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

Average Error: 0.2 → 0.2

Time: 5.7s

Precision: binary64

Math

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

↓

\[\left|{\left(\frac{\frac{1}{\sqrt{\pi}}}{1} \cdot \left(\left(\left(2 \cdot \left|x\right| + 0.66666666666666663 \cdot {\left(\left|x\right|\right)}^{3}\right) + \left({\left(\left|x\right|\right)}^{3} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \frac{1}{5}\right) + {\left(\left|x\right|\right)}^{6} \cdot \left(\left|x\right| \cdot \frac{1}{21}\right)\right)\right)}^{1}\right|\]

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) pow(((double) (((double) (((double) (1.0 / ((double) sqrt(((double) M_PI))))) / 1.0)) * ((double) (((double) (((double) (((double) (2.0 * ((double) fabs(x)))) + ((double) (0.6666666666666666 * ((double) pow(((double) fabs(x)), 3.0)))))) + ((double) (((double) (((double) pow(((double) fabs(x)), 3.0)) * ((double) (((double) fabs(x)) * ((double) fabs(x)))))) * ((double) (1.0 / 5.0)))))) + ((double) (((double) pow(((double) fabs(x)), 6.0)) * ((double) (((double) fabs(x)) * ((double) (1.0 / 21.0)))))))))), 1.0))));
}

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. Taylor expanded around 0 0.2

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \color{blue}{0.66666666666666663 \cdot {\left(\left|x\right|\right)}^{3}}\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
3. Using strategy rm

4. Applied add-sqr-sqrt 0.2

   \[\leadsto \left|\frac{1}{\sqrt{\color{blue}{\sqrt{\pi} \cdot \sqrt{\pi}}}} \cdot \left(\left(\left(2 \cdot \left|x\right| + 0.66666666666666663 \cdot {\left(\left|x\right|\right)}^{3}\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]

5. Applied sqrt-prod 0.2

   \[\leadsto \left|\frac{1}{\color{blue}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}} \cdot \left(\left(\left(2 \cdot \left|x\right| + 0.66666666666666663 \cdot {\left(\left|x\right|\right)}^{3}\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]

6. Applied add-sqr-sqrt 0.2

   \[\leadsto \left|\frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}} \cdot \left(\left(\left(2 \cdot \left|x\right| + 0.66666666666666663 \cdot {\left(\left|x\right|\right)}^{3}\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]

7. Applied times-frac 0.2

   \[\leadsto \left|\color{blue}{\left(\frac{\sqrt{1}}{\sqrt{\sqrt{\pi}}} \cdot \frac{\sqrt{1}}{\sqrt{\sqrt{\pi}}}\right)} \cdot \left(\left(\left(2 \cdot \left|x\right| + 0.66666666666666663 \cdot {\left(\left|x\right|\right)}^{3}\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]

8. Applied associate-*l* 0.5

   \[\leadsto \left|\color{blue}{\frac{\sqrt{1}}{\sqrt{\sqrt{\pi}}} \cdot \left(\frac{\sqrt{1}}{\sqrt{\sqrt{\pi}}} \cdot \left(\left(\left(2 \cdot \left|x\right| + 0.66666666666666663 \cdot {\left(\left|x\right|\right)}^{3}\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right)}\right|\]

9. Simplified 0.4

   \[\leadsto \left|\frac{\sqrt{1}}{\sqrt{\sqrt{\pi}}} \cdot \color{blue}{\frac{\left(\left(\left(\left({\left(\left|x\right|\right)}^{3} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \frac{1}{21} + \left(2 \cdot \left|x\right| + 0.66666666666666663 \cdot {\left(\left|x\right|\right)}^{3}\right)\right) + \left({\left(\left|x\right|\right)}^{3} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \frac{1}{5}\right) \cdot \sqrt{1}}{\sqrt{\sqrt{\pi}}}}\right|\]
10. Using strategy rm

11. Applied pow1 0.4

    \[\leadsto \left|\frac{\sqrt{1}}{\sqrt{\sqrt{\pi}}} \cdot \color{blue}{{\left(\frac{\left(\left(\left(\left({\left(\left|x\right|\right)}^{3} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \frac{1}{21} + \left(2 \cdot \left|x\right| + 0.66666666666666663 \cdot {\left(\left|x\right|\right)}^{3}\right)\right) + \left({\left(\left|x\right|\right)}^{3} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \frac{1}{5}\right) \cdot \sqrt{1}}{\sqrt{\sqrt{\pi}}}\right)}^{1}}\right|\]

12. Applied pow1 0.4

    \[\leadsto \left|\color{blue}{{\left(\frac{\sqrt{1}}{\sqrt{\sqrt{\pi}}}\right)}^{1}} \cdot {\left(\frac{\left(\left(\left(\left({\left(\left|x\right|\right)}^{3} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \frac{1}{21} + \left(2 \cdot \left|x\right| + 0.66666666666666663 \cdot {\left(\left|x\right|\right)}^{3}\right)\right) + \left({\left(\left|x\right|\right)}^{3} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \frac{1}{5}\right) \cdot \sqrt{1}}{\sqrt{\sqrt{\pi}}}\right)}^{1}\right|\]

13. Applied pow-prod-down 0.4

    \[\leadsto \left|\color{blue}{{\left(\frac{\sqrt{1}}{\sqrt{\sqrt{\pi}}} \cdot \frac{\left(\left(\left(\left({\left(\left|x\right|\right)}^{3} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \frac{1}{21} + \left(2 \cdot \left|x\right| + 0.66666666666666663 \cdot {\left(\left|x\right|\right)}^{3}\right)\right) + \left({\left(\left|x\right|\right)}^{3} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \frac{1}{5}\right) \cdot \sqrt{1}}{\sqrt{\sqrt{\pi}}}\right)}^{1}}\right|\]

14. Simplified 0.2

    \[\leadsto \left|{\color{blue}{\left(\frac{\frac{1}{\sqrt{\pi}}}{1} \cdot \left(\left(\left(2 \cdot \left|x\right| + 0.66666666666666663 \cdot {\left(\left|x\right|\right)}^{3}\right) + \left({\left(\left|x\right|\right)}^{3} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \frac{1}{5}\right) + {\left(\left|x\right|\right)}^{6} \cdot \left(\left|x\right| \cdot \frac{1}{21}\right)\right)\right)}}^{1}\right|\]

15. Final simplification 0.2

    \[\leadsto \left|{\left(\frac{\frac{1}{\sqrt{\pi}}}{1} \cdot \left(\left(\left(2 \cdot \left|x\right| + 0.66666666666666663 \cdot {\left(\left|x\right|\right)}^{3}\right) + \left({\left(\left|x\right|\right)}^{3} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \frac{1}{5}\right) + {\left(\left|x\right|\right)}^{6} \cdot \left(\left|x\right| \cdot \frac{1}{21}\right)\right)\right)}^{1}\right|\]

Reproduce

herbie shell --seed 2020155 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  :precision binary64
  (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)))))))