Jmat.Real.gamma, branch z greater than 0.5

Average Error: 94.0% → 96.7%

Time: 56.7s

Precision: binary64

Cost: 55364.00

\[z > 0.5\]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{771.3234287776531}{z + 2}\\ t_1 := \frac{12.507343278686905}{z + 4}\\ t_2 := \frac{-0.13857109526572012}{z + 5}\\ t_3 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_4 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_5 := \frac{-176.6150291621406}{z + 3}\\ t_6 := 0.9999999999998099 + \frac{676.5203681218851}{z}\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \left(\frac{\mathsf{fma}\left(z, -1259.1392167224028, \mathsf{fma}\left(676.5203681218851, z, 676.5203681218851\right)\right)}{\mathsf{fma}\left(z, z, z\right)} + t_0\right)\right) + \left(\left(\left(t_2 + t_4\right) + \left(t_5 + t_1\right)\right) + t_3\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(\sqrt[3]{{\pi}^{1.5}} \cdot e^{\mathsf{fma}\left(-\log \left(z + 6.5\right), 0.5 - z, -6.5 - z\right)}\right)\right) \cdot \left(\left(\left(\frac{t_6 \cdot t_6 + \frac{\frac{-1585431.567088306}{z + 1}}{z + 1}}{\frac{676.5203681218851}{z} + \left(0.9999999999998099 - \frac{-1259.1392167224028}{z + 1}\right)} + \left(t_5 + t_0\right)\right) + \left(t_2 + t_1\right)\right) + \left(t_4 + t_3\right)\right)\\ \end{array} \]

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 (/ 771.3234287776531 (+ z 2.0)))
        (t_1 (/ 12.507343278686905 (+ z 4.0)))
        (t_2 (/ -0.13857109526572012 (+ z 5.0)))
        (t_3 (/ 1.5056327351493116e-7 (+ z 7.0)))
        (t_4 (/ 9.984369578019572e-6 (+ z 6.0)))
        (t_5 (/ -176.6150291621406 (+ z 3.0)))
        (t_6 (+ 0.9999999999998099 (/ 676.5203681218851 z))))
   (if (<= (+ z -1.0) 140.0)
     (*
      (sqrt (* PI 2.0))
      (*
       (pow (+ z 6.5) (+ z -0.5))
       (*
        (/ (exp -6.5) (exp z))
        (+
         (+
          0.9999999999998099
          (+
           (/
            (fma
             z
             -1259.1392167224028
             (fma 676.5203681218851 z 676.5203681218851))
            (fma z z z))
           t_0))
         (+ (+ (+ t_2 t_4) (+ t_5 t_1)) t_3)))))
     (*
      (*
       (sqrt 2.0)
       (*
        (cbrt (pow PI 1.5))
        (exp (fma (- (log (+ z 6.5))) (- 0.5 z) (- -6.5 z)))))
      (+
       (+
        (+
         (/
          (+ (* t_6 t_6) (/ (/ -1585431.567088306 (+ z 1.0)) (+ z 1.0)))
          (+
           (/ 676.5203681218851 z)
           (- 0.9999999999998099 (/ -1259.1392167224028 (+ z 1.0)))))
         (+ t_5 t_0))
        (+ t_2 t_1))
       (+ t_4 t_3))))))

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 = 771.3234287776531 / (z + 2.0);
	double t_1 = 12.507343278686905 / (z + 4.0);
	double t_2 = -0.13857109526572012 / (z + 5.0);
	double t_3 = 1.5056327351493116e-7 / (z + 7.0);
	double t_4 = 9.984369578019572e-6 / (z + 6.0);
	double t_5 = -176.6150291621406 / (z + 3.0);
	double t_6 = 0.9999999999998099 + (676.5203681218851 / z);
	double tmp;
	if ((z + -1.0) <= 140.0) {
		tmp = sqrt((((double) M_PI) * 2.0)) * (pow((z + 6.5), (z + -0.5)) * ((exp(-6.5) / exp(z)) * ((0.9999999999998099 + ((fma(z, -1259.1392167224028, fma(676.5203681218851, z, 676.5203681218851)) / fma(z, z, z)) + t_0)) + (((t_2 + t_4) + (t_5 + t_1)) + t_3))));
	} else {
		tmp = (sqrt(2.0) * (cbrt(pow(((double) M_PI), 1.5)) * exp(fma(-log((z + 6.5)), (0.5 - z), (-6.5 - z))))) * ((((((t_6 * t_6) + ((-1585431.567088306 / (z + 1.0)) / (z + 1.0))) / ((676.5203681218851 / z) + (0.9999999999998099 - (-1259.1392167224028 / (z + 1.0))))) + (t_5 + t_0)) + (t_2 + t_1)) + (t_4 + t_3));
	}
	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(771.3234287776531 / Float64(z + 2.0))
	t_1 = Float64(12.507343278686905 / Float64(z + 4.0))
	t_2 = Float64(-0.13857109526572012 / Float64(z + 5.0))
	t_3 = Float64(1.5056327351493116e-7 / Float64(z + 7.0))
	t_4 = Float64(9.984369578019572e-6 / Float64(z + 6.0))
	t_5 = Float64(-176.6150291621406 / Float64(z + 3.0))
	t_6 = Float64(0.9999999999998099 + Float64(676.5203681218851 / z))
	tmp = 0.0
	if (Float64(z + -1.0) <= 140.0)
		tmp = Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(z + 6.5) ^ Float64(z + -0.5)) * Float64(Float64(exp(-6.5) / exp(z)) * Float64(Float64(0.9999999999998099 + Float64(Float64(fma(z, -1259.1392167224028, fma(676.5203681218851, z, 676.5203681218851)) / fma(z, z, z)) + t_0)) + Float64(Float64(Float64(t_2 + t_4) + Float64(t_5 + t_1)) + t_3)))));
	else
		tmp = Float64(Float64(sqrt(2.0) * Float64(cbrt((pi ^ 1.5)) * exp(fma(Float64(-log(Float64(z + 6.5))), Float64(0.5 - z), Float64(-6.5 - z))))) * Float64(Float64(Float64(Float64(Float64(Float64(t_6 * t_6) + Float64(Float64(-1585431.567088306 / Float64(z + 1.0)) / Float64(z + 1.0))) / Float64(Float64(676.5203681218851 / z) + Float64(0.9999999999998099 - Float64(-1259.1392167224028 / Float64(z + 1.0))))) + Float64(t_5 + t_0)) + Float64(t_2 + t_1)) + Float64(t_4 + t_3)));
	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[(771.3234287776531 / N[(z + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(12.507343278686905 / N[(z + 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-0.13857109526572012 / N[(z + 5.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.5056327351493116e-7 / N[(z + 7.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(9.984369578019572e-6 / N[(z + 6.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(-176.6150291621406 / N[(z + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(0.9999999999998099 + N[(676.5203681218851 / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z + -1.0), $MachinePrecision], 140.0], 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[(N[Exp[-6.5], $MachinePrecision] / N[Exp[z], $MachinePrecision]), $MachinePrecision] * N[(N[(0.9999999999998099 + N[(N[(N[(z * -1259.1392167224028 + N[(676.5203681218851 * z + 676.5203681218851), $MachinePrecision]), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$2 + t$95$4), $MachinePrecision] + N[(t$95$5 + t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Power[Pi, 1.5], $MachinePrecision], 1/3], $MachinePrecision] * N[Exp[N[((-N[Log[N[(z + 6.5), $MachinePrecision]], $MachinePrecision]) * N[(0.5 - z), $MachinePrecision] + N[(-6.5 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(t$95$6 * t$95$6), $MachinePrecision] + N[(N[(-1585431.567088306 / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(676.5203681218851 / z), $MachinePrecision] + N[(0.9999999999998099 - N[(-1259.1392167224028 / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$5 + t$95$0), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 96.5

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

   2. Simplified 96.7

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

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

   3. Applied egg-rr 96.6

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

   4. Applied egg-rr 96.5

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

   5. Simplified 97.0

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

      [Start] 96.5	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{676.5203681218851 \cdot \left(z + 1\right) + z \cdot -1259.1392167224028}{z \cdot \left(z + 1\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      +-commutative [=>] 96.5	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\color{blue}{z \cdot -1259.1392167224028 + 676.5203681218851 \cdot \left(z + 1\right)}}{z \cdot \left(z + 1\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      fma-def [=>] 96.8	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\color{blue}{\mathsf{fma}\left(z, -1259.1392167224028, 676.5203681218851 \cdot \left(z + 1\right)\right)}}{z \cdot \left(z + 1\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      distribute-lft-in [=>] 96.9	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\mathsf{fma}\left(z, -1259.1392167224028, \color{blue}{676.5203681218851 \cdot z + 676.5203681218851 \cdot 1}\right)}{z \cdot \left(z + 1\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      metadata-eval [=>] 96.9	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\mathsf{fma}\left(z, -1259.1392167224028, 676.5203681218851 \cdot z + \color{blue}{676.5203681218851}\right)}{z \cdot \left(z + 1\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      fma-def [=>] 96.9	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\mathsf{fma}\left(z, -1259.1392167224028, \color{blue}{\mathsf{fma}\left(676.5203681218851, z, 676.5203681218851\right)}\right)}{z \cdot \left(z + 1\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      distribute-rgt-in [=>] 96.9	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\mathsf{fma}\left(z, -1259.1392167224028, \mathsf{fma}\left(676.5203681218851, z, 676.5203681218851\right)\right)}{\color{blue}{z \cdot z + 1 \cdot z}}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      *-lft-identity [=>] 96.9	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\mathsf{fma}\left(z, -1259.1392167224028, \mathsf{fma}\left(676.5203681218851, z, 676.5203681218851\right)\right)}{z \cdot z + \color{blue}{z}}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      fma-def [=>] 97.0	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\mathsf{fma}\left(z, -1259.1392167224028, \mathsf{fma}\left(676.5203681218851, z, 676.5203681218851\right)\right)}{\color{blue}{\mathsf{fma}\left(z, z, z\right)}}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

   if 140 < (-.f64 z 1)

   1. Initial program 4.7

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

   2. Simplified 4.8

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

      [Start] 4.7

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

   3. Taylor expanded in z around -inf 4.8

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

   4. Simplified 88.0

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

      [Start] 4.8	\[ \left(\left(\sqrt{2} \cdot \left(e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right)} \cdot e^{-1 \cdot z - 6.5}\right)\right) \cdot \sqrt{\pi}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z - 0}\right) + \frac{-1259.1392167224028}{z - -1}\right) + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \]
      associate-*l* [=>] 4.8	\[ \color{blue}{\left(\sqrt{2} \cdot \left(\left(e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right)} \cdot e^{-1 \cdot z - 6.5}\right) \cdot \sqrt{\pi}\right)\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z - 0}\right) + \frac{-1259.1392167224028}{z - -1}\right) + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \]
      *-commutative [=>] 4.8	\[ \left(\sqrt{2} \cdot \color{blue}{\left(\sqrt{\pi} \cdot \left(e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right)} \cdot e^{-1 \cdot z - 6.5}\right)\right)}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z - 0}\right) + \frac{-1259.1392167224028}{z - -1}\right) + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \]
      prod-exp [=>] 87.9	\[ \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot \color{blue}{e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right) + \left(-1 \cdot z - 6.5\right)}}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z - 0}\right) + \frac{-1259.1392167224028}{z - -1}\right) + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \]
      associate-*r* [=>] 87.9	\[ \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot e^{\color{blue}{\left(-1 \cdot \log \left(6.5 - -1 \cdot z\right)\right) \cdot \left(-1 \cdot z + 0.5\right)} + \left(-1 \cdot z - 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z - 0}\right) + \frac{-1259.1392167224028}{z - -1}\right) + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \]
      fma-def [=>] 88.0	\[ \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot e^{\color{blue}{\mathsf{fma}\left(-1 \cdot \log \left(6.5 - -1 \cdot z\right), -1 \cdot z + 0.5, -1 \cdot z - 6.5\right)}}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z - 0}\right) + \frac{-1259.1392167224028}{z - -1}\right) + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \]

   5. Applied egg-rr 88.0

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

   6. Simplified 87.9

      \[\leadsto \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -6.5 - z\right)}\right)\right) \cdot \left(\left(\left(\color{blue}{\frac{\left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) \cdot \left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) - \frac{\frac{1585431.567088306}{z + 1}}{z + 1}}{\frac{676.5203681218851}{z} + \left(0.9999999999998099 - \frac{-1259.1392167224028}{z + 1}\right)}} + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \]
      Proof

      [Start] 88.0	\[ \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -6.5 - z\right)}\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) - \frac{-1259.1392167224028}{z + 1} \cdot \frac{-1259.1392167224028}{z + 1}}{\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) - \frac{-1259.1392167224028}{z + 1}} + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \]
      +-commutative [=>] 88.0	\[ \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -6.5 - z\right)}\right)\right) \cdot \left(\left(\left(\frac{\color{blue}{\left(\frac{676.5203681218851}{z} + 0.9999999999998099\right)} \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) - \frac{-1259.1392167224028}{z + 1} \cdot \frac{-1259.1392167224028}{z + 1}}{\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) - \frac{-1259.1392167224028}{z + 1}} + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \]
      +-commutative [=>] 88.0	\[ \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -6.5 - z\right)}\right)\right) \cdot \left(\left(\left(\frac{\left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) \cdot \color{blue}{\left(\frac{676.5203681218851}{z} + 0.9999999999998099\right)} - \frac{-1259.1392167224028}{z + 1} \cdot \frac{-1259.1392167224028}{z + 1}}{\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) - \frac{-1259.1392167224028}{z + 1}} + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \]
      associate-*l/ [=>] 88.0	\[ \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -6.5 - z\right)}\right)\right) \cdot \left(\left(\left(\frac{\left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) \cdot \left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) - \color{blue}{\frac{-1259.1392167224028 \cdot \frac{-1259.1392167224028}{z + 1}}{z + 1}}}{\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) - \frac{-1259.1392167224028}{z + 1}} + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \]
      associate-*r/ [=>] 87.9	\[ \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -6.5 - z\right)}\right)\right) \cdot \left(\left(\left(\frac{\left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) \cdot \left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) - \frac{\color{blue}{\frac{-1259.1392167224028 \cdot -1259.1392167224028}{z + 1}}}{z + 1}}{\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) - \frac{-1259.1392167224028}{z + 1}} + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \]
      metadata-eval [=>] 87.9	\[ \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -6.5 - z\right)}\right)\right) \cdot \left(\left(\left(\frac{\left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) \cdot \left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) - \frac{\frac{\color{blue}{1585431.567088306}}{z + 1}}{z + 1}}{\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) - \frac{-1259.1392167224028}{z + 1}} + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \]
      +-commutative [=>] 87.9	\[ \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -6.5 - z\right)}\right)\right) \cdot \left(\left(\left(\frac{\left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) \cdot \left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) - \frac{\frac{1585431.567088306}{z + 1}}{z + 1}}{\color{blue}{\left(\frac{676.5203681218851}{z} + 0.9999999999998099\right)} - \frac{-1259.1392167224028}{z + 1}} + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \]
      metadata-eval [<=] 87.9	\[ \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -6.5 - z\right)}\right)\right) \cdot \left(\left(\left(\frac{\left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) \cdot \left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) - \frac{\frac{1585431.567088306}{z + 1}}{z + 1}}{\left(\frac{\color{blue}{676.5203681218851 \cdot 1}}{z} + 0.9999999999998099\right) - \frac{-1259.1392167224028}{z + 1}} + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \]
      associate-*r/ [<=] 87.9	\[ \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -6.5 - z\right)}\right)\right) \cdot \left(\left(\left(\frac{\left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) \cdot \left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) - \frac{\frac{1585431.567088306}{z + 1}}{z + 1}}{\left(\color{blue}{676.5203681218851 \cdot \frac{1}{z}} + 0.9999999999998099\right) - \frac{-1259.1392167224028}{z + 1}} + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \]
      associate--l+ [=>] 87.9	\[ \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -6.5 - z\right)}\right)\right) \cdot \left(\left(\left(\frac{\left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) \cdot \left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) - \frac{\frac{1585431.567088306}{z + 1}}{z + 1}}{\color{blue}{676.5203681218851 \cdot \frac{1}{z} + \left(0.9999999999998099 - \frac{-1259.1392167224028}{z + 1}\right)}} + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \]
      associate-*r/ [=>] 87.9	\[ \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -6.5 - z\right)}\right)\right) \cdot \left(\left(\left(\frac{\left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) \cdot \left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) - \frac{\frac{1585431.567088306}{z + 1}}{z + 1}}{\color{blue}{\frac{676.5203681218851 \cdot 1}{z}} + \left(0.9999999999998099 - \frac{-1259.1392167224028}{z + 1}\right)} + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \]
      metadata-eval [=>] 87.9	\[ \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -6.5 - z\right)}\right)\right) \cdot \left(\left(\left(\frac{\left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) \cdot \left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) - \frac{\frac{1585431.567088306}{z + 1}}{z + 1}}{\frac{\color{blue}{676.5203681218851}}{z} + \left(0.9999999999998099 - \frac{-1259.1392167224028}{z + 1}\right)} + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \]

   7. Applied egg-rr 87.9

      \[\leadsto \left(\sqrt{2} \cdot \left(\color{blue}{{\left({\pi}^{1.5}\right)}^{0.3333333333333333}} \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -6.5 - z\right)}\right)\right) \cdot \left(\left(\left(\frac{\left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) \cdot \left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) - \frac{\frac{1585431.567088306}{z + 1}}{z + 1}}{\frac{676.5203681218851}{z} + \left(0.9999999999998099 - \frac{-1259.1392167224028}{z + 1}\right)} + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \]

   8. Simplified 87.9

      \[\leadsto \left(\sqrt{2} \cdot \left(\color{blue}{\sqrt[3]{{\pi}^{1.5}}} \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -6.5 - z\right)}\right)\right) \cdot \left(\left(\left(\frac{\left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) \cdot \left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) - \frac{\frac{1585431.567088306}{z + 1}}{z + 1}}{\frac{676.5203681218851}{z} + \left(0.9999999999998099 - \frac{-1259.1392167224028}{z + 1}\right)} + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \]
      Proof

      [Start] 87.9

      \[ \left(\sqrt{2} \cdot \left({\left({\pi}^{1.5}\right)}^{0.3333333333333333} \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -6.5 - z\right)}\right)\right) \cdot \left(\left(\left(\frac{\left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) \cdot \left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) - \frac{\frac{1585431.567088306}{z + 1}}{z + 1}}{\frac{676.5203681218851}{z} + \left(0.9999999999998099 - \frac{-1259.1392167224028}{z + 1}\right)} + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \]

      unpow1/3 [=>] 87.9

      \[ \left(\sqrt{2} \cdot \left(\color{blue}{\sqrt[3]{{\pi}^{1.5}}} \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -6.5 - z\right)}\right)\right) \cdot \left(\left(\left(\frac{\left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) \cdot \left(\frac{676.5203681218851}{z} + 0.9999999999998099\right) - \frac{\frac{1585431.567088306}{z + 1}}{z + 1}}{\frac{676.5203681218851}{z} + \left(0.9999999999998099 - \frac{-1259.1392167224028}{z + 1}\right)} + \left(\frac{771.3234287776531}{z - -2} + \frac{-176.6150291621406}{z - -3}\right)\right) + \left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 96.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \left(\frac{\mathsf{fma}\left(z, -1259.1392167224028, \mathsf{fma}\left(676.5203681218851, z, 676.5203681218851\right)\right)}{\mathsf{fma}\left(z, z, z\right)} + \frac{771.3234287776531}{z + 2}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(\sqrt[3]{{\pi}^{1.5}} \cdot e^{\mathsf{fma}\left(-\log \left(z + 6.5\right), 0.5 - z, -6.5 - z\right)}\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \frac{\frac{-1585431.567088306}{z + 1}}{z + 1}}{\frac{676.5203681218851}{z} + \left(0.9999999999998099 - \frac{-1259.1392167224028}{z + 1}\right)} + \left(\frac{-176.6150291621406}{z + 3} + \frac{771.3234287776531}{z + 2}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \frac{12.507343278686905}{z + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	96.4%
Cost	50436.00

\[\begin{array}{l} t_0 := \frac{-176.6150291621406}{z + 3}\\ t_1 := \frac{771.3234287776531}{z + 2}\\ t_2 := \frac{12.507343278686905}{z + 4}\\ t_3 := \frac{-0.13857109526572012}{z + 5}\\ t_4 := \frac{-1259.1392167224028}{z + 1}\\ t_5 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_6 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_7 := 0.9999999999998099 + \frac{676.5203681218851}{z}\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(\left(\left(t_3 + t_5\right) + \left(t_0 + t_2\right)\right) + t_6\right) + \left(0.9999999999998099 + \left(\frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\mathsf{fma}\left(t_4, t_4 + \frac{-676.5203681218851}{z}, \frac{\frac{457679.80848377093}{z}}{z}\right)} + t_1\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(\sqrt[3]{{\pi}^{1.5}} \cdot e^{\mathsf{fma}\left(-\log \left(z + 6.5\right), 0.5 - z, -6.5 - z\right)}\right)\right) \cdot \left(\left(\left(\frac{t_7 \cdot t_7 + \frac{\frac{-1585431.567088306}{z + 1}}{z + 1}}{\frac{676.5203681218851}{z} + \left(0.9999999999998099 - t_4\right)} + \left(t_0 + t_1\right)\right) + \left(t_3 + t_2\right)\right) + \left(t_5 + t_6\right)\right)\\ \end{array} \]

Alternative 2
Error	96.4%
Cost	50372.00

\[\begin{array}{l} t_0 := \frac{-176.6150291621406}{z + 3}\\ t_1 := \frac{771.3234287776531}{z + 2}\\ t_2 := \frac{12.507343278686905}{z + 4}\\ t_3 := \frac{-0.13857109526572012}{z + 5}\\ t_4 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_5 := \frac{-1259.1392167224028}{z + 1}\\ t_6 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_7 := 0.9999999999998099 + \frac{676.5203681218851}{z}\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(\left(\left(t_3 + t_6\right) + \left(t_0 + t_2\right)\right) + t_4\right) + \left(0.9999999999998099 + \left(\frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\mathsf{fma}\left(t_5, t_5 + \frac{-676.5203681218851}{z}, \frac{\frac{457679.80848377093}{z}}{z}\right)} + t_1\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{t_7 \cdot t_7 + \frac{\frac{-1585431.567088306}{z + 1}}{z + 1}}{\frac{676.5203681218851}{z} + \left(0.9999999999998099 - t_5\right)} + \left(t_0 + t_1\right)\right) + \left(t_3 + t_2\right)\right) + \left(t_6 + t_4\right)\right) \cdot \left(\sqrt{2} \cdot \left(e^{\mathsf{fma}\left(-\log \left(z + 6.5\right), 0.5 - z, -6.5 - z\right)} \cdot \sqrt{\pi}\right)\right)\\ \end{array} \]

Alternative 3
Error	96.4%
Cost	43972.00

\[\begin{array}{l} t_0 := \frac{-1259.1392167224028}{z + 1}\\ t_1 := 0.9999999999998099 + \frac{676.5203681218851}{z}\\ t_2 := \frac{12.507343278686905}{z + 4}\\ t_3 := \frac{-0.13857109526572012}{z + 5}\\ t_4 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_5 := \frac{-176.6150291621406}{z + 3}\\ t_6 := \frac{771.3234287776531}{z + 2}\\ t_7 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(t_6 + \left(\frac{676.5203681218851}{z} + t_0\right)\right)\right) + \left(\left(t_3 + t_7\right) + \left(t_5 + \left(t_2 + t_4\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{t_1 \cdot t_1 + \frac{\frac{-1585431.567088306}{z + 1}}{z + 1}}{\frac{676.5203681218851}{z} + \left(0.9999999999998099 - t_0\right)} + \left(t_5 + t_6\right)\right) + \left(t_3 + t_2\right)\right) + \left(t_7 + t_4\right)\right) \cdot \left(\sqrt{2} \cdot \left(e^{\mathsf{fma}\left(-\log \left(z + 6.5\right), 0.5 - z, -6.5 - z\right)} \cdot \sqrt{\pi}\right)\right)\\ \end{array} \]

Alternative 4
Error	96.4%
Cost	42564.00

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

Alternative 5
Error	96.4%
Cost	42564.00

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

Alternative 6
Error	96.4%
Cost	37636.00

\[\begin{array}{l} t_0 := \frac{-1259.1392167224028}{z + 1}\\ t_1 := 0.9999999999998099 + \frac{676.5203681218851}{z}\\ t_2 := \frac{12.507343278686905}{z + 4}\\ t_3 := \frac{-0.13857109526572012}{z + 5}\\ t_4 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_5 := \frac{-176.6150291621406}{z + 3}\\ t_6 := \frac{771.3234287776531}{z + 2}\\ t_7 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(\left(0.9999999999998099 + \left(t_6 + \left(\frac{676.5203681218851}{z} + t_0\right)\right)\right) + \left(\left(t_3 + t_7\right) + \left(t_5 + \left(t_2 + t_4\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{t_1 \cdot t_1 + \frac{\frac{-1585431.567088306}{z + 1}}{z + 1}}{\frac{676.5203681218851}{z} + \left(0.9999999999998099 - t_0\right)} + \left(t_5 + t_6\right)\right) + \left(t_3 + t_2\right)\right) + \left(t_7 + t_4\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot e^{\left(-6.5 - z\right) + \left(z + -0.5\right) \cdot \log \left(z + 6.5\right)}\right)\right)\\ \end{array} \]

Alternative 7
Error	96.4%
Cost	36164.00

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

Alternative 8
Error	96.4%
Cost	29828.00

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

Alternative 9
Error	96.4%
Cost	29700.00

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

Alternative 10
Error	94.0%
Cost	29504.00

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

Alternative 11
Error	94.1%
Cost	29504.00

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

Alternative 12
Error	94.1%
Cost	29504.00

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

Alternative 13
Error	94.1%
Cost	29504.00

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

Alternative 14
Error	26.9%
Cost	28736.00

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

Alternative 15
Error	25.7%
Cost	27200.00

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

Alternative 16
Error	20.5%
Cost	26880.00

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

Alternative 17
Error	19.7%
Cost	26816.00

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

Alternative 18
Error	19.8%
Cost	26816.00

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

Alternative 19
Error	19.4%
Cost	26756.00

\[\begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z \leq 4:\\ \;\;\;\;t_0 \cdot \left(676.5203681218851 \cdot \frac{e^{-6.5} \cdot \sqrt{0.15384615384615385}}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(0.9999999999998099 \cdot e^{\left(-6.5 - z\right) + \left(z + -0.5\right) \cdot \log \left(z + 6.5\right)}\right)\\ \end{array} \]

Alternative 20
Error	18.8%
Cost	26692.00

\[\begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z \leq 4:\\ \;\;\;\;t_0 \cdot \left(676.5203681218851 \cdot \frac{e^{-6.5} \cdot \sqrt{0.15384615384615385}}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(0.9999999999998099 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right)\right)\\ \end{array} \]

Alternative 21
Error	13.1%
Cost	19584.00

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

Reproduce

herbie shell --seed 2023093 
(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)))))