Average Error: 0.2 → 0.1
Time: 44.1s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\left({\left(\left|x\right|\right)}^{5} \cdot \frac{1}{5} + \left|x\right| \cdot \left(2 + \frac{2}{3} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right)\right) \cdot \sqrt{\frac{1}{\pi}} + \frac{\sqrt{\frac{1}{\pi}} \cdot {\left(\left|x\right|\right)}^{7}}{21}\right|\]
\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)}^{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) \cdot \sqrt{\frac{1}{\pi}} + \frac{\sqrt{\frac{1}{\pi}} \cdot {\left(\left|x\right|\right)}^{7}}{21}\right|
double f(double x) {
        double r7890227 = 1.0;
        double r7890228 = atan2(1.0, 0.0);
        double r7890229 = sqrt(r7890228);
        double r7890230 = r7890227 / r7890229;
        double r7890231 = 2.0;
        double r7890232 = x;
        double r7890233 = fabs(r7890232);
        double r7890234 = r7890231 * r7890233;
        double r7890235 = 3.0;
        double r7890236 = r7890231 / r7890235;
        double r7890237 = r7890233 * r7890233;
        double r7890238 = r7890237 * r7890233;
        double r7890239 = r7890236 * r7890238;
        double r7890240 = r7890234 + r7890239;
        double r7890241 = 5.0;
        double r7890242 = r7890227 / r7890241;
        double r7890243 = r7890238 * r7890233;
        double r7890244 = r7890243 * r7890233;
        double r7890245 = r7890242 * r7890244;
        double r7890246 = r7890240 + r7890245;
        double r7890247 = 21.0;
        double r7890248 = r7890227 / r7890247;
        double r7890249 = r7890244 * r7890233;
        double r7890250 = r7890249 * r7890233;
        double r7890251 = r7890248 * r7890250;
        double r7890252 = r7890246 + r7890251;
        double r7890253 = r7890230 * r7890252;
        double r7890254 = fabs(r7890253);
        return r7890254;
}
↓
double f(double x) {
        double r7890255 = x;
        double r7890256 = fabs(r7890255);
        double r7890257 = 5.0;
        double r7890258 = pow(r7890256, r7890257);
        double r7890259 = 0.2;
        double r7890260 = r7890258 * r7890259;
        double r7890261 = 2.0;
        double r7890262 = 0.6666666666666666;
        double r7890263 = r7890256 * r7890256;
        double r7890264 = r7890262 * r7890263;
        double r7890265 = r7890261 + r7890264;
        double r7890266 = r7890256 * r7890265;
        double r7890267 = r7890260 + r7890266;
        double r7890268 = 1.0;
        double r7890269 = atan2(1.0, 0.0);
        double r7890270 = r7890268 / r7890269;
        double r7890271 = sqrt(r7890270);
        double r7890272 = r7890267 * r7890271;
        double r7890273 = 7.0;
        double r7890274 = pow(r7890256, r7890273);
        double r7890275 = r7890271 * r7890274;
        double r7890276 = 21.0;
        double r7890277 = r7890275 / r7890276;
        double r7890278 = r7890272 + r7890277;
        double r7890279 = fabs(r7890278);
        return r7890279;
}

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

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

  4. Applied *-un-lft-identity0.2

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

  5. Applied times-frac0.2

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

  6. Applied associate-*l*0.2

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

  7. Taylor expanded around 0 0.1

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

  8. Simplified0.1

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

  10. Applied add-sqr-sqrt0.1

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

  11. Applied associate-/r*0.2

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

  13. Applied distribute-rgt-in0.2

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

  14. Simplified0.1

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

  15. Final simplification0.1

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

Reproduce

herbie shell --seed 2019164 
(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)))))))