?

Average Error: 6.06% → 2.69%
Time: 27.2s
Precision: binary64
Cost: 36292

?

\[z > 0.5\]
\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
↓
\[\begin{array}{l} t_0 := \frac{-176.6150291621406}{z + 3}\\ t_1 := \frac{12.507343278686905}{z + 4}\\ t_2 := \frac{-0.13857109526572012}{z + 5}\\ t_3 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_4 := \sqrt{\pi \cdot 2}\\ t_5 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;t_4 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \frac{1353.0407362437702 + z \cdot \left(282.6060996985028 - z \cdot -188.7045801771354\right)}{\left(z + 2\right) \cdot \mathsf{fma}\left(z, z, z\right)}\right) + \left(\left(\left(t_2 + t_3\right) + \left(t_0 + t_1\right)\right) + t_5\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(t_0 + \left(t_1 + \frac{771.3234287776531}{z + 2}\right)\right)\right) + \left(t_2 + \left(t_3 + t_5\right)\right)\right)\right) \cdot e^{\left(-6.5 - z\right) + \left(z + -0.5\right) \cdot \log \left(z + 6.5\right)}\right)\\ \end{array} \]
(FPCore (z)
 :precision binary64
 (*
  (*
   (* (sqrt (* PI 2.0)) (pow (+ (+ (- z 1.0) 7.0) 0.5) (+ (- z 1.0) 0.5)))
   (exp (- (+ (+ (- z 1.0) 7.0) 0.5))))
  (+
   (+
    (+
     (+
      (+
       (+
        (+
         (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- z 1.0) 1.0)))
         (/ -1259.1392167224028 (+ (- z 1.0) 2.0)))
        (/ 771.3234287776531 (+ (- z 1.0) 3.0)))
       (/ -176.6150291621406 (+ (- z 1.0) 4.0)))
      (/ 12.507343278686905 (+ (- z 1.0) 5.0)))
     (/ -0.13857109526572012 (+ (- z 1.0) 6.0)))
    (/ 9.984369578019572e-6 (+ (- z 1.0) 7.0)))
   (/ 1.5056327351493116e-7 (+ (- z 1.0) 8.0)))))
↓
(FPCore (z)
 :precision binary64
 (let* ((t_0 (/ -176.6150291621406 (+ z 3.0)))
        (t_1 (/ 12.507343278686905 (+ z 4.0)))
        (t_2 (/ -0.13857109526572012 (+ z 5.0)))
        (t_3 (/ 9.984369578019572e-6 (+ z 6.0)))
        (t_4 (sqrt (* PI 2.0)))
        (t_5 (/ 1.5056327351493116e-7 (+ z 7.0))))
   (if (<= (+ z -1.0) 140.0)
     (*
      t_4
      (*
       (pow (+ z 6.5) (+ z -0.5))
       (*
        (exp (- -6.5 z))
        (+
         (+
          0.9999999999998099
          (/
           (+
            1353.0407362437702
            (* z (- 282.6060996985028 (* z -188.7045801771354))))
           (* (+ z 2.0) (fma z z z))))
         (+ (+ (+ t_2 t_3) (+ t_0 t_1)) t_5)))))
     (*
      t_4
      (*
       (+
        (+ 0.9999999999998099 (/ 676.5203681218851 z))
        (+
         (+
          (/ -1259.1392167224028 (+ z 1.0))
          (+ t_0 (+ t_1 (/ 771.3234287776531 (+ z 2.0)))))
         (+ t_2 (+ t_3 t_5))))
       (exp (+ (- -6.5 z) (* (+ z -0.5) (log (+ z 6.5))))))))))
double code(double z) {
	return ((sqrt((((double) M_PI) * 2.0)) * pow((((z - 1.0) + 7.0) + 0.5), ((z - 1.0) + 0.5))) * exp(-(((z - 1.0) + 7.0) + 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / ((z - 1.0) + 1.0))) + (-1259.1392167224028 / ((z - 1.0) + 2.0))) + (771.3234287776531 / ((z - 1.0) + 3.0))) + (-176.6150291621406 / ((z - 1.0) + 4.0))) + (12.507343278686905 / ((z - 1.0) + 5.0))) + (-0.13857109526572012 / ((z - 1.0) + 6.0))) + (9.984369578019572e-6 / ((z - 1.0) + 7.0))) + (1.5056327351493116e-7 / ((z - 1.0) + 8.0)));
}
↓
double code(double z) {
	double t_0 = -176.6150291621406 / (z + 3.0);
	double t_1 = 12.507343278686905 / (z + 4.0);
	double t_2 = -0.13857109526572012 / (z + 5.0);
	double t_3 = 9.984369578019572e-6 / (z + 6.0);
	double t_4 = sqrt((((double) M_PI) * 2.0));
	double t_5 = 1.5056327351493116e-7 / (z + 7.0);
	double tmp;
	if ((z + -1.0) <= 140.0) {
		tmp = t_4 * (pow((z + 6.5), (z + -0.5)) * (exp((-6.5 - z)) * ((0.9999999999998099 + ((1353.0407362437702 + (z * (282.6060996985028 - (z * -188.7045801771354)))) / ((z + 2.0) * fma(z, z, z)))) + (((t_2 + t_3) + (t_0 + t_1)) + t_5))));
	} else {
		tmp = t_4 * (((0.9999999999998099 + (676.5203681218851 / z)) + (((-1259.1392167224028 / (z + 1.0)) + (t_0 + (t_1 + (771.3234287776531 / (z + 2.0))))) + (t_2 + (t_3 + t_5)))) * exp(((-6.5 - z) + ((z + -0.5) * log((z + 6.5))))));
	}
	return tmp;
}

function code(z)
	return Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (Float64(Float64(Float64(z - 1.0) + 7.0) + 0.5) ^ Float64(Float64(z - 1.0) + 0.5))) * exp(Float64(-Float64(Float64(Float64(z - 1.0) + 7.0) + 0.5)))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(Float64(z - 1.0) + 1.0))) + Float64(-1259.1392167224028 / Float64(Float64(z - 1.0) + 2.0))) + Float64(771.3234287776531 / Float64(Float64(z - 1.0) + 3.0))) + Float64(-176.6150291621406 / Float64(Float64(z - 1.0) + 4.0))) + Float64(12.507343278686905 / Float64(Float64(z - 1.0) + 5.0))) + Float64(-0.13857109526572012 / Float64(Float64(z - 1.0) + 6.0))) + Float64(9.984369578019572e-6 / Float64(Float64(z - 1.0) + 7.0))) + Float64(1.5056327351493116e-7 / Float64(Float64(z - 1.0) + 8.0))))
end
↓
function code(z)
	t_0 = Float64(-176.6150291621406 / Float64(z + 3.0))
	t_1 = Float64(12.507343278686905 / Float64(z + 4.0))
	t_2 = Float64(-0.13857109526572012 / Float64(z + 5.0))
	t_3 = Float64(9.984369578019572e-6 / Float64(z + 6.0))
	t_4 = sqrt(Float64(pi * 2.0))
	t_5 = Float64(1.5056327351493116e-7 / Float64(z + 7.0))
	tmp = 0.0
	if (Float64(z + -1.0) <= 140.0)
		tmp = Float64(t_4 * Float64((Float64(z + 6.5) ^ Float64(z + -0.5)) * Float64(exp(Float64(-6.5 - z)) * Float64(Float64(0.9999999999998099 + Float64(Float64(1353.0407362437702 + Float64(z * Float64(282.6060996985028 - Float64(z * -188.7045801771354)))) / Float64(Float64(z + 2.0) * fma(z, z, z)))) + Float64(Float64(Float64(t_2 + t_3) + Float64(t_0 + t_1)) + t_5)))));
	else
		tmp = Float64(t_4 * Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / z)) + Float64(Float64(Float64(-1259.1392167224028 / Float64(z + 1.0)) + Float64(t_0 + Float64(t_1 + Float64(771.3234287776531 / Float64(z + 2.0))))) + Float64(t_2 + Float64(t_3 + t_5)))) * exp(Float64(Float64(-6.5 - z) + Float64(Float64(z + -0.5) * log(Float64(z + 6.5)))))));
	end
	return tmp
end

code[z_] := N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[(N[(z - 1.0), $MachinePrecision] + 7.0), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(z - 1.0), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[(N[(z - 1.0), $MachinePrecision] + 7.0), $MachinePrecision] + 0.5), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(N[(z - 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(N[(z - 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(N[(z - 1.0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(z - 1.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(N[(z - 1.0), $MachinePrecision] + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(z - 1.0), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(N[(z - 1.0), $MachinePrecision] + 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(z - 1.0), $MachinePrecision] + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[z_] := Block[{t$95$0 = N[(-176.6150291621406 / N[(z + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(12.507343278686905 / N[(z + 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-0.13857109526572012 / N[(z + 5.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(9.984369578019572e-6 / N[(z + 6.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(1.5056327351493116e-7 / N[(z + 7.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z + -1.0), $MachinePrecision], 140.0], N[(t$95$4 * N[(N[Power[N[(z + 6.5), $MachinePrecision], N[(z + -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(-6.5 - z), $MachinePrecision]], $MachinePrecision] * N[(N[(0.9999999999998099 + N[(N[(1353.0407362437702 + N[(z * N[(282.6060996985028 - N[(z * -188.7045801771354), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(z + 2.0), $MachinePrecision] * N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$2 + t$95$3), $MachinePrecision] + N[(t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$4 * N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-1259.1392167224028 / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 + N[(t$95$1 + N[(771.3234287776531 / N[(z + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(t$95$3 + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(-6.5 - z), $MachinePrecision] + N[(N[(z + -0.5), $MachinePrecision] * N[Log[N[(z + 6.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)
↓
\begin{array}{l}
t_0 := \frac{-176.6150291621406}{z + 3}\\
t_1 := \frac{12.507343278686905}{z + 4}\\
t_2 := \frac{-0.13857109526572012}{z + 5}\\
t_3 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\
t_4 := \sqrt{\pi \cdot 2}\\
t_5 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\
\mathbf{if}\;z + -1 \leq 140:\\
\;\;\;\;t_4 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \frac{1353.0407362437702 + z \cdot \left(282.6060996985028 - z \cdot -188.7045801771354\right)}{\left(z + 2\right) \cdot \mathsf{fma}\left(z, z, z\right)}\right) + \left(\left(\left(t_2 + t_3\right) + \left(t_0 + t_1\right)\right) + t_5\right)\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;t_4 \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(t_0 + \left(t_1 + \frac{771.3234287776531}{z + 2}\right)\right)\right) + \left(t_2 + \left(t_3 + t_5\right)\right)\right)\right) \cdot e^{\left(-6.5 - z\right) + \left(z + -0.5\right) \cdot \log \left(z + 6.5\right)}\right)\\


\end{array}

Error?

Derivation?

  1. Split input into 2 regimes

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

    1. Initial program 3.43

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

    2. Simplified3.32

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

      [Start]3.43

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

      associate-*l* [=>]3.43

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

      associate-*l* [=>]3.44

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

    3. Applied egg-rr3.4

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

    4. Simplified3.21

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

      [Start]3.4

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

      +-commutative [=>]3.4

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

      fma-def [=>]3.15

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

      distribute-lft-in [=>]3.19

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

      metadata-eval [=>]3.19

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

      fma-def [=>]3.17

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

      distribute-rgt-in [=>]3.26

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

      *-lft-identity [=>]3.26

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

      fma-def [=>]3.21

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

    5. Applied egg-rr3.21

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

    6. Simplified3.09

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

      [Start]3.21

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

      *-lft-identity [=>]3.21

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

      *-lft-identity [<=]3.21

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

      *-lft-identity [=>]3.21

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

      fma-udef [=>]3.44

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

      fma-udef [=>]3.45

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

      associate-+r+ [=>]3.41

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

      +-commutative [=>]3.41

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

      *-commutative [=>]3.41

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

      distribute-lft-out [=>]3.09

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

      metadata-eval [=>]3.09

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

    7. Applied egg-rr3.13

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

    8. Taylor expanded in z around 0 2.41

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \frac{\color{blue}{1353.0407362437702 + \left(188.7045801771354 \cdot {z}^{2} + 282.6060996985028 \cdot z\right)}}{\left(z + 2\right) \cdot \mathsf{fma}\left(z, z, z\right)}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

    9. Simplified2.41

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \frac{\color{blue}{1353.0407362437702 + z \cdot \left(282.6060996985028 + z \cdot 188.7045801771354\right)}}{\left(z + 2\right) \cdot \mathsf{fma}\left(z, z, z\right)}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      Proof

      [Start]2.41

      \[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \frac{1353.0407362437702 + \left(188.7045801771354 \cdot {z}^{2} + 282.6060996985028 \cdot z\right)}{\left(z + 2\right) \cdot \mathsf{fma}\left(z, z, z\right)}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

      +-commutative [=>]2.41

      \[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \frac{1353.0407362437702 + \color{blue}{\left(282.6060996985028 \cdot z + 188.7045801771354 \cdot {z}^{2}\right)}}{\left(z + 2\right) \cdot \mathsf{fma}\left(z, z, z\right)}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

      *-commutative [=>]2.41

      \[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \frac{1353.0407362437702 + \left(\color{blue}{z \cdot 282.6060996985028} + 188.7045801771354 \cdot {z}^{2}\right)}{\left(z + 2\right) \cdot \mathsf{fma}\left(z, z, z\right)}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

      *-commutative [=>]2.41

      \[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \frac{1353.0407362437702 + \left(z \cdot 282.6060996985028 + \color{blue}{{z}^{2} \cdot 188.7045801771354}\right)}{\left(z + 2\right) \cdot \mathsf{fma}\left(z, z, z\right)}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

      unpow2 [=>]2.41

      \[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \frac{1353.0407362437702 + \left(z \cdot 282.6060996985028 + \color{blue}{\left(z \cdot z\right)} \cdot 188.7045801771354\right)}{\left(z + 2\right) \cdot \mathsf{fma}\left(z, z, z\right)}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

      associate-*l* [=>]2.39

      \[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \frac{1353.0407362437702 + \left(z \cdot 282.6060996985028 + \color{blue}{z \cdot \left(z \cdot 188.7045801771354\right)}\right)}{\left(z + 2\right) \cdot \mathsf{fma}\left(z, z, z\right)}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

      distribute-lft-out [=>]2.41

      \[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \frac{1353.0407362437702 + \color{blue}{z \cdot \left(282.6060996985028 + z \cdot 188.7045801771354\right)}}{\left(z + 2\right) \cdot \mathsf{fma}\left(z, z, z\right)}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

    if 140 < (-.f64 z 1)

    1. Initial program 95.74

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

    2. Simplified95.35

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

      [Start]95.74

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

      associate-*l* [=>]95.32

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

      associate-*l* [=>]95.32

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

    3. Taylor expanded in z around -inf 95.74

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

    4. Simplified12.13

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

      [Start]95.74

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

      div-exp [=>]12.13

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

      sub-neg [=>]12.13

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

      associate--r+ [=>]12.13

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

      mul-1-neg [=>]12.13

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

      remove-double-neg [=>]12.13

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

      associate--r+ [<=]12.13

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

      mul-1-neg [=>]12.13

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

      distribute-rgt-neg-in [=>]12.13

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

      sub-neg [=>]12.13

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

      mul-1-neg [=>]12.13

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

      remove-double-neg [=>]12.13

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

      +-commutative [=>]12.13

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

      mul-1-neg [=>]12.13

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

      unsub-neg [=>]12.13

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

  4. Final simplification2.69

    \[\leadsto \begin{array}{l} \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \frac{1353.0407362437702 + z \cdot \left(282.6060996985028 - z \cdot -188.7045801771354\right)}{\left(z + 2\right) \cdot \mathsf{fma}\left(z, z, z\right)}\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{12.507343278686905}{z + 4} + \frac{771.3234287776531}{z + 2}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\left(-6.5 - z\right) + \left(z + -0.5\right) \cdot \log \left(z + 6.5\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error3.41%
Cost67268
\[\begin{array}{l} t_0 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_1 := \frac{-176.6150291621406}{z + 3}\\ t_2 := \frac{12.507343278686905}{z + 4}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_5 := \frac{771.3234287776531}{z + 2}\\ t_6 := \left(z + -1\right) + 7\\ t_7 := \frac{-0.13857109526572012}{z + 5}\\ \mathbf{if}\;\left(\left(t_3 \cdot {\left(0.5 + t_6\right)}^{\left(\left(z + -1\right) + 0.5\right)}\right) \cdot e^{-0.5 + \left(-7 + \left(1 - z\right)\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 + \left(z + -1\right)}\right) + \frac{-1259.1392167224028}{\left(z + -1\right) + 2}\right) + \frac{771.3234287776531}{\left(z + -1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z + -1\right) + 4}\right) + \frac{12.507343278686905}{\left(z + -1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z + -1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t_6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8}\right) \leq 10^{+225}:\\ \;\;\;\;t_3 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(\left(\left(t_7 + t_0\right) + \left(t_1 + t_2\right)\right) + t_4\right) + \left(0.9999999999998099 + \left(t_5 + \frac{676.5203681218851 + z \cdot -582.6188486005177}{\mathsf{fma}\left(z, z, z\right)}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(t_1 + \left(t_2 + t_5\right)\right)\right) + \left(t_7 + \left(t_0 + t_4\right)\right)\right)\right) \cdot e^{\left(-6.5 - z\right) + \left(z + -0.5\right) \cdot \log \left(z + 6.5\right)}\right)\\ \end{array} \]
Alternative 2
Error3.36%
Cost36164
\[\begin{array}{l} t_0 := \frac{12.507343278686905}{z + 4}\\ t_1 := \frac{-0.13857109526572012}{z + 5}\\ t_2 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_5 := \frac{771.3234287776531}{z + 2}\\ t_6 := \frac{-176.6150291621406}{z + 3}\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;t_3 \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(0.9999999999998099 + \left(\left(t_5 + \frac{676.5203681218851 + z \cdot -582.6188486005177}{\mathsf{fma}\left(z, z, z\right)}\right) + \left(\left(t_1 + t_4\right) + \left(t_6 + \left(t_0 + t_2\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(t_6 + \left(t_0 + t_5\right)\right)\right) + \left(t_1 + \left(t_4 + t_2\right)\right)\right)\right) \cdot e^{\left(-6.5 - z\right) + \left(z + -0.5\right) \cdot \log \left(z + 6.5\right)}\right)\\ \end{array} \]
Alternative 3
Error3.65%
Cost30148
\[\begin{array}{l} t_0 := \frac{771.3234287776531}{z + 2}\\ t_1 := \frac{-176.6150291621406}{z + 3}\\ t_2 := \frac{12.507343278686905}{z + 4}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \frac{-0.13857109526572012}{z + 5}\\ t_5 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_6 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;t_3 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(\left(\left(t_4 + t_6\right) + \left(t_1 + t_2\right)\right) + t_5\right) + \left(0.9999999999998099 + \left(t_0 + \frac{676.5203681218851 \cdot \left(z + 1\right) + z \cdot -1259.1392167224028}{z \cdot \left(z + 1\right)}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(t_1 + \left(t_2 + t_0\right)\right)\right) + \left(t_4 + \left(t_6 + t_5\right)\right)\right)\right) \cdot e^{\left(-6.5 - z\right) + \left(z + -0.5\right) \cdot \log \left(z + 6.5\right)}\right)\\ \end{array} \]
Alternative 4
Error3.85%
Cost29700
\[\begin{array}{l} t_0 := \frac{-176.6150291621406}{z + 3}\\ t_1 := \frac{-0.13857109526572012}{z + 5}\\ t_2 := \frac{-1259.1392167224028}{z + 1}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \frac{12.507343278686905}{z + 4}\\ t_5 := \frac{771.3234287776531}{z + 2}\\ t_6 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ \mathbf{if}\;z \leq 145:\\ \;\;\;\;t_3 \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(t_5 + \left(t_2 + \left(t_0 + \frac{676.5203681218851}{z}\right)\right)\right)\right) + \left(t_6 + \left(t_1 + t_4\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(t_2 + \left(t_0 + \left(t_4 + t_5\right)\right)\right) + \left(t_1 + t_6\right)\right)\right) \cdot e^{\left(-6.5 - z\right) + \left(z + -0.5\right) \cdot \log \left(z + 6.5\right)}\right)\\ \end{array} \]
Alternative 5
Error5.98%
Cost29504
\[\sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right) + \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{771.3234287776531}{z + 2}\right)\right)\right)\right)\right) \]
Alternative 6
Error5.95%
Cost29504
\[\sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right) \]
Alternative 7
Error6.03%
Cost29504
\[\sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \frac{676.5203681218851}{z}\right)\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right)\right)\right) \]
Alternative 8
Error73.09%
Cost28992
\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(z + -1\right) + 7.5\right)}^{\left(z + -0.5\right)}\right) \cdot e^{-0.5 + \left(-6 - z\right)}\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right) + \left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right) + \left(0.9999999999998099 + \left(\frac{12.0895510149948}{z} + \frac{246.3374466535184}{z \cdot z}\right)\right)\right)\right) \]
Alternative 9
Error73.09%
Cost28736
\[\sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right) + \left(0.9999999999998099 + \left(\frac{12.0895510149948}{z} + \frac{246.3374466535184}{z \cdot z}\right)\right)\right)\right) \]
Alternative 10
Error74.37%
Cost27200
\[\sqrt{\pi \cdot 2} \cdot \left(\left(0.9999999999998099 + \left(\frac{24.458333333348836}{z} + \frac{197.000868054939}{z \cdot z}\right)\right) \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}}\right) \]
Alternative 11
Error77.82%
Cost27012
\[\begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z \leq 2.75:\\ \;\;\;\;\frac{\left(\left(t_0 \cdot 676.5203681218851\right) \cdot e^{-6.5}\right) \cdot \sqrt{0.15384615384615385}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(e^{\left(-6.5 - z\right) + \left(z + -0.5\right) \cdot \log \left(z + 6.5\right)} \cdot \left(0.9999999999998099 + \frac{24.458333333348836}{z}\right)\right)\\ \end{array} \]
Alternative 12
Error78.61%
Cost26948
\[\begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z \leq 2.75:\\ \;\;\;\;\frac{\left(\left(t_0 \cdot 676.5203681218851\right) \cdot e^{-6.5}\right) \cdot \sqrt{0.15384615384615385}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(0.9999999999998099 + \frac{24.458333333348836}{z}\right)\right)\\ \end{array} \]
Alternative 13
Error80.55%
Cost26756
\[\begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z \leq 3.95:\\ \;\;\;\;\frac{\left(\left(t_0 \cdot 676.5203681218851\right) \cdot e^{-6.5}\right) \cdot \sqrt{0.15384615384615385}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(0.9999999999998099 \cdot e^{\left(-6.5 - z\right) + \left(z + -0.5\right) \cdot \log \left(z + 6.5\right)}\right)\\ \end{array} \]
Alternative 14
Error81.15%
Cost26692
\[\begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z \leq 3.95:\\ \;\;\;\;\frac{\left(\left(t_0 \cdot 676.5203681218851\right) \cdot e^{-6.5}\right) \cdot \sqrt{0.15384615384615385}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(0.9999999999998099 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right)\right)\\ \end{array} \]
Alternative 15
Error86.83%
Cost19584
\[\frac{\sqrt{e^{-13} \cdot \left(\pi \cdot 140824.5564565449\right)}}{z} \]

Error

Reproduce?

herbie shell --seed 2023089 
(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)))))