Jmat.Real.gamma, branch z less than 0.5

Average Error: 1.7 → 0.5

Time: 2.2min

Precision: binary64

Cost: 49792

\[z \leq 0.5\]

Math

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

↓

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

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 (+ (- z) 7.5)))
   (*
    (/ PI (sin (* PI z)))
    (*
     (exp (- t_0))
     (*
      (* (sqrt (* PI 2.0)) (pow t_0 (+ (- z) 0.5)))
      (+
       (+
        (+
         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)))))))))

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 = -z + 7.5;
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (exp(-t_0) * ((sqrt((((double) M_PI) * 2.0)) * pow(t_0, (-z + 0.5))) * (((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))))));
}

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 = -z + 7.5;
	return (Math.PI / Math.sin((Math.PI * z))) * (Math.exp(-t_0) * ((Math.sqrt((Math.PI * 2.0)) * Math.pow(t_0, (-z + 0.5))) * (((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))))));
}

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 = -z + 7.5
	return (math.pi / math.sin((math.pi * z))) * (math.exp(-t_0) * ((math.sqrt((math.pi * 2.0)) * math.pow(t_0, (-z + 0.5))) * (((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))))))

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 = Float64(Float64(-z) + 7.5)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(exp(Float64(-t_0)) * Float64(Float64(sqrt(Float64(pi * 2.0)) * (t_0 ^ Float64(Float64(-z) + 0.5))) * Float64(Float64(Float64(0.9999999999998099 + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(-1259.1392167224028 / Float64(2.0 + Float64(-z))))) + Float64(Float64(Float64(771.3234287776531 / Float64(Float64(-z) + 3.0)) + Float64(-176.6150291621406 / Float64(Float64(-z) + 4.0))) + Float64(12.507343278686905 / Float64(Float64(-z) + 5.0)))) + Float64(Float64(Float64(-0.13857109526572012 / Float64(Float64(-z) + 6.0)) + Float64(9.984369578019572e-6 / Float64(Float64(-z) + 7.0))) + Float64(1.5056327351493116e-7 / Float64(Float64(-z) + 8.0)))))))
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)
	t_0 = -z + 7.5;
	tmp = (pi / sin((pi * z))) * (exp(-t_0) * ((sqrt((pi * 2.0)) * (t_0 ^ (-z + 0.5))) * (((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))))));
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[((-z) + 7.5), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-t$95$0)], $MachinePrecision] * N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$0, N[((-z) + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 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[(N[(771.3234287776531 / N[((-z) + 3.0), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[((-z) + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[((-z) + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.13857109526572012 / N[((-z) + 6.0), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[((-z) + 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[((-z) + 8.0), $MachinePrecision]), $MachinePrecision]), $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 0.5

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

   [Start] 1.7

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

   rational.json-simplify-2 [=>] 1.7

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

3. Final simplification 0.5

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

Alternatives

Alternative 1
Error	1.1
Cost	49280

\[\begin{array}{l} t_0 := \left(-z\right) + 7.5\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(e^{-t_0} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {t_0}^{\left(\left(-z\right) + 0.5\right)}\right) \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \left(-z\right)}\right)\right) + \left(\left(212.9540523020159 + z \cdot 74.66416387488323\right) + \frac{12.507343278686905}{\left(-z\right) + 5}\right)\right) + \left(\left(\frac{-0.13857109526572012}{\left(-z\right) + 6} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(-z\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(-z\right) + 8}\right)\right)\right)\right) \end{array} \]

Alternative 2
Error	1.1
Cost	41472

\[\begin{array}{l} t_0 := \left(-z\right) + 7.5\\ \left(\frac{1}{z} + {\pi}^{2} \cdot \left(0.16666666666666666 \cdot z\right)\right) \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {t_0}^{\left(\left(-z\right) + 0.5\right)}\right) \cdot \left(e^{-t_0} \cdot \left(\frac{-176.6150291621406}{\left(-z\right) + 4} + \left(\left(0.9999999999998099 + \left(447.4381671388014 \cdot z + 304.05856935323476\right)\right) + \left(0.49644474017195733 \cdot z + 2.4783749183520145\right)\right)\right)\right)\right) \end{array} \]

Alternative 3
Error	1.2
Cost	28288

\[\begin{array}{l} t_0 := \left(-z\right) + 7.5\\ \frac{1}{z} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {t_0}^{\left(\left(-z\right) + 0.5\right)}\right) \cdot \left(e^{-t_0} \cdot \left(\frac{-176.6150291621406}{\left(-z\right) + 4} + \left(\left(0.9999999999998099 + \left(447.4381671388014 \cdot z + 304.05856935323476\right)\right) + \left(0.49644474017195733 \cdot z + 2.4783749183520145\right)\right)\right)\right)\right) \end{array} \]

Alternative 4
Error	1.9
Cost	26112

\[263.3831869810514 \cdot \left(\sqrt{15} \cdot \left(\frac{e^{-7.5}}{z} \cdot \sqrt{\pi}\right)\right) \]

Alternative 5
Error	1.8
Cost	26112

\[263.3831869810514 \cdot \frac{\sqrt{15}}{\frac{z}{e^{-7.5} \cdot \sqrt{\pi}}} \]

Alternative 6
Error	2.3
Cost	19712

\[263.3831869810514 \cdot \left(e^{-7.5} \cdot \frac{\sqrt{15 \cdot \pi}}{z}\right) \]

Alternative 7
Error	2.1
Cost	19712

\[263.3831869810514 \cdot \frac{e^{-7.5} \cdot \sqrt{15 \cdot \pi}}{z} \]

Alternative 8
Error	2.0
Cost	19712

\[\frac{e^{-7.5}}{z} \cdot \left(263.3831869810514 \cdot \sqrt{15 \cdot \pi}\right) \]

Reproduce

herbie shell --seed 2023073 
(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))))))