Jmat.Real.gamma, branch z greater than 0.5

Average Error: 3.9 → 2.0

Time: 30.8s

Precision: binary64

Cost: 113348

\[z > 0.5\]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{12.507343278686905}{z + 4}\\ t_1 := \frac{-176.6150291621406}{z + 3}\\ t_2 := \sqrt{\pi \cdot 2}\\ t_3 := \frac{-0.13857109526572012}{z + 5}\\ t_4 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_5 := \frac{771.3234287776531}{z + 2}\\ t_6 := t_5 + \frac{\mathsf{fma}\left(z, -582.6188486005177, 676.5203681218851\right)}{\mathsf{fma}\left(z, z, z\right)}\\ t_7 := 0.9999999999998099 + t_6\\ t_8 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_9 := 0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(t_1 + \left(t_5 + t_0\right)\right) + \left(t_3 + \left(t_8 + t_4\right)\right)\right)\right)\right)\\ t_10 := t_3 + \left(t_8 + \left(t_0 + \left(t_1 + t_4\right)\right)\right)\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;t_2 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \frac{{t_7}^{3} + {t_10}^{3}}{\mathsf{fma}\left(t_10, \left(t_3 + t_8\right) + \left(\left(t_0 + t_1\right) + \left(\left(t_4 + -0.9999999999998099\right) - t_6\right)\right), t_7 \cdot t_7\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(\sqrt[3]{t_9 \cdot \left(t_9 \cdot t_9\right)} \cdot e^{\mathsf{fma}\left(-\log \left(z + 6.5\right), 0.5 - z, -6.5 - z\right)}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (z)
 :precision binary64
 (let* ((t_0 (/ 12.507343278686905 (+ z 4.0)))
        (t_1 (/ -176.6150291621406 (+ z 3.0)))
        (t_2 (sqrt (* PI 2.0)))
        (t_3 (/ -0.13857109526572012 (+ z 5.0)))
        (t_4 (/ 1.5056327351493116e-7 (+ z 7.0)))
        (t_5 (/ 771.3234287776531 (+ z 2.0)))
        (t_6
         (+ t_5 (/ (fma z -582.6188486005177 676.5203681218851) (fma z z z))))
        (t_7 (+ 0.9999999999998099 t_6))
        (t_8 (/ 9.984369578019572e-6 (+ z 6.0)))
        (t_9
         (+
          0.9999999999998099
          (+
           (/ 676.5203681218851 z)
           (+
            (/ -1259.1392167224028 (+ z 1.0))
            (+ (+ t_1 (+ t_5 t_0)) (+ t_3 (+ t_8 t_4)))))))
        (t_10 (+ t_3 (+ t_8 (+ t_0 (+ t_1 t_4))))))
   (if (<= (+ z -1.0) 140.0)
     (*
      t_2
      (*
       (pow (+ z 6.5) (+ z -0.5))
       (*
        (/ (exp -6.5) (exp z))
        (/
         (+ (pow t_7 3.0) (pow t_10 3.0))
         (fma
          t_10
          (+ (+ t_3 t_8) (+ (+ t_0 t_1) (- (+ t_4 -0.9999999999998099) t_6)))
          (* t_7 t_7))))))
     (*
      t_2
      (*
       (cbrt (* t_9 (* t_9 t_9)))
       (exp (fma (- (log (+ z 6.5))) (- 0.5 z) (- -6.5 z))))))))

C

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

↓

double code(double z) {
	double t_0 = 12.507343278686905 / (z + 4.0);
	double t_1 = -176.6150291621406 / (z + 3.0);
	double t_2 = sqrt((((double) M_PI) * 2.0));
	double t_3 = -0.13857109526572012 / (z + 5.0);
	double t_4 = 1.5056327351493116e-7 / (z + 7.0);
	double t_5 = 771.3234287776531 / (z + 2.0);
	double t_6 = t_5 + (fma(z, -582.6188486005177, 676.5203681218851) / fma(z, z, z));
	double t_7 = 0.9999999999998099 + t_6;
	double t_8 = 9.984369578019572e-6 / (z + 6.0);
	double t_9 = 0.9999999999998099 + ((676.5203681218851 / z) + ((-1259.1392167224028 / (z + 1.0)) + ((t_1 + (t_5 + t_0)) + (t_3 + (t_8 + t_4)))));
	double t_10 = t_3 + (t_8 + (t_0 + (t_1 + t_4)));
	double tmp;
	if ((z + -1.0) <= 140.0) {
		tmp = t_2 * (pow((z + 6.5), (z + -0.5)) * ((exp(-6.5) / exp(z)) * ((pow(t_7, 3.0) + pow(t_10, 3.0)) / fma(t_10, ((t_3 + t_8) + ((t_0 + t_1) + ((t_4 + -0.9999999999998099) - t_6))), (t_7 * t_7)))));
	} else {
		tmp = t_2 * (cbrt((t_9 * (t_9 * t_9))) * exp(fma(-log((z + 6.5)), (0.5 - z), (-6.5 - z))));
	}
	return tmp;
}

Julia

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

↓

function code(z)
	t_0 = Float64(12.507343278686905 / Float64(z + 4.0))
	t_1 = Float64(-176.6150291621406 / Float64(z + 3.0))
	t_2 = sqrt(Float64(pi * 2.0))
	t_3 = Float64(-0.13857109526572012 / Float64(z + 5.0))
	t_4 = Float64(1.5056327351493116e-7 / Float64(z + 7.0))
	t_5 = Float64(771.3234287776531 / Float64(z + 2.0))
	t_6 = Float64(t_5 + Float64(fma(z, -582.6188486005177, 676.5203681218851) / fma(z, z, z)))
	t_7 = Float64(0.9999999999998099 + t_6)
	t_8 = Float64(9.984369578019572e-6 / Float64(z + 6.0))
	t_9 = Float64(0.9999999999998099 + Float64(Float64(676.5203681218851 / z) + Float64(Float64(-1259.1392167224028 / Float64(z + 1.0)) + Float64(Float64(t_1 + Float64(t_5 + t_0)) + Float64(t_3 + Float64(t_8 + t_4))))))
	t_10 = Float64(t_3 + Float64(t_8 + Float64(t_0 + Float64(t_1 + t_4))))
	tmp = 0.0
	if (Float64(z + -1.0) <= 140.0)
		tmp = Float64(t_2 * Float64((Float64(z + 6.5) ^ Float64(z + -0.5)) * Float64(Float64(exp(-6.5) / exp(z)) * Float64(Float64((t_7 ^ 3.0) + (t_10 ^ 3.0)) / fma(t_10, Float64(Float64(t_3 + t_8) + Float64(Float64(t_0 + t_1) + Float64(Float64(t_4 + -0.9999999999998099) - t_6))), Float64(t_7 * t_7))))));
	else
		tmp = Float64(t_2 * Float64(cbrt(Float64(t_9 * Float64(t_9 * t_9))) * exp(fma(Float64(-log(Float64(z + 6.5))), Float64(0.5 - z), Float64(-6.5 - z)))));
	end
	return tmp
end

Wolfram

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

↓

code[z_] := Block[{t$95$0 = N[(12.507343278686905 / N[(z + 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-176.6150291621406 / N[(z + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(-0.13857109526572012 / N[(z + 5.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(1.5056327351493116e-7 / N[(z + 7.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(771.3234287776531 / N[(z + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 + N[(N[(z * -582.6188486005177 + 676.5203681218851), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(0.9999999999998099 + t$95$6), $MachinePrecision]}, Block[{t$95$8 = N[(9.984369578019572e-6 / N[(z + 6.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[(0.9999999999998099 + N[(N[(676.5203681218851 / z), $MachinePrecision] + N[(N[(-1259.1392167224028 / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 + N[(t$95$5 + t$95$0), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 + N[(t$95$8 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$10 = N[(t$95$3 + N[(t$95$8 + N[(t$95$0 + N[(t$95$1 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z + -1.0), $MachinePrecision], 140.0], N[(t$95$2 * 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[(N[Power[t$95$7, 3.0], $MachinePrecision] + N[Power[t$95$10, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$10 * N[(N[(t$95$3 + t$95$8), $MachinePrecision] + N[(N[(t$95$0 + t$95$1), $MachinePrecision] + N[(N[(t$95$4 + -0.9999999999998099), $MachinePrecision] - t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$7 * t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(N[Power[N[(t$95$9 * N[(t$95$9 * t$95$9), $MachinePrecision]), $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]]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 2.2

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

   2. Simplified 2.1

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

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

   3. Applied egg-rr 2.2

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

   4. Simplified 2.0

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

      [Start] 2.2	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{676.5203681218851 \cdot \left(z + 1\right) + z \cdot -1259.1392167224028}{z \cdot \left(z + 1\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      +-commutative [=>] 2.2	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\color{blue}{z \cdot -1259.1392167224028 + 676.5203681218851 \cdot \left(z + 1\right)}}{z \cdot \left(z + 1\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      fma-def [=>] 2.0	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\color{blue}{\mathsf{fma}\left(z, -1259.1392167224028, 676.5203681218851 \cdot \left(z + 1\right)\right)}}{z \cdot \left(z + 1\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      distribute-lft-in [=>] 2.0	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\mathsf{fma}\left(z, -1259.1392167224028, \color{blue}{676.5203681218851 \cdot z + 676.5203681218851 \cdot 1}\right)}{z \cdot \left(z + 1\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      metadata-eval [=>] 2.0	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\mathsf{fma}\left(z, -1259.1392167224028, 676.5203681218851 \cdot z + \color{blue}{676.5203681218851}\right)}{z \cdot \left(z + 1\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      fma-def [=>] 2.0	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\mathsf{fma}\left(z, -1259.1392167224028, \color{blue}{\mathsf{fma}\left(676.5203681218851, z, 676.5203681218851\right)}\right)}{z \cdot \left(z + 1\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      distribute-rgt-in [=>] 2.1	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\mathsf{fma}\left(z, -1259.1392167224028, \mathsf{fma}\left(676.5203681218851, z, 676.5203681218851\right)\right)}{\color{blue}{z \cdot z + 1 \cdot z}}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      *-lft-identity [=>] 2.1	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\mathsf{fma}\left(z, -1259.1392167224028, \mathsf{fma}\left(676.5203681218851, z, 676.5203681218851\right)\right)}{z \cdot z + \color{blue}{z}}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      fma-def [=>] 2.0	\[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\mathsf{fma}\left(z, -1259.1392167224028, \mathsf{fma}\left(676.5203681218851, z, 676.5203681218851\right)\right)}{\color{blue}{\mathsf{fma}\left(z, z, z\right)}}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

   5. Taylor expanded in z around 0 2.0

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\color{blue}{676.5203681218851 + -582.6188486005177 \cdot z}}{\mathsf{fma}\left(z, z, z\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

   6. Simplified 2.0

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{\color{blue}{676.5203681218851 + z \cdot -582.6188486005177}}{\mathsf{fma}\left(z, z, z\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]
      Proof

      [Start] 2.0

      \[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{676.5203681218851 + -582.6188486005177 \cdot z}{\mathsf{fma}\left(z, z, z\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

      *-commutative [=>] 2.0

      \[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{676.5203681218851 + \color{blue}{z \cdot -582.6188486005177}}{\mathsf{fma}\left(z, z, z\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

   7. Applied egg-rr 1.9

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\color{blue}{\frac{e^{-6.5}}{e^{z}}} \cdot \left(\left(0.9999999999998099 + \left(\frac{771.3234287776531}{2 + z} + \frac{676.5203681218851 + z \cdot -582.6188486005177}{\mathsf{fma}\left(z, z, z\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

   8. Applied egg-rr 1.9

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \color{blue}{\frac{{\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \frac{676.5203681218851 + z \cdot -582.6188486005177}{\mathsf{fma}\left(z, z, z\right)}\right)\right)}^{3} + {\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)}^{3}}{\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \frac{676.5203681218851 + z \cdot -582.6188486005177}{\mathsf{fma}\left(z, z, z\right)}\right)\right) \cdot \left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \frac{676.5203681218851 + z \cdot -582.6188486005177}{\mathsf{fma}\left(z, z, z\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) - \left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \frac{676.5203681218851 + z \cdot -582.6188486005177}{\mathsf{fma}\left(z, z, z\right)}\right)\right) \cdot \left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)}}\right)\right) \]

   9. Simplified 1.9

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \color{blue}{\frac{{\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \frac{\mathsf{fma}\left(z, -582.6188486005177, 676.5203681218851\right)}{\mathsf{fma}\left(z, z, z\right)}\right)\right)}^{3} + {\left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)}^{3}}{\mathsf{fma}\left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right), \left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{z + 7} - 0.9999999999998099\right) - \left(\frac{771.3234287776531}{z + 2} + \frac{\mathsf{fma}\left(z, -582.6188486005177, 676.5203681218851\right)}{\mathsf{fma}\left(z, z, z\right)}\right)\right)\right), \left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \frac{\mathsf{fma}\left(z, -582.6188486005177, 676.5203681218851\right)}{\mathsf{fma}\left(z, z, z\right)}\right)\right) \cdot \left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \frac{\mathsf{fma}\left(z, -582.6188486005177, 676.5203681218851\right)}{\mathsf{fma}\left(z, z, z\right)}\right)\right)\right)}}\right)\right) \]
      Proof

      [Start] 1.9

      \[ \sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \frac{{\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \frac{676.5203681218851 + z \cdot -582.6188486005177}{\mathsf{fma}\left(z, z, z\right)}\right)\right)}^{3} + {\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)}^{3}}{\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \frac{676.5203681218851 + z \cdot -582.6188486005177}{\mathsf{fma}\left(z, z, z\right)}\right)\right) \cdot \left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \frac{676.5203681218851 + z \cdot -582.6188486005177}{\mathsf{fma}\left(z, z, z\right)}\right)\right) + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) - \left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \frac{676.5203681218851 + z \cdot -582.6188486005177}{\mathsf{fma}\left(z, z, z\right)}\right)\right) \cdot \left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)}\right)\right) \]

   if 140 < (-.f64 z 1)

   1. Initial program 60.5

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

   2. Simplified 60.5

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

      [Start] 60.5	\[ \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      associate-*l* [=>] 60.5	\[ \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right)\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      associate-*l* [=>] 60.5	\[ \color{blue}{\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\right)} \]

   3. Taylor expanded in z around -inf 60.8

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

   4. Simplified 7.6

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

      [Start] 60.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \frac{e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right)}}{e^{6.5 - -1 \cdot z}}\right) \]
      div-exp [=>] 7.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \color{blue}{e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right) - \left(6.5 - -1 \cdot z\right)}}\right) \]
      associate-*r* [=>] 7.8	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\color{blue}{\left(-1 \cdot \log \left(6.5 - -1 \cdot z\right)\right) \cdot \left(-1 \cdot z + 0.5\right)} - \left(6.5 - -1 \cdot z\right)}\right) \]
      fma-neg [=>] 7.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\color{blue}{\mathsf{fma}\left(-1 \cdot \log \left(6.5 - -1 \cdot z\right), -1 \cdot z + 0.5, -\left(6.5 - -1 \cdot z\right)\right)}}\right) \]
      mul-1-neg [=>] 7.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(\color{blue}{-\log \left(6.5 - -1 \cdot z\right)}, -1 \cdot z + 0.5, -\left(6.5 - -1 \cdot z\right)\right)}\right) \]
      sub-neg [=>] 7.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \color{blue}{\left(6.5 + \left(--1 \cdot z\right)\right)}, -1 \cdot z + 0.5, -\left(6.5 - -1 \cdot z\right)\right)}\right) \]
      mul-1-neg [=>] 7.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + \left(-\color{blue}{\left(-z\right)}\right)\right), -1 \cdot z + 0.5, -\left(6.5 - -1 \cdot z\right)\right)}\right) \]
      remove-double-neg [=>] 7.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + \color{blue}{z}\right), -1 \cdot z + 0.5, -\left(6.5 - -1 \cdot z\right)\right)}\right) \]
      +-commutative [=>] 7.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), \color{blue}{0.5 + -1 \cdot z}, -\left(6.5 - -1 \cdot z\right)\right)}\right) \]
      mul-1-neg [=>] 7.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 + \color{blue}{\left(-z\right)}, -\left(6.5 - -1 \cdot z\right)\right)}\right) \]
      unsub-neg [=>] 7.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), \color{blue}{0.5 - z}, -\left(6.5 - -1 \cdot z\right)\right)}\right) \]
      sub-neg [=>] 7.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -\color{blue}{\left(6.5 + \left(--1 \cdot z\right)\right)}\right)}\right) \]
      +-commutative [=>] 7.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -\color{blue}{\left(\left(--1 \cdot z\right) + 6.5\right)}\right)}\right) \]
      mul-1-neg [=>] 7.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -\left(\left(-\color{blue}{\left(-z\right)}\right) + 6.5\right)\right)}\right) \]
      remove-double-neg [=>] 7.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -\left(\color{blue}{z} + 6.5\right)\right)}\right) \]
      neg-sub0 [=>] 7.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, \color{blue}{0 - \left(z + 6.5\right)}\right)}\right) \]
      +-commutative [=>] 7.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, 0 - \color{blue}{\left(6.5 + z\right)}\right)}\right) \]
      associate--r+ [=>] 7.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, \color{blue}{\left(0 - 6.5\right) - z}\right)}\right) \]
      metadata-eval [=>] 7.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, \color{blue}{-6.5} - z\right)}\right) \]

   5. Applied egg-rr 7.6

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

   6. Simplified 7.6

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

      [Start] 7.6

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

3. Recombined 2 regimes into one program.

4. Final simplification 2.0

   \[\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 \frac{{\left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \frac{\mathsf{fma}\left(z, -582.6188486005177, 676.5203681218851\right)}{\mathsf{fma}\left(z, z, z\right)}\right)\right)}^{3} + {\left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)}^{3}}{\mathsf{fma}\left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-176.6150291621406}{z + 3} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right), \left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\left(\frac{12.507343278686905}{z + 4} + \frac{-176.6150291621406}{z + 3}\right) + \left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{z + 7} + -0.9999999999998099\right) - \left(\frac{771.3234287776531}{z + 2} + \frac{\mathsf{fma}\left(z, -582.6188486005177, 676.5203681218851\right)}{\mathsf{fma}\left(z, z, z\right)}\right)\right)\right), \left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \frac{\mathsf{fma}\left(z, -582.6188486005177, 676.5203681218851\right)}{\mathsf{fma}\left(z, z, z\right)}\right)\right) \cdot \left(0.9999999999998099 + \left(\frac{771.3234287776531}{z + 2} + \frac{\mathsf{fma}\left(z, -582.6188486005177, 676.5203681218851\right)}{\mathsf{fma}\left(z, z, z\right)}\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(\sqrt[3]{\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{z + 2} + \frac{12.507343278686905}{z + 4}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{z + 2} + \frac{12.507343278686905}{z + 4}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right) \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{z + 2} + \frac{12.507343278686905}{z + 4}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right)\right)} \cdot e^{\mathsf{fma}\left(-\log \left(z + 6.5\right), 0.5 - z, -6.5 - z\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	2.0
Cost	48964

\[\begin{array}{l} t_0 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_1 := \frac{12.507343278686905}{z + 4}\\ t_2 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_3 := \frac{-176.6150291621406}{z + 3}\\ t_4 := \frac{771.3234287776531}{z + 2}\\ t_5 := \frac{-0.13857109526572012}{z + 5}\\ t_6 := 0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(t_3 + \left(t_4 + t_1\right)\right) + \left(t_5 + \left(t_0 + t_2\right)\right)\right)\right)\right)\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \left(t_4 + \frac{676.5203681218851 + z \cdot -582.6188486005177}{\mathsf{fma}\left(z, z, z\right)}\right)\right) + \left(t_2 + \left(\left(t_5 + t_0\right) + \left(t_1 + t_3\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(\sqrt[3]{t_6 \cdot \left(t_6 \cdot t_6\right)} \cdot e^{\mathsf{fma}\left(-\log \left(z + 6.5\right), 0.5 - z, -6.5 - z\right)}\right)\\ \end{array} \]

Alternative 2
Error	2.0
Cost	48708

\[\begin{array}{l} t_0 := \frac{12.507343278686905}{z + 4}\\ t_1 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_2 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \frac{-176.6150291621406}{z + 3}\\ t_5 := \frac{771.3234287776531}{z + 2}\\ t_6 := \frac{-0.13857109526572012}{z + 5}\\ t_7 := 0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(t_4 + \left(t_5 + t_0\right)\right) + \left(t_6 + \left(t_2 + t_1\right)\right)\right)\right)\right)\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;t_3 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \left(t_5 + \frac{676.5203681218851 + z \cdot -582.6188486005177}{\mathsf{fma}\left(z, z, z\right)}\right)\right) + \left(t_1 + \left(\left(t_6 + t_2\right) + \left(t_0 + t_4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(\sqrt[3]{t_7 \cdot \left(t_7 \cdot t_7\right)} \cdot e^{\mathsf{fma}\left(-\log \left(z + 6.5\right), 0.5 - z, -6.5 - z\right)}\right)\\ \end{array} \]

Alternative 3
Error	2.0
Cost	42564

\[\begin{array}{l} t_0 := \frac{12.507343278686905}{z + 4}\\ t_1 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_2 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \frac{-176.6150291621406}{z + 3}\\ t_5 := \frac{771.3234287776531}{z + 2}\\ t_6 := \frac{-0.13857109526572012}{z + 5}\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;t_3 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(\frac{e^{-6.5}}{e^{z}} \cdot \left(\left(0.9999999999998099 + \left(t_5 + \frac{676.5203681218851 + z \cdot -582.6188486005177}{\mathsf{fma}\left(z, z, z\right)}\right)\right) + \left(t_1 + \left(\left(t_6 + t_2\right) + \left(t_0 + t_4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(t_4 + \left(t_5 + t_0\right)\right) + \left(t_6 + \left(t_2 + t_1\right)\right)\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(z + 6.5\right), 0.5 - z, -6.5 - z\right)}\right)\\ \end{array} \]

Alternative 4
Error	2.1
Cost	36164

\[\begin{array}{l} t_0 := \frac{12.507343278686905}{z + 4}\\ t_1 := \frac{-176.6150291621406}{z + 3}\\ t_2 := \frac{-0.13857109526572012}{z + 5}\\ t_3 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_4 := \frac{771.3234287776531}{z + 2}\\ t_5 := \sqrt{\pi \cdot 2}\\ t_6 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;t_5 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(t_6 + \left(\left(t_2 + t_3\right) + \left(t_0 + t_1\right)\right)\right) + \left(0.9999999999998099 + \left(t_4 + \frac{676.5203681218851 + z \cdot -582.6188486005177}{z \cdot \left(z + 1\right)}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_5 \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(t_1 + \left(t_4 + t_0\right)\right) + \left(t_2 + \left(t_3 + t_6\right)\right)\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(z + 6.5\right), 0.5 - z, -6.5 - z\right)}\right)\\ \end{array} \]

Alternative 5
Error	2.1
Cost	29892

\[\begin{array}{l} t_0 := \frac{12.507343278686905}{z + 4}\\ t_1 := \frac{-176.6150291621406}{z + 3}\\ t_2 := \frac{-0.13857109526572012}{z + 5}\\ t_3 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_4 := \frac{771.3234287776531}{z + 2}\\ t_5 := \sqrt{\pi \cdot 2}\\ t_6 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;t_5 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(t_6 + \left(\left(t_2 + t_3\right) + \left(t_0 + t_1\right)\right)\right) + \left(0.9999999999998099 + \left(t_4 + \frac{676.5203681218851 + z \cdot -582.6188486005177}{z \cdot \left(z + 1\right)}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_5 \cdot \left(e^{\left(-6.5 - z\right) + \left(z + -0.5\right) \cdot \log \left(z + 6.5\right)} \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(t_4 + t_1\right)\right)\right)\right) + \left(\left(t_3 + t_6\right) + \left(t_2 + t_0\right)\right)\right)\right)\\ \end{array} \]

Alternative 6
Error	2.3
Cost	29700

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

Alternative 7
Error	3.9
Cost	29504

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

Alternative 8
Error	3.8
Cost	29504

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

Alternative 9
Error	3.8
Cost	29504

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

Alternative 10
Error	3.8
Cost	29504

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

Alternative 11
Error	3.8
Cost	29504

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

Alternative 12
Error	46.8
Cost	28736

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

Alternative 13
Error	47.6
Cost	27200

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

Alternative 14
Error	47.6
Cost	27200

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

Alternative 15
Error	49.9
Cost	27012

\[\begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z \leq 2.7:\\ \;\;\;\;\frac{\left(e^{-6.5} \cdot \left(t_0 \cdot 676.5203681218851\right)\right) \cdot \sqrt{0.15384615384615385}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\left(0.9999999999998099 + \frac{24.458333333348836}{z}\right) \cdot e^{-6.5 + \left(\left(z + -0.5\right) \cdot \log \left(z + 6.5\right) - z\right)}\right)\\ \end{array} \]

Alternative 16
Error	50.4
Cost	26948

\[\begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z \leq 2.7:\\ \;\;\;\;\frac{\left(e^{-6.5} \cdot \left(t_0 \cdot 676.5203681218851\right)\right) \cdot \sqrt{0.15384615384615385}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\left(0.9999999999998099 + \frac{24.458333333348836}{z}\right) \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}}\right)\\ \end{array} \]

Alternative 17
Error	51.6
Cost	26756

\[\begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z \leq 3.95:\\ \;\;\;\;\frac{\left(e^{-6.5} \cdot \left(t_0 \cdot 676.5203681218851\right)\right) \cdot \sqrt{0.15384615384615385}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(0.9999999999998099 \cdot e^{-6.5 + \left(\left(z + -0.5\right) \cdot \log \left(z + 6.5\right) - z\right)}\right)\\ \end{array} \]

Alternative 18
Error	52.0
Cost	26692

\[\begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z \leq 3.95:\\ \;\;\;\;\frac{\left(e^{-6.5} \cdot \left(t_0 \cdot 676.5203681218851\right)\right) \cdot \sqrt{0.15384615384615385}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(0.9999999999998099 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right)\right)\\ \end{array} \]

Alternative 19
Error	55.7
Cost	26240

\[\frac{\left(e^{-6.5} \cdot \left(\sqrt{\pi \cdot 2} \cdot 676.5203681218851\right)\right) \cdot \sqrt{0.15384615384615385}}{z} \]

Alternative 20
Error	55.7
Cost	19776

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

Alternative 21
Error	55.7
Cost	19712

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

Reproduce

herbie shell --seed 2023083 
(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)))))