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

Average Error: 2.8 → 1.3

Time: 17.0s

Precision: binary64

Cost: 47040

\[x \geq 0.5\]

Math

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

↓

\[\begin{array}{l} t_0 := {\left(e^{x}\right)}^{\left(2 \cdot \frac{x}{4}\right)}\\ \frac{t_0}{x \cdot \frac{\sqrt{\pi}}{t_0}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \end{array} \]

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
 (let* ((t_0 (pow (exp x) (* 2.0 (/ x 4.0)))))
   (*
    (/ t_0 (* x (/ (sqrt PI) t_0)))
    (+ (/ (+ 0.5 (/ 0.75 (* x x))) (* x x)) (+ 1.0 (/ 1.875 (pow x 6.0)))))))

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) {
	double t_0 = pow(exp(x), (2.0 * (x / 4.0)));
	return (t_0 / (x * (sqrt(((double) M_PI)) / t_0))) * (((0.5 + (0.75 / (x * x))) / (x * x)) + (1.0 + (1.875 / pow(x, 6.0))));
}

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) {
	double t_0 = Math.pow(Math.exp(x), (2.0 * (x / 4.0)));
	return (t_0 / (x * (Math.sqrt(Math.PI) / t_0))) * (((0.5 + (0.75 / (x * x))) / (x * x)) + (1.0 + (1.875 / Math.pow(x, 6.0))));
}

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):
	t_0 = math.pow(math.exp(x), (2.0 * (x / 4.0)))
	return (t_0 / (x * (math.sqrt(math.pi) / t_0))) * (((0.5 + (0.75 / (x * x))) / (x * x)) + (1.0 + (1.875 / math.pow(x, 6.0))))

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)
	t_0 = exp(x) ^ Float64(2.0 * Float64(x / 4.0))
	return Float64(Float64(t_0 / Float64(x * Float64(sqrt(pi) / t_0))) * Float64(Float64(Float64(0.5 + Float64(0.75 / Float64(x * x))) / Float64(x * x)) + Float64(1.0 + Float64(1.875 / (x ^ 6.0)))))
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)
	t_0 = exp(x) ^ (2.0 * (x / 4.0));
	tmp = (t_0 / (x * (sqrt(pi) / t_0))) * (((0.5 + (0.75 / (x * x))) / (x * x)) + (1.0 + (1.875 / (x ^ 6.0))));
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_] := Block[{t$95$0 = N[Power[N[Exp[x], $MachinePrecision], N[(2.0 * N[(x / 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(t$95$0 / N[(x * N[(N[Sqrt[Pi], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.5 + N[(0.75 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 1.8

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

4. Applied egg-rr 1.5

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

5. Simplified 1.4

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

   [Start] 1.5	\[ \left(\frac{{\left(e^{x}\right)}^{\left(\frac{x}{4}\right)}}{\frac{\sqrt{\pi}}{{\left(\sqrt{e^{x}}\right)}^{x}} \cdot x} \cdot {\left(e^{x}\right)}^{\left(\frac{x}{4}\right)}\right) \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]
   associate-*l/ [=>] 1.5	\[ \color{blue}{\frac{{\left(e^{x}\right)}^{\left(\frac{x}{4}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{x}{4}\right)}}{\frac{\sqrt{\pi}}{{\left(\sqrt{e^{x}}\right)}^{x}} \cdot x}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]
   pow-sqr [=>] 1.4	\[ \frac{\color{blue}{{\left(e^{x}\right)}^{\left(2 \cdot \frac{x}{4}\right)}}}{\frac{\sqrt{\pi}}{{\left(\sqrt{e^{x}}\right)}^{x}} \cdot x} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]
   *-commutative [=>] 1.4	\[ \frac{{\left(e^{x}\right)}^{\left(2 \cdot \frac{x}{4}\right)}}{\color{blue}{x \cdot \frac{\sqrt{\pi}}{{\left(\sqrt{e^{x}}\right)}^{x}}}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

6. Applied egg-rr 1.3

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

7. Simplified 1.3

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

   [Start] 1.3	\[ \frac{{\left(e^{x}\right)}^{\left(2 \cdot \frac{x}{4}\right)}}{x \cdot \frac{\sqrt{\pi}}{{\left(e^{x}\right)}^{\left(\frac{x}{4}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{x}{4}\right)}}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]
   pow-sqr [=>] 1.3	\[ \frac{{\left(e^{x}\right)}^{\left(2 \cdot \frac{x}{4}\right)}}{x \cdot \frac{\sqrt{\pi}}{\color{blue}{{\left(e^{x}\right)}^{\left(2 \cdot \frac{x}{4}\right)}}}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

8. Final simplification 1.3

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

Alternatives

Alternative 1
Error	1.3
Cost	40128

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

Alternative 2
Error	1.3
Cost	33664

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

Alternative 3
Error	1.3
Cost	33664

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

Alternative 4
Error	1.3
Cost	33664

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

Alternative 5
Error	1.3
Cost	33536

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

Alternative 6
Error	1.3
Cost	33536

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

Alternative 7
Error	2.7
Cost	27200

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

Alternative 8
Error	2.7
Cost	27200

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

Alternative 9
Error	2.7
Cost	27200

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

Alternative 10
Error	44.7
Cost	26560

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

Alternative 11
Error	48.3
Cost	26240

\[\sqrt{\frac{1}{\pi}} \cdot e^{x \cdot x - \mathsf{log1p}\left(x + -1\right)} \]

Alternative 12
Error	48.3
Cost	26112

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

Alternative 13
Error	48.3
Cost	26048

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

Alternative 14
Error	48.3
Cost	26048

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

Alternative 15
Error	48.3
Cost	25920

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

Alternative 16
Error	48.3
Cost	19840

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

Alternative 17
Error	48.3
Cost	19712

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

Alternative 18
Error	53.0
Cost	19584

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

Alternative 19
Error	56.5
Cost	13568

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

Alternative 20
Error	56.8
Cost	13184

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

Reproduce

herbie shell --seed 2023025 
(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)))))))