Jmat.Real.erfi, branch x less than or equal to 0.5

Average Error: 0.2 → 0.2

Time: 28.2s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r84327 = 1.0;
        double r84328 = atan2(1.0, 0.0);
        double r84329 = sqrt(r84328);
        double r84330 = r84327 / r84329;
        double r84331 = 2.0;
        double r84332 = x;
        double r84333 = fabs(r84332);
        double r84334 = r84331 * r84333;
        double r84335 = 3.0;
        double r84336 = r84331 / r84335;
        double r84337 = r84333 * r84333;
        double r84338 = r84337 * r84333;
        double r84339 = r84336 * r84338;
        double r84340 = r84334 + r84339;
        double r84341 = 5.0;
        double r84342 = r84327 / r84341;
        double r84343 = r84338 * r84333;
        double r84344 = r84343 * r84333;
        double r84345 = r84342 * r84344;
        double r84346 = r84340 + r84345;
        double r84347 = 21.0;
        double r84348 = r84327 / r84347;
        double r84349 = r84344 * r84333;
        double r84350 = r84349 * r84333;
        double r84351 = r84348 * r84350;
        double r84352 = r84346 + r84351;
        double r84353 = r84330 * r84352;
        double r84354 = fabs(r84353);
        return r84354;
}

↓

double f(double x) {
        double r84355 = 1.0;
        double r84356 = sqrt(r84355);
        double r84357 = atan2(1.0, 0.0);
        double r84358 = sqrt(r84357);
        double r84359 = r84356 / r84358;
        double r84360 = x;
        double r84361 = fabs(r84360);
        double r84362 = 3.0;
        double r84363 = pow(r84361, r84362);
        double r84364 = 2.0;
        double r84365 = 3.0;
        double r84366 = r84364 / r84365;
        double r84367 = r84363 * r84366;
        double r84368 = r84364 * r84361;
        double r84369 = r84367 + r84368;
        double r84370 = 5.0;
        double r84371 = pow(r84361, r84370);
        double r84372 = r84355 * r84371;
        double r84373 = 5.0;
        double r84374 = r84372 / r84373;
        double r84375 = 7.0;
        double r84376 = pow(r84361, r84375);
        double r84377 = 21.0;
        double r84378 = r84377 / r84355;
        double r84379 = r84376 / r84378;
        double r84380 = r84374 + r84379;
        double r84381 = r84369 + r84380;
        double r84382 = r84359 * r84381;
        double r84383 = r84356 * r84382;
        double r84384 = fabs(r84383);
        return r84384;
}

Error

Bits error versus x

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 pow1 0.2

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

4. Applied pow1 0.2

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

5. Applied pow1 0.2

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

6. Applied pow-prod-up 0.2

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

7. Applied pow-prod-up 0.2

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

8. Simplified 0.2

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

10. Applied *-un-lft-identity 0.2

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

11. Applied sqrt-prod 0.2

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

12. Applied add-sqr-sqrt 0.2

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

13. Applied times-frac 0.2

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

14. Applied associate-*l* 0.2

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

15. Simplified 0.2

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

16. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019325 
(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)))))))