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

Average Error: 0.2 → 0.2

Time: 8.0s

Precision: 64

Math

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

↓

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

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_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))))));
}

↓

double code(double x) {
	return fabs(((((((2.0 * pow(fabs(x), 3.0)) / 3.0) + (2.0 * fabs(x))) + ((pow(fabs(x), 3.0) * (fabs(x) * fabs(x))) * (1.0 / 5.0))) + ((1.0 * pow(fabs(x), 7.0)) / 21.0)) * (1.0 / 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 add-sqr-sqrt 0.2

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

4. Applied associate-*l* 0.5

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

6. Applied sqrt-div 0.5

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

7. Applied associate-*l/ 0.4

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

8. Applied sqrt-div 0.4

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

9. Applied frac-times 0.4

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

10. Simplified 0.4

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

11. Simplified 0.6

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

13. Applied *-un-lft-identity 0.6

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

14. Applied sqrt-prod 0.6

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

15. Applied times-frac 0.2

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

16. Simplified 0.2

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

17. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020100 
(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)))))))