Jmat.Real.gamma, branch z greater than 0.5

Average Error: 4.0 → 2.3

Time: 32.4s

Precision: binary64

Cost: 112452

\[z > 0.5\]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{12.507343278686905}{\left(z + -1\right) + 5}\\ t_1 := \frac{-0.13857109526572012}{\left(z + -1\right) + 6}\\ t_2 := \sqrt{\pi \cdot 2}\\ t_3 := \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8}\\ t_4 := \frac{-176.6150291621406}{\left(z + -1\right) + 4}\\ t_5 := \frac{771.3234287776531}{\left(z + -1\right) + 3}\\ t_6 := \sqrt[3]{t_2}\\ t_7 := \left(z + -1\right) + 7\\ t_8 := \left(t_2 \cdot {\left(t_7 + 0.5\right)}^{\left(\left(z + -1\right) + 0.5\right)}\right) \cdot e^{-0.5 + \left(\left(1 - z\right) + -7\right)}\\ t_9 := \frac{9.984369578019572 \cdot 10^{-6}}{t_7}\\ t_10 := \frac{-1259.1392167224028}{z + 1}\\ \mathbf{if}\;t_8 \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 + \left(z + -1\right)}\right) + \frac{-1259.1392167224028}{2 + \left(z + -1\right)}\right) + t_5\right) + t_4\right) + t_0\right) + t_1\right) + t_9\right) + t_3\right) \leq 10^{+234}:\\ \;\;\;\;t_8 \cdot \left(t_3 + \left(t_9 + \left(t_1 + \left(t_0 + \left(t_4 + \left(t_5 + \sqrt[3]{\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + t_10\right)\right) \cdot {\left(0.9999999999998099 + \frac{-676.5203681218851 + z \cdot 582.6188486005177}{z \cdot \left(-1 - z\right)}\right)}^{2}}\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_6 \cdot \left(t_6 \cdot t_6\right)\right) \cdot \left(\left(\frac{676.5203681218851}{z} + \left(0.9999999999998099 + \left(t_10 + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-176.6150291621406}{z + 3} + \frac{771.3234287776531}{2 + z}\right)\right) + \frac{12.507343278686905}{z + 4}\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(z + 6.5\right), 0.5 - z, -6.5 - z\right)}\right)\\ \end{array} \]

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 -1.0) 5.0)))
        (t_1 (/ -0.13857109526572012 (+ (+ z -1.0) 6.0)))
        (t_2 (sqrt (* PI 2.0)))
        (t_3 (/ 1.5056327351493116e-7 (+ (+ z -1.0) 8.0)))
        (t_4 (/ -176.6150291621406 (+ (+ z -1.0) 4.0)))
        (t_5 (/ 771.3234287776531 (+ (+ z -1.0) 3.0)))
        (t_6 (cbrt t_2))
        (t_7 (+ (+ z -1.0) 7.0))
        (t_8
         (*
          (* t_2 (pow (+ t_7 0.5) (+ (+ z -1.0) 0.5)))
          (exp (+ -0.5 (+ (- 1.0 z) -7.0)))))
        (t_9 (/ 9.984369578019572e-6 t_7))
        (t_10 (/ -1259.1392167224028 (+ z 1.0))))
   (if (<=
        (*
         t_8
         (+
          (+
           (+
            (+
             (+
              (+
               (+
                (+ 0.9999999999998099 (/ 676.5203681218851 (+ 1.0 (+ z -1.0))))
                (/ -1259.1392167224028 (+ 2.0 (+ z -1.0))))
               t_5)
              t_4)
             t_0)
            t_1)
           t_9)
          t_3))
        1e+234)
     (*
      t_8
      (+
       t_3
       (+
        t_9
        (+
         t_1
         (+
          t_0
          (+
           t_4
           (+
            t_5
            (cbrt
             (*
              (+ 0.9999999999998099 (+ (/ 676.5203681218851 z) t_10))
              (pow
               (+
                0.9999999999998099
                (/
                 (+ -676.5203681218851 (* z 582.6188486005177))
                 (* z (- -1.0 z))))
               2.0))))))))))
     (*
      (* t_6 (* t_6 t_6))
      (*
       (+
        (/ 676.5203681218851 z)
        (+
         0.9999999999998099
         (+
          t_10
          (+
           (+
            (+
             (/ -0.13857109526572012 (+ z 5.0))
             (+
              (/ 9.984369578019572e-6 (+ z 6.0))
              (/ 1.5056327351493116e-7 (+ z 7.0))))
            (+
             (/ -176.6150291621406 (+ z 3.0))
             (/ 771.3234287776531 (+ 2.0 z))))
           (/ 12.507343278686905 (+ z 4.0))))))
       (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 + -1.0) + 5.0);
	double t_1 = -0.13857109526572012 / ((z + -1.0) + 6.0);
	double t_2 = sqrt((((double) M_PI) * 2.0));
	double t_3 = 1.5056327351493116e-7 / ((z + -1.0) + 8.0);
	double t_4 = -176.6150291621406 / ((z + -1.0) + 4.0);
	double t_5 = 771.3234287776531 / ((z + -1.0) + 3.0);
	double t_6 = cbrt(t_2);
	double t_7 = (z + -1.0) + 7.0;
	double t_8 = (t_2 * pow((t_7 + 0.5), ((z + -1.0) + 0.5))) * exp((-0.5 + ((1.0 - z) + -7.0)));
	double t_9 = 9.984369578019572e-6 / t_7;
	double t_10 = -1259.1392167224028 / (z + 1.0);
	double tmp;
	if ((t_8 * ((((((((0.9999999999998099 + (676.5203681218851 / (1.0 + (z + -1.0)))) + (-1259.1392167224028 / (2.0 + (z + -1.0)))) + t_5) + t_4) + t_0) + t_1) + t_9) + t_3)) <= 1e+234) {
		tmp = t_8 * (t_3 + (t_9 + (t_1 + (t_0 + (t_4 + (t_5 + cbrt(((0.9999999999998099 + ((676.5203681218851 / z) + t_10)) * pow((0.9999999999998099 + ((-676.5203681218851 + (z * 582.6188486005177)) / (z * (-1.0 - z)))), 2.0)))))))));
	} else {
		tmp = (t_6 * (t_6 * t_6)) * (((676.5203681218851 / z) + (0.9999999999998099 + (t_10 + ((((-0.13857109526572012 / (z + 5.0)) + ((9.984369578019572e-6 / (z + 6.0)) + (1.5056327351493116e-7 / (z + 7.0)))) + ((-176.6150291621406 / (z + 3.0)) + (771.3234287776531 / (2.0 + z)))) + (12.507343278686905 / (z + 4.0)))))) * 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(Float64(z + -1.0) + 5.0))
	t_1 = Float64(-0.13857109526572012 / Float64(Float64(z + -1.0) + 6.0))
	t_2 = sqrt(Float64(pi * 2.0))
	t_3 = Float64(1.5056327351493116e-7 / Float64(Float64(z + -1.0) + 8.0))
	t_4 = Float64(-176.6150291621406 / Float64(Float64(z + -1.0) + 4.0))
	t_5 = Float64(771.3234287776531 / Float64(Float64(z + -1.0) + 3.0))
	t_6 = cbrt(t_2)
	t_7 = Float64(Float64(z + -1.0) + 7.0)
	t_8 = Float64(Float64(t_2 * (Float64(t_7 + 0.5) ^ Float64(Float64(z + -1.0) + 0.5))) * exp(Float64(-0.5 + Float64(Float64(1.0 - z) + -7.0))))
	t_9 = Float64(9.984369578019572e-6 / t_7)
	t_10 = Float64(-1259.1392167224028 / Float64(z + 1.0))
	tmp = 0.0
	if (Float64(t_8 * 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)))) + t_5) + t_4) + t_0) + t_1) + t_9) + t_3)) <= 1e+234)
		tmp = Float64(t_8 * Float64(t_3 + Float64(t_9 + Float64(t_1 + Float64(t_0 + Float64(t_4 + Float64(t_5 + cbrt(Float64(Float64(0.9999999999998099 + Float64(Float64(676.5203681218851 / z) + t_10)) * (Float64(0.9999999999998099 + Float64(Float64(-676.5203681218851 + Float64(z * 582.6188486005177)) / Float64(z * Float64(-1.0 - z)))) ^ 2.0))))))))));
	else
		tmp = Float64(Float64(t_6 * Float64(t_6 * t_6)) * Float64(Float64(Float64(676.5203681218851 / z) + Float64(0.9999999999998099 + Float64(t_10 + Float64(Float64(Float64(Float64(-0.13857109526572012 / Float64(z + 5.0)) + Float64(Float64(9.984369578019572e-6 / Float64(z + 6.0)) + Float64(1.5056327351493116e-7 / Float64(z + 7.0)))) + Float64(Float64(-176.6150291621406 / Float64(z + 3.0)) + Float64(771.3234287776531 / Float64(2.0 + z)))) + Float64(12.507343278686905 / Float64(z + 4.0)))))) * 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[(N[(z + -1.0), $MachinePrecision] + 5.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.13857109526572012 / N[(N[(z + -1.0), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(1.5056327351493116e-7 / N[(N[(z + -1.0), $MachinePrecision] + 8.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(-176.6150291621406 / N[(N[(z + -1.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(771.3234287776531 / N[(N[(z + -1.0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Power[t$95$2, 1/3], $MachinePrecision]}, Block[{t$95$7 = N[(N[(z + -1.0), $MachinePrecision] + 7.0), $MachinePrecision]}, Block[{t$95$8 = N[(N[(t$95$2 * N[Power[N[(t$95$7 + 0.5), $MachinePrecision], N[(N[(z + -1.0), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(-0.5 + N[(N[(1.0 - z), $MachinePrecision] + -7.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[(9.984369578019572e-6 / t$95$7), $MachinePrecision]}, Block[{t$95$10 = N[(-1259.1392167224028 / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$8 * 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] + t$95$5), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$9), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], 1e+234], N[(t$95$8 * N[(t$95$3 + N[(t$95$9 + N[(t$95$1 + N[(t$95$0 + N[(t$95$4 + N[(t$95$5 + N[Power[N[(N[(0.9999999999998099 + N[(N[(676.5203681218851 / z), $MachinePrecision] + t$95$10), $MachinePrecision]), $MachinePrecision] * N[Power[N[(0.9999999999998099 + N[(N[(-676.5203681218851 + N[(z * 582.6188486005177), $MachinePrecision]), $MachinePrecision] / N[(z * N[(-1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$6 * N[(t$95$6 * t$95$6), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(676.5203681218851 / z), $MachinePrecision] + N[(0.9999999999998099 + N[(t$95$10 + N[(N[(N[(N[(-0.13857109526572012 / N[(z + 5.0), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(z + 6.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(z + 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-176.6150291621406 / N[(z + 3.0), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(2.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(z + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 (*.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)))) < 1.00000000000000002e234

   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. Applied egg-rr 2.3

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

   3. Applied egg-rr 2.2

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

   4. Simplified 2.1

      \[\leadsto \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\sqrt[3]{\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z - -1}\right)\right) \cdot {\left(0.9999999999998099 + \color{blue}{\frac{-676.5203681218851 + z \cdot 582.6188486005177}{z \cdot \left(-1 - z\right)}}\right)}^{2}} + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      Proof

      [Start] 2.2

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

      associate-*r/ [=>] 2.2

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

   if 1.00000000000000002e234 < (*.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 55.3

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

   2. Simplified 54.9

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

      [Start] 55.3	\[ \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      associate-*l* [=>] 54.9	\[ \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right)\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      associate-*l* [=>] 54.9	\[ \color{blue}{\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\right)} \]

   3. Taylor expanded in z around -inf 55.7

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

   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] 55.7	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \frac{e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right)}}{e^{6.5 - -1 \cdot z}}\right) \]
      div-exp [=>] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot \color{blue}{e^{-1 \cdot \left(\log \left(6.5 - -1 \cdot z\right) \cdot \left(-1 \cdot z + 0.5\right)\right) - \left(6.5 - -1 \cdot z\right)}}\right) \]
      associate-*r* [=>] 7.9	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \cdot e^{\color{blue}{\left(-1 \cdot \log \left(6.5 - -1 \cdot z\right)\right) \cdot \left(-1 \cdot z + 0.5\right)} - \left(6.5 - -1 \cdot z\right)}\right) \]
      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

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

   6. Simplified 7.6

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

      [Start] 7.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(1 \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{z + 2} + \frac{12.507343278686905}{z + 4}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -6.5 - z\right)}\right) \]
      *-lft-identity [=>] 7.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{z + 2} + \frac{12.507343278686905}{z + 4}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\right)} \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -6.5 - z\right)}\right) \]
      +-commutative [=>] 7.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\left(\left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{z + 2} + \frac{12.507343278686905}{z + 4}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right) + 0.9999999999998099\right)} \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -6.5 - z\right)}\right) \]
      associate-+l+ [=>] 7.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\left(\frac{676.5203681218851}{z} + \left(\left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{z + 2} + \frac{12.507343278686905}{z + 4}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) + 0.9999999999998099\right)\right)} \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -6.5 - z\right)}\right) \]
      +-commutative [<=] 7.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\frac{676.5203681218851}{z} + \color{blue}{\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{z + 2} + \frac{12.507343278686905}{z + 4}\right)\right) + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)}\right) \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -6.5 - z\right)}\right) \]
      +-commutative [=>] 7.6	\[ \sqrt{\pi \cdot 2} \cdot \left(\left(\frac{676.5203681218851}{z} + \left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{z + 1} + \color{blue}{\left(\left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{z + 2} + \frac{12.507343278686905}{z + 4}\right)\right)\right)}\right)\right)\right) \cdot e^{\mathsf{fma}\left(-\log \left(6.5 + z\right), 0.5 - z, -6.5 - z\right)}\right) \]

   7. Applied egg-rr 7.6

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

4. Final simplification 2.3

   \[\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(\left(1 - z\right) + -7\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 10^{+234}:\\ \;\;\;\;\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(\left(1 - z\right) + -7\right)}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(z + -1\right) + 7} + \left(\frac{-0.13857109526572012}{\left(z + -1\right) + 6} + \left(\frac{12.507343278686905}{\left(z + -1\right) + 5} + \left(\frac{-176.6150291621406}{\left(z + -1\right) + 4} + \left(\frac{771.3234287776531}{\left(z + -1\right) + 3} + \sqrt[3]{\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right) \cdot {\left(0.9999999999998099 + \frac{-676.5203681218851 + z \cdot 582.6188486005177}{z \cdot \left(-1 - z\right)}\right)}^{2}}\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\sqrt{\pi \cdot 2}} \cdot \left(\sqrt[3]{\sqrt{\pi \cdot 2}} \cdot \sqrt[3]{\sqrt{\pi \cdot 2}}\right)\right) \cdot \left(\left(\frac{676.5203681218851}{z} + \left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{z + 1} + \left(\left(\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) + \left(\frac{-176.6150291621406}{z + 3} + \frac{771.3234287776531}{2 + z}\right)\right) + \frac{12.507343278686905}{z + 4}\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	2.2
Cost	81348

\[\begin{array}{l} t_0 := \left(z + -1\right) + 7\\ t_1 := \sqrt{\pi \cdot 2}\\ t_2 := \sqrt[3]{t_1}\\ t_3 := \frac{-1259.1392167224028}{z + 1}\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;\left(\left(t_1 \cdot {\left(t_0 + 0.5\right)}^{\left(\left(z + -1\right) + 0.5\right)}\right) \cdot e^{-0.5 + \left(\left(1 - z\right) + -7\right)}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{t_0} + \left(\frac{-0.13857109526572012}{\left(z + -1\right) + 6} + \left(\frac{12.507343278686905}{\left(z + -1\right) + 5} + \left(\frac{-176.6150291621406}{\left(z + -1\right) + 4} + \left(\frac{771.3234287776531}{\left(z + -1\right) + 3} + \sqrt[3]{\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + t_3\right)\right) \cdot {\left(0.9999999999998099 + \frac{-676.5203681218851 + z \cdot 582.6188486005177}{z \cdot \left(-1 - z\right)}\right)}^{2}}\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_2 \cdot \left(t_2 \cdot t_2\right)\right) \cdot \left(e^{\mathsf{fma}\left(-\log \left(z + 6.5\right), 0.5 - z, -6.5 - z\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(t_3 + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right)\right) + \left(0.9999999999998099 + \frac{676.5203681218851}{z}\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	2.2
Cost	44932

\[\begin{array}{l} t_0 := \left(z + -1\right) + 7\\ t_1 := \sqrt{\pi \cdot 2}\\ t_2 := \frac{-1259.1392167224028}{z + 1}\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;\left(\left(t_1 \cdot {\left(t_0 + 0.5\right)}^{\left(\left(z + -1\right) + 0.5\right)}\right) \cdot e^{-0.5 + \left(\left(1 - z\right) + -7\right)}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{t_0} + \left(\frac{-0.13857109526572012}{\left(z + -1\right) + 6} + \left(\frac{12.507343278686905}{\left(z + -1\right) + 5} + \left(\frac{-176.6150291621406}{\left(z + -1\right) + 4} + \left(\frac{771.3234287776531}{\left(z + -1\right) + 3} + \sqrt[3]{\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + t_2\right)\right) \cdot {\left(0.9999999999998099 + \frac{-676.5203681218851 + z \cdot 582.6188486005177}{z \cdot \left(-1 - z\right)}\right)}^{2}}\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(e^{\mathsf{fma}\left(-\log \left(z + 6.5\right), 0.5 - z, -6.5 - z\right)} \cdot \left(t_2 + \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{z + 7} + \left(\frac{-0.13857109526572012}{z + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right)\right)\right) + \left(\frac{-176.6150291621406}{z + 3} + \left(\frac{771.3234287776531}{2 + z} + \frac{12.507343278686905}{z + 4}\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 3
Error	2.3
Cost	36164

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

Alternative 4
Error	2.3
Cost	36164

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

Alternative 5
Error	2.3
Cost	36164

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

Alternative 6
Error	2.3
Cost	29828

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

Alternative 7
Error	4.0
Cost	29504

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

Alternative 8
Error	4.0
Cost	29504

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

Alternative 9
Error	4.0
Cost	29504

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

Alternative 10
Error	4.0
Cost	29504

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

Alternative 11
Error	3.9
Cost	29504

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

Alternative 12
Error	46.9
Cost	27264

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

Alternative 13
Error	47.7
Cost	27200

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

Alternative 14
Error	49.8
Cost	27012

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

Alternative 15
Error	50.4
Cost	26948

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

Alternative 16
Error	51.6
Cost	26756

\[\begin{array}{l} \mathbf{if}\;z \leq 4:\\ \;\;\;\;\frac{1}{z} \cdot \sqrt{\pi \cdot \left(140824.5564565449 \cdot e^{-13}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(0.9999999999998099 \cdot e^{\left(-6.5 - z\right) + \log \left(z + 6.5\right) \cdot \left(z + -0.5\right)}\right)\\ \end{array} \]

Alternative 17
Error	52.0
Cost	26692

\[\begin{array}{l} \mathbf{if}\;z \leq 4:\\ \;\;\;\;\frac{1}{z} \cdot \sqrt{\pi \cdot \left(140824.5564565449 \cdot e^{-13}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(0.9999999999998099 \cdot \left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-6.5 - z}\right)\right)\\ \end{array} \]

Alternative 18
Error	55.6
Cost	19584

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

Reproduce

herbie shell --seed 2023066 
(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)))))