Jmat.Real.gamma, branch z greater than 0.5

Average Accuracy: 93.9% → 96.4%

Time: 31.9s

Precision: binary64

Cost: 42308

\[z > 0.5\]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_1 := \frac{-1259.1392167224028}{z + 1}\\ t_2 := \frac{676.5203681218851}{z} + 0.9999999999998099\\ t_3 := \frac{-176.6150291621406}{z + 3}\\ t_4 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_5 := \sqrt{\pi \cdot 2}\\ t_6 := \frac{12.507343278686905}{z + 4}\\ t_7 := \frac{-0.13857109526572012}{z + 5}\\ t_8 := \frac{1259.1392167224028}{z + 1}\\ t_9 := \frac{771.3234287776531}{2 + z}\\ t_10 := t_9 + t_6\\ t_11 := \left(\frac{176.6150291621406}{z + 3} - t_10\right) + \left(\frac{0.13857109526572012}{z + 5} + \left(\frac{-9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{-1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;t_5 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \left(t_9 + \left(\frac{676.5203681218851}{z} + t_1\right)\right)\right) + \left(t_4 + \left(\left(t_7 + t_0\right) + \left(t_3 + t_6\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_5 \cdot \left(\frac{t_2 \cdot t_2 + \left(t_1 + \left(\left(t_3 + t_10\right) + \left(t_7 + \left(t_0 + t_4\right)\right)\right)\right) \cdot \left(t_11 + t_8\right)}{\left(t_2 + t_8\right) + t_11} \cdot e^{\mathsf{fma}\left(-\log \left(z + 6.5\right), 0.5 - z, -6.5 - z\right)}\right)\\ \end{array} \]

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 (/ 9.984369578019572e-6 (+ z 6.0)))
        (t_1 (/ -1259.1392167224028 (+ z 1.0)))
        (t_2 (+ (/ 676.5203681218851 z) 0.9999999999998099))
        (t_3 (/ -176.6150291621406 (+ z 3.0)))
        (t_4 (/ 1.5056327351493116e-7 (+ z 7.0)))
        (t_5 (sqrt (* PI 2.0)))
        (t_6 (/ 12.507343278686905 (+ z 4.0)))
        (t_7 (/ -0.13857109526572012 (+ z 5.0)))
        (t_8 (/ 1259.1392167224028 (+ z 1.0)))
        (t_9 (/ 771.3234287776531 (+ 2.0 z)))
        (t_10 (+ t_9 t_6))
        (t_11
         (+
          (- (/ 176.6150291621406 (+ z 3.0)) t_10)
          (+
           (/ 0.13857109526572012 (+ z 5.0))
           (+
            (/ -9.984369578019572e-6 (+ z 6.0))
            (/ -1.5056327351493116e-7 (+ z 7.0)))))))
   (if (<= (+ z -1.0) 140.0)
     (*
      t_5
      (*
       (pow (+ z 6.5) (+ z -0.5))
       (*
        (/ (exp -6.5) (exp z))
        (+
         (+ 0.9999999999998099 (+ t_9 (+ (/ 676.5203681218851 z) t_1)))
         (+ t_4 (+ (+ t_7 t_0) (+ t_3 t_6)))))))
     (*
      t_5
      (*
       (/
        (+
         (* t_2 t_2)
         (* (+ t_1 (+ (+ t_3 t_10) (+ t_7 (+ t_0 t_4)))) (+ t_11 t_8)))
        (+ (+ t_2 t_8) t_11))
       (exp (fma (- (log (+ z 6.5))) (- 0.5 z) (- -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 = 9.984369578019572e-6 / (z + 6.0);
	double t_1 = -1259.1392167224028 / (z + 1.0);
	double t_2 = (676.5203681218851 / z) + 0.9999999999998099;
	double t_3 = -176.6150291621406 / (z + 3.0);
	double t_4 = 1.5056327351493116e-7 / (z + 7.0);
	double t_5 = sqrt((((double) M_PI) * 2.0));
	double t_6 = 12.507343278686905 / (z + 4.0);
	double t_7 = -0.13857109526572012 / (z + 5.0);
	double t_8 = 1259.1392167224028 / (z + 1.0);
	double t_9 = 771.3234287776531 / (2.0 + z);
	double t_10 = t_9 + t_6;
	double t_11 = ((176.6150291621406 / (z + 3.0)) - t_10) + ((0.13857109526572012 / (z + 5.0)) + ((-9.984369578019572e-6 / (z + 6.0)) + (-1.5056327351493116e-7 / (z + 7.0))));
	double tmp;
	if ((z + -1.0) <= 140.0) {
		tmp = t_5 * (pow((z + 6.5), (z + -0.5)) * ((exp(-6.5) / exp(z)) * ((0.9999999999998099 + (t_9 + ((676.5203681218851 / z) + t_1))) + (t_4 + ((t_7 + t_0) + (t_3 + t_6))))));
	} else {
		tmp = t_5 * ((((t_2 * t_2) + ((t_1 + ((t_3 + t_10) + (t_7 + (t_0 + t_4)))) * (t_11 + t_8))) / ((t_2 + t_8) + t_11)) * exp(fma(-log((z + 6.5)), (0.5 - z), (-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(9.984369578019572e-6 / Float64(z + 6.0))
	t_1 = Float64(-1259.1392167224028 / Float64(z + 1.0))
	t_2 = Float64(Float64(676.5203681218851 / z) + 0.9999999999998099)
	t_3 = Float64(-176.6150291621406 / Float64(z + 3.0))
	t_4 = Float64(1.5056327351493116e-7 / Float64(z + 7.0))
	t_5 = sqrt(Float64(pi * 2.0))
	t_6 = Float64(12.507343278686905 / Float64(z + 4.0))
	t_7 = Float64(-0.13857109526572012 / Float64(z + 5.0))
	t_8 = Float64(1259.1392167224028 / Float64(z + 1.0))
	t_9 = Float64(771.3234287776531 / Float64(2.0 + z))
	t_10 = Float64(t_9 + t_6)
	t_11 = Float64(Float64(Float64(176.6150291621406 / Float64(z + 3.0)) - t_10) + Float64(Float64(0.13857109526572012 / Float64(z + 5.0)) + 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_5 * Float64((Float64(z + 6.5) ^ Float64(z + -0.5)) * Float64(Float64(exp(-6.5) / exp(z)) * Float64(Float64(0.9999999999998099 + Float64(t_9 + Float64(Float64(676.5203681218851 / z) + t_1))) + Float64(t_4 + Float64(Float64(t_7 + t_0) + Float64(t_3 + t_6)))))));
	else
		tmp = Float64(t_5 * Float64(Float64(Float64(Float64(t_2 * t_2) + Float64(Float64(t_1 + Float64(Float64(t_3 + t_10) + Float64(t_7 + Float64(t_0 + t_4)))) * Float64(t_11 + t_8))) / Float64(Float64(t_2 + t_8) + t_11)) * exp(fma(Float64(-log(Float64(z + 6.5))), Float64(0.5 - z), 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[(9.984369578019572e-6 / N[(z + 6.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1259.1392167224028 / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(676.5203681218851 / z), $MachinePrecision] + 0.9999999999998099), $MachinePrecision]}, Block[{t$95$3 = N[(-176.6150291621406 / N[(z + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(1.5056327351493116e-7 / N[(z + 7.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(12.507343278686905 / N[(z + 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(-0.13857109526572012 / N[(z + 5.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(1259.1392167224028 / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[(771.3234287776531 / N[(2.0 + z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$10 = N[(t$95$9 + t$95$6), $MachinePrecision]}, Block[{t$95$11 = N[(N[(N[(176.6150291621406 / N[(z + 3.0), $MachinePrecision]), $MachinePrecision] - t$95$10), $MachinePrecision] + N[(N[(0.13857109526572012 / N[(z + 5.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-9.984369578019572e-6 / N[(z + 6.0), $MachinePrecision]), $MachinePrecision] + N[(-1.5056327351493116e-7 / N[(z + 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z + -1.0), $MachinePrecision], 140.0], N[(t$95$5 * N[(N[Power[N[(z + 6.5), $MachinePrecision], N[(z + -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Exp[-6.5], $MachinePrecision] / N[Exp[z], $MachinePrecision]), $MachinePrecision] * N[(N[(0.9999999999998099 + N[(t$95$9 + N[(N[(676.5203681218851 / z), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 + N[(N[(t$95$7 + t$95$0), $MachinePrecision] + N[(t$95$3 + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$5 * N[(N[(N[(N[(t$95$2 * t$95$2), $MachinePrecision] + N[(N[(t$95$1 + N[(N[(t$95$3 + t$95$10), $MachinePrecision] + N[(t$95$7 + N[(t$95$0 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$11 + t$95$8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$2 + t$95$8), $MachinePrecision] + t$95$11), $MachinePrecision]), $MachinePrecision] * N[Exp[N[((-N[Log[N[(z + 6.5), $MachinePrecision]], $MachinePrecision]) * N[(0.5 - z), $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 96.5%

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

   2. Simplified 96.7%

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

      [Start] 96.5	\[ \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      associate-*l* [=>] 96.5	\[ \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right)\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      associate-*l* [=>] 96.5	\[ \color{blue}{\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\right)} \]

   3. Applied egg-rr 96.6%

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

      [Start] 96.7

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

      exp-diff [=>] 96.6

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

   if 140 < (-.f64 z 1)

   1. Initial program 3.1%

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

   2. Simplified 3.1%

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

      [Start] 3.1	\[ \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      associate-*l* [=>] 3.1	\[ \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right)\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      associate-*l* [=>] 3.1	\[ \color{blue}{\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\right)} \]

   3. Taylor expanded in z around -inf 2.7%

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

   4. Simplified 88.1%

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

      [Start] 2.7	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \frac{e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right)}}{e^{6.5 - -1 \cdot z}}\right) \]
      div-exp [=>] 87.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \color{blue}{e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right) - \left(6.5 - -1 \cdot z\right)}}\right) \]
      associate-*r* [=>] 87.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\color{blue}{\left(-1 \cdot \log \left(6.5 - -1 \cdot z\right)\right) \cdot \left(-1 \cdot z + 0.5\right)} - \left(6.5 - -1 \cdot z\right)}\right) \]
      fma-neg [=>] 88.1	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\color{blue}{\mathsf{fma}\left(-1 \cdot \log \left(6.5 - -1 \cdot z\right), -1 \cdot z + 0.5, -\left(6.5 - -1 \cdot z\right)\right)}}\right) \]
      mul-1-neg [=>] 88.1	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(\color{blue}{-\log \left(6.5 - -1 \cdot z\right)}, -1 \cdot z + 0.5, -\left(6.5 - -1 \cdot z\right)\right)}\right) \]
      sub-neg [=>] 88.1	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \color{blue}{\left(6.5 + \left(--1 \cdot z\right)\right)}, -1 \cdot z + 0.5, -\left(6.5 - -1 \cdot z\right)\right)}\right) \]
      mul-1-neg [=>] 88.1	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + \left(-\color{blue}{\left(-z\right)}\right)\right), -1 \cdot z + 0.5, -\left(6.5 - -1 \cdot z\right)\right)}\right) \]
      remove-double-neg [=>] 88.1	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + \color{blue}{z}\right), -1 \cdot z + 0.5, -\left(6.5 - -1 \cdot z\right)\right)}\right) \]
      +-commutative [=>] 88.1	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), \color{blue}{0.5 + -1 \cdot z}, -\left(6.5 - -1 \cdot z\right)\right)}\right) \]
      mul-1-neg [=>] 88.1	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 + \color{blue}{\left(-z\right)}, -\left(6.5 - -1 \cdot z\right)\right)}\right) \]
      unsub-neg [=>] 88.1	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), \color{blue}{0.5 - z}, -\left(6.5 - -1 \cdot z\right)\right)}\right) \]
      sub-neg [=>] 88.1	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -\color{blue}{\left(6.5 + \left(--1 \cdot z\right)\right)}\right)}\right) \]
      +-commutative [=>] 88.1	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -\color{blue}{\left(\left(--1 \cdot z\right) + 6.5\right)}\right)}\right) \]
      mul-1-neg [=>] 88.1	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -\left(\left(-\color{blue}{\left(-z\right)}\right) + 6.5\right)\right)}\right) \]
      remove-double-neg [=>] 88.1	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -\left(\color{blue}{z} + 6.5\right)\right)}\right) \]
      neg-sub0 [=>] 88.1	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, \color{blue}{0 - \left(z + 6.5\right)}\right)}\right) \]
      +-commutative [=>] 88.1	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, 0 - \color{blue}{\left(6.5 + z\right)}\right)}\right) \]
      associate--r+ [=>] 88.1	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, \color{blue}{\left(0 - 6.5\right) - z}\right)}\right) \]
      metadata-eval [=>] 88.1	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, \color{blue}{-6.5} - z\right)}\right) \]

   5. Applied egg-rr 88.1%

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

      [Start] 88.1

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

      flip-+ [=>] 88.1

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

   6. Simplified 88.1%

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

      [Start] 88.1

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

3. Recombined 2 regimes into one program.

4. Final simplification 96.4%

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

Alternatives

Alternative 1
Accuracy	96.4%
Cost	43072

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

Alternative 2
Accuracy	96.4%
Cost	42052

\[\begin{array}{l} t_0 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_1 := \frac{-176.6150291621406}{z + 3}\\ t_2 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \frac{12.507343278686905}{z + 4}\\ t_5 := \frac{771.3234287776531}{2 + z}\\ t_6 := \frac{-1259.1392167224028}{z + 1}\\ t_7 := \frac{-0.13857109526572012}{z + 5}\\ t_8 := t_1 + \left(\left(t_5 + t_4\right) + \left(t_7 + \left(t_0 + t_2\right)\right)\right)\\ t_9 := \frac{1259.1392167224028}{z + 1} - t_8\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;t_3 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \left(t_5 + \left(\frac{676.5203681218851}{z} + t_6\right)\right)\right) + \left(t_2 + \left(\left(t_7 + t_0\right) + \left(t_1 + t_4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(e^{\mathsf{fma}\left(-\log \left(z + 6.5\right), 0.5 - z, -6.5 - z\right)} \cdot \left(0.9999999999998099 + \frac{\frac{676.5203681218851}{z} \cdot \frac{676.5203681218851}{z} + \left(t_6 + t_8\right) \cdot t_9}{\frac{676.5203681218851}{z} + t_9}\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	96.4%
Cost	41924

\[\begin{array}{l} t_0 := \frac{-1259.1392167224028}{z + 1}\\ t_1 := \frac{-176.6150291621406}{z + 3}\\ t_2 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \frac{12.507343278686905}{z + 4}\\ t_5 := \frac{771.3234287776531}{2 + z}\\ t_6 := \frac{-0.13857109526572012}{z + 5}\\ t_7 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_8 := t_5 + t_4\\ t_9 := t_0 + \left(\left(t_1 + t_8\right) + \left(t_6 + \left(t_7 + t_2\right)\right)\right)\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;t_3 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \left(t_5 + \left(\frac{676.5203681218851}{z} + t_0\right)\right)\right) + \left(t_2 + \left(\left(t_6 + t_7\right) + \left(t_1 + t_4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(e^{\mathsf{fma}\left(-\log \left(z + 6.5\right), 0.5 - z, -6.5 - z\right)} \cdot \left(0.9999999999998099 + \frac{\frac{\frac{457679.80848377093}{z}}{z} - t_9 \cdot t_9}{\left(\frac{676.5203681218851}{z} + \frac{1259.1392167224028}{z + 1}\right) + \left(\left(\frac{176.6150291621406}{z + 3} - t_8\right) + \left(\frac{0.13857109526572012}{z + 5} + \left(\frac{-9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{-1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)}\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	96.4%
Cost	36164

\[\begin{array}{l} t_0 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_1 := \frac{-1259.1392167224028}{z + 1}\\ t_2 := \frac{-176.6150291621406}{z + 3}\\ t_3 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_4 := \sqrt{\pi \cdot 2}\\ t_5 := \frac{12.507343278686905}{z + 4}\\ t_6 := \frac{771.3234287776531}{2 + z}\\ t_7 := \frac{-0.13857109526572012}{z + 5}\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;t_4 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \left(t_6 + \left(\frac{676.5203681218851}{z} + t_1\right)\right)\right) + \left(t_3 + \left(\left(t_7 + t_0\right) + \left(t_2 + t_5\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot \left(\left(\left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) + \left(\left(t_7 + \left(t_0 + t_3\right)\right) + \left(t_1 + \left(t_2 + \left(t_6 + t_5\right)\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(z + -0.5, \mathsf{log1p}\left(z + 5.5\right), -6.5 - z\right)}\right)\\ \end{array} \]

Alternative 5
Accuracy	96.4%
Cost	36100

\[\begin{array}{l} t_0 := \frac{-176.6150291621406}{z + 3}\\ t_1 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_2 := \frac{12.507343278686905}{z + 4}\\ t_3 := \frac{771.3234287776531}{2 + z}\\ t_4 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_5 := \frac{-1259.1392167224028}{z + 1}\\ t_6 := \sqrt{\pi \cdot 2}\\ t_7 := \frac{-0.13857109526572012}{z + 5}\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;t_6 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\left(\left(0.9999999999998099 + \left(t_3 + \left(\frac{676.5203681218851}{z} + t_5\right)\right)\right) + \left(t_1 + \left(\left(t_7 + t_4\right) + \left(t_0 + t_2\right)\right)\right)\right) \cdot e^{-6.5 - z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_6 \cdot \left(\left(\left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) + \left(\left(t_7 + \left(t_4 + t_1\right)\right) + \left(t_5 + \left(t_0 + \left(t_3 + t_2\right)\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(z + -0.5, \mathsf{log1p}\left(z + 5.5\right), -6.5 - z\right)}\right)\\ \end{array} \]

Alternative 6
Accuracy	96.4%
Cost	29828

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

Alternative 7
Accuracy	94.0%
Cost	29504

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

Alternative 8
Accuracy	94.0%
Cost	29504

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

Alternative 9
Accuracy	94.0%
Cost	29504

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

Alternative 10
Accuracy	26.9%
Cost	28736

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

Alternative 11
Accuracy	25.6%
Cost	27200

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

Alternative 12
Accuracy	25.6%
Cost	27200

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

Alternative 13
Accuracy	21.3%
Cost	26948

\[\begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z \leq 2.76:\\ \;\;\;\;\frac{\left(e^{-6.5} \cdot \left(t_0 \cdot 676.5203681218851\right)\right) \cdot \sqrt{0.15384615384615385}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\left(0.9999999999998099 + \frac{24.458333333348836}{z}\right) \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}}\right)\\ \end{array} \]

Alternative 14
Accuracy	18.5%
Cost	26240

\[\sqrt{\pi \cdot 2} \cdot \left(0.9999999999998099 \cdot e^{z \cdot \left(\log z + -1\right)}\right) \]

Alternative 15
Accuracy	13.1%
Cost	19584

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

Reproduce

herbie shell --seed 2023131 
(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)))))