Jmat.Real.gamma, branch z greater than 0.5

Average Error: 3.9 → 3.5

Time: 31.1s

Precision: binary64

Cost: 43968

\[z > 0.5\]

Math

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

↓

\[\left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot \frac{\frac{1}{e^{z + 6.5}}}{{\left(z + 6.5\right)}^{\left(0.5 - z\right)}}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\frac{\left(-1258.1392167221638 + \frac{394153.5179299711}{z}\right) + \mathsf{fma}\left(z, 0.9999999999996197, \frac{-457679.80848377093}{z \cdot z}\right)}{\left(0.9999999999998099 - \frac{676.5203681218851}{z}\right) \cdot \left(1 + z\right)} + \frac{771.3234287776531}{\left(z + -1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z + -1\right) + 4}\right) + \frac{12.507343278686905}{\left(z + -1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z + -1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z + -1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8}\right) \]

FPCore

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

↓

(FPCore (z)
 :precision binary64
 (*
  (*
   (sqrt 2.0)
   (* (sqrt PI) (/ (/ 1.0 (exp (+ z 6.5))) (pow (+ z 6.5) (- 0.5 z)))))
  (+
   (+
    (+
     (+
      (+
       (+
        (/
         (+
          (+ -1258.1392167221638 (/ 394153.5179299711 z))
          (fma z 0.9999999999996197 (/ -457679.80848377093 (* z z))))
         (* (- 0.9999999999998099 (/ 676.5203681218851 z)) (+ 1.0 z)))
        (/ 771.3234287776531 (+ (+ z -1.0) 3.0)))
       (/ -176.6150291621406 (+ (+ z -1.0) 4.0)))
      (/ 12.507343278686905 (+ (+ z -1.0) 5.0)))
     (/ -0.13857109526572012 (+ (+ z -1.0) 6.0)))
    (/ 9.984369578019572e-6 (+ (+ z -1.0) 7.0)))
   (/ 1.5056327351493116e-7 (+ (+ z -1.0) 8.0)))))

C

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

↓

double code(double z) {
	return (sqrt(2.0) * (sqrt(((double) M_PI)) * ((1.0 / exp((z + 6.5))) / pow((z + 6.5), (0.5 - z))))) * (((((((((-1258.1392167221638 + (394153.5179299711 / z)) + fma(z, 0.9999999999996197, (-457679.80848377093 / (z * z)))) / ((0.9999999999998099 - (676.5203681218851 / z)) * (1.0 + z))) + (771.3234287776531 / ((z + -1.0) + 3.0))) + (-176.6150291621406 / ((z + -1.0) + 4.0))) + (12.507343278686905 / ((z + -1.0) + 5.0))) + (-0.13857109526572012 / ((z + -1.0) + 6.0))) + (9.984369578019572e-6 / ((z + -1.0) + 7.0))) + (1.5056327351493116e-7 / ((z + -1.0) + 8.0)));
}

Julia

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

↓

function code(z)
	return Float64(Float64(sqrt(2.0) * Float64(sqrt(pi) * Float64(Float64(1.0 / exp(Float64(z + 6.5))) / (Float64(z + 6.5) ^ Float64(0.5 - z))))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-1258.1392167221638 + Float64(394153.5179299711 / z)) + fma(z, 0.9999999999996197, Float64(-457679.80848377093 / Float64(z * z)))) / Float64(Float64(0.9999999999998099 - Float64(676.5203681218851 / z)) * Float64(1.0 + z))) + Float64(771.3234287776531 / Float64(Float64(z + -1.0) + 3.0))) + Float64(-176.6150291621406 / Float64(Float64(z + -1.0) + 4.0))) + Float64(12.507343278686905 / Float64(Float64(z + -1.0) + 5.0))) + Float64(-0.13857109526572012 / Float64(Float64(z + -1.0) + 6.0))) + Float64(9.984369578019572e-6 / Float64(Float64(z + -1.0) + 7.0))) + Float64(1.5056327351493116e-7 / Float64(Float64(z + -1.0) + 8.0))))
end

Wolfram

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

↓

code[z_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[(1.0 / N[Exp[N[(z + 6.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Power[N[(z + 6.5), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(N[(-1258.1392167221638 + N[(394153.5179299711 / z), $MachinePrecision]), $MachinePrecision] + N[(z * 0.9999999999996197 + N[(-457679.80848377093 / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.9999999999998099 - N[(676.5203681218851 / z), $MachinePrecision]), $MachinePrecision] * N[(1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(N[(z + -1.0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(z + -1.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(N[(z + -1.0), $MachinePrecision] + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(z + -1.0), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(N[(z + -1.0), $MachinePrecision] + 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(z + -1.0), $MachinePrecision] + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.9

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

2. Applied egg-rr 4.1

   \[\leadsto \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\frac{\left(0.9999999999996197 - \frac{457679.80848377093}{z \cdot z}\right) \cdot \left(z - -1\right) + \left(0.9999999999998099 - \frac{676.5203681218851}{z}\right) \cdot -1259.1392167224028}{\left(0.9999999999998099 - \frac{676.5203681218851}{z}\right) \cdot \left(z - -1\right)}} + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]

3. Taylor expanded in z around 0 3.8

   \[\leadsto \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\frac{\color{blue}{\left(0.9999999999996197 \cdot z + 394153.5179299711 \cdot \frac{1}{z}\right) - \left(1258.1392167221638 + 457679.80848377093 \cdot \frac{1}{{z}^{2}}\right)}}{\left(0.9999999999998099 - \frac{676.5203681218851}{z}\right) \cdot \left(z - -1\right)} + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]

4. Simplified 3.7

   \[\leadsto \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\frac{\color{blue}{\left(-1258.1392167221638 - \frac{-394153.5179299711}{z}\right) + \mathsf{fma}\left(z, 0.9999999999996197, \frac{-457679.80848377093}{z \cdot z}\right)}}{\left(0.9999999999998099 - \frac{676.5203681218851}{z}\right) \cdot \left(z - -1\right)} + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
   Proof
   (+.f64 (-.f64 -125813921672216381783490107119158199/100000000000000000000000000000000 (/.f64 -39415351792997107601807255991627/100000000000000000000000000 z)) (fma.f64 z 99999999999961980000000003613801/100000000000000000000000000000000 (/.f64 -45767980848377092942628957760201/100000000000000000000000000 (*.f64 z z)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (-.f64 (Rewrite<= metadata-eval (neg.f64 125813921672216381783490107119158199/100000000000000000000000000000000)) (/.f64 -39415351792997107601807255991627/100000000000000000000000000 z)) (fma.f64 z 99999999999961980000000003613801/100000000000000000000000000000000 (/.f64 -45767980848377092942628957760201/100000000000000000000000000 (*.f64 z z)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (-.f64 (neg.f64 125813921672216381783490107119158199/100000000000000000000000000000000) (/.f64 (Rewrite<= metadata-eval (*.f64 -39415351792997107601807255991627/100000000000000000000000000 1)) z)) (fma.f64 z 99999999999961980000000003613801/100000000000000000000000000000000 (/.f64 -45767980848377092942628957760201/100000000000000000000000000 (*.f64 z z)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (-.f64 (neg.f64 125813921672216381783490107119158199/100000000000000000000000000000000) (/.f64 (*.f64 (Rewrite<= metadata-eval (neg.f64 39415351792997107601807255991627/100000000000000000000000000)) 1) z)) (fma.f64 z 99999999999961980000000003613801/100000000000000000000000000000000 (/.f64 -45767980848377092942628957760201/100000000000000000000000000 (*.f64 z z)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (-.f64 (neg.f64 125813921672216381783490107119158199/100000000000000000000000000000000) (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 39415351792997107601807255991627/100000000000000000000000000) (/.f64 1 z)))) (fma.f64 z 99999999999961980000000003613801/100000000000000000000000000000000 (/.f64 -45767980848377092942628957760201/100000000000000000000000000 (*.f64 z z)))): 31 points increase in error, 37 points decrease in error
   (+.f64 (-.f64 (neg.f64 125813921672216381783490107119158199/100000000000000000000000000000000) (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 39415351792997107601807255991627/100000000000000000000000000 (/.f64 1 z))))) (fma.f64 z 99999999999961980000000003613801/100000000000000000000000000000000 (/.f64 -45767980848377092942628957760201/100000000000000000000000000 (*.f64 z z)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite=> sub-neg_binary64 (+.f64 (neg.f64 125813921672216381783490107119158199/100000000000000000000000000000000) (neg.f64 (neg.f64 (*.f64 39415351792997107601807255991627/100000000000000000000000000 (/.f64 1 z)))))) (fma.f64 z 99999999999961980000000003613801/100000000000000000000000000000000 (/.f64 -45767980848377092942628957760201/100000000000000000000000000 (*.f64 z z)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (neg.f64 125813921672216381783490107119158199/100000000000000000000000000000000) (Rewrite=> remove-double-neg_binary64 (*.f64 39415351792997107601807255991627/100000000000000000000000000 (/.f64 1 z)))) (fma.f64 z 99999999999961980000000003613801/100000000000000000000000000000000 (/.f64 -45767980848377092942628957760201/100000000000000000000000000 (*.f64 z z)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 39415351792997107601807255991627/100000000000000000000000000 (/.f64 1 z)) (neg.f64 125813921672216381783490107119158199/100000000000000000000000000000000))) (fma.f64 z 99999999999961980000000003613801/100000000000000000000000000000000 (/.f64 -45767980848377092942628957760201/100000000000000000000000000 (*.f64 z z)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 39415351792997107601807255991627/100000000000000000000000000 (/.f64 1 z)) 125813921672216381783490107119158199/100000000000000000000000000000000)) (fma.f64 z 99999999999961980000000003613801/100000000000000000000000000000000 (/.f64 -45767980848377092942628957760201/100000000000000000000000000 (*.f64 z z)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (-.f64 (*.f64 39415351792997107601807255991627/100000000000000000000000000 (/.f64 1 z)) 125813921672216381783490107119158199/100000000000000000000000000000000) (fma.f64 z 99999999999961980000000003613801/100000000000000000000000000000000 (/.f64 -45767980848377092942628957760201/100000000000000000000000000 (Rewrite<= unpow2_binary64 (pow.f64 z 2))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (-.f64 (*.f64 39415351792997107601807255991627/100000000000000000000000000 (/.f64 1 z)) 125813921672216381783490107119158199/100000000000000000000000000000000) (fma.f64 z 99999999999961980000000003613801/100000000000000000000000000000000 (/.f64 (Rewrite<= metadata-eval (neg.f64 45767980848377092942628957760201/100000000000000000000000000)) (pow.f64 z 2)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (-.f64 (*.f64 39415351792997107601807255991627/100000000000000000000000000 (/.f64 1 z)) 125813921672216381783490107119158199/100000000000000000000000000000000) (fma.f64 z 99999999999961980000000003613801/100000000000000000000000000000000 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 45767980848377092942628957760201/100000000000000000000000000 (pow.f64 z 2)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (-.f64 (*.f64 39415351792997107601807255991627/100000000000000000000000000 (/.f64 1 z)) 125813921672216381783490107119158199/100000000000000000000000000000000) (fma.f64 z 99999999999961980000000003613801/100000000000000000000000000000000 (neg.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 45767980848377092942628957760201/100000000000000000000000000 1)) (pow.f64 z 2))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (-.f64 (*.f64 39415351792997107601807255991627/100000000000000000000000000 (/.f64 1 z)) 125813921672216381783490107119158199/100000000000000000000000000000000) (fma.f64 z 99999999999961980000000003613801/100000000000000000000000000000000 (neg.f64 (Rewrite<= associate-*r/_binary64 (*.f64 45767980848377092942628957760201/100000000000000000000000000 (/.f64 1 (pow.f64 z 2))))))): 16 points increase in error, 15 points decrease in error
   (+.f64 (-.f64 (*.f64 39415351792997107601807255991627/100000000000000000000000000 (/.f64 1 z)) 125813921672216381783490107119158199/100000000000000000000000000000000) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 z 99999999999961980000000003613801/100000000000000000000000000000000) (*.f64 45767980848377092942628957760201/100000000000000000000000000 (/.f64 1 (pow.f64 z 2)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (-.f64 (*.f64 39415351792997107601807255991627/100000000000000000000000000 (/.f64 1 z)) 125813921672216381783490107119158199/100000000000000000000000000000000) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 99999999999961980000000003613801/100000000000000000000000000000000 z)) (*.f64 45767980848377092942628957760201/100000000000000000000000000 (/.f64 1 (pow.f64 z 2))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (-.f64 (*.f64 39415351792997107601807255991627/100000000000000000000000000 (/.f64 1 z)) 125813921672216381783490107119158199/100000000000000000000000000000000) (*.f64 99999999999961980000000003613801/100000000000000000000000000000000 z)) (*.f64 45767980848377092942628957760201/100000000000000000000000000 (/.f64 1 (pow.f64 z 2))))): 30 points increase in error, 37 points decrease in error
   (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 99999999999961980000000003613801/100000000000000000000000000000000 z) (-.f64 (*.f64 39415351792997107601807255991627/100000000000000000000000000 (/.f64 1 z)) 125813921672216381783490107119158199/100000000000000000000000000000000))) (*.f64 45767980848377092942628957760201/100000000000000000000000000 (/.f64 1 (pow.f64 z 2)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (*.f64 99999999999961980000000003613801/100000000000000000000000000000000 z) (*.f64 39415351792997107601807255991627/100000000000000000000000000 (/.f64 1 z))) 125813921672216381783490107119158199/100000000000000000000000000000000)) (*.f64 45767980848377092942628957760201/100000000000000000000000000 (/.f64 1 (pow.f64 z 2)))): 10 points increase in error, 11 points decrease in error
   (Rewrite<= associate--r+_binary64 (-.f64 (+.f64 (*.f64 99999999999961980000000003613801/100000000000000000000000000000000 z) (*.f64 39415351792997107601807255991627/100000000000000000000000000 (/.f64 1 z))) (+.f64 125813921672216381783490107119158199/100000000000000000000000000000000 (*.f64 45767980848377092942628957760201/100000000000000000000000000 (/.f64 1 (pow.f64 z 2)))))): 19 points increase in error, 17 points decrease in error

5. Taylor expanded in z around inf 3.7

   \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \left({\left(6.5 + z\right)}^{\left(z - 0.5\right)} \cdot e^{-\left(6.5 + z\right)}\right)\right) \cdot \sqrt{\pi}\right)} \cdot \left(\left(\left(\left(\left(\left(\frac{\left(-1258.1392167221638 - \frac{-394153.5179299711}{z}\right) + \mathsf{fma}\left(z, 0.9999999999996197, \frac{-457679.80848377093}{z \cdot z}\right)}{\left(0.9999999999998099 - \frac{676.5203681218851}{z}\right) \cdot \left(z - -1\right)} + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]

6. Simplified 3.5

   \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot \frac{e^{-6.5 - z}}{{\left(z - -6.5\right)}^{\left(0.5 - z\right)}}\right)\right)} \cdot \left(\left(\left(\left(\left(\left(\frac{\left(-1258.1392167221638 - \frac{-394153.5179299711}{z}\right) + \mathsf{fma}\left(z, 0.9999999999996197, \frac{-457679.80848377093}{z \cdot z}\right)}{\left(0.9999999999998099 - \frac{676.5203681218851}{z}\right) \cdot \left(z - -1\right)} + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
   Proof
   (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (-.f64 -13/2 z)) (pow.f64 (-.f64 z -13/2) (-.f64 1/2 z))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (Rewrite<= unsub-neg_binary64 (+.f64 -13/2 (neg.f64 z)))) (pow.f64 (-.f64 z -13/2) (-.f64 1/2 z))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (+.f64 (Rewrite<= metadata-eval (neg.f64 13/2)) (neg.f64 z))) (pow.f64 (-.f64 z -13/2) (-.f64 1/2 z))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 13/2 z)))) (pow.f64 (-.f64 z -13/2) (-.f64 1/2 z))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (neg.f64 (Rewrite=> +-commutative_binary64 (+.f64 z 13/2)))) (pow.f64 (-.f64 z -13/2) (-.f64 1/2 z))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (Rewrite=> distribute-neg-in_binary64 (+.f64 (neg.f64 z) (neg.f64 13/2)))) (pow.f64 (-.f64 z -13/2) (-.f64 1/2 z))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (+.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 z)) (neg.f64 13/2))) (pow.f64 (-.f64 z -13/2) (-.f64 1/2 z))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 -1 z) 13/2))) (pow.f64 (-.f64 z -13/2) (-.f64 1/2 z))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (pow.f64 (-.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 z)) -13/2) (-.f64 1/2 z))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (pow.f64 (Rewrite=> fma-neg_binary64 (fma.f64 1 z (neg.f64 -13/2))) (-.f64 1/2 z))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (pow.f64 (fma.f64 1 z (Rewrite=> metadata-eval 13/2)) (-.f64 1/2 z))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (pow.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 1 z) 13/2)) (-.f64 1/2 z))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (pow.f64 (+.f64 (Rewrite=> *-lft-identity_binary64 z) 13/2) (-.f64 1/2 z))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (pow.f64 (Rewrite<= +-commutative_binary64 (+.f64 13/2 z)) (-.f64 1/2 z))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (pow.f64 (+.f64 13/2 z) (Rewrite<= unsub-neg_binary64 (+.f64 1/2 (neg.f64 z))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (pow.f64 (+.f64 13/2 z) (+.f64 1/2 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 z))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (pow.f64 (+.f64 13/2 z) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 z) 1/2)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (+.f64 13/2 z)) (+.f64 (*.f64 -1 z) 1/2))))))): 95 points increase in error, 103 points decrease in error
   (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (exp.f64 (*.f64 (log.f64 (+.f64 13/2 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 z))))) (+.f64 (*.f64 -1 z) 1/2)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (exp.f64 (*.f64 (log.f64 (+.f64 13/2 (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 z))))) (+.f64 (*.f64 -1 z) 1/2)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (exp.f64 (*.f64 (log.f64 (Rewrite<= sub-neg_binary64 (-.f64 13/2 (*.f64 -1 z)))) (+.f64 (*.f64 -1 z) 1/2)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (Rewrite=> div-exp_binary64 (exp.f64 (-.f64 (-.f64 (*.f64 -1 z) 13/2) (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (+.f64 (*.f64 -1 z) 1/2))))))): 79 points increase in error, 71 points decrease in error
   (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (exp.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (-.f64 (*.f64 -1 z) 13/2) (neg.f64 (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (+.f64 (*.f64 -1 z) 1/2)))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (exp.f64 (+.f64 (-.f64 (*.f64 -1 z) 13/2) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (+.f64 (*.f64 -1 z) 1/2)))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (Rewrite<= prod-exp_binary64 (*.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (exp.f64 (*.f64 -1 (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (+.f64 (*.f64 -1 z) 1/2)))))))): 74 points increase in error, 70 points decrease in error
   (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (Rewrite<= *-commutative_binary64 (*.f64 (exp.f64 (*.f64 -1 (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (+.f64 (*.f64 -1 z) 1/2)))) (exp.f64 (-.f64 (*.f64 -1 z) 13/2)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 2) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (exp.f64 (*.f64 -1 (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (+.f64 (*.f64 -1 z) 1/2)))) (exp.f64 (-.f64 (*.f64 -1 z) 13/2))) (sqrt.f64 (PI.f64))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 2) (Rewrite=> associate-*l*_binary64 (*.f64 (exp.f64 (*.f64 -1 (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (+.f64 (*.f64 -1 z) 1/2)))) (*.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (sqrt.f64 (PI.f64)))))): 31 points increase in error, 33 points decrease in error
   (*.f64 (sqrt.f64 2) (*.f64 (exp.f64 (*.f64 -1 (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (+.f64 (*.f64 -1 z) 1/2)))) (*.f64 (exp.f64 (Rewrite=> sub-neg_binary64 (+.f64 (*.f64 -1 z) (neg.f64 13/2)))) (sqrt.f64 (PI.f64))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 2) (*.f64 (exp.f64 (*.f64 -1 (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (+.f64 (*.f64 -1 z) 1/2)))) (*.f64 (exp.f64 (+.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 z)) (neg.f64 13/2))) (sqrt.f64 (PI.f64))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 2) (*.f64 (exp.f64 (*.f64 -1 (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (+.f64 (*.f64 -1 z) 1/2)))) (*.f64 (exp.f64 (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 z 13/2)))) (sqrt.f64 (PI.f64))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 2) (*.f64 (exp.f64 (*.f64 -1 (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (+.f64 (*.f64 -1 z) 1/2)))) (*.f64 (exp.f64 (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 13/2 z)))) (sqrt.f64 (PI.f64))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sqrt.f64 2) (exp.f64 (*.f64 -1 (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (+.f64 (*.f64 -1 z) 1/2))))) (*.f64 (exp.f64 (neg.f64 (+.f64 13/2 z))) (sqrt.f64 (PI.f64))))): 47 points increase in error, 38 points decrease in error
   (*.f64 (Rewrite=> *-commutative_binary64 (*.f64 (exp.f64 (*.f64 -1 (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (+.f64 (*.f64 -1 z) 1/2)))) (sqrt.f64 2))) (*.f64 (exp.f64 (neg.f64 (+.f64 13/2 z))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (exp.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (+.f64 (*.f64 -1 z) 1/2))))) (sqrt.f64 2)) (*.f64 (exp.f64 (neg.f64 (+.f64 13/2 z))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (exp.f64 (Rewrite=> distribute-rgt-neg-in_binary64 (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (neg.f64 (+.f64 (*.f64 -1 z) 1/2))))) (sqrt.f64 2)) (*.f64 (exp.f64 (neg.f64 (+.f64 13/2 z))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (exp.f64 (*.f64 (log.f64 (Rewrite=> sub-neg_binary64 (+.f64 13/2 (neg.f64 (*.f64 -1 z))))) (neg.f64 (+.f64 (*.f64 -1 z) 1/2)))) (sqrt.f64 2)) (*.f64 (exp.f64 (neg.f64 (+.f64 13/2 z))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (exp.f64 (*.f64 (log.f64 (+.f64 13/2 (neg.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 z))))) (neg.f64 (+.f64 (*.f64 -1 z) 1/2)))) (sqrt.f64 2)) (*.f64 (exp.f64 (neg.f64 (+.f64 13/2 z))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (exp.f64 (*.f64 (log.f64 (+.f64 13/2 (Rewrite=> remove-double-neg_binary64 z))) (neg.f64 (+.f64 (*.f64 -1 z) 1/2)))) (sqrt.f64 2)) (*.f64 (exp.f64 (neg.f64 (+.f64 13/2 z))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (Rewrite=> exp-to-pow_binary64 (pow.f64 (+.f64 13/2 z) (neg.f64 (+.f64 (*.f64 -1 z) 1/2)))) (sqrt.f64 2)) (*.f64 (exp.f64 (neg.f64 (+.f64 13/2 z))) (sqrt.f64 (PI.f64)))): 103 points increase in error, 98 points decrease in error
   (*.f64 (*.f64 (pow.f64 (+.f64 13/2 z) (Rewrite=> distribute-neg-in_binary64 (+.f64 (neg.f64 (*.f64 -1 z)) (neg.f64 1/2)))) (sqrt.f64 2)) (*.f64 (exp.f64 (neg.f64 (+.f64 13/2 z))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (pow.f64 (+.f64 13/2 z) (+.f64 (neg.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 z))) (neg.f64 1/2))) (sqrt.f64 2)) (*.f64 (exp.f64 (neg.f64 (+.f64 13/2 z))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (pow.f64 (+.f64 13/2 z) (+.f64 (Rewrite=> remove-double-neg_binary64 z) (neg.f64 1/2))) (sqrt.f64 2)) (*.f64 (exp.f64 (neg.f64 (+.f64 13/2 z))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (pow.f64 (+.f64 13/2 z) (Rewrite<= sub-neg_binary64 (-.f64 z 1/2))) (sqrt.f64 2)) (*.f64 (exp.f64 (neg.f64 (+.f64 13/2 z))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 2) (pow.f64 (+.f64 13/2 z) (-.f64 z 1/2)))) (*.f64 (exp.f64 (neg.f64 (+.f64 13/2 z))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (pow.f64 (+.f64 13/2 z) (-.f64 z 1/2))) (exp.f64 (neg.f64 (+.f64 13/2 z)))) (sqrt.f64 (PI.f64)))): 47 points increase in error, 40 points decrease in error
   (*.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (sqrt.f64 2) (*.f64 (pow.f64 (+.f64 13/2 z) (-.f64 z 1/2)) (exp.f64 (neg.f64 (+.f64 13/2 z)))))) (sqrt.f64 (PI.f64))): 34 points increase in error, 41 points decrease in error

7. Applied egg-rr 3.5

   \[\leadsto \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot \frac{\color{blue}{\frac{1}{e^{z + 6.5}}}}{{\left(z - -6.5\right)}^{\left(0.5 - z\right)}}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\frac{\left(-1258.1392167221638 - \frac{-394153.5179299711}{z}\right) + \mathsf{fma}\left(z, 0.9999999999996197, \frac{-457679.80848377093}{z \cdot z}\right)}{\left(0.9999999999998099 - \frac{676.5203681218851}{z}\right) \cdot \left(z - -1\right)} + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]

8. Final simplification 3.5

   \[\leadsto \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot \frac{\frac{1}{e^{z + 6.5}}}{{\left(z + 6.5\right)}^{\left(0.5 - z\right)}}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\frac{\left(-1258.1392167221638 + \frac{394153.5179299711}{z}\right) + \mathsf{fma}\left(z, 0.9999999999996197, \frac{-457679.80848377093}{z \cdot z}\right)}{\left(0.9999999999998099 - \frac{676.5203681218851}{z}\right) \cdot \left(1 + z\right)} + \frac{771.3234287776531}{\left(z + -1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z + -1\right) + 4}\right) + \frac{12.507343278686905}{\left(z + -1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z + -1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z + -1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8}\right) \]

Alternatives

Alternative 1
Error	3.5
Cost	43840

\[\left(\left(\left(\left(\left(\left(\frac{\left(-1258.1392167221638 + \frac{394153.5179299711}{z}\right) + \mathsf{fma}\left(z, 0.9999999999996197, \frac{-457679.80848377093}{z \cdot z}\right)}{\left(0.9999999999998099 - \frac{676.5203681218851}{z}\right) \cdot \left(1 + z\right)} + \frac{771.3234287776531}{\left(z + -1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z + -1\right) + 4}\right) + \frac{12.507343278686905}{\left(z + -1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z + -1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z + -1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8}\right) \cdot \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot \frac{e^{-6.5 - z}}{{\left(z + 6.5\right)}^{\left(0.5 - z\right)}}\right)\right) \]

Alternative 2
Error	3.7
Cost	37696

\[\begin{array}{l} t_0 := \left(z + -1\right) + 7.5\\ \left(\left(\left(\left(\left(\left(\frac{\left(-1258.1392167221638 + \frac{394153.5179299711}{z}\right) + \mathsf{fma}\left(z, 0.9999999999996197, \frac{-457679.80848377093}{z \cdot z}\right)}{\left(0.9999999999998099 - \frac{676.5203681218851}{z}\right) \cdot \left(1 + z\right)} + \frac{771.3234287776531}{\left(z + -1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z + -1\right) + 4}\right) + \frac{12.507343278686905}{\left(z + -1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z + -1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z + -1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8}\right) \cdot \frac{\sqrt{2 \cdot \pi} \cdot {t_0}^{\left(z + -0.5\right)}}{e^{t_0}} \end{array} \]

Alternative 3
Error	3.8
Cost	31680

\[\begin{array}{l} t_0 := \frac{-771.3234287776531}{2 + z}\\ t_1 := \frac{1259.1392167224028}{1 + z}\\ \sqrt{2 \cdot \pi} \cdot \left(\frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}} \cdot \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{12.507343278686905}{z + 4} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \frac{0.9999999999996197 + \left(\frac{676.5203681218851}{z} + \left(\frac{771.3234287776531}{2 + z} + \frac{-1259.1392167224028}{1 + z}\right)\right) \cdot \left(\frac{-676.5203681218851}{z} + \left(t_0 + t_1\right)\right)}{0.9999999999998099 + \left(t_0 + \left(t_1 - \frac{676.5203681218851}{z}\right)\right)}\right)\right) \end{array} \]

Alternative 4
Error	3.8
Cost	31680

\[\begin{array}{l} t_0 := \frac{-676.5203681218851}{z} + \left(\frac{-771.3234287776531}{2 + z} + \frac{1259.1392167224028}{1 + z}\right)\\ \sqrt{2 \cdot \pi} \cdot \left(\left(\frac{0.9999999999996197 + \left(\frac{676.5203681218851}{z} + \left(\frac{771.3234287776531}{2 + z} + \frac{-1259.1392167224028}{1 + z}\right)\right) \cdot t_0}{0.9999999999998099 + t_0} + \left(\frac{-176.6150291621406}{z + 3} + \left(\left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{12.507343278686905}{z + 4} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}}\right) \end{array} \]

Alternative 5
Error	3.9
Cost	30912

\[\begin{array}{l} t_0 := \left(z + -1\right) + 7.5\\ \frac{\sqrt{2 \cdot \pi} \cdot {t_0}^{\left(z + -0.5\right)}}{e^{t_0}} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(z + -1\right) + 7} + \left(\frac{-0.13857109526572012}{\left(z + -1\right) + 6} + \left(\frac{12.507343278686905}{\left(z + -1\right) + 5} + \left(\frac{-176.6150291621406}{\left(z + -1\right) + 4} + \left(\frac{771.3234287776531}{\left(z + -1\right) + 3} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 + \left(z + -1\right)}\right) + \frac{-1259.1392167224028}{2 + \left(z + -1\right)}\right)\right)\right)\right)\right)\right)\right) \end{array} \]

Alternative 6
Error	3.9
Cost	30912

\[\begin{array}{l} t_0 := \left(z + -1\right) + 7\\ \left(\left(\sqrt{2 \cdot \pi} \cdot {\left(0.5 + t_0\right)}^{\left(0.5 + \left(z + -1\right)\right)}\right) \cdot e^{-0.5 + \left(\left(1 - z\right) + -7\right)}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{t_0} + \left(\frac{-0.13857109526572012}{\left(z + -1\right) + 6} + \left(\frac{12.507343278686905}{\left(z + -1\right) + 5} + \left(\frac{-176.6150291621406}{\left(z + -1\right) + 4} + \left(\frac{771.3234287776531}{\left(z + -1\right) + 3} + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{1 + z}\right)\right)\right)\right)\right)\right)\right)\right) \end{array} \]

Alternative 7
Error	3.9
Cost	29696

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

Alternative 8
Error	3.9
Cost	29504

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

Alternative 9
Error	3.9
Cost	29504

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

Alternative 10
Error	48.9
Cost	29252

\[\begin{array}{l} t_0 := \frac{-176.6150291621406}{z + 3} + \left(\left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{12.507343278686905}{z + 4} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\\ t_1 := \sqrt{2 \cdot \pi}\\ t_2 := \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}}\\ t_3 := 0.9999999999998099 + \frac{188.7045801771354}{z}\\ \mathbf{if}\;z \leq 11.5:\\ \;\;\;\;t_1 \cdot \left(t_2 \cdot \left(t_0 + t_3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(t_2 \cdot \left(t_0 + \left(t_3 + \frac{-283.5076408329034}{z \cdot z}\right)\right)\right)\\ \end{array} \]

Alternative 11
Error	50.0
Cost	28736

\[\sqrt{2 \cdot \pi} \cdot \left(\frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}} \cdot \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{12.507343278686905}{z + 4} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(0.9999999999998099 + \frac{188.7045801771354}{z}\right)\right)\right) \]

Alternative 12
Error	54.0
Cost	28612

\[\begin{array}{l} t_0 := \sqrt{2 \cdot \pi}\\ t_1 := \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{12.507343278686905}{z + 4} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ \mathbf{if}\;z \leq 2.794869546817802:\\ \;\;\;\;t_0 \cdot \left(\left(0.9999999999998099 + \left(t_1 + \left(\frac{529.8450874864218}{z \cdot z} + \frac{-176.6150291621406}{z}\right)\right)\right) \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}} \cdot \left(0.9999999999998099 + \left(t_1 + \left(z \cdot 19.623892129126734 + -58.8716763873802\right)\right)\right)\right)\\ \end{array} \]

Alternative 13
Error	53.5
Cost	28608

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

Alternative 14
Error	56.3
Cost	28420

\[\begin{array}{l} t_0 := \sqrt{2 \cdot \pi}\\ t_1 := \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\\ t_2 := \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{12.507343278686905}{z + 4} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ \mathbf{if}\;z \leq 2.953568479306968:\\ \;\;\;\;t_0 \cdot \left(\left(0.9999999999998099 + \left(t_2 + \left(\frac{529.8450874864218}{z \cdot z} + \frac{-176.6150291621406}{z}\right)\right)\right) \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(0.9999999999998099 + \left(t_2 + \left(z \cdot 19.623892129126734 + -58.8716763873802\right)\right)\right)\right)\\ \end{array} \]

Alternative 15
Error	59.5
Cost	28032

\[\sqrt{2 \cdot \pi} \cdot \left(\frac{\sqrt{0.15384615384615385}}{e^{6.5}} \cdot \left(0.9999999999998099 + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{12.507343278686905}{z + 4} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right) + \left(z \cdot 19.623892129126734 + -58.8716763873802\right)\right)\right)\right) \]

Alternative 16
Error	63.1
Cost	27904

\[\sqrt{2 \cdot \pi} \cdot \left(\frac{\sqrt{0.15384615384615385}}{e^{6.5}} \cdot \left(0.9999999999998099 + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{12.507343278686905}{z + 4} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right) + \frac{-176.6150291621406}{z}\right)\right)\right) \]

Alternative 17
Error	63.1
Cost	27648

\[\sqrt{2 \cdot \pi} \cdot \left(\frac{\sqrt{0.15384615384615385}}{e^{6.5}} \cdot \left(0.9999999999998099 + \left(-58.8716763873802 + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{z + 7} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{12.507343278686905}{z + 4} + \frac{9.984369578019572 \cdot 10^{-6}}{z}\right)\right)\right)\right)\right)\right) \]

Alternative 18
Error	63.1
Cost	27520

\[\sqrt{2 \cdot \pi} \cdot \left(\frac{\sqrt{0.15384615384615385}}{e^{6.5}} \cdot \left(0.9999999999998099 + \left(-58.8716763873802 + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{z + 7} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{12.507343278686905}{z + 4} + 1.6640615963365953 \cdot 10^{-6}\right)\right)\right)\right)\right)\right) \]

Reproduce

herbie shell --seed 2022300 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z greater than 0.5"
  :precision binary64
  :pre (> z 0.5)
  (* (* (* (sqrt (* PI 2.0)) (pow (+ (+ (- z 1.0) 7.0) 0.5) (+ (- z 1.0) 0.5))) (exp (- (+ (+ (- z 1.0) 7.0) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- z 1.0) 1.0))) (/ -1259.1392167224028 (+ (- z 1.0) 2.0))) (/ 771.3234287776531 (+ (- z 1.0) 3.0))) (/ -176.6150291621406 (+ (- z 1.0) 4.0))) (/ 12.507343278686905 (+ (- z 1.0) 5.0))) (/ -0.13857109526572012 (+ (- z 1.0) 6.0))) (/ 9.984369578019572e-6 (+ (- z 1.0) 7.0))) (/ 1.5056327351493116e-7 (+ (- z 1.0) 8.0)))))