Jmat.Real.gamma, branch z greater than 0.5

Average Error: 3.7 → 2.3

Time: 28.6s

Precision: binary64

Cost: 50372

\[z > 0.5\]

Math

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

↓

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

FPCore

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

↓

(FPCore (z)
 :precision binary64
 (let* ((t_0 (/ -176.6150291621406 (+ z 3.0)))
        (t_1 (/ 12.507343278686905 (+ z 4.0)))
        (t_2 (sqrt (* PI 2.0)))
        (t_3 (/ 771.3234287776531 (+ z 2.0)))
        (t_4 (/ -0.13857109526572012 (+ z 5.0)))
        (t_5 (/ -1259.1392167224028 (+ z 1.0)))
        (t_6
         (+
          (/ 9.984369578019572e-6 (+ z 6.0))
          (/ 1.5056327351493116e-7 (+ z 7.0)))))
   (if (<= (+ z -1.0) 140.0)
     (*
      t_2
      (*
       (+
        (+
         0.9999999999998099
         (+
          (/
           (+
            (/ 309629712.5173946 (pow z 3.0))
            (/ -1996279061.5505414 (pow (+ z 1.0) 3.0)))
           (fma
            t_5
            (+ t_5 (/ -676.5203681218851 z))
            (/ (/ 457679.80848377093 z) z)))
          (+ t_0 t_3)))
        (+ t_6 (+ t_1 t_4)))
       (* (pow (+ z 6.5) (+ z -0.5)) (exp (- -6.5 z)))))
     (*
      t_2
      (*
       (+
        (+ 0.9999999999998099 (/ 676.5203681218851 z))
        (+ (+ t_5 (+ t_0 (+ t_1 t_3))) (+ t_4 t_6)))
       (exp (+ (* (+ z -0.5) (log (+ z 6.5))) (- -6.5 z))))))))

C

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

↓

double code(double z) {
	double t_0 = -176.6150291621406 / (z + 3.0);
	double t_1 = 12.507343278686905 / (z + 4.0);
	double t_2 = sqrt((((double) M_PI) * 2.0));
	double t_3 = 771.3234287776531 / (z + 2.0);
	double t_4 = -0.13857109526572012 / (z + 5.0);
	double t_5 = -1259.1392167224028 / (z + 1.0);
	double t_6 = (9.984369578019572e-6 / (z + 6.0)) + (1.5056327351493116e-7 / (z + 7.0));
	double tmp;
	if ((z + -1.0) <= 140.0) {
		tmp = t_2 * (((0.9999999999998099 + ((((309629712.5173946 / pow(z, 3.0)) + (-1996279061.5505414 / pow((z + 1.0), 3.0))) / fma(t_5, (t_5 + (-676.5203681218851 / z)), ((457679.80848377093 / z) / z))) + (t_0 + t_3))) + (t_6 + (t_1 + t_4))) * (pow((z + 6.5), (z + -0.5)) * exp((-6.5 - z))));
	} else {
		tmp = t_2 * (((0.9999999999998099 + (676.5203681218851 / z)) + ((t_5 + (t_0 + (t_1 + t_3))) + (t_4 + t_6))) * exp((((z + -0.5) * log((z + 6.5))) + (-6.5 - z))));
	}
	return tmp;
}

Julia

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

↓

function code(z)
	t_0 = Float64(-176.6150291621406 / Float64(z + 3.0))
	t_1 = Float64(12.507343278686905 / Float64(z + 4.0))
	t_2 = sqrt(Float64(pi * 2.0))
	t_3 = Float64(771.3234287776531 / Float64(z + 2.0))
	t_4 = Float64(-0.13857109526572012 / Float64(z + 5.0))
	t_5 = Float64(-1259.1392167224028 / Float64(z + 1.0))
	t_6 = Float64(Float64(9.984369578019572e-6 / Float64(z + 6.0)) + Float64(1.5056327351493116e-7 / Float64(z + 7.0)))
	tmp = 0.0
	if (Float64(z + -1.0) <= 140.0)
		tmp = Float64(t_2 * Float64(Float64(Float64(0.9999999999998099 + Float64(Float64(Float64(Float64(309629712.5173946 / (z ^ 3.0)) + Float64(-1996279061.5505414 / (Float64(z + 1.0) ^ 3.0))) / fma(t_5, Float64(t_5 + Float64(-676.5203681218851 / z)), Float64(Float64(457679.80848377093 / z) / z))) + Float64(t_0 + t_3))) + Float64(t_6 + Float64(t_1 + t_4))) * Float64((Float64(z + 6.5) ^ Float64(z + -0.5)) * exp(Float64(-6.5 - z)))));
	else
		tmp = Float64(t_2 * Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / z)) + Float64(Float64(t_5 + Float64(t_0 + Float64(t_1 + t_3))) + Float64(t_4 + t_6))) * exp(Float64(Float64(Float64(z + -0.5) * log(Float64(z + 6.5))) + Float64(-6.5 - z)))));
	end
	return tmp
end

Wolfram

code[z_] := N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[(N[(z - 1.0), $MachinePrecision] + 7.0), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(z - 1.0), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[(N[(z - 1.0), $MachinePrecision] + 7.0), $MachinePrecision] + 0.5), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(N[(z - 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(N[(z - 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(N[(z - 1.0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(z - 1.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(N[(z - 1.0), $MachinePrecision] + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(z - 1.0), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(N[(z - 1.0), $MachinePrecision] + 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(z - 1.0), $MachinePrecision] + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[z_] := Block[{t$95$0 = N[(-176.6150291621406 / N[(z + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(12.507343278686905 / N[(z + 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(771.3234287776531 / N[(z + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(-0.13857109526572012 / N[(z + 5.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(-1259.1392167224028 / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(9.984369578019572e-6 / N[(z + 6.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(z + 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z + -1.0), $MachinePrecision], 140.0], N[(t$95$2 * N[(N[(N[(0.9999999999998099 + N[(N[(N[(N[(309629712.5173946 / N[Power[z, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-1996279061.5505414 / N[Power[N[(z + 1.0), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$5 * N[(t$95$5 + N[(-676.5203681218851 / z), $MachinePrecision]), $MachinePrecision] + N[(N[(457679.80848377093 / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$6 + N[(t$95$1 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(z + 6.5), $MachinePrecision], N[(z + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-6.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / z), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$5 + N[(t$95$0 + N[(t$95$1 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(N[(z + -0.5), $MachinePrecision] * N[Log[N[(z + 6.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(-6.5 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 2.2

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

   2. Simplified 2.1

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

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

   3. Applied egg-rr 2.3

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

   4. Simplified 2.2

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \left(\color{blue}{\frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\mathsf{fma}\left(\frac{-1259.1392167224028}{z + 1}, \frac{-1259.1392167224028}{z + 1} + \frac{-676.5203681218851}{z}, \frac{\frac{457679.80848377093}{z}}{z}\right)}} + \left(\frac{771.3234287776531}{z + 2} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      Proof

      [Start] 2.3	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{{\left(\frac{676.5203681218851}{z}\right)}^{3} + {\left(\frac{-1259.1392167224028}{z + 1}\right)}^{3}}{\frac{676.5203681218851}{z} \cdot \frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} \cdot \frac{-1259.1392167224028}{z + 1} - \frac{676.5203681218851}{z} \cdot \frac{-1259.1392167224028}{z + 1}\right)} + \left(\frac{771.3234287776531}{z + 2} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      cube-div [=>] 2.3	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{\color{blue}{\frac{{676.5203681218851}^{3}}{{z}^{3}}} + {\left(\frac{-1259.1392167224028}{z + 1}\right)}^{3}}{\frac{676.5203681218851}{z} \cdot \frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} \cdot \frac{-1259.1392167224028}{z + 1} - \frac{676.5203681218851}{z} \cdot \frac{-1259.1392167224028}{z + 1}\right)} + \left(\frac{771.3234287776531}{z + 2} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      metadata-eval [=>] 2.3	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{\frac{\color{blue}{309629712.5173946}}{{z}^{3}} + {\left(\frac{-1259.1392167224028}{z + 1}\right)}^{3}}{\frac{676.5203681218851}{z} \cdot \frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} \cdot \frac{-1259.1392167224028}{z + 1} - \frac{676.5203681218851}{z} \cdot \frac{-1259.1392167224028}{z + 1}\right)} + \left(\frac{771.3234287776531}{z + 2} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      cube-div [=>] 2.2	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{\frac{309629712.5173946}{{z}^{3}} + \color{blue}{\frac{{-1259.1392167224028}^{3}}{{\left(z + 1\right)}^{3}}}}{\frac{676.5203681218851}{z} \cdot \frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} \cdot \frac{-1259.1392167224028}{z + 1} - \frac{676.5203681218851}{z} \cdot \frac{-1259.1392167224028}{z + 1}\right)} + \left(\frac{771.3234287776531}{z + 2} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      metadata-eval [=>] 2.2	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{\frac{309629712.5173946}{{z}^{3}} + \frac{\color{blue}{-1996279061.5505414}}{{\left(z + 1\right)}^{3}}}{\frac{676.5203681218851}{z} \cdot \frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} \cdot \frac{-1259.1392167224028}{z + 1} - \frac{676.5203681218851}{z} \cdot \frac{-1259.1392167224028}{z + 1}\right)} + \left(\frac{771.3234287776531}{z + 2} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      +-commutative [=>] 2.2	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\color{blue}{\left(\frac{-1259.1392167224028}{z + 1} \cdot \frac{-1259.1392167224028}{z + 1} - \frac{676.5203681218851}{z} \cdot \frac{-1259.1392167224028}{z + 1}\right) + \frac{676.5203681218851}{z} \cdot \frac{676.5203681218851}{z}}} + \left(\frac{771.3234287776531}{z + 2} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      distribute-rgt-out-- [=>] 2.2	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\color{blue}{\frac{-1259.1392167224028}{z + 1} \cdot \left(\frac{-1259.1392167224028}{z + 1} - \frac{676.5203681218851}{z}\right)} + \frac{676.5203681218851}{z} \cdot \frac{676.5203681218851}{z}} + \left(\frac{771.3234287776531}{z + 2} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      fma-def [=>] 2.2	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\color{blue}{\mathsf{fma}\left(\frac{-1259.1392167224028}{z + 1}, \frac{-1259.1392167224028}{z + 1} - \frac{676.5203681218851}{z}, \frac{676.5203681218851}{z} \cdot \frac{676.5203681218851}{z}\right)}} + \left(\frac{771.3234287776531}{z + 2} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      sub-neg [=>] 2.2	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\mathsf{fma}\left(\frac{-1259.1392167224028}{z + 1}, \color{blue}{\frac{-1259.1392167224028}{z + 1} + \left(-\frac{676.5203681218851}{z}\right)}, \frac{676.5203681218851}{z} \cdot \frac{676.5203681218851}{z}\right)} + \left(\frac{771.3234287776531}{z + 2} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      distribute-neg-frac [=>] 2.2	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\mathsf{fma}\left(\frac{-1259.1392167224028}{z + 1}, \frac{-1259.1392167224028}{z + 1} + \color{blue}{\frac{-676.5203681218851}{z}}, \frac{676.5203681218851}{z} \cdot \frac{676.5203681218851}{z}\right)} + \left(\frac{771.3234287776531}{z + 2} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      metadata-eval [=>] 2.2	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\mathsf{fma}\left(\frac{-1259.1392167224028}{z + 1}, \frac{-1259.1392167224028}{z + 1} + \frac{\color{blue}{-676.5203681218851}}{z}, \frac{676.5203681218851}{z} \cdot \frac{676.5203681218851}{z}\right)} + \left(\frac{771.3234287776531}{z + 2} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      associate-*l/ [=>] 2.2	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\mathsf{fma}\left(\frac{-1259.1392167224028}{z + 1}, \frac{-1259.1392167224028}{z + 1} + \frac{-676.5203681218851}{z}, \color{blue}{\frac{676.5203681218851 \cdot \frac{676.5203681218851}{z}}{z}}\right)} + \left(\frac{771.3234287776531}{z + 2} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      associate-*r/ [=>] 2.2	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\mathsf{fma}\left(\frac{-1259.1392167224028}{z + 1}, \frac{-1259.1392167224028}{z + 1} + \frac{-676.5203681218851}{z}, \frac{\color{blue}{\frac{676.5203681218851 \cdot 676.5203681218851}{z}}}{z}\right)} + \left(\frac{771.3234287776531}{z + 2} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      metadata-eval [=>] 2.2	\[ \sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\mathsf{fma}\left(\frac{-1259.1392167224028}{z + 1}, \frac{-1259.1392167224028}{z + 1} + \frac{-676.5203681218851}{z}, \frac{\frac{\color{blue}{457679.80848377093}}{z}}{z}\right)} + \left(\frac{771.3234287776531}{z + 2} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

   if 140 < (-.f64 z 1)

   1. Initial program 60.4

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

   2. Simplified 59.8

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

      [Start] 60.4	\[ \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      associate-*l* [=>] 59.8	\[ \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right)\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      associate-*l* [=>] 59.8	\[ \color{blue}{\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\right)} \]

   3. Applied egg-rr 59.8

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

   4. Simplified 59.8

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

      [Start] 59.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{\sqrt[3]{\left(e^{z + 6.5} \cdot e^{z + 6.5}\right) \cdot e^{z + 6.5}}}\right) \]
      *-commutative [=>] 59.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{\sqrt[3]{\color{blue}{e^{z + 6.5} \cdot \left(e^{z + 6.5} \cdot e^{z + 6.5}\right)}}}\right) \]
      cube-unmult [=>] 59.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{\sqrt[3]{\color{blue}{{\left(e^{z + 6.5}\right)}^{3}}}}\right) \]
      +-commutative [=>] 59.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{\sqrt[3]{{\left(e^{\color{blue}{6.5 + z}}\right)}^{3}}}\right) \]

   5. Applied egg-rr 7.9

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

4. Final simplification 2.3

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

Alternatives

Alternative 1
Error	2.2
Cost	80068

\[\begin{array}{l} t_0 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_1 := \sqrt{\pi \cdot 2}\\ t_2 := \frac{-0.13857109526572012}{z + 5}\\ t_3 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_4 := \left(z + -1\right) + 7\\ t_5 := \frac{12.507343278686905}{z + 4}\\ t_6 := \frac{771.3234287776531}{z + 2}\\ t_7 := \frac{-176.6150291621406}{z + 3}\\ \mathbf{if}\;\left(\left(t_1 \cdot {\left(t_4 + 0.5\right)}^{\left(\left(z + -1\right) + 0.5\right)}\right) \cdot e^{-0.5 + \left(-7 + \left(1 - z\right)\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 + \left(z + -1\right)}\right) + \frac{-1259.1392167224028}{\left(z + -1\right) + 2}\right) + \frac{771.3234287776531}{\left(z + -1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z + -1\right) + 4}\right) + \frac{12.507343278686905}{\left(z + -1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z + -1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t_4}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8}\right) \leq 4 \cdot 10^{+233}:\\ \;\;\;\;t_1 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(t_6 + \frac{\mathsf{fma}\left(z, -1259.1392167224028, \mathsf{fma}\left(676.5203681218851, z, 676.5203681218851\right)\right)}{\mathsf{fma}\left(z, z, z\right)}\right)\right) + \left(t_0 + \left(\left(t_2 + t_3\right) + \left(t_7 + t_5\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(t_7 + \left(t_5 + t_6\right)\right)\right) + \left(t_2 + \left(t_3 + t_0\right)\right)\right)\right) \cdot e^{\left(z + -0.5\right) \cdot \log \left(z + 6.5\right) + \left(-6.5 - z\right)}\right)\\ \end{array} \]

Alternative 2
Error	2.3
Cost	70852

\[\begin{array}{l} t_0 := \frac{-176.6150291621406}{z + 3}\\ t_1 := \frac{771.3234287776531}{z + 2}\\ t_2 := \frac{176.6150291621406}{z + 3} + \frac{-771.3234287776531}{z + 2}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \frac{-0.13857109526572012}{z + 5}\\ t_5 := \left(z + -1\right) + 7\\ t_6 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_7 := \frac{-1259.1392167224028}{z + 1}\\ t_8 := t_7 + \frac{676.5203681218851}{z}\\ t_9 := \frac{12.507343278686905}{z + 4}\\ \mathbf{if}\;\left(\left(t_3 \cdot {\left(t_5 + 0.5\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_5}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8}\right) \leq 4 \cdot 10^{+233}:\\ \;\;\;\;\left(\left(t_3 \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(t_6 + \left(\frac{\left(0.9999999999998099 + \frac{\mathsf{fma}\left(676.5203681218851, z + 1, z \cdot -1259.1392167224028\right)}{z \cdot \left(z + 1\right)}\right) \cdot \left(0.9999999999998099 + t_8\right) + \left(t_0 + t_1\right) \cdot t_2}{0.9999999999998099 + \left(t_8 + t_2\right)} + \left(t_9 + t_4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(t_7 + \left(t_0 + \left(t_9 + t_1\right)\right)\right) + \left(t_4 + t_6\right)\right)\right) \cdot e^{\left(z + -0.5\right) \cdot \log \left(z + 6.5\right) + \left(-6.5 - z\right)}\right)\\ \end{array} \]

Alternative 3
Error	2.3
Cost	67268

\[\begin{array}{l} t_0 := \frac{-0.13857109526572012}{z + 5}\\ t_1 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_2 := \frac{12.507343278686905}{z + 4}\\ t_3 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_4 := \frac{-1259.1392167224028}{z + 1}\\ t_5 := \sqrt{\pi \cdot 2}\\ t_6 := \frac{-176.6150291621406}{z + 3}\\ t_7 := \left(z + -1\right) + 7\\ t_8 := \frac{771.3234287776531}{z + 2}\\ \mathbf{if}\;\left(\left(t_5 \cdot {\left(t_7 + 0.5\right)}^{\left(\left(z + -1\right) + 0.5\right)}\right) \cdot e^{-0.5 + \left(-7 + \left(1 - z\right)\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 + \left(z + -1\right)}\right) + \frac{-1259.1392167224028}{\left(z + -1\right) + 2}\right) + \frac{771.3234287776531}{\left(z + -1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z + -1\right) + 4}\right) + \frac{12.507343278686905}{\left(z + -1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z + -1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t_7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8}\right) \leq 4 \cdot 10^{+233}:\\ \;\;\;\;t_5 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(t_3 + \left(\left(t_0 + t_1\right) + \left(t_6 + t_2\right)\right)\right) + \left(0.9999999999998099 + \left(t_8 + \left(t_4 + \frac{676.5203681218851}{z}\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_5 \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(t_4 + \left(t_6 + \left(t_2 + t_8\right)\right)\right) + \left(t_0 + \left(t_1 + t_3\right)\right)\right)\right) \cdot e^{\left(z + -0.5\right) \cdot \log \left(z + 6.5\right) + \left(-6.5 - z\right)}\right)\\ \end{array} \]

Alternative 4
Error	2.3
Cost	33476

\[\begin{array}{l} t_0 := \frac{-1259.1392167224028}{z + 1}\\ t_1 := t_0 + \frac{676.5203681218851}{z}\\ t_2 := \frac{-0.13857109526572012}{z + 5}\\ t_3 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_4 := \sqrt{\pi \cdot 2}\\ t_5 := \frac{-176.6150291621406}{z + 3}\\ t_6 := \frac{12.507343278686905}{z + 4}\\ t_7 := \frac{771.3234287776531}{z + 2}\\ t_8 := t_5 + t_7\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;\left(\left(t_4 \cdot {\left(\left(z + -1\right) + 7.5\right)}^{\left(z + -0.5\right)}\right) \cdot e^{-0.5 + \left(-6 - z\right)}\right) \cdot \left(t_3 + \left(\left(t_6 + t_2\right) + \frac{\left(0.9999999999998099 + t_1\right) \cdot \left(0.9999999999998099 + \frac{z \cdot -1259.1392167224028 + \left(z + 1\right) \cdot 676.5203681218851}{z \cdot \left(z + 1\right)}\right) - t_8 \cdot t_8}{0.9999999999998099 + \left(t_1 + \left(\frac{176.6150291621406}{z + 3} + \frac{-771.3234287776531}{z + 2}\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(t_0 + \left(t_5 + \left(t_6 + t_7\right)\right)\right) + \left(t_2 + t_3\right)\right)\right) \cdot e^{\left(z + -0.5\right) \cdot \log \left(z + 6.5\right) + \left(-6.5 - z\right)}\right)\\ \end{array} \]

Alternative 5
Error	2.3
Cost	29828

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

Alternative 6
Error	3.1
Cost	29636

\[\begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z \leq 145:\\ \;\;\;\;t_0 \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right) + \left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right)\right) + \left(0.9999999999998099 + \left(\left(\frac{-1259.1392167224028}{z + 1} + \frac{676.5203681218851}{z}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{771.3234287776531}{z + 2}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\left(\left(0.9999999999998099 + \frac{24.458333333348836}{z}\right) + \frac{197.000868054939}{z \cdot z}\right) \cdot e^{\left(z + -0.5\right) \cdot \log \left(z + 6.5\right) + \left(-6.5 - z\right)}\right)\\ \end{array} \]

Alternative 7
Error	3.1
Cost	29636

\[\begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z \leq 145:\\ \;\;\;\;t_0 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{z + 7} + \left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{676.5203681218851}{z}\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\left(\left(0.9999999999998099 + \frac{24.458333333348836}{z}\right) + \frac{197.000868054939}{z \cdot z}\right) \cdot e^{\left(z + -0.5\right) \cdot \log \left(z + 6.5\right) + \left(-6.5 - z\right)}\right)\\ \end{array} \]

Alternative 8
Error	47.0
Cost	27264

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

Alternative 9
Error	47.6
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 10
Error	47.4
Cost	27200

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

Alternative 11
Error	50.3
Cost	26948

\[\begin{array}{l} \mathbf{if}\;z \leq 2.8:\\ \;\;\;\;\frac{1}{z} \cdot \sqrt{\pi \cdot \left(140824.5564565449 \cdot e^{-13}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(\left(0.9999999999998099 + \frac{24.458333333348836}{z}\right) \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right)\right)\\ \end{array} \]

Alternative 12
Error	51.9
Cost	26692

\[\begin{array}{l} \mathbf{if}\;z \leq 3.95:\\ \;\;\;\;\frac{1}{z} \cdot \sqrt{\pi \cdot \left(140824.5564565449 \cdot e^{-13}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(0.9999999999998099 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right)\right)\\ \end{array} \]

Alternative 13
Error	55.5
Cost	19584

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

Reproduce

herbie shell --seed 2023073 
(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)))))