Jmat.Real.gamma, branch z less than 0.5

Average Accuracy: 97.3% → 99.3%

Time: 1.5min

Precision: binary64

Cost: 59328

\[z \leq 0.5\]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{0.13880072788059747 + \left(z \cdot 1.8611992721194026 - z\right)}{\left(0.00147815209581367 - z \cdot 0.00147815209581367\right) \cdot \left(2 - z\right)}\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{\left(z + -1\right) + -6.5} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\sqrt[3]{t_0 \cdot \left(t_0 \cdot t_0\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \end{array} \]

FPCore

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

↓

(FPCore (z)
 :precision binary64
 (let* ((t_0
         (/
          (+ 0.13880072788059747 (- (* z 1.8611992721194026) z))
          (* (- 0.00147815209581367 (* z 0.00147815209581367)) (- 2.0 z)))))
   (*
    (/ PI (sin (* PI z)))
    (*
     (* (sqrt (* PI 2.0)) (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)))
     (*
      (exp (+ (+ z -1.0) -6.5))
      (+
       (+
        (+
         (/ -0.13857109526572012 (+ (- 1.0 z) 5.0))
         (+
          0.9999999999998099
          (+
           (cbrt (* t_0 (* t_0 t_0)))
           (* (/ 1.0 (- 3.0 z)) 771.3234287776531))))
        (+
         (/ -176.6150291621406 (+ (- 1.0 z) 3.0))
         (/ 12.507343278686905 (+ (- 1.0 z) 4.0))))
       (+
        (/ 9.984369578019572e-6 (- 1.0 (+ z -6.0)))
        (/ 1.5056327351493116e-7 (+ (- 1.0 z) 7.0)))))))))

C

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

↓

double code(double z) {
	double t_0 = (0.13880072788059747 + ((z * 1.8611992721194026) - z)) / ((0.00147815209581367 - (z * 0.00147815209581367)) * (2.0 - z));
	return (((double) M_PI) / sin((((double) M_PI) * z))) * ((sqrt((((double) M_PI) * 2.0)) * pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5))) * (exp(((z + -1.0) + -6.5)) * ((((-0.13857109526572012 / ((1.0 - z) + 5.0)) + (0.9999999999998099 + (cbrt((t_0 * (t_0 * t_0))) + ((1.0 / (3.0 - z)) * 771.3234287776531)))) + ((-176.6150291621406 / ((1.0 - z) + 3.0)) + (12.507343278686905 / ((1.0 - z) + 4.0)))) + ((9.984369578019572e-6 / (1.0 - (z + -6.0))) + (1.5056327351493116e-7 / ((1.0 - z) + 7.0))))));
}

Java

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

↓

public static double code(double z) {
	double t_0 = (0.13880072788059747 + ((z * 1.8611992721194026) - z)) / ((0.00147815209581367 - (z * 0.00147815209581367)) * (2.0 - z));
	return (Math.PI / Math.sin((Math.PI * z))) * ((Math.sqrt((Math.PI * 2.0)) * Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5))) * (Math.exp(((z + -1.0) + -6.5)) * ((((-0.13857109526572012 / ((1.0 - z) + 5.0)) + (0.9999999999998099 + (Math.cbrt((t_0 * (t_0 * t_0))) + ((1.0 / (3.0 - z)) * 771.3234287776531)))) + ((-176.6150291621406 / ((1.0 - z) + 3.0)) + (12.507343278686905 / ((1.0 - z) + 4.0)))) + ((9.984369578019572e-6 / (1.0 - (z + -6.0))) + (1.5056327351493116e-7 / ((1.0 - z) + 7.0))))));
}

Julia

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

↓

function code(z)
	t_0 = Float64(Float64(0.13880072788059747 + Float64(Float64(z * 1.8611992721194026) - z)) / Float64(Float64(0.00147815209581367 - Float64(z * 0.00147815209581367)) * Float64(2.0 - z)))
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(sqrt(Float64(pi * 2.0)) * (Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5))) * Float64(exp(Float64(Float64(z + -1.0) + -6.5)) * Float64(Float64(Float64(Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) + 5.0)) + Float64(0.9999999999998099 + Float64(cbrt(Float64(t_0 * Float64(t_0 * t_0))) + Float64(Float64(1.0 / Float64(3.0 - z)) * 771.3234287776531)))) + Float64(Float64(-176.6150291621406 / Float64(Float64(1.0 - z) + 3.0)) + Float64(12.507343278686905 / Float64(Float64(1.0 - z) + 4.0)))) + Float64(Float64(9.984369578019572e-6 / Float64(1.0 - Float64(z + -6.0))) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) + 7.0)))))))
end

Wolfram

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

↓

code[z_] := Block[{t$95$0 = N[(N[(0.13880072788059747 + N[(N[(z * 1.8611992721194026), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / N[(N[(0.00147815209581367 - N[(z * 0.00147815209581367), $MachinePrecision]), $MachinePrecision] * N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(N[(z + -1.0), $MachinePrecision] + -6.5), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] + 5.0), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(N[Power[N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[(1.0 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision] * 771.3234287776531), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(1.0 - N[(z + -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] + 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 97.3%

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

2. Simplified 97.8%

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

   [Start] 97.3

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

   associate-*l* [=>] 97.3

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

3. Applied egg-rr 99.3%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\color{blue}{\frac{\left(2 - z\right) + \left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot -1259.1392167224028}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)}} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   Proof

   [Start] 97.8	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   clear-num [=>] 97.8	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\left(\color{blue}{\frac{1}{\frac{1 - z}{676.5203681218851}}} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   frac-add [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\color{blue}{\frac{1 \cdot \left(\left(1 - z\right) + 1\right) + \frac{1 - z}{676.5203681218851} \cdot -1259.1392167224028}{\frac{1 - z}{676.5203681218851} \cdot \left(\left(1 - z\right) + 1\right)}} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   *-un-lft-identity [<=] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\color{blue}{\left(\left(1 - z\right) + 1\right)} + \frac{1 - z}{676.5203681218851} \cdot -1259.1392167224028}{\frac{1 - z}{676.5203681218851} \cdot \left(\left(1 - z\right) + 1\right)} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   +-commutative [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\color{blue}{\left(1 + \left(1 - z\right)\right)} + \frac{1 - z}{676.5203681218851} \cdot -1259.1392167224028}{\frac{1 - z}{676.5203681218851} \cdot \left(\left(1 - z\right) + 1\right)} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   associate-+r- [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\color{blue}{\left(\left(1 + 1\right) - z\right)} + \frac{1 - z}{676.5203681218851} \cdot -1259.1392167224028}{\frac{1 - z}{676.5203681218851} \cdot \left(\left(1 - z\right) + 1\right)} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   metadata-eval [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\left(\color{blue}{2} - z\right) + \frac{1 - z}{676.5203681218851} \cdot -1259.1392167224028}{\frac{1 - z}{676.5203681218851} \cdot \left(\left(1 - z\right) + 1\right)} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   div-sub [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\left(2 - z\right) + \color{blue}{\left(\frac{1}{676.5203681218851} - \frac{z}{676.5203681218851}\right)} \cdot -1259.1392167224028}{\frac{1 - z}{676.5203681218851} \cdot \left(\left(1 - z\right) + 1\right)} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   metadata-eval [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\left(2 - z\right) + \left(\color{blue}{0.00147815209581367} - \frac{z}{676.5203681218851}\right) \cdot -1259.1392167224028}{\frac{1 - z}{676.5203681218851} \cdot \left(\left(1 - z\right) + 1\right)} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   div-sub [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\left(2 - z\right) + \left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot -1259.1392167224028}{\color{blue}{\left(\frac{1}{676.5203681218851} - \frac{z}{676.5203681218851}\right)} \cdot \left(\left(1 - z\right) + 1\right)} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   metadata-eval [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\left(2 - z\right) + \left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot -1259.1392167224028}{\left(\color{blue}{0.00147815209581367} - \frac{z}{676.5203681218851}\right) \cdot \left(\left(1 - z\right) + 1\right)} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   +-commutative [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\left(2 - z\right) + \left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot -1259.1392167224028}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \color{blue}{\left(1 + \left(1 - z\right)\right)}} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   associate-+r- [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\left(2 - z\right) + \left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot -1259.1392167224028}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \color{blue}{\left(\left(1 + 1\right) - z\right)}} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   metadata-eval [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\left(2 - z\right) + \left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot -1259.1392167224028}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(\color{blue}{2} - z\right)} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]

4. Applied egg-rr 99.3%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\left(2 - z\right) + \left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot -1259.1392167224028}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \color{blue}{\frac{1}{3 - z} \cdot 771.3234287776531}\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   Proof

   [Start] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\left(2 - z\right) + \left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot -1259.1392167224028}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   div-inv [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\left(2 - z\right) + \left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot -1259.1392167224028}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \color{blue}{771.3234287776531 \cdot \frac{1}{\left(1 - z\right) + 2}}\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   *-commutative [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\left(2 - z\right) + \left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot -1259.1392167224028}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \color{blue}{\frac{1}{\left(1 - z\right) + 2} \cdot 771.3234287776531}\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   +-commutative [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\left(2 - z\right) + \left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot -1259.1392167224028}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{\color{blue}{2 + \left(1 - z\right)}} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   associate-+r- [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\left(2 - z\right) + \left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot -1259.1392167224028}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{\color{blue}{\left(2 + 1\right) - z}} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   metadata-eval [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\left(2 - z\right) + \left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot -1259.1392167224028}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{\color{blue}{3} - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]

5. Applied egg-rr 99.3%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\color{blue}{2 - \left(z + \left(1.8611992721194026 + 1259.1392167224028 \cdot \left(z \cdot -0.00147815209581367\right)\right)\right)}}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   Proof

   [Start] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\left(2 - z\right) + \left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot -1259.1392167224028}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   associate-+l- [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\color{blue}{2 - \left(z - \left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot -1259.1392167224028\right)}}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   *-commutative [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{2 - \left(z - \color{blue}{-1259.1392167224028 \cdot \left(0.00147815209581367 - \frac{z}{676.5203681218851}\right)}\right)}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   cancel-sign-sub-inv [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{2 - \color{blue}{\left(z + \left(--1259.1392167224028\right) \cdot \left(0.00147815209581367 - \frac{z}{676.5203681218851}\right)\right)}}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   metadata-eval [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{2 - \left(z + \color{blue}{1259.1392167224028} \cdot \left(0.00147815209581367 - \frac{z}{676.5203681218851}\right)\right)}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   metadata-eval [<=] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{2 - \left(z + \color{blue}{\sqrt{1585431.567088306}} \cdot \left(0.00147815209581367 - \frac{z}{676.5203681218851}\right)\right)}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   metadata-eval [<=] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{2 - \left(z + \sqrt{\color{blue}{-1259.1392167224028 \cdot -1259.1392167224028}} \cdot \left(0.00147815209581367 - \frac{z}{676.5203681218851}\right)\right)}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   sub-neg [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{2 - \left(z + \sqrt{-1259.1392167224028 \cdot -1259.1392167224028} \cdot \color{blue}{\left(0.00147815209581367 + \left(-\frac{z}{676.5203681218851}\right)\right)}\right)}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   distribute-lft-in [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{2 - \left(z + \color{blue}{\left(\sqrt{-1259.1392167224028 \cdot -1259.1392167224028} \cdot 0.00147815209581367 + \sqrt{-1259.1392167224028 \cdot -1259.1392167224028} \cdot \left(-\frac{z}{676.5203681218851}\right)\right)}\right)}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   metadata-eval [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{2 - \left(z + \left(\sqrt{\color{blue}{1585431.567088306}} \cdot 0.00147815209581367 + \sqrt{-1259.1392167224028 \cdot -1259.1392167224028} \cdot \left(-\frac{z}{676.5203681218851}\right)\right)\right)}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   metadata-eval [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{2 - \left(z + \left(\color{blue}{1259.1392167224028} \cdot 0.00147815209581367 + \sqrt{-1259.1392167224028 \cdot -1259.1392167224028} \cdot \left(-\frac{z}{676.5203681218851}\right)\right)\right)}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   metadata-eval [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{2 - \left(z + \left(\color{blue}{1.8611992721194026} + \sqrt{-1259.1392167224028 \cdot -1259.1392167224028} \cdot \left(-\frac{z}{676.5203681218851}\right)\right)\right)}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   metadata-eval [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{2 - \left(z + \left(1.8611992721194026 + \sqrt{\color{blue}{1585431.567088306}} \cdot \left(-\frac{z}{676.5203681218851}\right)\right)\right)}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   metadata-eval [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{2 - \left(z + \left(1.8611992721194026 + \color{blue}{1259.1392167224028} \cdot \left(-\frac{z}{676.5203681218851}\right)\right)\right)}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   div-inv [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{2 - \left(z + \left(1.8611992721194026 + 1259.1392167224028 \cdot \left(-\color{blue}{z \cdot \frac{1}{676.5203681218851}}\right)\right)\right)}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   metadata-eval [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{2 - \left(z + \left(1.8611992721194026 + 1259.1392167224028 \cdot \left(-z \cdot \color{blue}{0.00147815209581367}\right)\right)\right)}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   distribute-rgt-neg-in [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{2 - \left(z + \left(1.8611992721194026 + 1259.1392167224028 \cdot \color{blue}{\left(z \cdot \left(-0.00147815209581367\right)\right)}\right)\right)}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   metadata-eval [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{2 - \left(z + \left(1.8611992721194026 + 1259.1392167224028 \cdot \left(z \cdot \color{blue}{-0.00147815209581367}\right)\right)\right)}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]

6. Simplified 99.3%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\color{blue}{\left(0.13880072788059747 - z \cdot -1.8611992721194026\right) - z}}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   Proof

   [Start] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{2 - \left(z + \left(1.8611992721194026 + 1259.1392167224028 \cdot \left(z \cdot -0.00147815209581367\right)\right)\right)}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   +-commutative [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{2 - \color{blue}{\left(\left(1.8611992721194026 + 1259.1392167224028 \cdot \left(z \cdot -0.00147815209581367\right)\right) + z\right)}}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   associate--r+ [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\color{blue}{\left(2 - \left(1.8611992721194026 + 1259.1392167224028 \cdot \left(z \cdot -0.00147815209581367\right)\right)\right) - z}}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   associate--r+ [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\color{blue}{\left(\left(2 - 1.8611992721194026\right) - 1259.1392167224028 \cdot \left(z \cdot -0.00147815209581367\right)\right)} - z}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   metadata-eval [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\left(\color{blue}{0.13880072788059747} - 1259.1392167224028 \cdot \left(z \cdot -0.00147815209581367\right)\right) - z}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   *-commutative [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\left(0.13880072788059747 - \color{blue}{\left(z \cdot -0.00147815209581367\right) \cdot 1259.1392167224028}\right) - z}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   associate-*l* [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\left(0.13880072788059747 - \color{blue}{z \cdot \left(-0.00147815209581367 \cdot 1259.1392167224028\right)}\right) - z}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   metadata-eval [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\left(0.13880072788059747 - z \cdot \color{blue}{-1.8611992721194026}\right) - z}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]

7. Applied egg-rr 99.3%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\color{blue}{\sqrt[3]{\left(\frac{0.13880072788059747 - \left(z \cdot -1.8611992721194026 + z\right)}{\left(0.00147815209581367 - z \cdot 0.00147815209581367\right) \cdot \left(2 - z\right)} \cdot \frac{0.13880072788059747 - \left(z \cdot -1.8611992721194026 + z\right)}{\left(0.00147815209581367 - z \cdot 0.00147815209581367\right) \cdot \left(2 - z\right)}\right) \cdot \frac{0.13880072788059747 - \left(z \cdot -1.8611992721194026 + z\right)}{\left(0.00147815209581367 - z \cdot 0.00147815209581367\right) \cdot \left(2 - z\right)}}} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   Proof

   [Start] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\left(0.13880072788059747 - z \cdot -1.8611992721194026\right) - z}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   add-cbrt-cube [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\color{blue}{\sqrt[3]{\left(\frac{\left(0.13880072788059747 - z \cdot -1.8611992721194026\right) - z}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} \cdot \frac{\left(0.13880072788059747 - z \cdot -1.8611992721194026\right) - z}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)}\right) \cdot \frac{\left(0.13880072788059747 - z \cdot -1.8611992721194026\right) - z}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)}}} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   associate--l- [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\sqrt[3]{\left(\frac{\color{blue}{0.13880072788059747 - \left(z \cdot -1.8611992721194026 + z\right)}}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)} \cdot \frac{\left(0.13880072788059747 - z \cdot -1.8611992721194026\right) - z}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)}\right) \cdot \frac{\left(0.13880072788059747 - z \cdot -1.8611992721194026\right) - z}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)}} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   div-inv [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\sqrt[3]{\left(\frac{0.13880072788059747 - \left(z \cdot -1.8611992721194026 + z\right)}{\left(0.00147815209581367 - \color{blue}{z \cdot \frac{1}{676.5203681218851}}\right) \cdot \left(2 - z\right)} \cdot \frac{\left(0.13880072788059747 - z \cdot -1.8611992721194026\right) - z}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)}\right) \cdot \frac{\left(0.13880072788059747 - z \cdot -1.8611992721194026\right) - z}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)}} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   metadata-eval [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\sqrt[3]{\left(\frac{0.13880072788059747 - \left(z \cdot -1.8611992721194026 + z\right)}{\left(0.00147815209581367 - z \cdot \color{blue}{0.00147815209581367}\right) \cdot \left(2 - z\right)} \cdot \frac{\left(0.13880072788059747 - z \cdot -1.8611992721194026\right) - z}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)}\right) \cdot \frac{\left(0.13880072788059747 - z \cdot -1.8611992721194026\right) - z}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)}} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   associate--l- [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\sqrt[3]{\left(\frac{0.13880072788059747 - \left(z \cdot -1.8611992721194026 + z\right)}{\left(0.00147815209581367 - z \cdot 0.00147815209581367\right) \cdot \left(2 - z\right)} \cdot \frac{\color{blue}{0.13880072788059747 - \left(z \cdot -1.8611992721194026 + z\right)}}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)}\right) \cdot \frac{\left(0.13880072788059747 - z \cdot -1.8611992721194026\right) - z}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)}} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   div-inv [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\sqrt[3]{\left(\frac{0.13880072788059747 - \left(z \cdot -1.8611992721194026 + z\right)}{\left(0.00147815209581367 - z \cdot 0.00147815209581367\right) \cdot \left(2 - z\right)} \cdot \frac{0.13880072788059747 - \left(z \cdot -1.8611992721194026 + z\right)}{\left(0.00147815209581367 - \color{blue}{z \cdot \frac{1}{676.5203681218851}}\right) \cdot \left(2 - z\right)}\right) \cdot \frac{\left(0.13880072788059747 - z \cdot -1.8611992721194026\right) - z}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)}} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
   metadata-eval [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\sqrt[3]{\left(\frac{0.13880072788059747 - \left(z \cdot -1.8611992721194026 + z\right)}{\left(0.00147815209581367 - z \cdot 0.00147815209581367\right) \cdot \left(2 - z\right)} \cdot \frac{0.13880072788059747 - \left(z \cdot -1.8611992721194026 + z\right)}{\left(0.00147815209581367 - z \cdot \color{blue}{0.00147815209581367}\right) \cdot \left(2 - z\right)}\right) \cdot \frac{\left(0.13880072788059747 - z \cdot -1.8611992721194026\right) - z}{\left(0.00147815209581367 - \frac{z}{676.5203681218851}\right) \cdot \left(2 - z\right)}} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]

8. Final simplification 99.3%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{\left(z + -1\right) + -6.5} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\sqrt[3]{\frac{0.13880072788059747 + \left(z \cdot 1.8611992721194026 - z\right)}{\left(0.00147815209581367 - z \cdot 0.00147815209581367\right) \cdot \left(2 - z\right)} \cdot \left(\frac{0.13880072788059747 + \left(z \cdot 1.8611992721194026 - z\right)}{\left(0.00147815209581367 - z \cdot 0.00147815209581367\right) \cdot \left(2 - z\right)} \cdot \frac{0.13880072788059747 + \left(z \cdot 1.8611992721194026 - z\right)}{\left(0.00147815209581367 - z \cdot 0.00147815209581367\right) \cdot \left(2 - z\right)}\right)} + \frac{1}{3 - z} \cdot 771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \frac{12.507343278686905}{\left(1 - z\right) + 4}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]

Alternatives

Alternative 1
Accuracy	99.3%
Cost	50368

\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{\left(z + -1\right) + -6.5} \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(0.9999999999998099 + \left(\frac{\left(0.13880072788059747 + z \cdot 1.8611992721194026\right) - z}{\left(2 - z\right) \cdot \left(0.00147815209581367 - \frac{z}{676.5203681218851}\right)} - \frac{1}{3 - z} \cdot -771.3234287776531\right)\right)\right) + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right) \]

Alternative 2
Accuracy	99.3%
Cost	49984

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

Alternative 3
Accuracy	99.3%
Cost	49088

\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)\right)\right)\right) \]

Alternative 4
Accuracy	97.8%
Cost	47424

\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) + \left(263.4062807184368 + z \cdot 436.9000215473151\right)\right)\right) \]

Alternative 5
Accuracy	96.7%
Cost	26688

\[\frac{\sqrt{\pi \cdot 2}}{\frac{z \cdot 0.0037967495627271876}{e^{z + -7.5} \cdot {\left(z + 7.5\right)}^{\left(z + 0.5\right)}}} \]

Alternative 6
Accuracy	96.8%
Cost	26688

\[\frac{\frac{\left(\sqrt{\pi \cdot 2} \cdot e^{z + -7.5}\right) \cdot {\left(z + 7.5\right)}^{\left(z + 0.5\right)}}{0.0037967495627271876}}{z} \]

Alternative 7
Accuracy	96.0%
Cost	26368

\[\frac{1}{z} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot \left(e^{-7.5} \cdot \sqrt{7.5}\right)\right) \cdot 263.3831869810514\right) \]

Reproduce

herbie shell --seed 2023131 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z less than 0.5"
  :precision binary64
  :pre (<= z 0.5)
  (* (/ PI (sin (* PI z))) (* (* (* (sqrt (* PI 2.0)) (pow (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5) (+ (- (- 1.0 z) 1.0) 0.5))) (exp (- (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- (- 1.0 z) 1.0) 1.0))) (/ -1259.1392167224028 (+ (- (- 1.0 z) 1.0) 2.0))) (/ 771.3234287776531 (+ (- (- 1.0 z) 1.0) 3.0))) (/ -176.6150291621406 (+ (- (- 1.0 z) 1.0) 4.0))) (/ 12.507343278686905 (+ (- (- 1.0 z) 1.0) 5.0))) (/ -0.13857109526572012 (+ (- (- 1.0 z) 1.0) 6.0))) (/ 9.984369578019572e-6 (+ (- (- 1.0 z) 1.0) 7.0))) (/ 1.5056327351493116e-7 (+ (- (- 1.0 z) 1.0) 8.0))))))