Jmat.Real.erfi, branch x greater than or equal to 5

Average Accuracy: 95.6% → 98.1%

Time: 15.3s

Precision: binary64

Cost: 40000

\[x \geq 0.5\]

Math

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

↓

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

FPCore

(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) (/ (pow PI -0.5) x))
  (+
   1.0
   (+ (/ 1.875 (pow (fabs x) 6.0)) (/ (+ 0.5 (/ 0.75 (* x x))) (* x x))))))

C

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) * (pow(((double) M_PI), -0.5) / x)) * (1.0 + ((1.875 / pow(fabs(x), 6.0)) + ((0.5 + (0.75 / (x * x))) / (x * x))));
}

Java

public static double code(double x) {
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((Math.abs(x) * Math.abs(x)))) * ((((1.0 / Math.abs(x)) + ((1.0 / 2.0) * (((1.0 / Math.abs(x)) * (1.0 / Math.abs(x))) * (1.0 / Math.abs(x))))) + ((3.0 / 4.0) * (((((1.0 / Math.abs(x)) * (1.0 / Math.abs(x))) * (1.0 / Math.abs(x))) * (1.0 / Math.abs(x))) * (1.0 / Math.abs(x))))) + ((15.0 / 8.0) * (((((((1.0 / Math.abs(x)) * (1.0 / Math.abs(x))) * (1.0 / Math.abs(x))) * (1.0 / Math.abs(x))) * (1.0 / Math.abs(x))) * (1.0 / Math.abs(x))) * (1.0 / Math.abs(x)))));
}

↓

public static double code(double x) {
	return (Math.pow(Math.exp(x), x) * (Math.pow(Math.PI, -0.5) / x)) * (1.0 + ((1.875 / Math.pow(Math.abs(x), 6.0)) + ((0.5 + (0.75 / (x * x))) / (x * x))));
}

Python

def code(x):
	return ((1.0 / math.sqrt(math.pi)) * math.exp((math.fabs(x) * math.fabs(x)))) * ((((1.0 / math.fabs(x)) + ((1.0 / 2.0) * (((1.0 / math.fabs(x)) * (1.0 / math.fabs(x))) * (1.0 / math.fabs(x))))) + ((3.0 / 4.0) * (((((1.0 / math.fabs(x)) * (1.0 / math.fabs(x))) * (1.0 / math.fabs(x))) * (1.0 / math.fabs(x))) * (1.0 / math.fabs(x))))) + ((15.0 / 8.0) * (((((((1.0 / math.fabs(x)) * (1.0 / math.fabs(x))) * (1.0 / math.fabs(x))) * (1.0 / math.fabs(x))) * (1.0 / math.fabs(x))) * (1.0 / math.fabs(x))) * (1.0 / math.fabs(x)))))

↓

def code(x):
	return (math.pow(math.exp(x), x) * (math.pow(math.pi, -0.5) / x)) * (1.0 + ((1.875 / math.pow(math.fabs(x), 6.0)) + ((0.5 + (0.75 / (x * x))) / (x * x))))

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(Float64(Float64(Float64(1.0 / abs(x)) + Float64(Float64(1.0 / 2.0) * Float64(Float64(Float64(1.0 / abs(x)) * Float64(1.0 / abs(x))) * Float64(1.0 / abs(x))))) + Float64(Float64(3.0 / 4.0) * Float64(Float64(Float64(Float64(Float64(1.0 / abs(x)) * Float64(1.0 / abs(x))) * Float64(1.0 / abs(x))) * Float64(1.0 / abs(x))) * Float64(1.0 / abs(x))))) + Float64(Float64(15.0 / 8.0) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 / abs(x)) * Float64(1.0 / abs(x))) * Float64(1.0 / abs(x))) * Float64(1.0 / abs(x))) * Float64(1.0 / abs(x))) * Float64(1.0 / abs(x))) * Float64(1.0 / abs(x))))))
end

↓

function code(x)
	return Float64(Float64((exp(x) ^ x) * Float64((pi ^ -0.5) / x)) * Float64(1.0 + Float64(Float64(1.875 / (abs(x) ^ 6.0)) + Float64(Float64(0.5 + Float64(0.75 / Float64(x * x))) / Float64(x * x)))))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / sqrt(pi)) * exp((abs(x) * abs(x)))) * ((((1.0 / abs(x)) + ((1.0 / 2.0) * (((1.0 / abs(x)) * (1.0 / abs(x))) * (1.0 / abs(x))))) + ((3.0 / 4.0) * (((((1.0 / abs(x)) * (1.0 / abs(x))) * (1.0 / abs(x))) * (1.0 / abs(x))) * (1.0 / abs(x))))) + ((15.0 / 8.0) * (((((((1.0 / abs(x)) * (1.0 / abs(x))) * (1.0 / abs(x))) * (1.0 / abs(x))) * (1.0 / abs(x))) * (1.0 / abs(x))) * (1.0 / abs(x)))));
end

↓

function tmp = code(x)
	tmp = ((exp(x) ^ x) * ((pi ^ -0.5) / x)) * (1.0 + ((1.875 / (abs(x) ^ 6.0)) + ((0.5 + (0.75 / (x * x))) / (x * x))));
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 2.0), $MachinePrecision] * N[(N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * N[(N[(N[(N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] * N[(N[Power[Pi, -0.5], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(1.875 / N[Power[N[Abs[x], $MachinePrecision], 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 + N[(0.75 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.6%

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

2. Simplified 95.8%

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

   [Start] 95.6

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

   associate-+l+ [=>] 95.6

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

3. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{\left(0 + \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}}\right)} \cdot \left(1 + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]
   Proof

   [Start] 95.8	\[ \frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]
   add-log-exp [=>] 42.2	\[ \color{blue}{\log \left(e^{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\pi}}}\right)} \cdot \left(1 + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]
   *-un-lft-identity [=>] 42.2	\[ \log \color{blue}{\left(1 \cdot e^{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\pi}}}\right)} \cdot \left(1 + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]
   log-prod [=>] 42.2	\[ \color{blue}{\left(\log 1 + \log \left(e^{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\pi}}}\right)\right)} \cdot \left(1 + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]
   metadata-eval [=>] 42.2	\[ \left(\color{blue}{0} + \log \left(e^{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\pi}}}\right)\right) \cdot \left(1 + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]
   add-log-exp [<=] 95.8	\[ \left(0 + \color{blue}{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\pi}}}\right) \cdot \left(1 + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]
   associate-/r* [=>] 95.8	\[ \left(0 + \color{blue}{\frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}}}\right) \cdot \left(1 + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]
   add-sqr-sqrt [=>] 95.7	\[ \left(0 + \frac{\frac{e^{x \cdot x}}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}}{\sqrt{\pi}}\right) \cdot \left(1 + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]
   fabs-sqr [=>] 95.7	\[ \left(0 + \frac{\frac{e^{x \cdot x}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}}{\sqrt{\pi}}\right) \cdot \left(1 + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]
   add-sqr-sqrt [<=] 95.8	\[ \left(0 + \frac{\frac{e^{x \cdot x}}{\color{blue}{x}}}{\sqrt{\pi}}\right) \cdot \left(1 + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]
   associate-/r* [<=] 95.8	\[ \left(0 + \color{blue}{\frac{e^{x \cdot x}}{x \cdot \sqrt{\pi}}}\right) \cdot \left(1 + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]
   exp-prod [=>] 98.0	\[ \left(0 + \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{x \cdot \sqrt{\pi}}\right) \cdot \left(1 + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

4. Simplified 98.0%

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

   [Start] 98.0	\[ \left(0 + \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}}\right) \cdot \left(1 + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]
   +-lft-identity [=>] 98.0	\[ \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}}} \cdot \left(1 + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

5. Applied egg-rr 98.1%

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

   [Start] 98.0	\[ \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]
   div-inv [=>] 98.0	\[ \color{blue}{\left({\left(e^{x}\right)}^{x} \cdot \frac{1}{x \cdot \sqrt{\pi}}\right)} \cdot \left(1 + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]
   *-commutative [=>] 98.0	\[ \left({\left(e^{x}\right)}^{x} \cdot \frac{1}{\color{blue}{\sqrt{\pi} \cdot x}}\right) \cdot \left(1 + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]
   associate-/r* [=>] 98.1	\[ \left({\left(e^{x}\right)}^{x} \cdot \color{blue}{\frac{\frac{1}{\sqrt{\pi}}}{x}}\right) \cdot \left(1 + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]
   pow1/2 [=>] 98.1	\[ \left({\left(e^{x}\right)}^{x} \cdot \frac{\frac{1}{\color{blue}{{\pi}^{0.5}}}}{x}\right) \cdot \left(1 + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]
   pow-flip [=>] 98.1	\[ \left({\left(e^{x}\right)}^{x} \cdot \frac{\color{blue}{{\pi}^{\left(-0.5\right)}}}{x}\right) \cdot \left(1 + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]
   metadata-eval [=>] 98.1	\[ \left({\left(e^{x}\right)}^{x} \cdot \frac{{\pi}^{\color{blue}{-0.5}}}{x}\right) \cdot \left(1 + \left(\frac{1.875}{{\left(\left|x\right|\right)}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

6. Final simplification 98.1%

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

Alternatives

Alternative 1
Accuracy	98.1%
Cost	33664

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

Alternative 2
Accuracy	98.0%
Cost	33536

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

Alternative 3
Accuracy	95.8%
Cost	27328

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

Alternative 4
Accuracy	95.8%
Cost	27328

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

Alternative 5
Accuracy	24.6%
Cost	26048

\[\frac{{\left(e^{x}\right)}^{x}}{x} \cdot \sqrt{\frac{1}{\pi}} \]

Alternative 6
Accuracy	24.6%
Cost	19712

\[\frac{e^{x \cdot x}}{x} \cdot \sqrt{\frac{1}{\pi}} \]

Alternative 7
Accuracy	11.2%
Cost	19520

\[\frac{1.875}{\sqrt{\pi}} \cdot {x}^{-7} \]

Reproduce

herbie shell --seed 2023151 
(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)))))))