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

Percentage Accurate: 100.0% → 100.0%

Time: 9.6s

Alternatives: 2

Speedup: 3.0×

• 99.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t_0 \cdot t_0\right) \cdot t_0\\ t_2 := \left(t_1 \cdot t_0\right) \cdot t_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t_0 + \frac{1}{2} \cdot t_1\right) + \frac{3}{4} \cdot t_2\right) + \frac{15}{8} \cdot \left(\left(t_2 \cdot t_0\right) \cdot t_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x)))
        (t_1 (* (* t_0 t_0) t_0))
        (t_2 (* (* t_1 t_0) t_0)))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
    (+
     (+ (+ t_0 (* (/ 1.0 2.0) t_1)) (* (/ 3.0 4.0) t_2))
     (* (/ 15.0 8.0) (* (* t_2 t_0) t_0))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / sqrt(((double) M_PI))) * exp((fabs(x) * fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((Math.abs(x) * Math.abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	t_1 = (t_0 * t_0) * t_0
	t_2 = (t_1 * t_0) * t_0
	return ((1.0 / math.sqrt(math.pi)) * math.exp((math.fabs(x) * math.fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	t_1 = Float64(Float64(t_0 * t_0) * t_0)
	t_2 = Float64(Float64(t_1 * t_0) * t_0)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(Float64(Float64(t_0 + Float64(Float64(1.0 / 2.0) * t_1)) + Float64(Float64(3.0 / 4.0) * t_2)) + Float64(Float64(15.0 / 8.0) * Float64(Float64(t_2 * t_0) * t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	t_1 = (t_0 * t_0) * t_0;
	t_2 = (t_1 * t_0) * t_0;
	tmp = ((1.0 / sqrt(pi)) * exp((abs(x) * abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, 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[(t$95$0 + N[(N[(1.0 / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(t$95$2 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t_0 \cdot t_0\right) \cdot t_0\\ t_2 := \left(t_1 \cdot t_0\right) \cdot t_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t_0 + \frac{1}{2} \cdot t_1\right) + \frac{3}{4} \cdot t_2\right) + \frac{15}{8} \cdot \left(\left(t_2 \cdot t_0\right) \cdot t_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x)))
        (t_1 (* (* t_0 t_0) t_0))
        (t_2 (* (* t_1 t_0) t_0)))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
    (+
     (+ (+ t_0 (* (/ 1.0 2.0) t_1)) (* (/ 3.0 4.0) t_2))
     (* (/ 15.0 8.0) (* (* t_2 t_0) t_0))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / sqrt(((double) M_PI))) * exp((fabs(x) * fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((Math.abs(x) * Math.abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	t_1 = (t_0 * t_0) * t_0
	t_2 = (t_1 * t_0) * t_0
	return ((1.0 / math.sqrt(math.pi)) * math.exp((math.fabs(x) * math.fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	t_1 = Float64(Float64(t_0 * t_0) * t_0)
	t_2 = Float64(Float64(t_1 * t_0) * t_0)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(Float64(Float64(t_0 + Float64(Float64(1.0 / 2.0) * t_1)) + Float64(Float64(3.0 / 4.0) * t_2)) + Float64(Float64(15.0 / 8.0) * Float64(Float64(t_2 * t_0) * t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	t_1 = (t_0 * t_0) * t_0;
	t_2 = (t_1 * t_0) * t_0;
	tmp = ((1.0 / sqrt(pi)) * exp((abs(x) * abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, 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[(t$95$0 + N[(N[(1.0 / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(t$95$2 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Alternative 1: 100.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{x \cdot x}\right) \cdot \left(\left(\left(t_0 + 0.5 \cdot \frac{\frac{1}{x}}{x \cdot x}\right) + 0.75 \cdot {x}^{-5}\right) + 1.875 \cdot \left(t_0 \cdot \left(t_0 \cdot {\left({x}^{-0.5}\right)}^{10}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x))))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* x x)))
    (+
     (+ (+ t_0 (* 0.5 (/ (/ 1.0 x) (* x x)))) (* 0.75 (pow x -5.0)))
     (* 1.875 (* t_0 (* t_0 (pow (pow x -0.5) 10.0))))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	return ((1.0 / sqrt(((double) M_PI))) * exp((x * x))) * (((t_0 + (0.5 * ((1.0 / x) / (x * x)))) + (0.75 * pow(x, -5.0))) + (1.875 * (t_0 * (t_0 * pow(pow(x, -0.5), 10.0)))));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((x * x))) * (((t_0 + (0.5 * ((1.0 / x) / (x * x)))) + (0.75 * Math.pow(x, -5.0))) + (1.875 * (t_0 * (t_0 * Math.pow(Math.pow(x, -0.5), 10.0)))));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	return ((1.0 / math.sqrt(math.pi)) * math.exp((x * x))) * (((t_0 + (0.5 * ((1.0 / x) / (x * x)))) + (0.75 * math.pow(x, -5.0))) + (1.875 * (t_0 * (t_0 * math.pow(math.pow(x, -0.5), 10.0)))))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(x * x))) * Float64(Float64(Float64(t_0 + Float64(0.5 * Float64(Float64(1.0 / x) / Float64(x * x)))) + Float64(0.75 * (x ^ -5.0))) + Float64(1.875 * Float64(t_0 * Float64(t_0 * ((x ^ -0.5) ^ 10.0))))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	tmp = ((1.0 / sqrt(pi)) * exp((x * x))) * (((t_0 + (0.5 * ((1.0 / x) / (x * x)))) + (0.75 * (x ^ -5.0))) + (1.875 * (t_0 * (t_0 * ((x ^ -0.5) ^ 10.0)))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 + N[(0.5 * N[(N[(1.0 / x), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.75 * N[Power[x, -5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.875 * N[(t$95$0 * N[(t$95$0 * N[Power[N[Power[x, -0.5], $MachinePrecision], 10.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \color{blue}{\frac{1}{{\left(\left|x\right|\right)}^{5}}}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
3. Step-by-step derivation

   1. rem-square-sqrt 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \frac{1}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{5}}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   2. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \frac{1}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{5}}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   3. rem-square-sqrt 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \frac{1}{{\color{blue}{x}}^{5}}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   4. exp-to-pow 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \frac{1}{\color{blue}{e^{\log x \cdot 5}}}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   5. exp-neg 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \color{blue}{e^{-\log x \cdot 5}}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   6. distribute-lft-neg-out 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot e^{\color{blue}{\left(-\log x\right) \cdot 5}}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   7. *-commutative 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot e^{\color{blue}{5 \cdot \left(-\log x\right)}}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   8. neg-mul-1 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot e^{5 \cdot \color{blue}{\left(-1 \cdot \log x\right)}}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   9. associate-*r* 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot e^{\color{blue}{\left(5 \cdot -1\right) \cdot \log x}}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   10. metadata-eval 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot e^{\color{blue}{-5} \cdot \log x}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   11. log-pow 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot e^{\color{blue}{\log \left({x}^{-5}\right)}}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   12. rem-exp-log 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \color{blue}{{x}^{-5}}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

4. Simplified 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \color{blue}{{x}^{-5}}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
5. Step-by-step derivation

   1. associate-*l* 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left(\color{blue}{\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right)} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   2. pow3 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left(\left(\color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{3}} \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   3. pow2 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left(\left({\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{2}}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   4. pow-prod-up 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left(\color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{\left(3 + 2\right)}} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   5. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\color{blue}{\left(\sqrt{\frac{1}{\left|x\right|}} \cdot \sqrt{\frac{1}{\left|x\right|}}\right)}}^{\left(3 + 2\right)} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   6. pow2 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\color{blue}{\left({\left(\sqrt{\frac{1}{\left|x\right|}}\right)}^{2}\right)}}^{\left(3 + 2\right)} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   7. pow-pow 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left(\color{blue}{{\left(\sqrt{\frac{1}{\left|x\right|}}\right)}^{\left(2 \cdot \left(3 + 2\right)\right)}} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   8. inv-pow 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left(\sqrt{\color{blue}{{\left(\left|x\right|\right)}^{-1}}}\right)}^{\left(2 \cdot \left(3 + 2\right)\right)} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   9. sqrt-pow1 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\color{blue}{\left({\left(\left|x\right|\right)}^{\left(\frac{-1}{2}\right)}\right)}}^{\left(2 \cdot \left(3 + 2\right)\right)} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(\frac{-1}{2}\right)}\right)}^{\left(2 \cdot \left(3 + 2\right)\right)} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   11. fabs-sqr 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(\frac{-1}{2}\right)}\right)}^{\left(2 \cdot \left(3 + 2\right)\right)} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   12. add-sqr-sqrt 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({\color{blue}{x}}^{\left(\frac{-1}{2}\right)}\right)}^{\left(2 \cdot \left(3 + 2\right)\right)} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   13. metadata-eval 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{\color{blue}{-0.5}}\right)}^{\left(2 \cdot \left(3 + 2\right)\right)} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   14. metadata-eval 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{-0.5}\right)}^{\left(2 \cdot \color{blue}{5}\right)} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   15. metadata-eval 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{-0.5}\right)}^{\color{blue}{10}} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left(\color{blue}{{\left({x}^{-0.5}\right)}^{10}} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
7. Step-by-step derivation

   1. un-div-inv 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\color{blue}{\frac{\frac{1}{\left|x\right|}}{\left|x\right|}} \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{-0.5}\right)}^{10} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   2. frac-times 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \color{blue}{\frac{\frac{1}{\left|x\right|} \cdot 1}{\left|x\right| \cdot \left|x\right|}}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{-0.5}\right)}^{10} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   3. *-rgt-identity 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\color{blue}{\frac{1}{\left|x\right|}}}{\left|x\right| \cdot \left|x\right|}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{-0.5}\right)}^{10} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   4. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}}{\left|x\right| \cdot \left|x\right|}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{-0.5}\right)}^{10} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   5. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}}{\left|x\right| \cdot \left|x\right|}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{-0.5}\right)}^{10} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   6. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{\color{blue}{x}}}{\left|x\right| \cdot \left|x\right|}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{-0.5}\right)}^{10} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   7. pow2 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{\color{blue}{{\left(\left|x\right|\right)}^{2}}}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{-0.5}\right)}^{10} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{2}}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{-0.5}\right)}^{10} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   9. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{2}}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{-0.5}\right)}^{10} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{{\color{blue}{x}}^{2}}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{-0.5}\right)}^{10} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   11. pow2 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{\color{blue}{x \cdot x}}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{-0.5}\right)}^{10} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

8. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \color{blue}{\frac{\frac{1}{x}}{x \cdot x}}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{-0.5}\right)}^{10} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

9. Final simplification 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{x \cdot x}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{\frac{1}{x}}{x \cdot x}\right) + 0.75 \cdot {x}^{-5}\right) + 1.875 \cdot \left(\frac{1}{\left|x\right|} \cdot \left(\frac{1}{\left|x\right|} \cdot {\left({x}^{-0.5}\right)}^{10}\right)\right)\right) \]

Alternative 2: 100.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{x \cdot x}\right) \cdot \left(\left(\left(t_0 + 0.5 \cdot \frac{\frac{1}{x}}{x \cdot x}\right) + 0.75 \cdot {x}^{-5}\right) + 1.875 \cdot \left(t_0 \cdot {x}^{-6}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x))))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* x x)))
    (+
     (+ (+ t_0 (* 0.5 (/ (/ 1.0 x) (* x x)))) (* 0.75 (pow x -5.0)))
     (* 1.875 (* t_0 (pow x -6.0)))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	return ((1.0 / sqrt(((double) M_PI))) * exp((x * x))) * (((t_0 + (0.5 * ((1.0 / x) / (x * x)))) + (0.75 * pow(x, -5.0))) + (1.875 * (t_0 * pow(x, -6.0))));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((x * x))) * (((t_0 + (0.5 * ((1.0 / x) / (x * x)))) + (0.75 * Math.pow(x, -5.0))) + (1.875 * (t_0 * Math.pow(x, -6.0))));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	return ((1.0 / math.sqrt(math.pi)) * math.exp((x * x))) * (((t_0 + (0.5 * ((1.0 / x) / (x * x)))) + (0.75 * math.pow(x, -5.0))) + (1.875 * (t_0 * math.pow(x, -6.0))))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(x * x))) * Float64(Float64(Float64(t_0 + Float64(0.5 * Float64(Float64(1.0 / x) / Float64(x * x)))) + Float64(0.75 * (x ^ -5.0))) + Float64(1.875 * Float64(t_0 * (x ^ -6.0)))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	tmp = ((1.0 / sqrt(pi)) * exp((x * x))) * (((t_0 + (0.5 * ((1.0 / x) / (x * x)))) + (0.75 * (x ^ -5.0))) + (1.875 * (t_0 * (x ^ -6.0))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 + N[(0.5 * N[(N[(1.0 / x), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.75 * N[Power[x, -5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.875 * N[(t$95$0 * N[Power[x, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \color{blue}{\frac{1}{{\left(\left|x\right|\right)}^{5}}}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
3. Step-by-step derivation

   1. rem-square-sqrt 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \frac{1}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{5}}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   2. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \frac{1}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{5}}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   3. rem-square-sqrt 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \frac{1}{{\color{blue}{x}}^{5}}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   4. exp-to-pow 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \frac{1}{\color{blue}{e^{\log x \cdot 5}}}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   5. exp-neg 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \color{blue}{e^{-\log x \cdot 5}}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   6. distribute-lft-neg-out 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot e^{\color{blue}{\left(-\log x\right) \cdot 5}}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   7. *-commutative 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot e^{\color{blue}{5 \cdot \left(-\log x\right)}}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   8. neg-mul-1 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot e^{5 \cdot \color{blue}{\left(-1 \cdot \log x\right)}}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   9. associate-*r* 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot e^{\color{blue}{\left(5 \cdot -1\right) \cdot \log x}}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   10. metadata-eval 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot e^{\color{blue}{-5} \cdot \log x}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   11. log-pow 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot e^{\color{blue}{\log \left({x}^{-5}\right)}}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   12. rem-exp-log 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \color{blue}{{x}^{-5}}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

4. Simplified 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \color{blue}{{x}^{-5}}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
5. Step-by-step derivation

   1. associate-*l* 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left(\color{blue}{\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right)} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   2. pow3 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left(\left(\color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{3}} \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   3. pow2 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left(\left({\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{2}}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   4. pow-prod-up 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left(\color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{\left(3 + 2\right)}} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   5. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\color{blue}{\left(\sqrt{\frac{1}{\left|x\right|}} \cdot \sqrt{\frac{1}{\left|x\right|}}\right)}}^{\left(3 + 2\right)} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   6. pow2 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\color{blue}{\left({\left(\sqrt{\frac{1}{\left|x\right|}}\right)}^{2}\right)}}^{\left(3 + 2\right)} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   7. pow-pow 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left(\color{blue}{{\left(\sqrt{\frac{1}{\left|x\right|}}\right)}^{\left(2 \cdot \left(3 + 2\right)\right)}} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   8. inv-pow 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left(\sqrt{\color{blue}{{\left(\left|x\right|\right)}^{-1}}}\right)}^{\left(2 \cdot \left(3 + 2\right)\right)} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   9. sqrt-pow1 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\color{blue}{\left({\left(\left|x\right|\right)}^{\left(\frac{-1}{2}\right)}\right)}}^{\left(2 \cdot \left(3 + 2\right)\right)} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(\frac{-1}{2}\right)}\right)}^{\left(2 \cdot \left(3 + 2\right)\right)} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   11. fabs-sqr 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(\frac{-1}{2}\right)}\right)}^{\left(2 \cdot \left(3 + 2\right)\right)} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   12. add-sqr-sqrt 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({\color{blue}{x}}^{\left(\frac{-1}{2}\right)}\right)}^{\left(2 \cdot \left(3 + 2\right)\right)} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   13. metadata-eval 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{\color{blue}{-0.5}}\right)}^{\left(2 \cdot \left(3 + 2\right)\right)} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   14. metadata-eval 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{-0.5}\right)}^{\left(2 \cdot \color{blue}{5}\right)} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   15. metadata-eval 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{-0.5}\right)}^{\color{blue}{10}} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left(\color{blue}{{\left({x}^{-0.5}\right)}^{10}} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
7. Step-by-step derivation

   1. un-div-inv 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\color{blue}{\frac{\frac{1}{\left|x\right|}}{\left|x\right|}} \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{-0.5}\right)}^{10} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   2. frac-times 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \color{blue}{\frac{\frac{1}{\left|x\right|} \cdot 1}{\left|x\right| \cdot \left|x\right|}}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{-0.5}\right)}^{10} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   3. *-rgt-identity 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\color{blue}{\frac{1}{\left|x\right|}}}{\left|x\right| \cdot \left|x\right|}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{-0.5}\right)}^{10} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   4. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}}{\left|x\right| \cdot \left|x\right|}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{-0.5}\right)}^{10} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   5. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}}{\left|x\right| \cdot \left|x\right|}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{-0.5}\right)}^{10} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   6. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{\color{blue}{x}}}{\left|x\right| \cdot \left|x\right|}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{-0.5}\right)}^{10} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   7. pow2 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{\color{blue}{{\left(\left|x\right|\right)}^{2}}}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{-0.5}\right)}^{10} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{2}}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{-0.5}\right)}^{10} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   9. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{2}}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{-0.5}\right)}^{10} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{{\color{blue}{x}}^{2}}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{-0.5}\right)}^{10} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   11. pow2 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{\color{blue}{x \cdot x}}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{-0.5}\right)}^{10} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

8. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \color{blue}{\frac{\frac{1}{x}}{x \cdot x}}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\left({\left({x}^{-0.5}\right)}^{10} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

9. Taylor expanded in x around 0 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\color{blue}{\frac{1}{\left|x\right| \cdot {x}^{5}}} \cdot \frac{1}{\left|x\right|}\right)\right) \]
10. Step-by-step derivation

    1. *-commutative 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\frac{1}{\color{blue}{{x}^{5} \cdot \left|x\right|}} \cdot \frac{1}{\left|x\right|}\right)\right) \]

    2. metadata-eval 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\frac{1}{{x}^{\color{blue}{\left(4 + 1\right)}} \cdot \left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right) \]

    3. pow-plus 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\frac{1}{\color{blue}{\left({x}^{4} \cdot x\right)} \cdot \left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right) \]

    4. metadata-eval 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\frac{1}{\left({x}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot x\right) \cdot \left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right) \]

    5. pow-sqr 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\frac{1}{\left(\color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} \cdot x\right) \cdot \left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right) \]

    6. unpow2 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\frac{1}{\left(\left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{2}\right) \cdot x\right) \cdot \left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right) \]

    7. unpow2 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\frac{1}{\left(\left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot x\right) \cdot \left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right) \]

    8. rem-square-sqrt 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\frac{1}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|} \cdot \frac{1}{\left|x\right|}\right)\right) \]

    9. fabs-sqr 100.0%

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\frac{1}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}} \cdot \frac{1}{\left|x\right|}\right)\right) \]

    10. rem-square-sqrt 100.0%

        \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\frac{1}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \color{blue}{x}} \cdot \frac{1}{\left|x\right|}\right)\right) \]

    11. associate-*r* 100.0%

        \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\frac{1}{\color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)}} \cdot \frac{1}{\left|x\right|}\right)\right) \]

    12. unpow3 100.0%

        \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\frac{1}{\color{blue}{{\left(x \cdot x\right)}^{3}}} \cdot \frac{1}{\left|x\right|}\right)\right) \]

    13. cube-prod 100.0%

        \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\frac{1}{\color{blue}{{x}^{3} \cdot {x}^{3}}} \cdot \frac{1}{\left|x\right|}\right)\right) \]

    14. associate-/r* 100.0%

        \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\color{blue}{\frac{\frac{1}{{x}^{3}}}{{x}^{3}}} \cdot \frac{1}{\left|x\right|}\right)\right) \]

    15. metadata-eval 100.0%

        \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\frac{\frac{\color{blue}{{1}^{3}}}{{x}^{3}}}{{x}^{3}} \cdot \frac{1}{\left|x\right|}\right)\right) \]

    16. cube-div 100.0%

        \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\frac{\color{blue}{{\left(\frac{1}{x}\right)}^{3}}}{{x}^{3}} \cdot \frac{1}{\left|x\right|}\right)\right) \]

    17. unpow-1 100.0%

        \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\frac{{\color{blue}{\left({x}^{-1}\right)}}^{3}}{{x}^{3}} \cdot \frac{1}{\left|x\right|}\right)\right) \]

    18. exp-to-pow 100.0%

        \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\frac{{\color{blue}{\left(e^{\log x \cdot -1}\right)}}^{3}}{{x}^{3}} \cdot \frac{1}{\left|x\right|}\right)\right) \]

    19. exp-prod 100.0%

        \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\frac{\color{blue}{e^{\left(\log x \cdot -1\right) \cdot 3}}}{{x}^{3}} \cdot \frac{1}{\left|x\right|}\right)\right) \]

    20. associate-*l* 100.0%

        \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\frac{e^{\color{blue}{\log x \cdot \left(-1 \cdot 3\right)}}}{{x}^{3}} \cdot \frac{1}{\left|x\right|}\right)\right) \]

    21. metadata-eval 100.0%

        \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\frac{e^{\log x \cdot \color{blue}{-3}}}{{x}^{3}} \cdot \frac{1}{\left|x\right|}\right)\right) \]

    22. exp-to-pow 100.0%

        \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\frac{\color{blue}{{x}^{-3}}}{{x}^{3}} \cdot \frac{1}{\left|x\right|}\right)\right) \]

11. Simplified 100.0%

    \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{3}{4} \cdot {x}^{-5}\right) + \frac{15}{8} \cdot \left(\color{blue}{{x}^{-6}} \cdot \frac{1}{\left|x\right|}\right)\right) \]

12. Final simplification 100.0%

    \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{x \cdot x}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{\frac{1}{x}}{x \cdot x}\right) + 0.75 \cdot {x}^{-5}\right) + 1.875 \cdot \left(\frac{1}{\left|x\right|} \cdot {x}^{-6}\right)\right) \]

Reproduce

herbie shell --seed 2023174 
(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)))))))