Average Error: 2.8 → 1.4
Time: 9.5s
Precision: binary64
\[x \ge 0.5\]
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[\frac{\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot {\left(\frac{1}{\left|x\right|}\right)}^{3}\right) \cdot \left(\frac{3}{4} + \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{15}{8}\right) + \frac{1}{\frac{2}{{\left(\frac{1}{\left|x\right|}\right)}^{3}}}\right) + \frac{1}{\left|x\right|}\right) \cdot 1}{\frac{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}{\sqrt{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)}} \cdot \sqrt{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)}}}}\]
\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)
↓
\frac{\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot {\left(\frac{1}{\left|x\right|}\right)}^{3}\right) \cdot \left(\frac{3}{4} + \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{15}{8}\right) + \frac{1}{\frac{2}{{\left(\frac{1}{\left|x\right|}\right)}^{3}}}\right) + \frac{1}{\left|x\right|}\right) \cdot 1}{\frac{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}{\sqrt{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)}} \cdot \sqrt{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)}}}}
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) (((double) (((double) (((double) (((double) (((double) (((double) (((double) (1.0 / ((double) fabs(x)))) * ((double) (1.0 / ((double) fabs(x)))))) * ((double) pow(((double) (1.0 / ((double) fabs(x)))), 3.0)))) * ((double) (((double) (3.0 / 4.0)) + ((double) (((double) (((double) (1.0 / ((double) fabs(x)))) * ((double) (1.0 / ((double) fabs(x)))))) * ((double) (15.0 / 8.0)))))))) + ((double) (1.0 / ((double) (2.0 / ((double) pow(((double) (1.0 / ((double) fabs(x)))), 3.0)))))))) + ((double) (1.0 / ((double) fabs(x)))))) * 1.0)) / ((double) (((double) (((double) sqrt(((double) sqrt(((double) M_PI))))) * ((double) sqrt(((double) sqrt(((double) M_PI))))))) / ((double) (((double) sqrt(((double) pow(((double) exp(((double) fabs(x)))), ((double) fabs(x)))))) * ((double) sqrt(((double) pow(((double) exp(((double) fabs(x)))), ((double) fabs(x))))))))))));
}

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. Simplified2.8

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

  4. Applied add-log-exp2.8

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

  5. Applied exp-to-pow1.4

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

  7. Applied add-sqr-sqrt1.4

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

  8. Applied sqrt-prod1.4

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

  10. Applied add-sqr-sqrt1.4

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

  11. Final simplification1.4

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

Reproduce

herbie shell --seed 2020171 
(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)))))))