Jmat.Real.gamma, branch z greater than 0.5

Percentage Accurate: 1.2% → 99.2%

Time: 49.3s

Precision: binary64

Cost: 31556

• 24.8% of points produce a very large (infinite) output. You may want to add a precondition.

\[z > 0.5\]

Math

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

↓

\[\begin{array}{l} t_0 := 7 + \left(z + -1\right)\\ t_1 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z + -1 \leq 200:\\ \;\;\;\;\left(\left(t_1 \cdot {\left(0.5 + t_0\right)}^{\left(0.5 + \left(z + -1\right)\right)}\right) \cdot e^{\left(\left(1 - z\right) - 7\right) - 0.5}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 + \left(z + -1\right)}\right) + \frac{-1259.1392167224028}{2 + \left(z + -1\right)}\right) + 771.3234287776531 \cdot \frac{1}{z - -2}\right) + \frac{-176.6150291621406}{\left(z + -1\right) + 4}\right) + \frac{12.507343278686905}{5 + \left(z + -1\right)}\right) + \frac{-0.13857109526572012}{6 + \left(z + -1\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t_0}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{\left(z + 1\right) \cdot e^{6.5}}\right) \cdot 4.099123286189028\\ \end{array} \]

FPCore

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

↓

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ 7.0 (+ z -1.0))) (t_1 (sqrt (* PI 2.0))))
   (if (<= (+ z -1.0) 200.0)
     (*
      (*
       (* t_1 (pow (+ 0.5 t_0) (+ 0.5 (+ z -1.0))))
       (exp (- (- (- 1.0 z) 7.0) 0.5)))
      (+
       (+
        (+
         (+
          (+
           (+
            (+
             (+ 0.9999999999998099 (/ 676.5203681218851 (+ 1.0 (+ z -1.0))))
             (/ -1259.1392167224028 (+ 2.0 (+ z -1.0))))
            (* 771.3234287776531 (/ 1.0 (- z -2.0))))
           (/ -176.6150291621406 (+ (+ z -1.0) 4.0)))
          (/ 12.507343278686905 (+ 5.0 (+ z -1.0))))
         (/ -0.13857109526572012 (+ 6.0 (+ z -1.0))))
        (/ 9.984369578019572e-6 t_0))
       (/ 1.5056327351493116e-7 (+ (+ z -1.0) 8.0))))
     (*
      (* t_1 (/ (pow (+ z 6.5) (+ z -0.5)) (* (+ z 1.0) (exp 6.5))))
      4.099123286189028))))

C

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

↓

double code(double z) {
	double t_0 = 7.0 + (z + -1.0);
	double t_1 = sqrt((((double) M_PI) * 2.0));
	double tmp;
	if ((z + -1.0) <= 200.0) {
		tmp = ((t_1 * pow((0.5 + t_0), (0.5 + (z + -1.0)))) * exp((((1.0 - z) - 7.0) - 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / (1.0 + (z + -1.0)))) + (-1259.1392167224028 / (2.0 + (z + -1.0)))) + (771.3234287776531 * (1.0 / (z - -2.0)))) + (-176.6150291621406 / ((z + -1.0) + 4.0))) + (12.507343278686905 / (5.0 + (z + -1.0)))) + (-0.13857109526572012 / (6.0 + (z + -1.0)))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / ((z + -1.0) + 8.0)));
	} else {
		tmp = (t_1 * (pow((z + 6.5), (z + -0.5)) / ((z + 1.0) * exp(6.5)))) * 4.099123286189028;
	}
	return tmp;
}

Java

public static double code(double z) {
	return ((Math.sqrt((Math.PI * 2.0)) * Math.pow((((z - 1.0) + 7.0) + 0.5), ((z - 1.0) + 0.5))) * Math.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)));
}

↓

public static double code(double z) {
	double t_0 = 7.0 + (z + -1.0);
	double t_1 = Math.sqrt((Math.PI * 2.0));
	double tmp;
	if ((z + -1.0) <= 200.0) {
		tmp = ((t_1 * Math.pow((0.5 + t_0), (0.5 + (z + -1.0)))) * Math.exp((((1.0 - z) - 7.0) - 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / (1.0 + (z + -1.0)))) + (-1259.1392167224028 / (2.0 + (z + -1.0)))) + (771.3234287776531 * (1.0 / (z - -2.0)))) + (-176.6150291621406 / ((z + -1.0) + 4.0))) + (12.507343278686905 / (5.0 + (z + -1.0)))) + (-0.13857109526572012 / (6.0 + (z + -1.0)))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / ((z + -1.0) + 8.0)));
	} else {
		tmp = (t_1 * (Math.pow((z + 6.5), (z + -0.5)) / ((z + 1.0) * Math.exp(6.5)))) * 4.099123286189028;
	}
	return tmp;
}

Python

def code(z):
	return ((math.sqrt((math.pi * 2.0)) * math.pow((((z - 1.0) + 7.0) + 0.5), ((z - 1.0) + 0.5))) * math.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)))

↓

def code(z):
	t_0 = 7.0 + (z + -1.0)
	t_1 = math.sqrt((math.pi * 2.0))
	tmp = 0
	if (z + -1.0) <= 200.0:
		tmp = ((t_1 * math.pow((0.5 + t_0), (0.5 + (z + -1.0)))) * math.exp((((1.0 - z) - 7.0) - 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / (1.0 + (z + -1.0)))) + (-1259.1392167224028 / (2.0 + (z + -1.0)))) + (771.3234287776531 * (1.0 / (z - -2.0)))) + (-176.6150291621406 / ((z + -1.0) + 4.0))) + (12.507343278686905 / (5.0 + (z + -1.0)))) + (-0.13857109526572012 / (6.0 + (z + -1.0)))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / ((z + -1.0) + 8.0)))
	else:
		tmp = (t_1 * (math.pow((z + 6.5), (z + -0.5)) / ((z + 1.0) * math.exp(6.5)))) * 4.099123286189028
	return tmp

Julia

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

↓

function code(z)
	t_0 = Float64(7.0 + Float64(z + -1.0))
	t_1 = sqrt(Float64(pi * 2.0))
	tmp = 0.0
	if (Float64(z + -1.0) <= 200.0)
		tmp = Float64(Float64(Float64(t_1 * (Float64(0.5 + t_0) ^ Float64(0.5 + Float64(z + -1.0)))) * exp(Float64(Float64(Float64(1.0 - z) - 7.0) - 0.5))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 + Float64(z + -1.0)))) + Float64(-1259.1392167224028 / Float64(2.0 + Float64(z + -1.0)))) + Float64(771.3234287776531 * Float64(1.0 / Float64(z - -2.0)))) + Float64(-176.6150291621406 / Float64(Float64(z + -1.0) + 4.0))) + Float64(12.507343278686905 / Float64(5.0 + Float64(z + -1.0)))) + Float64(-0.13857109526572012 / Float64(6.0 + Float64(z + -1.0)))) + Float64(9.984369578019572e-6 / t_0)) + Float64(1.5056327351493116e-7 / Float64(Float64(z + -1.0) + 8.0))));
	else
		tmp = Float64(Float64(t_1 * Float64((Float64(z + 6.5) ^ Float64(z + -0.5)) / Float64(Float64(z + 1.0) * exp(6.5)))) * 4.099123286189028);
	end
	return tmp
end

MATLAB

function tmp = code(z)
	tmp = ((sqrt((pi * 2.0)) * ((((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)));
end

↓

function tmp_2 = code(z)
	t_0 = 7.0 + (z + -1.0);
	t_1 = sqrt((pi * 2.0));
	tmp = 0.0;
	if ((z + -1.0) <= 200.0)
		tmp = ((t_1 * ((0.5 + t_0) ^ (0.5 + (z + -1.0)))) * exp((((1.0 - z) - 7.0) - 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / (1.0 + (z + -1.0)))) + (-1259.1392167224028 / (2.0 + (z + -1.0)))) + (771.3234287776531 * (1.0 / (z - -2.0)))) + (-176.6150291621406 / ((z + -1.0) + 4.0))) + (12.507343278686905 / (5.0 + (z + -1.0)))) + (-0.13857109526572012 / (6.0 + (z + -1.0)))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / ((z + -1.0) + 8.0)));
	else
		tmp = (t_1 * (((z + 6.5) ^ (z + -0.5)) / ((z + 1.0) * exp(6.5)))) * 4.099123286189028;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[z_] := Block[{t$95$0 = N[(7.0 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(z + -1.0), $MachinePrecision], 200.0], N[(N[(N[(t$95$1 * N[Power[N[(0.5 + t$95$0), $MachinePrecision], N[(0.5 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(N[(1.0 - z), $MachinePrecision] - 7.0), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 * N[(1.0 / N[(z - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(z + -1.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(5.0 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(6.0 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(z + -1.0), $MachinePrecision] + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(N[Power[N[(z + 6.5), $MachinePrecision], N[(z + -0.5), $MachinePrecision]], $MachinePrecision] / N[(N[(z + 1.0), $MachinePrecision] * N[Exp[6.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.099123286189028), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z 1) < 200

   1. Initial program 97.2%

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

   2. Applied egg-rr 97.6%

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

      [Start] 97.2%	\[ \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      div-inv [=>] 97.6%	\[ \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \color{blue}{771.3234287776531 \cdot \frac{1}{\left(z - 1\right) + 3}}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      associate-+l- [=>] 97.6%	\[ \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + 771.3234287776531 \cdot \frac{1}{\color{blue}{z - \left(1 - 3\right)}}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      metadata-eval [=>] 97.6%	\[ \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + 771.3234287776531 \cdot \frac{1}{z - \color{blue}{-2}}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]

   if 200 < (-.f64 z 1)

   1. Initial program 0.0%

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

   2. Simplified 0.0%

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

      [Start] 0.0%	\[ \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      *-commutative [=>] 0.0%	\[ \color{blue}{\left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right)} \]
      associate-*r* [=>] 0.0%	\[ \color{blue}{\left(\left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right)\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}} \]
      exp-neg [=>] 0.0%	\[ \left(\left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right)\right) \cdot \color{blue}{\frac{1}{e^{\left(\left(z - 1\right) + 7\right) + 0.5}}} \]

   3. Taylor expanded in z around inf 0.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}}\right) \cdot \left(0.9999999999998099 + \left(\color{blue}{\frac{12.0895510149948}{z}} + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right) \]

   4. Taylor expanded in z around 0 0.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}}\right) \cdot \left(0.9999999999998099 + \left(\frac{12.0895510149948}{z} + \color{blue}{3.0991232861892177}\right)\right) \]

   5. Taylor expanded in z around 0 100.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{\color{blue}{e^{6.5} + e^{6.5} \cdot z}}\right) \cdot \left(0.9999999999998099 + \left(\frac{12.0895510149948}{z} + 3.0991232861892177\right)\right) \]

   6. Simplified 100.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{\color{blue}{\left(z + 1\right) \cdot e^{6.5}}}\right) \cdot \left(0.9999999999998099 + \left(\frac{12.0895510149948}{z} + 3.0991232861892177\right)\right) \]
      Step-by-step derivation

      [Start] 100.0%	\[ \left(\sqrt{\pi \cdot 2} \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{6.5} + e^{6.5} \cdot z}\right) \cdot \left(0.9999999999998099 + \left(\frac{12.0895510149948}{z} + 3.0991232861892177\right)\right) \]
      *-commutative [=>] 100.0%	\[ \left(\sqrt{\pi \cdot 2} \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{6.5} + \color{blue}{z \cdot e^{6.5}}}\right) \cdot \left(0.9999999999998099 + \left(\frac{12.0895510149948}{z} + 3.0991232861892177\right)\right) \]
      distribute-rgt1-in [=>] 100.0%	\[ \left(\sqrt{\pi \cdot 2} \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{\color{blue}{\left(z + 1\right) \cdot e^{6.5}}}\right) \cdot \left(0.9999999999998099 + \left(\frac{12.0895510149948}{z} + 3.0991232861892177\right)\right) \]

   7. Taylor expanded in z around inf 100.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{\left(z + 1\right) \cdot e^{6.5}}\right) \cdot \color{blue}{4.099123286189028} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + -1 \leq 200:\\ \;\;\;\;\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(0.5 + \left(7 + \left(z + -1\right)\right)\right)}^{\left(0.5 + \left(z + -1\right)\right)}\right) \cdot e^{\left(\left(1 - z\right) - 7\right) - 0.5}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 + \left(z + -1\right)}\right) + \frac{-1259.1392167224028}{2 + \left(z + -1\right)}\right) + 771.3234287776531 \cdot \frac{1}{z - -2}\right) + \frac{-176.6150291621406}{\left(z + -1\right) + 4}\right) + \frac{12.507343278686905}{5 + \left(z + -1\right)}\right) + \frac{-0.13857109526572012}{6 + \left(z + -1\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 + \left(z + -1\right)}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{\left(z + 1\right) \cdot e^{6.5}}\right) \cdot 4.099123286189028\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.2%
Cost	31556

\[\begin{array}{l} t_0 := 7 + \left(z + -1\right)\\ t_1 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z + -1 \leq 200:\\ \;\;\;\;\left(\left(t_1 \cdot {\left(0.5 + t_0\right)}^{\left(0.5 + \left(z + -1\right)\right)}\right) \cdot e^{\left(\left(1 - z\right) - 7\right) - 0.5}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 + \left(z + -1\right)}\right) + \frac{-1259.1392167224028}{2 + \left(z + -1\right)}\right) + 771.3234287776531 \cdot \frac{1}{z - -2}\right) + \frac{-176.6150291621406}{\left(z + -1\right) + 4}\right) + \frac{12.507343278686905}{5 + \left(z + -1\right)}\right) + \frac{-0.13857109526572012}{6 + \left(z + -1\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t_0}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{\left(z + 1\right) \cdot e^{6.5}}\right) \cdot 4.099123286189028\\ \end{array} \]

Alternative 2
Accuracy	99.9%
Cost	35904

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

Alternative 3
Accuracy	99.2%
Cost	29636

\[\begin{array}{l} t_0 := {\left(z + 6.5\right)}^{\left(z + -0.5\right)}\\ t_1 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z \leq 200:\\ \;\;\;\;t_1 \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right) + \left(\left(\frac{12.507343278686905}{z - -4} + \frac{-0.13857109526572012}{z + 5}\right) + \left(\frac{771.3234287776531}{2 + z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) \cdot \left(t_0 \cdot e^{-6.5 - z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 \cdot \frac{t_0}{\left(z + 1\right) \cdot e^{6.5}}\right) \cdot 4.099123286189028\\ \end{array} \]

Alternative 4
Accuracy	99.2%
Cost	29636

\[\begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ t_1 := {\left(z + 6.5\right)}^{\left(z + -0.5\right)}\\ \mathbf{if}\;z \leq 200:\\ \;\;\;\;\left(t_0 \cdot \left(t_1 \cdot e^{-6.5 - z}\right)\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) + \left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{676.5203681218851}{z} + \left(\frac{-176.6150291621406}{z + 3} + \frac{771.3234287776531}{z - -2}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \frac{t_1}{\left(z + 1\right) \cdot e^{6.5}}\right) \cdot 4.099123286189028\\ \end{array} \]

Alternative 5
Accuracy	99.2%
Cost	29636

\[\begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ t_1 := {\left(z + 6.5\right)}^{\left(z + -0.5\right)}\\ \mathbf{if}\;z \leq 200:\\ \;\;\;\;\left(t_0 \cdot \frac{t_1}{e^{z + 6.5}}\right) \cdot \left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{z} + \left(\frac{-1259.1392167224028}{z + 1} + \left(\frac{771.3234287776531}{2 + z} + \frac{-176.6150291621406}{z + 3}\right)\right)\right) + \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \frac{t_1}{\left(z + 1\right) \cdot e^{6.5}}\right) \cdot 4.099123286189028\\ \end{array} \]

Alternative 6
Accuracy	98.3%
Cost	26944

\[\left(\sqrt{\pi \cdot 2} \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{\left(z + 1\right) \cdot e^{6.5}}\right) \cdot \left(4.099123286189028 + \frac{12.0895510149948}{z}\right) \]

Alternative 7
Accuracy	98.3%
Cost	26816

\[\left(\sqrt{\pi \cdot 2} \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{\left(z + 1\right) \cdot e^{6.5}}\right) \cdot \frac{12.0895510149948}{z} \]

Alternative 8
Accuracy	98.3%
Cost	26688

\[\left(\sqrt{\pi \cdot 2} \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{\left(z + 1\right) \cdot e^{6.5}}\right) \cdot 4.099123286189028 \]

Alternative 9
Accuracy	6.2%
Cost	26496

\[\left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + \left(z \cdot 1085.1560852655925 - 929.2500554347672\right)\right) \]

Alternative 10
Accuracy	6.2%
Cost	26368

\[\left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \cdot \left(0.9999999999998099 + z \cdot 1085.1560852655925\right) \]

Alternative 11
Accuracy	3.2%
Cost	26112

\[4.099123286189028 \cdot \left(\sqrt{\pi \cdot 2} \cdot \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\right) \]

Reproduce

herbie shell --seed 2023272 
(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)))))