Average Error: 2.8 → 1.3
Time: 6.6s
Precision: binary64
\[x \geq 0.5\]
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[\frac{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)}}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}} \cdot \left(\left(e^{\log \left(\frac{1}{\left|x\right|} + \frac{0.5}{{\left(\left|x\right|\right)}^{3}}\right)} + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right) + \frac{1.875}{{\left(\left|x\right|\right)}^{7}}\right)\]
\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(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)}}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}} \cdot \left(\left(e^{\log \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)
(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)) (* (sqrt (sqrt PI)) (sqrt (sqrt PI))))
  (+
   (+
    (exp (log (+ (/ 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)))))
double code(double x) {
	return ((double) (((double) ((1.0 / ((double) sqrt(((double) M_PI)))) * ((double) exp(((double) (((double) fabs(x)) * ((double) fabs(x)))))))) * ((double) (((double) (((double) ((1.0 / ((double) fabs(x))) + ((double) ((1.0 / 2.0) * ((double) (((double) ((1.0 / ((double) fabs(x))) * (1.0 / ((double) fabs(x))))) * (1.0 / ((double) fabs(x))))))))) + ((double) ((3.0 / 4.0) * ((double) (((double) (((double) (((double) ((1.0 / ((double) fabs(x))) * (1.0 / ((double) fabs(x))))) * (1.0 / ((double) fabs(x))))) * (1.0 / ((double) fabs(x))))) * (1.0 / ((double) fabs(x))))))))) + ((double) ((15.0 / 8.0) * ((double) (((double) (((double) (((double) (((double) (((double) ((1.0 / ((double) fabs(x))) * (1.0 / ((double) fabs(x))))) * (1.0 / ((double) fabs(x))))) * (1.0 / ((double) fabs(x))))) * (1.0 / ((double) fabs(x))))) * (1.0 / ((double) fabs(x))))) * (1.0 / ((double) fabs(x)))))))))));
}
↓
double code(double x) {
	return ((double) ((((double) pow(((double) exp(((double) fabs(x)))), ((double) fabs(x)))) / ((double) (((double) sqrt(((double) sqrt(((double) M_PI))))) * ((double) sqrt(((double) sqrt(((double) M_PI)))))))) * ((double) (((double) (((double) exp(((double) log(((double) ((1.0 / ((double) fabs(x))) + (0.5 / ((double) pow(((double) fabs(x)), 3.0))))))))) + (0.75 / ((double) pow(((double) fabs(x)), 5.0))))) + (1.875 / ((double) pow(((double) fabs(x)), 7.0)))))));
}

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 add-exp-log_binary641.3

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

  9. Applied add-sqr-sqrt_binary641.3

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

  10. Applied sqrt-prod_binary641.3

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

  11. Final simplification1.3

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

Reproduce

herbie shell --seed 2020219 
(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)))))))