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

Average Error: 0.2 → 0.3

Time: 4.4s

Precision: binary64

\[x \le 0.5\]

Math

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

↓

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

C

double code(double x) {
	return ((double) fabs(((double) (((double) (1.0 / ((double) sqrt(((double) M_PI))))) * ((double) (((double) (((double) (((double) (2.0 * ((double) fabs(x)))) + ((double) (((double) (2.0 / 3.0)) * ((double) (((double) (((double) fabs(x)) * ((double) fabs(x)))) * ((double) fabs(x)))))))) + ((double) (((double) (1.0 / 5.0)) * ((double) (((double) (((double) (((double) (((double) fabs(x)) * ((double) fabs(x)))) * ((double) fabs(x)))) * ((double) fabs(x)))) * ((double) fabs(x)))))))) + ((double) (((double) (1.0 / 21.0)) * ((double) (((double) (((double) (((double) (((double) (((double) (((double) fabs(x)) * ((double) fabs(x)))) * ((double) fabs(x)))) * ((double) fabs(x)))) * ((double) fabs(x)))) * ((double) fabs(x)))) * ((double) fabs(x))))))))))));
}

↓

double code(double x) {
	return ((double) fabs(((double) (((double) (1.0 / ((double) sqrt(((double) sqrt(((double) M_PI))))))) * ((double) (1.0 * ((double) (((double) (((double) (2.0 * ((double) (((double) pow(((double) fabs(x)), 3.0)) / 3.0)))) + ((double) (((double) (2.0 * ((double) fabs(x)))) + ((double) (((double) (1.0 * ((double) (((double) pow(((double) fabs(x)), 5.0)) / 5.0)))) + ((double) (((double) pow(((double) fabs(x)), 3.5)) * ((double) (1.0 * ((double) (((double) pow(((double) fabs(x)), 3.5)) / 21.0)))))))))))) / ((double) sqrt(((double) sqrt(((double) M_PI)))))))))))));
}

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 associate-*r* 0.1

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

4. Simplified 0.1

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

6. Applied add-sqr-sqrt 0.1

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

7. Applied sqrt-prod 0.1

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

8. Applied *-un-lft-identity 0.1

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

9. Applied times-frac 0.1

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

10. Applied associate-*l* 0.5

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

11. Simplified 0.3

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

13. Applied sqr-pow 0.3

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

14. Applied associate-*r* 0.3

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

15. Simplified 0.3

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

16. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020184 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  :precision binary64
  :pre (<= x 0.5)
  (fabs (* (/ 1.0 (sqrt PI)) (+ (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1.0 5.0) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1.0 21.0) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))))