Jmat.Real.gamma, branch z greater than 0.5

Average Error: 3.9 → 2.2

Time: 30.7s

Precision: binary64

Cost: 62020

\[z > 0.5\]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{-176.6150291621406}{z + 3}\\ t_1 := \frac{12.507343278686905}{z + 4}\\ t_2 := \frac{-0.13857109526572012}{z + 5}\\ t_3 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_4 := \sqrt{\pi \cdot 2}\\ t_5 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ \mathbf{if}\;z + -1 \leq 145:\\ \;\;\;\;t_4 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \frac{\frac{\mathsf{fma}\left(z + 2, \mathsf{fma}\left(z, -582.6188486005177, 676.5203681218851\right), 771.3234287776531 \cdot \mathsf{fma}\left(z, z, z\right)\right)}{\mathsf{fma}\left(z, z, z\right)}}{z + 2}\right) + \left(\left(\left(t_2 + t_3\right) + \left(t_0 + t_1\right)\right) + t_5\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(t_0 + \left(t_1 + \frac{771.3234287776531}{z + 2}\right)\right)\right) + \left(t_2 + \left(t_3 + t_5\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(\log \left(z + 6.5\right), z + -0.5, -6.5 - z\right)}\right)\\ \end{array} \]

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 (/ -0.13857109526572012 (+ z 5.0)))
        (t_3 (/ 9.984369578019572e-6 (+ z 6.0)))
        (t_4 (sqrt (* PI 2.0)))
        (t_5 (/ 1.5056327351493116e-7 (+ z 7.0))))
   (if (<= (+ z -1.0) 145.0)
     (*
      t_4
      (*
       (pow (+ z 6.5) (+ z -0.5))
       (*
        (/ (exp -6.5) (exp z))
        (+
         (+
          0.9999999999998099
          (/
           (/
            (fma
             (+ z 2.0)
             (fma z -582.6188486005177 676.5203681218851)
             (* 771.3234287776531 (fma z z z)))
            (fma z z z))
           (+ z 2.0)))
         (+ (+ (+ t_2 t_3) (+ t_0 t_1)) t_5)))))
     (*
      t_4
      (*
       (+
        (+ 0.9999999999998099 (/ 676.5203681218851 z))
        (+
         (+
          (/ -1259.1392167224028 (+ z 1.0))
          (+ t_0 (+ t_1 (/ 771.3234287776531 (+ z 2.0)))))
         (+ t_2 (+ t_3 t_5))))
       (exp (fma (log (+ z 6.5)) (+ z -0.5) (- -6.5 z))))))))

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 = -0.13857109526572012 / (z + 5.0);
	double t_3 = 9.984369578019572e-6 / (z + 6.0);
	double t_4 = sqrt((((double) M_PI) * 2.0));
	double t_5 = 1.5056327351493116e-7 / (z + 7.0);
	double tmp;
	if ((z + -1.0) <= 145.0) {
		tmp = t_4 * (pow((z + 6.5), (z + -0.5)) * ((exp(-6.5) / exp(z)) * ((0.9999999999998099 + ((fma((z + 2.0), fma(z, -582.6188486005177, 676.5203681218851), (771.3234287776531 * fma(z, z, z))) / fma(z, z, z)) / (z + 2.0))) + (((t_2 + t_3) + (t_0 + t_1)) + t_5))));
	} else {
		tmp = t_4 * (((0.9999999999998099 + (676.5203681218851 / z)) + (((-1259.1392167224028 / (z + 1.0)) + (t_0 + (t_1 + (771.3234287776531 / (z + 2.0))))) + (t_2 + (t_3 + t_5)))) * exp(fma(log((z + 6.5)), (z + -0.5), (-6.5 - z))));
	}
	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 = Float64(-0.13857109526572012 / Float64(z + 5.0))
	t_3 = Float64(9.984369578019572e-6 / Float64(z + 6.0))
	t_4 = sqrt(Float64(pi * 2.0))
	t_5 = Float64(1.5056327351493116e-7 / Float64(z + 7.0))
	tmp = 0.0
	if (Float64(z + -1.0) <= 145.0)
		tmp = Float64(t_4 * Float64((Float64(z + 6.5) ^ Float64(z + -0.5)) * Float64(Float64(exp(-6.5) / exp(z)) * Float64(Float64(0.9999999999998099 + Float64(Float64(fma(Float64(z + 2.0), fma(z, -582.6188486005177, 676.5203681218851), Float64(771.3234287776531 * fma(z, z, z))) / fma(z, z, z)) / Float64(z + 2.0))) + Float64(Float64(Float64(t_2 + t_3) + Float64(t_0 + t_1)) + t_5)))));
	else
		tmp = Float64(t_4 * Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / z)) + Float64(Float64(Float64(-1259.1392167224028 / Float64(z + 1.0)) + Float64(t_0 + Float64(t_1 + Float64(771.3234287776531 / Float64(z + 2.0))))) + Float64(t_2 + Float64(t_3 + t_5)))) * exp(fma(log(Float64(z + 6.5)), Float64(z + -0.5), Float64(-6.5 - z)))));
	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[(-0.13857109526572012 / N[(z + 5.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(9.984369578019572e-6 / N[(z + 6.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(1.5056327351493116e-7 / N[(z + 7.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z + -1.0), $MachinePrecision], 145.0], N[(t$95$4 * N[(N[Power[N[(z + 6.5), $MachinePrecision], N[(z + -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Exp[-6.5], $MachinePrecision] / N[Exp[z], $MachinePrecision]), $MachinePrecision] * N[(N[(0.9999999999998099 + N[(N[(N[(N[(z + 2.0), $MachinePrecision] * N[(z * -582.6188486005177 + 676.5203681218851), $MachinePrecision] + N[(771.3234287776531 * N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / N[(z + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$2 + t$95$3), $MachinePrecision] + N[(t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$4 * N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-1259.1392167224028 / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 + N[(t$95$1 + N[(771.3234287776531 / N[(z + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(t$95$3 + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Log[N[(z + 6.5), $MachinePrecision]], $MachinePrecision] * N[(z + -0.5), $MachinePrecision] + N[(-6.5 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 2.5

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

   2. Simplified 2.4

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

      [Start] 2.5	\[ \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      associate-*l* [=>] 2.5	\[ \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right)\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      associate-*l* [=>] 2.5	\[ \color{blue}{\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\right)} \]

   3. Applied egg-rr 2.4

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

   4. Simplified 2.3

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

      [Start] 2.4	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{676.5203681218851 \cdot \left(z + 1\right) + z \cdot -1259.1392167224028}{z \cdot \left(z + 1\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      +-commutative [=>] 2.4	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\color{blue}{z \cdot -1259.1392167224028 + 676.5203681218851 \cdot \left(z + 1\right)}}{z \cdot \left(z + 1\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      fma-def [=>] 2.3	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\color{blue}{\mathsf{fma}\left(z, -1259.1392167224028, 676.5203681218851 \cdot \left(z + 1\right)\right)}}{z \cdot \left(z + 1\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      distribute-lft-in [=>] 2.3	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\mathsf{fma}\left(z, -1259.1392167224028, \color{blue}{676.5203681218851 \cdot z + 676.5203681218851 \cdot 1}\right)}{z \cdot \left(z + 1\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      metadata-eval [=>] 2.3	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\mathsf{fma}\left(z, -1259.1392167224028, 676.5203681218851 \cdot z + \color{blue}{676.5203681218851}\right)}{z \cdot \left(z + 1\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      fma-def [=>] 2.3	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\mathsf{fma}\left(z, -1259.1392167224028, \color{blue}{\mathsf{fma}\left(676.5203681218851, z, 676.5203681218851\right)}\right)}{z \cdot \left(z + 1\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      distribute-rgt-in [=>] 2.3	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\mathsf{fma}\left(z, -1259.1392167224028, \mathsf{fma}\left(676.5203681218851, z, 676.5203681218851\right)\right)}{\color{blue}{z \cdot z + 1 \cdot z}}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      *-lft-identity [=>] 2.3	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\mathsf{fma}\left(z, -1259.1392167224028, \mathsf{fma}\left(676.5203681218851, z, 676.5203681218851\right)\right)}{z \cdot z + \color{blue}{z}}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      fma-def [=>] 2.3	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\mathsf{fma}\left(z, -1259.1392167224028, \mathsf{fma}\left(676.5203681218851, z, 676.5203681218851\right)\right)}{\color{blue}{\mathsf{fma}\left(z, z, z\right)}}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

   5. Applied egg-rr 2.2

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

   6. Applied egg-rr 2.2

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

   7. Simplified 2.1

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \color{blue}{\left(\frac{771.3234287776531}{z + 2} + \frac{676.5203681218851 + z \cdot -582.6188486005177}{\mathsf{fma}\left(z, z, z\right)}\right)}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      Proof

      [Start] 2.2	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + 1 \cdot \left(\frac{771.3234287776531}{z + 2} + \frac{\mathsf{fma}\left(z, -1259.1392167224028, \mathsf{fma}\left(676.5203681218851, z, 676.5203681218851\right)\right)}{\mathsf{fma}\left(z, z, z\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      *-lft-identity [=>] 2.2	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \color{blue}{\left(\frac{771.3234287776531}{z + 2} + \frac{\mathsf{fma}\left(z, -1259.1392167224028, \mathsf{fma}\left(676.5203681218851, z, 676.5203681218851\right)\right)}{\mathsf{fma}\left(z, z, z\right)}\right)}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      *-lft-identity [<=] 2.2	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \color{blue}{1 \cdot \frac{\mathsf{fma}\left(z, -1259.1392167224028, \mathsf{fma}\left(676.5203681218851, z, 676.5203681218851\right)\right)}{\mathsf{fma}\left(z, z, z\right)}}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      *-lft-identity [=>] 2.2	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \color{blue}{\frac{\mathsf{fma}\left(z, -1259.1392167224028, \mathsf{fma}\left(676.5203681218851, z, 676.5203681218851\right)\right)}{\mathsf{fma}\left(z, z, z\right)}}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      fma-udef [=>] 2.4	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \frac{\color{blue}{z \cdot -1259.1392167224028 + \mathsf{fma}\left(676.5203681218851, z, 676.5203681218851\right)}}{\mathsf{fma}\left(z, z, z\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      fma-udef [=>] 2.4	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \frac{z \cdot -1259.1392167224028 + \color{blue}{\left(676.5203681218851 \cdot z + 676.5203681218851\right)}}{\mathsf{fma}\left(z, z, z\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      associate-+r+ [=>] 2.4	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \frac{\color{blue}{\left(z \cdot -1259.1392167224028 + 676.5203681218851 \cdot z\right) + 676.5203681218851}}{\mathsf{fma}\left(z, z, z\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      +-commutative [=>] 2.4	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \frac{\color{blue}{676.5203681218851 + \left(z \cdot -1259.1392167224028 + 676.5203681218851 \cdot z\right)}}{\mathsf{fma}\left(z, z, z\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      *-commutative [=>] 2.4	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \frac{676.5203681218851 + \left(z \cdot -1259.1392167224028 + \color{blue}{z \cdot 676.5203681218851}\right)}{\mathsf{fma}\left(z, z, z\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      distribute-lft-out [=>] 2.1	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \frac{676.5203681218851 + \color{blue}{z \cdot \left(-1259.1392167224028 + 676.5203681218851\right)}}{\mathsf{fma}\left(z, z, z\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      metadata-eval [=>] 2.1	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \frac{676.5203681218851 + z \cdot \color{blue}{-582.6188486005177}}{\mathsf{fma}\left(z, z, z\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

   8. Applied egg-rr 2.2

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \color{blue}{\frac{771.3234287776531 \cdot \mathsf{fma}\left(z, z, z\right) + \left(z + 2\right) \cdot \left(676.5203681218851 + z \cdot -582.6188486005177\right)}{\left(z + 2\right) \cdot \mathsf{fma}\left(z, z, z\right)}}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

   9. Simplified 2.1

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \color{blue}{\frac{\frac{\mathsf{fma}\left(z + 2, \mathsf{fma}\left(z, -582.6188486005177, 676.5203681218851\right), 771.3234287776531 \cdot \mathsf{fma}\left(z, z, z\right)\right)}{\mathsf{fma}\left(z, z, z\right)}}{z + 2}}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      Proof

      [Start] 2.2	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \frac{771.3234287776531 \cdot \mathsf{fma}\left(z, z, z\right) + \left(z + 2\right) \cdot \left(676.5203681218851 + z \cdot -582.6188486005177\right)}{\left(z + 2\right) \cdot \mathsf{fma}\left(z, z, z\right)}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      *-commutative [=>] 2.2	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \frac{771.3234287776531 \cdot \mathsf{fma}\left(z, z, z\right) + \left(z + 2\right) \cdot \left(676.5203681218851 + z \cdot -582.6188486005177\right)}{\color{blue}{\mathsf{fma}\left(z, z, z\right) \cdot \left(z + 2\right)}}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      associate-/r* [=>] 2.1	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \color{blue}{\frac{\frac{771.3234287776531 \cdot \mathsf{fma}\left(z, z, z\right) + \left(z + 2\right) \cdot \left(676.5203681218851 + z \cdot -582.6188486005177\right)}{\mathsf{fma}\left(z, z, z\right)}}{z + 2}}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      +-commutative [=>] 2.1	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \frac{\frac{\color{blue}{\left(z + 2\right) \cdot \left(676.5203681218851 + z \cdot -582.6188486005177\right) + 771.3234287776531 \cdot \mathsf{fma}\left(z, z, z\right)}}{\mathsf{fma}\left(z, z, z\right)}}{z + 2}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      fma-def [=>] 2.1	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \frac{\frac{\color{blue}{\mathsf{fma}\left(z + 2, 676.5203681218851 + z \cdot -582.6188486005177, 771.3234287776531 \cdot \mathsf{fma}\left(z, z, z\right)\right)}}{\mathsf{fma}\left(z, z, z\right)}}{z + 2}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      +-commutative [=>] 2.1	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \frac{\frac{\mathsf{fma}\left(z + 2, \color{blue}{z \cdot -582.6188486005177 + 676.5203681218851}, 771.3234287776531 \cdot \mathsf{fma}\left(z, z, z\right)\right)}{\mathsf{fma}\left(z, z, z\right)}}{z + 2}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      fma-def [=>] 2.1	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \frac{\frac{\mathsf{fma}\left(z + 2, \color{blue}{\mathsf{fma}\left(z, -582.6188486005177, 676.5203681218851\right)}, 771.3234287776531 \cdot \mathsf{fma}\left(z, z, z\right)\right)}{\mathsf{fma}\left(z, z, z\right)}}{z + 2}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

   if 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

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

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

   3. Taylor expanded in z around inf 64.0

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

   4. Simplified 64.0

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

      [Start] 64.0	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \frac{{\left(6.5 + z\right)}^{\left(z - 0.5\right)}}{e^{6.5 + z}}\right) \]
      +-commutative [<=] 64.0	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \frac{{\color{blue}{\left(z + 6.5\right)}}^{\left(z - 0.5\right)}}{e^{6.5 + z}}\right) \]
      sqr-pow [=>] 64.0	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \frac{\color{blue}{{\left(z + 6.5\right)}^{\left(\frac{z - 0.5}{2}\right)} \cdot {\left(z + 6.5\right)}^{\left(\frac{z - 0.5}{2}\right)}}}{e^{6.5 + z}}\right) \]
      exp-sum [=>] 64.0	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \frac{{\left(z + 6.5\right)}^{\left(\frac{z - 0.5}{2}\right)} \cdot {\left(z + 6.5\right)}^{\left(\frac{z - 0.5}{2}\right)}}{\color{blue}{e^{6.5} \cdot e^{z}}}\right) \]
      sqr-pow [<=] 64.0	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \frac{\color{blue}{{\left(z + 6.5\right)}^{\left(z - 0.5\right)}}}{e^{6.5} \cdot e^{z}}\right) \]
      sub-neg [=>] 64.0	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \frac{{\left(z + 6.5\right)}^{\color{blue}{\left(z + \left(-0.5\right)\right)}}}{e^{6.5} \cdot e^{z}}\right) \]
      metadata-eval [=>] 64.0	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \frac{{\left(z + 6.5\right)}^{\left(z + \color{blue}{-0.5}\right)}}{e^{6.5} \cdot e^{z}}\right) \]
      *-lft-identity [<=] 64.0	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \frac{\color{blue}{1 \cdot {\left(z + 6.5\right)}^{\left(z + -0.5\right)}}}{e^{6.5} \cdot e^{z}}\right) \]
      *-commutative [<=] 64.0	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \frac{1 \cdot {\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{\color{blue}{e^{z} \cdot e^{6.5}}}\right) \]
      exp-sum [<=] 64.0	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \frac{1 \cdot {\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{\color{blue}{e^{z + 6.5}}}\right) \]
      associate-*l/ [<=] 64.0	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \color{blue}{\left(\frac{1}{e^{z + 6.5}} \cdot {\left(z + 6.5\right)}^{\left(z + -0.5\right)}\right)}\right) \]
      *-commutative [<=] 64.0	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \color{blue}{\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \frac{1}{e^{z + 6.5}}\right)}\right) \]
      rec-exp [=>] 64.0	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \color{blue}{e^{-\left(z + 6.5\right)}}\right)\right) \]
      neg-sub0 [=>] 64.0	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{\color{blue}{0 - \left(z + 6.5\right)}}\right)\right) \]
      +-commutative [=>] 64.0	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{0 - \color{blue}{\left(6.5 + z\right)}}\right)\right) \]
      associate--r+ [=>] 64.0	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{\color{blue}{\left(0 - 6.5\right) - z}}\right)\right) \]
      metadata-eval [=>] 64.0	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{\color{blue}{-6.5} - z}\right)\right) \]

   5. Taylor expanded in z around inf 64.0

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

   6. Simplified 7.5

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

      [Start] 64.0	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \left({\left(6.5 + z\right)}^{\left(z - 0.5\right)} \cdot e^{-\left(6.5 + z\right)}\right)\right) \]
      sub-neg [=>] 64.0	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \left({\left(6.5 + z\right)}^{\color{blue}{\left(z + \left(-0.5\right)\right)}} \cdot e^{-\left(6.5 + z\right)}\right)\right) \]
      metadata-eval [=>] 64.0	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \left({\left(6.5 + z\right)}^{\left(z + \color{blue}{-0.5}\right)} \cdot e^{-\left(6.5 + z\right)}\right)\right) \]
      exp-to-pow [<=] 64.0	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \left(\color{blue}{e^{\log \left(6.5 + z\right) \cdot \left(z + -0.5\right)}} \cdot e^{-\left(6.5 + z\right)}\right)\right) \]
      +-commutative [=>] 64.0	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \left(e^{\log \color{blue}{\left(z + 6.5\right)} \cdot \left(z + -0.5\right)} \cdot e^{-\left(6.5 + z\right)}\right)\right) \]
      *-commutative [<=] 64.0	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \left(e^{\color{blue}{\left(z + -0.5\right) \cdot \log \left(z + 6.5\right)}} \cdot e^{-\left(6.5 + z\right)}\right)\right) \]
      distribute-neg-in [=>] 64.0	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \left(e^{\left(z + -0.5\right) \cdot \log \left(z + 6.5\right)} \cdot e^{\color{blue}{\left(-6.5\right) + \left(-z\right)}}\right)\right) \]
      metadata-eval [=>] 64.0	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \left(e^{\left(z + -0.5\right) \cdot \log \left(z + 6.5\right)} \cdot e^{\color{blue}{-6.5} + \left(-z\right)}\right)\right) \]
      sub-neg [<=] 64.0	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \left(e^{\left(z + -0.5\right) \cdot \log \left(z + 6.5\right)} \cdot e^{\color{blue}{-6.5 - z}}\right)\right) \]
      exp-sum [<=] 7.7	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \color{blue}{e^{\left(z + -0.5\right) \cdot \log \left(z + 6.5\right) + \left(-6.5 - z\right)}}\right) \]
      *-commutative [=>] 7.7	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\color{blue}{\log \left(z + 6.5\right) \cdot \left(z + -0.5\right)} + \left(-6.5 - z\right)}\right) \]
      fma-def [=>] 7.5	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\color{blue}{\mathsf{fma}\left(\log \left(z + 6.5\right), z + -0.5, -6.5 - z\right)}}\right) \]
      +-commutative [<=] 7.5	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(6.5 + z\right)}, z + -0.5, -6.5 - z\right)}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + -1 \leq 145:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \frac{\frac{\mathsf{fma}\left(z + 2, \mathsf{fma}\left(z, -582.6188486005177, 676.5203681218851\right), 771.3234287776531 \cdot \mathsf{fma}\left(z, z, z\right)\right)}{\mathsf{fma}\left(z, z, z\right)}}{z + 2}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{12.507343278686905}{z + 4} + \frac{771.3234287776531}{z + 2}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(\log \left(z + 6.5\right), z + -0.5, -6.5 - z\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	2.3
Cost	42564

\[\begin{array}{l} t_0 := \frac{-176.6150291621406}{z + 3}\\ t_1 := \frac{12.507343278686905}{z + 4}\\ t_2 := \frac{-0.13857109526572012}{z + 5}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_5 := \frac{771.3234287776531}{z + 2}\\ t_6 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ \mathbf{if}\;z + -1 \leq 145:\\ \;\;\;\;t_3 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(0.9999999999998099 + \left(t_5 + \left(\frac{676.5203681218851 + z \cdot -582.6188486005177}{\mathsf{fma}\left(z, z, z\right)} + \left(t_2 + \left(t_6 + \left(\left(t_0 + t_1\right) + t_4\right)\right)\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(t_0 + \left(t_1 + t_5\right)\right)\right) + \left(t_2 + \left(t_6 + t_4\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(\log \left(z + 6.5\right), z + -0.5, -6.5 - z\right)}\right)\\ \end{array} \]

Alternative 2
Error	2.4
Cost	42436

\[\begin{array}{l} t_0 := \frac{771.3234287776531}{z + 2}\\ t_1 := \frac{-0.13857109526572012}{z + 5}\\ t_2 := \sqrt{\pi \cdot 2}\\ t_3 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_4 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_5 := \frac{12.507343278686905}{z + 4}\\ t_6 := \frac{-176.6150291621406}{z + 3}\\ \mathbf{if}\;z + -1 \leq 145:\\ \;\;\;\;t_2 \cdot \left(\left(0.9999999999998099 + \left(\left(t_0 + \frac{\mathsf{fma}\left(z, -582.6188486005177, 676.5203681218851\right)}{\mathsf{fma}\left(z, z, z\right)}\right) + \left(t_4 + \left(t_6 + \left(\left(t_1 + t_3\right) + t_5\right)\right)\right)\right)\right) \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(t_6 + \left(t_5 + t_0\right)\right)\right) + \left(t_1 + \left(t_3 + t_4\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(\log \left(z + 6.5\right), z + -0.5, -6.5 - z\right)}\right)\\ \end{array} \]

Alternative 3
Error	2.4
Cost	36164

\[\begin{array}{l} t_0 := \frac{-176.6150291621406}{z + 3}\\ t_1 := \sqrt{\pi \cdot 2}\\ t_2 := \frac{12.507343278686905}{z + 4}\\ t_3 := \frac{-0.13857109526572012}{z + 5}\\ t_4 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_5 := \frac{771.3234287776531}{z + 2}\\ t_6 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ \mathbf{if}\;z + -1 \leq 145:\\ \;\;\;\;t_1 \cdot \left(\left(0.9999999999998099 + \left(t_5 + \left(\frac{676.5203681218851 + z \cdot -582.6188486005177}{\mathsf{fma}\left(z, z, z\right)} + \left(t_3 + \left(t_6 + \left(\left(t_0 + t_2\right) + t_4\right)\right)\right)\right)\right)\right) \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(t_0 + \left(t_2 + t_5\right)\right)\right) + \left(t_3 + \left(t_6 + t_4\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(\log \left(z + 6.5\right), z + -0.5, -6.5 - z\right)}\right)\\ \end{array} \]

Alternative 4
Error	2.4
Cost	36164

\[\begin{array}{l} t_0 := \frac{771.3234287776531}{z + 2}\\ t_1 := \frac{-176.6150291621406}{z + 3}\\ t_2 := \frac{12.507343278686905}{z + 4}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \frac{-0.13857109526572012}{z + 5}\\ t_5 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_6 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ \mathbf{if}\;z + -1 \leq 145:\\ \;\;\;\;t_3 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(\left(\left(t_4 + t_6\right) + \left(t_1 + t_2\right)\right) + t_5\right) + \left(0.9999999999998099 + \left(t_0 + \frac{676.5203681218851 + z \cdot -582.6188486005177}{\mathsf{fma}\left(z, z, z\right)}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(t_1 + \left(t_2 + t_0\right)\right)\right) + \left(t_4 + \left(t_6 + t_5\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(\log \left(z + 6.5\right), z + -0.5, -6.5 - z\right)}\right)\\ \end{array} \]

Alternative 5
Error	2.5
Cost	36100

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

Alternative 6
Error	2.5
Cost	36100

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

Alternative 7
Error	2.5
Cost	29828

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

Alternative 8
Error	2.4
Cost	29700

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

Alternative 9
Error	3.8
Cost	29504

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

Alternative 10
Error	3.8
Cost	29504

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

Alternative 11
Error	3.8
Cost	29504

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

Alternative 12
Error	46.8
Cost	28992

\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(z + -1\right) + 7.5\right)}^{\left(z + -0.5\right)}\right) \cdot e^{-0.5 + \left(-6 - z\right)}\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right) + \left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right) + \left(0.9999999999998099 + \left(\frac{12.0895510149948}{z} + \frac{246.3374466535184}{z \cdot z}\right)\right)\right)\right) \]

Alternative 13
Error	46.8
Cost	28992

\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(z + -1\right) + 7.5\right)}^{\left(z + -0.5\right)}\right) \cdot e^{-0.5 + \left(-6 - z\right)}\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right) + \left(\left(\left(0.9999999999998099 + \frac{12.0895510149948}{z}\right) + \frac{246.3374466535184}{z \cdot z}\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right) \]

Alternative 14
Error	46.8
Cost	28736

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

Alternative 15
Error	46.8
Cost	28736

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

Alternative 16
Error	46.8
Cost	28736

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

Alternative 17
Error	47.6
Cost	27200

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

Alternative 18
Error	51.4
Cost	26816

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

Alternative 19
Error	51.6
Cost	26756

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

Alternative 20
Error	52.0
Cost	26692

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

Alternative 21
Error	55.6
Cost	19584

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

Reproduce

herbie shell --seed 2023046 
(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)))))