Average Error: 2.8 → 1.2
Time: 7.7s
Precision: binary64
\[x \ge 0.5\]
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[1 \cdot \frac{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)} \cdot \left(\frac{1}{\left|x\right|} + \left(1 \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{2} + \left(3 \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{5}}{4} + 15 \cdot \frac{\frac{1}{{\left(\left|x\right|\right)}^{7}}}{8}\right)\right)\right)}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}\]
\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)
↓
1 \cdot \frac{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)} \cdot \left(\frac{1}{\left|x\right|} + \left(1 \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{2} + \left(3 \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{5}}{4} + 15 \cdot \frac{\frac{1}{{\left(\left|x\right|\right)}^{7}}}{8}\right)\right)\right)}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}
double code(double x) {
	return ((double) (((double) (((double) (1.0 / ((double) sqrt(((double) M_PI))))) * ((double) exp(((double) (((double) fabs(x)) * ((double) fabs(x)))))))) * ((double) (((double) (((double) (((double) (1.0 / ((double) fabs(x)))) + ((double) (((double) (1.0 / 2.0)) * ((double) (((double) (((double) (1.0 / ((double) fabs(x)))) * ((double) (1.0 / ((double) fabs(x)))))) * ((double) (1.0 / ((double) fabs(x)))))))))) + ((double) (((double) (3.0 / 4.0)) * ((double) (((double) (((double) (((double) (((double) (1.0 / ((double) fabs(x)))) * ((double) (1.0 / ((double) fabs(x)))))) * ((double) (1.0 / ((double) fabs(x)))))) * ((double) (1.0 / ((double) fabs(x)))))) * ((double) (1.0 / ((double) fabs(x)))))))))) + ((double) (((double) (15.0 / 8.0)) * ((double) (((double) (((double) (((double) (((double) (((double) (((double) (1.0 / ((double) fabs(x)))) * ((double) (1.0 / ((double) fabs(x)))))) * ((double) (1.0 / ((double) fabs(x)))))) * ((double) (1.0 / ((double) fabs(x)))))) * ((double) (1.0 / ((double) fabs(x)))))) * ((double) (1.0 / ((double) fabs(x)))))) * ((double) (1.0 / ((double) fabs(x))))))))))));
}
↓
double code(double x) {
	return ((double) (1.0 * ((double) (((double) (((double) pow(((double) exp(((double) fabs(x)))), ((double) fabs(x)))) * ((double) (((double) (1.0 / ((double) fabs(x)))) + ((double) (((double) (1.0 * ((double) (((double) pow(((double) (1.0 / ((double) fabs(x)))), 3.0)) / 2.0)))) + ((double) (((double) (3.0 * ((double) (((double) pow(((double) (1.0 / ((double) fabs(x)))), 5.0)) / 4.0)))) + ((double) (15.0 * ((double) (((double) (1.0 / ((double) pow(((double) fabs(x)), 7.0)))) / 8.0)))))))))))) / ((double) (((double) sqrt(((double) sqrt(((double) M_PI))))) * ((double) sqrt(((double) sqrt(((double) M_PI)))))))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 2.8

    \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]

  2. Simplified1.4

    \[\leadsto \color{blue}{1 \cdot \frac{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)} \cdot \left(\frac{1}{\left|x\right|} + \left(1 \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{2} + \left(3 \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{5}}{4} + 15 \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{7}}{8}\right)\right)\right)}{\sqrt{\pi}}}\]

  3. Taylor expanded around 0 1.3

    \[\leadsto 1 \cdot \frac{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)} \cdot \left(\frac{1}{\left|x\right|} + \left(1 \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{2} + \left(3 \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{5}}{4} + 15 \cdot \frac{\color{blue}{\frac{1}{{\left(\left|x\right|\right)}^{7}}}}{8}\right)\right)\right)}{\sqrt{\pi}}\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt1.3

    \[\leadsto 1 \cdot \frac{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)} \cdot \left(\frac{1}{\left|x\right|} + \left(1 \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{2} + \left(3 \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{5}}{4} + 15 \cdot \frac{\frac{1}{{\left(\left|x\right|\right)}^{7}}}{8}\right)\right)\right)}{\sqrt{\color{blue}{\sqrt{\pi} \cdot \sqrt{\pi}}}}\]

  6. Applied sqrt-prod1.2

    \[\leadsto 1 \cdot \frac{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)} \cdot \left(\frac{1}{\left|x\right|} + \left(1 \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{2} + \left(3 \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{5}}{4} + 15 \cdot \frac{\frac{1}{{\left(\left|x\right|\right)}^{7}}}{8}\right)\right)\right)}{\color{blue}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}}\]

  7. Final simplification1.2

    \[\leadsto 1 \cdot \frac{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)} \cdot \left(\frac{1}{\left|x\right|} + \left(1 \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{3}}{2} + \left(3 \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^{5}}{4} + 15 \cdot \frac{\frac{1}{{\left(\left|x\right|\right)}^{7}}}{8}\right)\right)\right)}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x greater than or equal to 5"
  :precision binary64
  :pre (>= x 0.5)
  (* (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x)))) (+ (+ (+ (/ 1.0 (fabs x)) (* (/ 1.0 2.0) (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 3.0 4.0) (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 15.0 8.0) (* (* (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))))