Jmat.Real.gamma, branch z greater than 0.5

Average Error: 3.7 → 2.1

Time: 36.0s

Precision: binary64

Cost: 36736

\[z > 0.5\]

Math

\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]

↓

\[\begin{array}{l} t_0 := -\left(0.5 + \left(-z\right)\right)\\ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \frac{-176.6150291621406}{z + 3}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{771.3234287776531}{z + 2}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(1 \cdot \left(\left(e^{-6.5 - z} \cdot {\left(\frac{6.5 + z}{4}\right)}^{t_0}\right) \cdot {4}^{t_0}\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (* (sqrt (* PI 2.0)) (pow (+ (+ (- z 1.0) 7.0) 0.5) (+ (- z 1.0) 0.5)))
   (exp (- (+ (+ (- z 1.0) 7.0) 0.5))))
  (+
   (+
    (+
     (+
      (+
       (+
        (+
         (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- z 1.0) 1.0)))
         (/ -1259.1392167224028 (+ (- z 1.0) 2.0)))
        (/ 771.3234287776531 (+ (- z 1.0) 3.0)))
       (/ -176.6150291621406 (+ (- z 1.0) 4.0)))
      (/ 12.507343278686905 (+ (- z 1.0) 5.0)))
     (/ -0.13857109526572012 (+ (- z 1.0) 6.0)))
    (/ 9.984369578019572e-6 (+ (- z 1.0) 7.0)))
   (/ 1.5056327351493116e-7 (+ (- z 1.0) 8.0)))))

↓

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (+ 0.5 (- z)))))
   (*
    (sqrt (* PI 2.0))
    (*
     (+
      (+
       0.9999999999998099
       (+
        (+ (/ 676.5203681218851 z) (/ -176.6150291621406 (+ z 3.0)))
        (+ (/ 1259.1392167224028 (- -1.0 z)) (/ 771.3234287776531 (+ z 2.0)))))
      (+
       (+
        (/ 9.984369578019572e-6 (+ z 6.0))
        (+
         (/ 12.507343278686905 (+ z 4.0))
         (/ -0.13857109526572012 (+ z 5.0))))
       (/ 1.5056327351493116e-7 (+ z 7.0))))
     (*
      1.0
      (* (* (exp (- -6.5 z)) (pow (/ (+ 6.5 z) 4.0) t_0)) (pow 4.0 t_0)))))))

C

double code(double z) {
	return ((sqrt((((double) M_PI) * 2.0)) * pow((((z - 1.0) + 7.0) + 0.5), ((z - 1.0) + 0.5))) * exp(-(((z - 1.0) + 7.0) + 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / ((z - 1.0) + 1.0))) + (-1259.1392167224028 / ((z - 1.0) + 2.0))) + (771.3234287776531 / ((z - 1.0) + 3.0))) + (-176.6150291621406 / ((z - 1.0) + 4.0))) + (12.507343278686905 / ((z - 1.0) + 5.0))) + (-0.13857109526572012 / ((z - 1.0) + 6.0))) + (9.984369578019572e-6 / ((z - 1.0) + 7.0))) + (1.5056327351493116e-7 / ((z - 1.0) + 8.0)));
}

↓

double code(double z) {
	double t_0 = -(0.5 + -z);
	return sqrt((((double) M_PI) * 2.0)) * (((0.9999999999998099 + (((676.5203681218851 / z) + (-176.6150291621406 / (z + 3.0))) + ((1259.1392167224028 / (-1.0 - z)) + (771.3234287776531 / (z + 2.0))))) + (((9.984369578019572e-6 / (z + 6.0)) + ((12.507343278686905 / (z + 4.0)) + (-0.13857109526572012 / (z + 5.0)))) + (1.5056327351493116e-7 / (z + 7.0)))) * (1.0 * ((exp((-6.5 - z)) * pow(((6.5 + z) / 4.0), t_0)) * pow(4.0, t_0))));
}

Java

public static double code(double z) {
	return ((Math.sqrt((Math.PI * 2.0)) * Math.pow((((z - 1.0) + 7.0) + 0.5), ((z - 1.0) + 0.5))) * Math.exp(-(((z - 1.0) + 7.0) + 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / ((z - 1.0) + 1.0))) + (-1259.1392167224028 / ((z - 1.0) + 2.0))) + (771.3234287776531 / ((z - 1.0) + 3.0))) + (-176.6150291621406 / ((z - 1.0) + 4.0))) + (12.507343278686905 / ((z - 1.0) + 5.0))) + (-0.13857109526572012 / ((z - 1.0) + 6.0))) + (9.984369578019572e-6 / ((z - 1.0) + 7.0))) + (1.5056327351493116e-7 / ((z - 1.0) + 8.0)));
}

↓

public static double code(double z) {
	double t_0 = -(0.5 + -z);
	return Math.sqrt((Math.PI * 2.0)) * (((0.9999999999998099 + (((676.5203681218851 / z) + (-176.6150291621406 / (z + 3.0))) + ((1259.1392167224028 / (-1.0 - z)) + (771.3234287776531 / (z + 2.0))))) + (((9.984369578019572e-6 / (z + 6.0)) + ((12.507343278686905 / (z + 4.0)) + (-0.13857109526572012 / (z + 5.0)))) + (1.5056327351493116e-7 / (z + 7.0)))) * (1.0 * ((Math.exp((-6.5 - z)) * Math.pow(((6.5 + z) / 4.0), t_0)) * Math.pow(4.0, t_0))));
}

Python

def code(z):
	return ((math.sqrt((math.pi * 2.0)) * math.pow((((z - 1.0) + 7.0) + 0.5), ((z - 1.0) + 0.5))) * math.exp(-(((z - 1.0) + 7.0) + 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / ((z - 1.0) + 1.0))) + (-1259.1392167224028 / ((z - 1.0) + 2.0))) + (771.3234287776531 / ((z - 1.0) + 3.0))) + (-176.6150291621406 / ((z - 1.0) + 4.0))) + (12.507343278686905 / ((z - 1.0) + 5.0))) + (-0.13857109526572012 / ((z - 1.0) + 6.0))) + (9.984369578019572e-6 / ((z - 1.0) + 7.0))) + (1.5056327351493116e-7 / ((z - 1.0) + 8.0)))

↓

def code(z):
	t_0 = -(0.5 + -z)
	return math.sqrt((math.pi * 2.0)) * (((0.9999999999998099 + (((676.5203681218851 / z) + (-176.6150291621406 / (z + 3.0))) + ((1259.1392167224028 / (-1.0 - z)) + (771.3234287776531 / (z + 2.0))))) + (((9.984369578019572e-6 / (z + 6.0)) + ((12.507343278686905 / (z + 4.0)) + (-0.13857109526572012 / (z + 5.0)))) + (1.5056327351493116e-7 / (z + 7.0)))) * (1.0 * ((math.exp((-6.5 - z)) * math.pow(((6.5 + z) / 4.0), t_0)) * math.pow(4.0, t_0))))

Julia

function code(z)
	return Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (Float64(Float64(Float64(z - 1.0) + 7.0) + 0.5) ^ Float64(Float64(z - 1.0) + 0.5))) * exp(Float64(-Float64(Float64(Float64(z - 1.0) + 7.0) + 0.5)))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(Float64(z - 1.0) + 1.0))) + Float64(-1259.1392167224028 / Float64(Float64(z - 1.0) + 2.0))) + Float64(771.3234287776531 / Float64(Float64(z - 1.0) + 3.0))) + Float64(-176.6150291621406 / Float64(Float64(z - 1.0) + 4.0))) + Float64(12.507343278686905 / Float64(Float64(z - 1.0) + 5.0))) + Float64(-0.13857109526572012 / Float64(Float64(z - 1.0) + 6.0))) + Float64(9.984369578019572e-6 / Float64(Float64(z - 1.0) + 7.0))) + Float64(1.5056327351493116e-7 / Float64(Float64(z - 1.0) + 8.0))))
end

↓

function code(z)
	t_0 = Float64(-Float64(0.5 + Float64(-z)))
	return Float64(sqrt(Float64(pi * 2.0)) * Float64(Float64(Float64(0.9999999999998099 + Float64(Float64(Float64(676.5203681218851 / z) + Float64(-176.6150291621406 / Float64(z + 3.0))) + Float64(Float64(1259.1392167224028 / Float64(-1.0 - z)) + Float64(771.3234287776531 / Float64(z + 2.0))))) + Float64(Float64(Float64(9.984369578019572e-6 / Float64(z + 6.0)) + Float64(Float64(12.507343278686905 / Float64(z + 4.0)) + Float64(-0.13857109526572012 / Float64(z + 5.0)))) + Float64(1.5056327351493116e-7 / Float64(z + 7.0)))) * Float64(1.0 * Float64(Float64(exp(Float64(-6.5 - z)) * (Float64(Float64(6.5 + z) / 4.0) ^ t_0)) * (4.0 ^ t_0)))))
end

MATLAB

function tmp = code(z)
	tmp = ((sqrt((pi * 2.0)) * ((((z - 1.0) + 7.0) + 0.5) ^ ((z - 1.0) + 0.5))) * exp(-(((z - 1.0) + 7.0) + 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / ((z - 1.0) + 1.0))) + (-1259.1392167224028 / ((z - 1.0) + 2.0))) + (771.3234287776531 / ((z - 1.0) + 3.0))) + (-176.6150291621406 / ((z - 1.0) + 4.0))) + (12.507343278686905 / ((z - 1.0) + 5.0))) + (-0.13857109526572012 / ((z - 1.0) + 6.0))) + (9.984369578019572e-6 / ((z - 1.0) + 7.0))) + (1.5056327351493116e-7 / ((z - 1.0) + 8.0)));
end

↓

function tmp = code(z)
	t_0 = -(0.5 + -z);
	tmp = sqrt((pi * 2.0)) * (((0.9999999999998099 + (((676.5203681218851 / z) + (-176.6150291621406 / (z + 3.0))) + ((1259.1392167224028 / (-1.0 - z)) + (771.3234287776531 / (z + 2.0))))) + (((9.984369578019572e-6 / (z + 6.0)) + ((12.507343278686905 / (z + 4.0)) + (-0.13857109526572012 / (z + 5.0)))) + (1.5056327351493116e-7 / (z + 7.0)))) * (1.0 * ((exp((-6.5 - z)) * (((6.5 + z) / 4.0) ^ t_0)) * (4.0 ^ t_0))));
end

Wolfram

code[z_] := N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[(N[(z - 1.0), $MachinePrecision] + 7.0), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(z - 1.0), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[(N[(z - 1.0), $MachinePrecision] + 7.0), $MachinePrecision] + 0.5), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(N[(z - 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(N[(z - 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(N[(z - 1.0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(z - 1.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(N[(z - 1.0), $MachinePrecision] + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(z - 1.0), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(N[(z - 1.0), $MachinePrecision] + 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(z - 1.0), $MachinePrecision] + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[z_] := Block[{t$95$0 = (-N[(0.5 + (-z)), $MachinePrecision])}, N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(0.9999999999998099 + N[(N[(N[(676.5203681218851 / z), $MachinePrecision] + N[(-176.6150291621406 / N[(z + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1259.1392167224028 / N[(-1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(z + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(9.984369578019572e-6 / N[(z + 6.0), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(z + 4.0), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(z + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(z + 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 * N[(N[(N[Exp[N[(-6.5 - z), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[(6.5 + z), $MachinePrecision] / 4.0), $MachinePrecision], t$95$0], $MachinePrecision]), $MachinePrecision] * N[Power[4.0, t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 3.7

   \[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]

2. Simplified 3.9

   \[\leadsto \color{blue}{\sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right)\right)} \]
   Proof

   [Start] 3.7	\[ \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
   rational.json-simplify-2 [=>] 3.7	\[ \color{blue}{\left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right)} \]
   rational.json-simplify-2 [=>] 3.7	\[ \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \cdot \color{blue}{\left(e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right)\right)} \]
   rational.json-simplify-43 [=>] 3.7	\[ \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \cdot \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right)\right)} \]
   rational.json-simplify-2 [=>] 3.7	\[ \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \cdot \color{blue}{\left(\left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \sqrt{\pi \cdot 2}\right)} \]

3. Taylor expanded in z around -inf 5.1

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \color{blue}{\left(e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right)} \cdot e^{-1 \cdot z - 6.5}\right)}\right) \]

4. Simplified 3.8

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \color{blue}{e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(z - -6.5\right)\right) + \left(-6.5 - z\right)}}\right) \]
   Proof

   [Start] 5.1	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right)} \cdot e^{-1 \cdot z - 6.5}\right)\right) \]
   exponential.json-simplify-3 [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \color{blue}{e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right) + \left(-1 \cdot z - 6.5\right)}}\right) \]
   rational.json-simplify-2 [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot e^{-1 \cdot \color{blue}{\left(\left(-1 \cdot z + 0.5\right) \cdot \log \left(6.5 - -1 \cdot z\right)\right)} + \left(-1 \cdot z - 6.5\right)}\right) \]
   rational.json-simplify-43 [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot e^{\color{blue}{\left(-1 \cdot z + 0.5\right) \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot -1\right)} + \left(-1 \cdot z - 6.5\right)}\right) \]
   rational.json-simplify-1 [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot e^{\color{blue}{\left(0.5 + -1 \cdot z\right)} \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot -1\right) + \left(-1 \cdot z - 6.5\right)}\right) \]
   rational.json-simplify-2 [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot e^{\left(0.5 + \color{blue}{z \cdot -1}\right) \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot -1\right) + \left(-1 \cdot z - 6.5\right)}\right) \]
   rational.json-simplify-8 [<=] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot e^{\left(0.5 + \color{blue}{\left(-z\right)}\right) \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot -1\right) + \left(-1 \cdot z - 6.5\right)}\right) \]
   rational.json-simplify-9 [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot e^{\left(0.5 + \left(-z\right)\right) \cdot \color{blue}{\left(-\log \left(6.5 - -1 \cdot z\right)\right)} + \left(-1 \cdot z - 6.5\right)}\right) \]
   rational.json-simplify-2 [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(6.5 - \color{blue}{z \cdot -1}\right)\right) + \left(-1 \cdot z - 6.5\right)}\right) \]
   rational.json-simplify-8 [<=] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(6.5 - \color{blue}{\left(-z\right)}\right)\right) + \left(-1 \cdot z - 6.5\right)}\right) \]
   rational.json-simplify-12 [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(6.5 - \color{blue}{\left(0 - z\right)}\right)\right) + \left(-1 \cdot z - 6.5\right)}\right) \]
   rational.json-simplify-45 [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \color{blue}{\left(z - \left(0 - 6.5\right)\right)}\right) + \left(-1 \cdot z - 6.5\right)}\right) \]
   metadata-eval [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(z - \color{blue}{-6.5}\right)\right) + \left(-1 \cdot z - 6.5\right)}\right) \]
   rational.json-simplify-2 [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(z - -6.5\right)\right) + \left(\color{blue}{z \cdot -1} - 6.5\right)}\right) \]
   rational.json-simplify-8 [<=] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(z - -6.5\right)\right) + \left(\color{blue}{\left(-z\right)} - 6.5\right)}\right) \]
   rational.json-simplify-12 [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(z - -6.5\right)\right) + \left(\color{blue}{\left(0 - z\right)} - 6.5\right)}\right) \]
   rational.json-simplify-42 [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(z - -6.5\right)\right) + \color{blue}{\left(\left(0 - 6.5\right) - z\right)}}\right) \]
   metadata-eval [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(z - -6.5\right)\right) + \left(\color{blue}{-6.5} - z\right)}\right) \]

5. Applied egg-rr 2.4

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \color{blue}{\left(\left({4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)} + {4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right) \cdot \left({\left(\frac{z + 6.5}{4}\right)}^{\left(-\left(0.5 + \left(-z\right)\right)\right)} \cdot \left(0.5 \cdot e^{-6.5 - z}\right)\right)\right)}\right) \]

6. Simplified 2.4

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \color{blue}{\left(1 \cdot \left(\left(e^{-6.5 - z} \cdot {\left(\frac{6.5 + z}{4}\right)}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right) \cdot {4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right)\right)}\right) \]
   Proof

   [Start] 2.4	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(\left({4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)} + {4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right) \cdot \left({\left(\frac{z + 6.5}{4}\right)}^{\left(-\left(0.5 + \left(-z\right)\right)\right)} \cdot \left(0.5 \cdot e^{-6.5 - z}\right)\right)\right)\right) \]
   rational.json-simplify-43 [=>] 2.4	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(\left({4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)} + {4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right) \cdot \color{blue}{\left(0.5 \cdot \left(e^{-6.5 - z} \cdot {\left(\frac{z + 6.5}{4}\right)}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right)\right)}\right)\right) \]
   rational.json-simplify-53 [=>] 2.4	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \color{blue}{\left(\left(0.5 + 0.5\right) \cdot \left(\left(e^{-6.5 - z} \cdot {\left(\frac{z + 6.5}{4}\right)}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right) \cdot {4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right)\right)}\right) \]
   metadata-eval [=>] 2.4	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(\color{blue}{1} \cdot \left(\left(e^{-6.5 - z} \cdot {\left(\frac{z + 6.5}{4}\right)}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right) \cdot {4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right)\right)\right) \]
   rational.json-simplify-1 [<=] 2.4	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(1 \cdot \left(\left(e^{-6.5 - z} \cdot {\left(\frac{\color{blue}{6.5 + z}}{4}\right)}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right) \cdot {4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right)\right)\right) \]

7. Applied egg-rr 2.2

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(\color{blue}{\left(\left(\frac{771.3234287776531}{z + 2} + \left(0.9999999999998099 + \left(\frac{1259.1392167224028}{-1 - z} + \left(\frac{676.5203681218851}{z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right)\right) - 0\right)} + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(1 \cdot \left(\left(e^{-6.5 - z} \cdot {\left(\frac{6.5 + z}{4}\right)}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right) \cdot {4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right)\right)\right) \]

8. Simplified 2.1

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(\color{blue}{\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \frac{-176.6150291621406}{z + 3}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{771.3234287776531}{z + 2}\right)\right)\right)} + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(1 \cdot \left(\left(e^{-6.5 - z} \cdot {\left(\frac{6.5 + z}{4}\right)}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right) \cdot {4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right)\right)\right) \]
   Proof

   [Start] 2.2	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\left(\frac{771.3234287776531}{z + 2} + \left(0.9999999999998099 + \left(\frac{1259.1392167224028}{-1 - z} + \left(\frac{676.5203681218851}{z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right)\right) - 0\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(1 \cdot \left(\left(e^{-6.5 - z} \cdot {\left(\frac{6.5 + z}{4}\right)}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right) \cdot {4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right)\right)\right) \]
   rational.json-simplify-5 [=>] 2.2	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\color{blue}{\left(\frac{771.3234287776531}{z + 2} + \left(0.9999999999998099 + \left(\frac{1259.1392167224028}{-1 - z} + \left(\frac{676.5203681218851}{z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right)\right)} + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(1 \cdot \left(\left(e^{-6.5 - z} \cdot {\left(\frac{6.5 + z}{4}\right)}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right) \cdot {4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right)\right)\right) \]
   rational.json-simplify-41 [=>] 2.1	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\color{blue}{\left(0.9999999999998099 + \left(\left(\frac{1259.1392167224028}{-1 - z} + \left(\frac{676.5203681218851}{z} + \frac{-176.6150291621406}{z + 3}\right)\right) + \frac{771.3234287776531}{z + 2}\right)\right)} + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(1 \cdot \left(\left(e^{-6.5 - z} \cdot {\left(\frac{6.5 + z}{4}\right)}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right) \cdot {4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right)\right)\right) \]
   rational.json-simplify-1 [=>] 2.1	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \color{blue}{\left(\frac{771.3234287776531}{z + 2} + \left(\frac{1259.1392167224028}{-1 - z} + \left(\frac{676.5203681218851}{z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right)}\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(1 \cdot \left(\left(e^{-6.5 - z} \cdot {\left(\frac{6.5 + z}{4}\right)}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right) \cdot {4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right)\right)\right) \]
   rational.json-simplify-1 [=>] 2.1	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \color{blue}{\left(\left(\frac{676.5203681218851}{z} + \frac{-176.6150291621406}{z + 3}\right) + \frac{1259.1392167224028}{-1 - z}\right)}\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(1 \cdot \left(\left(e^{-6.5 - z} \cdot {\left(\frac{6.5 + z}{4}\right)}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right) \cdot {4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right)\right)\right) \]
   rational.json-simplify-41 [=>] 2.1	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \color{blue}{\left(\left(\frac{676.5203681218851}{z} + \frac{-176.6150291621406}{z + 3}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{771.3234287776531}{z + 2}\right)\right)}\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(1 \cdot \left(\left(e^{-6.5 - z} \cdot {\left(\frac{6.5 + z}{4}\right)}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right) \cdot {4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right)\right)\right) \]

9. Final simplification 2.1

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \frac{-176.6150291621406}{z + 3}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{771.3234287776531}{z + 2}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(1 \cdot \left(\left(e^{-6.5 - z} \cdot {\left(\frac{6.5 + z}{4}\right)}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right) \cdot {4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right)\right)\right) \]

Alternatives

Alternative 1
Error	2.2
Cost	61956

\[\begin{array}{l} t_0 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_1 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_2 := \frac{771.3234287776531}{2 + z}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \frac{-176.6150291621406}{z + 3}\\ t_5 := \left(z - 1\right) + 7\\ t_6 := \frac{-0.13857109526572012}{z + 5}\\ t_7 := z + \left(z + 2\right)\\ t_8 := t_5 + 0.5\\ t_9 := \frac{12.507343278686905}{z + 4}\\ \mathbf{if}\;\left(\left(t_3 \cdot {t_8}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-t_8}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t_5}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \leq 4 \cdot 10^{+233}:\\ \;\;\;\;t_3 \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{z \cdot 0.00147815209581367 + t_7 \cdot -0.00039709667792059005}{\frac{-t_7}{2518.2784334448056} \cdot \left(z \cdot 0.00147815209581367\right)} + t_2\right)\right) + \left(\left(t_4 + t_9\right) + t_6\right)\right) + \left(t_0 + t_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(\left(\left(t_2 + \left(\frac{1259.1392167224028}{-1 - z} + \left(\frac{676.5203681218851}{z} + \left(t_4 + 0.9999999999998099\right)\right)\right)\right) + \left(\left(t_0 + \left(t_9 + t_6\right)\right) + t_1\right)\right) \cdot e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(z - -6.5\right)\right) + \left(-6.5 - z\right)}\right)\\ \end{array} \]

Alternative 2
Error	2.9
Cost	36672

\[\begin{array}{l} t_0 := 0.5 + \left(-z\right)\\ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1259.1392167224028}{-1 - z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(1 \cdot \left(\left(e^{-6.5 - z} \cdot {0.5}^{t_0}\right) \cdot {\left(0.5 \cdot \left(6.5 + z\right)\right)}^{\left(-t_0\right)}\right)\right)\right) \end{array} \]

Alternative 3
Error	2.4
Cost	29892

\[\begin{array}{l} t_0 := \frac{771.3234287776531}{2 + z}\\ t_1 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_2 := \frac{-176.6150291621406}{z + 3}\\ t_3 := \frac{12.507343278686905}{z + 4}\\ t_4 := \frac{-0.13857109526572012}{z + 5}\\ t_5 := \sqrt{\pi \cdot 2}\\ t_6 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ \mathbf{if}\;z \leq 145:\\ \;\;\;\;t_5 \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \frac{2518.2784334448056}{-\left(z + \left(z + 2\right)\right)}\right) + t_0\right)\right) + \left(\left(t_2 + t_3\right) + t_4\right)\right) + \left(t_1 + t_6\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_5 \cdot \left(\left(\left(t_0 + \left(\frac{1259.1392167224028}{-1 - z} + \left(\frac{676.5203681218851}{z} + \left(t_2 + 0.9999999999998099\right)\right)\right)\right) + \left(\left(t_1 + \left(t_3 + t_4\right)\right) + t_6\right)\right) \cdot e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(z - -6.5\right)\right) + \left(-6.5 - z\right)}\right)\\ \end{array} \]

Alternative 4
Error	2.4
Cost	29828

\[\begin{array}{l} t_0 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_1 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_2 := \frac{-0.13857109526572012}{z + 5}\\ t_3 := \frac{12.507343278686905}{z + 4}\\ t_4 := \sqrt{\pi \cdot 2}\\ t_5 := \frac{-176.6150291621406}{z + 3}\\ \mathbf{if}\;z \leq 145:\\ \;\;\;\;t_4 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \frac{0.9999999999998099 + \left(\left(\frac{771.3234287776531}{z + 2} + \left(\frac{676.5203681218851}{z} + \frac{1259.1392167224028}{-1 - z}\right)\right) + \left(t_0 + \left(\left(t_5 + \left(t_2 + t_3\right)\right) + t_1\right)\right)\right)}{e^{z + 6.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot \left(e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(z - -6.5\right)\right) + \left(-6.5 - z\right)} \cdot \left(\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(t_5 + t_3\right) + t_2\right)\right) + \left(t_0 + t_1\right)\right)\right)\\ \end{array} \]

Alternative 5
Error	2.4
Cost	29828

\[\begin{array}{l} t_0 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_1 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_2 := \frac{771.3234287776531}{z + 2}\\ t_3 := \frac{12.507343278686905}{z + 4}\\ t_4 := \sqrt{\pi \cdot 2}\\ t_5 := \frac{-176.6150291621406}{z + 3}\\ t_6 := \frac{1259.1392167224028}{-1 - z}\\ t_7 := \frac{-0.13857109526572012}{z + 5}\\ \mathbf{if}\;z \leq 145:\\ \;\;\;\;t_4 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \frac{0.9999999999998099 + \left(\left(t_2 + \left(\frac{676.5203681218851}{z} + t_6\right)\right) + \left(t_0 + \left(\left(t_5 + \left(t_7 + t_3\right)\right) + t_1\right)\right)\right)}{e^{z + 6.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot \left(\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + t_5\right) + \left(t_6 + t_2\right)\right)\right) + \left(\left(t_0 + \left(t_3 + t_7\right)\right) + t_1\right)\right) \cdot e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(z - -6.5\right)\right) + \left(-6.5 - z\right)}\right)\\ \end{array} \]

Alternative 6
Error	2.4
Cost	29828

\[\begin{array}{l} t_0 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_1 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_2 := \frac{12.507343278686905}{z + 4}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \frac{-176.6150291621406}{z + 3}\\ t_5 := \frac{1259.1392167224028}{-1 - z}\\ t_6 := \frac{-0.13857109526572012}{z + 5}\\ \mathbf{if}\;z \leq 145:\\ \;\;\;\;t_3 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \frac{0.9999999999998099 + \left(\left(\frac{771.3234287776531}{z + 2} + \left(\frac{676.5203681218851}{z} + t_5\right)\right) + \left(t_0 + \left(\left(t_4 + \left(t_6 + t_2\right)\right) + t_1\right)\right)\right)}{e^{z + 6.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(t_5 + \left(\frac{676.5203681218851}{z} + \left(t_4 + 0.9999999999998099\right)\right)\right)\right) + \left(\left(t_0 + \left(t_2 + t_6\right)\right) + t_1\right)\right) \cdot e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(z - -6.5\right)\right) + \left(-6.5 - z\right)}\right)\\ \end{array} \]

Alternative 7
Error	2.4
Cost	29700

\[\begin{array}{l} t_0 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_1 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_2 := \frac{12.507343278686905}{z + 4}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \frac{-176.6150291621406}{z + 3}\\ t_5 := \frac{1259.1392167224028}{-1 - z}\\ t_6 := \frac{-0.13857109526572012}{z + 5}\\ \mathbf{if}\;z \leq 145:\\ \;\;\;\;t_3 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \frac{0.9999999999998099 + \left(\left(\frac{771.3234287776531}{z + 2} + \left(\frac{676.5203681218851}{z} + t_5\right)\right) + \left(t_0 + \left(\left(t_4 + \left(t_6 + t_2\right)\right) + t_1\right)\right)\right)}{e^{z + 6.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(\left(\left(\frac{771.3234287776531}{2 + z} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(t_5 + t_4\right)\right)\right) + \left(\left(t_0 + \left(t_2 + t_6\right)\right) + t_1\right)\right) \cdot e^{\log \left(6.5 + z\right) \cdot \left(z - 0.5\right) - \left(6.5 + z\right)}\right)\\ \end{array} \]

Alternative 8
Error	3.6
Cost	29504

\[\sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]

Alternative 9
Error	3.7
Cost	29504

\[\sqrt{\pi \cdot 2} \cdot \left(\left(\left(\left(\frac{771.3234287776531}{z + 2} + \left(\frac{-1259.1392167224028}{z - -1} + \left(0.9999999999998099 + \frac{676.5203681218851}{z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{12.507343278686905}{z + 4}\right)\right) \cdot \frac{{\left(6.5 + z\right)}^{\left(z + -0.5\right)}}{e^{6.5 + z}}\right) \]

Alternative 10
Error	3.6
Cost	29504

\[\sqrt{\pi \cdot 2} \cdot \left(\frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}} \cdot \left(\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]

Alternative 11
Error	3.6
Cost	29504

\[\sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \frac{0.9999999999998099 + \left(\left(\frac{771.3234287776531}{z + 2} + \left(\frac{676.5203681218851}{z} + \frac{1259.1392167224028}{-1 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)}{e^{z + 6.5}}\right) \]

Alternative 12
Error	50.0
Cost	28864

\[\sqrt{\pi \cdot 2} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{-0.13857109526572012}{z + 5} + \frac{-176.6150291621406}{z + 3}\right)\right) + 188.7045801771354 \cdot \frac{1}{z}\right)\right)\right) \cdot \frac{{\left(6.5 + z\right)}^{\left(z + -0.5\right)}}{e^{6.5 + z}}\right) \]

Alternative 13
Error	50.0
Cost	28804

\[\begin{array}{l} \mathbf{if}\;z \leq 3:\\ \;\;\;\;676.5203681218851 \cdot \left(\frac{1}{-\frac{z}{\sqrt{\pi \cdot 0.3076923076923077}}} \cdot \left(-e^{-6.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{1}{z} \cdot 12.0895510149948\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(z - -6.5\right)\right) + \left(-6.5 - z\right)}\right)\\ \end{array} \]

Alternative 14
Error	50.3
Cost	28164

\[\begin{array}{l} \mathbf{if}\;z \leq 2.8:\\ \;\;\;\;676.5203681218851 \cdot \left(\frac{1}{-\frac{z}{\sqrt{\pi \cdot 0.3076923076923077}}} \cdot \left(-e^{-6.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-\left(z + 6.5\right)} \cdot \left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\left(0.9999999999998099 + \frac{24.596894293681704}{z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\\ \end{array} \]

Alternative 15
Error	50.3
Cost	27076

\[\begin{array}{l} \mathbf{if}\;z \leq 2.8:\\ \;\;\;\;676.5203681218851 \cdot \left(\frac{1}{-\frac{z}{\sqrt{\pi \cdot 0.3076923076923077}}} \cdot \left(-e^{-6.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \frac{0.9999999999998099 + 24.458333333348836 \cdot \frac{1}{z}}{e^{z + 6.5}}\right)\\ \end{array} \]

Alternative 16
Error	51.9
Cost	26692

\[\begin{array}{l} \mathbf{if}\;z \leq 3.95:\\ \;\;\;\;676.5203681218851 \cdot \left(\frac{1}{-\frac{z}{\sqrt{\pi \cdot 0.3076923076923077}}} \cdot \left(-e^{-6.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(0.9999999999998099 \cdot \frac{{\left(6.5 + z\right)}^{\left(z - 0.5\right)}}{e^{6.5 + z}}\right)\\ \end{array} \]

Alternative 17
Error	52.1
Cost	26688

\[\sqrt{\pi \cdot 2} \cdot \left(201.07336249548945 \cdot \frac{{\left(6.5 + z\right)}^{\left(z - 0.5\right)}}{e^{6.5 + z} \cdot z}\right) \]

Alternative 18
Error	55.5
Cost	19712

\[676.5203681218851 \cdot \left(\sqrt{0.3076923076923077 \cdot \pi} \cdot \frac{e^{-6.5}}{z}\right) \]

Reproduce

herbie shell --seed 2023073 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z greater than 0.5"
  :precision binary64
  :pre (> z 0.5)
  (* (* (* (sqrt (* PI 2.0)) (pow (+ (+ (- z 1.0) 7.0) 0.5) (+ (- z 1.0) 0.5))) (exp (- (+ (+ (- z 1.0) 7.0) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- z 1.0) 1.0))) (/ -1259.1392167224028 (+ (- z 1.0) 2.0))) (/ 771.3234287776531 (+ (- z 1.0) 3.0))) (/ -176.6150291621406 (+ (- z 1.0) 4.0))) (/ 12.507343278686905 (+ (- z 1.0) 5.0))) (/ -0.13857109526572012 (+ (- z 1.0) 6.0))) (/ 9.984369578019572e-6 (+ (- z 1.0) 7.0))) (/ 1.5056327351493116e-7 (+ (- z 1.0) 8.0)))))