Jmat.Real.gamma, branch z greater than 0.5

Average Accuracy: 94.0% → 96.4%

Time: 33.6s

Precision: binary64

Cost: 68484

\[z > 0.5\]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{-176.6150291621406}{z + 3}\\ t_1 := \frac{771.3234287776531}{z + 2}\\ t_2 := \frac{12.507343278686905}{z + 4}\\ t_3 := \frac{-0.13857109526572012}{z + 5}\\ t_4 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_5 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_6 := \sqrt[3]{\frac{1259.1392167224028}{-1 - z}}\\ \mathbf{if}\;z + -1 \leq 142:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\sqrt[3]{{\left(e^{-6.5 - z}\right)}^{3}} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + t_1\right)\right) + \left(\left(\left(t_3 + t_5\right) + \left(t_0 + t_2\right)\right) + t_4\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\mathsf{fma}\left(-\log \left(z + 6.5\right), 0.5 - z, -6.5 - z\right)} \cdot \left(\sqrt{2} \cdot \sqrt{\pi}\right)\right) \cdot \left(\left(\left(\mathsf{fma}\left({t_6}^{2}, t_6, 0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(t_0 + t_1\right)\right) + \left(t_3 + t_2\right)\right) + \left(t_5 + t_4\right)\right)\\ \end{array} \]

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 (/ 771.3234287776531 (+ z 2.0)))
        (t_2 (/ 12.507343278686905 (+ z 4.0)))
        (t_3 (/ -0.13857109526572012 (+ z 5.0)))
        (t_4 (/ 1.5056327351493116e-7 (+ z 7.0)))
        (t_5 (/ 9.984369578019572e-6 (+ z 6.0)))
        (t_6 (cbrt (/ 1259.1392167224028 (- -1.0 z)))))
   (if (<= (+ z -1.0) 142.0)
     (*
      (sqrt (* PI 2.0))
      (*
       (pow (+ z 6.5) (+ z -0.5))
       (*
        (cbrt (pow (exp (- -6.5 z)) 3.0))
        (+
         (+
          0.9999999999998099
          (+
           (+ (/ 676.5203681218851 z) (/ -1259.1392167224028 (+ z 1.0)))
           t_1))
         (+ (+ (+ t_3 t_5) (+ t_0 t_2)) t_4)))))
     (*
      (*
       (exp (fma (- (log (+ z 6.5))) (- 0.5 z) (- -6.5 z)))
       (* (sqrt 2.0) (sqrt PI)))
      (+
       (+
        (+
         (fma (pow t_6 2.0) t_6 (+ 0.9999999999998099 (/ 676.5203681218851 z)))
         (+ t_0 t_1))
        (+ t_3 t_2))
       (+ t_5 t_4))))))

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 = 771.3234287776531 / (z + 2.0);
	double t_2 = 12.507343278686905 / (z + 4.0);
	double t_3 = -0.13857109526572012 / (z + 5.0);
	double t_4 = 1.5056327351493116e-7 / (z + 7.0);
	double t_5 = 9.984369578019572e-6 / (z + 6.0);
	double t_6 = cbrt((1259.1392167224028 / (-1.0 - z)));
	double tmp;
	if ((z + -1.0) <= 142.0) {
		tmp = sqrt((((double) M_PI) * 2.0)) * (pow((z + 6.5), (z + -0.5)) * (cbrt(pow(exp((-6.5 - z)), 3.0)) * ((0.9999999999998099 + (((676.5203681218851 / z) + (-1259.1392167224028 / (z + 1.0))) + t_1)) + (((t_3 + t_5) + (t_0 + t_2)) + t_4))));
	} else {
		tmp = (exp(fma(-log((z + 6.5)), (0.5 - z), (-6.5 - z))) * (sqrt(2.0) * sqrt(((double) M_PI)))) * (((fma(pow(t_6, 2.0), t_6, (0.9999999999998099 + (676.5203681218851 / z))) + (t_0 + t_1)) + (t_3 + t_2)) + (t_5 + t_4));
	}
	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(771.3234287776531 / Float64(z + 2.0))
	t_2 = Float64(12.507343278686905 / Float64(z + 4.0))
	t_3 = Float64(-0.13857109526572012 / Float64(z + 5.0))
	t_4 = Float64(1.5056327351493116e-7 / Float64(z + 7.0))
	t_5 = Float64(9.984369578019572e-6 / Float64(z + 6.0))
	t_6 = cbrt(Float64(1259.1392167224028 / Float64(-1.0 - z)))
	tmp = 0.0
	if (Float64(z + -1.0) <= 142.0)
		tmp = Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(z + 6.5) ^ Float64(z + -0.5)) * Float64(cbrt((exp(Float64(-6.5 - z)) ^ 3.0)) * Float64(Float64(0.9999999999998099 + Float64(Float64(Float64(676.5203681218851 / z) + Float64(-1259.1392167224028 / Float64(z + 1.0))) + t_1)) + Float64(Float64(Float64(t_3 + t_5) + Float64(t_0 + t_2)) + t_4)))));
	else
		tmp = Float64(Float64(exp(fma(Float64(-log(Float64(z + 6.5))), Float64(0.5 - z), Float64(-6.5 - z))) * Float64(sqrt(2.0) * sqrt(pi))) * Float64(Float64(Float64(fma((t_6 ^ 2.0), t_6, Float64(0.9999999999998099 + Float64(676.5203681218851 / z))) + Float64(t_0 + t_1)) + Float64(t_3 + t_2)) + Float64(t_5 + t_4)));
	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[(771.3234287776531 / N[(z + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(12.507343278686905 / N[(z + 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-0.13857109526572012 / N[(z + 5.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(1.5056327351493116e-7 / N[(z + 7.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(9.984369578019572e-6 / N[(z + 6.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Power[N[(1259.1392167224028 / N[(-1.0 - z), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(z + -1.0), $MachinePrecision], 142.0], N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(z + 6.5), $MachinePrecision], N[(z + -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[Power[N[Exp[N[(-6.5 - z), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision] * N[(N[(0.9999999999998099 + N[(N[(N[(676.5203681218851 / z), $MachinePrecision] + N[(-1259.1392167224028 / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$3 + t$95$5), $MachinePrecision] + N[(t$95$0 + t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[((-N[Log[N[(z + 6.5), $MachinePrecision]], $MachinePrecision]) * N[(0.5 - z), $MachinePrecision] + N[(-6.5 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[Power[t$95$6, 2.0], $MachinePrecision] * t$95$6 + N[(0.9999999999998099 + N[(676.5203681218851 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 + t$95$2), $MachinePrecision]), $MachinePrecision] + N[(t$95$5 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z 1) < 142

   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 96.6%

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

      [Start] 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%

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

   4. Simplified 96.6%

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

      [Start] 96.6	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\sqrt[3]{\left(e^{-6.5 - z} \cdot e^{-6.5 - z}\right) \cdot e^{-6.5 - z}} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      *-commutative [=>] 96.6	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\sqrt[3]{\color{blue}{e^{-6.5 - z} \cdot \left(e^{-6.5 - z} \cdot e^{-6.5 - z}\right)}} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      cube-unmult [=>] 96.6	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\sqrt[3]{\color{blue}{{\left(e^{-6.5 - z}\right)}^{3}}} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

   if 142 < (-.f64 z 1)

   1. Initial program 0.0%

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

   2. Simplified 0.0%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(z + -1\right) + 7.5\right)}^{\left(z - 0.5\right)}\right) \cdot e^{\left(-\left(z - -6\right)\right) + -0.5}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z - 0}\right) + \frac{-1259.1392167224028}{z - -1}\right) + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right)} \]
      Proof

      [Start] 0.0

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

   3. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{e^{\log \left(\sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{\left(-\left(z + 6\right)\right) + -0.5}\right)\right)}} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z - 0}\right) + \frac{-1259.1392167224028}{z - -1}\right) + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \]

   4. Taylor expanded in z around -inf 0.0%

      \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \left(e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right)} \cdot e^{-1 \cdot z - 6.5}\right)\right) \cdot \sqrt{\pi}\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z - 0}\right) + \frac{-1259.1392167224028}{z - -1}\right) + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \]

   5. Simplified 88.6%

      \[\leadsto \color{blue}{\left(e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -6.5 - z\right)} \cdot \left(\sqrt{2} \cdot \sqrt{\pi}\right)\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z - 0}\right) + \frac{-1259.1392167224028}{z - -1}\right) + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \]
      Proof

      [Start] 0.0	\[ \left(\left(\sqrt{2} \cdot \left(e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right)} \cdot e^{-1 \cdot z - 6.5}\right)\right) \cdot \sqrt{\pi}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z - 0}\right) + \frac{-1259.1392167224028}{z - -1}\right) + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \]
      *-commutative [=>] 0.0	\[ \left(\color{blue}{\left(\left(e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right)} \cdot e^{-1 \cdot z - 6.5}\right) \cdot \sqrt{2}\right)} \cdot \sqrt{\pi}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z - 0}\right) + \frac{-1259.1392167224028}{z - -1}\right) + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \]
      associate-*l* [=>] 0.0	\[ \color{blue}{\left(\left(e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right)} \cdot e^{-1 \cdot z - 6.5}\right) \cdot \left(\sqrt{2} \cdot \sqrt{\pi}\right)\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z - 0}\right) + \frac{-1259.1392167224028}{z - -1}\right) + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \]

   6. Applied egg-rr 88.6%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + -1 \leq 142:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\sqrt[3]{{\left(e^{-6.5 - z}\right)}^{3}} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + \frac{771.3234287776531}{z + 2}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\mathsf{fma}\left(-\log \left(z + 6.5\right), 0.5 - z, -6.5 - z\right)} \cdot \left(\sqrt{2} \cdot \sqrt{\pi}\right)\right) \cdot \left(\left(\left(\mathsf{fma}\left({\left(\sqrt[3]{\frac{1259.1392167224028}{-1 - z}}\right)}^{2}, \sqrt[3]{\frac{1259.1392167224028}{-1 - z}}, 0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{771.3234287776531}{z + 2}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	96.4%
Cost	48964

\[\begin{array}{l} t_0 := \frac{-176.6150291621406}{z + 3}\\ t_1 := \frac{771.3234287776531}{z + 2}\\ t_2 := \frac{12.507343278686905}{z + 4}\\ t_3 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_4 := \frac{-0.13857109526572012}{z + 5}\\ t_5 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ \mathbf{if}\;z + -1 \leq 142:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\sqrt[3]{{\left(e^{-6.5 - z}\right)}^{3}} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + t_1\right)\right) + \left(\left(\left(t_4 + t_3\right) + \left(t_0 + t_2\right)\right) + t_5\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\mathsf{fma}\left(-\log \left(z + 6.5\right), 0.5 - z, -6.5 - z\right)} \cdot \left(\sqrt{2} \cdot \sqrt{\pi}\right)\right) \cdot \left(\left(\left(\mathsf{fma}\left(-1259.1392167224028, \frac{-1}{-1 - z}, 0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(t_0 + t_1\right)\right) + \left(t_4 + t_2\right)\right) + \left(t_3 + t_5\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	96.4%
Cost	42628

\[\begin{array}{l} t_0 := \frac{-1259.1392167224028}{z + 1}\\ t_1 := \frac{-176.6150291621406}{z + 3}\\ t_2 := \frac{771.3234287776531}{z + 2}\\ t_3 := \frac{12.507343278686905}{z + 4}\\ t_4 := \sqrt{\pi \cdot 2}\\ t_5 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_6 := \frac{-0.13857109526572012}{z + 5}\\ t_7 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ \mathbf{if}\;z + -1 \leq 142:\\ \;\;\;\;t_4 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\sqrt[3]{{\left(e^{-6.5 - z}\right)}^{3}} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + t_0\right) + t_2\right)\right) + \left(\left(\left(t_6 + t_5\right) + \left(t_1 + t_3\right)\right) + t_7\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot \left(e^{\mathsf{fma}\left(-\log \left(z + 6.5\right), 0.5 - z, -6.5 - z\right)} \cdot \left(\frac{676.5203681218851}{z} + \left(\left(0.9999999999998099 + \left(t_0 + \left(\left(t_6 + t_3\right) + \left(t_1 + t_2\right)\right)\right)\right) + \left(t_5 + t_7\right)\right)\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	96.4%
Cost	36164

\[\begin{array}{l} t_0 := \frac{771.3234287776531}{z + 2}\\ t_1 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_2 := \frac{-0.13857109526572012}{z + 5}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \frac{-176.6150291621406}{z + 3}\\ t_5 := \frac{-1259.1392167224028}{z + 1}\\ t_6 := \frac{12.507343278686905}{z + 4}\\ \mathbf{if}\;z + -1 \leq 142:\\ \;\;\;\;t_3 \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(t_5 + t_0\right)\right)\right) + \left(t_2 + \left(\left(t_4 + t_6\right) + t_1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(e^{\mathsf{fma}\left(-\log \left(z + 6.5\right), 0.5 - z, -6.5 - z\right)} \cdot \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(t_5 + \left(t_4 + \left(t_6 + t_0\right)\right)\right) + \left(t_2 + t_1\right)\right)\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	96.4%
Cost	36164

\[\begin{array}{l} t_0 := \frac{771.3234287776531}{z + 2}\\ t_1 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_2 := \frac{-0.13857109526572012}{z + 5}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \frac{-176.6150291621406}{z + 3}\\ t_5 := \frac{-1259.1392167224028}{z + 1}\\ t_6 := \frac{12.507343278686905}{z + 4}\\ \mathbf{if}\;z + -1 \leq 142:\\ \;\;\;\;t_3 \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(t_5 + t_0\right)\right)\right) + \left(t_2 + \left(\left(t_4 + t_6\right) + t_1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(e^{\mathsf{fma}\left(-\log \left(z + 6.5\right), 0.5 - z, -6.5 - z\right)} \cdot \left(\frac{676.5203681218851}{z} + \left(\left(0.9999999999998099 + \left(t_5 + \left(\left(t_2 + t_6\right) + \left(t_4 + t_0\right)\right)\right)\right) + t_1\right)\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	96.4%
Cost	36100

\[\begin{array}{l} t_0 := \frac{771.3234287776531}{z + 2}\\ t_1 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_2 := \frac{-0.13857109526572012}{z + 5}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \frac{-176.6150291621406}{z + 3}\\ t_5 := \frac{-1259.1392167224028}{z + 1}\\ t_6 := \frac{12.507343278686905}{z + 4}\\ \mathbf{if}\;z + -1 \leq 142:\\ \;\;\;\;t_3 \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(t_5 + t_0\right)\right)\right) + \left(t_2 + \left(\left(t_4 + t_6\right) + t_1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(e^{\mathsf{fma}\left(\log \left(z + 6.5\right), z + -0.5, -6.5 - z\right)} \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + t_5\right) + \left(t_4 + t_0\right)\right)\right) + \left(\left(t_2 + t_6\right) + t_1\right)\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	96.3%
Cost	29828

\[\begin{array}{l} t_0 := \frac{771.3234287776531}{z + 2}\\ t_1 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_2 := \frac{-0.13857109526572012}{z + 5}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \frac{-176.6150291621406}{z + 3}\\ t_5 := \frac{-1259.1392167224028}{z + 1}\\ t_6 := \frac{12.507343278686905}{z + 4}\\ \mathbf{if}\;z + -1 \leq 142:\\ \;\;\;\;t_3 \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(t_5 + t_0\right)\right)\right) + \left(t_2 + \left(\left(t_4 + t_6\right) + t_1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(t_5 + \left(t_4 + \left(t_6 + t_0\right)\right)\right) + \left(t_2 + t_1\right)\right)\right) \cdot e^{\left(-6.5 - z\right) + \left(z + -0.5\right) \cdot \log \left(z + 6.5\right)}\right)\\ \end{array} \]

Alternative 7
Accuracy	96.3%
Cost	29700

\[\begin{array}{l} t_0 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_1 := \frac{-0.13857109526572012}{z + 5}\\ t_2 := \sqrt{\pi \cdot 2}\\ t_3 := \frac{-176.6150291621406}{z + 3}\\ t_4 := \frac{-1259.1392167224028}{z + 1}\\ t_5 := \frac{12.507343278686905}{z + 4}\\ t_6 := \frac{771.3234287776531}{z + 2}\\ \mathbf{if}\;z \leq 143:\\ \;\;\;\;t_2 \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(t_4 + t_6\right)\right)\right) + \left(t_1 + \left(\left(t_3 + t_5\right) + t_0\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + t_4\right) + \left(t_3 + t_6\right)\right)\right) + \left(\left(t_1 + t_5\right) + t_0\right)\right) \cdot e^{\left(-6.5 - z\right) + \left(z + -0.5\right) \cdot \log \left(z + 6.5\right)}\right)\\ \end{array} \]

Alternative 8
Accuracy	94.0%
Cost	29504

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

Alternative 9
Accuracy	94.1%
Cost	29504

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

Alternative 10
Accuracy	26.8%
Cost	28736

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

Alternative 11
Accuracy	25.5%
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	25.5%
Cost	27200

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

Alternative 13
Accuracy	19.6%
Cost	26816

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

Alternative 14
Accuracy	19.9%
Cost	26756

\[\begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z \leq 14.5:\\ \;\;\;\;t_0 \cdot \left(0.9999999999998099 \cdot e^{z \cdot \left(-1 + \log z\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(0.9999999999998099 \cdot e^{\left(-6.5 - z\right) + \left(z + -0.5\right) \cdot \log \left(z + 6.5\right)}\right)\\ \end{array} \]

Alternative 15
Accuracy	18.5%
Cost	26240

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

Alternative 16
Accuracy	13.2%
Cost	19712

\[\frac{1}{z} \cdot \sqrt{\pi \cdot \left(140824.5564565449 \cdot e^{-13}\right)} \]

Alternative 17
Accuracy	13.2%
Cost	19584

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

Reproduce

herbie shell --seed 2023129 
(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)))))