Average Error: 2.8 → 1.2
Time: 2.7min
Precision: binary64
Cost: 65856
\[x \geq 0.5\]
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[\frac{1}{\sqrt{\sqrt{\pi}}} \cdot \left(\left(1 + \left(\log \left(e^{\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}}\right) + \frac{1.875}{{x}^{6}}\right)\right) \cdot \frac{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\sqrt{\pi}}}}{\left|x\right|}\right)\]
\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{1}{\sqrt{\sqrt{\pi}}} \cdot \left(\left(1 + \left(\log \left(e^{\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}}\right) + \frac{1.875}{{x}^{6}}\right)\right) \cdot \frac{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\sqrt{\pi}}}}{\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
 (*
  (/ 1.0 (sqrt (sqrt PI)))
  (*
   (+
    1.0
    (+ (log (exp (/ (+ 0.5 (/ 0.75 (* x x))) (* x x)))) (/ 1.875 (pow x 6.0))))
   (/ (/ (pow (exp x) x) (sqrt (sqrt 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 (1.0 / sqrt(sqrt((double) M_PI))) * ((1.0 + (log(exp((0.5 + (0.75 / (x * x))) / (x * x))) + (1.875 / pow(x, 6.0)))) * ((pow(exp(x), x) / sqrt(sqrt((double) M_PI))) / fabs(x)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error1.2
Cost53184
\[\frac{1}{\sqrt{\sqrt{\pi}}} \cdot \left(\frac{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\sqrt{\pi}}}}{\left|x\right|} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \left(0.5 + \frac{0.75}{x \cdot x}\right) \cdot \frac{1}{x \cdot x}\right)\right)\right)\]
Alternative 2
Error1.2
Cost53056
\[\frac{1}{\sqrt{\sqrt{\pi}}} \cdot \left(\frac{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\sqrt{\pi}}}}{\left|x\right|} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\frac{0.5 + \frac{0.75}{x \cdot x}}{x}}{x}\right)\right)\right)\]
Alternative 3
Error1.2
Cost53056
\[\frac{1}{\sqrt{\sqrt{\pi}}} \cdot \left(\frac{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\sqrt{\pi}}}}{\left|x\right|} \cdot \left(1 + \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right)\]
Alternative 4
Error1.3
Cost33600
\[\frac{\left(1 + \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}}}{\left|x\right|}\]
Alternative 5
Error2.7
Cost27264
\[\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)}{\left|x\right|}\]
Alternative 6
Error2.7
Cost27264
\[\left(1 + \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right) \cdot \frac{\frac{e^{x \cdot x}}{\sqrt{\pi}}}{\left|x\right|}\]
Alternative 7
Error43.8
Cost26880
\[\frac{\frac{e^{x \cdot x}}{\sqrt{\pi}}}{\left|x\right|} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right)\]
Alternative 8
Error47.5
Cost32833
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.2655903347466368:\\ \;\;\;\;\left(1 + \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right) \cdot \frac{\frac{1 + x \cdot x}{\sqrt{\pi}}}{\left|x\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \frac{{\left(e^{x}\right)}^{x}}{\left|x\right|}\\ \end{array}\]
Alternative 9
Error55.5
Cost20992
\[\left(1 + \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right) \cdot \frac{\frac{1 + x \cdot x}{\sqrt{\pi}}}{\left|x\right|}\]
Alternative 10
Error56.2
Cost20736
\[\left(1 + \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right) \cdot \frac{\frac{1}{\sqrt{\pi}}}{\left|x\right|}\]
Alternative 11
Error56.6
Cost64
\[1\]

Error

Time

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 *-un-lft-identity_binary64_21242.7

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

  5. Applied add-sqr-sqrt_binary64_21462.6

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

  6. Applied *-un-lft-identity_binary64_21242.6

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

  7. Applied times-frac_binary64_21302.6

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

  8. Applied times-frac_binary64_21302.6

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

  9. Applied associate-*l*_binary64_20652.6

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

  10. Simplified1.2

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

  12. Applied add-log-exp_binary64_21631.2

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

  13. Simplified1.2

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

  14. Final simplification1.2

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

Reproduce

herbie shell --seed 2021065 
(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)))))))