Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.4% → 98.3%

Time: 57.9s

Alternatives: 11

Speedup: 1.5×

• could not determine a ground truth

  1. z = -5.853391114518271e+158

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, 1.1× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ PI (sin (* PI z)))
  (*
   (* (sqrt (* PI 2.0)) (* (exp (+ z -7.5)) (pow (- 7.5 z) (- 0.5 z))))
   (+
    (/ 676.5203681218851 (- 1.0 z))
    (+
     (+
      (+
       (+ 0.9999999999998099 (/ 771.3234287776531 (- 3.0 z)))
       (/ -1259.1392167224028 (- 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))) * ((sqrt((((double) M_PI) * 2.0)) * (exp((z + -7.5)) * pow((7.5 - z), (0.5 - z)))) * ((676.5203681218851 / (1.0 - z)) + ((((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + (-1259.1392167224028 / (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.sqrt((Math.PI * 2.0)) * (Math.exp((z + -7.5)) * Math.pow((7.5 - z), (0.5 - z)))) * ((676.5203681218851 / (1.0 - z)) + ((((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + (-1259.1392167224028 / (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.sqrt((math.pi * 2.0)) * (math.exp((z + -7.5)) * math.pow((7.5 - z), (0.5 - z)))) * ((676.5203681218851 / (1.0 - z)) + ((((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + (-1259.1392167224028 / (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(sqrt(Float64(pi * 2.0)) * Float64(exp(Float64(z + -7.5)) * (Float64(7.5 - z) ^ Float64(0.5 - z)))) * Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(771.3234287776531 / Float64(3.0 - z))) + Float64(-1259.1392167224028 / Float64(2.0 - 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(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))) * ((sqrt((pi * 2.0)) * (exp((z + -7.5)) * ((7.5 - z) ^ (0.5 - z)))) * ((676.5203681218851 / (1.0 - z)) + ((((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + (-1259.1392167224028 / (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[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(0.9999999999998099 + N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $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[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

4. Applied egg-rr 97.3%

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

6. Simplified 98.1%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)} \]

7. Taylor expanded in z around inf 98.1%

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

   1. exp-to-pow 98.1%

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

   2. *-commutative 98.1%

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

   3. sub-neg 98.1%

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

   4. metadata-eval 98.1%

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

   5. +-commutative 98.1%

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

9. Simplified 98.1%

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

10. Final simplification 98.1%

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

Alternative 2: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

3. Simplified 95.6%

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

   1. pow1 95.6%

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

6. Applied egg-rr 95.5%

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

8. Simplified 97.7%

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

10. Applied egg-rr 97.8%

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

12. Simplified 97.8%

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

13. Final simplification 97.8%

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

Alternative 3: 97.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

3. Simplified 95.6%

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

   1. pow1 95.6%

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

6. Applied egg-rr 95.5%

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

8. Simplified 97.7%

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

9. Final simplification 97.7%

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

Alternative 4: 97.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot \left(e^{z + -7.5} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ PI (sin (* PI z)))
  (*
   (* (sqrt (* PI 2.0)) (* (exp (+ z -7.5)) (pow (- 7.5 z) (- 0.5 z))))
   (+
    (/ 676.5203681218851 (- 1.0 z))
    (+
     (+
      (+
       (+ 0.9999999999998099 (/ 771.3234287776531 (- 3.0 z)))
       (/ -1259.1392167224028 (- 2.0 z)))
      (/ -176.6150291621406 (- 4.0 z)))
     (+
      2.4783749183520145
      (*
       z
       (+
        0.49644474017195733
        (* z (+ 0.09941724278406093 (* z 0.01990483129967024)))))))))))

C

double code(double z) {
	return (((double) M_PI) / sin((((double) M_PI) * z))) * ((sqrt((((double) M_PI) * 2.0)) * (exp((z + -7.5)) * pow((7.5 - z), (0.5 - z)))) * ((676.5203681218851 / (1.0 - z)) + ((((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + (-1259.1392167224028 / (2.0 - z))) + (-176.6150291621406 / (4.0 - z))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * (0.09941724278406093 + (z * 0.01990483129967024)))))))));
}

Java

public static double code(double z) {
	return (Math.PI / Math.sin((Math.PI * z))) * ((Math.sqrt((Math.PI * 2.0)) * (Math.exp((z + -7.5)) * Math.pow((7.5 - z), (0.5 - z)))) * ((676.5203681218851 / (1.0 - z)) + ((((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + (-1259.1392167224028 / (2.0 - z))) + (-176.6150291621406 / (4.0 - z))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * (0.09941724278406093 + (z * 0.01990483129967024)))))))));
}

Python

def code(z):
	return (math.pi / math.sin((math.pi * z))) * ((math.sqrt((math.pi * 2.0)) * (math.exp((z + -7.5)) * math.pow((7.5 - z), (0.5 - z)))) * ((676.5203681218851 / (1.0 - z)) + ((((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + (-1259.1392167224028 / (2.0 - z))) + (-176.6150291621406 / (4.0 - z))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * (0.09941724278406093 + (z * 0.01990483129967024)))))))))

Julia

function code(z)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64(exp(Float64(z + -7.5)) * (Float64(7.5 - z) ^ Float64(0.5 - z)))) * Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(771.3234287776531 / Float64(3.0 - z))) + Float64(-1259.1392167224028 / Float64(2.0 - z))) + Float64(-176.6150291621406 / Float64(4.0 - z))) + Float64(2.4783749183520145 + Float64(z * Float64(0.49644474017195733 + Float64(z * Float64(0.09941724278406093 + Float64(z * 0.01990483129967024))))))))))
end

MATLAB

function tmp = code(z)
	tmp = (pi / sin((pi * z))) * ((sqrt((pi * 2.0)) * (exp((z + -7.5)) * ((7.5 - z) ^ (0.5 - z)))) * ((676.5203681218851 / (1.0 - z)) + ((((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + (-1259.1392167224028 / (2.0 - z))) + (-176.6150291621406 / (4.0 - z))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * (0.09941724278406093 + (z * 0.01990483129967024)))))))));
end

Wolfram

code[z_] := N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(0.9999999999998099 + N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.4783749183520145 + N[(z * N[(0.49644474017195733 + N[(z * N[(0.09941724278406093 + N[(z * 0.01990483129967024), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

4. Applied egg-rr 97.3%

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

6. Simplified 98.1%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)} \]

7. Taylor expanded in z around inf 98.1%

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

   1. exp-to-pow 98.1%

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

   2. *-commutative 98.1%

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

   3. sub-neg 98.1%

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

   4. metadata-eval 98.1%

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

   5. +-commutative 98.1%

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

9. Simplified 98.1%

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

10. Taylor expanded in z around 0 97.5%

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left(e^{-7.5 + z} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \color{blue}{\left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + 0.01990483129967024 \cdot z\right)\right)\right)}\right)\right)\right) \]
11. Step-by-step derivation

    1. *-commutative 97.5%

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left(e^{-7.5 + z} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + \color{blue}{z \cdot 0.01990483129967024}\right)\right)\right)\right)\right)\right) \]

12. Simplified 97.5%

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left(e^{-7.5 + z} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \color{blue}{\left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)}\right)\right)\right) \]

13. Final simplification 97.5%

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot \left(e^{z + -7.5} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)\right)\right)\right) \]
14. Add Preprocessing

Alternative 5: 97.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot \left(e^{z + -7.5} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ PI (sin (* PI z)))
  (*
   (* (sqrt (* PI 2.0)) (* (exp (+ z -7.5)) (pow (- 7.5 z) (- 0.5 z))))
   (+
    (/ 676.5203681218851 (- 1.0 z))
    (+
     (+
      (+
       (+ 0.9999999999998099 (/ 771.3234287776531 (- 3.0 z)))
       (/ -1259.1392167224028 (- 2.0 z)))
      (/ -176.6150291621406 (- 4.0 z)))
     (+
      2.4783749183520145
      (* z (+ 0.49644474017195733 (* z 0.09941724278406093)))))))))

C

double code(double z) {
	return (((double) M_PI) / sin((((double) M_PI) * z))) * ((sqrt((((double) M_PI) * 2.0)) * (exp((z + -7.5)) * pow((7.5 - z), (0.5 - z)))) * ((676.5203681218851 / (1.0 - z)) + ((((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + (-1259.1392167224028 / (2.0 - z))) + (-176.6150291621406 / (4.0 - z))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * 0.09941724278406093)))))));
}

Java

public static double code(double z) {
	return (Math.PI / Math.sin((Math.PI * z))) * ((Math.sqrt((Math.PI * 2.0)) * (Math.exp((z + -7.5)) * Math.pow((7.5 - z), (0.5 - z)))) * ((676.5203681218851 / (1.0 - z)) + ((((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + (-1259.1392167224028 / (2.0 - z))) + (-176.6150291621406 / (4.0 - z))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * 0.09941724278406093)))))));
}

Python

def code(z):
	return (math.pi / math.sin((math.pi * z))) * ((math.sqrt((math.pi * 2.0)) * (math.exp((z + -7.5)) * math.pow((7.5 - z), (0.5 - z)))) * ((676.5203681218851 / (1.0 - z)) + ((((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + (-1259.1392167224028 / (2.0 - z))) + (-176.6150291621406 / (4.0 - z))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * 0.09941724278406093)))))))

Julia

function code(z)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64(exp(Float64(z + -7.5)) * (Float64(7.5 - z) ^ Float64(0.5 - z)))) * Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(771.3234287776531 / Float64(3.0 - z))) + Float64(-1259.1392167224028 / Float64(2.0 - z))) + Float64(-176.6150291621406 / Float64(4.0 - z))) + Float64(2.4783749183520145 + Float64(z * Float64(0.49644474017195733 + Float64(z * 0.09941724278406093))))))))
end

MATLAB

function tmp = code(z)
	tmp = (pi / sin((pi * z))) * ((sqrt((pi * 2.0)) * (exp((z + -7.5)) * ((7.5 - z) ^ (0.5 - z)))) * ((676.5203681218851 / (1.0 - z)) + ((((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + (-1259.1392167224028 / (2.0 - z))) + (-176.6150291621406 / (4.0 - z))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * 0.09941724278406093)))))));
end

Wolfram

code[z_] := N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(0.9999999999998099 + N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.4783749183520145 + N[(z * N[(0.49644474017195733 + N[(z * 0.09941724278406093), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

4. Applied egg-rr 97.3%

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

6. Simplified 98.1%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)} \]

7. Taylor expanded in z around inf 98.1%

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

   1. exp-to-pow 98.1%

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

   2. *-commutative 98.1%

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

   3. sub-neg 98.1%

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

   4. metadata-eval 98.1%

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

   5. +-commutative 98.1%

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

9. Simplified 98.1%

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

10. Taylor expanded in z around 0 97.2%

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left(e^{-7.5 + z} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \color{blue}{\left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + 0.09941724278406093 \cdot z\right)\right)}\right)\right)\right) \]
11. Step-by-step derivation

    1. *-commutative 97.2%

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left(e^{-7.5 + z} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + \color{blue}{z \cdot 0.09941724278406093}\right)\right)\right)\right)\right) \]

12. Simplified 97.2%

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left(e^{-7.5 + z} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \color{blue}{\left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)}\right)\right)\right) \]

13. Final simplification 97.2%

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot \left(e^{z + -7.5} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)\right)\right)\right) \]
14. Add Preprocessing

Alternative 6: 97.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z - 7.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ PI (sin (* PI z)))
  (*
   (* (sqrt (* PI 2.0)) (* (pow (- 7.5 z) (- 0.5 z)) (exp (- z 7.5))))
   (+
    (/ 676.5203681218851 (- 1.0 z))
    (+
     (+
      (+
       (+ 0.9999999999998099 (/ 771.3234287776531 (- 3.0 z)))
       (/ -1259.1392167224028 (- 2.0 z)))
      (/ -176.6150291621406 (- 4.0 z)))
     (+ 2.4783749183520145 (* z 0.49644474017195733)))))))

C

double code(double z) {
	return (((double) M_PI) / sin((((double) M_PI) * z))) * ((sqrt((((double) M_PI) * 2.0)) * (pow((7.5 - z), (0.5 - z)) * exp((z - 7.5)))) * ((676.5203681218851 / (1.0 - z)) + ((((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + (-1259.1392167224028 / (2.0 - z))) + (-176.6150291621406 / (4.0 - z))) + (2.4783749183520145 + (z * 0.49644474017195733)))));
}

Java

public static double code(double z) {
	return (Math.PI / Math.sin((Math.PI * z))) * ((Math.sqrt((Math.PI * 2.0)) * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z - 7.5)))) * ((676.5203681218851 / (1.0 - z)) + ((((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + (-1259.1392167224028 / (2.0 - z))) + (-176.6150291621406 / (4.0 - z))) + (2.4783749183520145 + (z * 0.49644474017195733)))));
}

Python

def code(z):
	return (math.pi / math.sin((math.pi * z))) * ((math.sqrt((math.pi * 2.0)) * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z - 7.5)))) * ((676.5203681218851 / (1.0 - z)) + ((((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + (-1259.1392167224028 / (2.0 - z))) + (-176.6150291621406 / (4.0 - z))) + (2.4783749183520145 + (z * 0.49644474017195733)))))

Julia

function code(z)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z - 7.5)))) * Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(771.3234287776531 / Float64(3.0 - z))) + Float64(-1259.1392167224028 / Float64(2.0 - z))) + Float64(-176.6150291621406 / Float64(4.0 - z))) + Float64(2.4783749183520145 + Float64(z * 0.49644474017195733))))))
end

MATLAB

function tmp = code(z)
	tmp = (pi / sin((pi * z))) * ((sqrt((pi * 2.0)) * (((7.5 - z) ^ (0.5 - z)) * exp((z - 7.5)))) * ((676.5203681218851 / (1.0 - z)) + ((((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + (-1259.1392167224028 / (2.0 - z))) + (-176.6150291621406 / (4.0 - z))) + (2.4783749183520145 + (z * 0.49644474017195733)))));
end

Wolfram

code[z_] := N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z - 7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(0.9999999999998099 + N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.4783749183520145 + N[(z * 0.49644474017195733), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

4. Applied egg-rr 97.3%

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

6. Simplified 98.1%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)} \]

7. Taylor expanded in z around 0 96.9%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \color{blue}{\left(2.4783749183520145 + 0.49644474017195733 \cdot z\right)}\right)\right)\right) \]
8. Step-by-step derivation

   1. *-commutative 96.9%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(2.4783749183520145 + \color{blue}{z \cdot 0.49644474017195733}\right)\right)\right)\right) \]

9. Simplified 96.9%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \color{blue}{\left(2.4783749183520145 + z \cdot 0.49644474017195733\right)}\right)\right)\right) \]

10. Taylor expanded in z around inf 96.9%

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \color{blue}{e^{z - 7.5}}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right)\right) \]

11. Final simplification 96.9%

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z - 7.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right)\right) \]
12. Add Preprocessing

Alternative 7: 97.1% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (* (sqrt (* PI 2.0)) (* (exp (+ z -7.5)) (pow (- 7.5 z) (- 0.5 z))))
   (+
    (/ 676.5203681218851 (- 1.0 z))
    (+
     (+
      (+
       (+ 0.9999999999998099 (/ 771.3234287776531 (- 3.0 z)))
       (/ -1259.1392167224028 (- 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)))))))
  (/ 1.0 z)))

C

double code(double z) {
	return ((sqrt((((double) M_PI) * 2.0)) * (exp((z + -7.5)) * pow((7.5 - z), (0.5 - z)))) * ((676.5203681218851 / (1.0 - z)) + ((((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + (-1259.1392167224028 / (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))))))) * (1.0 / z);
}

Java

public static double code(double z) {
	return ((Math.sqrt((Math.PI * 2.0)) * (Math.exp((z + -7.5)) * Math.pow((7.5 - z), (0.5 - z)))) * ((676.5203681218851 / (1.0 - z)) + ((((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + (-1259.1392167224028 / (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))))))) * (1.0 / z);
}

Python

def code(z):
	return ((math.sqrt((math.pi * 2.0)) * (math.exp((z + -7.5)) * math.pow((7.5 - z), (0.5 - z)))) * ((676.5203681218851 / (1.0 - z)) + ((((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + (-1259.1392167224028 / (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))))))) * (1.0 / z)

Julia

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

MATLAB

function tmp = code(z)
	tmp = ((sqrt((pi * 2.0)) * (exp((z + -7.5)) * ((7.5 - z) ^ (0.5 - z)))) * ((676.5203681218851 / (1.0 - z)) + ((((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + (-1259.1392167224028 / (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))))))) * (1.0 / z);
end

Wolfram

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

Derivation

1. Initial program 96.2%

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

4. Applied egg-rr 97.3%

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

6. Simplified 98.1%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)} \]

7. Taylor expanded in z around inf 98.1%

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

   1. exp-to-pow 98.1%

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

   2. *-commutative 98.1%

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

   3. sub-neg 98.1%

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

   4. metadata-eval 98.1%

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

   5. +-commutative 98.1%

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

9. Simplified 98.1%

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

10. Taylor expanded in z around 0 96.4%

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

11. Final simplification 96.4%

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

Alternative 8: 97.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{z} \cdot \left(\left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(z + -1\right) + -6.5}\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ 1.0 z)
  (*
   (+
    (/ 676.5203681218851 (- 1.0 z))
    (+
     (+
      (+
       (+ 0.9999999999998099 (/ 771.3234287776531 (- 3.0 z)))
       (/ -1259.1392167224028 (- 2.0 z)))
      (/ -176.6150291621406 (- 4.0 z)))
     (+ 2.4783749183520145 (* z 0.49644474017195733))))
   (*
    (sqrt (* PI 2.0))
    (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ (+ z -1.0) -6.5)))))))

C

double code(double z) {
	return (1.0 / z) * (((676.5203681218851 / (1.0 - z)) + ((((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + (-1259.1392167224028 / (2.0 - z))) + (-176.6150291621406 / (4.0 - z))) + (2.4783749183520145 + (z * 0.49644474017195733)))) * (sqrt((((double) M_PI) * 2.0)) * (pow((7.5 - z), (0.5 - z)) * exp(((z + -1.0) + -6.5)))));
}

Java

public static double code(double z) {
	return (1.0 / z) * (((676.5203681218851 / (1.0 - z)) + ((((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + (-1259.1392167224028 / (2.0 - z))) + (-176.6150291621406 / (4.0 - z))) + (2.4783749183520145 + (z * 0.49644474017195733)))) * (Math.sqrt((Math.PI * 2.0)) * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp(((z + -1.0) + -6.5)))));
}

Python

def code(z):
	return (1.0 / z) * (((676.5203681218851 / (1.0 - z)) + ((((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + (-1259.1392167224028 / (2.0 - z))) + (-176.6150291621406 / (4.0 - z))) + (2.4783749183520145 + (z * 0.49644474017195733)))) * (math.sqrt((math.pi * 2.0)) * (math.pow((7.5 - z), (0.5 - z)) * math.exp(((z + -1.0) + -6.5)))))

Julia

function code(z)
	return Float64(Float64(1.0 / z) * Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(771.3234287776531 / Float64(3.0 - z))) + Float64(-1259.1392167224028 / Float64(2.0 - z))) + Float64(-176.6150291621406 / Float64(4.0 - z))) + Float64(2.4783749183520145 + Float64(z * 0.49644474017195733)))) * Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(Float64(z + -1.0) + -6.5))))))
end

MATLAB

function tmp = code(z)
	tmp = (1.0 / z) * (((676.5203681218851 / (1.0 - z)) + ((((0.9999999999998099 + (771.3234287776531 / (3.0 - z))) + (-1259.1392167224028 / (2.0 - z))) + (-176.6150291621406 / (4.0 - z))) + (2.4783749183520145 + (z * 0.49644474017195733)))) * (sqrt((pi * 2.0)) * (((7.5 - z) ^ (0.5 - z)) * exp(((z + -1.0) + -6.5)))));
end

Wolfram

code[z_] := N[(N[(1.0 / z), $MachinePrecision] * N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(0.9999999999998099 + N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.4783749183520145 + N[(z * 0.49644474017195733), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(z + -1.0), $MachinePrecision] + -6.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

4. Applied egg-rr 97.3%

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

6. Simplified 98.1%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)} \]

7. Taylor expanded in z around 0 96.9%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \color{blue}{\left(2.4783749183520145 + 0.49644474017195733 \cdot z\right)}\right)\right)\right) \]
8. Step-by-step derivation

   1. *-commutative 96.9%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(2.4783749183520145 + \color{blue}{z \cdot 0.49644474017195733}\right)\right)\right)\right) \]

9. Simplified 96.9%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \color{blue}{\left(2.4783749183520145 + z \cdot 0.49644474017195733\right)}\right)\right)\right) \]

10. Taylor expanded in z around 0 96.4%

    \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right)\right) \]

11. Final simplification 96.4%

    \[\leadsto \frac{1}{z} \cdot \left(\left(\frac{676.5203681218851}{1 - z} + \left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(z + -1\right) + -6.5}\right)\right)\right) \]
12. 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 96.2%

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

3. Simplified 95.6%

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

   1. pow1 95.6%

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

6. Applied egg-rr 95.5%

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

8. Simplified 97.7%

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

9. Taylor expanded in z around 0 96.3%

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

10. Taylor expanded in z around 0 95.2%

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

    1. associate-*r/ 95.5%

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

    2. associate-*r* 95.5%

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

12. Simplified 95.5%

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

13. Final simplification 95.5%

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

Alternative 10: 95.7% accurate, 1.7× speedup

Math

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

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(263.3831869810514 * Float64(Float64(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[(263.3831869810514 * N[(N[(N[Exp[-7.5], $MachinePrecision] * N[Sqrt[7.5], $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

3. Simplified 95.6%

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

   1. pow1 95.6%

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

6. Applied egg-rr 95.5%

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

8. Simplified 97.7%

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

9. Taylor expanded in z around 0 96.3%

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

10. Taylor expanded in z around 0 95.2%

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

11. Final simplification 95.2%

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

Alternative 11: 95.6% accurate, 1.7× speedup

Math

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

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(263.3831869810514 * Float64(exp(-7.5) * Float64(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[(263.3831869810514 * N[(N[Exp[-7.5], $MachinePrecision] * N[(N[Sqrt[7.5], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

3. Simplified 95.4%

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

5. Taylor expanded in z around 0 93.9%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \color{blue}{\frac{263.3831869810514}{z}} \]
6. Step-by-step derivation

   1. associate-*r/ 94.5%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot 263.3831869810514}{z}} \]

   2. associate-+l- 94.5%

      \[\leadsto \frac{\left(\sqrt{\pi \cdot 2} \cdot \left({\color{blue}{\left(1 - \left(z - 6.5\right)\right)}}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot 263.3831869810514}{z} \]

   3. associate-+l- 94.5%

      \[\leadsto \frac{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(1 - \left(z - 6.5\right)\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\color{blue}{\left(1 - \left(z - 6.5\right)\right)}}\right)\right) \cdot 263.3831869810514}{z} \]

7. Applied egg-rr 94.5%

   \[\leadsto \color{blue}{\frac{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(1 - \left(z - 6.5\right)\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(1 - \left(z - 6.5\right)\right)}\right)\right) \cdot 263.3831869810514}{z}} \]
8. Step-by-step derivation

   1. associate-*r/ 93.9%

      \[\leadsto \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(1 - \left(z - 6.5\right)\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(1 - \left(z - 6.5\right)\right)}\right)\right) \cdot \frac{263.3831869810514}{z}} \]

   2. associate-*l* 94.0%

      \[\leadsto \color{blue}{\sqrt{\pi \cdot 2} \cdot \left(\left({\left(1 - \left(z - 6.5\right)\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(1 - \left(z - 6.5\right)\right)}\right) \cdot \frac{263.3831869810514}{z}\right)} \]

   3. *-commutative 94.0%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \pi}} \cdot \left(\left({\left(1 - \left(z - 6.5\right)\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(1 - \left(z - 6.5\right)\right)}\right) \cdot \frac{263.3831869810514}{z}\right) \]

   4. associate--r- 94.0%

      \[\leadsto \sqrt{2 \cdot \pi} \cdot \left(\left({\color{blue}{\left(\left(1 - z\right) + 6.5\right)}}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(1 - \left(z - 6.5\right)\right)}\right) \cdot \frac{263.3831869810514}{z}\right) \]

   5. associate--r- 94.0%

      \[\leadsto \sqrt{2 \cdot \pi} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\color{blue}{\left(\left(1 - z\right) + 6.5\right)}}\right) \cdot \frac{263.3831869810514}{z}\right) \]

   6. distribute-neg-in 94.0%

      \[\leadsto \sqrt{2 \cdot \pi} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\color{blue}{\left(-\left(1 - z\right)\right) + \left(-6.5\right)}}\right) \cdot \frac{263.3831869810514}{z}\right) \]

   7. metadata-eval 94.0%

      \[\leadsto \sqrt{2 \cdot \pi} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(-\left(1 - z\right)\right) + \color{blue}{-6.5}}\right) \cdot \frac{263.3831869810514}{z}\right) \]

9. Simplified 94.0%

   \[\leadsto \color{blue}{\sqrt{2 \cdot \pi} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right) \cdot \frac{263.3831869810514}{z}\right)} \]

10. Taylor expanded in z around 0 95.2%

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

    1. associate-*r/ 95.1%

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

12. Simplified 95.1%

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

13. Final simplification 95.1%

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

Reproduce

herbie shell --seed 2024184 
(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))))))