Average Error: 2.8 → 1.3
Time: 7.9s
Precision: binary64
\[x \geq 0.5\]
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[{\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} + \frac{0.5}{{\left(\left|x\right|\right)}^{3}}\right) + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right) + \frac{1.875}{{\left(\left|x\right|\right)}^{7}}\right) \cdot \frac{{\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)}}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}\right)\]
\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)
↓
{\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} + \frac{0.5}{{\left(\left|x\right|\right)}^{3}}\right) + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right) + \frac{1.875}{{\left(\left|x\right|\right)}^{7}}\right) \cdot \frac{{\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)}}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}\right)
(FPCore (x)
 :precision binary64
 (*
  (* (/ 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)))))))
↓
(FPCore (x)
 :precision binary64
 (*
  (pow (exp (fabs x)) (/ (fabs x) 2.0))
  (*
   (+
    (+
     (+ (/ 1.0 (fabs x)) (/ 0.5 (pow (fabs x) 3.0)))
     (/ 0.75 (pow (fabs x) 5.0)))
    (/ 1.875 (pow (fabs x) 7.0)))
   (/
    (pow (exp (fabs x)) (/ (fabs x) 2.0))
    (* (sqrt (sqrt PI)) (sqrt (sqrt PI)))))))
double code(double x) {
	return ((1.0 / sqrt((double) M_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)))));
}
↓
double code(double x) {
	return pow(exp(fabs(x)), (fabs(x) / 2.0)) * (((((1.0 / fabs(x)) + (0.5 / pow(fabs(x), 3.0))) + (0.75 / pow(fabs(x), 5.0))) + (1.875 / pow(fabs(x), 7.0))) * (pow(exp(fabs(x)), (fabs(x) / 2.0)) / (sqrt(sqrt((double) M_PI)) * sqrt(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. Simplified2.7

    \[\leadsto \color{blue}{\frac{e^{{\left(\left|x\right|\right)}^{2}}}{\sqrt{\pi}} \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{0.5}{{\left(\left|x\right|\right)}^{3}}\right) + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right) + \frac{1.875}{{\left(\left|x\right|\right)}^{7}}\right)}\]
  3. Using strategy rm

  4. Applied unpow2_binary642.7

    \[\leadsto \frac{e^{\color{blue}{\left|x\right| \cdot \left|x\right|}}}{\sqrt{\pi}} \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{0.5}{{\left(\left|x\right|\right)}^{3}}\right) + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right) + \frac{1.875}{{\left(\left|x\right|\right)}^{7}}\right)\]

  5. Applied exp-prod_binary641.3

    \[\leadsto \frac{\color{blue}{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)}}}{\sqrt{\pi}} \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{0.5}{{\left(\left|x\right|\right)}^{3}}\right) + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right) + \frac{1.875}{{\left(\left|x\right|\right)}^{7}}\right)\]
  6. Using strategy rm

  7. Applied *-un-lft-identity_binary641.3

    \[\leadsto \frac{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)}}{\color{blue}{1 \cdot \sqrt{\pi}}} \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{0.5}{{\left(\left|x\right|\right)}^{3}}\right) + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right) + \frac{1.875}{{\left(\left|x\right|\right)}^{7}}\right)\]

  8. Applied sqr-pow_binary641.3

    \[\leadsto \frac{\color{blue}{{\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)} \cdot {\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)}}}{1 \cdot \sqrt{\pi}} \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{0.5}{{\left(\left|x\right|\right)}^{3}}\right) + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right) + \frac{1.875}{{\left(\left|x\right|\right)}^{7}}\right)\]

  9. Applied times-frac_binary641.3

    \[\leadsto \color{blue}{\left(\frac{{\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)}}{1} \cdot \frac{{\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)}}{\sqrt{\pi}}\right)} \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{0.5}{{\left(\left|x\right|\right)}^{3}}\right) + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right) + \frac{1.875}{{\left(\left|x\right|\right)}^{7}}\right)\]

  10. Applied associate-*l*_binary641.3

    \[\leadsto \color{blue}{\frac{{\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)}}{1} \cdot \left(\frac{{\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)}}{\sqrt{\pi}} \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{0.5}{{\left(\left|x\right|\right)}^{3}}\right) + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right) + \frac{1.875}{{\left(\left|x\right|\right)}^{7}}\right)\right)}\]

  11. Simplified1.3

    \[\leadsto \frac{{\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)}}{1} \cdot \color{blue}{\left(\left(\left(\left(\frac{1}{\left|x\right|} + \frac{0.5}{{\left(\left|x\right|\right)}^{3}}\right) + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right) + \frac{1.875}{{\left(\left|x\right|\right)}^{7}}\right) \cdot \frac{{\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)}}{\sqrt{\pi}}\right)}\]
  12. Using strategy rm

  13. Applied add-sqr-sqrt_binary641.3

    \[\leadsto \frac{{\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)}}{1} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} + \frac{0.5}{{\left(\left|x\right|\right)}^{3}}\right) + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right) + \frac{1.875}{{\left(\left|x\right|\right)}^{7}}\right) \cdot \frac{{\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)}}{\color{blue}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}}\right)\]

  14. Final simplification1.3

    \[\leadsto {\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} + \frac{0.5}{{\left(\left|x\right|\right)}^{3}}\right) + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right) + \frac{1.875}{{\left(\left|x\right|\right)}^{7}}\right) \cdot \frac{{\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)}}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}\right)\]

Reproduce

herbie shell --seed 2020233 
(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)))))))