Jmat.Real.gamma, branch z greater than 0.5

Average Error: 4.2 → 2.5

Time: 30.8s

Precision: binary64

Cost: 36740

\[z > 0.5\]

Math

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

↓

\[\begin{array}{l} t_0 := \left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\\ t_1 := \sqrt{2 \cdot \pi}\\ \mathbf{if}\;z - 1 \leq 145:\\ \;\;\;\;t_0 \cdot \left({\left(6.5 + z\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot t_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\left(e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(6.5 + z\right)\right) + \left(-13 - \left(z + z\right)\right)} \cdot e^{6.5 + z}\right) \cdot t_1\right)\\ \end{array} \]

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
         (+
          (+
           0.9999999999998099
           (+
            (/ 676.5203681218851 z)
            (+
             (/ -1259.1392167224028 (+ z 1.0))
             (/ 771.3234287776531 (+ 2.0 z)))))
          (+
           (+
            (/ -176.6150291621406 (+ z 3.0))
            (+
             (/ 9.984369578019572e-6 (+ z 6.0))
             (/ 1.5056327351493116e-7 (+ z 7.0))))
           (+
            (/ -0.13857109526572012 (+ z 5.0))
            (/ 12.507343278686905 (+ z 4.0))))))
        (t_1 (sqrt (* 2.0 PI))))
   (if (<= (- z 1.0) 145.0)
     (* t_0 (* (pow (+ 6.5 z) (+ z -0.5)) (* (exp (- -6.5 z)) t_1)))
     (*
      t_0
      (*
       (*
        (exp (+ (* (+ 0.5 (- z)) (- (log (+ 6.5 z)))) (- -13.0 (+ z z))))
        (exp (+ 6.5 z)))
       t_1)))))

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 = (0.9999999999998099 + ((676.5203681218851 / z) + ((-1259.1392167224028 / (z + 1.0)) + (771.3234287776531 / (2.0 + z))))) + (((-176.6150291621406 / (z + 3.0)) + ((9.984369578019572e-6 / (z + 6.0)) + (1.5056327351493116e-7 / (z + 7.0)))) + ((-0.13857109526572012 / (z + 5.0)) + (12.507343278686905 / (z + 4.0))));
	double t_1 = sqrt((2.0 * ((double) M_PI)));
	double tmp;
	if ((z - 1.0) <= 145.0) {
		tmp = t_0 * (pow((6.5 + z), (z + -0.5)) * (exp((-6.5 - z)) * t_1));
	} else {
		tmp = t_0 * ((exp((((0.5 + -z) * -log((6.5 + z))) + (-13.0 - (z + z)))) * exp((6.5 + z))) * t_1);
	}
	return tmp;
}

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 = (0.9999999999998099 + ((676.5203681218851 / z) + ((-1259.1392167224028 / (z + 1.0)) + (771.3234287776531 / (2.0 + z))))) + (((-176.6150291621406 / (z + 3.0)) + ((9.984369578019572e-6 / (z + 6.0)) + (1.5056327351493116e-7 / (z + 7.0)))) + ((-0.13857109526572012 / (z + 5.0)) + (12.507343278686905 / (z + 4.0))));
	double t_1 = Math.sqrt((2.0 * Math.PI));
	double tmp;
	if ((z - 1.0) <= 145.0) {
		tmp = t_0 * (Math.pow((6.5 + z), (z + -0.5)) * (Math.exp((-6.5 - z)) * t_1));
	} else {
		tmp = t_0 * ((Math.exp((((0.5 + -z) * -Math.log((6.5 + z))) + (-13.0 - (z + z)))) * Math.exp((6.5 + z))) * t_1);
	}
	return tmp;
}

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 = (0.9999999999998099 + ((676.5203681218851 / z) + ((-1259.1392167224028 / (z + 1.0)) + (771.3234287776531 / (2.0 + z))))) + (((-176.6150291621406 / (z + 3.0)) + ((9.984369578019572e-6 / (z + 6.0)) + (1.5056327351493116e-7 / (z + 7.0)))) + ((-0.13857109526572012 / (z + 5.0)) + (12.507343278686905 / (z + 4.0))))
	t_1 = math.sqrt((2.0 * math.pi))
	tmp = 0
	if (z - 1.0) <= 145.0:
		tmp = t_0 * (math.pow((6.5 + z), (z + -0.5)) * (math.exp((-6.5 - z)) * t_1))
	else:
		tmp = t_0 * ((math.exp((((0.5 + -z) * -math.log((6.5 + z))) + (-13.0 - (z + z)))) * math.exp((6.5 + z))) * t_1)
	return tmp

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(0.9999999999998099 + Float64(Float64(676.5203681218851 / z) + Float64(Float64(-1259.1392167224028 / Float64(z + 1.0)) + Float64(771.3234287776531 / Float64(2.0 + z))))) + Float64(Float64(Float64(-176.6150291621406 / Float64(z + 3.0)) + Float64(Float64(9.984369578019572e-6 / Float64(z + 6.0)) + Float64(1.5056327351493116e-7 / Float64(z + 7.0)))) + Float64(Float64(-0.13857109526572012 / Float64(z + 5.0)) + Float64(12.507343278686905 / Float64(z + 4.0)))))
	t_1 = sqrt(Float64(2.0 * pi))
	tmp = 0.0
	if (Float64(z - 1.0) <= 145.0)
		tmp = Float64(t_0 * Float64((Float64(6.5 + z) ^ Float64(z + -0.5)) * Float64(exp(Float64(-6.5 - z)) * t_1)));
	else
		tmp = Float64(t_0 * Float64(Float64(exp(Float64(Float64(Float64(0.5 + Float64(-z)) * Float64(-log(Float64(6.5 + z)))) + Float64(-13.0 - Float64(z + z)))) * exp(Float64(6.5 + z))) * t_1));
	end
	return tmp
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_2 = code(z)
	t_0 = (0.9999999999998099 + ((676.5203681218851 / z) + ((-1259.1392167224028 / (z + 1.0)) + (771.3234287776531 / (2.0 + z))))) + (((-176.6150291621406 / (z + 3.0)) + ((9.984369578019572e-6 / (z + 6.0)) + (1.5056327351493116e-7 / (z + 7.0)))) + ((-0.13857109526572012 / (z + 5.0)) + (12.507343278686905 / (z + 4.0))));
	t_1 = sqrt((2.0 * pi));
	tmp = 0.0;
	if ((z - 1.0) <= 145.0)
		tmp = t_0 * (((6.5 + z) ^ (z + -0.5)) * (exp((-6.5 - z)) * t_1));
	else
		tmp = t_0 * ((exp((((0.5 + -z) * -log((6.5 + z))) + (-13.0 - (z + z)))) * exp((6.5 + z))) * t_1);
	end
	tmp_2 = tmp;
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[(N[(0.9999999999998099 + N[(N[(676.5203681218851 / z), $MachinePrecision] + N[(N[(-1259.1392167224028 / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(2.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-176.6150291621406 / N[(z + 3.0), $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] + N[(N[(-0.13857109526572012 / N[(z + 5.0), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(z + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(z - 1.0), $MachinePrecision], 145.0], N[(t$95$0 * N[(N[Power[N[(6.5 + z), $MachinePrecision], N[(z + -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(-6.5 - z), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(N[Exp[N[(N[(N[(0.5 + (-z)), $MachinePrecision] * (-N[Log[N[(6.5 + z), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] + N[(-13.0 - N[(z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(6.5 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z 1) < 145

   1. Initial program 2.4

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

   2. Simplified 2.4

      \[\leadsto \color{blue}{{\left(\left(z + -1\right) + 7.5\right)}^{\left(z + -0.5\right)} \cdot \left(\left(\left(\left(\left(\left(\frac{676.5203681218851}{z + 0} + \frac{-1259.1392167224028}{z + 1}\right) + \left(0.9999999999998099 + \frac{771.3234287776531}{2 + z}\right)\right) + \frac{12.507343278686905}{z + 4}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(z + -1\right) + 7.5\right)}\right)\right)} \]
      Proof

      [Start] 2.4	\[ \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      rational_best-simplify-2 [=>] 2.4	\[ \color{blue}{\left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right)} \]
      rational_best-simplify-2 [=>] 2.4	\[ \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \cdot \color{blue}{\left(e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right)\right)} \]
      rational_best-simplify-2 [=>] 2.4	\[ \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \cdot \left(e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)} \cdot \color{blue}{\left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot \sqrt{\pi \cdot 2}\right)}\right) \]

   3. Applied egg-rr 2.4

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

   4. Simplified 2.4

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

      [Start] 2.4	\[ 0 + \left(\left(\left(\left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right)\right) + \frac{12.507343278686905}{z + 4}\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right)\right) \]
      rational_best-simplify-1 [=>] 2.4	\[ \color{blue}{\left(\left(\left(\left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right)\right) + \frac{12.507343278686905}{z + 4}\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right)\right) + 0} \]
      rational_best-simplify-4 [=>] 2.4	\[ \color{blue}{\left(\left(\left(\left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right)\right) + \frac{12.507343278686905}{z + 4}\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right)\right)} \]

   5. Applied egg-rr 2.4

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

   6. Simplified 2.4

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

      [Start] 2.4	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left(e^{-\left(z + 6.5\right)} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \sqrt{2 \cdot \pi}\right) + 0\right) \]
      rational_best-simplify-4 [=>] 2.4	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \color{blue}{\left(e^{-\left(z + 6.5\right)} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \sqrt{2 \cdot \pi}\right)\right)} \]
      rational_best-simplify-44 [=>] 2.4	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \color{blue}{\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-\left(z + 6.5\right)} \cdot \sqrt{2 \cdot \pi}\right)\right)} \]
      rational_best-simplify-1 [=>] 2.4	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left({\color{blue}{\left(6.5 + z\right)}}^{\left(z + -0.5\right)} \cdot \left(e^{-\left(z + 6.5\right)} \cdot \sqrt{2 \cdot \pi}\right)\right) \]
      rational_best-simplify-11 [=>] 2.4	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left({\left(6.5 + z\right)}^{\left(z + -0.5\right)} \cdot \left(e^{\color{blue}{0 - \left(z + 6.5\right)}} \cdot \sqrt{2 \cdot \pi}\right)\right) \]
      metadata-eval [<=] 2.4	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left({\left(6.5 + z\right)}^{\left(z + -0.5\right)} \cdot \left(e^{0 - \left(z + \color{blue}{\left(6.5 - 0\right)}\right)} \cdot \sqrt{2 \cdot \pi}\right)\right) \]
      rational_best-simplify-46 [<=] 2.4	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left({\left(6.5 + z\right)}^{\left(z + -0.5\right)} \cdot \left(e^{0 - \color{blue}{\left(6.5 - \left(0 - z\right)\right)}} \cdot \sqrt{2 \cdot \pi}\right)\right) \]
      rational_best-simplify-45 [=>] 2.4	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left({\left(6.5 + z\right)}^{\left(z + -0.5\right)} \cdot \left(e^{0 - \color{blue}{\left(z - \left(0 - 6.5\right)\right)}} \cdot \sqrt{2 \cdot \pi}\right)\right) \]
      metadata-eval [=>] 2.4	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left({\left(6.5 + z\right)}^{\left(z + -0.5\right)} \cdot \left(e^{0 - \left(z - \color{blue}{-6.5}\right)} \cdot \sqrt{2 \cdot \pi}\right)\right) \]
      rational_best-simplify-45 [=>] 2.4	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left({\left(6.5 + z\right)}^{\left(z + -0.5\right)} \cdot \left(e^{\color{blue}{-6.5 - \left(z - 0\right)}} \cdot \sqrt{2 \cdot \pi}\right)\right) \]
      rational_best-simplify-6 [=>] 2.4	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left({\left(6.5 + z\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - \color{blue}{z}} \cdot \sqrt{2 \cdot \pi}\right)\right) \]

   if 145 < (-.f64 z 1)

   1. Initial program 64.0

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

   2. Simplified 64.0

      \[\leadsto \color{blue}{{\left(\left(z + -1\right) + 7.5\right)}^{\left(z + -0.5\right)} \cdot \left(\left(\left(\left(\left(\left(\frac{676.5203681218851}{z + 0} + \frac{-1259.1392167224028}{z + 1}\right) + \left(0.9999999999998099 + \frac{771.3234287776531}{2 + z}\right)\right) + \frac{12.507343278686905}{z + 4}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(z + -1\right) + 7.5\right)}\right)\right)} \]
      Proof

      [Start] 64.0	\[ \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      rational_best-simplify-2 [=>] 64.0	\[ \color{blue}{\left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right)} \]
      rational_best-simplify-2 [=>] 64.0	\[ \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \cdot \color{blue}{\left(e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right)\right)} \]
      rational_best-simplify-2 [=>] 64.0	\[ \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \cdot \left(e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)} \cdot \color{blue}{\left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot \sqrt{\pi \cdot 2}\right)}\right) \]

   3. Applied egg-rr 64.0

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

   4. Simplified 64.0

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

      [Start] 64.0	\[ 0 + \left(\left(\left(\left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right)\right) + \frac{12.507343278686905}{z + 4}\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right)\right) \]
      rational_best-simplify-1 [=>] 64.0	\[ \color{blue}{\left(\left(\left(\left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right)\right) + \frac{12.507343278686905}{z + 4}\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right)\right) + 0} \]
      rational_best-simplify-4 [=>] 64.0	\[ \color{blue}{\left(\left(\left(\left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right)\right) + \frac{12.507343278686905}{z + 4}\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right)\right)} \]

   5. Applied egg-rr 64.0

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

   6. Simplified 64.0

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

      [Start] 64.0	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left(e^{-\left(z + 6.5\right)} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \sqrt{2 \cdot \pi}\right) + 0\right) \]
      rational_best-simplify-4 [=>] 64.0	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \color{blue}{\left(e^{-\left(z + 6.5\right)} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \sqrt{2 \cdot \pi}\right)\right)} \]
      rational_best-simplify-44 [=>] 64.0	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \color{blue}{\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-\left(z + 6.5\right)} \cdot \sqrt{2 \cdot \pi}\right)\right)} \]
      rational_best-simplify-1 [=>] 64.0	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left({\color{blue}{\left(6.5 + z\right)}}^{\left(z + -0.5\right)} \cdot \left(e^{-\left(z + 6.5\right)} \cdot \sqrt{2 \cdot \pi}\right)\right) \]
      rational_best-simplify-11 [=>] 64.0	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left({\left(6.5 + z\right)}^{\left(z + -0.5\right)} \cdot \left(e^{\color{blue}{0 - \left(z + 6.5\right)}} \cdot \sqrt{2 \cdot \pi}\right)\right) \]
      metadata-eval [<=] 64.0	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left({\left(6.5 + z\right)}^{\left(z + -0.5\right)} \cdot \left(e^{0 - \left(z + \color{blue}{\left(6.5 - 0\right)}\right)} \cdot \sqrt{2 \cdot \pi}\right)\right) \]
      rational_best-simplify-46 [<=] 64.0	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left({\left(6.5 + z\right)}^{\left(z + -0.5\right)} \cdot \left(e^{0 - \color{blue}{\left(6.5 - \left(0 - z\right)\right)}} \cdot \sqrt{2 \cdot \pi}\right)\right) \]
      rational_best-simplify-45 [=>] 64.0	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left({\left(6.5 + z\right)}^{\left(z + -0.5\right)} \cdot \left(e^{0 - \color{blue}{\left(z - \left(0 - 6.5\right)\right)}} \cdot \sqrt{2 \cdot \pi}\right)\right) \]
      metadata-eval [=>] 64.0	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left({\left(6.5 + z\right)}^{\left(z + -0.5\right)} \cdot \left(e^{0 - \left(z - \color{blue}{-6.5}\right)} \cdot \sqrt{2 \cdot \pi}\right)\right) \]
      rational_best-simplify-45 [=>] 64.0	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left({\left(6.5 + z\right)}^{\left(z + -0.5\right)} \cdot \left(e^{\color{blue}{-6.5 - \left(z - 0\right)}} \cdot \sqrt{2 \cdot \pi}\right)\right) \]
      rational_best-simplify-6 [=>] 64.0	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left({\left(6.5 + z\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - \color{blue}{z}} \cdot \sqrt{2 \cdot \pi}\right)\right) \]

   7. Taylor expanded in z around -inf 64.0

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

   8. Simplified 7.8

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

      [Start] 64.0	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left(\left(\sqrt{2} \cdot \left(e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right)} \cdot e^{-1 \cdot z - 6.5}\right)\right) \cdot \sqrt{\pi}\right) \]
      rational_best-simplify-2 [=>] 64.0	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \color{blue}{\left(\sqrt{\pi} \cdot \left(\sqrt{2} \cdot \left(e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right)} \cdot e^{-1 \cdot z - 6.5}\right)\right)\right)} \]
      rational_best-simplify-2 [=>] 64.0	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left(\sqrt{\pi} \cdot \color{blue}{\left(\left(e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right)} \cdot e^{-1 \cdot z - 6.5}\right) \cdot \sqrt{2}\right)}\right) \]
      rational_best-simplify-44 [=>] 64.0	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \color{blue}{\left(\left(e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right)} \cdot e^{-1 \cdot z - 6.5}\right) \cdot \left(\sqrt{\pi} \cdot \sqrt{2}\right)\right)} \]

   9. Applied egg-rr 64.0

      \[\leadsto \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left(\color{blue}{\left(e^{z + 6.5} \cdot \left(e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(z + 6.5\right)\right)} \cdot e^{-13 - \left(z + z\right)}\right)\right)} \cdot \sqrt{2 \cdot \pi}\right) \]

   10. Simplified 7.6

       \[\leadsto \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left(\color{blue}{\left(e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(6.5 + z\right)\right) + \left(-13 - \left(z + z\right)\right)} \cdot e^{6.5 + z}\right)} \cdot \sqrt{2 \cdot \pi}\right) \]
       Proof

       [Start] 64.0	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left(\left(e^{z + 6.5} \cdot \left(e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(z + 6.5\right)\right)} \cdot e^{-13 - \left(z + z\right)}\right)\right) \cdot \sqrt{2 \cdot \pi}\right) \]
       rational_best-simplify-2 [=>] 64.0	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left(\color{blue}{\left(\left(e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(z + 6.5\right)\right)} \cdot e^{-13 - \left(z + z\right)}\right) \cdot e^{z + 6.5}\right)} \cdot \sqrt{2 \cdot \pi}\right) \]
       exponential-simplify-1 [<=] 7.6	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left(\left(\color{blue}{e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(z + 6.5\right)\right) + \left(-13 - \left(z + z\right)\right)}} \cdot e^{z + 6.5}\right) \cdot \sqrt{2 \cdot \pi}\right) \]
       rational_best-simplify-1 [<=] 7.6	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left(\left(e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \color{blue}{\left(6.5 + z\right)}\right) + \left(-13 - \left(z + z\right)\right)} \cdot e^{z + 6.5}\right) \cdot \sqrt{2 \cdot \pi}\right) \]
       rational_best-simplify-1 [<=] 7.6	\[ \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left(\left(e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(6.5 + z\right)\right) + \left(-13 - \left(z + z\right)\right)} \cdot e^{\color{blue}{6.5 + z}}\right) \cdot \sqrt{2 \cdot \pi}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z - 1 \leq 145:\\ \;\;\;\;\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left({\left(6.5 + z\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \sqrt{2 \cdot \pi}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left(\left(e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(6.5 + z\right)\right) + \left(-13 - \left(z + z\right)\right)} \cdot e^{6.5 + z}\right) \cdot \sqrt{2 \cdot \pi}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	2.5
Cost	30340

\[\begin{array}{l} t_0 := \left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{2 + z}\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\\ t_1 := \sqrt{2 \cdot \pi}\\ \mathbf{if}\;z - 1 \leq 145:\\ \;\;\;\;t_0 \cdot \left({\left(6.5 + z\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot t_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(e^{\left(\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(6.5 + z\right)\right) + \left(-13 - \left(z + z\right)\right)\right) + \left(6.5 + z\right)} \cdot t_1\right)\\ \end{array} \]

Alternative 2
Error	2.4
Cost	29828

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

Alternative 3
Error	2.4
Cost	29764

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

Alternative 4
Error	2.4
Cost	29700

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

Alternative 5
Error	4.1
Cost	29504

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

Alternative 6
Error	4.1
Cost	29504

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

Alternative 7
Error	4.1
Cost	29504

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

Alternative 8
Error	49.4
Cost	28928

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

Alternative 9
Error	49.9
Cost	28740

\[\begin{array}{l} t_0 := \frac{-0.13857109526572012}{z + 5}\\ t_1 := \frac{12.507343278686905}{z + 4}\\ t_2 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_3 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ \mathbf{if}\;z \leq 5:\\ \;\;\;\;\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(z \cdot 1066.3083595279895 - 873.4775023335762\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(t_2 + t_3\right)\right) + \left(t_0 + t_1\right)\right)\right) \cdot \left(e^{-6.5} \cdot \sqrt{0.3076923076923077 \cdot \pi}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{12.0895510149948}{z}\right) + \left(t_2 + \left(t_3 + \left(t_1 + t_0\right)\right)\right)\right) \cdot e^{\left(0.5 + \left(-z\right)\right) \cdot \left(-\log \left(6.5 + z\right)\right) + \left(-6.5 + \left(-z\right)\right)}\right)\\ \end{array} \]

Alternative 10
Error	50.0
Cost	28736

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

Alternative 11
Error	51.4
Cost	27008

\[\sqrt{\pi \cdot 2} \cdot \left(e^{-\left(z + 6.5\right)} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(0.9999999999998099 + 24.458333333348836 \cdot \frac{1}{z}\right)\right)\right) \]

Alternative 12
Error	51.4
Cost	26884

\[\begin{array}{l} \mathbf{if}\;z \leq 5.8:\\ \;\;\;\;\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(z \cdot 1066.3083595279895 - 873.4775023335762\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left(e^{-6.5} \cdot \sqrt{0.3076923076923077 \cdot \pi}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(0.9999999999998099 \cdot e^{\log \left(z + 6.5\right) \cdot \left(-\left(0.5 + \left(-z\right)\right)\right) + \left(-6.5 - z\right)}\right)\\ \end{array} \]

Alternative 13
Error	51.9
Cost	26692

\[\begin{array}{l} \mathbf{if}\;z \leq 5.8:\\ \;\;\;\;\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(z \cdot 1066.3083595279895 - 873.4775023335762\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left(e^{-6.5} \cdot \sqrt{0.3076923076923077 \cdot \pi}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(0.9999999999998099 \cdot \left({\left(z + 6.5\right)}^{\left(z - 0.5\right)} \cdot e^{-6.5 - z}\right)\right)\\ \end{array} \]

Alternative 14
Error	55.4
Cost	22144

\[\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(z \cdot 1066.3083595279895 - 873.4775023335762\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \cdot \left(e^{-6.5} \cdot \sqrt{0.3076923076923077 \cdot \pi}\right) \]

Alternative 15
Error	55.4
Cost	20224

\[\left(\left(676.5203681218851 \cdot \frac{1}{z} + z \cdot 1085.1560852655925\right) - 928.2500554347674\right) \cdot \left(e^{-6.5} \cdot \sqrt{0.3076923076923077 \cdot \pi}\right) \]

Alternative 16
Error	55.7
Cost	19712

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

Alternative 17
Error	57.5
Cost	19584

\[0.9999999999998099 \cdot \left(e^{-6.5} \cdot \sqrt{0.3076923076923077 \cdot \pi}\right) \]

Reproduce

herbie shell --seed 2023092 
(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)))))