Average Error: 2.8 → 1.3
Time: 9.2s
Precision: binary64
\[x \geq 0.5\]
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[\left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot \sqrt{\frac{\frac{1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)}{\left|\sqrt[3]{\pi}\right|}}{\sqrt{\sqrt[3]{\pi}} \cdot \left|x\right|}}\right) \cdot \left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot \sqrt{\frac{\frac{1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)}{\left|\sqrt{\sqrt[3]{\pi}} \cdot \sqrt{\sqrt[3]{\pi}}\right|}}{\sqrt{\sqrt[3]{\pi}} \cdot \left|x\right|}}\right)\]
\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)
↓
\left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot \sqrt{\frac{\frac{1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)}{\left|\sqrt[3]{\pi}\right|}}{\sqrt{\sqrt[3]{\pi}} \cdot \left|x\right|}}\right) \cdot \left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot \sqrt{\frac{\frac{1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)}{\left|\sqrt{\sqrt[3]{\pi}} \cdot \sqrt{\sqrt[3]{\pi}}\right|}}{\sqrt{\sqrt[3]{\pi}} \cdot \left|x\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
 (*
  (*
   (pow (exp x) (/ x 2.0))
   (sqrt
    (/
     (/
      (+ 1.0 (+ (/ 1.875 (pow x 6.0)) (/ (+ 0.5 (/ 0.75 (* x x))) (* x x))))
      (fabs (cbrt PI)))
     (* (sqrt (cbrt PI)) (fabs x)))))
  (*
   (pow (exp x) (/ x 2.0))
   (sqrt
    (/
     (/
      (+ 1.0 (+ (/ 1.875 (pow x 6.0)) (/ (+ 0.5 (/ 0.75 (* x x))) (* x x))))
      (fabs (* (sqrt (cbrt PI)) (sqrt (cbrt PI)))))
     (* (sqrt (cbrt PI)) (fabs x)))))))
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(x), (x / 2.0)) * sqrt(((1.0 + ((1.875 / pow(x, 6.0)) + ((0.5 + (0.75 / (x * x))) / (x * x)))) / fabs(cbrt((double) M_PI))) / (sqrt(cbrt((double) M_PI)) * fabs(x)))) * (pow(exp(x), (x / 2.0)) * sqrt(((1.0 + ((1.875 / pow(x, 6.0)) + ((0.5 + (0.75 / (x * x))) / (x * x)))) / fabs(sqrt(cbrt((double) M_PI)) * sqrt(cbrt((double) M_PI)))) / (sqrt(cbrt((double) M_PI)) * 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. Simplified2.7

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

  4. Applied add-cube-cbrt_binary64_18182.9

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

  5. Applied sqrt-prod_binary64_17992.7

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

  6. Applied *-un-lft-identity_binary64_17832.7

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

  7. Applied times-frac_binary64_17892.7

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

  8. Applied associate-/l*_binary64_17282.7

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

  9. Simplified1.3

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

  10. Simplified1.3

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

  12. Applied add-sqr-sqrt_binary64_18051.4

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

  13. Applied sqr-pow_binary64_17551.4

    \[\leadsto \color{blue}{\left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}\right)} \cdot \left(\sqrt{\frac{\frac{1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)}{\left|\sqrt[3]{\pi}\right|}}{\sqrt{\sqrt[3]{\pi}} \cdot \left|x\right|}} \cdot \sqrt{\frac{\frac{1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)}{\left|\sqrt[3]{\pi}\right|}}{\sqrt{\sqrt[3]{\pi}} \cdot \left|x\right|}}\right)\]

  14. Applied unswap-sqr_binary64_17511.3

    \[\leadsto \color{blue}{\left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot \sqrt{\frac{\frac{1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)}{\left|\sqrt[3]{\pi}\right|}}{\sqrt{\sqrt[3]{\pi}} \cdot \left|x\right|}}\right) \cdot \left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot \sqrt{\frac{\frac{1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)}{\left|\sqrt[3]{\pi}\right|}}{\sqrt{\sqrt[3]{\pi}} \cdot \left|x\right|}}\right)}\]
  15. Using strategy rm

  16. Applied add-sqr-sqrt_binary64_18051.3

    \[\leadsto \left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot \sqrt{\frac{\frac{1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)}{\left|\sqrt[3]{\pi}\right|}}{\sqrt{\sqrt[3]{\pi}} \cdot \left|x\right|}}\right) \cdot \left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot \sqrt{\frac{\frac{1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)}{\left|\color{blue}{\sqrt{\sqrt[3]{\pi}} \cdot \sqrt{\sqrt[3]{\pi}}}\right|}}{\sqrt{\sqrt[3]{\pi}} \cdot \left|x\right|}}\right)\]

  17. Final simplification1.3

    \[\leadsto \left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot \sqrt{\frac{\frac{1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)}{\left|\sqrt[3]{\pi}\right|}}{\sqrt{\sqrt[3]{\pi}} \cdot \left|x\right|}}\right) \cdot \left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot \sqrt{\frac{\frac{1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)}{\left|\sqrt{\sqrt[3]{\pi}} \cdot \sqrt{\sqrt[3]{\pi}}\right|}}{\sqrt{\sqrt[3]{\pi}} \cdot \left|x\right|}}\right)\]

Reproduce

herbie shell --seed 2020342 
(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)))))))