r124974 = r124973 * r124965;
        double r124975 = 0.047619047619047616;
        double r124976 = 7.0;
        double r124977 = pow(r124965, r124976);
        double r124978 = r124975 * r124977;
        double r124979 = r124974 + r124978;
        double r124980 = r124972 + r124979;
        double r124981 = r124968 + r124980;
        double r124982 = r124962 * r124981;
        double r124983 = r124958 * r124982;
        double r124984 = fabs(r124983);
        return r124984;
}

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

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019294 
(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)))))))