Jmat.Real.gamma, branch z less than 0.5

Average Accuracy: 97.3% → 99.3%

Time: 1.7min

Precision: binary64

Cost: 62016

\[z \leq 0.5\]

Math

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

↓

\[\left(\left(\left(\frac{\mathsf{fma}\left(z, -676.5203681218851, \mathsf{fma}\left(z, 1259.1392167224028, 93.9015195213674\right)\right)}{\left(z + -1\right) \cdot \left(z + -2\right)} + \left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{12.507343278686905}{5 - z}\right) - \left(\left(\frac{0.13857109526572012}{6 - z} + \left(\frac{-9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{-1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) + \frac{176.6150291621406}{4 - z}\right)\right)\right) \cdot e^{z + -7.5}\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{2 \cdot \pi}\right) \]

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
 (*
  (*
   (*
    (+
     (/
      (fma z -676.5203681218851 (fma z 1259.1392167224028 93.9015195213674))
      (* (+ z -1.0) (+ z -2.0)))
     (-
      (+
       (+ 0.9999999999998099 (/ 771.3234287776531 (- 3.0 z)))
       (/ 12.507343278686905 (- 5.0 z)))
      (+
       (+
        (/ 0.13857109526572012 (- 6.0 z))
        (+
         (/ -9.984369578019572e-6 (- 7.0 z))
         (/ -1.5056327351493116e-7 (- 8.0 z))))
       (/ 176.6150291621406 (- 4.0 z)))))
    (exp (+ z -7.5)))
   (/ PI (sin (* z PI))))
  (* (pow (- 7.5 z) (- 0.5 z)) (sqrt (* 2.0 PI)))))

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) {
	return ((((fma(z, -676.5203681218851, fma(z, 1259.1392167224028, 93.9015195213674)) / ((z + -1.0) * (z + -2.0))) + (((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + (12.507343278686905 / (5.0 - z))) - (((0.13857109526572012 / (6.0 - z)) + ((-9.984369578019572e-6 / (7.0 - z)) + (-1.5056327351493116e-7 / (8.0 - z)))) + (176.6150291621406 / (4.0 - z))))) * exp((z + -7.5))) * (((double) M_PI) / sin((z * ((double) M_PI))))) * (pow((7.5 - z), (0.5 - z)) * sqrt((2.0 * ((double) M_PI))));
}

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)
	return Float64(Float64(Float64(Float64(Float64(fma(z, -676.5203681218851, fma(z, 1259.1392167224028, 93.9015195213674)) / Float64(Float64(z + -1.0) * Float64(z + -2.0))) + Float64(Float64(Float64(0.9999999999998099 + Float64(771.3234287776531 / Float64(3.0 - z))) + Float64(12.507343278686905 / Float64(5.0 - z))) - Float64(Float64(Float64(0.13857109526572012 / Float64(6.0 - z)) + Float64(Float64(-9.984369578019572e-6 / Float64(7.0 - z)) + Float64(-1.5056327351493116e-7 / Float64(8.0 - z)))) + Float64(176.6150291621406 / Float64(4.0 - z))))) * exp(Float64(z + -7.5))) * Float64(pi / sin(Float64(z * pi)))) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * sqrt(Float64(2.0 * pi))))
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_] := N[(N[(N[(N[(N[(N[(z * -676.5203681218851 + N[(z * 1259.1392167224028 + 93.9015195213674), $MachinePrecision]), $MachinePrecision] / N[(N[(z + -1.0), $MachinePrecision] * N[(z + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.9999999999998099 + N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(-9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(Pi / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * Pi), $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 98.4%

   \[\leadsto \color{blue}{\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{2 - z} + \left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)} \]
   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 97.3%

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

   [Start] 98.4	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{2 - z} + \left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   *-un-lft-identity [=>] 98.4	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\color{blue}{1 \cdot \left(0.9999999999998099 + \left(\frac{-1259.1392167224028}{2 - z} + \left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right)\right)\right)} + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   +-commutative [=>] 98.4	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(1 \cdot \color{blue}{\left(\left(\frac{-1259.1392167224028}{2 - z} + \left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right)\right) + 0.9999999999998099\right)} + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   associate-+l+ [=>] 97.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(1 \cdot \color{blue}{\left(\frac{-1259.1392167224028}{2 - z} + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + 0.9999999999998099\right)\right)} + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]

4. Simplified 99.3%

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

   [Start] 97.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(1 \cdot \left(\frac{-1259.1392167224028}{2 - z} + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + 0.9999999999998099\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   *-lft-identity [=>] 97.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\color{blue}{\left(\frac{-1259.1392167224028}{2 - z} + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{771.3234287776531}{3 - z}\right) + 0.9999999999998099\right)\right)} + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   associate-+l+ [=>] 99.2	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} + \color{blue}{\left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + 0.9999999999998099\right)\right)}\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   associate-+r+ [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\color{blue}{\left(\left(\frac{-1259.1392167224028}{2 - z} + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + 0.9999999999998099\right)\right)} + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   +-commutative [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\frac{-1259.1392167224028}{2 - z} + \frac{676.5203681218851}{1 - z}\right) + \color{blue}{\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)}\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]

5. Applied egg-rr 99.3%

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

   [Start] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\frac{-1259.1392167224028}{2 - z} + \frac{676.5203681218851}{1 - z}\right) + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   frac-2neg [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\color{blue}{\frac{--1259.1392167224028}{-\left(2 - z\right)}} + \frac{676.5203681218851}{1 - z}\right) + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   frac-2neg [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\frac{--1259.1392167224028}{-\left(2 - z\right)} + \color{blue}{\frac{-676.5203681218851}{-\left(1 - z\right)}}\right) + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   frac-add [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\color{blue}{\frac{\left(--1259.1392167224028\right) \cdot \left(-\left(1 - z\right)\right) + \left(-\left(2 - z\right)\right) \cdot \left(-676.5203681218851\right)}{\left(-\left(2 - z\right)\right) \cdot \left(-\left(1 - z\right)\right)}} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   metadata-eval [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{\color{blue}{1259.1392167224028} \cdot \left(-\left(1 - z\right)\right) + \left(-\left(2 - z\right)\right) \cdot \left(-676.5203681218851\right)}{\left(-\left(2 - z\right)\right) \cdot \left(-\left(1 - z\right)\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   neg-sub0 [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{1259.1392167224028 \cdot \color{blue}{\left(0 - \left(1 - z\right)\right)} + \left(-\left(2 - z\right)\right) \cdot \left(-676.5203681218851\right)}{\left(-\left(2 - z\right)\right) \cdot \left(-\left(1 - z\right)\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   metadata-eval [<=] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{1259.1392167224028 \cdot \left(\color{blue}{\log 1} - \left(1 - z\right)\right) + \left(-\left(2 - z\right)\right) \cdot \left(-676.5203681218851\right)}{\left(-\left(2 - z\right)\right) \cdot \left(-\left(1 - z\right)\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   associate--r- [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{1259.1392167224028 \cdot \color{blue}{\left(\left(\log 1 - 1\right) + z\right)} + \left(-\left(2 - z\right)\right) \cdot \left(-676.5203681218851\right)}{\left(-\left(2 - z\right)\right) \cdot \left(-\left(1 - z\right)\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   metadata-eval [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{1259.1392167224028 \cdot \left(\left(\color{blue}{0} - 1\right) + z\right) + \left(-\left(2 - z\right)\right) \cdot \left(-676.5203681218851\right)}{\left(-\left(2 - z\right)\right) \cdot \left(-\left(1 - z\right)\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   metadata-eval [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{1259.1392167224028 \cdot \left(\color{blue}{-1} + z\right) + \left(-\left(2 - z\right)\right) \cdot \left(-676.5203681218851\right)}{\left(-\left(2 - z\right)\right) \cdot \left(-\left(1 - z\right)\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   neg-sub0 [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{1259.1392167224028 \cdot \left(-1 + z\right) + \color{blue}{\left(0 - \left(2 - z\right)\right)} \cdot \left(-676.5203681218851\right)}{\left(-\left(2 - z\right)\right) \cdot \left(-\left(1 - z\right)\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   metadata-eval [<=] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{1259.1392167224028 \cdot \left(-1 + z\right) + \left(\color{blue}{\log 1} - \left(2 - z\right)\right) \cdot \left(-676.5203681218851\right)}{\left(-\left(2 - z\right)\right) \cdot \left(-\left(1 - z\right)\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   associate--r- [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{1259.1392167224028 \cdot \left(-1 + z\right) + \color{blue}{\left(\left(\log 1 - 2\right) + z\right)} \cdot \left(-676.5203681218851\right)}{\left(-\left(2 - z\right)\right) \cdot \left(-\left(1 - z\right)\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   metadata-eval [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{1259.1392167224028 \cdot \left(-1 + z\right) + \left(\left(\color{blue}{0} - 2\right) + z\right) \cdot \left(-676.5203681218851\right)}{\left(-\left(2 - z\right)\right) \cdot \left(-\left(1 - z\right)\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   metadata-eval [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{1259.1392167224028 \cdot \left(-1 + z\right) + \left(\color{blue}{-2} + z\right) \cdot \left(-676.5203681218851\right)}{\left(-\left(2 - z\right)\right) \cdot \left(-\left(1 - z\right)\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   metadata-eval [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{1259.1392167224028 \cdot \left(-1 + z\right) + \left(-2 + z\right) \cdot \color{blue}{-676.5203681218851}}{\left(-\left(2 - z\right)\right) \cdot \left(-\left(1 - z\right)\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]

6. Simplified 99.3%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\color{blue}{\frac{z \cdot -676.5203681218851 + \left(z \cdot 1259.1392167224028 + 93.9015195213674\right)}{\left(z + -1\right) \cdot \left(z + -2\right)}} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   Proof

   [Start] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{1259.1392167224028 \cdot \left(-1 + z\right) + \left(-2 + z\right) \cdot -676.5203681218851}{\left(-2 + z\right) \cdot \left(-1 + z\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   *-commutative [<=] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{1259.1392167224028 \cdot \left(-1 + z\right) + \left(-2 + z\right) \cdot -676.5203681218851}{\color{blue}{\left(-1 + z\right) \cdot \left(-2 + z\right)}} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   *-commutative [<=] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{1259.1392167224028 \cdot \left(-1 + z\right) + \color{blue}{-676.5203681218851 \cdot \left(-2 + z\right)}}{\left(-1 + z\right) \cdot \left(-2 + z\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   distribute-lft-in [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{1259.1392167224028 \cdot \left(-1 + z\right) + \color{blue}{\left(-676.5203681218851 \cdot -2 + -676.5203681218851 \cdot z\right)}}{\left(-1 + z\right) \cdot \left(-2 + z\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   associate-+r+ [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{\color{blue}{\left(1259.1392167224028 \cdot \left(-1 + z\right) + -676.5203681218851 \cdot -2\right) + -676.5203681218851 \cdot z}}{\left(-1 + z\right) \cdot \left(-2 + z\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   +-commutative [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{\color{blue}{-676.5203681218851 \cdot z + \left(1259.1392167224028 \cdot \left(-1 + z\right) + -676.5203681218851 \cdot -2\right)}}{\left(-1 + z\right) \cdot \left(-2 + z\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   *-commutative [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{\color{blue}{z \cdot -676.5203681218851} + \left(1259.1392167224028 \cdot \left(-1 + z\right) + -676.5203681218851 \cdot -2\right)}{\left(-1 + z\right) \cdot \left(-2 + z\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   distribute-rgt-in [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{z \cdot -676.5203681218851 + \left(\color{blue}{\left(-1 \cdot 1259.1392167224028 + z \cdot 1259.1392167224028\right)} + -676.5203681218851 \cdot -2\right)}{\left(-1 + z\right) \cdot \left(-2 + z\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   metadata-eval [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{z \cdot -676.5203681218851 + \left(\left(\color{blue}{-1259.1392167224028} + z \cdot 1259.1392167224028\right) + -676.5203681218851 \cdot -2\right)}{\left(-1 + z\right) \cdot \left(-2 + z\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   +-commutative [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{z \cdot -676.5203681218851 + \left(\color{blue}{\left(z \cdot 1259.1392167224028 + -1259.1392167224028\right)} + -676.5203681218851 \cdot -2\right)}{\left(-1 + z\right) \cdot \left(-2 + z\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   associate-+l+ [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{z \cdot -676.5203681218851 + \color{blue}{\left(z \cdot 1259.1392167224028 + \left(-1259.1392167224028 + -676.5203681218851 \cdot -2\right)\right)}}{\left(-1 + z\right) \cdot \left(-2 + z\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   metadata-eval [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{z \cdot -676.5203681218851 + \left(z \cdot 1259.1392167224028 + \left(-1259.1392167224028 + \color{blue}{1353.0407362437702}\right)\right)}{\left(-1 + z\right) \cdot \left(-2 + z\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   metadata-eval [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{z \cdot -676.5203681218851 + \left(z \cdot 1259.1392167224028 + \color{blue}{93.9015195213674}\right)}{\left(-1 + z\right) \cdot \left(-2 + z\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   +-commutative [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{z \cdot -676.5203681218851 + \left(z \cdot 1259.1392167224028 + 93.9015195213674\right)}{\color{blue}{\left(z + -1\right)} \cdot \left(-2 + z\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]
   +-commutative [=>] 99.3	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{z \cdot -676.5203681218851 + \left(z \cdot 1259.1392167224028 + 93.9015195213674\right)}{\left(z + -1\right) \cdot \color{blue}{\left(z + -2\right)}} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]

7. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{\pi \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{\mathsf{fma}\left(z, -676.5203681218851, \mathsf{fma}\left(z, 1259.1392167224028, 93.9015195213674\right)\right)}{\left(z + -1\right) \cdot \left(z + -2\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)\right)}{\sin \left(\pi \cdot z\right)}} \]
   Proof

   [Start] 99.3

   \[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{z \cdot -676.5203681218851 + \left(z \cdot 1259.1392167224028 + 93.9015195213674\right)}{\left(z + -1\right) \cdot \left(z + -2\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \]

   associate-*l/ [=>] 98.8

   \[ \color{blue}{\frac{\pi \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{z \cdot -676.5203681218851 + \left(z \cdot 1259.1392167224028 + 93.9015195213674\right)}{\left(z + -1\right) \cdot \left(z + -2\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)}{\sin \left(\pi \cdot z\right)}} \]

8. Simplified 99.3%

   \[\leadsto \color{blue}{\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \left(\left(e^{-7.5 + z} \cdot \left(\frac{\mathsf{fma}\left(z, -676.5203681218851, \mathsf{fma}\left(z, 1259.1392167224028, 93.9015195213674\right)\right)}{\left(z + -1\right) \cdot \left(z + -2\right)} + \left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{12.507343278686905}{5 - z}\right) + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right)} \]
   Proof

   [Start] 98.8	\[ \frac{\pi \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{\mathsf{fma}\left(z, -676.5203681218851, \mathsf{fma}\left(z, 1259.1392167224028, 93.9015195213674\right)\right)}{\left(z + -1\right) \cdot \left(z + -2\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)\right)}{\sin \left(\pi \cdot z\right)} \]
   associate-*l/ [<=] 99.3	\[ \color{blue}{\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{\mathsf{fma}\left(z, -676.5203681218851, \mathsf{fma}\left(z, 1259.1392167224028, 93.9015195213674\right)\right)}{\left(z + -1\right) \cdot \left(z + -2\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)\right)} \]
   *-commutative [=>] 99.3	\[ \color{blue}{\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{\mathsf{fma}\left(z, -676.5203681218851, \mathsf{fma}\left(z, 1259.1392167224028, 93.9015195213674\right)\right)}{\left(z + -1\right) \cdot \left(z + -2\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}} \]
   associate-*l* [=>] 99.3	\[ \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(e^{z + -7.5} \cdot \left(\left(\frac{\mathsf{fma}\left(z, -676.5203681218851, \mathsf{fma}\left(z, 1259.1392167224028, 93.9015195213674\right)\right)}{\left(z + -1\right) \cdot \left(z + -2\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)} \]

9. Final simplification 99.3%

   \[\leadsto \left(\left(\left(\frac{\mathsf{fma}\left(z, -676.5203681218851, \mathsf{fma}\left(z, 1259.1392167224028, 93.9015195213674\right)\right)}{\left(z + -1\right) \cdot \left(z + -2\right)} + \left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{12.507343278686905}{5 - z}\right) - \left(\left(\frac{0.13857109526572012}{6 - z} + \left(\frac{-9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{-1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) + \frac{176.6150291621406}{4 - z}\right)\right)\right) \cdot e^{z + -7.5}\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{2 \cdot \pi}\right) \]

Alternatives

Alternative 1
Accuracy	99.3%
Cost	49472

\[\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{z \cdot -676.5203681218851 + \left(93.9015195213674 - z \cdot -1259.1392167224028\right)}{\left(z + -1\right) \cdot \left(z + -2\right)}\right) - \left(\left(\frac{-9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{-1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{-12.507343278686905}{5 - z} + \left(\frac{0.13857109526572012}{6 - z} + \frac{176.6150291621406}{4 - z}\right)\right)\right)\right)\right)\right) \]

Alternative 2
Accuracy	98.4%
Cost	49088

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

Alternative 3
Accuracy	98.4%
Cost	49088

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

Alternative 4
Accuracy	99.3%
Cost	49088

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

Alternative 5
Accuracy	98.2%
Cost	48704

\[\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{z \cdot -676.5203681218851 + \left(93.9015195213674 - z \cdot -1259.1392167224028\right)}{\left(z + -1\right) \cdot \left(z + -2\right)}\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(z \cdot -10.541994788577025 + -41.67538381734206\right)\right)\right)\right)\right) \]

Alternative 6
Accuracy	98.2%
Cost	48320

\[\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(z \cdot -10.541994788577025 + -41.67538381734206\right)\right) - \left(\left(-0.9999999999998099 + \frac{-771.3234287776531}{3 - z}\right) + \left(\frac{1259.1392167224028}{2 - z} + \frac{-676.5203681218851}{1 - z}\right)\right)\right)\right)\right) \]

Alternative 7
Accuracy	98.3%
Cost	47680

\[\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \left(\left(263.4062807184368 - z \cdot \left(z \cdot -545.0359493463282 + -436.9000215473151\right)\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right)\right)\right)\right) \]

Alternative 8
Accuracy	97.8%
Cost	47424

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

Alternative 9
Accuracy	97.8%
Cost	47424

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

Alternative 10
Accuracy	96.2%
Cost	32640

\[263.3831869810514 \cdot \left(\sqrt{\pi} \cdot \left(\frac{\sqrt{2}}{z} \cdot \sqrt{7.5 \cdot e^{-15}}\right)\right) \]

Alternative 11
Accuracy	96.9%
Cost	32640

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

Alternative 12
Accuracy	96.9%
Cost	32640

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

Alternative 13
Accuracy	97.1%
Cost	32640

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

Alternative 14
Accuracy	23.9%
Cost	19968

\[\sqrt{\pi \cdot \left(\left(-7.5 \cdot \left(e^{-15} \cdot \frac{-2}{z \cdot z}\right)\right) \cdot 69370.70318429549\right)} \]

Alternative 15
Accuracy	23.8%
Cost	19840

\[\sqrt{\frac{e^{-15} \cdot 15}{z \cdot z} \cdot \left(\pi \cdot 69370.70318429549\right)} \]

Reproduce

herbie shell --seed 2023143 
(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))))))