Average Error: 0.2 → 0.2
Time: 16.4s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{21} \cdot \sqrt[3]{21}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt[3]{21}} \cdot {\left(\left|x\right|\right)}^{6}\right)\right) \cdot \left|x\right|\right)\right|\]
\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) + \left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{21} \cdot \sqrt[3]{21}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt[3]{21}} \cdot {\left(\left|x\right|\right)}^{6}\right)\right) \cdot \left|x\right|\right)\right|
double f(double x) {
        double r147304 = 1.0;
        double r147305 = atan2(1.0, 0.0);
        double r147306 = sqrt(r147305);
        double r147307 = r147304 / r147306;
        double r147308 = 2.0;
        double r147309 = x;
        double r147310 = fabs(r147309);
        double r147311 = r147308 * r147310;
        double r147312 = 3.0;
        double r147313 = r147308 / r147312;
        double r147314 = r147310 * r147310;
        double r147315 = r147314 * r147310;
        double r147316 = r147313 * r147315;
        double r147317 = r147311 + r147316;
        double r147318 = 5.0;
        double r147319 = r147304 / r147318;
        double r147320 = r147315 * r147310;
        double r147321 = r147320 * r147310;
        double r147322 = r147319 * r147321;
        double r147323 = r147317 + r147322;
        double r147324 = 21.0;
        double r147325 = r147304 / r147324;
        double r147326 = r147321 * r147310;
        double r147327 = r147326 * r147310;
        double r147328 = r147325 * r147327;
        double r147329 = r147323 + r147328;
        double r147330 = r147307 * r147329;
        double r147331 = fabs(r147330);
        return r147331;
}
↓
double f(double x) {
        double r147332 = 1.0;
        double r147333 = atan2(1.0, 0.0);
        double r147334 = sqrt(r147333);
        double r147335 = r147332 / r147334;
        double r147336 = 2.0;
        double r147337 = x;
        double r147338 = fabs(r147337);
        double r147339 = r147336 * r147338;
        double r147340 = 3.0;
        double r147341 = r147336 / r147340;
        double r147342 = r147338 * r147338;
        double r147343 = r147342 * r147338;
        double r147344 = r147341 * r147343;
        double r147345 = r147339 + r147344;
        double r147346 = 5.0;
        double r147347 = r147332 / r147346;
        double r147348 = r147343 * r147338;
        double r147349 = r147348 * r147338;
        double r147350 = r147347 * r147349;
        double r147351 = r147345 + r147350;
        double r147352 = cbrt(r147332);
        double r147353 = r147352 * r147352;
        double r147354 = 21.0;
        double r147355 = cbrt(r147354);
        double r147356 = r147355 * r147355;
        double r147357 = r147353 / r147356;
        double r147358 = r147352 / r147355;
        double r147359 = 6.0;
        double r147360 = pow(r147338, r147359);
        double r147361 = r147358 * r147360;
        double r147362 = r147357 * r147361;
        double r147363 = r147362 * r147338;
        double r147364 = r147351 + r147363;
        double r147365 = r147335 * r147364;
        double r147366 = fabs(r147365);
        return r147366;
}

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

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

  6. Applied add-cube-cbrt0.2

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

  7. Applied add-cube-cbrt0.2

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

  8. Applied times-frac0.2

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

  9. Applied associate-*l*0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(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)))))))