Average Error: 0.2 → 0.2
Time: 40.8s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{\frac{21}{{\left(\left|x\right|\right)}^{6}}} \cdot \left|x\right|\right)\right|\]
\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(\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}{\frac{21}{{\left(\left|x\right|\right)}^{6}}} \cdot \left|x\right|\right)\right|
double f(double x) {
        double r143138 = 1.0;
        double r143139 = atan2(1.0, 0.0);
        double r143140 = sqrt(r143139);
        double r143141 = r143138 / r143140;
        double r143142 = 2.0;
        double r143143 = x;
        double r143144 = fabs(r143143);
        double r143145 = r143142 * r143144;
        double r143146 = 3.0;
        double r143147 = r143142 / r143146;
        double r143148 = r143144 * r143144;
        double r143149 = r143148 * r143144;
        double r143150 = r143147 * r143149;
        double r143151 = r143145 + r143150;
        double r143152 = 5.0;
        double r143153 = r143138 / r143152;
        double r143154 = r143149 * r143144;
        double r143155 = r143154 * r143144;
        double r143156 = r143153 * r143155;
        double r143157 = r143151 + r143156;
        double r143158 = 21.0;
        double r143159 = r143138 / r143158;
        double r143160 = r143155 * r143144;
        double r143161 = r143160 * r143144;
        double r143162 = r143159 * r143161;
        double r143163 = r143157 + r143162;
        double r143164 = r143141 * r143163;
        double r143165 = fabs(r143164);
        return r143165;
}
↓
double f(double x) {
        double r143166 = 1.0;
        double r143167 = atan2(1.0, 0.0);
        double r143168 = sqrt(r143167);
        double r143169 = r143166 / r143168;
        double r143170 = 2.0;
        double r143171 = x;
        double r143172 = fabs(r143171);
        double r143173 = r143170 * r143172;
        double r143174 = 3.0;
        double r143175 = r143170 / r143174;
        double r143176 = r143172 * r143172;
        double r143177 = r143176 * r143172;
        double r143178 = r143175 * r143177;
        double r143179 = r143173 + r143178;
        double r143180 = 5.0;
        double r143181 = r143166 / r143180;
        double r143182 = r143177 * r143172;
        double r143183 = r143182 * r143172;
        double r143184 = r143181 * r143183;
        double r143185 = r143179 + r143184;
        double r143186 = 21.0;
        double r143187 = 6.0;
        double r143188 = pow(r143172, r143187);
        double r143189 = r143186 / r143188;
        double r143190 = r143166 / r143189;
        double r143191 = r143190 * r143172;
        double r143192 = r143185 + r143191;
        double r143193 = r143169 * r143192;
        double r143194 = fabs(r143193);
        return r143194;
}

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 associate-*r*0.2

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019306 
(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)))))))