Average Error: 2.8 → 1.2
Time: 21.9s
Precision: binary64
\[x \geq 0.5\]
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
↓
\[\begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ {\left(e^{x}\right)}^{x} \cdot \left(\frac{t_0}{{x}^{3}} \cdot \left(\frac{1.875}{{x}^{4}} + 0.5\right) + \frac{t_0}{x} \cdot \left(1 + \frac{0.75}{{x}^{4}}\right)\right) \end{array} \]
\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)
↓
\begin{array}{l}
t_0 := \sqrt{\frac{1}{\pi}}\\
{\left(e^{x}\right)}^{x} \cdot \left(\frac{t_0}{{x}^{3}} \cdot \left(\frac{1.875}{{x}^{4}} + 0.5\right) + \frac{t_0}{x} \cdot \left(1 + \frac{0.75}{{x}^{4}}\right)\right)
\end{array}

(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
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (*
    (pow (exp x) x)
    (+
     (* (/ t_0 (pow x 3.0)) (+ (/ 1.875 (pow x 4.0)) 0.5))
     (* (/ t_0 x) (+ 1.0 (/ 0.75 (pow x 4.0))))))))
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) {
	double t_0 = sqrt(1.0 / ((double) M_PI));
	return pow(exp(x), x) * (((t_0 / pow(x, 3.0)) * ((1.875 / pow(x, 4.0)) + 0.5)) + ((t_0 / x) * (1.0 + (0.75 / pow(x, 4.0)))));
}

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

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

  3. Applied add-log-exp_binary642.7

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

  4. Applied exp-to-pow_binary641.3

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

  5. Applied add-log-exp_binary641.3

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

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

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

  7. Applied div-inv_binary641.2

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

  8. Applied times-frac_binary641.2

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

  9. Applied associate-*l*_binary641.2

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

  10. Taylor expanded in x around 0 1.2

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{1} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \frac{1}{\left|x\right|} + \left(0.5 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \frac{1}{\left|x\right| \cdot {x}^{2}}\right) + \left(1.875 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \frac{1}{\left|x\right| \cdot {x}^{6}}\right) + 0.75 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \frac{1}{\left|x\right| \cdot {x}^{4}}\right)\right)\right)\right)} \]

  11. Simplified1.2

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{1} \cdot \color{blue}{\left(\frac{\sqrt{\frac{1}{\pi}}}{{x}^{3}} \cdot \left(\frac{1.875}{{x}^{4}} + 0.5\right) + \frac{\sqrt{\frac{1}{\pi}}}{x} \cdot \left(\frac{0.75}{{x}^{4}} + 1\right)\right)} \]

  12. Final simplification1.2

    \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\frac{\sqrt{\frac{1}{\pi}}}{{x}^{3}} \cdot \left(\frac{1.875}{{x}^{4}} + 0.5\right) + \frac{\sqrt{\frac{1}{\pi}}}{x} \cdot \left(1 + \frac{0.75}{{x}^{4}}\right)\right) \]

Reproduce

herbie shell --seed 2022088 
(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)))))))