Jmat.Real.gamma, branch z greater than 0.5

Average Error: 4.0 → 2.3

Time: 33.7s

Precision: binary64

Cost: 56836

\[z > 0.5\]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{-176.6150291621406}{z + 3}\\ t_1 := \frac{12.507343278686905}{z + 4}\\ t_2 := \sqrt{\pi \cdot 2}\\ t_3 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_4 := \frac{771.3234287776531}{z + 2}\\ t_5 := \frac{-0.13857109526572012}{z + 5}\\ t_6 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;t_2 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \sqrt[3]{{\left(t_4 + \frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\frac{\frac{1585431.567088306}{z + 1} + \frac{851833.326413742}{z}}{z + 1} + \frac{\frac{457679.80848377093}{z}}{z}}\right)}^{3}}\right) + \left(\left(\left(t_5 + t_3\right) + \left(t_0 + t_1\right)\right) + t_6\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(\left(\frac{676.5203681218851}{z} + \left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(t_5 + t_1\right) + \left(t_4 + t_0\right)\right)\right)\right) + \left(t_3 + t_6\right)\right)\right) \cdot e^{\left(-6.5 - z\right) + \log \left(z + 6.5\right) \cdot \left(z + -0.5\right)}\right)\\ \end{array} \]

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 (/ -176.6150291621406 (+ z 3.0)))
        (t_1 (/ 12.507343278686905 (+ z 4.0)))
        (t_2 (sqrt (* PI 2.0)))
        (t_3 (/ 9.984369578019572e-6 (+ z 6.0)))
        (t_4 (/ 771.3234287776531 (+ z 2.0)))
        (t_5 (/ -0.13857109526572012 (+ z 5.0)))
        (t_6 (/ 1.5056327351493116e-7 (+ z 7.0))))
   (if (<= (+ z -1.0) 140.0)
     (*
      t_2
      (*
       (pow (+ z 6.5) (+ z -0.5))
       (*
        (exp (- -6.5 z))
        (+
         (+
          0.9999999999998099
          (cbrt
           (pow
            (+
             t_4
             (/
              (+
               (/ 309629712.5173946 (pow z 3.0))
               (/ -1996279061.5505414 (pow (+ z 1.0) 3.0)))
              (+
               (/
                (+ (/ 1585431.567088306 (+ z 1.0)) (/ 851833.326413742 z))
                (+ z 1.0))
               (/ (/ 457679.80848377093 z) z))))
            3.0)))
         (+ (+ (+ t_5 t_3) (+ t_0 t_1)) t_6)))))
     (*
      t_2
      (*
       (+
        (/ 676.5203681218851 z)
        (+
         (+
          0.9999999999998099
          (+ (/ -1259.1392167224028 (+ z 1.0)) (+ (+ t_5 t_1) (+ t_4 t_0))))
         (+ t_3 t_6)))
       (exp (+ (- -6.5 z) (* (log (+ z 6.5)) (+ z -0.5)))))))))

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 = -176.6150291621406 / (z + 3.0);
	double t_1 = 12.507343278686905 / (z + 4.0);
	double t_2 = sqrt((((double) M_PI) * 2.0));
	double t_3 = 9.984369578019572e-6 / (z + 6.0);
	double t_4 = 771.3234287776531 / (z + 2.0);
	double t_5 = -0.13857109526572012 / (z + 5.0);
	double t_6 = 1.5056327351493116e-7 / (z + 7.0);
	double tmp;
	if ((z + -1.0) <= 140.0) {
		tmp = t_2 * (pow((z + 6.5), (z + -0.5)) * (exp((-6.5 - z)) * ((0.9999999999998099 + cbrt(pow((t_4 + (((309629712.5173946 / pow(z, 3.0)) + (-1996279061.5505414 / pow((z + 1.0), 3.0))) / ((((1585431.567088306 / (z + 1.0)) + (851833.326413742 / z)) / (z + 1.0)) + ((457679.80848377093 / z) / z)))), 3.0))) + (((t_5 + t_3) + (t_0 + t_1)) + t_6))));
	} else {
		tmp = t_2 * (((676.5203681218851 / z) + ((0.9999999999998099 + ((-1259.1392167224028 / (z + 1.0)) + ((t_5 + t_1) + (t_4 + t_0)))) + (t_3 + t_6))) * exp(((-6.5 - z) + (log((z + 6.5)) * (z + -0.5)))));
	}
	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 = -176.6150291621406 / (z + 3.0);
	double t_1 = 12.507343278686905 / (z + 4.0);
	double t_2 = Math.sqrt((Math.PI * 2.0));
	double t_3 = 9.984369578019572e-6 / (z + 6.0);
	double t_4 = 771.3234287776531 / (z + 2.0);
	double t_5 = -0.13857109526572012 / (z + 5.0);
	double t_6 = 1.5056327351493116e-7 / (z + 7.0);
	double tmp;
	if ((z + -1.0) <= 140.0) {
		tmp = t_2 * (Math.pow((z + 6.5), (z + -0.5)) * (Math.exp((-6.5 - z)) * ((0.9999999999998099 + Math.cbrt(Math.pow((t_4 + (((309629712.5173946 / Math.pow(z, 3.0)) + (-1996279061.5505414 / Math.pow((z + 1.0), 3.0))) / ((((1585431.567088306 / (z + 1.0)) + (851833.326413742 / z)) / (z + 1.0)) + ((457679.80848377093 / z) / z)))), 3.0))) + (((t_5 + t_3) + (t_0 + t_1)) + t_6))));
	} else {
		tmp = t_2 * (((676.5203681218851 / z) + ((0.9999999999998099 + ((-1259.1392167224028 / (z + 1.0)) + ((t_5 + t_1) + (t_4 + t_0)))) + (t_3 + t_6))) * Math.exp(((-6.5 - z) + (Math.log((z + 6.5)) * (z + -0.5)))));
	}
	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(-176.6150291621406 / Float64(z + 3.0))
	t_1 = Float64(12.507343278686905 / Float64(z + 4.0))
	t_2 = sqrt(Float64(pi * 2.0))
	t_3 = Float64(9.984369578019572e-6 / Float64(z + 6.0))
	t_4 = Float64(771.3234287776531 / Float64(z + 2.0))
	t_5 = Float64(-0.13857109526572012 / Float64(z + 5.0))
	t_6 = Float64(1.5056327351493116e-7 / Float64(z + 7.0))
	tmp = 0.0
	if (Float64(z + -1.0) <= 140.0)
		tmp = Float64(t_2 * Float64((Float64(z + 6.5) ^ Float64(z + -0.5)) * Float64(exp(Float64(-6.5 - z)) * Float64(Float64(0.9999999999998099 + cbrt((Float64(t_4 + Float64(Float64(Float64(309629712.5173946 / (z ^ 3.0)) + Float64(-1996279061.5505414 / (Float64(z + 1.0) ^ 3.0))) / Float64(Float64(Float64(Float64(1585431.567088306 / Float64(z + 1.0)) + Float64(851833.326413742 / z)) / Float64(z + 1.0)) + Float64(Float64(457679.80848377093 / z) / z)))) ^ 3.0))) + Float64(Float64(Float64(t_5 + t_3) + Float64(t_0 + t_1)) + t_6)))));
	else
		tmp = Float64(t_2 * Float64(Float64(Float64(676.5203681218851 / z) + Float64(Float64(0.9999999999998099 + Float64(Float64(-1259.1392167224028 / Float64(z + 1.0)) + Float64(Float64(t_5 + t_1) + Float64(t_4 + t_0)))) + Float64(t_3 + t_6))) * exp(Float64(Float64(-6.5 - z) + Float64(log(Float64(z + 6.5)) * Float64(z + -0.5))))));
	end
	return 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[(-176.6150291621406 / N[(z + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(12.507343278686905 / N[(z + 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(9.984369578019572e-6 / N[(z + 6.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(771.3234287776531 / N[(z + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(-0.13857109526572012 / N[(z + 5.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(1.5056327351493116e-7 / N[(z + 7.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z + -1.0), $MachinePrecision], 140.0], N[(t$95$2 * N[(N[Power[N[(z + 6.5), $MachinePrecision], N[(z + -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(-6.5 - z), $MachinePrecision]], $MachinePrecision] * N[(N[(0.9999999999998099 + N[Power[N[Power[N[(t$95$4 + N[(N[(N[(309629712.5173946 / N[Power[z, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-1996279061.5505414 / N[Power[N[(z + 1.0), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(1585431.567088306 / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] + N[(851833.326413742 / z), $MachinePrecision]), $MachinePrecision] / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(457679.80848377093 / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$5 + t$95$3), $MachinePrecision] + N[(t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(N[(N[(676.5203681218851 / z), $MachinePrecision] + N[(N[(0.9999999999998099 + N[(N[(-1259.1392167224028 / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$5 + t$95$1), $MachinePrecision] + N[(t$95$4 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(-6.5 - z), $MachinePrecision] + N[(N[Log[N[(z + 6.5), $MachinePrecision]], $MachinePrecision] * N[(z + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z 1) < 140

   1. Initial program 2.2

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

   2. Simplified 2.1

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

      [Start] 2.2	\[ \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      associate-*l* [=>] 2.2	\[ \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right)\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      associate-*l* [=>] 2.2	\[ \color{blue}{\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\right)} \]

   3. Applied egg-rr 2.3

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

   4. Simplified 2.1

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \color{blue}{\frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\mathsf{fma}\left(\frac{676.5203681218851}{z}, \frac{676.5203681218851}{z}, \frac{-1259.1392167224028}{z + 1} \cdot \left(\frac{-1259.1392167224028}{z + 1} - \frac{676.5203681218851}{z}\right)\right)}}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      Proof

      [Start] 2.3	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{{\left(\frac{676.5203681218851}{z}\right)}^{3} + {\left(\frac{-1259.1392167224028}{z + 1}\right)}^{3}}{\frac{676.5203681218851}{z} \cdot \frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} \cdot \frac{-1259.1392167224028}{z + 1} - \frac{676.5203681218851}{z} \cdot \frac{-1259.1392167224028}{z + 1}\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      cube-div [=>] 2.3	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\color{blue}{\frac{{676.5203681218851}^{3}}{{z}^{3}}} + {\left(\frac{-1259.1392167224028}{z + 1}\right)}^{3}}{\frac{676.5203681218851}{z} \cdot \frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} \cdot \frac{-1259.1392167224028}{z + 1} - \frac{676.5203681218851}{z} \cdot \frac{-1259.1392167224028}{z + 1}\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      metadata-eval [=>] 2.3	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\frac{\color{blue}{309629712.5173946}}{{z}^{3}} + {\left(\frac{-1259.1392167224028}{z + 1}\right)}^{3}}{\frac{676.5203681218851}{z} \cdot \frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} \cdot \frac{-1259.1392167224028}{z + 1} - \frac{676.5203681218851}{z} \cdot \frac{-1259.1392167224028}{z + 1}\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      cube-div [=>] 2.1	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\frac{309629712.5173946}{{z}^{3}} + \color{blue}{\frac{{-1259.1392167224028}^{3}}{{\left(z + 1\right)}^{3}}}}{\frac{676.5203681218851}{z} \cdot \frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} \cdot \frac{-1259.1392167224028}{z + 1} - \frac{676.5203681218851}{z} \cdot \frac{-1259.1392167224028}{z + 1}\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      metadata-eval [=>] 2.1	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\frac{309629712.5173946}{{z}^{3}} + \frac{\color{blue}{-1996279061.5505414}}{{\left(z + 1\right)}^{3}}}{\frac{676.5203681218851}{z} \cdot \frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} \cdot \frac{-1259.1392167224028}{z + 1} - \frac{676.5203681218851}{z} \cdot \frac{-1259.1392167224028}{z + 1}\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      fma-def [=>] 2.1	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\color{blue}{\mathsf{fma}\left(\frac{676.5203681218851}{z}, \frac{676.5203681218851}{z}, \frac{-1259.1392167224028}{z + 1} \cdot \frac{-1259.1392167224028}{z + 1} - \frac{676.5203681218851}{z} \cdot \frac{-1259.1392167224028}{z + 1}\right)}}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      distribute-rgt-out-- [=>] 2.1	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\mathsf{fma}\left(\frac{676.5203681218851}{z}, \frac{676.5203681218851}{z}, \color{blue}{\frac{-1259.1392167224028}{z + 1} \cdot \left(\frac{-1259.1392167224028}{z + 1} - \frac{676.5203681218851}{z}\right)}\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

   5. Applied egg-rr 2.2

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \color{blue}{\sqrt[3]{\left(\left(\frac{771.3234287776531}{z + 2} + \frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\mathsf{fma}\left(\frac{676.5203681218851}{z}, \frac{676.5203681218851}{z}, \frac{-1259.1392167224028}{z + 1} \cdot \left(\frac{-1259.1392167224028}{z + 1} - \frac{676.5203681218851}{z}\right)\right)}\right) \cdot \left(\frac{771.3234287776531}{z + 2} + \frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\mathsf{fma}\left(\frac{676.5203681218851}{z}, \frac{676.5203681218851}{z}, \frac{-1259.1392167224028}{z + 1} \cdot \left(\frac{-1259.1392167224028}{z + 1} - \frac{676.5203681218851}{z}\right)\right)}\right)\right) \cdot \left(\frac{771.3234287776531}{z + 2} + \frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\mathsf{fma}\left(\frac{676.5203681218851}{z}, \frac{676.5203681218851}{z}, \frac{-1259.1392167224028}{z + 1} \cdot \left(\frac{-1259.1392167224028}{z + 1} - \frac{676.5203681218851}{z}\right)\right)}\right)}}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

   6. Simplified 2.1

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \color{blue}{\sqrt[3]{{\left(\frac{771.3234287776531}{z + 2} + \frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\frac{\frac{1585431.567088306}{z + 1} + \frac{851833.326413742}{z}}{z + 1} + \frac{\frac{457679.80848377093}{z}}{z}}\right)}^{3}}}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      Proof

      [Start] 2.2

      \[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \sqrt[3]{\left(\left(\frac{771.3234287776531}{z + 2} + \frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\mathsf{fma}\left(\frac{676.5203681218851}{z}, \frac{676.5203681218851}{z}, \frac{-1259.1392167224028}{z + 1} \cdot \left(\frac{-1259.1392167224028}{z + 1} - \frac{676.5203681218851}{z}\right)\right)}\right) \cdot \left(\frac{771.3234287776531}{z + 2} + \frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\mathsf{fma}\left(\frac{676.5203681218851}{z}, \frac{676.5203681218851}{z}, \frac{-1259.1392167224028}{z + 1} \cdot \left(\frac{-1259.1392167224028}{z + 1} - \frac{676.5203681218851}{z}\right)\right)}\right)\right) \cdot \left(\frac{771.3234287776531}{z + 2} + \frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\mathsf{fma}\left(\frac{676.5203681218851}{z}, \frac{676.5203681218851}{z}, \frac{-1259.1392167224028}{z + 1} \cdot \left(\frac{-1259.1392167224028}{z + 1} - \frac{676.5203681218851}{z}\right)\right)}\right)}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

      *-commutative [=>] 2.2

      \[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \sqrt[3]{\color{blue}{\left(\frac{771.3234287776531}{z + 2} + \frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\mathsf{fma}\left(\frac{676.5203681218851}{z}, \frac{676.5203681218851}{z}, \frac{-1259.1392167224028}{z + 1} \cdot \left(\frac{-1259.1392167224028}{z + 1} - \frac{676.5203681218851}{z}\right)\right)}\right) \cdot \left(\left(\frac{771.3234287776531}{z + 2} + \frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\mathsf{fma}\left(\frac{676.5203681218851}{z}, \frac{676.5203681218851}{z}, \frac{-1259.1392167224028}{z + 1} \cdot \left(\frac{-1259.1392167224028}{z + 1} - \frac{676.5203681218851}{z}\right)\right)}\right) \cdot \left(\frac{771.3234287776531}{z + 2} + \frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\mathsf{fma}\left(\frac{676.5203681218851}{z}, \frac{676.5203681218851}{z}, \frac{-1259.1392167224028}{z + 1} \cdot \left(\frac{-1259.1392167224028}{z + 1} - \frac{676.5203681218851}{z}\right)\right)}\right)\right)}}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

   if 140 < (-.f64 z 1)

   1. Initial program 61.5

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

   2. Simplified 61.5

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

      [Start] 61.5	\[ \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      associate-*l* [=>] 61.5	\[ \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right)\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      associate-*l* [=>] 61.5	\[ \color{blue}{\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\right)} \]

   3. Applied egg-rr 61.5

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

   4. Simplified 61.5

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

      [Start] 61.5	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(1 \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{z + 2} + \frac{12.507343278686905}{z + 4}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right)\right) \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}}\right) \]
      *-lft-identity [=>] 61.5	\[ \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{z + 2} + \frac{12.507343278686905}{z + 4}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right)} \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}}\right) \]
      +-commutative [=>] 61.5	\[ \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\left(\left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{z + 2} + \frac{12.507343278686905}{z + 4}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right) + 0.9999999999998099\right)} \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}}\right) \]
      associate-+l+ [=>] 61.5	\[ \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\left(\frac{676.5203681218851}{z} + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{z + 2} + \frac{12.507343278686905}{z + 4}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) + 0.9999999999998099\right)\right)} \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}}\right) \]
      +-commutative [<=] 61.5	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\frac{676.5203681218851}{z} + \color{blue}{\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{z + 2} + \frac{12.507343278686905}{z + 4}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)}\right) \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}}\right) \]
      associate-+r+ [=>] 61.5	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\frac{676.5203681218851}{z} + \left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{z + 1} + \color{blue}{\left(\left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{z + 2} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)}\right)\right)\right) \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}}\right) \]

   5. Taylor expanded in z around -inf 61.7

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

   6. Simplified 7.9

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

      [Start] 61.7	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\frac{676.5203681218851}{z} + \left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{771.3234287776531}{z + 2}\right) + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \cdot \frac{e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right)}}{e^{6.5 - -1 \cdot z}}\right) \]
      div-exp [=>] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\frac{676.5203681218851}{z} + \left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{771.3234287776531}{z + 2}\right) + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \cdot \color{blue}{e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right) - \left(6.5 - -1 \cdot z\right)}}\right) \]
      sub-neg [=>] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\frac{676.5203681218851}{z} + \left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{771.3234287776531}{z + 2}\right) + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \cdot e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right) - \color{blue}{\left(6.5 + \left(--1 \cdot z\right)\right)}}\right) \]
      associate--r+ [=>] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\frac{676.5203681218851}{z} + \left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{771.3234287776531}{z + 2}\right) + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \cdot e^{\color{blue}{\left(-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right) - 6.5\right) - \left(--1 \cdot z\right)}}\right) \]
      mul-1-neg [=>] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\frac{676.5203681218851}{z} + \left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{771.3234287776531}{z + 2}\right) + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \cdot e^{\left(-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right) - 6.5\right) - \left(-\color{blue}{\left(-z\right)}\right)}\right) \]
      remove-double-neg [=>] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\frac{676.5203681218851}{z} + \left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{771.3234287776531}{z + 2}\right) + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \cdot e^{\left(-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right) - 6.5\right) - \color{blue}{z}}\right) \]
      associate--r+ [<=] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\frac{676.5203681218851}{z} + \left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{771.3234287776531}{z + 2}\right) + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \cdot e^{\color{blue}{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right) - \left(6.5 + z\right)}}\right) \]
      mul-1-neg [=>] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\frac{676.5203681218851}{z} + \left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{771.3234287776531}{z + 2}\right) + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \cdot e^{\color{blue}{\left(-\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right)} - \left(6.5 + z\right)}\right) \]
      distribute-rgt-neg-in [=>] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\frac{676.5203681218851}{z} + \left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{771.3234287776531}{z + 2}\right) + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \cdot e^{\color{blue}{\log \left(6.5 - -1 \cdot z\right) \cdot \left(-\left(-1 \cdot z + 0.5\right)\right)} - \left(6.5 + z\right)}\right) \]
      sub-neg [=>] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\frac{676.5203681218851}{z} + \left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{771.3234287776531}{z + 2}\right) + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \cdot e^{\log \color{blue}{\left(6.5 + \left(--1 \cdot z\right)\right)} \cdot \left(-\left(-1 \cdot z + 0.5\right)\right) - \left(6.5 + z\right)}\right) \]
      mul-1-neg [=>] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\frac{676.5203681218851}{z} + \left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{771.3234287776531}{z + 2}\right) + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \cdot e^{\log \left(6.5 + \left(-\color{blue}{\left(-z\right)}\right)\right) \cdot \left(-\left(-1 \cdot z + 0.5\right)\right) - \left(6.5 + z\right)}\right) \]
      remove-double-neg [=>] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\frac{676.5203681218851}{z} + \left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{771.3234287776531}{z + 2}\right) + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \cdot e^{\log \left(6.5 + \color{blue}{z}\right) \cdot \left(-\left(-1 \cdot z + 0.5\right)\right) - \left(6.5 + z\right)}\right) \]
      +-commutative [=>] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\frac{676.5203681218851}{z} + \left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{771.3234287776531}{z + 2}\right) + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \cdot e^{\log \left(6.5 + z\right) \cdot \left(-\color{blue}{\left(0.5 + -1 \cdot z\right)}\right) - \left(6.5 + z\right)}\right) \]
      mul-1-neg [=>] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\frac{676.5203681218851}{z} + \left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{771.3234287776531}{z + 2}\right) + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \cdot e^{\log \left(6.5 + z\right) \cdot \left(-\left(0.5 + \color{blue}{\left(-z\right)}\right)\right) - \left(6.5 + z\right)}\right) \]
      unsub-neg [=>] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\frac{676.5203681218851}{z} + \left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{771.3234287776531}{z + 2}\right) + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \cdot e^{\log \left(6.5 + z\right) \cdot \left(-\color{blue}{\left(0.5 - z\right)}\right) - \left(6.5 + z\right)}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \sqrt[3]{{\left(\frac{771.3234287776531}{z + 2} + \frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\frac{\frac{1585431.567088306}{z + 1} + \frac{851833.326413742}{z}}{z + 1} + \frac{\frac{457679.80848377093}{z}}{z}}\right)}^{3}}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(\left(\frac{676.5203681218851}{z} + \left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right) + \left(\frac{771.3234287776531}{z + 2} + \frac{-176.6150291621406}{z + 3}\right)\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \cdot e^{\left(-6.5 - z\right) + \log \left(z + 6.5\right) \cdot \left(z + -0.5\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	2.2
Cost	51780

\[\begin{array}{l} t_0 := \frac{-176.6150291621406}{z + 3}\\ t_1 := \frac{\frac{1585431.567088306}{z + 1} + \frac{851833.326413742}{z}}{z + 1} + \frac{\frac{457679.80848377093}{z}}{z}\\ t_2 := \frac{12.507343278686905}{z + 4}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \frac{-0.13857109526572012}{z + 5}\\ t_5 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_6 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;t_3 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(\left(\left(t_4 + t_6\right) + \left(t_0 + t_2\right)\right) + t_5\right) + \left(0.9999999999998099 + \frac{\mathsf{fma}\left(771.3234287776531, t_1, \left(z + 2\right) \cdot \left(\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}\right)\right)}{\left(z + 2\right) \cdot t_1}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(\left(\frac{676.5203681218851}{z} + \left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(t_4 + t_2\right) + \left(\frac{771.3234287776531}{z + 2} + t_0\right)\right)\right)\right) + \left(t_6 + t_5\right)\right)\right) \cdot e^{\left(-6.5 - z\right) + \log \left(z + 6.5\right) \cdot \left(z + -0.5\right)}\right)\\ \end{array} \]

Alternative 2
Error	2.3
Cost	50500

\[\begin{array}{l} t_0 := \frac{771.3234287776531}{z + 2}\\ t_1 := \frac{12.507343278686905}{z + 4}\\ t_2 := \sqrt{\pi \cdot 2}\\ t_3 := \frac{-0.13857109526572012}{z + 5}\\ t_4 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_5 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_6 := \frac{-176.6150291621406}{z + 3}\\ t_7 := \frac{-1259.1392167224028}{z + 1}\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;t_2 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(\left(\left(t_3 + t_5\right) + \left(t_6 + t_1\right)\right) + t_4\right) + \left(0.9999999999998099 + \left(t_0 + \frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\mathsf{fma}\left(\frac{676.5203681218851}{z}, \frac{676.5203681218851}{z}, t_7 \cdot \left(t_7 + \frac{-676.5203681218851}{z}\right)\right)}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(\left(\frac{676.5203681218851}{z} + \left(\left(0.9999999999998099 + \left(t_7 + \left(\left(t_3 + t_1\right) + \left(t_0 + t_6\right)\right)\right)\right) + \left(t_5 + t_4\right)\right)\right) \cdot e^{\left(-6.5 - z\right) + \log \left(z + 6.5\right) \cdot \left(z + -0.5\right)}\right)\\ \end{array} \]

Alternative 3
Error	2.2
Cost	48964

\[\begin{array}{l} t_0 := \frac{771.3234287776531}{z + 2}\\ t_1 := \frac{-176.6150291621406}{z + 3}\\ t_2 := \frac{12.507343278686905}{z + 4}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \frac{-0.13857109526572012}{z + 5}\\ t_5 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_6 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;t_3 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(\left(\left(t_4 + t_6\right) + \left(t_1 + t_2\right)\right) + t_5\right) + \left(0.9999999999998099 + \left(t_0 + \frac{\mathsf{fma}\left(z, -1259.1392167224028, \mathsf{fma}\left(676.5203681218851, z, 676.5203681218851\right)\right)}{\mathsf{fma}\left(z, z, z\right)}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(\left(\frac{676.5203681218851}{z} + \left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(t_4 + t_2\right) + \left(t_0 + t_1\right)\right)\right)\right) + \left(t_6 + t_5\right)\right)\right) \cdot e^{\left(-6.5 - z\right) + \log \left(z + 6.5\right) \cdot \left(z + -0.5\right)}\right)\\ \end{array} \]

Alternative 4
Error	2.3
Cost	36420

\[\begin{array}{l} t_0 := \frac{771.3234287776531}{z + 2}\\ t_1 := \frac{-176.6150291621406}{z + 3}\\ t_2 := \frac{12.507343278686905}{z + 4}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \frac{-0.13857109526572012}{z + 5}\\ t_5 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_6 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;t_3 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(\left(\left(t_4 + t_6\right) + \left(t_1 + t_2\right)\right) + t_5\right) + \left(0.9999999999998099 + \left(t_0 + \frac{\mathsf{fma}\left(676.5203681218851, z + 1, z \cdot -1259.1392167224028\right)}{z \cdot \left(z + 1\right)}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(\left(\frac{676.5203681218851}{z} + \left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(t_4 + t_2\right) + \left(t_0 + t_1\right)\right)\right)\right) + \left(t_6 + t_5\right)\right)\right) \cdot e^{\left(-6.5 - z\right) + \log \left(z + 6.5\right) \cdot \left(z + -0.5\right)}\right)\\ \end{array} \]

Alternative 5
Error	2.3
Cost	29828

\[\begin{array}{l} t_0 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_1 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_2 := \frac{-176.6150291621406}{z + 3}\\ t_3 := \frac{-1259.1392167224028}{z + 1}\\ t_4 := \frac{-0.13857109526572012}{z + 5}\\ t_5 := \sqrt{\pi \cdot 2}\\ t_6 := \frac{771.3234287776531}{z + 2}\\ t_7 := \frac{12.507343278686905}{z + 4}\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;t_5 \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(t_6 + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + t_3\right)\right)\right) + \left(t_1 + \left(t_2 + \left(\left(t_4 + t_0\right) + t_7\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_5 \cdot \left(\left(\frac{676.5203681218851}{z} + \left(\left(0.9999999999998099 + \left(t_3 + \left(\left(t_4 + t_7\right) + \left(t_6 + t_2\right)\right)\right)\right) + \left(t_0 + t_1\right)\right)\right) \cdot e^{\left(-6.5 - z\right) + \log \left(z + 6.5\right) \cdot \left(z + -0.5\right)}\right)\\ \end{array} \]

Alternative 6
Error	2.4
Cost	29700

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

Alternative 7
Error	4.0
Cost	29504

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

Alternative 8
Error	4.0
Cost	29504

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

Alternative 9
Error	4.0
Cost	29504

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

Alternative 10
Error	46.8
Cost	28736

\[\sqrt{\pi \cdot 2} \cdot \left(\frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}} \cdot \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(0.9999999999998099 + \left(\frac{246.3374466535184}{z \cdot z} + \frac{12.0895510149948}{z}\right)\right)\right)\right) \]

Alternative 11
Error	47.6
Cost	27200

\[\sqrt{\pi \cdot 2} \cdot \left(\frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}} \cdot \left(0.9999999999998099 + \left(\frac{24.458333333348836}{z} + \frac{197.000868054939}{z \cdot z}\right)\right)\right) \]

Alternative 12
Error	47.6
Cost	27200

\[\sqrt{\pi \cdot 2} \cdot \left(\frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}} \cdot \left(\left(0.9999999999998099 + \frac{24.458333333348836}{z}\right) + \frac{197.000868054939}{z \cdot z}\right)\right) \]

Alternative 13
Error	50.4
Cost	26948

\[\begin{array}{l} \mathbf{if}\;z \leq 2.75:\\ \;\;\;\;\sqrt{140824.5564565449 \cdot \left(\pi \cdot \frac{e^{-13}}{z \cdot z}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(\frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}} \cdot \left(0.9999999999998099 + \frac{24.458333333348836}{z}\right)\right)\\ \end{array} \]

Alternative 14
Error	51.6
Cost	26756

\[\begin{array}{l} \mathbf{if}\;z \leq 4:\\ \;\;\;\;\sqrt{140824.5564565449 \cdot \left(\pi \cdot \frac{e^{-13}}{z \cdot z}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(0.9999999999998099 \cdot e^{\left(-6.5 - z\right) + \log \left(z + 6.5\right) \cdot \left(z + -0.5\right)}\right)\\ \end{array} \]

Alternative 15
Error	52.0
Cost	26692

\[\begin{array}{l} \mathbf{if}\;z \leq 4:\\ \;\;\;\;\sqrt{140824.5564565449 \cdot \left(\pi \cdot \frac{e^{-13}}{z \cdot z}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(0.9999999999998099 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right)\right)\\ \end{array} \]

Alternative 16
Error	55.6
Cost	19712

\[\sqrt{140824.5564565449 \cdot \left(\pi \cdot \frac{e^{-13}}{z \cdot z}\right)} \]

Reproduce

herbie shell --seed 2023053 
(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)))))