Average Error: 2.8 → 1.2
Time: 7.6s
Precision: binary64
\[x \geq 0.5\]
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[\frac{1}{\sqrt{\pi}} \cdot \left({\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)} \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{1}{2} \cdot \left({\left({\left(\sqrt{\frac{1}{\left|x\right|}}\right)}^{\left({\left(\sqrt[3]{3}\right)}^{2}\right)}\right)}^{\left(\sqrt[3]{3}\right)} \cdot {\left({\left(\sqrt{\frac{1}{\left|x\right|}}\right)}^{\left({\left(\sqrt[3]{3}\right)}^{2}\right)}\right)}^{\left(\sqrt[3]{3}\right)}\right) + \left(\frac{3}{4} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{5} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right)\right)\right)\]
\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{1}{\sqrt{\pi}} \cdot \left({\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)} \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{1}{2} \cdot \left({\left({\left(\sqrt{\frac{1}{\left|x\right|}}\right)}^{\left({\left(\sqrt[3]{3}\right)}^{2}\right)}\right)}^{\left(\sqrt[3]{3}\right)} \cdot {\left({\left(\sqrt{\frac{1}{\left|x\right|}}\right)}^{\left({\left(\sqrt[3]{3}\right)}^{2}\right)}\right)}^{\left(\sqrt[3]{3}\right)}\right) + \left(\frac{3}{4} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{5} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right)\right)\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
 (*
  (/ 1.0 (sqrt PI))
  (*
   (pow (exp (fabs x)) (fabs x))
   (+
    (/ 1.0 (fabs x))
    (+
     (*
      (/ 1.0 2.0)
      (*
       (pow (pow (sqrt (/ 1.0 (fabs x))) (pow (cbrt 3.0) 2.0)) (cbrt 3.0))
       (pow (pow (sqrt (/ 1.0 (fabs x))) (pow (cbrt 3.0) 2.0)) (cbrt 3.0))))
     (+
      (* (/ 3.0 4.0) (pow (/ 1.0 (fabs x)) 5.0))
      (* (/ 15.0 8.0) (/ 1.0 (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) ((1.0 / ((double) sqrt(((double) M_PI)))) * ((double) (((double) pow(((double) exp(((double) fabs(x)))), ((double) fabs(x)))) * ((double) ((1.0 / ((double) fabs(x))) + ((double) (((double) ((1.0 / 2.0) * ((double) (((double) pow(((double) pow(((double) sqrt((1.0 / ((double) fabs(x))))), ((double) pow(((double) cbrt(3.0)), 2.0)))), ((double) cbrt(3.0)))) * ((double) pow(((double) pow(((double) sqrt((1.0 / ((double) fabs(x))))), ((double) pow(((double) cbrt(3.0)), 2.0)))), ((double) cbrt(3.0)))))))) + ((double) (((double) ((3.0 / 4.0) * ((double) pow((1.0 / ((double) fabs(x))), 5.0)))) + ((double) ((15.0 / 8.0) * (1.0 / ((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. Simplified1.4

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

  3. Taylor expanded around 0 1.2

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

  5. Applied add-cube-cbrt1.2

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

  6. Applied pow-unpow1.2

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

  8. Applied add-sqr-sqrt1.2

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

  9. Applied unpow-prod-down1.2

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

  10. Applied unpow-prod-down1.2

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

  11. Simplified1.2

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

  12. Simplified1.2

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

  13. Final simplification1.2

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

Reproduce

herbie shell --seed 2020198 
(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)))))))