Average Error: 0.2 → 0.2
Time: 28.8s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\frac{1}{5} \cdot \left(\left|x\right| \cdot \left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(\left|x\right| \cdot 2 + \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \frac{2}{3}\right)\right) + \frac{1}{21} \cdot \left(\left|x\right| \cdot \left(\left(\left(\left|x\right| \cdot \left(\left(\sqrt{\left|x\right| \cdot \left|x\right|} \cdot \sqrt{\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|}\right) \cdot \sqrt{\left|x\right|}\right)\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right)\right|\]
\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|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\frac{1}{5} \cdot \left(\left|x\right| \cdot \left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(\left|x\right| \cdot 2 + \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \frac{2}{3}\right)\right) + \frac{1}{21} \cdot \left(\left|x\right| \cdot \left(\left(\left(\left|x\right| \cdot \left(\left(\sqrt{\left|x\right| \cdot \left|x\right|} \cdot \sqrt{\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|}\right) \cdot \sqrt{\left|x\right|}\right)\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right)\right|
double f(double x) {
        double r2278183 = 1.0;
        double r2278184 = atan2(1.0, 0.0);
        double r2278185 = sqrt(r2278184);
        double r2278186 = r2278183 / r2278185;
        double r2278187 = 2.0;
        double r2278188 = x;
        double r2278189 = fabs(r2278188);
        double r2278190 = r2278187 * r2278189;
        double r2278191 = 3.0;
        double r2278192 = r2278187 / r2278191;
        double r2278193 = r2278189 * r2278189;
        double r2278194 = r2278193 * r2278189;
        double r2278195 = r2278192 * r2278194;
        double r2278196 = r2278190 + r2278195;
        double r2278197 = 5.0;
        double r2278198 = r2278183 / r2278197;
        double r2278199 = r2278194 * r2278189;
        double r2278200 = r2278199 * r2278189;
        double r2278201 = r2278198 * r2278200;
        double r2278202 = r2278196 + r2278201;
        double r2278203 = 21.0;
        double r2278204 = r2278183 / r2278203;
        double r2278205 = r2278200 * r2278189;
        double r2278206 = r2278205 * r2278189;
        double r2278207 = r2278204 * r2278206;
        double r2278208 = r2278202 + r2278207;
        double r2278209 = r2278186 * r2278208;
        double r2278210 = fabs(r2278209);
        return r2278210;
}
↓
double f(double x) {
        double r2278211 = 1.0;
        double r2278212 = atan2(1.0, 0.0);
        double r2278213 = sqrt(r2278212);
        double r2278214 = r2278211 / r2278213;
        double r2278215 = 0.2;
        double r2278216 = x;
        double r2278217 = fabs(r2278216);
        double r2278218 = r2278217 * r2278217;
        double r2278219 = r2278218 * r2278217;
        double r2278220 = r2278219 * r2278217;
        double r2278221 = r2278217 * r2278220;
        double r2278222 = r2278215 * r2278221;
        double r2278223 = 2.0;
        double r2278224 = r2278217 * r2278223;
        double r2278225 = 0.6666666666666666;
        double r2278226 = r2278219 * r2278225;
        double r2278227 = r2278224 + r2278226;
        double r2278228 = r2278222 + r2278227;
        double r2278229 = 0.047619047619047616;
        double r2278230 = sqrt(r2278218);
        double r2278231 = sqrt(r2278219);
        double r2278232 = r2278230 * r2278231;
        double r2278233 = sqrt(r2278217);
        double r2278234 = r2278232 * r2278233;
        double r2278235 = r2278217 * r2278234;
        double r2278236 = r2278235 * r2278217;
        double r2278237 = r2278236 * r2278217;
        double r2278238 = r2278217 * r2278237;
        double r2278239 = r2278229 * r2278238;
        double r2278240 = r2278228 + r2278239;
        double r2278241 = r2278214 * r2278240;
        double r2278242 = fabs(r2278241);
        return r2278242;
}

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. Using strategy rm

  3. Applied add-sqr-sqrt0.2

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

  5. Applied sqrt-prod0.2

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

  6. Applied associate-*r*0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(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)))))))