Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.4% → 98.3%

Time: 1.8min

Alternatives: 9

Speedup: 1.4×

• could not determine a ground truth

  1. z = -3.8981061180876074e+99

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: 96.4% 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: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{\left(\left(z + -1\right) - -1\right) - 7.5}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 - \left(\frac{676.5203681218851}{z + -1} - \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \frac{\frac{\frac{31192.868525943773}{4 - z}}{4 - z} + \frac{\frac{594939.8317813153}{3 - z}}{z - 3}}{\frac{771.3234287776531}{z - 3} + \frac{176.6150291621406}{z - 4}}\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (/ PI (sin (* PI z)))
   (*
    (*
     (* (sqrt PI) (sqrt 2.0))
     (pow (+ (+ (- 1.0 z) -1.0) 7.5) (- (- 1.0 z) 0.5)))
    (exp (- (- (+ z -1.0) -1.0) 7.5))))
  (+
   (+
    (+
     (-
      0.9999999999998099
      (-
       (/ 676.5203681218851 (+ z -1.0))
       (/ -1259.1392167224028 (- (- 1.0 z) -1.0))))
     (/
      (+
       (/ (/ 31192.868525943773 (- 4.0 z)) (- 4.0 z))
       (/ (/ 594939.8317813153 (- 3.0 z)) (- z 3.0)))
      (+ (/ 771.3234287776531 (- z 3.0)) (/ 176.6150291621406 (- z 4.0)))))
    (+
     (/ 12.507343278686905 (- (- 1.0 z) -4.0))
     (/ -0.13857109526572012 (- (- 1.0 z) -5.0))))
   (+
    (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0))
    (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0))))))

C

double code(double z) {
	return ((((double) M_PI) / sin((((double) M_PI) * z))) * (((sqrt(((double) M_PI)) * sqrt(2.0)) * pow((((1.0 - z) + -1.0) + 7.5), ((1.0 - z) - 0.5))) * exp((((z + -1.0) - -1.0) - 7.5)))) * ((((0.9999999999998099 - ((676.5203681218851 / (z + -1.0)) - (-1259.1392167224028 / ((1.0 - z) - -1.0)))) + ((((31192.868525943773 / (4.0 - z)) / (4.0 - z)) + ((594939.8317813153 / (3.0 - z)) / (z - 3.0))) / ((771.3234287776531 / (z - 3.0)) + (176.6150291621406 / (z - 4.0))))) + ((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0)))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))));
}

Java

public static double code(double z) {
	return ((Math.PI / Math.sin((Math.PI * z))) * (((Math.sqrt(Math.PI) * Math.sqrt(2.0)) * Math.pow((((1.0 - z) + -1.0) + 7.5), ((1.0 - z) - 0.5))) * Math.exp((((z + -1.0) - -1.0) - 7.5)))) * ((((0.9999999999998099 - ((676.5203681218851 / (z + -1.0)) - (-1259.1392167224028 / ((1.0 - z) - -1.0)))) + ((((31192.868525943773 / (4.0 - z)) / (4.0 - z)) + ((594939.8317813153 / (3.0 - z)) / (z - 3.0))) / ((771.3234287776531 / (z - 3.0)) + (176.6150291621406 / (z - 4.0))))) + ((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0)))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))));
}

Python

def code(z):
	return ((math.pi / math.sin((math.pi * z))) * (((math.sqrt(math.pi) * math.sqrt(2.0)) * math.pow((((1.0 - z) + -1.0) + 7.5), ((1.0 - z) - 0.5))) * math.exp((((z + -1.0) - -1.0) - 7.5)))) * ((((0.9999999999998099 - ((676.5203681218851 / (z + -1.0)) - (-1259.1392167224028 / ((1.0 - z) - -1.0)))) + ((((31192.868525943773 / (4.0 - z)) / (4.0 - z)) + ((594939.8317813153 / (3.0 - z)) / (z - 3.0))) / ((771.3234287776531 / (z - 3.0)) + (176.6150291621406 / (z - 4.0))))) + ((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0)))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))))

Julia

function code(z)
	return Float64(Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(sqrt(pi) * sqrt(2.0)) * (Float64(Float64(Float64(1.0 - z) + -1.0) + 7.5) ^ Float64(Float64(1.0 - z) - 0.5))) * exp(Float64(Float64(Float64(z + -1.0) - -1.0) - 7.5)))) * Float64(Float64(Float64(Float64(0.9999999999998099 - Float64(Float64(676.5203681218851 / Float64(z + -1.0)) - Float64(-1259.1392167224028 / Float64(Float64(1.0 - z) - -1.0)))) + Float64(Float64(Float64(Float64(31192.868525943773 / Float64(4.0 - z)) / Float64(4.0 - z)) + Float64(Float64(594939.8317813153 / Float64(3.0 - z)) / Float64(z - 3.0))) / Float64(Float64(771.3234287776531 / Float64(z - 3.0)) + Float64(176.6150291621406 / Float64(z - 4.0))))) + Float64(Float64(12.507343278686905 / Float64(Float64(1.0 - z) - -4.0)) + Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) - -5.0)))) + Float64(Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0)) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0)))))
end

MATLAB

function tmp = code(z)
	tmp = ((pi / sin((pi * z))) * (((sqrt(pi) * sqrt(2.0)) * ((((1.0 - z) + -1.0) + 7.5) ^ ((1.0 - z) - 0.5))) * exp((((z + -1.0) - -1.0) - 7.5)))) * ((((0.9999999999998099 - ((676.5203681218851 / (z + -1.0)) - (-1259.1392167224028 / ((1.0 - z) - -1.0)))) + ((((31192.868525943773 / (4.0 - z)) / (4.0 - z)) + ((594939.8317813153 / (3.0 - z)) / (z - 3.0))) / ((771.3234287776531 / (z - 3.0)) + (176.6150291621406 / (z - 4.0))))) + ((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0)))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))));
end

Wolfram

code[z_] := N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(1.0 - z), $MachinePrecision] + -1.0), $MachinePrecision] + 7.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(N[(z + -1.0), $MachinePrecision] - -1.0), $MachinePrecision] - 7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.9999999999998099 - N[(N[(676.5203681218851 / N[(z + -1.0), $MachinePrecision]), $MachinePrecision] - N[(-1259.1392167224028 / N[(N[(1.0 - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(31192.868525943773 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(594939.8317813153 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision] / N[(z - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(771.3234287776531 / N[(z - 3.0), $MachinePrecision]), $MachinePrecision] + N[(176.6150291621406 / N[(z - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] - -4.0), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] - -5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.4%

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

3. Simplified 98.9%

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

6. Applied egg-rr 98.9%

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

   1. associate-*l/ 98.9%

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

   2. associate-*r/ 98.9%

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

   3. metadata-eval 98.7%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \frac{\frac{\frac{\color{blue}{594939.8317813153}}{3 + \left(-z\right)}}{3 + \left(-z\right)} - \frac{-176.6150291621406}{4 - z} \cdot \frac{-176.6150291621406}{4 - z}}{\frac{771.3234287776531}{3 + \left(-z\right)} - \frac{-176.6150291621406}{4 - z}}\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   4. unsub-neg 98.7%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \frac{\frac{\frac{594939.8317813153}{\color{blue}{3 - z}}}{3 + \left(-z\right)} - \frac{-176.6150291621406}{4 - z} \cdot \frac{-176.6150291621406}{4 - z}}{\frac{771.3234287776531}{3 + \left(-z\right)} - \frac{-176.6150291621406}{4 - z}}\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   5. unsub-neg 98.7%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \frac{\frac{\frac{594939.8317813153}{3 - z}}{\color{blue}{3 - z}} - \frac{-176.6150291621406}{4 - z} \cdot \frac{-176.6150291621406}{4 - z}}{\frac{771.3234287776531}{3 + \left(-z\right)} - \frac{-176.6150291621406}{4 - z}}\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   6. associate-*l/ 98.7%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \frac{\frac{\frac{594939.8317813153}{3 - z}}{3 - z} - \color{blue}{\frac{-176.6150291621406 \cdot \frac{-176.6150291621406}{4 - z}}{4 - z}}}{\frac{771.3234287776531}{3 + \left(-z\right)} - \frac{-176.6150291621406}{4 - z}}\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   7. associate-*r/ 98.7%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \frac{\frac{\frac{594939.8317813153}{3 - z}}{3 - z} - \frac{\color{blue}{\frac{-176.6150291621406 \cdot -176.6150291621406}{4 - z}}}{4 - z}}{\frac{771.3234287776531}{3 + \left(-z\right)} - \frac{-176.6150291621406}{4 - z}}\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   8. metadata-eval 98.7%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \frac{\frac{\frac{594939.8317813153}{3 - z}}{3 - z} - \frac{\frac{\color{blue}{31192.868525943773}}{4 - z}}{4 - z}}{\frac{771.3234287776531}{3 + \left(-z\right)} - \frac{-176.6150291621406}{4 - z}}\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   9. sub-neg 98.7%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \frac{\frac{\frac{594939.8317813153}{3 - z}}{3 - z} - \frac{\frac{31192.868525943773}{4 - z}}{4 - z}}{\color{blue}{\frac{771.3234287776531}{3 + \left(-z\right)} + \left(-\frac{-176.6150291621406}{4 - z}\right)}}\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   10. unsub-neg 98.7%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \frac{\frac{\frac{594939.8317813153}{3 - z}}{3 - z} - \frac{\frac{31192.868525943773}{4 - z}}{4 - z}}{\frac{771.3234287776531}{\color{blue}{3 - z}} + \left(-\frac{-176.6150291621406}{4 - z}\right)}\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   11. distribute-neg-frac 98.7%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \frac{\frac{\frac{594939.8317813153}{3 - z}}{3 - z} - \frac{\frac{31192.868525943773}{4 - z}}{4 - z}}{\frac{771.3234287776531}{3 - z} + \color{blue}{\frac{--176.6150291621406}{4 - z}}}\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   12. metadata-eval 98.7%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \frac{\frac{\frac{594939.8317813153}{3 - z}}{3 - z} - \frac{\frac{31192.868525943773}{4 - z}}{4 - z}}{\frac{771.3234287776531}{3 - z} + \frac{\color{blue}{176.6150291621406}}{4 - z}}\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

8. Simplified 98.7%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \color{blue}{\frac{\frac{\frac{594939.8317813153}{3 - z}}{3 - z} - \frac{\frac{31192.868525943773}{4 - z}}{4 - z}}{\frac{771.3234287776531}{3 - z} + \frac{176.6150291621406}{4 - z}}}\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
9. Step-by-step derivation

   1. sqrt-prod 99.3%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \frac{\frac{\frac{594939.8317813153}{3 - z}}{3 - z} - \frac{\frac{31192.868525943773}{4 - z}}{4 - z}}{\frac{771.3234287776531}{3 - z} + \frac{176.6150291621406}{4 - z}}\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

10. Applied egg-rr 99.3%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \frac{\frac{\frac{594939.8317813153}{3 - z}}{3 - z} - \frac{\frac{31192.868525943773}{4 - z}}{4 - z}}{\frac{771.3234287776531}{3 - z} + \frac{176.6150291621406}{4 - z}}\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

11. Final simplification 99.3%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{\left(\left(z + -1\right) - -1\right) - 7.5}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 - \left(\frac{676.5203681218851}{z + -1} - \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \frac{\frac{\frac{31192.868525943773}{4 - z}}{4 - z} + \frac{\frac{594939.8317813153}{3 - z}}{z - 3}}{\frac{771.3234287776531}{z - 3} + \frac{176.6150291621406}{z - 4}}\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
12. Add Preprocessing

Alternative 2: 97.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \sqrt{\pi \cdot 2}\\ t_1 := \frac{676.5203681218851}{z + -1} + \frac{-1259.1392167224028}{z - 2}\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{-8}:\\ \;\;\;\;\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \left(0.9999999999998099 - t\_1\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \frac{-0.13857109526572012}{6 - z}\right)\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{z} \cdot \left(t\_0 \cdot \left(\frac{771.3234287776531}{z - 3} + \left(\left(t\_1 + \left(\left(\frac{-0.13857109526572012}{z - 6} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - 7} + \frac{12.507343278686905}{z - 5}\right)\right) + \left(\frac{-176.6150291621406}{z - 4} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - 8}\right)\right)\right) - 0.9999999999998099\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0
         (* (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5))) (sqrt (* PI 2.0))))
        (t_1
         (+
          (/ 676.5203681218851 (+ z -1.0))
          (/ -1259.1392167224028 (- z 2.0)))))
   (if (<= z -1.9e-8)
     (*
      (*
       (/ PI (sin (* PI z)))
       (+
        (+
         (+
          (/ 1.5056327351493116e-7 (- 8.0 z))
          (/ 9.984369578019572e-6 (- 7.0 z)))
         (+ (/ 771.3234287776531 (- 3.0 z)) (- 0.9999999999998099 t_1)))
        (+
         (+ (/ -176.6150291621406 (- 4.0 z)) (/ 12.507343278686905 (- 5.0 z)))
         (/ -0.13857109526572012 (- 6.0 z)))))
      t_0)
     (*
      (/ -1.0 z)
      (*
       t_0
       (+
        (/ 771.3234287776531 (- z 3.0))
        (-
         (+
          t_1
          (+
           (+
            (/ -0.13857109526572012 (- z 6.0))
            (+
             (/ 9.984369578019572e-6 (- z 7.0))
             (/ 12.507343278686905 (- z 5.0))))
           (+
            (/ -176.6150291621406 (- z 4.0))
            (/ 1.5056327351493116e-7 (- z 8.0)))))
         0.9999999999998099)))))))

C

double code(double z) {
	double t_0 = (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5))) * sqrt((((double) M_PI) * 2.0));
	double t_1 = (676.5203681218851 / (z + -1.0)) + (-1259.1392167224028 / (z - 2.0));
	double tmp;
	if (z <= -1.9e-8) {
		tmp = ((((double) M_PI) / sin((((double) M_PI) * z))) * ((((1.5056327351493116e-7 / (8.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + ((771.3234287776531 / (3.0 - z)) + (0.9999999999998099 - t_1))) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + (-0.13857109526572012 / (6.0 - z))))) * t_0;
	} else {
		tmp = (-1.0 / z) * (t_0 * ((771.3234287776531 / (z - 3.0)) + ((t_1 + (((-0.13857109526572012 / (z - 6.0)) + ((9.984369578019572e-6 / (z - 7.0)) + (12.507343278686905 / (z - 5.0)))) + ((-176.6150291621406 / (z - 4.0)) + (1.5056327351493116e-7 / (z - 8.0))))) - 0.9999999999998099)));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5))) * Math.sqrt((Math.PI * 2.0));
	double t_1 = (676.5203681218851 / (z + -1.0)) + (-1259.1392167224028 / (z - 2.0));
	double tmp;
	if (z <= -1.9e-8) {
		tmp = ((Math.PI / Math.sin((Math.PI * z))) * ((((1.5056327351493116e-7 / (8.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + ((771.3234287776531 / (3.0 - z)) + (0.9999999999998099 - t_1))) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + (-0.13857109526572012 / (6.0 - z))))) * t_0;
	} else {
		tmp = (-1.0 / z) * (t_0 * ((771.3234287776531 / (z - 3.0)) + ((t_1 + (((-0.13857109526572012 / (z - 6.0)) + ((9.984369578019572e-6 / (z - 7.0)) + (12.507343278686905 / (z - 5.0)))) + ((-176.6150291621406 / (z - 4.0)) + (1.5056327351493116e-7 / (z - 8.0))))) - 0.9999999999998099)));
	}
	return tmp;
}

Python

def code(z):
	t_0 = (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5))) * math.sqrt((math.pi * 2.0))
	t_1 = (676.5203681218851 / (z + -1.0)) + (-1259.1392167224028 / (z - 2.0))
	tmp = 0
	if z <= -1.9e-8:
		tmp = ((math.pi / math.sin((math.pi * z))) * ((((1.5056327351493116e-7 / (8.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + ((771.3234287776531 / (3.0 - z)) + (0.9999999999998099 - t_1))) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + (-0.13857109526572012 / (6.0 - z))))) * t_0
	else:
		tmp = (-1.0 / z) * (t_0 * ((771.3234287776531 / (z - 3.0)) + ((t_1 + (((-0.13857109526572012 / (z - 6.0)) + ((9.984369578019572e-6 / (z - 7.0)) + (12.507343278686905 / (z - 5.0)))) + ((-176.6150291621406 / (z - 4.0)) + (1.5056327351493116e-7 / (z - 8.0))))) - 0.9999999999998099)))
	return tmp

Julia

function code(z)
	t_0 = Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5))) * sqrt(Float64(pi * 2.0)))
	t_1 = Float64(Float64(676.5203681218851 / Float64(z + -1.0)) + Float64(-1259.1392167224028 / Float64(z - 2.0)))
	tmp = 0.0
	if (z <= -1.9e-8)
		tmp = Float64(Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) + Float64(9.984369578019572e-6 / Float64(7.0 - z))) + Float64(Float64(771.3234287776531 / Float64(3.0 - z)) + Float64(0.9999999999998099 - t_1))) + Float64(Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) + Float64(12.507343278686905 / Float64(5.0 - z))) + Float64(-0.13857109526572012 / Float64(6.0 - z))))) * t_0);
	else
		tmp = Float64(Float64(-1.0 / z) * Float64(t_0 * Float64(Float64(771.3234287776531 / Float64(z - 3.0)) + Float64(Float64(t_1 + Float64(Float64(Float64(-0.13857109526572012 / Float64(z - 6.0)) + Float64(Float64(9.984369578019572e-6 / Float64(z - 7.0)) + Float64(12.507343278686905 / Float64(z - 5.0)))) + Float64(Float64(-176.6150291621406 / Float64(z - 4.0)) + Float64(1.5056327351493116e-7 / Float64(z - 8.0))))) - 0.9999999999998099))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5))) * sqrt((pi * 2.0));
	t_1 = (676.5203681218851 / (z + -1.0)) + (-1259.1392167224028 / (z - 2.0));
	tmp = 0.0;
	if (z <= -1.9e-8)
		tmp = ((pi / sin((pi * z))) * ((((1.5056327351493116e-7 / (8.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + ((771.3234287776531 / (3.0 - z)) + (0.9999999999998099 - t_1))) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + (-0.13857109526572012 / (6.0 - z))))) * t_0;
	else
		tmp = (-1.0 / z) * (t_0 * ((771.3234287776531 / (z - 3.0)) + ((t_1 + (((-0.13857109526572012 / (z - 6.0)) + ((9.984369578019572e-6 / (z - 7.0)) + (12.507343278686905 / (z - 5.0)))) + ((-176.6150291621406 / (z - 4.0)) + (1.5056327351493116e-7 / (z - 8.0))))) - 0.9999999999998099)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(676.5203681218851 / N[(z + -1.0), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(z - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.9e-8], N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(-1.0 / z), $MachinePrecision] * N[(t$95$0 * N[(N[(771.3234287776531 / N[(z - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 + N[(N[(N[(-0.13857109526572012 / N[(z - 6.0), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(z - 7.0), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(z - 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-176.6150291621406 / N[(z - 4.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(z - 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.9999999999998099), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.90000000000000014e-8

   1. Initial program 96.3%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in z around inf 96.0%

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

      1. exp-to-pow 96.0%

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

      2. sub-neg 96.0%

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

      3. metadata-eval 96.0%

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

   7. Simplified 96.0%

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

   if -1.90000000000000014e-8 < z

   1. Initial program 97.4%

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

   4. Applied egg-rr 97.8%

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

   6. Simplified 99.3%

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

   8. Applied egg-rr 99.3%

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

   10. Simplified 99.3%

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

   11. Taylor expanded in z around 0 99.1%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-8}:\\ \;\;\;\;\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \left(0.9999999999998099 - \left(\frac{676.5203681218851}{z + -1} + \frac{-1259.1392167224028}{z - 2}\right)\right)\right)\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \frac{-0.13857109526572012}{6 - z}\right)\right)\right) \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \sqrt{\pi \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{z} \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\frac{771.3234287776531}{z - 3} + \left(\left(\left(\frac{676.5203681218851}{z + -1} + \frac{-1259.1392167224028}{z - 2}\right) + \left(\left(\frac{-0.13857109526572012}{z - 6} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - 7} + \frac{12.507343278686905}{z - 5}\right)\right) + \left(\frac{-176.6150291621406}{z - 4} + \frac{1.5056327351493116 \cdot 10^{-7}}{z - 8}\right)\right)\right) - 0.9999999999998099\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z - 7.5}\right) \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{93.9015195213674 + z \cdot 582.6188486005177}{\left(1 - z\right) \cdot \left(2 - z\right)}\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ PI (sin (* PI z)))
  (*
   (* (* (pow (- 7.5 z) (- 0.5 z)) (exp (- z 7.5))) (sqrt (* PI 2.0)))
   (+
    (+
     (+
      (+ 0.9999999999998099 (/ 771.3234287776531 (- 3.0 z)))
      (/ (+ 93.9015195213674 (* z 582.6188486005177)) (* (- 1.0 z) (- 2.0 z))))
     (+
      (+ (/ -176.6150291621406 (- 4.0 z)) (/ 12.507343278686905 (- 5.0 z)))
      (+
       (/ -0.13857109526572012 (- 6.0 z))
       (/ 9.984369578019572e-6 (- 7.0 z)))))
    (/ 1.5056327351493116e-7 (- 8.0 z))))))

C

double code(double z) {
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (((pow((7.5 - z), (0.5 - z)) * exp((z - 7.5))) * sqrt((((double) M_PI) * 2.0))) * ((((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + ((93.9015195213674 + (z * 582.6188486005177)) / ((1.0 - z) * (2.0 - z)))) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + ((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))))) + (1.5056327351493116e-7 / (8.0 - z))));
}

Java

public static double code(double z) {
	return (Math.PI / Math.sin((Math.PI * z))) * (((Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z - 7.5))) * Math.sqrt((Math.PI * 2.0))) * ((((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + ((93.9015195213674 + (z * 582.6188486005177)) / ((1.0 - z) * (2.0 - z)))) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + ((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))))) + (1.5056327351493116e-7 / (8.0 - z))));
}

Python

def code(z):
	return (math.pi / math.sin((math.pi * z))) * (((math.pow((7.5 - z), (0.5 - z)) * math.exp((z - 7.5))) * math.sqrt((math.pi * 2.0))) * ((((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + ((93.9015195213674 + (z * 582.6188486005177)) / ((1.0 - z) * (2.0 - z)))) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + ((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))))) + (1.5056327351493116e-7 / (8.0 - z))))

Julia

function code(z)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z - 7.5))) * sqrt(Float64(pi * 2.0))) * Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(771.3234287776531 / Float64(3.0 - z))) + Float64(Float64(93.9015195213674 + Float64(z * 582.6188486005177)) / Float64(Float64(1.0 - z) * Float64(2.0 - z)))) + Float64(Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) + Float64(12.507343278686905 / Float64(5.0 - z))) + Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(9.984369578019572e-6 / Float64(7.0 - z))))) + Float64(1.5056327351493116e-7 / Float64(8.0 - z)))))
end

MATLAB

function tmp = code(z)
	tmp = (pi / sin((pi * z))) * (((((7.5 - z) ^ (0.5 - z)) * exp((z - 7.5))) * sqrt((pi * 2.0))) * ((((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + ((93.9015195213674 + (z * 582.6188486005177)) / ((1.0 - z) * (2.0 - z)))) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + ((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))))) + (1.5056327351493116e-7 / (8.0 - z))));
end

Wolfram

code[z_] := N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z - 7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.9999999999998099 + N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(93.9015195213674 + N[(z * 582.6188486005177), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - z), $MachinePrecision] * N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.4%

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

4. Applied egg-rr 97.7%

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

6. Simplified 99.2%

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

   1. frac-add 99.2%

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

   2. fma-define 99.2%

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

8. Applied egg-rr 99.2%

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

9. Taylor expanded in z around 0 99.2%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-\left(7.5 - z\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{771.3234287776531}{3 - z} + 0.9999999999998099\right) + \frac{\color{blue}{93.9015195213674 + 582.6188486005177 \cdot z}}{\left(1 - z\right) \cdot \left(2 - z\right)}\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\frac{-0.13857109526572012}{\left(-z\right) + 6} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(-z\right) + 7}\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(-z\right) + 8}\right)\right) \]

10. Final simplification 99.2%

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z - 7.5}\right) \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{93.9015195213674 + z \cdot 582.6188486005177}{\left(1 - z\right) \cdot \left(2 - z\right)}\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) \]
11. Add Preprocessing

Alternative 4: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.4%

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

4. Applied egg-rr 97.7%

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

6. Simplified 98.4%

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

   1. *-un-lft-identity 98.4%

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

   2. +-commutative 98.4%

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

8. Applied egg-rr 98.4%

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

   1. *-lft-identity 98.4%

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

   2. +-commutative 98.4%

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

   3. associate-+r+ 99.2%

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

   4. +-commutative 99.2%

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

10. Simplified 99.2%

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

11. Final simplification 99.2%

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

Alternative 5: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.4%

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

4. Applied egg-rr 97.7%

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

6. Simplified 99.2%

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

7. Taylor expanded in z around inf 99.2%

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

   1. exp-to-pow 99.2%

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

   2. sub-neg 99.2%

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

   3. metadata-eval 99.2%

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

   4. +-commutative 99.2%

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

9. Simplified 99.2%

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

10. Final simplification 99.2%

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

Alternative 6: 97.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z - 7.5}\right) \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(\left(z \cdot -10.53814559148631 - 41.65228863479777\right) - \left(\frac{-0.13857109526572012}{z - 6} + \frac{9.984369578019572 \cdot 10^{-6}}{z - 7}\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ PI (sin (* PI z)))
  (*
   (* (* (pow (- 7.5 z) (- 0.5 z)) (exp (- z 7.5))) (sqrt (* PI 2.0)))
   (+
    (/ 1.5056327351493116e-7 (- 8.0 z))
    (+
     (+
      (+ 0.9999999999998099 (/ 771.3234287776531 (- 3.0 z)))
      (+ (/ 676.5203681218851 (- 1.0 z)) (/ -1259.1392167224028 (- 2.0 z))))
     (-
      (- (* z -10.53814559148631) 41.65228863479777)
      (+
       (/ -0.13857109526572012 (- z 6.0))
       (/ 9.984369578019572e-6 (- z 7.0)))))))))

C

double code(double z) {
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (((pow((7.5 - z), (0.5 - z)) * exp((z - 7.5))) * sqrt((((double) M_PI) * 2.0))) * ((1.5056327351493116e-7 / (8.0 - z)) + (((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (((z * -10.53814559148631) - 41.65228863479777) - ((-0.13857109526572012 / (z - 6.0)) + (9.984369578019572e-6 / (z - 7.0)))))));
}

Java

public static double code(double z) {
	return (Math.PI / Math.sin((Math.PI * z))) * (((Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z - 7.5))) * Math.sqrt((Math.PI * 2.0))) * ((1.5056327351493116e-7 / (8.0 - z)) + (((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (((z * -10.53814559148631) - 41.65228863479777) - ((-0.13857109526572012 / (z - 6.0)) + (9.984369578019572e-6 / (z - 7.0)))))));
}

Python

def code(z):
	return (math.pi / math.sin((math.pi * z))) * (((math.pow((7.5 - z), (0.5 - z)) * math.exp((z - 7.5))) * math.sqrt((math.pi * 2.0))) * ((1.5056327351493116e-7 / (8.0 - z)) + (((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (((z * -10.53814559148631) - 41.65228863479777) - ((-0.13857109526572012 / (z - 6.0)) + (9.984369578019572e-6 / (z - 7.0)))))))

Julia

function code(z)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z - 7.5))) * sqrt(Float64(pi * 2.0))) * Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) + Float64(Float64(Float64(0.9999999999998099 + Float64(771.3234287776531 / Float64(3.0 - z))) + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(-1259.1392167224028 / Float64(2.0 - z)))) + Float64(Float64(Float64(z * -10.53814559148631) - 41.65228863479777) - Float64(Float64(-0.13857109526572012 / Float64(z - 6.0)) + Float64(9.984369578019572e-6 / Float64(z - 7.0))))))))
end

MATLAB

function tmp = code(z)
	tmp = (pi / sin((pi * z))) * (((((7.5 - z) ^ (0.5 - z)) * exp((z - 7.5))) * sqrt((pi * 2.0))) * ((1.5056327351493116e-7 / (8.0 - z)) + (((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (((z * -10.53814559148631) - 41.65228863479777) - ((-0.13857109526572012 / (z - 6.0)) + (9.984369578019572e-6 / (z - 7.0)))))));
end

Wolfram

code[z_] := N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z - 7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.9999999999998099 + N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(z * -10.53814559148631), $MachinePrecision] - 41.65228863479777), $MachinePrecision] - N[(N[(-0.13857109526572012 / N[(z - 6.0), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(z - 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.4%

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

4. Applied egg-rr 97.7%

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

6. Simplified 99.2%

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

7. Taylor expanded in z around 0 97.3%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-\left(7.5 - z\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{771.3234287776531}{3 - z} + 0.9999999999998099\right) + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(\color{blue}{\left(-10.53814559148631 \cdot z - 41.65228863479777\right)} + \left(\frac{-0.13857109526572012}{\left(-z\right) + 6} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(-z\right) + 7}\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(-z\right) + 8}\right)\right) \]

8. Final simplification 97.3%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z - 7.5}\right) \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(\left(z \cdot -10.53814559148631 - 41.65228863479777\right) - \left(\frac{-0.13857109526572012}{z - 6} + \frac{9.984369578019572 \cdot 10^{-6}}{z - 7}\right)\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 7: 97.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.4%

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

4. Applied egg-rr 97.7%

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

6. Simplified 99.2%

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

8. Applied egg-rr 99.2%

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

10. Simplified 99.2%

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

11. Taylor expanded in z around 0 97.0%

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

12. Final simplification 97.0%

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

Alternative 8: 95.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.4%

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

4. Applied egg-rr 97.7%

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

6. Simplified 98.4%

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

7. Taylor expanded in z around 0 95.2%

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

8. Taylor expanded in z around 0 94.9%

   \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\sqrt{2 \cdot \pi} \cdot \left(263.3831869810514 \cdot \left(e^{-7.5} \cdot \sqrt{7.5}\right)\right)\right) \]
9. Step-by-step derivation

   1. associate-*l/ 94.9%

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

   2. *-un-lft-identity 94.9%

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

10. Applied egg-rr 94.9%

    \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \pi} \cdot \left(263.3831869810514 \cdot \left(e^{-7.5} \cdot \sqrt{7.5}\right)\right)}{z}} \]
11. Step-by-step derivation

    1. associate-/l* 95.8%

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

    2. *-commutative 95.8%

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

    3. *-commutative 95.8%

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

    4. associate-*l* 94.8%

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

12. Simplified 94.8%

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

13. Final simplification 94.8%

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

Alternative 9: 96.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.4%

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

4. Applied egg-rr 97.7%

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

6. Simplified 98.4%

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

7. Taylor expanded in z around 0 95.2%

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

8. Taylor expanded in z around 0 94.9%

   \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\sqrt{2 \cdot \pi} \cdot \left(263.3831869810514 \cdot \left(e^{-7.5} \cdot \sqrt{7.5}\right)\right)\right) \]
9. Step-by-step derivation

   1. associate-*l/ 94.9%

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

   2. *-un-lft-identity 94.9%

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

10. Applied egg-rr 94.9%

    \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \pi} \cdot \left(263.3831869810514 \cdot \left(e^{-7.5} \cdot \sqrt{7.5}\right)\right)}{z}} \]
11. Step-by-step derivation

    1. associate-/l* 95.8%

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

    2. *-commutative 95.8%

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

    3. associate-*r* 95.8%

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

12. Simplified 95.8%

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

13. Final simplification 95.8%

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

Reproduce

herbie shell --seed 2024041 
(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))))))