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

Average Error: 2.8 → 1.2

Time: 16.2s

Precision: binary64

Cost: 46272

\[x \geq 0.5\]

Math

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

↓

\[\left(\left(\left(\frac{1.875}{{x}^{7}} + \frac{0.75}{{x}^{5}}\right) + \left(\frac{0.5}{{x}^{3}} + \frac{1}{x}\right)\right) \cdot {\left(e^{x}\right)}^{x}\right) \cdot {\pi}^{-0.5} \]

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
 (*
  (*
   (+
    (+ (/ 1.875 (pow x 7.0)) (/ 0.75 (pow x 5.0)))
    (+ (/ 0.5 (pow x 3.0)) (/ 1.0 x)))
   (pow (exp x) x))
  (pow PI -0.5)))

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 ((((1.875 / pow(x, 7.0)) + (0.75 / pow(x, 5.0))) + ((0.5 / pow(x, 3.0)) + (1.0 / x))) * pow(exp(x), x)) * pow(((double) M_PI), -0.5);
}

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 ((((1.875 / Math.pow(x, 7.0)) + (0.75 / Math.pow(x, 5.0))) + ((0.5 / Math.pow(x, 3.0)) + (1.0 / x))) * Math.pow(Math.exp(x), x)) * Math.pow(Math.PI, -0.5);
}

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 ((((1.875 / math.pow(x, 7.0)) + (0.75 / math.pow(x, 5.0))) + ((0.5 / math.pow(x, 3.0)) + (1.0 / x))) * math.pow(math.exp(x), x)) * math.pow(math.pi, -0.5)

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(Float64(Float64(Float64(1.875 / (x ^ 7.0)) + Float64(0.75 / (x ^ 5.0))) + Float64(Float64(0.5 / (x ^ 3.0)) + Float64(1.0 / x))) * (exp(x) ^ x)) * (pi ^ -0.5))
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 = ((((1.875 / (x ^ 7.0)) + (0.75 / (x ^ 5.0))) + ((0.5 / (x ^ 3.0)) + (1.0 / x))) * (exp(x) ^ x)) * (pi ^ -0.5);
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[(N[(N[(1.875 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(0.75 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision] * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]

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. Simplified 1.3

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

   [Start] 2.8	\[ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
   distribute-lft-in [=>] 2.8	\[ \color{blue}{\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)} \]
   +-commutative [=>] 2.8	\[ \color{blue}{\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)} \]

3. Applied egg-rr 1.2

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\left|x\right| \cdot \color{blue}{{\left({\pi}^{0.25}\right)}^{2}}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

4. Applied egg-rr 1.3

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

5. Applied egg-rr 1.2

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

6. Taylor expanded in x around 0 1.2

   \[\leadsto \left(\color{blue}{\left(0.75 \cdot \frac{1}{{x}^{5}} + \left(\frac{1}{x} + \left(0.5 \cdot \frac{1}{{x}^{3}} + 1.875 \cdot \frac{1}{{x}^{7}}\right)\right)\right)} \cdot {\left(e^{x}\right)}^{x}\right) \cdot {\pi}^{-0.5} \]

7. Simplified 1.2

   \[\leadsto \left(\color{blue}{\left(\left(\frac{1.875}{{x}^{7}} + \frac{0.75}{{x}^{5}}\right) + \left(\frac{0.5}{{x}^{3}} + \frac{1}{x}\right)\right)} \cdot {\left(e^{x}\right)}^{x}\right) \cdot {\pi}^{-0.5} \]
   Proof

   [Start] 1.2	\[ \left(\left(0.75 \cdot \frac{1}{{x}^{5}} + \left(\frac{1}{x} + \left(0.5 \cdot \frac{1}{{x}^{3}} + 1.875 \cdot \frac{1}{{x}^{7}}\right)\right)\right) \cdot {\left(e^{x}\right)}^{x}\right) \cdot {\pi}^{-0.5} \]
   associate-+r+ [=>] 1.2	\[ \left(\left(0.75 \cdot \frac{1}{{x}^{5}} + \color{blue}{\left(\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right) + 1.875 \cdot \frac{1}{{x}^{7}}\right)}\right) \cdot {\left(e^{x}\right)}^{x}\right) \cdot {\pi}^{-0.5} \]
   +-commutative [<=] 1.2	\[ \left(\left(0.75 \cdot \frac{1}{{x}^{5}} + \left(\color{blue}{\left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)} + 1.875 \cdot \frac{1}{{x}^{7}}\right)\right) \cdot {\left(e^{x}\right)}^{x}\right) \cdot {\pi}^{-0.5} \]
   +-commutative [=>] 1.2	\[ \left(\left(0.75 \cdot \frac{1}{{x}^{5}} + \color{blue}{\left(1.875 \cdot \frac{1}{{x}^{7}} + \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)\right)}\right) \cdot {\left(e^{x}\right)}^{x}\right) \cdot {\pi}^{-0.5} \]
   associate-+l+ [<=] 1.2	\[ \left(\color{blue}{\left(\left(0.75 \cdot \frac{1}{{x}^{5}} + 1.875 \cdot \frac{1}{{x}^{7}}\right) + \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)\right)} \cdot {\left(e^{x}\right)}^{x}\right) \cdot {\pi}^{-0.5} \]
   +-commutative [=>] 1.2	\[ \left(\left(\color{blue}{\left(1.875 \cdot \frac{1}{{x}^{7}} + 0.75 \cdot \frac{1}{{x}^{5}}\right)} + \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)\right) \cdot {\left(e^{x}\right)}^{x}\right) \cdot {\pi}^{-0.5} \]
   associate-*r/ [=>] 1.2	\[ \left(\left(\left(\color{blue}{\frac{1.875 \cdot 1}{{x}^{7}}} + 0.75 \cdot \frac{1}{{x}^{5}}\right) + \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)\right) \cdot {\left(e^{x}\right)}^{x}\right) \cdot {\pi}^{-0.5} \]
   metadata-eval [=>] 1.2	\[ \left(\left(\left(\frac{\color{blue}{1.875}}{{x}^{7}} + 0.75 \cdot \frac{1}{{x}^{5}}\right) + \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)\right) \cdot {\left(e^{x}\right)}^{x}\right) \cdot {\pi}^{-0.5} \]
   associate-*r/ [=>] 1.2	\[ \left(\left(\left(\frac{1.875}{{x}^{7}} + \color{blue}{\frac{0.75 \cdot 1}{{x}^{5}}}\right) + \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)\right) \cdot {\left(e^{x}\right)}^{x}\right) \cdot {\pi}^{-0.5} \]
   metadata-eval [=>] 1.2	\[ \left(\left(\left(\frac{1.875}{{x}^{7}} + \frac{\color{blue}{0.75}}{{x}^{5}}\right) + \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)\right) \cdot {\left(e^{x}\right)}^{x}\right) \cdot {\pi}^{-0.5} \]
   associate-*r/ [=>] 1.2	\[ \left(\left(\left(\frac{1.875}{{x}^{7}} + \frac{0.75}{{x}^{5}}\right) + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{3}}} + \frac{1}{x}\right)\right) \cdot {\left(e^{x}\right)}^{x}\right) \cdot {\pi}^{-0.5} \]
   metadata-eval [=>] 1.2	\[ \left(\left(\left(\frac{1.875}{{x}^{7}} + \frac{0.75}{{x}^{5}}\right) + \left(\frac{\color{blue}{0.5}}{{x}^{3}} + \frac{1}{x}\right)\right) \cdot {\left(e^{x}\right)}^{x}\right) \cdot {\pi}^{-0.5} \]

8. Final simplification 1.2

   \[\leadsto \left(\left(\left(\frac{1.875}{{x}^{7}} + \frac{0.75}{{x}^{5}}\right) + \left(\frac{0.5}{{x}^{3}} + \frac{1}{x}\right)\right) \cdot {\left(e^{x}\right)}^{x}\right) \cdot {\pi}^{-0.5} \]

Alternatives

Alternative 1
Error	1.2
Cost	40064

\[\frac{{\left(e^{x}\right)}^{x}}{x \cdot {\left({\pi}^{0.25}\right)}^{2}} \cdot \left(\frac{0.5 + \frac{\frac{0.75}{x}}{x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

Alternative 2
Error	1.2
Cost	39936

\[{\pi}^{-0.5} \cdot \left({\left(e^{x}\right)}^{x} \cdot \frac{\left(\frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}\right) + \left(1 + 1.875 \cdot {x}^{-6}\right)}{x}\right) \]

Alternative 3
Error	1.2
Cost	33728

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

Alternative 4
Error	1.3
Cost	33536

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

Alternative 5
Error	1.3
Cost	33536

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

Alternative 6
Error	1.3
Cost	33536

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

Alternative 7
Error	1.3
Cost	33536

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

Alternative 8
Error	2.7
Cost	27264

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

Alternative 9
Error	2.7
Cost	27200

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

Alternative 10
Error	2.6
Cost	27200

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

Alternative 11
Error	44.7
Cost	26496

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

Alternative 12
Error	48.3
Cost	26112

\[{\pi}^{-0.5} \cdot \left(\frac{1}{x} \cdot {\left(e^{x}\right)}^{x}\right) \]

Alternative 13
Error	48.3
Cost	25984

\[{\pi}^{-0.5} \cdot \frac{{\left(e^{x}\right)}^{x}}{x} \]

Alternative 14
Error	48.3
Cost	19648

\[{\pi}^{-0.5} \cdot \frac{e^{x \cdot x}}{x} \]

Alternative 15
Error	56.8
Cost	19584

\[\frac{1.875}{{x}^{7}} \cdot {\pi}^{-0.5} \]

Alternative 16
Error	57.1
Cost	13312

\[\frac{\frac{0.5}{x}}{x \cdot \left(x \cdot \sqrt{\pi}\right)} \]

Reproduce

herbie shell --seed 2023083 
(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)))))))