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

Average Error: 0.2 → 0.2

Time: 57.5s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r5532411 = 1.0;
        double r5532412 = atan2(1.0, 0.0);
        double r5532413 = sqrt(r5532412);
        double r5532414 = r5532411 / r5532413;
        double r5532415 = 2.0;
        double r5532416 = x;
        double r5532417 = fabs(r5532416);
        double r5532418 = r5532415 * r5532417;
        double r5532419 = 3.0;
        double r5532420 = r5532415 / r5532419;
        double r5532421 = r5532417 * r5532417;
        double r5532422 = r5532421 * r5532417;
        double r5532423 = r5532420 * r5532422;
        double r5532424 = r5532418 + r5532423;
        double r5532425 = 5.0;
        double r5532426 = r5532411 / r5532425;
        double r5532427 = r5532422 * r5532417;
        double r5532428 = r5532427 * r5532417;
        double r5532429 = r5532426 * r5532428;
        double r5532430 = r5532424 + r5532429;
        double r5532431 = 21.0;
        double r5532432 = r5532411 / r5532431;
        double r5532433 = r5532428 * r5532417;
        double r5532434 = r5532433 * r5532417;
        double r5532435 = r5532432 * r5532434;
        double r5532436 = r5532430 + r5532435;
        double r5532437 = r5532414 * r5532436;
        double r5532438 = fabs(r5532437);
        return r5532438;
}

↓

double f(double x) {
        double r5532439 = 1.0;
        double r5532440 = atan2(1.0, 0.0);
        double r5532441 = r5532439 / r5532440;
        double r5532442 = sqrt(r5532441);
        double r5532443 = 0.2;
        double r5532444 = x;
        double r5532445 = fabs(r5532444);
        double r5532446 = 5.0;
        double r5532447 = pow(r5532445, r5532446);
        double r5532448 = 2.0;
        double r5532449 = 0.6666666666666666;
        double r5532450 = r5532445 * r5532445;
        double r5532451 = r5532450 * r5532445;
        double r5532452 = 21.0;
        double r5532453 = cbrt(r5532452);
        double r5532454 = r5532453 * r5532453;
        double r5532455 = r5532439 / r5532454;
        double r5532456 = 7.0;
        double r5532457 = pow(r5532445, r5532456);
        double r5532458 = r5532457 / r5532453;
        double r5532459 = r5532455 * r5532458;
        double r5532460 = fma(r5532449, r5532451, r5532459);
        double r5532461 = fma(r5532445, r5532448, r5532460);
        double r5532462 = fma(r5532443, r5532447, r5532461);
        double r5532463 = r5532442 * r5532462;
        double r5532464 = fabs(r5532463);
        return r5532464;
}

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 add-sqr-sqrt 0.2

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

4. Applied *-un-lft-identity 0.2

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

5. Applied times-frac 0.2

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

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 0.2

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

8. Simplified 0.2

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

10. Applied add-cube-cbrt 0.3

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

11. Applied *-un-lft-identity 0.3

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

12. Applied unpow-prod-down 0.3

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

13. Applied times-frac 0.2

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

14. Simplified 0.2

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

15. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(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)))))))