Jmat.Real.gamma, branch z greater than 0.5

Average Accuracy: 93.7% → 96.4%

Time: 32.1s

Precision: binary64

Cost: 62020

\[z > 0.5\]

Math

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

↓

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

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

code[z_] := Block[{t$95$0 = N[(-1259.1392167224028 / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(771.3234287776531 / N[(z + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-176.6150291621406 / N[(z + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(-0.13857109526572012 / N[(z + 5.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(9.984369578019572e-6 / N[(z + 6.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(z + 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Power[t$95$0, 1/3], $MachinePrecision]}, If[LessEqual[N[(z + -1.0), $MachinePrecision], 140.0], N[(N[(t$95$3 * N[(N[Power[N[(z + 6.5), $MachinePrecision], N[(z - 0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-z) - 6.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 + N[(0.9999999999998099 + N[(N[(676.5203681218851 / z), $MachinePrecision] + N[(t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(12.507343278686905 / N[(z + 4.0), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * N[(N[(t$95$5 + N[(N[(0.9999999999998099 + N[(N[Power[t$95$6, 2.0], $MachinePrecision] * t$95$6 + N[(676.5203681218851 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 + N[(12.507343278686905 / N[(z - -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Log[N[(z + 6.5), $MachinePrecision]], $MachinePrecision] * N[(z + -0.5), $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 96.5%

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

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

   3. Applied egg-rr 96.4%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z - 0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\left(\frac{-176.6150291621406}{z + 3} + \left(0.9999999999998099 + \frac{676.5203681218851}{z}\right)\right) + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right)\right)} - 1\right)} + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
      Proof

      [Start] 96.5	\[ \left(\sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z - 0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right)\right) \cdot \left(\left(\left(\frac{-176.6150291621406}{z + 3} + \left(0.9999999999998099 + \frac{676.5203681218851}{z}\right)\right) + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right)\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
      expm1-log1p-u [=>] 96.4	\[ \left(\sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z - 0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{-176.6150291621406}{z + 3} + \left(0.9999999999998099 + \frac{676.5203681218851}{z}\right)\right) + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right)\right)\right)} + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
      expm1-udef [=>] 96.4	\[ \left(\sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z - 0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\left(\frac{-176.6150291621406}{z + 3} + \left(0.9999999999998099 + \frac{676.5203681218851}{z}\right)\right) + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right)\right)} - 1\right)} + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]

   4. Simplified 96.7%

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

      [Start] 96.4	\[ \left(\sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z - 0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right)\right) \cdot \left(\left(e^{\mathsf{log1p}\left(\left(\frac{-176.6150291621406}{z + 3} + \left(0.9999999999998099 + \frac{676.5203681218851}{z}\right)\right) + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right)\right)} - 1\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
      expm1-def [=>] 96.4	\[ \left(\sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z - 0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{-176.6150291621406}{z + 3} + \left(0.9999999999998099 + \frac{676.5203681218851}{z}\right)\right) + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right)\right)\right)} + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
      expm1-log1p [=>] 96.5	\[ \left(\sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z - 0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(\left(\frac{-176.6150291621406}{z + 3} + \left(0.9999999999998099 + \frac{676.5203681218851}{z}\right)\right) + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right)\right)} + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
      associate-+l+ [=>] 96.5	\[ \left(\sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z - 0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(\frac{-176.6150291621406}{z + 3} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right)\right)\right)} + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]
      associate-+l+ [=>] 96.7	\[ \left(\sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z - 0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right)\right) \cdot \left(\left(\frac{-176.6150291621406}{z + 3} + \color{blue}{\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right)\right)\right)}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]

   if 140 < (-.f64 z 1)

   1. Initial program 4.0%

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

   2. Simplified 4.8%

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

      [Start] 4.0	\[ \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      associate-*l* [=>] 4.8	\[ \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right)\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      associate-*l* [=>] 4.8	\[ \color{blue}{\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\right)} \]

   3. Taylor expanded in z around inf 4.8%

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

   4. Simplified 88.2%

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

      [Start] 4.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right) + \left(\left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{771.3234287776531}{2 + z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left({\left(6.5 + z\right)}^{\left(z - 0.5\right)} \cdot e^{-\left(6.5 + z\right)}\right)\right) \]
      sub-neg [=>] 4.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right) + \left(\left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{771.3234287776531}{2 + z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left({\left(6.5 + z\right)}^{\color{blue}{\left(z + \left(-0.5\right)\right)}} \cdot e^{-\left(6.5 + z\right)}\right)\right) \]
      remove-double-neg [<=] 4.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right) + \left(\left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{771.3234287776531}{2 + z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left({\left(6.5 + z\right)}^{\left(\color{blue}{\left(-\left(-z\right)\right)} + \left(-0.5\right)\right)} \cdot e^{-\left(6.5 + z\right)}\right)\right) \]
      mul-1-neg [<=] 4.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right) + \left(\left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{771.3234287776531}{2 + z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left({\left(6.5 + z\right)}^{\left(\left(-\color{blue}{-1 \cdot z}\right) + \left(-0.5\right)\right)} \cdot e^{-\left(6.5 + z\right)}\right)\right) \]
      distribute-neg-in [<=] 4.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right) + \left(\left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{771.3234287776531}{2 + z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left({\left(6.5 + z\right)}^{\color{blue}{\left(-\left(-1 \cdot z + 0.5\right)\right)}} \cdot e^{-\left(6.5 + z\right)}\right)\right) \]
      exp-to-pow [<=] 4.4	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right) + \left(\left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{771.3234287776531}{2 + z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(\color{blue}{e^{\log \left(6.5 + z\right) \cdot \left(-\left(-1 \cdot z + 0.5\right)\right)}} \cdot e^{-\left(6.5 + z\right)}\right)\right) \]
      *-lft-identity [<=] 4.4	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right) + \left(\left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{771.3234287776531}{2 + z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(e^{\log \left(6.5 + \color{blue}{1 \cdot z}\right) \cdot \left(-\left(-1 \cdot z + 0.5\right)\right)} \cdot e^{-\left(6.5 + z\right)}\right)\right) \]
      metadata-eval [<=] 4.4	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right) + \left(\left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{771.3234287776531}{2 + z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(e^{\log \left(6.5 + \color{blue}{\left(--1\right)} \cdot z\right) \cdot \left(-\left(-1 \cdot z + 0.5\right)\right)} \cdot e^{-\left(6.5 + z\right)}\right)\right) \]
      cancel-sign-sub-inv [<=] 4.4	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right) + \left(\left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{771.3234287776531}{2 + z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(e^{\log \color{blue}{\left(6.5 - -1 \cdot z\right)} \cdot \left(-\left(-1 \cdot z + 0.5\right)\right)} \cdot e^{-\left(6.5 + z\right)}\right)\right) \]
      distribute-rgt-neg-in [<=] 4.4	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right) + \left(\left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{771.3234287776531}{2 + z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(e^{\color{blue}{-\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)}} \cdot e^{-\left(6.5 + z\right)}\right)\right) \]
      mul-1-neg [<=] 4.4	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right) + \left(\left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{771.3234287776531}{2 + z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(e^{\color{blue}{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right)}} \cdot e^{-\left(6.5 + z\right)}\right)\right) \]
      +-commutative [=>] 4.4	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right) + \left(\left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{771.3234287776531}{2 + z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right)} \cdot e^{-\color{blue}{\left(z + 6.5\right)}}\right)\right) \]
      distribute-neg-in [=>] 4.4	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right) + \left(\left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{771.3234287776531}{2 + z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right)} \cdot e^{\color{blue}{\left(-z\right) + \left(-6.5\right)}}\right)\right) \]
      mul-1-neg [<=] 4.4	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right) + \left(\left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{771.3234287776531}{2 + z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right)} \cdot e^{\color{blue}{-1 \cdot z} + \left(-6.5\right)}\right)\right) \]
      sub-neg [<=] 4.4	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right) + \left(\left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{771.3234287776531}{2 + z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right)} \cdot e^{\color{blue}{-1 \cdot z - 6.5}}\right)\right) \]
      prod-exp [=>] 87.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right) + \left(\left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{771.3234287776531}{2 + z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \color{blue}{e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right) + \left(-1 \cdot z - 6.5\right)}}\right) \]

   5. Applied egg-rr 88.2%

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

      [Start] 88.2	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right) + \left(\left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{771.3234287776531}{2 + z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot e^{\mathsf{fma}\left(\log \left(6.5 + z\right), z + -0.5, -6.5 - z\right)}\right) \]
      +-commutative [=>] 88.2	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\left(0.9999999999998099 + \color{blue}{\left(\frac{-1259.1392167224028}{z + 1} + \frac{676.5203681218851}{z}\right)}\right) + \left(\left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{771.3234287776531}{2 + z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot e^{\mathsf{fma}\left(\log \left(6.5 + z\right), z + -0.5, -6.5 - z\right)}\right) \]
      add-cube-cbrt [=>] 88.2	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\color{blue}{\left(\sqrt[3]{\frac{-1259.1392167224028}{z + 1}} \cdot \sqrt[3]{\frac{-1259.1392167224028}{z + 1}}\right) \cdot \sqrt[3]{\frac{-1259.1392167224028}{z + 1}}} + \frac{676.5203681218851}{z}\right)\right) + \left(\left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{771.3234287776531}{2 + z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot e^{\mathsf{fma}\left(\log \left(6.5 + z\right), z + -0.5, -6.5 - z\right)}\right) \]
      fma-def [=>] 88.2	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\left(0.9999999999998099 + \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{-1259.1392167224028}{z + 1}} \cdot \sqrt[3]{\frac{-1259.1392167224028}{z + 1}}, \sqrt[3]{\frac{-1259.1392167224028}{z + 1}}, \frac{676.5203681218851}{z}\right)}\right) + \left(\left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{771.3234287776531}{2 + z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot e^{\mathsf{fma}\left(\log \left(6.5 + z\right), z + -0.5, -6.5 - z\right)}\right) \]
      pow2 [=>] 88.2	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\left(0.9999999999998099 + \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\frac{-1259.1392167224028}{z + 1}}\right)}^{2}}, \sqrt[3]{\frac{-1259.1392167224028}{z + 1}}, \frac{676.5203681218851}{z}\right)\right) + \left(\left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{771.3234287776531}{2 + z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot e^{\mathsf{fma}\left(\log \left(6.5 + z\right), z + -0.5, -6.5 - z\right)}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 96.4%

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

Alternatives

Alternative 1
Accuracy	96.4%
Cost	67268

\[\begin{array}{l} t_0 := \frac{-1259.1392167224028}{z + 1}\\ t_1 := \frac{771.3234287776531}{z + 2}\\ t_2 := \frac{-0.13857109526572012}{z + 5}\\ t_3 := \left(z + -1\right) + 7\\ t_4 := \sqrt{\pi \cdot 2}\\ t_5 := \frac{12.507343278686905}{z + 4}\\ t_6 := \frac{-176.6150291621406}{z + 3}\\ t_7 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ \mathbf{if}\;\left(\left(t_4 \cdot {\left(0.5 + t_3\right)}^{\left(\left(z + -1\right) + 0.5\right)}\right) \cdot e^{\left(\left(\left(-z\right) - -1\right) - 7\right) - 0.5}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 + \left(z + -1\right)}\right) + \frac{-1259.1392167224028}{\left(z + -1\right) + 2}\right) + \frac{771.3234287776531}{\left(z + -1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z + -1\right) + 4}\right) + \frac{12.507343278686905}{\left(z + -1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z + -1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t_3}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8}\right) \leq 5 \cdot 10^{+235}:\\ \;\;\;\;\left(t_4 \cdot \left({\left(z + 6.5\right)}^{\left(z - 0.5\right)} \cdot e^{\left(-z\right) - 6.5}\right)\right) \cdot \left(\left(t_6 + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(t_0 + t_1\right)\right)\right)\right) + \left(\left(t_5 + t_2\right) + t_7\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot \left(e^{\mathsf{fma}\left(\log \left(z + 6.5\right), z + -0.5, -6.5 - z\right)} \cdot \left(t_7 + \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + t_0\right) + \left(t_5 + \left(t_2 + \left(t_6 + t_1\right)\right)\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	96.4%
Cost	67268

\[\begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ t_1 := \left(z + -1\right) + 7\\ \mathbf{if}\;\left(\left(t_0 \cdot {\left(0.5 + t_1\right)}^{\left(\left(z + -1\right) + 0.5\right)}\right) \cdot e^{\left(\left(\left(-z\right) - -1\right) - 7\right) - 0.5}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 + \left(z + -1\right)}\right) + \frac{-1259.1392167224028}{\left(z + -1\right) + 2}\right) + \frac{771.3234287776531}{\left(z + -1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z + -1\right) + 4}\right) + \frac{12.507343278686905}{\left(z + -1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z + -1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t_1}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8}\right) \leq 5 \cdot 10^{+235}:\\ \;\;\;\;\left(t_0 \cdot \left({\left(z + 6.5\right)}^{\left(z - 0.5\right)} \cdot e^{\left(-z\right) - 6.5}\right)\right) \cdot \left(\left(\frac{-176.6150291621406}{z + 3} + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right)\right)\right)\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot e^{\mathsf{fma}\left(z + -0.5, \log \left(z + 6.5\right), -6.5 - z\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z - -1}\right)\right) + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	96.4%
Cost	60996

\[\begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ t_1 := \left(z + -1\right) + 7\\ \mathbf{if}\;\left(\left(t_0 \cdot {\left(0.5 + t_1\right)}^{\left(\left(z + -1\right) + 0.5\right)}\right) \cdot e^{\left(\left(\left(-z\right) - -1\right) - 7\right) - 0.5}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 + \left(z + -1\right)}\right) + \frac{-1259.1392167224028}{\left(z + -1\right) + 2}\right) + \frac{771.3234287776531}{\left(z + -1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z + -1\right) + 4}\right) + \frac{12.507343278686905}{\left(z + -1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z + -1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t_1}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8}\right) \leq 5 \cdot 10^{+235}:\\ \;\;\;\;\left(t_0 \cdot \left({\left(z + 6.5\right)}^{\left(z - 0.5\right)} \cdot e^{\left(-z\right) - 6.5}\right)\right) \cdot \left(\left(\frac{-176.6150291621406}{z + 3} + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right)\right)\right)\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z - -1}\right)\right) + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \cdot \left(t_0 \cdot e^{\log \left(z + 6.5\right) \cdot \left(z + -0.5\right) - \left(z + 6.5\right)}\right)\\ \end{array} \]

Alternative 4
Accuracy	96.4%
Cost	36100

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

Alternative 5
Accuracy	96.3%
Cost	29700

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

Alternative 6
Accuracy	96.4%
Cost	29700

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

Alternative 7
Accuracy	96.4%
Cost	29700

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

Alternative 8
Accuracy	93.7%
Cost	29504

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

Alternative 9
Accuracy	93.7%
Cost	29504

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

Alternative 10
Accuracy	93.8%
Cost	29504

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

Alternative 11
Accuracy	26.9%
Cost	28864

\[\left(\sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z - 0.5\right)} \cdot \frac{1}{e^{z + 6.5}}\right)\right) \cdot \left(\left(\left(\frac{12.507343278686905}{z + 4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\left(0.9999999999998099 + \frac{12.0895510149948}{z}\right) + \frac{246.3374466535184}{z \cdot z}\right)\right) \]

Alternative 12
Accuracy	26.9%
Cost	28800

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

Alternative 13
Accuracy	26.9%
Cost	28736

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

Alternative 14
Accuracy	26.9%
Cost	28736

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

Alternative 15
Accuracy	25.6%
Cost	27968

\[\sqrt{\pi \cdot 2} \cdot \left(\left(e^{-6.5 - z} \cdot {\left(z + 6.5\right)}^{\left(z + -0.5\right)}\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right) + \left(0.9999999999998099 + \left(\frac{24.458323198415986}{z} + \frac{197.0009290150994}{z \cdot z}\right)\right)\right)\right) \]

Alternative 16
Accuracy	19.7%
Cost	27584

\[\sqrt{\pi \cdot 2} \cdot \left(\left(e^{-6.5 - z} \cdot {\left(z + 6.5\right)}^{\left(z + -0.5\right)}\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right) + \left(0.9999999999998099 + \frac{24.458323198415986}{z}\right)\right)\right) \]

Alternative 17
Accuracy	16.2%
Cost	27456

\[\sqrt{\pi \cdot 2} \cdot \left(\left(e^{-6.5 - z} \cdot {\left(z + 6.5\right)}^{\left(z + -0.5\right)}\right) \cdot \left(\frac{676.5203681218851}{z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \]

Alternative 18
Accuracy	16.5%
Cost	27328

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

Alternative 19
Accuracy	12.6%
Cost	27200

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

Alternative 20
Accuracy	10.2%
Cost	26752

\[\sqrt{\pi \cdot 2} \cdot \left(\left(e^{-6.5} \cdot \sqrt{0.15384615384615385}\right) \cdot \left(\left(1.0000016855704452 + z \cdot -2.804163192570842 \cdot 10^{-7}\right) + \left(z \cdot z\right) \cdot 4.6662893212665185 \cdot 10^{-8}\right)\right) \]

Alternative 21
Accuracy	10.2%
Cost	26752

\[\sqrt{\pi \cdot 2} \cdot \left(\left(0.9999999999998099 + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{z + 7} + \frac{9.984369578019572 \cdot 10^{-6}}{z}\right)\right) \cdot \left(e^{-6.5} \cdot \sqrt{0.15384615384615385}\right)\right) \]

Alternative 22
Accuracy	10.2%
Cost	25984

\[\sqrt{\pi \cdot 2} \cdot \sqrt{e^{-13} \cdot 0.15384667248365105} \]

Reproduce

herbie shell --seed 2023146 
(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)))))