Jmat.Real.gamma, branch z greater than 0.5

Average Error: 3.8 → 2.1

Time: 36.1s

Precision: binary64

Cost: 39040

\[z > 0.5\]

Math

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

↓

\[\begin{array}{l} t_0 := -\left(0.5 + \left(-z\right)\right)\\ t_1 := \left(z + 2\right) \cdot 0.0012964730004179177\\ t_2 := \frac{z \cdot 0.00147815209581367}{\left(z \cdot 0.00147815209581367 + \left(z + 1\right) \cdot -0.0007941933558411801\right) \cdot \frac{-1259.1392167224028}{z + 1}}\\ \sqrt{\pi \cdot 2} \cdot \left(\left(1 \cdot \left(\left(e^{-6.5 - z} \cdot {\left(\frac{6.5 + z}{4}\right)}^{t_0}\right) \cdot {4}^{t_0}\right)\right) \cdot \left(\left(\left(0.9999999999998099 + \frac{t_2 + t_1}{t_2 \cdot t_1}\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \end{array} \]

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))))
        (t_1 (* (+ z 2.0) 0.0012964730004179177))
        (t_2
         (/
          (* z 0.00147815209581367)
          (*
           (+ (* z 0.00147815209581367) (* (+ z 1.0) -0.0007941933558411801))
           (/ -1259.1392167224028 (+ z 1.0))))))
   (*
    (sqrt (* PI 2.0))
    (*
     (* 1.0 (* (* (exp (- -6.5 z)) (pow (/ (+ 6.5 z) 4.0) t_0)) (pow 4.0 t_0)))
     (+
      (+
       (+ 0.9999999999998099 (/ (+ t_2 t_1) (* t_2 t_1)))
       (+
        (+ (/ -176.6150291621406 (+ z 3.0)) (/ 12.507343278686905 (+ z 4.0)))
        (/ -0.13857109526572012 (+ z 5.0))))
      (+
       (/ 9.984369578019572e-6 (+ z 6.0))
       (/ 1.5056327351493116e-7 (+ z 7.0))))))))

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);
	double t_1 = (z + 2.0) * 0.0012964730004179177;
	double t_2 = (z * 0.00147815209581367) / (((z * 0.00147815209581367) + ((z + 1.0) * -0.0007941933558411801)) * (-1259.1392167224028 / (z + 1.0)));
	return sqrt((((double) M_PI) * 2.0)) * ((1.0 * ((exp((-6.5 - z)) * pow(((6.5 + z) / 4.0), t_0)) * pow(4.0, t_0))) * (((0.9999999999998099 + ((t_2 + t_1) / (t_2 * t_1))) + (((-176.6150291621406 / (z + 3.0)) + (12.507343278686905 / (z + 4.0))) + (-0.13857109526572012 / (z + 5.0)))) + ((9.984369578019572e-6 / (z + 6.0)) + (1.5056327351493116e-7 / (z + 7.0)))));
}

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);
	double t_1 = (z + 2.0) * 0.0012964730004179177;
	double t_2 = (z * 0.00147815209581367) / (((z * 0.00147815209581367) + ((z + 1.0) * -0.0007941933558411801)) * (-1259.1392167224028 / (z + 1.0)));
	return Math.sqrt((Math.PI * 2.0)) * ((1.0 * ((Math.exp((-6.5 - z)) * Math.pow(((6.5 + z) / 4.0), t_0)) * Math.pow(4.0, t_0))) * (((0.9999999999998099 + ((t_2 + t_1) / (t_2 * t_1))) + (((-176.6150291621406 / (z + 3.0)) + (12.507343278686905 / (z + 4.0))) + (-0.13857109526572012 / (z + 5.0)))) + ((9.984369578019572e-6 / (z + 6.0)) + (1.5056327351493116e-7 / (z + 7.0)))));
}

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)
	t_1 = (z + 2.0) * 0.0012964730004179177
	t_2 = (z * 0.00147815209581367) / (((z * 0.00147815209581367) + ((z + 1.0) * -0.0007941933558411801)) * (-1259.1392167224028 / (z + 1.0)))
	return math.sqrt((math.pi * 2.0)) * ((1.0 * ((math.exp((-6.5 - z)) * math.pow(((6.5 + z) / 4.0), t_0)) * math.pow(4.0, t_0))) * (((0.9999999999998099 + ((t_2 + t_1) / (t_2 * t_1))) + (((-176.6150291621406 / (z + 3.0)) + (12.507343278686905 / (z + 4.0))) + (-0.13857109526572012 / (z + 5.0)))) + ((9.984369578019572e-6 / (z + 6.0)) + (1.5056327351493116e-7 / (z + 7.0)))))

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)))
	t_1 = Float64(Float64(z + 2.0) * 0.0012964730004179177)
	t_2 = Float64(Float64(z * 0.00147815209581367) / Float64(Float64(Float64(z * 0.00147815209581367) + Float64(Float64(z + 1.0) * -0.0007941933558411801)) * Float64(-1259.1392167224028 / Float64(z + 1.0))))
	return Float64(sqrt(Float64(pi * 2.0)) * Float64(Float64(1.0 * Float64(Float64(exp(Float64(-6.5 - z)) * (Float64(Float64(6.5 + z) / 4.0) ^ t_0)) * (4.0 ^ t_0))) * Float64(Float64(Float64(0.9999999999998099 + Float64(Float64(t_2 + t_1) / Float64(t_2 * t_1))) + Float64(Float64(Float64(-176.6150291621406 / Float64(z + 3.0)) + Float64(12.507343278686905 / Float64(z + 4.0))) + Float64(-0.13857109526572012 / Float64(z + 5.0)))) + Float64(Float64(9.984369578019572e-6 / Float64(z + 6.0)) + Float64(1.5056327351493116e-7 / Float64(z + 7.0))))))
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);
	t_1 = (z + 2.0) * 0.0012964730004179177;
	t_2 = (z * 0.00147815209581367) / (((z * 0.00147815209581367) + ((z + 1.0) * -0.0007941933558411801)) * (-1259.1392167224028 / (z + 1.0)));
	tmp = sqrt((pi * 2.0)) * ((1.0 * ((exp((-6.5 - z)) * (((6.5 + z) / 4.0) ^ t_0)) * (4.0 ^ t_0))) * (((0.9999999999998099 + ((t_2 + t_1) / (t_2 * t_1))) + (((-176.6150291621406 / (z + 3.0)) + (12.507343278686905 / (z + 4.0))) + (-0.13857109526572012 / (z + 5.0)))) + ((9.984369578019572e-6 / (z + 6.0)) + (1.5056327351493116e-7 / (z + 7.0)))));
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])}, Block[{t$95$1 = N[(N[(z + 2.0), $MachinePrecision] * 0.0012964730004179177), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * 0.00147815209581367), $MachinePrecision] / N[(N[(N[(z * 0.00147815209581367), $MachinePrecision] + N[(N[(z + 1.0), $MachinePrecision] * -0.0007941933558411801), $MachinePrecision]), $MachinePrecision] * N[(-1259.1392167224028 / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(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] * N[(N[(N[(0.9999999999998099 + N[(N[(t$95$2 + t$95$1), $MachinePrecision] / N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-176.6150291621406 / N[(z + 3.0), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(z + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(z + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(z + 6.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(z + 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 3.8

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

2. Simplified 3.7

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

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

3. Applied egg-rr 3.9

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(\left(0.9999999999998099 + \left(\color{blue}{\frac{-1259.1392167224028}{\frac{z + 1}{\frac{676.5203681218851}{\frac{z}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}}} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]

4. Taylor expanded in z around -inf 5.2

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\left(e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right)} \cdot e^{-1 \cdot z - 6.5}\right)} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{\frac{z + 1}{\frac{676.5203681218851}{\frac{z}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]

5. Simplified 3.8

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(z - -6.5\right)\right) + \left(-6.5 - z\right)}} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{\frac{z + 1}{\frac{676.5203681218851}{\frac{z}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
   Proof

   [Start] 5.2	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right)} \cdot e^{-1 \cdot z - 6.5}\right) \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{\frac{z + 1}{\frac{676.5203681218851}{\frac{z}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
   exponential.json-simplify-3 [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right) + \left(-1 \cdot z - 6.5\right)}} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{\frac{z + 1}{\frac{676.5203681218851}{\frac{z}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
   rational.json-simplify-2 [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(e^{-1 \cdot \color{blue}{\left(\left(-1 \cdot z + 0.5\right) \cdot \log \left(6.5 - -1 \cdot z\right)\right)} + \left(-1 \cdot z - 6.5\right)} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{\frac{z + 1}{\frac{676.5203681218851}{\frac{z}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
   rational.json-simplify-43 [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(e^{\color{blue}{\left(-1 \cdot z + 0.5\right) \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot -1\right)} + \left(-1 \cdot z - 6.5\right)} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{\frac{z + 1}{\frac{676.5203681218851}{\frac{z}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
   rational.json-simplify-1 [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(e^{\color{blue}{\left(0.5 + -1 \cdot z\right)} \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot -1\right) + \left(-1 \cdot z - 6.5\right)} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{\frac{z + 1}{\frac{676.5203681218851}{\frac{z}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
   rational.json-simplify-2 [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(e^{\left(0.5 + \color{blue}{z \cdot -1}\right) \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot -1\right) + \left(-1 \cdot z - 6.5\right)} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{\frac{z + 1}{\frac{676.5203681218851}{\frac{z}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
   rational.json-simplify-8 [<=] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(e^{\left(0.5 + \color{blue}{\left(-z\right)}\right) \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot -1\right) + \left(-1 \cdot z - 6.5\right)} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{\frac{z + 1}{\frac{676.5203681218851}{\frac{z}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
   rational.json-simplify-9 [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(e^{\left(0.5 + \left(-z\right)\right) \cdot \color{blue}{\left(-\log \left(6.5 - -1 \cdot z\right)\right)} + \left(-1 \cdot z - 6.5\right)} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{\frac{z + 1}{\frac{676.5203681218851}{\frac{z}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
   rational.json-simplify-2 [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(6.5 - \color{blue}{z \cdot -1}\right)\right) + \left(-1 \cdot z - 6.5\right)} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{\frac{z + 1}{\frac{676.5203681218851}{\frac{z}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
   rational.json-simplify-8 [<=] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(6.5 - \color{blue}{\left(-z\right)}\right)\right) + \left(-1 \cdot z - 6.5\right)} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{\frac{z + 1}{\frac{676.5203681218851}{\frac{z}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
   rational.json-simplify-12 [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(6.5 - \color{blue}{\left(0 - z\right)}\right)\right) + \left(-1 \cdot z - 6.5\right)} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{\frac{z + 1}{\frac{676.5203681218851}{\frac{z}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
   rational.json-simplify-45 [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \color{blue}{\left(z - \left(0 - 6.5\right)\right)}\right) + \left(-1 \cdot z - 6.5\right)} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{\frac{z + 1}{\frac{676.5203681218851}{\frac{z}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
   metadata-eval [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(z - \color{blue}{-6.5}\right)\right) + \left(-1 \cdot z - 6.5\right)} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{\frac{z + 1}{\frac{676.5203681218851}{\frac{z}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
   rational.json-simplify-2 [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(z - -6.5\right)\right) + \left(\color{blue}{z \cdot -1} - 6.5\right)} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{\frac{z + 1}{\frac{676.5203681218851}{\frac{z}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
   rational.json-simplify-8 [<=] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(z - -6.5\right)\right) + \left(\color{blue}{\left(-z\right)} - 6.5\right)} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{\frac{z + 1}{\frac{676.5203681218851}{\frac{z}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
   rational.json-simplify-12 [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(z - -6.5\right)\right) + \left(\color{blue}{\left(0 - z\right)} - 6.5\right)} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{\frac{z + 1}{\frac{676.5203681218851}{\frac{z}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
   rational.json-simplify-42 [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(z - -6.5\right)\right) + \color{blue}{\left(\left(0 - 6.5\right) - z\right)}} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{\frac{z + 1}{\frac{676.5203681218851}{\frac{z}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
   metadata-eval [=>] 3.8	\[ \sqrt{\pi \cdot 2} \cdot \left(e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(z - -6.5\right)\right) + \left(\color{blue}{-6.5} - z\right)} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{\frac{z + 1}{\frac{676.5203681218851}{\frac{z}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]

6. Applied egg-rr 2.3

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\left(\left({4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)} + {4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right) \cdot \left({\left(\frac{z + 6.5}{4}\right)}^{\left(-\left(0.5 + \left(-z\right)\right)\right)} \cdot \left(0.5 \cdot e^{-6.5 - z}\right)\right)\right)} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{\frac{z + 1}{\frac{676.5203681218851}{\frac{z}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]

7. Simplified 2.3

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\left(1 \cdot \left(\left(e^{-6.5 - z} \cdot {\left(\frac{6.5 + z}{4}\right)}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right) \cdot {4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right)\right)} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{\frac{z + 1}{\frac{676.5203681218851}{\frac{z}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
   Proof

   [Start] 2.3	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left({4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)} + {4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right) \cdot \left({\left(\frac{z + 6.5}{4}\right)}^{\left(-\left(0.5 + \left(-z\right)\right)\right)} \cdot \left(0.5 \cdot e^{-6.5 - z}\right)\right)\right) \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{\frac{z + 1}{\frac{676.5203681218851}{\frac{z}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
   rational.json-simplify-43 [=>] 2.3	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left({4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)} + {4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right) \cdot \color{blue}{\left(0.5 \cdot \left(e^{-6.5 - z} \cdot {\left(\frac{z + 6.5}{4}\right)}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right)\right)}\right) \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{\frac{z + 1}{\frac{676.5203681218851}{\frac{z}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
   rational.json-simplify-53 [=>] 2.3	\[ \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\left(\left(0.5 + 0.5\right) \cdot \left(\left(e^{-6.5 - z} \cdot {\left(\frac{z + 6.5}{4}\right)}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right) \cdot {4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right)\right)} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{\frac{z + 1}{\frac{676.5203681218851}{\frac{z}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
   metadata-eval [=>] 2.3	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\color{blue}{1} \cdot \left(\left(e^{-6.5 - z} \cdot {\left(\frac{z + 6.5}{4}\right)}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right) \cdot {4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right)\right) \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{\frac{z + 1}{\frac{676.5203681218851}{\frac{z}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
   rational.json-simplify-1 [<=] 2.3	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(1 \cdot \left(\left(e^{-6.5 - z} \cdot {\left(\frac{\color{blue}{6.5 + z}}{4}\right)}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right) \cdot {4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right)\right) \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{\frac{z + 1}{\frac{676.5203681218851}{\frac{z}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]

8. Applied egg-rr 2.1

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(1 \cdot \left(\left(e^{-6.5 - z} \cdot {\left(\frac{6.5 + z}{4}\right)}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right) \cdot {4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right)\right) \cdot \left(\left(\left(0.9999999999998099 + \color{blue}{\frac{\frac{z \cdot 0.00147815209581367}{\left(z \cdot 0.00147815209581367 + \left(z + 1\right) \cdot -0.0007941933558411801\right) \cdot \frac{-1259.1392167224028}{z + 1}} + \left(z + 2\right) \cdot 0.0012964730004179177}{\frac{z \cdot 0.00147815209581367}{\left(z \cdot 0.00147815209581367 + \left(z + 1\right) \cdot -0.0007941933558411801\right) \cdot \frac{-1259.1392167224028}{z + 1}} \cdot \left(\left(z + 2\right) \cdot 0.0012964730004179177\right)}}\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]

9. Final simplification 2.1

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(1 \cdot \left(\left(e^{-6.5 - z} \cdot {\left(\frac{6.5 + z}{4}\right)}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right) \cdot {4}^{\left(-\left(0.5 + \left(-z\right)\right)\right)}\right)\right) \cdot \left(\left(\left(0.9999999999998099 + \frac{\frac{z \cdot 0.00147815209581367}{\left(z \cdot 0.00147815209581367 + \left(z + 1\right) \cdot -0.0007941933558411801\right) \cdot \frac{-1259.1392167224028}{z + 1}} + \left(z + 2\right) \cdot 0.0012964730004179177}{\frac{z \cdot 0.00147815209581367}{\left(z \cdot 0.00147815209581367 + \left(z + 1\right) \cdot -0.0007941933558411801\right) \cdot \frac{-1259.1392167224028}{z + 1}} \cdot \left(\left(z + 2\right) \cdot 0.0012964730004179177\right)}\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]

Alternatives

Alternative 1
Error	2.2
Cost	38784

\[\begin{array}{l} t_0 := -\left(0.5 + \left(-z\right)\right)\\ t_1 := \left(z + 2\right) \cdot 0.0012964730004179177\\ t_2 := \frac{\frac{z}{z \cdot 0.00147815209581367 + \left(z + 1\right) \cdot -0.0007941933558411801}}{\frac{-851833.326413742}{z + 1}}\\ \sqrt{\pi \cdot 2} \cdot \left(\left(1 \cdot \left(\left(e^{-6.5 - z} \cdot {\left(\frac{6.5 + z}{4}\right)}^{t_0}\right) \cdot {4}^{t_0}\right)\right) \cdot \left(\left(\left(0.9999999999998099 + \frac{t_2 + t_1}{t_2 \cdot t_1}\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \end{array} \]

Alternative 2
Error	2.2
Cost	37248

\[\begin{array}{l} t_0 := -\left(0.5 + \left(-z\right)\right)\\ \sqrt{\pi \cdot 2} \cdot \left(\left(1 \cdot \left(\left(e^{-6.5 - z} \cdot {\left(\frac{6.5 + z}{4}\right)}^{t_0}\right) \cdot {4}^{t_0}\right)\right) \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{z \cdot 0.00147815209581367 + \left(z + 1\right) \cdot -0.0007941933558411801}{z \cdot \left(\left(z + 1\right) \cdot -1.1739385734179321 \cdot 10^{-6}\right)} + \frac{771.3234287776531}{z + 2}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \end{array} \]

Alternative 3
Error	2.2
Cost	37120

\[\begin{array}{l} t_0 := -\left(0.5 + \left(-z\right)\right)\\ \sqrt{\pi \cdot 2} \cdot \left(\left(1 \cdot \left(\left(e^{-6.5 - z} \cdot {\left(\frac{6.5 + z}{4}\right)}^{t_0}\right) \cdot {4}^{t_0}\right)\right) \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{\frac{z + 1}{\frac{676.5203681218851}{\frac{z}{0.0006839587399724899 \cdot z - 0.0007941933558411801}}}} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \end{array} \]

Alternative 4
Error	3.7
Cost	36736

\[\sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-\left(z + 5.5\right)} \cdot e^{-1}\right)\right) \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{\frac{1259.1392167224028}{\frac{z + 1}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}{z \cdot -0.00147815209581367} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]

Alternative 5
Error	3.7
Cost	30272

\[\sqrt{\pi \cdot 2} \cdot \left(\frac{1}{\frac{e^{z + 6.5}}{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{\frac{1259.1392167224028}{\frac{z + 1}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}{z \cdot -0.00147815209581367} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]

Alternative 6
Error	3.7
Cost	30144

\[\sqrt{\pi \cdot 2} \cdot \left(\frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{\frac{1259.1392167224028}{\frac{z + 1}{\frac{-1 - z}{1259.1392167224028} + z \cdot 0.00147815209581367}}}{z \cdot -0.00147815209581367} + \frac{771.3234287776531}{2 + z}\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]

Alternative 7
Error	3.9
Cost	29504

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

Alternative 8
Error	3.7
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

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

Alternative 10
Error	49.9
Cost	28864

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

Alternative 11
Error	50.3
Cost	28292

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

Alternative 12
Error	50.3
Cost	27076

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

Alternative 13
Error	51.6
Cost	26884

\[\begin{array}{l} \mathbf{if}\;z \leq 4:\\ \;\;\;\;676.5203681218851 \cdot \left(\frac{\sqrt{\pi \cdot 0.3076923076923077}}{z} \cdot e^{-6.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(0.9999999999998099 \cdot e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(z - -6.5\right)\right) + \left(-6.5 - z\right)}\right)\\ \end{array} \]

Alternative 14
Error	51.9
Cost	26692

\[\begin{array}{l} \mathbf{if}\;z \leq 4:\\ \;\;\;\;676.5203681218851 \cdot \left(\frac{\sqrt{\pi \cdot 0.3076923076923077}}{z} \cdot e^{-6.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(0.9999999999998099 \cdot \left({\left(z + 6.5\right)}^{\left(z - 0.5\right)} \cdot e^{-6.5 - z}\right)\right)\\ \end{array} \]

Alternative 15
Error	55.6
Cost	19712

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

Reproduce

herbie shell --seed 2023075 
(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)))))