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

Average Error: 0.3 → 0.3

Time: 6.3s

Precision: 64

Math

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

↓

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

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

Error

Bits error versus x

Derivation

1. Initial program 0.3

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

3. Applied div-inv 0.3

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

4. Applied associate-*l* 0.3

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

5. Simplified 0.2

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

7. Applied associate-/l* 0.1

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

9. Applied add-cube-cbrt 1.1

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

10. Applied sqrt-prod 0.1

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

11. Applied *-un-lft-identity 0.1

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

12. Applied times-frac 0.1

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

13. Applied associate-*l* 0.3

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

14. Simplified 0.3

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

15. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020102 +o rules:numerics
(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)))))))