Jmat.Real.gamma, branch z greater than 0.5

Average Error: 3.9 → 3.7

Time: 32.7s

Precision: binary64

Cost: 48960

\[z > 0.5\]

Math

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

↓

\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(z + \left(-1 + 7.5\right)\right)}^{\left(z + -0.5\right)}\right) \cdot e^{-0.5 - \left(z - -6\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(\frac{-1259.1392167224028}{\mathsf{fma}\left(z, z, z\right)}, \mathsf{fma}\left(-1 - z, 0.5372879814536304, z\right), \frac{771.3234287776531}{z + 2}\right) + 0.9999999999998099\right) + \frac{-176.6150291621406}{z - -3}\right) + \frac{12.507343278686905}{z - -4}\right) + \frac{-0.13857109526572012}{z - -5}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{z - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right) \]

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 (* PI 2.0)) (pow (+ z (+ -1.0 7.5)) (+ z -0.5)))
   (exp (- -0.5 (- z -6.0))))
  (+
   (+
    (+
     (+
      (+
       (+
        (fma
         (/ -1259.1392167224028 (fma z z z))
         (fma (- -1.0 z) 0.5372879814536304 z)
         (/ 771.3234287776531 (+ z 2.0)))
        0.9999999999998099)
       (/ -176.6150291621406 (- z -3.0)))
      (/ 12.507343278686905 (- z -4.0)))
     (/ -0.13857109526572012 (- z -5.0)))
    (/ 9.984369578019572e-6 (- z -6.0)))
   (/ 1.5056327351493116e-7 (- z -7.0)))))

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((((double) M_PI) * 2.0)) * pow((z + (-1.0 + 7.5)), (z + -0.5))) * exp((-0.5 - (z - -6.0)))) * ((((((fma((-1259.1392167224028 / fma(z, z, z)), fma((-1.0 - z), 0.5372879814536304, z), (771.3234287776531 / (z + 2.0))) + 0.9999999999998099) + (-176.6150291621406 / (z - -3.0))) + (12.507343278686905 / (z - -4.0))) + (-0.13857109526572012 / (z - -5.0))) + (9.984369578019572e-6 / (z - -6.0))) + (1.5056327351493116e-7 / (z - -7.0)));
}

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(Float64(sqrt(Float64(pi * 2.0)) * (Float64(z + Float64(-1.0 + 7.5)) ^ Float64(z + -0.5))) * exp(Float64(-0.5 - Float64(z - -6.0)))) * Float64(Float64(Float64(Float64(Float64(Float64(fma(Float64(-1259.1392167224028 / fma(z, z, z)), fma(Float64(-1.0 - z), 0.5372879814536304, z), Float64(771.3234287776531 / Float64(z + 2.0))) + 0.9999999999998099) + Float64(-176.6150291621406 / Float64(z - -3.0))) + Float64(12.507343278686905 / Float64(z - -4.0))) + Float64(-0.13857109526572012 / Float64(z - -5.0))) + Float64(9.984369578019572e-6 / Float64(z - -6.0))) + Float64(1.5056327351493116e-7 / Float64(z - -7.0))))
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[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(z + N[(-1.0 + 7.5), $MachinePrecision]), $MachinePrecision], N[(z + -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(-0.5 - N[(z - -6.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-1259.1392167224028 / N[(z * z + z), $MachinePrecision]), $MachinePrecision] * N[(N[(-1.0 - z), $MachinePrecision] * 0.5372879814536304 + z), $MachinePrecision] + N[(771.3234287776531 / N[(z + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.9999999999998099), $MachinePrecision] + N[(-176.6150291621406 / N[(z - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(z - -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(z - -5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(z - -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(z - -7.0), $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. Simplified 3.9

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

3. Applied egg-rr 3.8

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

4. Applied egg-rr 3.9

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

5. Applied egg-rr 3.8

   \[\leadsto \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(z + \left(-1 + 7.5\right)\right)}^{\left(z + -0.5\right)}\right) \cdot e^{-0.5 - \left(z - -6\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\mathsf{fma}\left(\left(-1 - z\right) \cdot 0.5372879814536304 + z, \frac{1259.1392167224028}{z \cdot \left(-1 - z\right)}, 0.9999999999998099 + \frac{-771.3234287776531}{-2 - z}\right)} + \frac{-176.6150291621406}{z - -3}\right) + \frac{12.507343278686905}{z - -4}\right) + \frac{-0.13857109526572012}{z - -5}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{z - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right) \]

6. Simplified 3.8

   \[\leadsto \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(z + \left(-1 + 7.5\right)\right)}^{\left(z + -0.5\right)}\right) \cdot e^{-0.5 - \left(z - -6\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-1 - z, 0.5372879814536304, z\right), \frac{1259.1392167224028}{z \cdot \left(-1 - z\right)}, 0.9999999999998099 + \frac{-771.3234287776531}{-2 - z}\right)} + \frac{-176.6150291621406}{z - -3}\right) + \frac{12.507343278686905}{z - -4}\right) + \frac{-0.13857109526572012}{z - -5}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{z - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right) \]
   Proof

7. Applied egg-rr 3.7

   \[\leadsto \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(z + \left(-1 + 7.5\right)\right)}^{\left(z + -0.5\right)}\right) \cdot e^{-0.5 - \left(z - -6\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\mathsf{fma}\left(\frac{-1259.1392167224028}{\mathsf{fma}\left(z, z, z\right)}, \mathsf{fma}\left(-1 - z, 0.5372879814536304, z\right), \frac{771.3234287776531}{z + 2}\right) + 0.9999999999998099\right)} + \frac{-176.6150291621406}{z - -3}\right) + \frac{12.507343278686905}{z - -4}\right) + \frac{-0.13857109526572012}{z - -5}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{z - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right) \]

Alternatives

Alternative 1
Error	3.8
Cost	42496

\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(z + 6.5\right)}^{\left(z + -0.5\right)}\right) \cdot e^{-\left(z + 6.5\right)}\right) \cdot \left(\left(\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-1 - z, 0.5372879814536304, z\right), \frac{1259.1392167224028}{z \cdot \left(-1 - z\right)}, 0.9999999999998099 + \frac{-771.3234287776531}{-2 - z}\right) + \frac{-176.6150291621406}{z - -3}\right) + \frac{12.507343278686905}{z - -4}\right) + \frac{-0.13857109526572012}{z - -5}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{z - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right) \]

Alternative 2
Error	3.8
Cost	42496

\[\left(\left({\left(z + 6.5\right)}^{\left(z + -0.5\right)} \cdot e^{-\left(z + 6.5\right)}\right) \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-1 - z, 0.5372879814536304, z\right), \frac{1259.1392167224028}{z \cdot \left(-1 - z\right)}, 0.9999999999998099 + \frac{-771.3234287776531}{-2 - z}\right) + \frac{-176.6150291621406}{z - -3}\right) + \frac{12.507343278686905}{z - -4}\right) + \frac{-0.13857109526572012}{z - -5}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{z - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right) \]

Alternative 3
Error	3.8
Cost	36544

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

Alternative 4
Error	3.8
Cost	36416

\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(z + \left(-1 + 7.5\right)\right)}^{\left(z + -0.5\right)}\right) \cdot e^{-0.5 - \left(z - -6\right)}\right) \cdot \left(\left(\left(\left(\left(\mathsf{fma}\left(\frac{1259.1392167224028}{z \cdot \left(-1 - z\right)}, \left(-1 - z\right) \cdot 0.5372879814536304 + z, 0.9999999999998099 + \frac{-771.3234287776531}{-2 - z}\right) + \frac{-176.6150291621406}{z - -3}\right) + \frac{12.507343278686905}{z - -4}\right) + \frac{-0.13857109526572012}{z - -5}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{z - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z - -7}\right) \]

Alternative 5
Error	3.9
Cost	30080

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

Alternative 6
Error	3.9
Cost	30080

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

Alternative 7
Error	3.9
Cost	29568

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

Alternative 8
Error	3.8
Cost	29568

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

Alternative 9
Error	3.8
Cost	29568

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

Alternative 10
Error	53.1
Cost	28864

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

Alternative 11
Error	63.1
Cost	28544

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

Reproduce

herbie shell --seed 2023010 
(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)))))