Jmat.Real.gamma, branch z less than 0.5

Average Error: 1.7 → 0.5

Time: 1.9min

Precision: binary64

Cost: 62976

\[z \leq 0.5\]

Math

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

↓

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

FPCore

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

↓

(FPCore (z)
 :precision binary64
 (let* ((t_0 (exp (- 7.5 z))))
   (*
    (+
     (+
      (+
       (+
        0.9999999999998099
        (+
         (/ 676.5203681218851 (- 1.0 z))
         (/ -1259.1392167224028 (+ 2.0 (- z)))))
       (+
        (/ 771.3234287776531 (+ (- z) 3.0))
        (/ -176.6150291621406 (+ (- z) 4.0))))
      (+
       (/ 12.507343278686905 (+ (- z) 5.0))
       (/ -0.13857109526572012 (+ (- z) 6.0))))
     (+
      (/ 9.984369578019572e-6 (+ (- z) 7.0))
      (/ 1.5056327351493116e-7 (+ (- z) 8.0))))
    (*
     PI
     (/
      (*
       (pow (+ (- z) 7.5) (+ (- z) 0.5))
       (/ (/ (/ (sqrt (* 2.0 PI)) t_0) t_0) (exp (+ z -7.5))))
      (sin (* PI z)))))))

C

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

↓

double code(double z) {
	double t_0 = exp((7.5 - z));
	return ((((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 + -z)))) + ((771.3234287776531 / (-z + 3.0)) + (-176.6150291621406 / (-z + 4.0)))) + ((12.507343278686905 / (-z + 5.0)) + (-0.13857109526572012 / (-z + 6.0)))) + ((9.984369578019572e-6 / (-z + 7.0)) + (1.5056327351493116e-7 / (-z + 8.0)))) * (((double) M_PI) * ((pow((-z + 7.5), (-z + 0.5)) * (((sqrt((2.0 * ((double) M_PI))) / t_0) / t_0) / exp((z + -7.5)))) / sin((((double) M_PI) * z))));
}

Java

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

↓

public static double code(double z) {
	double t_0 = Math.exp((7.5 - z));
	return ((((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 + -z)))) + ((771.3234287776531 / (-z + 3.0)) + (-176.6150291621406 / (-z + 4.0)))) + ((12.507343278686905 / (-z + 5.0)) + (-0.13857109526572012 / (-z + 6.0)))) + ((9.984369578019572e-6 / (-z + 7.0)) + (1.5056327351493116e-7 / (-z + 8.0)))) * (Math.PI * ((Math.pow((-z + 7.5), (-z + 0.5)) * (((Math.sqrt((2.0 * Math.PI)) / t_0) / t_0) / Math.exp((z + -7.5)))) / Math.sin((Math.PI * z))));
}

Python

def code(z):
	return (math.pi / math.sin((math.pi * z))) * (((math.sqrt((math.pi * 2.0)) * math.pow(((((1.0 - z) - 1.0) + 7.0) + 0.5), (((1.0 - z) - 1.0) + 0.5))) * math.exp(-((((1.0 - z) - 1.0) + 7.0) + 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / (((1.0 - z) - 1.0) + 1.0))) + (-1259.1392167224028 / (((1.0 - z) - 1.0) + 2.0))) + (771.3234287776531 / (((1.0 - z) - 1.0) + 3.0))) + (-176.6150291621406 / (((1.0 - z) - 1.0) + 4.0))) + (12.507343278686905 / (((1.0 - z) - 1.0) + 5.0))) + (-0.13857109526572012 / (((1.0 - z) - 1.0) + 6.0))) + (9.984369578019572e-6 / (((1.0 - z) - 1.0) + 7.0))) + (1.5056327351493116e-7 / (((1.0 - z) - 1.0) + 8.0))))

↓

def code(z):
	t_0 = math.exp((7.5 - z))
	return ((((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 + -z)))) + ((771.3234287776531 / (-z + 3.0)) + (-176.6150291621406 / (-z + 4.0)))) + ((12.507343278686905 / (-z + 5.0)) + (-0.13857109526572012 / (-z + 6.0)))) + ((9.984369578019572e-6 / (-z + 7.0)) + (1.5056327351493116e-7 / (-z + 8.0)))) * (math.pi * ((math.pow((-z + 7.5), (-z + 0.5)) * (((math.sqrt((2.0 * math.pi)) / t_0) / t_0) / math.exp((z + -7.5)))) / math.sin((math.pi * z))))

Julia

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

↓

function code(z)
	t_0 = exp(Float64(7.5 - z))
	return Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(-1259.1392167224028 / Float64(2.0 + Float64(-z))))) + Float64(Float64(771.3234287776531 / Float64(Float64(-z) + 3.0)) + Float64(-176.6150291621406 / Float64(Float64(-z) + 4.0)))) + Float64(Float64(12.507343278686905 / Float64(Float64(-z) + 5.0)) + Float64(-0.13857109526572012 / Float64(Float64(-z) + 6.0)))) + Float64(Float64(9.984369578019572e-6 / Float64(Float64(-z) + 7.0)) + Float64(1.5056327351493116e-7 / Float64(Float64(-z) + 8.0)))) * Float64(pi * Float64(Float64((Float64(Float64(-z) + 7.5) ^ Float64(Float64(-z) + 0.5)) * Float64(Float64(Float64(sqrt(Float64(2.0 * pi)) / t_0) / t_0) / exp(Float64(z + -7.5)))) / sin(Float64(pi * z)))))
end

MATLAB

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

↓

function tmp = code(z)
	t_0 = exp((7.5 - z));
	tmp = ((((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 + -z)))) + ((771.3234287776531 / (-z + 3.0)) + (-176.6150291621406 / (-z + 4.0)))) + ((12.507343278686905 / (-z + 5.0)) + (-0.13857109526572012 / (-z + 6.0)))) + ((9.984369578019572e-6 / (-z + 7.0)) + (1.5056327351493116e-7 / (-z + 8.0)))) * (pi * ((((-z + 7.5) ^ (-z + 0.5)) * (((sqrt((2.0 * pi)) / t_0) / t_0) / exp((z + -7.5)))) / sin((pi * z))));
end

Wolfram

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

↓

code[z_] := Block[{t$95$0 = N[Exp[N[(7.5 - z), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(N[(0.9999999999998099 + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 + (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[((-z) + 3.0), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[((-z) + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[((-z) + 5.0), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[((-z) + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[((-z) + 7.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[((-z) + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(N[(N[Power[N[((-z) + 7.5), $MachinePrecision], N[((-z) + 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 1.7

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

2. Simplified 0.7

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

   [Start] 1.7

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

   rational_best-simplify-50 [=>] 1.6

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

3. Applied egg-rr 0.5

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

4. Simplified 0.5

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

   [Start] 0.5	\[ \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \left(-z\right)}\right)\right) + \left(\frac{771.3234287776531}{\left(-z\right) + 3} + \frac{-176.6150291621406}{\left(-z\right) + 4}\right)\right) + \left(\frac{12.507343278686905}{\left(-z\right) + 5} + \frac{-0.13857109526572012}{\left(-z\right) + 6}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(-z\right) + 7} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(-z\right) + 8}\right)\right) \cdot \left(\pi \cdot \frac{{\left(\left(-z\right) + 7.5\right)}^{\left(\left(-z\right) + 0.5\right)} \cdot \frac{\frac{\frac{\sqrt{2 \cdot \pi}}{\frac{e^{7.5 - z}}{2}}}{e^{7.5 - z}}}{2 \cdot e^{-7.5 + z}}}{\sin \left(\pi \cdot z\right)}\right) \]
   rational_best-simplify-54 [=>] 0.5	\[ \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \left(-z\right)}\right)\right) + \left(\frac{771.3234287776531}{\left(-z\right) + 3} + \frac{-176.6150291621406}{\left(-z\right) + 4}\right)\right) + \left(\frac{12.507343278686905}{\left(-z\right) + 5} + \frac{-0.13857109526572012}{\left(-z\right) + 6}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(-z\right) + 7} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(-z\right) + 8}\right)\right) \cdot \left(\pi \cdot \frac{{\left(\left(-z\right) + 7.5\right)}^{\left(\left(-z\right) + 0.5\right)} \cdot \color{blue}{\frac{\frac{\frac{\frac{\sqrt{2 \cdot \pi}}{\frac{e^{7.5 - z}}{2}}}{e^{7.5 - z}}}{2}}{e^{-7.5 + z}}}}{\sin \left(\pi \cdot z\right)}\right) \]
   rational_best-simplify-49 [=>] 0.5	\[ \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \left(-z\right)}\right)\right) + \left(\frac{771.3234287776531}{\left(-z\right) + 3} + \frac{-176.6150291621406}{\left(-z\right) + 4}\right)\right) + \left(\frac{12.507343278686905}{\left(-z\right) + 5} + \frac{-0.13857109526572012}{\left(-z\right) + 6}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(-z\right) + 7} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(-z\right) + 8}\right)\right) \cdot \left(\pi \cdot \frac{{\left(\left(-z\right) + 7.5\right)}^{\left(\left(-z\right) + 0.5\right)} \cdot \frac{\color{blue}{\frac{\frac{\frac{\sqrt{2 \cdot \pi}}{\frac{e^{7.5 - z}}{2}}}{2}}{e^{7.5 - z}}}}{e^{-7.5 + z}}}{\sin \left(\pi \cdot z\right)}\right) \]
   rational_best-simplify-49 [=>] 0.5	\[ \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \left(-z\right)}\right)\right) + \left(\frac{771.3234287776531}{\left(-z\right) + 3} + \frac{-176.6150291621406}{\left(-z\right) + 4}\right)\right) + \left(\frac{12.507343278686905}{\left(-z\right) + 5} + \frac{-0.13857109526572012}{\left(-z\right) + 6}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(-z\right) + 7} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(-z\right) + 8}\right)\right) \cdot \left(\pi \cdot \frac{{\left(\left(-z\right) + 7.5\right)}^{\left(\left(-z\right) + 0.5\right)} \cdot \frac{\frac{\color{blue}{\frac{\frac{\sqrt{2 \cdot \pi}}{2}}{\frac{e^{7.5 - z}}{2}}}}{e^{7.5 - z}}}{e^{-7.5 + z}}}{\sin \left(\pi \cdot z\right)}\right) \]
   rational_best-simplify-88 [=>] 0.5	\[ \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \left(-z\right)}\right)\right) + \left(\frac{771.3234287776531}{\left(-z\right) + 3} + \frac{-176.6150291621406}{\left(-z\right) + 4}\right)\right) + \left(\frac{12.507343278686905}{\left(-z\right) + 5} + \frac{-0.13857109526572012}{\left(-z\right) + 6}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(-z\right) + 7} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(-z\right) + 8}\right)\right) \cdot \left(\pi \cdot \frac{{\left(\left(-z\right) + 7.5\right)}^{\left(\left(-z\right) + 0.5\right)} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2 \cdot \pi}}{e^{7.5 - z}} - \frac{0}{2}}}{e^{7.5 - z}}}{e^{-7.5 + z}}}{\sin \left(\pi \cdot z\right)}\right) \]
   metadata-eval [=>] 0.5	\[ \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \left(-z\right)}\right)\right) + \left(\frac{771.3234287776531}{\left(-z\right) + 3} + \frac{-176.6150291621406}{\left(-z\right) + 4}\right)\right) + \left(\frac{12.507343278686905}{\left(-z\right) + 5} + \frac{-0.13857109526572012}{\left(-z\right) + 6}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(-z\right) + 7} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(-z\right) + 8}\right)\right) \cdot \left(\pi \cdot \frac{{\left(\left(-z\right) + 7.5\right)}^{\left(\left(-z\right) + 0.5\right)} \cdot \frac{\frac{\frac{\sqrt{2 \cdot \pi}}{e^{7.5 - z}} - \color{blue}{0}}{e^{7.5 - z}}}{e^{-7.5 + z}}}{\sin \left(\pi \cdot z\right)}\right) \]
   rational_best-simplify-9 [=>] 0.5	\[ \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \left(-z\right)}\right)\right) + \left(\frac{771.3234287776531}{\left(-z\right) + 3} + \frac{-176.6150291621406}{\left(-z\right) + 4}\right)\right) + \left(\frac{12.507343278686905}{\left(-z\right) + 5} + \frac{-0.13857109526572012}{\left(-z\right) + 6}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(-z\right) + 7} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(-z\right) + 8}\right)\right) \cdot \left(\pi \cdot \frac{{\left(\left(-z\right) + 7.5\right)}^{\left(\left(-z\right) + 0.5\right)} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2 \cdot \pi}}{e^{7.5 - z}}}}{e^{7.5 - z}}}{e^{-7.5 + z}}}{\sin \left(\pi \cdot z\right)}\right) \]
   rational_best-simplify-3 [=>] 0.5	\[ \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \left(-z\right)}\right)\right) + \left(\frac{771.3234287776531}{\left(-z\right) + 3} + \frac{-176.6150291621406}{\left(-z\right) + 4}\right)\right) + \left(\frac{12.507343278686905}{\left(-z\right) + 5} + \frac{-0.13857109526572012}{\left(-z\right) + 6}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(-z\right) + 7} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(-z\right) + 8}\right)\right) \cdot \left(\pi \cdot \frac{{\left(\left(-z\right) + 7.5\right)}^{\left(\left(-z\right) + 0.5\right)} \cdot \frac{\frac{\frac{\sqrt{2 \cdot \pi}}{e^{7.5 - z}}}{e^{7.5 - z}}}{e^{\color{blue}{z + -7.5}}}}{\sin \left(\pi \cdot z\right)}\right) \]

5. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.5
Cost	56448

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

Alternative 2
Error	0.5
Cost	49152

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

Alternative 3
Error	0.8
Cost	49088

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

Alternative 4
Error	1.3
Cost	48832

\[\pi \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} - \left(\left(\left(\frac{676.5203681218851}{z + -1} + \left(-0.9999999999998099 + \frac{771.3234287776531}{z - 3}\right)\right) - \left(\frac{-1259.1392167224028}{2 - z} + \frac{-176.6150291621406}{4 - z}\right)\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{z + -8} + \left(z \cdot -0.49644453405676175 - 2.4783734731930944\right)\right)\right)\right) \cdot \frac{\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(\left(-z\right) + 0.5\right)} \cdot e^{z + -7.5}\right)}{\sin \left(\pi \cdot z\right)}\right) \]

Alternative 5
Error	1.4
Cost	48320

\[\begin{array}{l} t_0 := \left(-z\right) + 7.5\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot \left({t_0}^{\left(\left(-z\right) + 0.5\right)} \cdot e^{-t_0}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\frac{12.507343278686905}{\left(-z\right) + 5} + \frac{-0.13857109526572012}{\left(-z\right) + 6}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(-z\right) + 7} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(-z\right) + 8}\right)\right)\right) \end{array} \]

Alternative 6
Error	1.6
Cost	48064

\[\pi \cdot \left(\left(\left(\left(260.9048120626994 + z \cdot 436.3997278161676\right) + \left(\frac{12.507343278686905}{5 - z} - \frac{0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(-z\right) + 7} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(-z\right) + 8}\right)\right) \cdot \frac{\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(\left(-z\right) + 0.5\right)} \cdot e^{z + -7.5}\right)}{\sin \left(\pi \cdot z\right)}\right) \]

Alternative 7
Error	1.7
Cost	46784

\[\pi \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} - \left(-436.8961723525769 \cdot z - 263.3831855547129\right)\right) \cdot \frac{\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(\left(-z\right) + 0.5\right)} \cdot e^{z + -7.5}\right)}{\sin \left(\pi \cdot z\right)}\right) \]

Alternative 8
Error	1.7
Cost	46400

\[\pi \cdot \left(\left(263.3831869810514 + 436.8961725563396 \cdot z\right) \cdot \frac{\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(\left(-z\right) + 0.5\right)} \cdot e^{z + -7.5}\right)}{\sin \left(\pi \cdot z\right)}\right) \]

Alternative 9
Error	2.3
Cost	46272

\[\begin{array}{l} t_0 := \left(-z\right) + 7.5\\ 263.3831869810514 \cdot \left(\pi \cdot \frac{{t_0}^{\left(\left(-z\right) + 0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-t_0}\right)}{\sin \left(\pi \cdot z\right)}\right) \end{array} \]

Alternative 10
Error	2.3
Cost	46144

\[\pi \cdot \left(\frac{\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(\left(-z\right) + 0.5\right)} \cdot e^{z + -7.5}\right)}{\sin \left(\pi \cdot z\right)} \cdot 263.3831869810514\right) \]

Alternative 11
Error	54.6
Cost	46016

\[3.4783749183518244 \cdot \frac{{\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\pi \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{-7.5 + z}\right)\right)}{\sin \left(z \cdot \pi\right)} \]

Alternative 12
Error	54.7
Cost	45568

\[\pi \cdot \left(3.4783749183518244 \cdot \frac{\sqrt{\pi \cdot 2} \cdot \left(e^{-7.5} \cdot \sqrt{7.5}\right)}{\sin \left(\pi \cdot z\right)}\right) \]

Alternative 13
Error	54.7
Cost	26112

\[3.4783749183518244 \cdot \left(\frac{e^{-7.5} \cdot \sqrt{15}}{z} \cdot \sqrt{\pi}\right) \]

Reproduce

herbie shell --seed 2023100 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z less than 0.5"
  :precision binary64
  :pre (<= z 0.5)
  (* (/ PI (sin (* PI z))) (* (* (* (sqrt (* PI 2.0)) (pow (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5) (+ (- (- 1.0 z) 1.0) 0.5))) (exp (- (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- (- 1.0 z) 1.0) 1.0))) (/ -1259.1392167224028 (+ (- (- 1.0 z) 1.0) 2.0))) (/ 771.3234287776531 (+ (- (- 1.0 z) 1.0) 3.0))) (/ -176.6150291621406 (+ (- (- 1.0 z) 1.0) 4.0))) (/ 12.507343278686905 (+ (- (- 1.0 z) 1.0) 5.0))) (/ -0.13857109526572012 (+ (- (- 1.0 z) 1.0) 6.0))) (/ 9.984369578019572e-6 (+ (- (- 1.0 z) 1.0) 7.0))) (/ 1.5056327351493116e-7 (+ (- (- 1.0 z) 1.0) 8.0))))))