Average Error: 0.2 → 0.2
Time: 56.9s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\left({\left(\left|x\right|\right)}^{7} \cdot \frac{1}{21} + \left(\frac{1}{5} \cdot \left(\left|x\right| \cdot \left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(\left|x\right| \cdot 2 + \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \frac{2}{3}\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right|\]
\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(\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)\right) \cdot \frac{1}{\sqrt{\pi}}\right|
double f(double x) {
        double r5859185 = 1.0;
        double r5859186 = atan2(1.0, 0.0);
        double r5859187 = sqrt(r5859186);
        double r5859188 = r5859185 / r5859187;
        double r5859189 = 2.0;
        double r5859190 = x;
        double r5859191 = fabs(r5859190);
        double r5859192 = r5859189 * r5859191;
        double r5859193 = 3.0;
        double r5859194 = r5859189 / r5859193;
        double r5859195 = r5859191 * r5859191;
        double r5859196 = r5859195 * r5859191;
        double r5859197 = r5859194 * r5859196;
        double r5859198 = r5859192 + r5859197;
        double r5859199 = 5.0;
        double r5859200 = r5859185 / r5859199;
        double r5859201 = r5859196 * r5859191;
        double r5859202 = r5859201 * r5859191;
        double r5859203 = r5859200 * r5859202;
        double r5859204 = r5859198 + r5859203;
        double r5859205 = 21.0;
        double r5859206 = r5859185 / r5859205;
        double r5859207 = r5859202 * r5859191;
        double r5859208 = r5859207 * r5859191;
        double r5859209 = r5859206 * r5859208;
        double r5859210 = r5859204 + r5859209;
        double r5859211 = r5859188 * r5859210;
        double r5859212 = fabs(r5859211);
        return r5859212;
}
↓
double f(double x) {
        double r5859213 = x;
        double r5859214 = fabs(r5859213);
        double r5859215 = 7.0;
        double r5859216 = pow(r5859214, r5859215);
        double r5859217 = 0.047619047619047616;
        double r5859218 = r5859216 * r5859217;
        double r5859219 = 0.2;
        double r5859220 = r5859214 * r5859214;
        double r5859221 = r5859220 * r5859214;
        double r5859222 = r5859221 * r5859214;
        double r5859223 = r5859214 * r5859222;
        double r5859224 = r5859219 * r5859223;
        double r5859225 = 2.0;
        double r5859226 = r5859214 * r5859225;
        double r5859227 = 0.6666666666666666;
        double r5859228 = r5859221 * r5859227;
        double r5859229 = r5859226 + r5859228;
        double r5859230 = r5859224 + r5859229;
        double r5859231 = r5859218 + r5859230;
        double r5859232 = 1.0;
        double r5859233 = atan2(1.0, 0.0);
        double r5859234 = sqrt(r5859233);
        double r5859235 = r5859232 / r5859234;
        double r5859236 = r5859231 * r5859235;
        double r5859237 = fabs(r5859236);
        return r5859237;
}

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

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019130 
(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)))))))