Jmat.Real.gamma, branch z less than 0.5

Average Error: 1.7 → 0.5

Time: 1.1min

Precision: binary64

Cost: 49536

\[z \leq 0.5\]

Math

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

↓

\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{-1259.1392167224028}{2 + \left(-z\right)} + \left(\frac{-0.13857109526572012}{\left(-z\right) + 6} + \left(\left(\frac{12.507343278686905}{\left(-z\right) + 5} + \frac{771.3234287776531}{3 + \left(-z\right)}\right) + \left(\frac{-176.6150291621406}{\left(-z\right) + 4} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \left(e^{-7.5 + z} \cdot \sqrt{2 \cdot \pi}\right)\right) + 0\right) \]

FPCore

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

↓

(FPCore (z)
 :precision binary64
 (*
  (/ PI (sin (* PI z)))
  (+
   (*
    (pow (- 7.5 z) (- 0.5 z))
    (*
     (+
      0.9999999999998099
      (+
       (/ 676.5203681218851 (- 1.0 z))
       (+
        (/ -1259.1392167224028 (+ 2.0 (- z)))
        (+
         (/ -0.13857109526572012 (+ (- z) 6.0))
         (+
          (+
           (/ 12.507343278686905 (+ (- z) 5.0))
           (/ 771.3234287776531 (+ 3.0 (- z))))
          (+
           (/ -176.6150291621406 (+ (- z) 4.0))
           (+
            (/ 1.5056327351493116e-7 (- 8.0 z))
            (/ 9.984369578019572e-6 (- 7.0 z)))))))))
     (* (exp (+ -7.5 z)) (sqrt (* 2.0 PI)))))
   0.0)))

C

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

↓

double code(double z) {
	return (((double) M_PI) / sin((((double) M_PI) * z))) * ((pow((7.5 - z), (0.5 - z)) * ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 + -z)) + ((-0.13857109526572012 / (-z + 6.0)) + (((12.507343278686905 / (-z + 5.0)) + (771.3234287776531 / (3.0 + -z))) + ((-176.6150291621406 / (-z + 4.0)) + ((1.5056327351493116e-7 / (8.0 - z)) + (9.984369578019572e-6 / (7.0 - z))))))))) * (exp((-7.5 + z)) * sqrt((2.0 * ((double) M_PI)))))) + 0.0);
}

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) {
	return (Math.PI / Math.sin((Math.PI * z))) * ((Math.pow((7.5 - z), (0.5 - z)) * ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 + -z)) + ((-0.13857109526572012 / (-z + 6.0)) + (((12.507343278686905 / (-z + 5.0)) + (771.3234287776531 / (3.0 + -z))) + ((-176.6150291621406 / (-z + 4.0)) + ((1.5056327351493116e-7 / (8.0 - z)) + (9.984369578019572e-6 / (7.0 - z))))))))) * (Math.exp((-7.5 + z)) * Math.sqrt((2.0 * Math.PI))))) + 0.0);
}

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):
	return (math.pi / math.sin((math.pi * z))) * ((math.pow((7.5 - z), (0.5 - z)) * ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 + -z)) + ((-0.13857109526572012 / (-z + 6.0)) + (((12.507343278686905 / (-z + 5.0)) + (771.3234287776531 / (3.0 + -z))) + ((-176.6150291621406 / (-z + 4.0)) + ((1.5056327351493116e-7 / (8.0 - z)) + (9.984369578019572e-6 / (7.0 - z))))))))) * (math.exp((-7.5 + z)) * math.sqrt((2.0 * math.pi))))) + 0.0)

Julia

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

↓

function code(z)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * Float64(Float64(0.9999999999998099 + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(-1259.1392167224028 / Float64(2.0 + Float64(-z))) + Float64(Float64(-0.13857109526572012 / Float64(Float64(-z) + 6.0)) + Float64(Float64(Float64(12.507343278686905 / Float64(Float64(-z) + 5.0)) + Float64(771.3234287776531 / Float64(3.0 + Float64(-z)))) + Float64(Float64(-176.6150291621406 / Float64(Float64(-z) + 4.0)) + Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) + Float64(9.984369578019572e-6 / Float64(7.0 - z))))))))) * Float64(exp(Float64(-7.5 + z)) * sqrt(Float64(2.0 * pi))))) + 0.0))
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)
	tmp = (pi / sin((pi * z))) * ((((7.5 - z) ^ (0.5 - z)) * ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 + -z)) + ((-0.13857109526572012 / (-z + 6.0)) + (((12.507343278686905 / (-z + 5.0)) + (771.3234287776531 / (3.0 + -z))) + ((-176.6150291621406 / (-z + 4.0)) + ((1.5056327351493116e-7 / (8.0 - z)) + (9.984369578019572e-6 / (7.0 - z))))))))) * (exp((-7.5 + z)) * sqrt((2.0 * pi))))) + 0.0);
end

Wolfram

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

↓

code[z_] := N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[(N[(0.9999999999998099 + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(-1259.1392167224028 / N[(2.0 + (-z)), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[((-z) + 6.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(12.507343278686905 / N[((-z) + 5.0), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(3.0 + (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-176.6150291621406 / N[((-z) + 4.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(-7.5 + z), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.0), $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 1.7

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

   [Start] 1.7

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

3. Taylor expanded in z around inf 1.7

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

4. Applied egg-rr 1.0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \left(\frac{-1259.1392167224028}{2 + \left(-z\right)} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{12.507343278686905}{\left(-z\right) + 5} + \left(\left(\left(\frac{-176.6150291621406}{\left(-z\right) + 4} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{-0.13857109526572012}{\left(-z\right) + 6}\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \frac{771.3234287776531}{\left(-z\right) + 3}\right)\right)\right)\right)\right) + 0\right)} \]

5. Simplified 0.5

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{-1259.1392167224028}{2 + \left(-z\right)} + \left(\frac{-0.13857109526572012}{\left(-z\right) + 6} + \left(\left(\frac{12.507343278686905}{\left(-z\right) + 5} + \frac{771.3234287776531}{3 + \left(-z\right)}\right) + \left(\frac{-176.6150291621406}{\left(-z\right) + 4} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \left(e^{-7.5 + z} \cdot \sqrt{2 \cdot \pi}\right)\right) + 0\right)} \]
   Proof

   [Start] 1.0

   \[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \left(\frac{-1259.1392167224028}{2 + \left(-z\right)} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{12.507343278686905}{\left(-z\right) + 5} + \left(\left(\left(\frac{-176.6150291621406}{\left(-z\right) + 4} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{-0.13857109526572012}{\left(-z\right) + 6}\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \frac{771.3234287776531}{\left(-z\right) + 3}\right)\right)\right)\right)\right) + 0\right) \]

6. Final simplification 0.5

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{-1259.1392167224028}{2 + \left(-z\right)} + \left(\frac{-0.13857109526572012}{\left(-z\right) + 6} + \left(\left(\frac{12.507343278686905}{\left(-z\right) + 5} + \frac{771.3234287776531}{3 + \left(-z\right)}\right) + \left(\frac{-176.6150291621406}{\left(-z\right) + 4} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \left(e^{-7.5 + z} \cdot \sqrt{2 \cdot \pi}\right)\right) + 0\right) \]

Alternatives

Alternative 1
Error	1.1
Cost	48128

\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{2 + \left(-z\right)} + \frac{771.3234287776531}{3 + \left(-z\right)}\right) + \left(z \cdot 0.49644474017195733 + \frac{676.5203681218851}{1 - z}\right)\right) + \left(3.4783749183518244 + \frac{-176.6150291621406}{\left(-z\right) + 4}\right)\right) \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right)\right) + 0\right) \]

Alternative 2
Error	1.3
Cost	42304

\[\begin{array}{l} t_0 := 7.5 + \left(-z\right)\\ \left(\frac{1}{z} + {\pi}^{2} \cdot \left(z \cdot 0.16666666666666666\right)\right) \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {t_0}^{\left(\left(-z\right) + 0.5\right)}\right) \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(-z\right) + 8} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(-z\right) + 7}\right) + \left(\left(\frac{-0.13857109526572012}{\left(-z\right) + 6} + \left(260.9048120626994 + z \cdot 436.3997278161676\right)\right) + \frac{12.507343278686905}{\left(-z\right) + 5}\right)\right) \cdot e^{-t_0}\right)\right) \end{array} \]

Alternative 3
Error	1.4
Cost	29120

\[\begin{array}{l} t_0 := 7.5 + \left(-z\right)\\ \frac{1}{z} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {t_0}^{\left(\left(-z\right) + 0.5\right)}\right) \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(-z\right) + 8} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(-z\right) + 7}\right) + \left(\left(\frac{-0.13857109526572012}{\left(-z\right) + 6} + \left(260.9048120626994 + z \cdot 436.3997278161676\right)\right) + \frac{12.507343278686905}{\left(-z\right) + 5}\right)\right) \cdot e^{-t_0}\right)\right) \end{array} \]

Alternative 4
Error	2.5
Cost	26944

\[263.3831869810514 \cdot \left(\sqrt{2 \cdot \pi} \cdot \left(e^{-7.5 + z} \cdot \left(\frac{1}{z} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right)\right) + 0 \]

Alternative 5
Error	2.2
Cost	26944

\[\sqrt{2 \cdot \pi} \cdot \left(\frac{1}{z} \cdot \left(\left(263.3831869810514 \cdot e^{-7.5 + z}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right) + 0 \]

Alternative 6
Error	2.5
Cost	26816

\[263.3831869810514 \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot \left(e^{-7.5} \cdot \frac{1}{z}\right)\right)\right) + 0 \]

Alternative 7
Error	2.6
Cost	26368

\[\frac{1}{z} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot \sqrt{7.5}\right) \cdot \left(263.3831869810514 \cdot e^{-7.5}\right)\right) \]

Reproduce

herbie shell --seed 2023053 
(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))))))