Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.5% → 96.7%

Time: 11.1s

Alternatives: 10

Speedup: 1.7×

• could not determine a ground truth

  1. z = -2286033365029.523

Specification

\[z \leq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := t\_0 + 7\\ t_2 := t\_1 + 0.5\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {t\_2}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{-t\_2}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{t\_0 + 1}\right) + \frac{-1259.1392167224028}{t\_0 + 2}\right) + \frac{771.3234287776531}{t\_0 + 3}\right) + \frac{-176.6150291621406}{t\_0 + 4}\right) + \frac{12.507343278686905}{t\_0 + 5}\right) + \frac{-0.13857109526572012}{t\_0 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_1}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)) (t_1 (+ t_0 7.0)) (t_2 (+ t_1 0.5)))
   (*
    (/ PI (sin (* PI z)))
    (*
     (* (* (sqrt (* PI 2.0)) (pow t_2 (+ t_0 0.5))) (exp (- t_2)))
     (+
      (+
       (+
        (+
         (+
          (+
           (+
            (+ 0.9999999999998099 (/ 676.5203681218851 (+ t_0 1.0)))
            (/ -1259.1392167224028 (+ t_0 2.0)))
           (/ 771.3234287776531 (+ t_0 3.0)))
          (/ -176.6150291621406 (+ t_0 4.0)))
         (/ 12.507343278686905 (+ t_0 5.0)))
        (/ -0.13857109526572012 (+ t_0 6.0)))
       (/ 9.984369578019572e-6 t_1))
      (/ 1.5056327351493116e-7 (+ t_0 8.0)))))))

C

double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = t_0 + 7.0;
	double t_2 = t_1 + 0.5;
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (((sqrt((((double) M_PI) * 2.0)) * pow(t_2, (t_0 + 0.5))) * exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = t_0 + 7.0;
	double t_2 = t_1 + 0.5;
	return (Math.PI / Math.sin((Math.PI * z))) * (((Math.sqrt((Math.PI * 2.0)) * Math.pow(t_2, (t_0 + 0.5))) * Math.exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
}

Python

def code(z):
	t_0 = (1.0 - z) - 1.0
	t_1 = t_0 + 7.0
	t_2 = t_1 + 0.5
	return (math.pi / math.sin((math.pi * z))) * (((math.sqrt((math.pi * 2.0)) * math.pow(t_2, (t_0 + 0.5))) * math.exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - 1.0)
	t_1 = Float64(t_0 + 7.0)
	t_2 = Float64(t_1 + 0.5)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (t_2 ^ Float64(t_0 + 0.5))) * exp(Float64(-t_2))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(t_0 + 1.0))) + Float64(-1259.1392167224028 / Float64(t_0 + 2.0))) + Float64(771.3234287776531 / Float64(t_0 + 3.0))) + Float64(-176.6150291621406 / Float64(t_0 + 4.0))) + Float64(12.507343278686905 / Float64(t_0 + 5.0))) + Float64(-0.13857109526572012 / Float64(t_0 + 6.0))) + Float64(9.984369578019572e-6 / t_1)) + Float64(1.5056327351493116e-7 / Float64(t_0 + 8.0)))))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - 1.0;
	t_1 = t_0 + 7.0;
	t_2 = t_1 + 0.5;
	tmp = (pi / sin((pi * z))) * (((sqrt((pi * 2.0)) * (t_2 ^ (t_0 + 0.5))) * exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 7.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + 0.5), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$2, N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$2)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(t$95$0 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(t$95$0 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(t$95$0 + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(t$95$0 + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(t$95$0 + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Initial Program: 96.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := t\_0 + 7\\ t_2 := t\_1 + 0.5\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {t\_2}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{-t\_2}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{t\_0 + 1}\right) + \frac{-1259.1392167224028}{t\_0 + 2}\right) + \frac{771.3234287776531}{t\_0 + 3}\right) + \frac{-176.6150291621406}{t\_0 + 4}\right) + \frac{12.507343278686905}{t\_0 + 5}\right) + \frac{-0.13857109526572012}{t\_0 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_1}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)) (t_1 (+ t_0 7.0)) (t_2 (+ t_1 0.5)))
   (*
    (/ PI (sin (* PI z)))
    (*
     (* (* (sqrt (* PI 2.0)) (pow t_2 (+ t_0 0.5))) (exp (- t_2)))
     (+
      (+
       (+
        (+
         (+
          (+
           (+
            (+ 0.9999999999998099 (/ 676.5203681218851 (+ t_0 1.0)))
            (/ -1259.1392167224028 (+ t_0 2.0)))
           (/ 771.3234287776531 (+ t_0 3.0)))
          (/ -176.6150291621406 (+ t_0 4.0)))
         (/ 12.507343278686905 (+ t_0 5.0)))
        (/ -0.13857109526572012 (+ t_0 6.0)))
       (/ 9.984369578019572e-6 t_1))
      (/ 1.5056327351493116e-7 (+ t_0 8.0)))))))

C

double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = t_0 + 7.0;
	double t_2 = t_1 + 0.5;
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (((sqrt((((double) M_PI) * 2.0)) * pow(t_2, (t_0 + 0.5))) * exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = t_0 + 7.0;
	double t_2 = t_1 + 0.5;
	return (Math.PI / Math.sin((Math.PI * z))) * (((Math.sqrt((Math.PI * 2.0)) * Math.pow(t_2, (t_0 + 0.5))) * Math.exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
}

Python

def code(z):
	t_0 = (1.0 - z) - 1.0
	t_1 = t_0 + 7.0
	t_2 = t_1 + 0.5
	return (math.pi / math.sin((math.pi * z))) * (((math.sqrt((math.pi * 2.0)) * math.pow(t_2, (t_0 + 0.5))) * math.exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - 1.0)
	t_1 = Float64(t_0 + 7.0)
	t_2 = Float64(t_1 + 0.5)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (t_2 ^ Float64(t_0 + 0.5))) * exp(Float64(-t_2))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(t_0 + 1.0))) + Float64(-1259.1392167224028 / Float64(t_0 + 2.0))) + Float64(771.3234287776531 / Float64(t_0 + 3.0))) + Float64(-176.6150291621406 / Float64(t_0 + 4.0))) + Float64(12.507343278686905 / Float64(t_0 + 5.0))) + Float64(-0.13857109526572012 / Float64(t_0 + 6.0))) + Float64(9.984369578019572e-6 / t_1)) + Float64(1.5056327351493116e-7 / Float64(t_0 + 8.0)))))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - 1.0;
	t_1 = t_0 + 7.0;
	t_2 = t_1 + 0.5;
	tmp = (pi / sin((pi * z))) * (((sqrt((pi * 2.0)) * (t_2 ^ (t_0 + 0.5))) * exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 7.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + 0.5), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$2, N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$2)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(t$95$0 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(t$95$0 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(t$95$0 + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(t$95$0 + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(t$95$0 + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Alternative 1: 96.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(\frac{1}{z} - 1\right) - 1\\ t_1 := 7.5 + -1 \cdot z\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(e^{-t\_1} \cdot \left(e^{\log t\_1 \cdot \left(0.5 + -1 \cdot z\right)} \cdot \sqrt{2 \cdot \pi}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{t\_0 + 1}\right) + \frac{-1259.1392167224028}{t\_0 + 2}\right) + \frac{771.3234287776531}{t\_0 + 3}\right) + \frac{-176.6150291621406}{t\_0 + 4}\right) + \frac{12.507343278686905}{t\_0 + 5}\right) + \frac{-0.13857109526572012}{t\_0 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_0 + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (* z (- (/ 1.0 z) 1.0)) 1.0)) (t_1 (+ 7.5 (* -1.0 z))))
   (*
    (/ PI (sin (* PI z)))
    (*
     (*
      (exp (- t_1))
      (* (exp (* (log t_1) (+ 0.5 (* -1.0 z)))) (sqrt (* 2.0 PI))))
     (+
      (+
       (+
        (+
         (+
          (+
           (+
            (+ 0.9999999999998099 (/ 676.5203681218851 (+ t_0 1.0)))
            (/ -1259.1392167224028 (+ t_0 2.0)))
           (/ 771.3234287776531 (+ t_0 3.0)))
          (/ -176.6150291621406 (+ t_0 4.0)))
         (/ 12.507343278686905 (+ t_0 5.0)))
        (/ -0.13857109526572012 (+ t_0 6.0)))
       (/ 9.984369578019572e-6 (+ t_0 7.0)))
      (/ 1.5056327351493116e-7 (+ t_0 8.0)))))))

C

double code(double z) {
	double t_0 = (z * ((1.0 / z) - 1.0)) - 1.0;
	double t_1 = 7.5 + (-1.0 * z);
	return (((double) M_PI) / sin((((double) M_PI) * z))) * ((exp(-t_1) * (exp((log(t_1) * (0.5 + (-1.0 * z)))) * sqrt((2.0 * ((double) M_PI))))) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / (t_0 + 7.0))) + (1.5056327351493116e-7 / (t_0 + 8.0))));
}

Java

public static double code(double z) {
	double t_0 = (z * ((1.0 / z) - 1.0)) - 1.0;
	double t_1 = 7.5 + (-1.0 * z);
	return (Math.PI / Math.sin((Math.PI * z))) * ((Math.exp(-t_1) * (Math.exp((Math.log(t_1) * (0.5 + (-1.0 * z)))) * Math.sqrt((2.0 * Math.PI)))) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / (t_0 + 7.0))) + (1.5056327351493116e-7 / (t_0 + 8.0))));
}

Python

def code(z):
	t_0 = (z * ((1.0 / z) - 1.0)) - 1.0
	t_1 = 7.5 + (-1.0 * z)
	return (math.pi / math.sin((math.pi * z))) * ((math.exp(-t_1) * (math.exp((math.log(t_1) * (0.5 + (-1.0 * z)))) * math.sqrt((2.0 * math.pi)))) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / (t_0 + 7.0))) + (1.5056327351493116e-7 / (t_0 + 8.0))))

Julia

function code(z)
	t_0 = Float64(Float64(z * Float64(Float64(1.0 / z) - 1.0)) - 1.0)
	t_1 = Float64(7.5 + Float64(-1.0 * z))
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(exp(Float64(-t_1)) * Float64(exp(Float64(log(t_1) * Float64(0.5 + Float64(-1.0 * z)))) * sqrt(Float64(2.0 * pi)))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(t_0 + 1.0))) + Float64(-1259.1392167224028 / Float64(t_0 + 2.0))) + Float64(771.3234287776531 / Float64(t_0 + 3.0))) + Float64(-176.6150291621406 / Float64(t_0 + 4.0))) + Float64(12.507343278686905 / Float64(t_0 + 5.0))) + Float64(-0.13857109526572012 / Float64(t_0 + 6.0))) + Float64(9.984369578019572e-6 / Float64(t_0 + 7.0))) + Float64(1.5056327351493116e-7 / Float64(t_0 + 8.0)))))
end

MATLAB

function tmp = code(z)
	t_0 = (z * ((1.0 / z) - 1.0)) - 1.0;
	t_1 = 7.5 + (-1.0 * z);
	tmp = (pi / sin((pi * z))) * ((exp(-t_1) * (exp((log(t_1) * (0.5 + (-1.0 * z)))) * sqrt((2.0 * pi)))) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / (t_0 + 7.0))) + (1.5056327351493116e-7 / (t_0 + 8.0))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(z * N[(N[(1.0 / z), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(7.5 + N[(-1.0 * z), $MachinePrecision]), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Exp[(-t$95$1)], $MachinePrecision] * N[(N[Exp[N[(N[Log[t$95$1], $MachinePrecision] * N[(0.5 + N[(-1.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(t$95$0 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(t$95$0 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(t$95$0 + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(t$95$0 + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(t$95$0 + 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(t$95$0 + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 96.5%

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

2. Taylor expanded in z around inf

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 96.5%

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

5. Taylor expanded in z around inf

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

7. Applied rewrites 96.4%

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

8. Taylor expanded in z around inf

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

10. Applied rewrites 96.4%

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

11. Taylor expanded in z around inf

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

13. Applied rewrites 96.7%

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

14. Taylor expanded in z around inf

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

16. Applied rewrites 96.7%

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

17. Taylor expanded in z around inf

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

19. Applied rewrites 96.7%

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

20. Taylor expanded in z around inf

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

22. Applied rewrites 96.7%

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

23. Taylor expanded in z around inf

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

25. Applied rewrites 96.7%

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

26. Taylor expanded in z around inf

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

28. Applied rewrites 96.7%

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

29. Taylor expanded in z around inf

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

31. Applied rewrites 96.7%

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

32. Taylor expanded in z around inf

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 8}\right)\right) \]

34. Applied rewrites 96.7%

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

35. Taylor expanded in z around -inf

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\color{blue}{\left(e^{\mathsf{neg}\left(\left(\frac{15}{2} + -1 \cdot z\right)\right)} \cdot \left(e^{\log \left(\frac{15}{2} + -1 \cdot z\right) \cdot \left(\frac{1}{2} + -1 \cdot z\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 8}\right)\right) \]

37. Applied rewrites 96.7%

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

Alternative 2: 96.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(\frac{1}{z} - 1\right) - 1\\ t_1 := 7.5 + -1 \cdot z\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(e^{-t\_1} \cdot \left(e^{\log t\_1 \cdot \left(0.5 + -1 \cdot z\right)} \cdot \sqrt{2 \cdot \pi}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{t\_0 + 1}\right) + \frac{-1259.1392167224028}{t\_0 + 2}\right) + \frac{771.3234287776531}{t\_0 + 3}\right) + \frac{-176.6150291621406}{t\_0 + 4}\right) + \frac{12.507343278686905}{t\_0 + 5}\right) + \frac{-0.13857109526572012}{t\_0 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_0 + 7}\right) + \left(1.8820409189366395 \cdot 10^{-8} + 2.3525511486707994 \cdot 10^{-9} \cdot z\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (* z (- (/ 1.0 z) 1.0)) 1.0)) (t_1 (+ 7.5 (* -1.0 z))))
   (*
    (/ PI (sin (* PI z)))
    (*
     (*
      (exp (- t_1))
      (* (exp (* (log t_1) (+ 0.5 (* -1.0 z)))) (sqrt (* 2.0 PI))))
     (+
      (+
       (+
        (+
         (+
          (+
           (+
            (+ 0.9999999999998099 (/ 676.5203681218851 (+ t_0 1.0)))
            (/ -1259.1392167224028 (+ t_0 2.0)))
           (/ 771.3234287776531 (+ t_0 3.0)))
          (/ -176.6150291621406 (+ t_0 4.0)))
         (/ 12.507343278686905 (+ t_0 5.0)))
        (/ -0.13857109526572012 (+ t_0 6.0)))
       (/ 9.984369578019572e-6 (+ t_0 7.0)))
      (+ 1.8820409189366395e-8 (* 2.3525511486707994e-9 z)))))))

C

double code(double z) {
	double t_0 = (z * ((1.0 / z) - 1.0)) - 1.0;
	double t_1 = 7.5 + (-1.0 * z);
	return (((double) M_PI) / sin((((double) M_PI) * z))) * ((exp(-t_1) * (exp((log(t_1) * (0.5 + (-1.0 * z)))) * sqrt((2.0 * ((double) M_PI))))) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / (t_0 + 7.0))) + (1.8820409189366395e-8 + (2.3525511486707994e-9 * z))));
}

Java

public static double code(double z) {
	double t_0 = (z * ((1.0 / z) - 1.0)) - 1.0;
	double t_1 = 7.5 + (-1.0 * z);
	return (Math.PI / Math.sin((Math.PI * z))) * ((Math.exp(-t_1) * (Math.exp((Math.log(t_1) * (0.5 + (-1.0 * z)))) * Math.sqrt((2.0 * Math.PI)))) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / (t_0 + 7.0))) + (1.8820409189366395e-8 + (2.3525511486707994e-9 * z))));
}

Python

def code(z):
	t_0 = (z * ((1.0 / z) - 1.0)) - 1.0
	t_1 = 7.5 + (-1.0 * z)
	return (math.pi / math.sin((math.pi * z))) * ((math.exp(-t_1) * (math.exp((math.log(t_1) * (0.5 + (-1.0 * z)))) * math.sqrt((2.0 * math.pi)))) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / (t_0 + 7.0))) + (1.8820409189366395e-8 + (2.3525511486707994e-9 * z))))

Julia

function code(z)
	t_0 = Float64(Float64(z * Float64(Float64(1.0 / z) - 1.0)) - 1.0)
	t_1 = Float64(7.5 + Float64(-1.0 * z))
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(exp(Float64(-t_1)) * Float64(exp(Float64(log(t_1) * Float64(0.5 + Float64(-1.0 * z)))) * sqrt(Float64(2.0 * pi)))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(t_0 + 1.0))) + Float64(-1259.1392167224028 / Float64(t_0 + 2.0))) + Float64(771.3234287776531 / Float64(t_0 + 3.0))) + Float64(-176.6150291621406 / Float64(t_0 + 4.0))) + Float64(12.507343278686905 / Float64(t_0 + 5.0))) + Float64(-0.13857109526572012 / Float64(t_0 + 6.0))) + Float64(9.984369578019572e-6 / Float64(t_0 + 7.0))) + Float64(1.8820409189366395e-8 + Float64(2.3525511486707994e-9 * z)))))
end

MATLAB

function tmp = code(z)
	t_0 = (z * ((1.0 / z) - 1.0)) - 1.0;
	t_1 = 7.5 + (-1.0 * z);
	tmp = (pi / sin((pi * z))) * ((exp(-t_1) * (exp((log(t_1) * (0.5 + (-1.0 * z)))) * sqrt((2.0 * pi)))) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / (t_0 + 7.0))) + (1.8820409189366395e-8 + (2.3525511486707994e-9 * z))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(z * N[(N[(1.0 / z), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(7.5 + N[(-1.0 * z), $MachinePrecision]), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Exp[(-t$95$1)], $MachinePrecision] * N[(N[Exp[N[(N[Log[t$95$1], $MachinePrecision] * N[(0.5 + N[(-1.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(t$95$0 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(t$95$0 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(t$95$0 + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(t$95$0 + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(t$95$0 + 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.8820409189366395e-8 + N[(2.3525511486707994e-9 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 96.5%

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

2. Taylor expanded in z around inf

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 96.5%

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

5. Taylor expanded in z around inf

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

7. Applied rewrites 96.4%

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

8. Taylor expanded in z around inf

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

10. Applied rewrites 96.4%

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

11. Taylor expanded in z around inf

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

13. Applied rewrites 96.7%

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

14. Taylor expanded in z around inf

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

16. Applied rewrites 96.7%

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

17. Taylor expanded in z around inf

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

19. Applied rewrites 96.7%

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

20. Taylor expanded in z around inf

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

22. Applied rewrites 96.7%

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

23. Taylor expanded in z around inf

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

25. Applied rewrites 96.7%

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

26. Taylor expanded in z around inf

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

28. Applied rewrites 96.7%

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

29. Taylor expanded in z around inf

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

31. Applied rewrites 96.7%

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

32. Taylor expanded in z around inf

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 8}\right)\right) \]

34. Applied rewrites 96.7%

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

35. Taylor expanded in z around -inf

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\color{blue}{\left(e^{\mathsf{neg}\left(\left(\frac{15}{2} + -1 \cdot z\right)\right)} \cdot \left(e^{\log \left(\frac{15}{2} + -1 \cdot z\right) \cdot \left(\frac{1}{2} + -1 \cdot z\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 8}\right)\right) \]

37. Applied rewrites 96.7%

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

38. Taylor expanded in z around 0

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(e^{-\left(\frac{15}{2} + -1 \cdot z\right)} \cdot \left(e^{\log \left(\frac{15}{2} + -1 \cdot z\right) \cdot \left(\frac{1}{2} + -1 \cdot z\right)} \cdot \sqrt{2 \cdot \pi}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7}\right) + \color{blue}{\left(\frac{3764081837873279}{200000000000000000000000} + \frac{3764081837873279}{1600000000000000000000000} \cdot z\right)}\right)\right) \]

40. Applied rewrites 96.6%

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(e^{-\left(7.5 + -1 \cdot z\right)} \cdot \left(e^{\log \left(7.5 + -1 \cdot z\right) \cdot \left(0.5 + -1 \cdot z\right)} \cdot \sqrt{2 \cdot \pi}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7}\right) + \color{blue}{\left(1.8820409189366395 \cdot 10^{-8} + 2.3525511486707994 \cdot 10^{-9} \cdot z\right)}\right)\right) \]
41. Add Preprocessing

Alternative 3: 96.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := 7.5 + -1 \cdot z\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {t\_1}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{-t\_1}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{t\_0 + 1}\right) + \frac{-1259.1392167224028}{t\_0 + 2}\right) + \frac{771.3234287776531}{t\_0 + 3}\right) + \frac{-176.6150291621406}{t\_0 + 4}\right) + \frac{12.507343278686905}{t\_0 + 5}\right) + \frac{-0.13857109526572012}{t\_0 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_0 + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)) (t_1 (+ 7.5 (* -1.0 z))))
   (*
    (/ PI (sin (* PI z)))
    (*
     (* (* (sqrt (* PI 2.0)) (pow t_1 (+ t_0 0.5))) (exp (- t_1)))
     (+
      (+
       (+
        (+
         (+
          (+
           (+
            (+ 0.9999999999998099 (/ 676.5203681218851 (+ t_0 1.0)))
            (/ -1259.1392167224028 (+ t_0 2.0)))
           (/ 771.3234287776531 (+ t_0 3.0)))
          (/ -176.6150291621406 (+ t_0 4.0)))
         (/ 12.507343278686905 (+ t_0 5.0)))
        (/ -0.13857109526572012 (+ t_0 6.0)))
       (/ 9.984369578019572e-6 (+ t_0 7.0)))
      (/ 1.5056327351493116e-7 (+ t_0 8.0)))))))

C

double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = 7.5 + (-1.0 * z);
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (((sqrt((((double) M_PI) * 2.0)) * pow(t_1, (t_0 + 0.5))) * exp(-t_1)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / (t_0 + 7.0))) + (1.5056327351493116e-7 / (t_0 + 8.0))));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = 7.5 + (-1.0 * z);
	return (Math.PI / Math.sin((Math.PI * z))) * (((Math.sqrt((Math.PI * 2.0)) * Math.pow(t_1, (t_0 + 0.5))) * Math.exp(-t_1)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / (t_0 + 7.0))) + (1.5056327351493116e-7 / (t_0 + 8.0))));
}

Python

def code(z):
	t_0 = (1.0 - z) - 1.0
	t_1 = 7.5 + (-1.0 * z)
	return (math.pi / math.sin((math.pi * z))) * (((math.sqrt((math.pi * 2.0)) * math.pow(t_1, (t_0 + 0.5))) * math.exp(-t_1)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / (t_0 + 7.0))) + (1.5056327351493116e-7 / (t_0 + 8.0))))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - 1.0)
	t_1 = Float64(7.5 + Float64(-1.0 * z))
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (t_1 ^ Float64(t_0 + 0.5))) * exp(Float64(-t_1))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(t_0 + 1.0))) + Float64(-1259.1392167224028 / Float64(t_0 + 2.0))) + Float64(771.3234287776531 / Float64(t_0 + 3.0))) + Float64(-176.6150291621406 / Float64(t_0 + 4.0))) + Float64(12.507343278686905 / Float64(t_0 + 5.0))) + Float64(-0.13857109526572012 / Float64(t_0 + 6.0))) + Float64(9.984369578019572e-6 / Float64(t_0 + 7.0))) + Float64(1.5056327351493116e-7 / Float64(t_0 + 8.0)))))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - 1.0;
	t_1 = 7.5 + (-1.0 * z);
	tmp = (pi / sin((pi * z))) * (((sqrt((pi * 2.0)) * (t_1 ^ (t_0 + 0.5))) * exp(-t_1)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / (t_0 + 7.0))) + (1.5056327351493116e-7 / (t_0 + 8.0))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(7.5 + N[(-1.0 * z), $MachinePrecision]), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$1, N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$1)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(t$95$0 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(t$95$0 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(t$95$0 + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(t$95$0 + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(t$95$0 + 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(t$95$0 + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 96.5%

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

2. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\color{blue}{\left(\frac{15}{2} + -1 \cdot z\right)}}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 96.5%

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

5. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\frac{15}{2} + -1 \cdot z\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\color{blue}{\left(\frac{15}{2} + -1 \cdot z\right)}}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

7. Applied rewrites 96.5%

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

Alternative 4: 96.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 \cdot z + 7\\ t_1 := t\_0 + 0.5\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {t\_1}^{\left(-1 \cdot z + 0.5\right)}\right) \cdot e^{-t\_1}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{-1 \cdot z + 1}\right) + \frac{-1259.1392167224028}{-1 \cdot z + 2}\right) + \frac{771.3234287776531}{-1 \cdot z + 3}\right) + \frac{-176.6150291621406}{-1 \cdot z + 4}\right) + \frac{12.507343278686905}{-1 \cdot z + 5}\right) + \frac{-0.13857109526572012}{-1 \cdot z + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_0}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{-1 \cdot z + 8}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ (* -1.0 z) 7.0)) (t_1 (+ t_0 0.5)))
   (*
    (/ PI (sin (* PI z)))
    (*
     (* (* (sqrt (* PI 2.0)) (pow t_1 (+ (* -1.0 z) 0.5))) (exp (- t_1)))
     (+
      (+
       (+
        (+
         (+
          (+
           (+
            (+ 0.9999999999998099 (/ 676.5203681218851 (+ (* -1.0 z) 1.0)))
            (/ -1259.1392167224028 (+ (* -1.0 z) 2.0)))
           (/ 771.3234287776531 (+ (* -1.0 z) 3.0)))
          (/ -176.6150291621406 (+ (* -1.0 z) 4.0)))
         (/ 12.507343278686905 (+ (* -1.0 z) 5.0)))
        (/ -0.13857109526572012 (+ (* -1.0 z) 6.0)))
       (/ 9.984369578019572e-6 t_0))
      (/ 1.5056327351493116e-7 (+ (* -1.0 z) 8.0)))))))

C

double code(double z) {
	double t_0 = (-1.0 * z) + 7.0;
	double t_1 = t_0 + 0.5;
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (((sqrt((((double) M_PI) * 2.0)) * pow(t_1, ((-1.0 * z) + 0.5))) * exp(-t_1)) * ((((((((0.9999999999998099 + (676.5203681218851 / ((-1.0 * z) + 1.0))) + (-1259.1392167224028 / ((-1.0 * z) + 2.0))) + (771.3234287776531 / ((-1.0 * z) + 3.0))) + (-176.6150291621406 / ((-1.0 * z) + 4.0))) + (12.507343278686905 / ((-1.0 * z) + 5.0))) + (-0.13857109526572012 / ((-1.0 * z) + 6.0))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / ((-1.0 * z) + 8.0))));
}

Java

public static double code(double z) {
	double t_0 = (-1.0 * z) + 7.0;
	double t_1 = t_0 + 0.5;
	return (Math.PI / Math.sin((Math.PI * z))) * (((Math.sqrt((Math.PI * 2.0)) * Math.pow(t_1, ((-1.0 * z) + 0.5))) * Math.exp(-t_1)) * ((((((((0.9999999999998099 + (676.5203681218851 / ((-1.0 * z) + 1.0))) + (-1259.1392167224028 / ((-1.0 * z) + 2.0))) + (771.3234287776531 / ((-1.0 * z) + 3.0))) + (-176.6150291621406 / ((-1.0 * z) + 4.0))) + (12.507343278686905 / ((-1.0 * z) + 5.0))) + (-0.13857109526572012 / ((-1.0 * z) + 6.0))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / ((-1.0 * z) + 8.0))));
}

Python

def code(z):
	t_0 = (-1.0 * z) + 7.0
	t_1 = t_0 + 0.5
	return (math.pi / math.sin((math.pi * z))) * (((math.sqrt((math.pi * 2.0)) * math.pow(t_1, ((-1.0 * z) + 0.5))) * math.exp(-t_1)) * ((((((((0.9999999999998099 + (676.5203681218851 / ((-1.0 * z) + 1.0))) + (-1259.1392167224028 / ((-1.0 * z) + 2.0))) + (771.3234287776531 / ((-1.0 * z) + 3.0))) + (-176.6150291621406 / ((-1.0 * z) + 4.0))) + (12.507343278686905 / ((-1.0 * z) + 5.0))) + (-0.13857109526572012 / ((-1.0 * z) + 6.0))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / ((-1.0 * z) + 8.0))))

Julia

function code(z)
	t_0 = Float64(Float64(-1.0 * z) + 7.0)
	t_1 = Float64(t_0 + 0.5)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (t_1 ^ Float64(Float64(-1.0 * z) + 0.5))) * exp(Float64(-t_1))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(Float64(-1.0 * z) + 1.0))) + Float64(-1259.1392167224028 / Float64(Float64(-1.0 * z) + 2.0))) + Float64(771.3234287776531 / Float64(Float64(-1.0 * z) + 3.0))) + Float64(-176.6150291621406 / Float64(Float64(-1.0 * z) + 4.0))) + Float64(12.507343278686905 / Float64(Float64(-1.0 * z) + 5.0))) + Float64(-0.13857109526572012 / Float64(Float64(-1.0 * z) + 6.0))) + Float64(9.984369578019572e-6 / t_0)) + Float64(1.5056327351493116e-7 / Float64(Float64(-1.0 * z) + 8.0)))))
end

MATLAB

function tmp = code(z)
	t_0 = (-1.0 * z) + 7.0;
	t_1 = t_0 + 0.5;
	tmp = (pi / sin((pi * z))) * (((sqrt((pi * 2.0)) * (t_1 ^ ((-1.0 * z) + 0.5))) * exp(-t_1)) * ((((((((0.9999999999998099 + (676.5203681218851 / ((-1.0 * z) + 1.0))) + (-1259.1392167224028 / ((-1.0 * z) + 2.0))) + (771.3234287776531 / ((-1.0 * z) + 3.0))) + (-176.6150291621406 / ((-1.0 * z) + 4.0))) + (12.507343278686905 / ((-1.0 * z) + 5.0))) + (-0.13857109526572012 / ((-1.0 * z) + 6.0))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / ((-1.0 * z) + 8.0))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(-1.0 * z), $MachinePrecision] + 7.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 0.5), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$1, N[(N[(-1.0 * z), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$1)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(N[(-1.0 * z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(N[(-1.0 * z), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(N[(-1.0 * z), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(-1.0 * z), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(N[(-1.0 * z), $MachinePrecision] + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(-1.0 * z), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(-1.0 * z), $MachinePrecision] + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 96.5%

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

2. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\color{blue}{-1 \cdot z} + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 96.5%

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

5. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}^{\left(\color{blue}{-1 \cdot z} + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

7. Applied rewrites 96.5%

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

8. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}^{\left(-1 \cdot z + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\color{blue}{-1 \cdot z} + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

10. Applied rewrites 96.5%

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

11. Taylor expanded in z around 0

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}^{\left(-1 \cdot z + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\color{blue}{-1 \cdot z} + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

13. Applied rewrites 96.5%

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

14. Taylor expanded in z around 0

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}^{\left(-1 \cdot z + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{-1 \cdot z + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\color{blue}{-1 \cdot z} + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

16. Applied rewrites 96.5%

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

17. Taylor expanded in z around 0

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}^{\left(-1 \cdot z + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{-1 \cdot z + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{-1 \cdot z + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\color{blue}{-1 \cdot z} + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

19. Applied rewrites 96.5%

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

20. Taylor expanded in z around 0

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}^{\left(-1 \cdot z + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{-1 \cdot z + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{-1 \cdot z + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{-1 \cdot z + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\color{blue}{-1 \cdot z} + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

22. Applied rewrites 96.5%

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

23. Taylor expanded in z around 0

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}^{\left(-1 \cdot z + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{-1 \cdot z + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{-1 \cdot z + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{-1 \cdot z + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{-1 \cdot z + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\color{blue}{-1 \cdot z} + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

25. Applied rewrites 96.5%

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

26. Taylor expanded in z around 0

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}^{\left(-1 \cdot z + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{-1 \cdot z + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{-1 \cdot z + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{-1 \cdot z + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{-1 \cdot z + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{-1 \cdot z + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\color{blue}{-1 \cdot z} + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

28. Applied rewrites 96.5%

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

29. Taylor expanded in z around 0

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}^{\left(-1 \cdot z + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{-1 \cdot z + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{-1 \cdot z + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{-1 \cdot z + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{-1 \cdot z + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{-1 \cdot z + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{-1 \cdot z + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\color{blue}{-1 \cdot z} + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

31. Applied rewrites 96.5%

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

32. Taylor expanded in z around 0

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}^{\left(-1 \cdot z + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{-1 \cdot z + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{-1 \cdot z + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{-1 \cdot z + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{-1 \cdot z + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{-1 \cdot z + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{-1 \cdot z + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{-1 \cdot z + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\color{blue}{-1 \cdot z} + 8}\right)\right) \]

34. Applied rewrites 96.5%

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

Alternative 5: 96.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := \left(t\_0 + 7\right) + 0.5\\ \frac{1 + 0.16666666666666666 \cdot \left({z}^{2} \cdot {\pi}^{2}\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {t\_1}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{-t\_1}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)) (t_1 (+ (+ t_0 7.0) 0.5)))
   (*
    (/ (+ 1.0 (* 0.16666666666666666 (* (pow z 2.0) (pow PI 2.0)))) z)
    (*
     (* (* (sqrt (* PI 2.0)) (pow t_1 (+ t_0 0.5))) (exp (- t_1)))
     (+
      263.3831869810514
      (*
       z
       (+
        436.8961725563396
        (* z (+ 545.0353078428827 (* 606.6766809167608 z))))))))))

C

double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = (t_0 + 7.0) + 0.5;
	return ((1.0 + (0.16666666666666666 * (pow(z, 2.0) * pow(((double) M_PI), 2.0)))) / z) * (((sqrt((((double) M_PI) * 2.0)) * pow(t_1, (t_0 + 0.5))) * exp(-t_1)) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z)))))));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = (t_0 + 7.0) + 0.5;
	return ((1.0 + (0.16666666666666666 * (Math.pow(z, 2.0) * Math.pow(Math.PI, 2.0)))) / z) * (((Math.sqrt((Math.PI * 2.0)) * Math.pow(t_1, (t_0 + 0.5))) * Math.exp(-t_1)) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z)))))));
}

Python

def code(z):
	t_0 = (1.0 - z) - 1.0
	t_1 = (t_0 + 7.0) + 0.5
	return ((1.0 + (0.16666666666666666 * (math.pow(z, 2.0) * math.pow(math.pi, 2.0)))) / z) * (((math.sqrt((math.pi * 2.0)) * math.pow(t_1, (t_0 + 0.5))) * math.exp(-t_1)) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z)))))))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - 1.0)
	t_1 = Float64(Float64(t_0 + 7.0) + 0.5)
	return Float64(Float64(Float64(1.0 + Float64(0.16666666666666666 * Float64((z ^ 2.0) * (pi ^ 2.0)))) / z) * Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (t_1 ^ Float64(t_0 + 0.5))) * exp(Float64(-t_1))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * Float64(545.0353078428827 + Float64(606.6766809167608 * z))))))))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - 1.0;
	t_1 = (t_0 + 7.0) + 0.5;
	tmp = ((1.0 + (0.16666666666666666 * ((z ^ 2.0) * (pi ^ 2.0)))) / z) * (((sqrt((pi * 2.0)) * (t_1 ^ (t_0 + 0.5))) * exp(-t_1)) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z)))))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + 7.0), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(N[(N[(1.0 + N[(0.16666666666666666 * N[(N[Power[z, 2.0], $MachinePrecision] * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$1, N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$1)], $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * N[(545.0353078428827 + N[(606.6766809167608 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 96.5%

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

2. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)}\right) \]

4. Applied rewrites 96.7%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)}\right) \]

5. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1 + \frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]

7. Applied rewrites 96.6%

   \[\leadsto \color{blue}{\frac{1 + 0.16666666666666666 \cdot \left({z}^{2} \cdot {\pi}^{2}\right)}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)\right) \]
8. Add Preprocessing

Alternative 6: 96.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := \left(t\_0 + 7\right) + 0.5\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {t\_1}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{-t\_1}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)) (t_1 (+ (+ t_0 7.0) 0.5)))
   (*
    (/ PI (sin (* PI z)))
    (*
     (* (* (sqrt (* PI 2.0)) (pow t_1 (+ t_0 0.5))) (exp (- t_1)))
     (+
      263.3831869810514
      (* z (+ 436.8961725563396 (* 545.0353078428827 z))))))))

C

double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = (t_0 + 7.0) + 0.5;
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (((sqrt((((double) M_PI) * 2.0)) * pow(t_1, (t_0 + 0.5))) * exp(-t_1)) * (263.3831869810514 + (z * (436.8961725563396 + (545.0353078428827 * z)))));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = (t_0 + 7.0) + 0.5;
	return (Math.PI / Math.sin((Math.PI * z))) * (((Math.sqrt((Math.PI * 2.0)) * Math.pow(t_1, (t_0 + 0.5))) * Math.exp(-t_1)) * (263.3831869810514 + (z * (436.8961725563396 + (545.0353078428827 * z)))));
}

Python

def code(z):
	t_0 = (1.0 - z) - 1.0
	t_1 = (t_0 + 7.0) + 0.5
	return (math.pi / math.sin((math.pi * z))) * (((math.sqrt((math.pi * 2.0)) * math.pow(t_1, (t_0 + 0.5))) * math.exp(-t_1)) * (263.3831869810514 + (z * (436.8961725563396 + (545.0353078428827 * z)))))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - 1.0)
	t_1 = Float64(Float64(t_0 + 7.0) + 0.5)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (t_1 ^ Float64(t_0 + 0.5))) * exp(Float64(-t_1))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(545.0353078428827 * z))))))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - 1.0;
	t_1 = (t_0 + 7.0) + 0.5;
	tmp = (pi / sin((pi * z))) * (((sqrt((pi * 2.0)) * (t_1 ^ (t_0 + 0.5))) * exp(-t_1)) * (263.3831869810514 + (z * (436.8961725563396 + (545.0353078428827 * z)))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + 7.0), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$1, N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$1)], $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(545.0353078428827 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 96.5%

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

2. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)}\right) \]

4. Applied rewrites 96.6%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\right)}\right) \]
5. Add Preprocessing

Alternative 7: 96.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 7.5 + -1 \cdot z\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(e^{-t\_0} \cdot \left(e^{\log t\_0 \cdot \left(0.5 + -1 \cdot z\right)} \cdot \sqrt{2 \cdot \pi}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ 7.5 (* -1.0 z))))
   (*
    (/ PI (sin (* PI z)))
    (*
     (*
      (exp (- t_0))
      (* (exp (* (log t_0) (+ 0.5 (* -1.0 z)))) (sqrt (* 2.0 PI))))
     (+
      263.3831869810514
      (* z (+ 436.8961725563396 (* 545.0353078428827 z))))))))

C

double code(double z) {
	double t_0 = 7.5 + (-1.0 * z);
	return (((double) M_PI) / sin((((double) M_PI) * z))) * ((exp(-t_0) * (exp((log(t_0) * (0.5 + (-1.0 * z)))) * sqrt((2.0 * ((double) M_PI))))) * (263.3831869810514 + (z * (436.8961725563396 + (545.0353078428827 * z)))));
}

Java

public static double code(double z) {
	double t_0 = 7.5 + (-1.0 * z);
	return (Math.PI / Math.sin((Math.PI * z))) * ((Math.exp(-t_0) * (Math.exp((Math.log(t_0) * (0.5 + (-1.0 * z)))) * Math.sqrt((2.0 * Math.PI)))) * (263.3831869810514 + (z * (436.8961725563396 + (545.0353078428827 * z)))));
}

Python

def code(z):
	t_0 = 7.5 + (-1.0 * z)
	return (math.pi / math.sin((math.pi * z))) * ((math.exp(-t_0) * (math.exp((math.log(t_0) * (0.5 + (-1.0 * z)))) * math.sqrt((2.0 * math.pi)))) * (263.3831869810514 + (z * (436.8961725563396 + (545.0353078428827 * z)))))

Julia

function code(z)
	t_0 = Float64(7.5 + Float64(-1.0 * z))
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(exp(Float64(-t_0)) * Float64(exp(Float64(log(t_0) * Float64(0.5 + Float64(-1.0 * z)))) * sqrt(Float64(2.0 * pi)))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(545.0353078428827 * z))))))
end

MATLAB

function tmp = code(z)
	t_0 = 7.5 + (-1.0 * z);
	tmp = (pi / sin((pi * z))) * ((exp(-t_0) * (exp((log(t_0) * (0.5 + (-1.0 * z)))) * sqrt((2.0 * pi)))) * (263.3831869810514 + (z * (436.8961725563396 + (545.0353078428827 * z)))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(7.5 + N[(-1.0 * z), $MachinePrecision]), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Exp[(-t$95$0)], $MachinePrecision] * N[(N[Exp[N[(N[Log[t$95$0], $MachinePrecision] * N[(0.5 + N[(-1.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(545.0353078428827 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 96.5%

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

2. Taylor expanded in z around inf

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 96.5%

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

5. Taylor expanded in z around inf

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

7. Applied rewrites 96.4%

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

8. Taylor expanded in z around inf

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

10. Applied rewrites 96.4%

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

11. Taylor expanded in z around inf

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

13. Applied rewrites 96.7%

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

14. Taylor expanded in z around inf

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

16. Applied rewrites 96.7%

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

17. Taylor expanded in z around inf

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

19. Applied rewrites 96.7%

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

20. Taylor expanded in z around inf

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

22. Applied rewrites 96.7%

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

23. Taylor expanded in z around inf

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

25. Applied rewrites 96.7%

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

26. Taylor expanded in z around inf

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

28. Applied rewrites 96.7%

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

29. Taylor expanded in z around inf

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

31. Applied rewrites 96.7%

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

32. Taylor expanded in z around inf

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\color{blue}{z \cdot \left(\frac{1}{z} - 1\right)} - 1\right) + 8}\right)\right) \]

34. Applied rewrites 96.7%

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

35. Taylor expanded in z around -inf

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\color{blue}{\left(e^{\mathsf{neg}\left(\left(\frac{15}{2} + -1 \cdot z\right)\right)} \cdot \left(e^{\log \left(\frac{15}{2} + -1 \cdot z\right) \cdot \left(\frac{1}{2} + -1 \cdot z\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(z \cdot \left(\frac{1}{z} - 1\right) - 1\right) + 8}\right)\right) \]

37. Applied rewrites 96.7%

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

38. Taylor expanded in z around 0

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(e^{-\left(\frac{15}{2} + -1 \cdot z\right)} \cdot \left(e^{\log \left(\frac{15}{2} + -1 \cdot z\right) \cdot \left(\frac{1}{2} + -1 \cdot z\right)} \cdot \sqrt{2 \cdot \pi}\right)\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)}\right) \]

40. Applied rewrites 96.6%

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(e^{-\left(7.5 + -1 \cdot z\right)} \cdot \left(e^{\log \left(7.5 + -1 \cdot z\right) \cdot \left(0.5 + -1 \cdot z\right)} \cdot \sqrt{2 \cdot \pi}\right)\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\right)}\right) \]
41. Add Preprocessing

Alternative 8: 96.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := \left(t\_0 + 7\right) + 0.5\\ \frac{1}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {t\_1}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{-t\_1}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)) (t_1 (+ (+ t_0 7.0) 0.5)))
   (*
    (/ 1.0 z)
    (*
     (* (* (sqrt (* PI 2.0)) (pow t_1 (+ t_0 0.5))) (exp (- t_1)))
     (+
      263.3831869810514
      (*
       z
       (+
        436.8961725563396
        (* z (+ 545.0353078428827 (* 606.6766809167608 z))))))))))

C

double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = (t_0 + 7.0) + 0.5;
	return (1.0 / z) * (((sqrt((((double) M_PI) * 2.0)) * pow(t_1, (t_0 + 0.5))) * exp(-t_1)) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z)))))));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = (t_0 + 7.0) + 0.5;
	return (1.0 / z) * (((Math.sqrt((Math.PI * 2.0)) * Math.pow(t_1, (t_0 + 0.5))) * Math.exp(-t_1)) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z)))))));
}

Python

def code(z):
	t_0 = (1.0 - z) - 1.0
	t_1 = (t_0 + 7.0) + 0.5
	return (1.0 / z) * (((math.sqrt((math.pi * 2.0)) * math.pow(t_1, (t_0 + 0.5))) * math.exp(-t_1)) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z)))))))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - 1.0)
	t_1 = Float64(Float64(t_0 + 7.0) + 0.5)
	return Float64(Float64(1.0 / z) * Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (t_1 ^ Float64(t_0 + 0.5))) * exp(Float64(-t_1))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * Float64(545.0353078428827 + Float64(606.6766809167608 * z))))))))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - 1.0;
	t_1 = (t_0 + 7.0) + 0.5;
	tmp = (1.0 / z) * (((sqrt((pi * 2.0)) * (t_1 ^ (t_0 + 0.5))) * exp(-t_1)) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z)))))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + 7.0), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(N[(1.0 / z), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$1, N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$1)], $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * N[(545.0353078428827 + N[(606.6766809167608 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 96.5%

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

2. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)}\right) \]

4. Applied rewrites 96.7%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)}\right) \]

5. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]

7. Applied rewrites 96.1%

   \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)\right) \]
8. Add Preprocessing

Alternative 9: 96.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := \left(t\_0 + 7\right) + 0.5\\ \frac{1}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {t\_1}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{-t\_1}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)) (t_1 (+ (+ t_0 7.0) 0.5)))
   (*
    (/ 1.0 z)
    (*
     (* (* (sqrt (* PI 2.0)) (pow t_1 (+ t_0 0.5))) (exp (- t_1)))
     (+
      263.3831869810514
      (* z (+ 436.8961725563396 (* 545.0353078428827 z))))))))

C

double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = (t_0 + 7.0) + 0.5;
	return (1.0 / z) * (((sqrt((((double) M_PI) * 2.0)) * pow(t_1, (t_0 + 0.5))) * exp(-t_1)) * (263.3831869810514 + (z * (436.8961725563396 + (545.0353078428827 * z)))));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	double t_1 = (t_0 + 7.0) + 0.5;
	return (1.0 / z) * (((Math.sqrt((Math.PI * 2.0)) * Math.pow(t_1, (t_0 + 0.5))) * Math.exp(-t_1)) * (263.3831869810514 + (z * (436.8961725563396 + (545.0353078428827 * z)))));
}

Python

def code(z):
	t_0 = (1.0 - z) - 1.0
	t_1 = (t_0 + 7.0) + 0.5
	return (1.0 / z) * (((math.sqrt((math.pi * 2.0)) * math.pow(t_1, (t_0 + 0.5))) * math.exp(-t_1)) * (263.3831869810514 + (z * (436.8961725563396 + (545.0353078428827 * z)))))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - 1.0)
	t_1 = Float64(Float64(t_0 + 7.0) + 0.5)
	return Float64(Float64(1.0 / z) * Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (t_1 ^ Float64(t_0 + 0.5))) * exp(Float64(-t_1))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(545.0353078428827 * z))))))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - 1.0;
	t_1 = (t_0 + 7.0) + 0.5;
	tmp = (1.0 / z) * (((sqrt((pi * 2.0)) * (t_1 ^ (t_0 + 0.5))) * exp(-t_1)) * (263.3831869810514 + (z * (436.8961725563396 + (545.0353078428827 * z)))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + 7.0), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(N[(1.0 / z), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$1, N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$1)], $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(545.0353078428827 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 96.5%

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

2. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)}\right) \]

4. Applied rewrites 96.6%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\right)}\right) \]

5. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot z\right)\right)\right) \]

7. Applied rewrites 96.1%

   \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\right)\right) \]
8. Add Preprocessing

Alternative 10: 95.4% accurate, 7.5× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (* 263.3831869810514 (/ (* (exp -7.5) (* (sqrt 7.5) (sqrt (* 2.0 PI)))) z)))

C

double code(double z) {
	return 263.3831869810514 * ((exp(-7.5) * (sqrt(7.5) * sqrt((2.0 * ((double) M_PI))))) / z);
}

Java

public static double code(double z) {
	return 263.3831869810514 * ((Math.exp(-7.5) * (Math.sqrt(7.5) * Math.sqrt((2.0 * Math.PI)))) / z);
}

Python

def code(z):
	return 263.3831869810514 * ((math.exp(-7.5) * (math.sqrt(7.5) * math.sqrt((2.0 * math.pi)))) / z)

Julia

function code(z)
	return Float64(263.3831869810514 * Float64(Float64(exp(-7.5) * Float64(sqrt(7.5) * sqrt(Float64(2.0 * pi)))) / z))
end

MATLAB

function tmp = code(z)
	tmp = 263.3831869810514 * ((exp(-7.5) * (sqrt(7.5) * sqrt((2.0 * pi)))) / z);
end

Wolfram

code[z_] := N[(263.3831869810514 * N[(N[(N[Exp[-7.5], $MachinePrecision] * N[(N[Sqrt[7.5], $MachinePrecision] * N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.5%

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

2. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)}{z}} \]

4. Applied rewrites 95.4%

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

5. Taylor expanded in z around 0

   \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{\frac{15}{2}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)}{\color{blue}{z}} \]

7. Applied rewrites 95.4%

   \[\leadsto 263.3831869810514 \cdot \frac{e^{-7.5} \cdot \left(\sqrt{7.5} \cdot \sqrt{2 \cdot \pi}\right)}{\color{blue}{z}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025160 
(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))))))