Jmat.Real.gamma, branch z greater than 0.5

Average Error: 3.9 → 2.3

Time: 34.3s

Precision: binary64

Cost: 81540

\[z > 0.5\]

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{-0.13857109526572012}{z + 5}\\ t_1 := \frac{-176.6150291621406}{z + 3}\\ t_2 := \frac{771.3234287776531}{2 + z}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_5 := \left(z + -1\right) + 7\\ t_6 := \frac{12.507343278686905}{z + 4}\\ t_7 := \frac{-1259.1392167224028}{z + 1}\\ t_8 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ \mathbf{if}\;\left(\left(t_3 \cdot {\left(t_5 + 0.5\right)}^{\left(\left(z + -1\right) + 0.5\right)}\right) \cdot e^{-0.5 + \left(-7 + \left(1 - z\right)\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 + \left(z + -1\right)}\right) + \frac{-1259.1392167224028}{2 + \left(z + -1\right)}\right) + \frac{771.3234287776531}{\left(z + -1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z + -1\right) + 4}\right) + \frac{12.507343278686905}{\left(z + -1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z + -1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t_5}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8}\right) \leq 5 \cdot 10^{+243}:\\ \;\;\;\;t_3 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(0.9999999999998099 + \left(\left(t_2 + \frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\frac{457679.80848377093}{z \cdot z} + \left({t_7}^{2} + \frac{851833.326413742}{z \cdot \left(z + 1\right)}\right)}\right) + \left(\left(t_0 + t_4\right) + \left(\left(t_1 + t_6\right) + t_8\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(t_7 + \left(t_1 + \left(t_2 + t_6\right)\right)\right) + \left(t_0 + \left(t_4 + t_8\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(z + -0.5, \mathsf{log1p}\left(z + 5.5\right), -6.5\right) - \mathsf{expm1}\left(\mathsf{log1p}\left(z\right)\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 (/ -0.13857109526572012 (+ z 5.0)))
        (t_1 (/ -176.6150291621406 (+ z 3.0)))
        (t_2 (/ 771.3234287776531 (+ 2.0 z)))
        (t_3 (sqrt (* PI 2.0)))
        (t_4 (/ 9.984369578019572e-6 (+ z 6.0)))
        (t_5 (+ (+ z -1.0) 7.0))
        (t_6 (/ 12.507343278686905 (+ z 4.0)))
        (t_7 (/ -1259.1392167224028 (+ z 1.0)))
        (t_8 (/ 1.5056327351493116e-7 (+ z 7.0))))
   (if (<=
        (*
         (*
          (* t_3 (pow (+ t_5 0.5) (+ (+ z -1.0) 0.5)))
          (exp (+ -0.5 (+ -7.0 (- 1.0 z)))))
         (+
          (+
           (+
            (+
             (+
              (+
               (+
                (+ 0.9999999999998099 (/ 676.5203681218851 (+ 1.0 (+ z -1.0))))
                (/ -1259.1392167224028 (+ 2.0 (+ z -1.0))))
               (/ 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 t_5))
          (/ 1.5056327351493116e-7 (+ (+ z -1.0) 8.0))))
        5e+243)
     (*
      t_3
      (*
       (pow (+ z 6.5) (+ z -0.5))
       (*
        (exp (- -6.5 z))
        (+
         0.9999999999998099
         (+
          (+
           t_2
           (/
            (+
             (/ 309629712.5173946 (pow z 3.0))
             (/ -1996279061.5505414 (pow (+ z 1.0) 3.0)))
            (+
             (/ 457679.80848377093 (* z z))
             (+ (pow t_7 2.0) (/ 851833.326413742 (* z (+ z 1.0)))))))
          (+ (+ t_0 t_4) (+ (+ t_1 t_6) t_8)))))))
     (*
      t_3
      (*
       (+
        (+ 0.9999999999998099 (/ 676.5203681218851 z))
        (+ (+ t_7 (+ t_1 (+ t_2 t_6))) (+ t_0 (+ t_4 t_8))))
       (exp (- (fma (+ z -0.5) (log1p (+ z 5.5)) -6.5) (expm1 (log1p 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 = -0.13857109526572012 / (z + 5.0);
	double t_1 = -176.6150291621406 / (z + 3.0);
	double t_2 = 771.3234287776531 / (2.0 + z);
	double t_3 = sqrt((((double) M_PI) * 2.0));
	double t_4 = 9.984369578019572e-6 / (z + 6.0);
	double t_5 = (z + -1.0) + 7.0;
	double t_6 = 12.507343278686905 / (z + 4.0);
	double t_7 = -1259.1392167224028 / (z + 1.0);
	double t_8 = 1.5056327351493116e-7 / (z + 7.0);
	double tmp;
	if ((((t_3 * pow((t_5 + 0.5), ((z + -1.0) + 0.5))) * exp((-0.5 + (-7.0 + (1.0 - z))))) * ((((((((0.9999999999998099 + (676.5203681218851 / (1.0 + (z + -1.0)))) + (-1259.1392167224028 / (2.0 + (z + -1.0)))) + (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 / t_5)) + (1.5056327351493116e-7 / ((z + -1.0) + 8.0)))) <= 5e+243) {
		tmp = t_3 * (pow((z + 6.5), (z + -0.5)) * (exp((-6.5 - z)) * (0.9999999999998099 + ((t_2 + (((309629712.5173946 / pow(z, 3.0)) + (-1996279061.5505414 / pow((z + 1.0), 3.0))) / ((457679.80848377093 / (z * z)) + (pow(t_7, 2.0) + (851833.326413742 / (z * (z + 1.0))))))) + ((t_0 + t_4) + ((t_1 + t_6) + t_8))))));
	} else {
		tmp = t_3 * (((0.9999999999998099 + (676.5203681218851 / z)) + ((t_7 + (t_1 + (t_2 + t_6))) + (t_0 + (t_4 + t_8)))) * exp((fma((z + -0.5), log1p((z + 5.5)), -6.5) - expm1(log1p(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(-0.13857109526572012 / Float64(z + 5.0))
	t_1 = Float64(-176.6150291621406 / Float64(z + 3.0))
	t_2 = Float64(771.3234287776531 / Float64(2.0 + z))
	t_3 = sqrt(Float64(pi * 2.0))
	t_4 = Float64(9.984369578019572e-6 / Float64(z + 6.0))
	t_5 = Float64(Float64(z + -1.0) + 7.0)
	t_6 = Float64(12.507343278686905 / Float64(z + 4.0))
	t_7 = Float64(-1259.1392167224028 / Float64(z + 1.0))
	t_8 = Float64(1.5056327351493116e-7 / Float64(z + 7.0))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * (Float64(t_5 + 0.5) ^ Float64(Float64(z + -1.0) + 0.5))) * exp(Float64(-0.5 + Float64(-7.0 + Float64(1.0 - z))))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 + Float64(z + -1.0)))) + Float64(-1259.1392167224028 / Float64(2.0 + Float64(z + -1.0)))) + 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 / t_5)) + Float64(1.5056327351493116e-7 / Float64(Float64(z + -1.0) + 8.0)))) <= 5e+243)
		tmp = Float64(t_3 * Float64((Float64(z + 6.5) ^ Float64(z + -0.5)) * Float64(exp(Float64(-6.5 - z)) * Float64(0.9999999999998099 + Float64(Float64(t_2 + Float64(Float64(Float64(309629712.5173946 / (z ^ 3.0)) + Float64(-1996279061.5505414 / (Float64(z + 1.0) ^ 3.0))) / Float64(Float64(457679.80848377093 / Float64(z * z)) + Float64((t_7 ^ 2.0) + Float64(851833.326413742 / Float64(z * Float64(z + 1.0))))))) + Float64(Float64(t_0 + t_4) + Float64(Float64(t_1 + t_6) + t_8)))))));
	else
		tmp = Float64(t_3 * Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / z)) + Float64(Float64(t_7 + Float64(t_1 + Float64(t_2 + t_6))) + Float64(t_0 + Float64(t_4 + t_8)))) * exp(Float64(fma(Float64(z + -0.5), log1p(Float64(z + 5.5)), -6.5) - expm1(log1p(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[(-0.13857109526572012 / N[(z + 5.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-176.6150291621406 / N[(z + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(771.3234287776531 / N[(2.0 + z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(9.984369578019572e-6 / N[(z + 6.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(z + -1.0), $MachinePrecision] + 7.0), $MachinePrecision]}, Block[{t$95$6 = N[(12.507343278686905 / N[(z + 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(-1259.1392167224028 / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(1.5056327351493116e-7 / N[(z + 7.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[Power[N[(t$95$5 + 0.5), $MachinePrecision], N[(N[(z + -1.0), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(-0.5 + N[(-7.0 + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 + N[(z + -1.0), $MachinePrecision]), $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 / t$95$5), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(z + -1.0), $MachinePrecision] + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+243], N[(t$95$3 * N[(N[Power[N[(z + 6.5), $MachinePrecision], N[(z + -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(-6.5 - z), $MachinePrecision]], $MachinePrecision] * N[(0.9999999999998099 + N[(N[(t$95$2 + N[(N[(N[(309629712.5173946 / N[Power[z, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-1996279061.5505414 / N[Power[N[(z + 1.0), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(457679.80848377093 / N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(N[Power[t$95$7, 2.0], $MachinePrecision] + N[(851833.326413742 / N[(z * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 + t$95$4), $MachinePrecision] + N[(N[(t$95$1 + t$95$6), $MachinePrecision] + t$95$8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / z), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$7 + N[(t$95$1 + N[(t$95$2 + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 + N[(t$95$4 + t$95$8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(N[(z + -0.5), $MachinePrecision] * N[Log[1 + N[(z + 5.5), $MachinePrecision]], $MachinePrecision] + -6.5), $MachinePrecision] - N[(Exp[N[Log[1 + z], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 (sqrt.f64 (*.f64 (PI.f64) 2)) (pow.f64 (+.f64 (+.f64 (-.f64 z 1) 7) 1/2) (+.f64 (-.f64 z 1) 1/2))) (exp.f64 (neg.f64 (+.f64 (+.f64 (-.f64 z 1) 7) 1/2)))) (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 9999999999998099/10000000000000000 (/.f64 6765203681218851/10000000000000 (+.f64 (-.f64 z 1) 1))) (/.f64 -3147848041806007/2500000000000 (+.f64 (-.f64 z 1) 2))) (/.f64 7713234287776531/10000000000000 (+.f64 (-.f64 z 1) 3))) (/.f64 -883075145810703/5000000000000 (+.f64 (-.f64 z 1) 4))) (/.f64 2501468655737381/200000000000000 (+.f64 (-.f64 z 1) 5))) (/.f64 -3464277381643003/25000000000000000 (+.f64 (-.f64 z 1) 6))) (/.f64 2496092394504893/250000000000000000000 (+.f64 (-.f64 z 1) 7))) (/.f64 3764081837873279/25000000000000000000000 (+.f64 (-.f64 z 1) 8)))) < 5.00000000000000037e243

   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 2.3

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

   4. Applied egg-rr 2.1

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \color{blue}{{\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(0.9999999999998099 + \left(\left(\frac{771.3234287776531}{z + 2} + \frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\frac{457679.80848377093}{z \cdot z} + \left({\left(\frac{-1259.1392167224028}{z + 1}\right)}^{2} - \frac{-851833.326413742}{z \cdot \left(z + 1\right)}\right)}\right) + \left(\left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right) + \left(\left(\frac{-176.6150291621406}{z + 3} + \frac{12.507343278686905}{z + 4}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right)}^{1}} \]

   if 5.00000000000000037e243 < (*.f64 (*.f64 (*.f64 (sqrt.f64 (*.f64 (PI.f64) 2)) (pow.f64 (+.f64 (+.f64 (-.f64 z 1) 7) 1/2) (+.f64 (-.f64 z 1) 1/2))) (exp.f64 (neg.f64 (+.f64 (+.f64 (-.f64 z 1) 7) 1/2)))) (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 9999999999998099/10000000000000000 (/.f64 6765203681218851/10000000000000 (+.f64 (-.f64 z 1) 1))) (/.f64 -3147848041806007/2500000000000 (+.f64 (-.f64 z 1) 2))) (/.f64 7713234287776531/10000000000000 (+.f64 (-.f64 z 1) 3))) (/.f64 -883075145810703/5000000000000 (+.f64 (-.f64 z 1) 4))) (/.f64 2501468655737381/200000000000000 (+.f64 (-.f64 z 1) 5))) (/.f64 -3464277381643003/25000000000000000 (+.f64 (-.f64 z 1) 6))) (/.f64 2496092394504893/250000000000000000000 (+.f64 (-.f64 z 1) 7))) (/.f64 3764081837873279/25000000000000000000000 (+.f64 (-.f64 z 1) 8))))

   1. Initial program 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0	\[ \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      associate-*l* [=>] 64.0	\[ \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right)\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      associate-*l* [=>] 64.0	\[ \color{blue}{\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\right)} \]

   3. Taylor expanded in z around -inf 64.0

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

   4. Simplified 7.9

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

      [Start] 64.0	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \frac{e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right)}}{e^{6.5 - -1 \cdot z}}\right) \]
      div-exp [=>] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \color{blue}{e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right) - \left(6.5 - -1 \cdot z\right)}}\right) \]
      sub-neg [=>] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right) - \color{blue}{\left(6.5 + \left(--1 \cdot z\right)\right)}}\right) \]
      associate--r+ [=>] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\color{blue}{\left(-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right) - 6.5\right) - \left(--1 \cdot z\right)}}\right) \]
      mul-1-neg [=>] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\left(-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right) - 6.5\right) - \left(-\color{blue}{\left(-z\right)}\right)}\right) \]
      remove-double-neg [=>] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\left(-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right) - 6.5\right) - \color{blue}{z}}\right) \]
      associate--r+ [<=] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\color{blue}{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right) - \left(6.5 + z\right)}}\right) \]
      mul-1-neg [=>] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\color{blue}{\left(-\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right)} - \left(6.5 + z\right)}\right) \]
      distribute-rgt-neg-in [=>] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\color{blue}{\log \left(6.5 - -1 \cdot z\right) \cdot \left(-\left(-1 \cdot z + 0.5\right)\right)} - \left(6.5 + z\right)}\right) \]
      sub-neg [=>] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\log \color{blue}{\left(6.5 + \left(--1 \cdot z\right)\right)} \cdot \left(-\left(-1 \cdot z + 0.5\right)\right) - \left(6.5 + z\right)}\right) \]
      mul-1-neg [=>] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\log \left(6.5 + \left(-\color{blue}{\left(-z\right)}\right)\right) \cdot \left(-\left(-1 \cdot z + 0.5\right)\right) - \left(6.5 + z\right)}\right) \]
      remove-double-neg [=>] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\log \left(6.5 + \color{blue}{z}\right) \cdot \left(-\left(-1 \cdot z + 0.5\right)\right) - \left(6.5 + z\right)}\right) \]
      +-commutative [=>] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\log \left(6.5 + z\right) \cdot \left(-\color{blue}{\left(0.5 + -1 \cdot z\right)}\right) - \left(6.5 + z\right)}\right) \]
      mul-1-neg [=>] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\log \left(6.5 + z\right) \cdot \left(-\left(0.5 + \color{blue}{\left(-z\right)}\right)\right) - \left(6.5 + z\right)}\right) \]
      unsub-neg [=>] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\log \left(6.5 + z\right) \cdot \left(-\color{blue}{\left(0.5 - z\right)}\right) - \left(6.5 + z\right)}\right) \]

   5. Applied egg-rr 8.2

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\color{blue}{\left(\mathsf{fma}\left(\mathsf{log1p}\left(z + 5.5\right), z + -0.5, -6.5\right) - e^{\mathsf{log1p}\left(z\right)}\right) + 1}}\right) \]

   6. Simplified 8.3

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\color{blue}{\mathsf{fma}\left(-0.5 + z, \mathsf{log1p}\left(z + 5.5\right), -6.5\right) - \mathsf{expm1}\left(\mathsf{log1p}\left(z\right)\right)}}\right) \]
      Proof

      [Start] 8.2	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\left(\mathsf{fma}\left(\mathsf{log1p}\left(z + 5.5\right), z + -0.5, -6.5\right) - e^{\mathsf{log1p}\left(z\right)}\right) + 1}\right) \]
      associate-+l- [=>] 8.2	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(z + 5.5\right), z + -0.5, -6.5\right) - \left(e^{\mathsf{log1p}\left(z\right)} - 1\right)}}\right) \]
      fma-udef [=>] 8.2	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\color{blue}{\left(\mathsf{log1p}\left(z + 5.5\right) \cdot \left(z + -0.5\right) + -6.5\right)} - \left(e^{\mathsf{log1p}\left(z\right)} - 1\right)}\right) \]
      *-commutative [=>] 8.2	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\left(\color{blue}{\left(z + -0.5\right) \cdot \mathsf{log1p}\left(z + 5.5\right)} + -6.5\right) - \left(e^{\mathsf{log1p}\left(z\right)} - 1\right)}\right) \]
      fma-def [=>] 8.2	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\color{blue}{\mathsf{fma}\left(z + -0.5, \mathsf{log1p}\left(z + 5.5\right), -6.5\right)} - \left(e^{\mathsf{log1p}\left(z\right)} - 1\right)}\right) \]
      +-commutative [=>] 8.2	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(\color{blue}{-0.5 + z}, \mathsf{log1p}\left(z + 5.5\right), -6.5\right) - \left(e^{\mathsf{log1p}\left(z\right)} - 1\right)}\right) \]
      expm1-def [=>] 8.3	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-0.5 + z, \mathsf{log1p}\left(z + 5.5\right), -6.5\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(z\right)\right)}}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\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^{-0.5 + \left(-7 + \left(1 - z\right)\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 + \left(z + -1\right)}\right) + \frac{-1259.1392167224028}{2 + \left(z + -1\right)}\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) \leq 5 \cdot 10^{+243}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(0.9999999999998099 + \left(\left(\frac{771.3234287776531}{2 + z} + \frac{\frac{309629712.5173946}{{z}^{3}} + \frac{-1996279061.5505414}{{\left(z + 1\right)}^{3}}}{\frac{457679.80848377093}{z \cdot z} + \left({\left(\frac{-1259.1392167224028}{z + 1}\right)}^{2} + \frac{851833.326413742}{z \cdot \left(z + 1\right)}\right)}\right) + \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)\right)\\ \mathbf{else}:\\ \;\;\;\;\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(z + -0.5, \mathsf{log1p}\left(z + 5.5\right), -6.5\right) - \mathsf{expm1}\left(\mathsf{log1p}\left(z\right)\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	2.3
Cost	48964

\[\begin{array}{l} t_0 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_1 := \frac{-0.13857109526572012}{z + 5}\\ t_2 := \frac{-176.6150291621406}{z + 3}\\ t_3 := \frac{12.507343278686905}{z + 4}\\ t_4 := \sqrt{\pi \cdot 2}\\ t_5 := \frac{771.3234287776531}{2 + z}\\ \mathbf{if}\;z + -1 \leq 142:\\ \;\;\;\;t_4 \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{\mathsf{fma}\left(z, -1259.1392167224028, \mathsf{fma}\left(676.5203681218851, z, 676.5203681218851\right)\right)}{\mathsf{fma}\left(z, z, z\right)} + \left(t_5 + t_2\right)\right)\right) + \left(t_0 + \left(t_1 + t_3\right)\right)\right) \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(t_2 + \left(t_5 + t_3\right)\right)\right) + \left(t_1 + t_0\right)\right)\right) \cdot e^{\mathsf{fma}\left(z + -0.5, \mathsf{log1p}\left(z + 5.5\right), -6.5\right) - \mathsf{expm1}\left(\mathsf{log1p}\left(z\right)\right)}\right)\\ \end{array} \]

Alternative 2
Error	2.4
Cost	48900

\[\begin{array}{l} t_0 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_1 := \frac{-176.6150291621406}{z + 3}\\ t_2 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \frac{-0.13857109526572012}{z + 5}\\ t_5 := \frac{771.3234287776531}{2 + z}\\ t_6 := \frac{12.507343278686905}{z + 4}\\ \mathbf{if}\;z + -1 \leq 142:\\ \;\;\;\;t_3 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(t_5 + \frac{676.5203681218851 \cdot \left(z + 1\right) + z \cdot -1259.1392167224028}{z \cdot \left(z + 1\right)}\right)\right) + \left(t_2 + \left(\left(t_4 + t_0\right) + \left(t_1 + t_6\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(t_1 + \left(t_5 + t_6\right)\right)\right) + \left(t_4 + \left(t_0 + t_2\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(z + -0.5, \mathsf{log1p}\left(z + 5.5\right), -6.5\right) - \mathsf{expm1}\left(\mathsf{log1p}\left(z\right)\right)}\right)\\ \end{array} \]

Alternative 3
Error	2.4
Cost	42756

\[\begin{array}{l} t_0 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_1 := \frac{-176.6150291621406}{z + 3}\\ t_2 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \frac{-0.13857109526572012}{z + 5}\\ t_5 := \frac{771.3234287776531}{2 + z}\\ t_6 := \frac{12.507343278686905}{z + 4}\\ \mathbf{if}\;z + -1 \leq 142:\\ \;\;\;\;t_3 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(t_5 + \frac{676.5203681218851 \cdot \left(z + 1\right) + z \cdot -1259.1392167224028}{z \cdot \left(z + 1\right)}\right)\right) + \left(t_2 + \left(\left(t_4 + t_0\right) + \left(t_1 + t_6\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(t_1 + \left(t_5 + t_6\right)\right)\right) + \left(t_4 + \left(t_0 + t_2\right)\right)\right)\right) \cdot e^{1 + \left(\left(z + -0.5\right) \cdot \mathsf{log1p}\left(z + 5.5\right) - e^{\log \left(z + 7.5\right)}\right)}\right)\\ \end{array} \]

Alternative 4
Error	2.4
Cost	42500

\[\begin{array}{l} t_0 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_1 := \frac{-176.6150291621406}{z + 3}\\ t_2 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \frac{-0.13857109526572012}{z + 5}\\ t_5 := \frac{771.3234287776531}{2 + z}\\ t_6 := \frac{12.507343278686905}{z + 4}\\ \mathbf{if}\;z + -1 \leq 142:\\ \;\;\;\;t_3 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(t_5 + \frac{676.5203681218851 \cdot \left(z + 1\right) + z \cdot -1259.1392167224028}{z \cdot \left(z + 1\right)}\right)\right) + \left(t_2 + \left(\left(t_4 + t_0\right) + \left(t_1 + t_6\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(t_4 + \left(t_0 + t_2\right)\right) + \left(t_1 + \left(t_5 + t_6\right)\right)\right)\right)\right)\right) \cdot {e}^{\left(\mathsf{fma}\left(\mathsf{log1p}\left(z + 5.5\right), z + -0.5, -6.5\right) - z\right)}\right)\\ \end{array} \]

Alternative 5
Error	2.4
Cost	36100

\[\begin{array}{l} t_0 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_1 := \frac{-176.6150291621406}{z + 3}\\ t_2 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \frac{-0.13857109526572012}{z + 5}\\ t_5 := \frac{771.3234287776531}{2 + z}\\ t_6 := \frac{12.507343278686905}{z + 4}\\ \mathbf{if}\;z + -1 \leq 142:\\ \;\;\;\;t_3 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(t_5 + \frac{676.5203681218851 \cdot \left(z + 1\right) + z \cdot -1259.1392167224028}{z \cdot \left(z + 1\right)}\right)\right) + \left(t_2 + \left(\left(t_4 + t_0\right) + \left(t_1 + t_6\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(t_4 + \left(t_0 + t_2\right)\right) + \left(t_1 + \left(t_5 + t_6\right)\right)\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(z + -0.5, \mathsf{log1p}\left(z + 5.5\right), -6.5 - z\right)}\right)\\ \end{array} \]

Alternative 6
Error	2.4
Cost	30468

\[\begin{array}{l} t_0 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_1 := \frac{-176.6150291621406}{z + 3}\\ t_2 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_3 := \sqrt{\pi \cdot 2}\\ t_4 := \frac{-0.13857109526572012}{z + 5}\\ t_5 := \frac{771.3234287776531}{2 + z}\\ t_6 := \frac{12.507343278686905}{z + 4}\\ \mathbf{if}\;z + -1 \leq 142:\\ \;\;\;\;t_3 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(t_5 + \frac{676.5203681218851 \cdot \left(z + 1\right) + z \cdot -1259.1392167224028}{z \cdot \left(z + 1\right)}\right)\right) + \left(t_2 + \left(\left(t_4 + t_0\right) + \left(t_1 + t_6\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(t_1 + \left(t_5 + t_6\right)\right)\right) + \left(t_4 + \left(t_0 + t_2\right)\right)\right)\right) \cdot e^{\left(\frac{-42.25}{6.5 - z} - \mathsf{log1p}\left(z + 5.5\right) \cdot \left(0.5 - z\right)\right) + \frac{z}{\frac{6.5 - z}{z}}}\right)\\ \end{array} \]

Alternative 7
Error	2.5
Cost	30340

\[\begin{array}{l} t_0 := \frac{-176.6150291621406}{z + 3}\\ t_1 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_2 := \sqrt{\pi \cdot 2}\\ t_3 := \frac{-0.13857109526572012}{z + 5}\\ t_4 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_5 := \frac{771.3234287776531}{2 + z}\\ t_6 := \frac{12.507343278686905}{z + 4}\\ \mathbf{if}\;z \leq 144:\\ \;\;\;\;t_2 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(t_5 + \frac{676.5203681218851 \cdot \left(z + 1\right) + z \cdot -1259.1392167224028}{z \cdot \left(z + 1\right)}\right)\right) + \left(t_1 + \left(\left(t_3 + t_4\right) + \left(t_0 + t_6\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(t_3 + \left(t_4 + t_1\right)\right) + \left(t_0 + \left(t_5 + t_6\right)\right)\right)\right)\right)\right) \cdot e^{\left(\frac{-42.25}{6.5 - z} - \mathsf{log1p}\left(z + 5.5\right) \cdot \left(0.5 - z\right)\right) + \frac{z}{\frac{6.5 - z}{z}}}\right)\\ \end{array} \]

Alternative 8
Error	2.5
Cost	30020

\[\begin{array}{l} t_0 := \frac{-176.6150291621406}{z + 3}\\ t_1 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ t_2 := \sqrt{\pi \cdot 2}\\ t_3 := \frac{-0.13857109526572012}{z + 5}\\ t_4 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_5 := \frac{771.3234287776531}{2 + z}\\ t_6 := \frac{12.507343278686905}{z + 4}\\ \mathbf{if}\;z \leq 144:\\ \;\;\;\;t_2 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot \left(e^{-6.5 - z} \cdot \left(\left(0.9999999999998099 + \left(t_5 + \frac{676.5203681218851 \cdot \left(z + 1\right) + z \cdot -1259.1392167224028}{z \cdot \left(z + 1\right)}\right)\right) + \left(t_1 + \left(\left(t_3 + t_4\right) + \left(t_0 + t_6\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(\left(0.9999999999998099 + \left(\left(\left(t_3 + \left(t_4 + t_1\right)\right) + \left(t_0 + \left(t_5 + t_6\right)\right)\right) + \left(\frac{-1259.1392167224028}{z + 1} + \frac{676.5203681218851}{z}\right)\right)\right) \cdot e^{\left(-6.5 - z\right) + \left(z + -0.5\right) \cdot \log \left(z + 6.5\right)}\right)\\ \end{array} \]

Alternative 9
Error	2.4
Cost	29700

\[\begin{array}{l} t_0 := \frac{771.3234287776531}{2 + z}\\ t_1 := \frac{12.507343278686905}{z + 4}\\ t_2 := \frac{-1259.1392167224028}{z + 1}\\ t_3 := \frac{-0.13857109526572012}{z + 5}\\ t_4 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_5 := \frac{-176.6150291621406}{z + 3}\\ t_6 := \sqrt{\pi \cdot 2}\\ t_7 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ \mathbf{if}\;z \leq 144:\\ \;\;\;\;t_6 \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(t_3 + \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(t_0 + t_2\right)\right)\right) + \left(t_4 + \left(t_5 + \left(t_1 + t_7\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_6 \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(t_2 + \left(\left(t_3 + \left(t_4 + t_7\right)\right) + \left(t_5 + \left(t_0 + t_1\right)\right)\right)\right)\right)\right) \cdot e^{\left(-6.5 - z\right) - \mathsf{log1p}\left(z + 5.5\right) \cdot \left(0.5 - z\right)}\right)\\ \end{array} \]

Alternative 10
Error	2.4
Cost	29700

\[\begin{array}{l} t_0 := \frac{771.3234287776531}{2 + z}\\ t_1 := \frac{12.507343278686905}{z + 4}\\ t_2 := \frac{-1259.1392167224028}{z + 1}\\ t_3 := \frac{-0.13857109526572012}{z + 5}\\ t_4 := \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\\ t_5 := \frac{-176.6150291621406}{z + 3}\\ t_6 := \sqrt{\pi \cdot 2}\\ t_7 := \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ \mathbf{if}\;z \leq 144:\\ \;\;\;\;t_6 \cdot \left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right) \cdot \left(t_3 + \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(t_0 + t_2\right)\right)\right) + \left(t_4 + \left(t_5 + \left(t_1 + t_7\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_6 \cdot \left(\left(0.9999999999998099 + \left(\left(\left(t_3 + \left(t_4 + t_7\right)\right) + \left(t_5 + \left(t_0 + t_1\right)\right)\right) + \left(t_2 + \frac{676.5203681218851}{z}\right)\right)\right) \cdot e^{\left(-6.5 - z\right) + \left(z + -0.5\right) \cdot \log \left(z + 6.5\right)}\right)\\ \end{array} \]

Alternative 11
Error	3.9
Cost	29504

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

Alternative 12
Error	3.8
Cost	29504

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

Alternative 13
Error	3.8
Cost	29504

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

Alternative 14
Error	46.8
Cost	28736

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

Alternative 15
Error	47.0
Cost	27264

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

Alternative 16
Error	47.6
Cost	27200

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

Alternative 17
Error	49.8
Cost	27012

\[\begin{array}{l} \mathbf{if}\;z \leq 2.8:\\ \;\;\;\;\frac{1}{z} \cdot \sqrt{\pi \cdot \left(140824.5564565449 \cdot e^{-13}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(e^{\left(-6.5 - z\right) + \left(z + -0.5\right) \cdot \log \left(z + 6.5\right)} \cdot \left(0.9999999999998099 + \frac{24.458333333348836}{z}\right)\right)\\ \end{array} \]

Alternative 18
Error	50.3
Cost	26948

\[\begin{array}{l} \mathbf{if}\;z \leq 2.8:\\ \;\;\;\;\frac{1}{z} \cdot \sqrt{\pi \cdot \left(140824.5564565449 \cdot e^{-13}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(\frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}} \cdot \left(0.9999999999998099 + \frac{24.458333333348836}{z}\right)\right)\\ \end{array} \]

Alternative 19
Error	51.6
Cost	26756

\[\begin{array}{l} \mathbf{if}\;z \leq 4:\\ \;\;\;\;\frac{\sqrt{e^{-13} \cdot \left(\pi \cdot 140824.5564565449\right)}}{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(0.9999999999998099 \cdot e^{\left(-6.5 - z\right) + \left(z + -0.5\right) \cdot \log \left(z + 6.5\right)}\right)\\ \end{array} \]

Alternative 20
Error	52.0
Cost	26692

\[\begin{array}{l} \mathbf{if}\;z \leq 4:\\ \;\;\;\;\frac{\sqrt{e^{-13} \cdot \left(\pi \cdot 140824.5564565449\right)}}{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(0.9999999999998099 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right)\right)\\ \end{array} \]

Alternative 21
Error	55.6
Cost	19584

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

Reproduce

herbie shell --seed 2023060 
(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)))))