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

Average Error: 0.2 → 0.2

Time: 6.6s

Precision: 64

Math

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

↓

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

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(pow((((fabs(x) * (2.0 + (((2.0 / 3.0) * (fabs(x) * fabs(x))) + (((1.0 / 5.0) * pow(fabs(x), 3.0)) * fabs(x))))) + ((1.0 * pow(fabs(x), 7.0)) / 21.0)) * (1.0 / sqrt(((double) M_PI)))), 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. Using strategy rm

3. Applied pow1 0.2

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

4. Applied pow1 0.2

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

5. Applied pow-prod-down 0.2

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

6. Simplified 0.6

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

8. Applied *-un-lft-identity 0.6

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

9. Applied sqrt-prod 0.6

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

10. Applied times-frac 0.2

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

11. Simplified 0.2

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

12. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020075 
(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)))))))