Jmat.Real.gamma, branch z greater than 0.5

Average Error: 4.2 → 2.0

Time: 31.8s

Precision: binary64

Cost: 42564

\[z > 0.5\]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\\ t_1 := \sqrt{\pi \cdot 2}\\ t_2 := \frac{771.3234287776531}{2 + z}\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;t_1 \cdot \left(\frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \left(t_2 + \frac{\mathsf{fma}\left(-582.6188486005177, z, 676.5203681218851\right)}{z \cdot \left(z + 1\right)}\right)\right) + t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(e^{\mathsf{fma}\left(\log \left(z + 6.5\right), z + -0.5, -6.5 - z\right)} \cdot \left(t_0 + \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + t_2\right)\right)\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (z)
 :precision binary64
 (let* ((t_0
         (+
          (/ -0.13857109526572012 (+ z 5.0))
          (+
           (/ 9.984369578019572e-6 (+ z 6.0))
           (+
            (/ 12.507343278686905 (+ z 4.0))
            (+
             (/ -176.6150291621406 (+ z 3.0))
             (/ 1.5056327351493116e-7 (+ z 7.0)))))))
        (t_1 (sqrt (* PI 2.0)))
        (t_2 (/ 771.3234287776531 (+ 2.0 z))))
   (if (<= (+ z -1.0) 140.0)
     (*
      t_1
      (*
       (/ (* (pow (+ z 6.5) (+ z -0.5)) (exp -6.5)) (exp z))
       (+
        (+
         0.9999999999998099
         (+
          t_2
          (/ (fma -582.6188486005177 z 676.5203681218851) (* z (+ z 1.0)))))
        t_0)))
     (*
      t_1
      (*
       (exp (fma (log (+ z 6.5)) (+ z -0.5) (- -6.5 z)))
       (+
        t_0
        (+
         0.9999999999998099
         (+
          (+ (/ 676.5203681218851 z) (/ -1259.1392167224028 (+ z 1.0)))
          t_2))))))))

C

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

↓

double code(double z) {
	double t_0 = (-0.13857109526572012 / (z + 5.0)) + ((9.984369578019572e-6 / (z + 6.0)) + ((12.507343278686905 / (z + 4.0)) + ((-176.6150291621406 / (z + 3.0)) + (1.5056327351493116e-7 / (z + 7.0)))));
	double t_1 = sqrt((((double) M_PI) * 2.0));
	double t_2 = 771.3234287776531 / (2.0 + z);
	double tmp;
	if ((z + -1.0) <= 140.0) {
		tmp = t_1 * (((pow((z + 6.5), (z + -0.5)) * exp(-6.5)) / exp(z)) * ((0.9999999999998099 + (t_2 + (fma(-582.6188486005177, z, 676.5203681218851) / (z * (z + 1.0))))) + t_0));
	} else {
		tmp = t_1 * (exp(fma(log((z + 6.5)), (z + -0.5), (-6.5 - z))) * (t_0 + (0.9999999999998099 + (((676.5203681218851 / z) + (-1259.1392167224028 / (z + 1.0))) + t_2))));
	}
	return tmp;
}

Julia

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

↓

function code(z)
	t_0 = Float64(Float64(-0.13857109526572012 / Float64(z + 5.0)) + Float64(Float64(9.984369578019572e-6 / Float64(z + 6.0)) + Float64(Float64(12.507343278686905 / Float64(z + 4.0)) + Float64(Float64(-176.6150291621406 / Float64(z + 3.0)) + Float64(1.5056327351493116e-7 / Float64(z + 7.0))))))
	t_1 = sqrt(Float64(pi * 2.0))
	t_2 = Float64(771.3234287776531 / Float64(2.0 + z))
	tmp = 0.0
	if (Float64(z + -1.0) <= 140.0)
		tmp = Float64(t_1 * Float64(Float64(Float64((Float64(z + 6.5) ^ Float64(z + -0.5)) * exp(-6.5)) / exp(z)) * Float64(Float64(0.9999999999998099 + Float64(t_2 + Float64(fma(-582.6188486005177, z, 676.5203681218851) / Float64(z * Float64(z + 1.0))))) + t_0)));
	else
		tmp = Float64(t_1 * Float64(exp(fma(log(Float64(z + 6.5)), Float64(z + -0.5), Float64(-6.5 - z))) * Float64(t_0 + Float64(0.9999999999998099 + Float64(Float64(Float64(676.5203681218851 / z) + Float64(-1259.1392167224028 / Float64(z + 1.0))) + t_2)))));
	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[(N[(-0.13857109526572012 / N[(z + 5.0), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(z + 6.0), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(z + 4.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-176.6150291621406 / N[(z + 3.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(z + 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(771.3234287776531 / N[(2.0 + z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z + -1.0), $MachinePrecision], 140.0], N[(t$95$1 * N[(N[(N[(N[Power[N[(z + 6.5), $MachinePrecision], N[(z + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[-6.5], $MachinePrecision]), $MachinePrecision] / N[Exp[z], $MachinePrecision]), $MachinePrecision] * N[(N[(0.9999999999998099 + N[(t$95$2 + N[(N[(-582.6188486005177 * z + 676.5203681218851), $MachinePrecision] / N[(z * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[Exp[N[(N[Log[N[(z + 6.5), $MachinePrecision]], $MachinePrecision] * N[(z + -0.5), $MachinePrecision] + N[(-6.5 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 + N[(0.9999999999998099 + N[(N[(N[(676.5203681218851 / z), $MachinePrecision] + N[(-1259.1392167224028 / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $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 2.2

      \[\leadsto \color{blue}{\sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)} \]
      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 2.2

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

   4. Simplified 2.2

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

      [Start] 2.2	\[ \sqrt{\pi \cdot 2} \cdot {\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)}^{1} \]
      unpow1 [=>] 2.2	\[ \sqrt{\pi \cdot 2} \cdot \color{blue}{\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)} \]
      associate-*r* [=>] 2.1	\[ \sqrt{\pi \cdot 2} \cdot \color{blue}{\left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)} \]
      +-commutative [=>] 2.1	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{\color{blue}{z + 2}} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
      associate-+l+ [=>] 2.1	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \color{blue}{\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)}\right)\right) \]
      associate-+l+ [=>] 2.1	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \color{blue}{\left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)}\right)\right) \]

   5. Applied egg-rr 2.2

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

   6. Simplified 2.2

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

      [Start] 2.2

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

      fma-def [=>] 2.2

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

   7. Taylor expanded in z around 0 2.0

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \frac{\color{blue}{676.5203681218851 + -582.6188486005177 \cdot z}}{z \cdot \left(z + 1\right)}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 1.9

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \frac{\color{blue}{\mathsf{fma}\left(-582.6188486005177, z, 676.5203681218851\right)}}{z \cdot \left(z + 1\right)}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right) \]
      Proof

      [Start] 2.0	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \frac{676.5203681218851 + -582.6188486005177 \cdot z}{z \cdot \left(z + 1\right)}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right) \]
      +-commutative [=>] 2.0	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \frac{\color{blue}{-582.6188486005177 \cdot z + 676.5203681218851}}{z \cdot \left(z + 1\right)}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right) \]
      fma-def [=>] 1.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \frac{\color{blue}{\mathsf{fma}\left(-582.6188486005177, z, 676.5203681218851\right)}}{z \cdot \left(z + 1\right)}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 1.8

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5}}{e^{z}}} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \frac{\mathsf{fma}\left(-582.6188486005177, z, 676.5203681218851\right)}{z \cdot \left(z + 1\right)}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right) \]

   if 140 < (-.f64 z 1)

   1. Initial program 62.3

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

   2. Simplified 61.6

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

      [Start] 62.3	\[ \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      associate-*l* [=>] 61.6	\[ \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right)\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      associate-*l* [=>] 61.6	\[ \color{blue}{\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\right)} \]

   3. Applied egg-rr 61.6

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

   4. Simplified 61.6

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

      [Start] 61.6	\[ \sqrt{\pi \cdot 2} \cdot {\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)}^{1} \]
      unpow1 [=>] 61.6	\[ \sqrt{\pi \cdot 2} \cdot \color{blue}{\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)} \]
      associate-*r* [=>] 61.6	\[ \sqrt{\pi \cdot 2} \cdot \color{blue}{\left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)} \]
      +-commutative [=>] 61.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{\color{blue}{z + 2}} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
      associate-+l+ [=>] 61.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \color{blue}{\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)}\right)\right) \]
      associate-+l+ [=>] 61.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \color{blue}{\left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)}\right)\right) \]

   5. Taylor expanded in z around inf 61.6

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

   6. Simplified 7.6

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

      [Start] 61.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(6.5 + z\right)}^{\left(z - 0.5\right)} \cdot e^{-\left(6.5 + z\right)}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right) \]
      sub-neg [=>] 61.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(6.5 + z\right)}^{\color{blue}{\left(z + \left(-0.5\right)\right)}} \cdot e^{-\left(6.5 + z\right)}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right) \]
      metadata-eval [=>] 61.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(6.5 + z\right)}^{\left(z + \color{blue}{-0.5}\right)} \cdot e^{-\left(6.5 + z\right)}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right) \]
      exp-to-pow [<=] 61.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\color{blue}{e^{\log \left(6.5 + z\right) \cdot \left(z + -0.5\right)}} \cdot e^{-\left(6.5 + z\right)}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right) \]
      +-commutative [=>] 61.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(e^{\log \color{blue}{\left(z + 6.5\right)} \cdot \left(z + -0.5\right)} \cdot e^{-\left(6.5 + z\right)}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right) \]
      *-commutative [<=] 61.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(e^{\color{blue}{\left(z + -0.5\right) \cdot \log \left(z + 6.5\right)}} \cdot e^{-\left(6.5 + z\right)}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right) \]
      distribute-neg-in [=>] 61.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(e^{\left(z + -0.5\right) \cdot \log \left(z + 6.5\right)} \cdot e^{\color{blue}{\left(-6.5\right) + \left(-z\right)}}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right) \]
      metadata-eval [=>] 61.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(e^{\left(z + -0.5\right) \cdot \log \left(z + 6.5\right)} \cdot e^{\color{blue}{-6.5} + \left(-z\right)}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right) \]
      sub-neg [<=] 61.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(e^{\left(z + -0.5\right) \cdot \log \left(z + 6.5\right)} \cdot e^{\color{blue}{-6.5 - z}}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right) \]
      exp-sum [<=] 7.7	\[ \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{e^{\left(z + -0.5\right) \cdot \log \left(z + 6.5\right) + \left(-6.5 - z\right)}} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right) \]
      *-commutative [=>] 7.7	\[ \sqrt{\pi \cdot 2} \cdot \left(e^{\color{blue}{\log \left(z + 6.5\right) \cdot \left(z + -0.5\right)} + \left(-6.5 - z\right)} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right) \]
      fma-def [=>] 7.6	\[ \sqrt{\pi \cdot 2} \cdot \left(e^{\color{blue}{\mathsf{fma}\left(\log \left(z + 6.5\right), z + -0.5, -6.5 - z\right)}} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right) \]
      +-commutative [<=] 7.6	\[ \sqrt{\pi \cdot 2} \cdot \left(e^{\mathsf{fma}\left(\log \color{blue}{\left(6.5 + z\right)}, z + -0.5, -6.5 - z\right)} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(\frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\mathsf{fma}\left(-582.6188486005177, z, 676.5203681218851\right)}{z \cdot \left(z + 1\right)}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(e^{\mathsf{fma}\left(\log \left(z + 6.5\right), z + -0.5, -6.5 - z\right)} \cdot \left(\left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) + \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{2 + z}\right)\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	2.2
Cost	43072

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

Alternative 2
Error	2.1
Cost	36804

\[\begin{array}{l} t_0 := \frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\\ t_1 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;t_1 \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(t_0 + \left(0.9999999999998099 + \frac{\mathsf{fma}\left(-771.3234287776531, z \cdot \left(-1 - z\right), \left(2 + z\right) \cdot \left(676.5203681218851 + z \cdot -582.6188486005177\right)\right)}{\left(z \cdot \left(z + 1\right)\right) \cdot \left(2 + z\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(e^{\mathsf{fma}\left(\log \left(z + 6.5\right), z + -0.5, -6.5 - z\right)} \cdot \left(t_0 + \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{2 + z}\right)\right)\right)\right)\\ \end{array} \]

Alternative 3
Error	2.2
Cost	36100

\[\begin{array}{l} t_0 := \frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\\ t_1 := \sqrt{\pi \cdot 2}\\ t_2 := z \cdot \left(z + 1\right)\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;t_1 \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(t_0 + \left(0.9999999999998099 + \frac{\left(2 + z\right) \cdot \left(676.5203681218851 + z \cdot -582.6188486005177\right) + 771.3234287776531 \cdot t_2}{t_2 \cdot \left(2 + z\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(e^{\mathsf{fma}\left(\log \left(z + 6.5\right), z + -0.5, -6.5 - z\right)} \cdot \left(t_0 + \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{2 + z}\right)\right)\right)\right)\\ \end{array} \]

Alternative 4
Error	2.2
Cost	30532

\[\begin{array}{l} t_0 := \frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\\ t_1 := \sqrt{\pi \cdot 2}\\ t_2 := z \cdot \left(z + 1\right)\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;t_1 \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(t_0 + \left(0.9999999999998099 + \frac{\left(2 + z\right) \cdot \left(676.5203681218851 + z \cdot -582.6188486005177\right) + 771.3234287776531 \cdot t_2}{t_2 \cdot \left(2 + z\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\left(t_0 + \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{2 + z}\right)\right)\right) \cdot e^{\left(-6.5 - z\right) + \left(z + -0.5\right) \cdot \log \left(z + 6.5\right)}\right)\\ \end{array} \]

Alternative 5
Error	2.2
Cost	29764

\[\begin{array}{l} t_0 := \frac{771.3234287776531}{2 + z}\\ t_1 := \frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\\ t_2 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z \leq 140:\\ \;\;\;\;t_2 \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(t_1 + \left(0.9999999999998099 + \left(t_0 + \frac{676.5203681218851 + z \cdot -582.6188486005177}{z \cdot \left(z + 1\right)}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(\left(t_1 + \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + t_0\right)\right)\right) \cdot e^{\left(-6.5 - z\right) + \left(z + -0.5\right) \cdot \log \left(z + 6.5\right)}\right)\\ \end{array} \]

Alternative 6
Error	2.3
Cost	29700

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

Alternative 7
Error	2.3
Cost	29700

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

Alternative 8
Error	4.1
Cost	29504

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

Alternative 9
Error	4.1
Cost	29504

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

Alternative 10
Error	4.1
Cost	29504

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

Alternative 11
Error	46.1
Cost	28800

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

Alternative 12
Error	46.9
Cost	28736

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

Alternative 13
Error	46.9
Cost	28736

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

Alternative 14
Error	47.7
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 15
Error	47.7
Cost	27200

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

Alternative 16
Error	50.4
Cost	26948

\[\begin{array}{l} \mathbf{if}\;z \leq 2.75:\\ \;\;\;\;\sqrt{e^{-13 - \log \left(6.5 \cdot \left(z \cdot z\right)\right)} \cdot \left(\pi \cdot 915359.6169675419\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(\frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}} \cdot \left(0.9999999999998099 + \frac{24.458333333348836}{z}\right)\right)\\ \end{array} \]

Alternative 17
Error	52.0
Cost	26692

\[\begin{array}{l} \mathbf{if}\;z \leq 4:\\ \;\;\;\;\sqrt{e^{-13 - \log \left(6.5 \cdot \left(z \cdot z\right)\right)} \cdot \left(\pi \cdot 915359.6169675419\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(0.9999999999998099 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right)\right)\\ \end{array} \]

Alternative 18
Error	55.6
Cost	19584

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

Reproduce

herbie shell --seed 2023187 
(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)))))