Average Error: 0.2 → 0.2
Time: 7.2s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|1 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.66666666666666663 \cdot {\left(\left|x\right|\right)}^{3}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(0.20000000000000001 \cdot {\left(\left|x\right|\right)}^{5} + \left(2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)\right|\]
\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(0.66666666666666663 \cdot {\left(\left|x\right|\right)}^{3}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(0.20000000000000001 \cdot {\left(\left|x\right|\right)}^{5} + \left(2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)\right|
double f(double x) {
        double r239201 = 1.0;
        double r239202 = atan2(1.0, 0.0);
        double r239203 = sqrt(r239202);
        double r239204 = r239201 / r239203;
        double r239205 = 2.0;
        double r239206 = x;
        double r239207 = fabs(r239206);
        double r239208 = r239205 * r239207;
        double r239209 = 3.0;
        double r239210 = r239205 / r239209;
        double r239211 = r239207 * r239207;
        double r239212 = r239211 * r239207;
        double r239213 = r239210 * r239212;
        double r239214 = r239208 + r239213;
        double r239215 = 5.0;
        double r239216 = r239201 / r239215;
        double r239217 = r239212 * r239207;
        double r239218 = r239217 * r239207;
        double r239219 = r239216 * r239218;
        double r239220 = r239214 + r239219;
        double r239221 = 21.0;
        double r239222 = r239201 / r239221;
        double r239223 = r239218 * r239207;
        double r239224 = r239223 * r239207;
        double r239225 = r239222 * r239224;
        double r239226 = r239220 + r239225;
        double r239227 = r239204 * r239226;
        double r239228 = fabs(r239227);
        return r239228;
}
↓
double f(double x) {
        double r239229 = 1.0;
        double r239230 = 1.0;
        double r239231 = atan2(1.0, 0.0);
        double r239232 = r239230 / r239231;
        double r239233 = sqrt(r239232);
        double r239234 = 0.6666666666666666;
        double r239235 = x;
        double r239236 = fabs(r239235);
        double r239237 = 3.0;
        double r239238 = pow(r239236, r239237);
        double r239239 = r239234 * r239238;
        double r239240 = r239233 * r239239;
        double r239241 = 0.2;
        double r239242 = 5.0;
        double r239243 = pow(r239236, r239242);
        double r239244 = r239241 * r239243;
        double r239245 = 2.0;
        double r239246 = r239245 * r239236;
        double r239247 = 0.047619047619047616;
        double r239248 = 7.0;
        double r239249 = pow(r239236, r239248);
        double r239250 = r239247 * r239249;
        double r239251 = r239246 + r239250;
        double r239252 = r239244 + r239251;
        double r239253 = r239233 * r239252;
        double r239254 = r239240 + r239253;
        double r239255 = r239229 * r239254;
        double r239256 = fabs(r239255);
        return r239256;
}

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

    \[\leadsto \left|\color{blue}{1 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.66666666666666663 \cdot {\left(\left|x\right|\right)}^{3} + \left(0.20000000000000001 \cdot {\left(\left|x\right|\right)}^{5} + \left(2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)\right)}\right|\]
  3. Using strategy rm

  4. Applied distribute-lft-in0.2

    \[\leadsto \left|1 \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.66666666666666663 \cdot {\left(\left|x\right|\right)}^{3}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(0.20000000000000001 \cdot {\left(\left|x\right|\right)}^{5} + \left(2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)}\right|\]

  5. Final simplification0.2

    \[\leadsto \left|1 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.66666666666666663 \cdot {\left(\left|x\right|\right)}^{3}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(0.20000000000000001 \cdot {\left(\left|x\right|\right)}^{5} + \left(2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)\right|\]

Reproduce

herbie shell --seed 2020018 
(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)))))))