Jmat.Real.gamma, branch z greater than 0.5

Average Accuracy: 94.1% → 94.6%

Time: 29.7s

Precision: binary64

Cost: 64000

\[z > 0.5\]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{-1259.1392167224028}{z + 1}\\ t_1 := \frac{771.3234287776531}{2 + z}\\ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \frac{1}{e^{z + 6.5}}\right) \cdot \left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, -1259.1392167224028, \mathsf{fma}\left(z, 676.5203681218851, 676.5203681218851\right)\right)}{\mathsf{fma}\left(z, z, z\right)} + t_1, t_1 + \left(\frac{676.5203681218851}{z} + t_0\right), \frac{-31192.868525943773}{{\left(z + 3\right)}^{2}}\right)}{\frac{676.5203681218851}{z} + \left(\left(t_1 + t_0\right) + \frac{176.6150291621406}{z + 3}\right)}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \end{array} \]

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 (/ -1259.1392167224028 (+ z 1.0)))
        (t_1 (/ 771.3234287776531 (+ 2.0 z))))
   (*
    (sqrt (* PI 2.0))
    (*
     (* (pow (+ z 6.5) (+ z -0.5)) (/ 1.0 (exp (+ z 6.5))))
     (+
      (+
       0.9999999999998099
       (/
        (fma
         (+
          (/
           (fma
            z
            -1259.1392167224028
            (fma z 676.5203681218851 676.5203681218851))
           (fma z z z))
          t_1)
         (+ t_1 (+ (/ 676.5203681218851 z) t_0))
         (/ -31192.868525943773 (pow (+ z 3.0) 2.0)))
        (+
         (/ 676.5203681218851 z)
         (+ (+ t_1 t_0) (/ 176.6150291621406 (+ z 3.0))))))
      (+
       (+ (/ 12.507343278686905 (+ z 4.0)) (/ -0.13857109526572012 (+ z 5.0)))
       (+
        (/ 9.984369578019572e-6 (+ z 6.0))
        (/ 1.5056327351493116e-7 (+ z 7.0)))))))))

C

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

↓

double code(double z) {
	double t_0 = -1259.1392167224028 / (z + 1.0);
	double t_1 = 771.3234287776531 / (2.0 + z);
	return sqrt((((double) M_PI) * 2.0)) * ((pow((z + 6.5), (z + -0.5)) * (1.0 / exp((z + 6.5)))) * ((0.9999999999998099 + (fma(((fma(z, -1259.1392167224028, fma(z, 676.5203681218851, 676.5203681218851)) / fma(z, z, z)) + t_1), (t_1 + ((676.5203681218851 / z) + t_0)), (-31192.868525943773 / pow((z + 3.0), 2.0))) / ((676.5203681218851 / z) + ((t_1 + t_0) + (176.6150291621406 / (z + 3.0)))))) + (((12.507343278686905 / (z + 4.0)) + (-0.13857109526572012 / (z + 5.0))) + ((9.984369578019572e-6 / (z + 6.0)) + (1.5056327351493116e-7 / (z + 7.0))))));
}

Julia

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

↓

function code(z)
	t_0 = Float64(-1259.1392167224028 / Float64(z + 1.0))
	t_1 = Float64(771.3234287776531 / Float64(2.0 + z))
	return Float64(sqrt(Float64(pi * 2.0)) * Float64(Float64((Float64(z + 6.5) ^ Float64(z + -0.5)) * Float64(1.0 / exp(Float64(z + 6.5)))) * Float64(Float64(0.9999999999998099 + Float64(fma(Float64(Float64(fma(z, -1259.1392167224028, fma(z, 676.5203681218851, 676.5203681218851)) / fma(z, z, z)) + t_1), Float64(t_1 + Float64(Float64(676.5203681218851 / z) + t_0)), Float64(-31192.868525943773 / (Float64(z + 3.0) ^ 2.0))) / Float64(Float64(676.5203681218851 / z) + Float64(Float64(t_1 + t_0) + Float64(176.6150291621406 / Float64(z + 3.0)))))) + Float64(Float64(Float64(12.507343278686905 / Float64(z + 4.0)) + Float64(-0.13857109526572012 / Float64(z + 5.0))) + Float64(Float64(9.984369578019572e-6 / Float64(z + 6.0)) + Float64(1.5056327351493116e-7 / Float64(z + 7.0)))))))
end

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[(-1259.1392167224028 / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(771.3234287776531 / N[(2.0 + z), $MachinePrecision]), $MachinePrecision]}, N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Power[N[(z + 6.5), $MachinePrecision], N[(z + -0.5), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Exp[N[(z + 6.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.9999999999998099 + N[(N[(N[(N[(N[(z * -1259.1392167224028 + N[(z * 676.5203681218851 + 676.5203681218851), $MachinePrecision]), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] * N[(t$95$1 + N[(N[(676.5203681218851 / z), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] + N[(-31192.868525943773 / N[Power[N[(z + 3.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(676.5203681218851 / z), $MachinePrecision] + N[(N[(t$95$1 + t$95$0), $MachinePrecision] + N[(176.6150291621406 / N[(z + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(12.507343278686905 / N[(z + 4.0), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(z + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(z + 6.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(z + 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 94.1%

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

2. Simplified 94.2%

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

   [Start] 94.1	\[ \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
   associate-*l* [=>] 94.2	\[ \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right)\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
   associate-*l* [=>] 94.2	\[ \color{blue}{\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\right)} \]

3. Applied egg-rr 94.3%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \color{blue}{\frac{\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) \cdot \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) - \frac{31192.868525943773}{{\left(z + 3\right)}^{2}}}{\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) - \frac{-176.6150291621406}{z + 3}}}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   Proof

   [Start] 94.2	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \left(\frac{771.3234287776531}{z + 2} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   associate-+r+ [=>] 94.3	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \color{blue}{\left(\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) + \frac{-176.6150291621406}{z + 3}\right)}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   flip-+ [=>] 94.3	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \color{blue}{\frac{\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) \cdot \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) - \frac{-176.6150291621406}{z + 3} \cdot \frac{-176.6150291621406}{z + 3}}{\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) - \frac{-176.6150291621406}{z + 3}}}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   frac-times [=>] 94.3	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \frac{\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) \cdot \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) - \color{blue}{\frac{-176.6150291621406 \cdot -176.6150291621406}{\left(z + 3\right) \cdot \left(z + 3\right)}}}{\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) - \frac{-176.6150291621406}{z + 3}}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   metadata-eval [=>] 94.3	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \frac{\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) \cdot \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) - \frac{\color{blue}{31192.868525943773}}{\left(z + 3\right) \cdot \left(z + 3\right)}}{\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) - \frac{-176.6150291621406}{z + 3}}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   pow1 [=>] 94.3	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \frac{\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) \cdot \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) - \frac{31192.868525943773}{\color{blue}{{\left(z + 3\right)}^{1}} \cdot \left(z + 3\right)}}{\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) - \frac{-176.6150291621406}{z + 3}}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   pow1 [=>] 94.3	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \frac{\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) \cdot \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) - \frac{31192.868525943773}{{\left(z + 3\right)}^{1} \cdot \color{blue}{{\left(z + 3\right)}^{1}}}}{\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) - \frac{-176.6150291621406}{z + 3}}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   pow-sqr [=>] 94.3	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \frac{\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) \cdot \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) - \frac{31192.868525943773}{\color{blue}{{\left(z + 3\right)}^{\left(2 \cdot 1\right)}}}}{\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) - \frac{-176.6150291621406}{z + 3}}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   metadata-eval [=>] 94.3	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \frac{\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) \cdot \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) - \frac{31192.868525943773}{{\left(z + 3\right)}^{\color{blue}{2}}}}{\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) - \frac{-176.6150291621406}{z + 3}}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

4. Simplified 94.3%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \color{blue}{\frac{\mathsf{fma}\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \frac{-31192.868525943773}{{\left(z + 3\right)}^{2}}\right)}{\frac{676.5203681218851}{z} + \left(\left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right) + \frac{176.6150291621406}{z + 3}\right)}}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   Proof

   [Start] 94.3	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \frac{\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) \cdot \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) - \frac{31192.868525943773}{{\left(z + 3\right)}^{2}}}{\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) - \frac{-176.6150291621406}{z + 3}}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   fma-neg [=>] 94.3	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, -\frac{31192.868525943773}{{\left(z + 3\right)}^{2}}\right)}}{\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) - \frac{-176.6150291621406}{z + 3}}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   distribute-neg-frac [=>] 94.3	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \color{blue}{\frac{-31192.868525943773}{{\left(z + 3\right)}^{2}}}\right)}{\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) - \frac{-176.6150291621406}{z + 3}}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   metadata-eval [=>] 94.3	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \frac{\color{blue}{-31192.868525943773}}{{\left(z + 3\right)}^{2}}\right)}{\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) - \frac{-176.6150291621406}{z + 3}}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   sub-neg [=>] 94.3	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \frac{-31192.868525943773}{{\left(z + 3\right)}^{2}}\right)}{\color{blue}{\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right) + \left(-\frac{-176.6150291621406}{z + 3}\right)}}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   associate-+l+ [=>] 94.3	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \frac{-31192.868525943773}{{\left(z + 3\right)}^{2}}\right)}{\color{blue}{\left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right)\right)} + \left(-\frac{-176.6150291621406}{z + 3}\right)}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   associate-+l+ [=>] 94.3	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \frac{-31192.868525943773}{{\left(z + 3\right)}^{2}}\right)}{\color{blue}{\frac{676.5203681218851}{z} + \left(\left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right) + \left(-\frac{-176.6150291621406}{z + 3}\right)\right)}}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   distribute-neg-frac [=>] 94.3	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \frac{-31192.868525943773}{{\left(z + 3\right)}^{2}}\right)}{\frac{676.5203681218851}{z} + \left(\left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right) + \color{blue}{\frac{--176.6150291621406}{z + 3}}\right)}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   metadata-eval [=>] 94.3	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \frac{-31192.868525943773}{{\left(z + 3\right)}^{2}}\right)}{\frac{676.5203681218851}{z} + \left(\left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right) + \frac{\color{blue}{176.6150291621406}}{z + 3}\right)}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

5. Applied egg-rr 94.5%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(\color{blue}{\frac{676.5203681218851 \cdot \left(z + 1\right) + z \cdot -1259.1392167224028}{z \cdot \left(z + 1\right)}} + \frac{771.3234287776531}{z + 2}, \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \frac{-31192.868525943773}{{\left(z + 3\right)}^{2}}\right)}{\frac{676.5203681218851}{z} + \left(\left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right) + \frac{176.6150291621406}{z + 3}\right)}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   Proof

   [Start] 94.3

   \[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \frac{-31192.868525943773}{{\left(z + 3\right)}^{2}}\right)}{\frac{676.5203681218851}{z} + \left(\left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right) + \frac{176.6150291621406}{z + 3}\right)}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

   frac-add [=>] 94.5

   \[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(\color{blue}{\frac{676.5203681218851 \cdot \left(z + 1\right) + z \cdot -1259.1392167224028}{z \cdot \left(z + 1\right)}} + \frac{771.3234287776531}{z + 2}, \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \frac{-31192.868525943773}{{\left(z + 3\right)}^{2}}\right)}{\frac{676.5203681218851}{z} + \left(\left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right) + \frac{176.6150291621406}{z + 3}\right)}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

6. Simplified 94.6%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(\color{blue}{\frac{\mathsf{fma}\left(z, -1259.1392167224028, \mathsf{fma}\left(z, 676.5203681218851, 676.5203681218851\right)\right)}{\mathsf{fma}\left(z, z, z\right)}} + \frac{771.3234287776531}{z + 2}, \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \frac{-31192.868525943773}{{\left(z + 3\right)}^{2}}\right)}{\frac{676.5203681218851}{z} + \left(\left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right) + \frac{176.6150291621406}{z + 3}\right)}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   Proof

   [Start] 94.5	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(\frac{676.5203681218851 \cdot \left(z + 1\right) + z \cdot -1259.1392167224028}{z \cdot \left(z + 1\right)} + \frac{771.3234287776531}{z + 2}, \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \frac{-31192.868525943773}{{\left(z + 3\right)}^{2}}\right)}{\frac{676.5203681218851}{z} + \left(\left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right) + \frac{176.6150291621406}{z + 3}\right)}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   +-commutative [=>] 94.5	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(\frac{\color{blue}{z \cdot -1259.1392167224028 + 676.5203681218851 \cdot \left(z + 1\right)}}{z \cdot \left(z + 1\right)} + \frac{771.3234287776531}{z + 2}, \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \frac{-31192.868525943773}{{\left(z + 3\right)}^{2}}\right)}{\frac{676.5203681218851}{z} + \left(\left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right) + \frac{176.6150291621406}{z + 3}\right)}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   fma-def [=>] 94.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(\frac{\color{blue}{\mathsf{fma}\left(z, -1259.1392167224028, 676.5203681218851 \cdot \left(z + 1\right)\right)}}{z \cdot \left(z + 1\right)} + \frac{771.3234287776531}{z + 2}, \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \frac{-31192.868525943773}{{\left(z + 3\right)}^{2}}\right)}{\frac{676.5203681218851}{z} + \left(\left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right) + \frac{176.6150291621406}{z + 3}\right)}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   distribute-rgt-in [=>] 94.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, -1259.1392167224028, \color{blue}{z \cdot 676.5203681218851 + 1 \cdot 676.5203681218851}\right)}{z \cdot \left(z + 1\right)} + \frac{771.3234287776531}{z + 2}, \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \frac{-31192.868525943773}{{\left(z + 3\right)}^{2}}\right)}{\frac{676.5203681218851}{z} + \left(\left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right) + \frac{176.6150291621406}{z + 3}\right)}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   metadata-eval [=>] 94.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, -1259.1392167224028, z \cdot 676.5203681218851 + \color{blue}{676.5203681218851}\right)}{z \cdot \left(z + 1\right)} + \frac{771.3234287776531}{z + 2}, \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \frac{-31192.868525943773}{{\left(z + 3\right)}^{2}}\right)}{\frac{676.5203681218851}{z} + \left(\left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right) + \frac{176.6150291621406}{z + 3}\right)}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   fma-def [=>] 94.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, -1259.1392167224028, \color{blue}{\mathsf{fma}\left(z, 676.5203681218851, 676.5203681218851\right)}\right)}{z \cdot \left(z + 1\right)} + \frac{771.3234287776531}{z + 2}, \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \frac{-31192.868525943773}{{\left(z + 3\right)}^{2}}\right)}{\frac{676.5203681218851}{z} + \left(\left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right) + \frac{176.6150291621406}{z + 3}\right)}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   distribute-rgt-in [=>] 94.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, -1259.1392167224028, \mathsf{fma}\left(z, 676.5203681218851, 676.5203681218851\right)\right)}{\color{blue}{z \cdot z + 1 \cdot z}} + \frac{771.3234287776531}{z + 2}, \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \frac{-31192.868525943773}{{\left(z + 3\right)}^{2}}\right)}{\frac{676.5203681218851}{z} + \left(\left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right) + \frac{176.6150291621406}{z + 3}\right)}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   *-lft-identity [=>] 94.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, -1259.1392167224028, \mathsf{fma}\left(z, 676.5203681218851, 676.5203681218851\right)\right)}{z \cdot z + \color{blue}{z}} + \frac{771.3234287776531}{z + 2}, \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \frac{-31192.868525943773}{{\left(z + 3\right)}^{2}}\right)}{\frac{676.5203681218851}{z} + \left(\left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right) + \frac{176.6150291621406}{z + 3}\right)}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   fma-def [=>] 94.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, -1259.1392167224028, \mathsf{fma}\left(z, 676.5203681218851, 676.5203681218851\right)\right)}{\color{blue}{\mathsf{fma}\left(z, z, z\right)}} + \frac{771.3234287776531}{z + 2}, \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \frac{-31192.868525943773}{{\left(z + 3\right)}^{2}}\right)}{\frac{676.5203681218851}{z} + \left(\left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right) + \frac{176.6150291621406}{z + 3}\right)}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

7. Applied egg-rr 94.6%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \color{blue}{\frac{1}{e^{z + 6.5}}}\right) \cdot \left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, -1259.1392167224028, \mathsf{fma}\left(z, 676.5203681218851, 676.5203681218851\right)\right)}{\mathsf{fma}\left(z, z, z\right)} + \frac{771.3234287776531}{z + 2}, \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \frac{-31192.868525943773}{{\left(z + 3\right)}^{2}}\right)}{\frac{676.5203681218851}{z} + \left(\left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right) + \frac{176.6150291621406}{z + 3}\right)}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
   Proof

   [Start] 94.6

   \[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, -1259.1392167224028, \mathsf{fma}\left(z, 676.5203681218851, 676.5203681218851\right)\right)}{\mathsf{fma}\left(z, z, z\right)} + \frac{771.3234287776531}{z + 2}, \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \frac{-31192.868525943773}{{\left(z + 3\right)}^{2}}\right)}{\frac{676.5203681218851}{z} + \left(\left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right) + \frac{176.6150291621406}{z + 3}\right)}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

   exp-neg [=>] 94.6

   \[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \color{blue}{\frac{1}{e^{z + 6.5}}}\right) \cdot \left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, -1259.1392167224028, \mathsf{fma}\left(z, 676.5203681218851, 676.5203681218851\right)\right)}{\mathsf{fma}\left(z, z, z\right)} + \frac{771.3234287776531}{z + 2}, \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}, \frac{-31192.868525943773}{{\left(z + 3\right)}^{2}}\right)}{\frac{676.5203681218851}{z} + \left(\left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right) + \frac{176.6150291621406}{z + 3}\right)}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

8. Final simplification 94.6%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \frac{1}{e^{z + 6.5}}\right) \cdot \left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, -1259.1392167224028, \mathsf{fma}\left(z, 676.5203681218851, 676.5203681218851\right)\right)}{\mathsf{fma}\left(z, z, z\right)} + \frac{771.3234287776531}{2 + z}, \frac{771.3234287776531}{2 + z} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right), \frac{-31192.868525943773}{{\left(z + 3\right)}^{2}}\right)}{\frac{676.5203681218851}{z} + \left(\left(\frac{771.3234287776531}{2 + z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{176.6150291621406}{z + 3}\right)}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

Alternatives

Alternative 1
Accuracy	94.6%
Cost	63872

\[\begin{array}{l} t_0 := \frac{771.3234287776531}{2 + z}\\ t_1 := \frac{-1259.1392167224028}{z + 1}\\ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, -1259.1392167224028, \mathsf{fma}\left(z, 676.5203681218851, 676.5203681218851\right)\right)}{\mathsf{fma}\left(z, z, z\right)} + t_0, t_0 + \left(\frac{676.5203681218851}{z} + t_1\right), \frac{-31192.868525943773}{{\left(z + 3\right)}^{2}}\right)}{\frac{676.5203681218851}{z} + \left(\left(t_0 + t_1\right) + \frac{176.6150291621406}{z + 3}\right)}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right)\right) \end{array} \]

Alternative 2
Accuracy	94.3%
Cost	44672

\[\begin{array}{l} t_0 := \frac{771.3234287776531}{2 + z}\\ t_1 := \frac{-1259.1392167224028}{z + 1}\\ t_2 := t_0 + \left(\frac{676.5203681218851}{z} + t_1\right)\\ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(0.9999999999998099 + \frac{\mathsf{fma}\left(t_2, t_2, \frac{-31192.868525943773}{{\left(z + 3\right)}^{2}}\right)}{\frac{676.5203681218851}{z} + \left(\left(t_0 + t_1\right) + \frac{176.6150291621406}{z + 3}\right)}\right)\right) \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right)\right) \end{array} \]

Alternative 3
Accuracy	94.2%
Cost	35904

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

Alternative 4
Accuracy	93.9%
Cost	29504

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

Alternative 5
Accuracy	94.3%
Cost	29504

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

Alternative 6
Accuracy	28.0%
Cost	28800

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

Alternative 7
Accuracy	27.0%
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 8
Accuracy	25.7%
Cost	27200

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

Alternative 9
Accuracy	21.4%
Cost	26948

\[\begin{array}{l} \mathbf{if}\;z \leq 2.8:\\ \;\;\;\;\sqrt{140824.5564565449 \cdot \left(\frac{\pi}{z \cdot z} \cdot e^{-13}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(\frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}} \cdot \left(0.9999999999998099 + \frac{24.458333333348836}{z}\right)\right)\\ \end{array} \]

Alternative 10
Accuracy	19.3%
Cost	26756

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

Alternative 11
Accuracy	18.8%
Cost	26692

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

Alternative 12
Accuracy	13.0%
Cost	19712

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

Reproduce

herbie shell --seed 2023133 
(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)))))