Average Error: 0.2 → 0.2
Time: 11.7s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{6}\right) \cdot \left|x\right|\right) \cdot \frac{1}{\sqrt{\pi}}\right|\]
\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(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}{21} \cdot {\left(\left|x\right|\right)}^{6}\right) \cdot \left|x\right|\right) \cdot \frac{1}{\sqrt{\pi}}\right|
double f(double x) {
        double r189299 = 1.0;
        double r189300 = atan2(1.0, 0.0);
        double r189301 = sqrt(r189300);
        double r189302 = r189299 / r189301;
        double r189303 = 2.0;
        double r189304 = x;
        double r189305 = fabs(r189304);
        double r189306 = r189303 * r189305;
        double r189307 = 3.0;
        double r189308 = r189303 / r189307;
        double r189309 = r189305 * r189305;
        double r189310 = r189309 * r189305;
        double r189311 = r189308 * r189310;
        double r189312 = r189306 + r189311;
        double r189313 = 5.0;
        double r189314 = r189299 / r189313;
        double r189315 = r189310 * r189305;
        double r189316 = r189315 * r189305;
        double r189317 = r189314 * r189316;
        double r189318 = r189312 + r189317;
        double r189319 = 21.0;
        double r189320 = r189299 / r189319;
        double r189321 = r189316 * r189305;
        double r189322 = r189321 * r189305;
        double r189323 = r189320 * r189322;
        double r189324 = r189318 + r189323;
        double r189325 = r189302 * r189324;
        double r189326 = fabs(r189325);
        return r189326;
}
↓
double f(double x) {
        double r189327 = 2.0;
        double r189328 = x;
        double r189329 = fabs(r189328);
        double r189330 = r189327 * r189329;
        double r189331 = 3.0;
        double r189332 = r189327 / r189331;
        double r189333 = r189329 * r189329;
        double r189334 = r189333 * r189329;
        double r189335 = r189332 * r189334;
        double r189336 = r189330 + r189335;
        double r189337 = 1.0;
        double r189338 = 5.0;
        double r189339 = r189337 / r189338;
        double r189340 = r189334 * r189329;
        double r189341 = r189340 * r189329;
        double r189342 = r189339 * r189341;
        double r189343 = r189336 + r189342;
        double r189344 = 21.0;
        double r189345 = r189337 / r189344;
        double r189346 = 6.0;
        double r189347 = pow(r189329, r189346);
        double r189348 = r189345 * r189347;
        double r189349 = r189348 * r189329;
        double r189350 = r189343 + r189349;
        double r189351 = atan2(1.0, 0.0);
        double r189352 = sqrt(r189351);
        double r189353 = r189337 / r189352;
        double r189354 = r189350 * r189353;
        double r189355 = fabs(r189354);
        return r189355;
}

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 *-commutative0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2020039 
(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)))))))