Jmat.Real.gamma, branch z less than 0.5

Average Error: 1.7 → 0.4

Time: 1.9min

Precision: binary64

Cost: 70016

\[z \leq 0.5\]

Math

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

↓

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

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 (+ (- 1.0 z) 6.5)))
   (*
    (*
     (pow t_0 (+ (- 1.0 z) -0.5))
     (+
      (+
       (/ 9.984369578019572e-6 (+ (- 1.0 z) 6.0))
       (/ 1.5056327351493116e-7 (+ (- 1.0 z) 7.0)))
      (+
       (+
        (+
         (/ -176.6150291621406 (+ 3.0 (- 1.0 z)))
         (+ 0.9999999999998099 (/ 676.5203681218851 (- 1.0 z))))
        (+
         (/ 12.507343278686905 (+ (- 1.0 z) 4.0))
         (/ -0.13857109526572012 (+ (- 1.0 z) 5.0))))
       (+
        (/ -1259.1392167224028 (+ 1.0 (- 1.0 z)))
        (/ 771.3234287776531 (+ 2.0 (- 1.0 z)))))))
    (*
     (/ PI (sin (* PI z)))
     (*
      (* (/ 1.0 (sqrt (* PI 2.0))) (sqrt (* PI (* 4.0 PI))))
      (exp (- t_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 = (1.0 - z) + 6.5;
	return (pow(t_0, ((1.0 - z) + -0.5)) * (((9.984369578019572e-6 / ((1.0 - z) + 6.0)) + (1.5056327351493116e-7 / ((1.0 - z) + 7.0))) + ((((-176.6150291621406 / (3.0 + (1.0 - z))) + (0.9999999999998099 + (676.5203681218851 / (1.0 - z)))) + ((12.507343278686905 / ((1.0 - z) + 4.0)) + (-0.13857109526572012 / ((1.0 - z) + 5.0)))) + ((-1259.1392167224028 / (1.0 + (1.0 - z))) + (771.3234287776531 / (2.0 + (1.0 - z))))))) * ((((double) M_PI) / sin((((double) M_PI) * z))) * (((1.0 / sqrt((((double) M_PI) * 2.0))) * sqrt((((double) M_PI) * (4.0 * ((double) M_PI))))) * exp(-t_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 = (1.0 - z) + 6.5;
	return (Math.pow(t_0, ((1.0 - z) + -0.5)) * (((9.984369578019572e-6 / ((1.0 - z) + 6.0)) + (1.5056327351493116e-7 / ((1.0 - z) + 7.0))) + ((((-176.6150291621406 / (3.0 + (1.0 - z))) + (0.9999999999998099 + (676.5203681218851 / (1.0 - z)))) + ((12.507343278686905 / ((1.0 - z) + 4.0)) + (-0.13857109526572012 / ((1.0 - z) + 5.0)))) + ((-1259.1392167224028 / (1.0 + (1.0 - z))) + (771.3234287776531 / (2.0 + (1.0 - z))))))) * ((Math.PI / Math.sin((Math.PI * z))) * (((1.0 / Math.sqrt((Math.PI * 2.0))) * Math.sqrt((Math.PI * (4.0 * Math.PI)))) * Math.exp(-t_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 = (1.0 - z) + 6.5
	return (math.pow(t_0, ((1.0 - z) + -0.5)) * (((9.984369578019572e-6 / ((1.0 - z) + 6.0)) + (1.5056327351493116e-7 / ((1.0 - z) + 7.0))) + ((((-176.6150291621406 / (3.0 + (1.0 - z))) + (0.9999999999998099 + (676.5203681218851 / (1.0 - z)))) + ((12.507343278686905 / ((1.0 - z) + 4.0)) + (-0.13857109526572012 / ((1.0 - z) + 5.0)))) + ((-1259.1392167224028 / (1.0 + (1.0 - z))) + (771.3234287776531 / (2.0 + (1.0 - z))))))) * ((math.pi / math.sin((math.pi * z))) * (((1.0 / math.sqrt((math.pi * 2.0))) * math.sqrt((math.pi * (4.0 * math.pi)))) * math.exp(-t_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(1.0 - z) + 6.5)
	return Float64(Float64((t_0 ^ Float64(Float64(1.0 - z) + -0.5)) * Float64(Float64(Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) + 6.0)) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) + 7.0))) + Float64(Float64(Float64(Float64(-176.6150291621406 / Float64(3.0 + Float64(1.0 - z))) + Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 - z)))) + Float64(Float64(12.507343278686905 / Float64(Float64(1.0 - z) + 4.0)) + Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) + 5.0)))) + Float64(Float64(-1259.1392167224028 / Float64(1.0 + Float64(1.0 - z))) + Float64(771.3234287776531 / Float64(2.0 + Float64(1.0 - z))))))) * Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(1.0 / sqrt(Float64(pi * 2.0))) * sqrt(Float64(pi * Float64(4.0 * pi)))) * exp(Float64(-t_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 = (1.0 - z) + 6.5;
	tmp = ((t_0 ^ ((1.0 - z) + -0.5)) * (((9.984369578019572e-6 / ((1.0 - z) + 6.0)) + (1.5056327351493116e-7 / ((1.0 - z) + 7.0))) + ((((-176.6150291621406 / (3.0 + (1.0 - z))) + (0.9999999999998099 + (676.5203681218851 / (1.0 - z)))) + ((12.507343278686905 / ((1.0 - z) + 4.0)) + (-0.13857109526572012 / ((1.0 - z) + 5.0)))) + ((-1259.1392167224028 / (1.0 + (1.0 - z))) + (771.3234287776531 / (2.0 + (1.0 - z))))))) * ((pi / sin((pi * z))) * (((1.0 / sqrt((pi * 2.0))) * sqrt((pi * (4.0 * pi)))) * exp(-t_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[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision]}, N[(N[(N[Power[t$95$0, N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] + 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(-176.6150291621406 / N[(3.0 + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1259.1392167224028 / N[(1.0 + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(2.0 + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(Pi * N[(4.0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$0)], $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 0.9

   \[\leadsto \color{blue}{\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{1 + \left(1 - z\right)} + \frac{771.3234287776531}{2 + \left(1 - z\right)}\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right)\right) + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)} \]
   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_best-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. Applied egg-rr 1.5

   \[\leadsto \color{blue}{\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right)\right) + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\left(\frac{-1259.1392167224028}{1 + \left(1 - z\right)} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right) + 0\right)} \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \]

4. Simplified 1.0

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

   [Start] 1.5	\[ \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right)\right) + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\left(\frac{-1259.1392167224028}{1 + \left(1 - z\right)} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right) + 0\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \]
   rational_best-simplify-4 [=>] 1.5	\[ \color{blue}{\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right)\right) + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\left(\frac{-1259.1392167224028}{1 + \left(1 - z\right)} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right)} \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \]
   rational_best-simplify-43 [=>] 1.5	\[ \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right) + \left(\left(\frac{-1259.1392167224028}{1 + \left(1 - z\right)} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right)\right) + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right)\right)\right)}\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \]
   rational_best-simplify-1 [=>] 1.5	\[ \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right) + \color{blue}{\left(\left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right)\right) + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{-1259.1392167224028}{1 + \left(1 - z\right)} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right)}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \]

5. Applied egg-rr 0.4

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

6. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.5
Cost	50432

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

Alternative 2
Error	0.7
Cost	50240

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

Alternative 3
Error	0.5
Cost	50240

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

Alternative 4
Error	0.7
Cost	50112

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

Alternative 5
Error	1.3
Cost	49792

\[\begin{array}{l} t_0 := \left(1 - z\right) + 6.5\\ \left(\sqrt{\pi \cdot 2} \cdot {t_0}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{1 + \left(1 - z\right)} + \frac{771.3234287776531}{2 + \left(1 - z\right)}\right) + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(z \cdot -10.53814559148631 - 41.65228863479777\right)\right)\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{-t_0}\right)\right) \end{array} \]

Alternative 6
Error	1.3
Cost	49600

\[\begin{array}{l} t_0 := \left(1 - z\right) + 6.5\\ \left(\sqrt{\pi \cdot 2} \cdot {t_0}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 + \left(-z\right)}\right) + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(z \cdot -10.53814559148631 - 41.65228863479777\right)\right)\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{-t_0}\right)\right) \end{array} \]

Alternative 7
Error	1.4
Cost	48768

\[\begin{array}{l} t_0 := \left(1 - z\right) + 6.5\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot \left({t_0}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-t_0}\right)\right) \cdot \left(\left(\frac{771.3234287776531}{2 + \left(1 - z\right)} + \left(6.298471125885741 + z \cdot 351.1974183497981\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \end{array} \]

Alternative 8
Error	1.8
Cost	47744

\[\begin{array}{l} t_0 := \left(1 - z\right) + 6.5\\ \left({t_0}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right) + \left(263.3831855358925 + 436.8961723502244 \cdot z\right)\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-t_0}\right)\right) \end{array} \]

Alternative 9
Error	1.9
Cost	45440

\[263.3831869810514 \cdot \left(\frac{e^{-7.5} \cdot \sqrt{15}}{\sin \left(z \cdot \pi\right)} \cdot \sqrt{{\pi}^{3}}\right) \]

Alternative 10
Error	55.0
Cost	26112

\[0.9999999999998099 \cdot \left(\frac{e^{-7.5} \cdot \sqrt{15}}{z} \cdot \sqrt{\pi}\right) \]

Alternative 11
Error	2.2
Cost	26112

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

Alternative 12
Error	2.1
Cost	26112

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

Reproduce

herbie shell --seed 2023092 
(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))))))