Average Error: 0.2 → 0.2
Time: 25.9s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\left({\left(\left|x\right|\right)}^{7} \cdot \frac{1}{21} + \left({\left(\left|x\right|\right)}^{5} \cdot \frac{1}{5} + \left(\frac{2}{3} \cdot \left(\left|x\right| \cdot \left|x\right|\right) + 2\right) \cdot \left|x\right|\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right|\]
\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({\left(\left|x\right|\right)}^{7} \cdot \frac{1}{21} + \left({\left(\left|x\right|\right)}^{5} \cdot \frac{1}{5} + \left(\frac{2}{3} \cdot \left(\left|x\right| \cdot \left|x\right|\right) + 2\right) \cdot \left|x\right|\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right|
double f(double x) {
        double r5422162 = 1.0;
        double r5422163 = atan2(1.0, 0.0);
        double r5422164 = sqrt(r5422163);
        double r5422165 = r5422162 / r5422164;
        double r5422166 = 2.0;
        double r5422167 = x;
        double r5422168 = fabs(r5422167);
        double r5422169 = r5422166 * r5422168;
        double r5422170 = 3.0;
        double r5422171 = r5422166 / r5422170;
        double r5422172 = r5422168 * r5422168;
        double r5422173 = r5422172 * r5422168;
        double r5422174 = r5422171 * r5422173;
        double r5422175 = r5422169 + r5422174;
        double r5422176 = 5.0;
        double r5422177 = r5422162 / r5422176;
        double r5422178 = r5422173 * r5422168;
        double r5422179 = r5422178 * r5422168;
        double r5422180 = r5422177 * r5422179;
        double r5422181 = r5422175 + r5422180;
        double r5422182 = 21.0;
        double r5422183 = r5422162 / r5422182;
        double r5422184 = r5422179 * r5422168;
        double r5422185 = r5422184 * r5422168;
        double r5422186 = r5422183 * r5422185;
        double r5422187 = r5422181 + r5422186;
        double r5422188 = r5422165 * r5422187;
        double r5422189 = fabs(r5422188);
        return r5422189;
}
↓
double f(double x) {
        double r5422190 = x;
        double r5422191 = fabs(r5422190);
        double r5422192 = 7.0;
        double r5422193 = pow(r5422191, r5422192);
        double r5422194 = 0.047619047619047616;
        double r5422195 = r5422193 * r5422194;
        double r5422196 = 5.0;
        double r5422197 = pow(r5422191, r5422196);
        double r5422198 = 0.2;
        double r5422199 = r5422197 * r5422198;
        double r5422200 = 0.6666666666666666;
        double r5422201 = r5422191 * r5422191;
        double r5422202 = r5422200 * r5422201;
        double r5422203 = 2.0;
        double r5422204 = r5422202 + r5422203;
        double r5422205 = r5422204 * r5422191;
        double r5422206 = r5422199 + r5422205;
        double r5422207 = r5422195 + r5422206;
        double r5422208 = 1.0;
        double r5422209 = atan2(1.0, 0.0);
        double r5422210 = r5422208 / r5422209;
        double r5422211 = sqrt(r5422210);
        double r5422212 = r5422207 * r5422211;
        double r5422213 = fabs(r5422212);
        return r5422213;
}

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

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

  5. Applied *-commutative0.2

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

  6. Final simplification0.2

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

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)))))))