Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.2% → 98.1%

Time: 2.2min

Alternatives: 14

Speedup: 1.1×

• could not determine a ground truth

  1. z = -6.065203706857379e+135

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: 96.2% 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: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{676.5203681218851}{1 - z}\\ t_1 := 0.9999999999998099 + t\_0\\ \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \sqrt[3]{{\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\mathsf{fma}\left(-1, 7 - z, -0.5\right)}\right)}^{3}}\right)\right) \cdot \left(\left(\left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right) - \left(\left(\frac{771.3234287776531}{-2 - \left(1 - z\right)} + \frac{-176.6150291621406}{-3 - \left(1 - z\right)}\right) - \frac{t\_1 \cdot t\_1 + \frac{\frac{1585431.567088306}{2 - z}}{z - 2}}{0.9999999999998099 + \left(t\_0 + \frac{1259.1392167224028}{2 - z}\right)}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (/ 676.5203681218851 (- 1.0 z))) (t_1 (+ 0.9999999999998099 t_0)))
   (*
    (*
     (/ PI (sin (* PI z)))
     (*
      (sqrt (* PI 2.0))
      (cbrt
       (pow
        (* (pow (- 7.5 z) (- 0.5 z)) (exp (fma -1.0 (- 7.0 z) -0.5)))
        3.0))))
    (+
     (-
      (+
       (/ 12.507343278686905 (- (- 1.0 z) -4.0))
       (/ -0.13857109526572012 (- (- 1.0 z) -5.0)))
      (-
       (+
        (/ 771.3234287776531 (- -2.0 (- 1.0 z)))
        (/ -176.6150291621406 (- -3.0 (- 1.0 z))))
       (/
        (+ (* t_1 t_1) (/ (/ 1585431.567088306 (- 2.0 z)) (- z 2.0)))
        (+ 0.9999999999998099 (+ t_0 (/ 1259.1392167224028 (- 2.0 z)))))))
     (+
      (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0))
      (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0)))))))

C

double code(double z) {
	double t_0 = 676.5203681218851 / (1.0 - z);
	double t_1 = 0.9999999999998099 + t_0;
	return ((((double) M_PI) / sin((((double) M_PI) * z))) * (sqrt((((double) M_PI) * 2.0)) * cbrt(pow((pow((7.5 - z), (0.5 - z)) * exp(fma(-1.0, (7.0 - z), -0.5))), 3.0)))) * ((((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0))) - (((771.3234287776531 / (-2.0 - (1.0 - z))) + (-176.6150291621406 / (-3.0 - (1.0 - z)))) - (((t_1 * t_1) + ((1585431.567088306 / (2.0 - z)) / (z - 2.0))) / (0.9999999999998099 + (t_0 + (1259.1392167224028 / (2.0 - z))))))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))));
}

Julia

function code(z)
	t_0 = Float64(676.5203681218851 / Float64(1.0 - z))
	t_1 = Float64(0.9999999999998099 + t_0)
	return Float64(Float64(Float64(pi / sin(Float64(pi * z))) * Float64(sqrt(Float64(pi * 2.0)) * cbrt((Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(fma(-1.0, Float64(7.0 - z), -0.5))) ^ 3.0)))) * Float64(Float64(Float64(Float64(12.507343278686905 / Float64(Float64(1.0 - z) - -4.0)) + Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) - -5.0))) - Float64(Float64(Float64(771.3234287776531 / Float64(-2.0 - Float64(1.0 - z))) + Float64(-176.6150291621406 / Float64(-3.0 - Float64(1.0 - z)))) - Float64(Float64(Float64(t_1 * t_1) + Float64(Float64(1585431.567088306 / Float64(2.0 - z)) / Float64(z - 2.0))) / Float64(0.9999999999998099 + Float64(t_0 + Float64(1259.1392167224028 / Float64(2.0 - z))))))) + Float64(Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0)) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0)))))
end

Wolfram

code[z_] := Block[{t$95$0 = N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.9999999999998099 + t$95$0), $MachinePrecision]}, N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[Power[N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-1.0 * N[(7.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] - -4.0), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] - -5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(771.3234287776531 / N[(-2.0 - N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(-3.0 - N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] + N[(N[(1585431.567088306 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] / N[(z - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.9999999999998099 + N[(t$95$0 + N[(1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 96.6%

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

3. Simplified 98.2%

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

   1. flip-+ 98.2%

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

6. Applied egg-rr 98.2%

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

   1. +-commutative 98.2%

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

   2. +-commutative 98.2%

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

   3. associate-*l/ 98.2%

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

   4. associate-*r/ 98.2%

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

   5. metadata-eval 98.2%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{\color{blue}{1585431.567088306}}{1 - \left(z - 1\right)}}{1 - \left(z - 1\right)}}{\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   6. associate--r- 98.2%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{\color{blue}{\left(1 - z\right) + 1}}}{1 - \left(z - 1\right)}}{\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   7. +-commutative 98.2%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{\color{blue}{1 + \left(1 - z\right)}}}{1 - \left(z - 1\right)}}{\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   8. associate-+r- 98.2%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{\color{blue}{\left(1 + 1\right) - z}}}{1 - \left(z - 1\right)}}{\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   9. metadata-eval 98.2%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{\color{blue}{2} - z}}{1 - \left(z - 1\right)}}{\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   10. associate--r- 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{\color{blue}{\left(1 - z\right) + 1}}}{\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   11. +-commutative 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{\color{blue}{1 + \left(1 - z\right)}}}{\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   12. associate-+r- 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{\color{blue}{\left(1 + 1\right) - z}}}{\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   13. metadata-eval 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{\color{blue}{2} - z}}{\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   14. +-commutative 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{\color{blue}{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)} - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

8. Simplified 98.2%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\color{blue}{\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
9. Step-by-step derivation

   1. add-cbrt-cube 98.5%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \color{blue}{\sqrt[3]{\left(\left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right) \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right) \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)}}\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

10. Applied egg-rr 98.4%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \color{blue}{\sqrt[3]{\left(\left(e^{\mathsf{fma}\left(-1, \left(1 - z\right) + 6, -0.5\right)} \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{\mathsf{fma}\left(-1, \left(1 - z\right) + 6, -0.5\right)} \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right)\right) \cdot \left(e^{\mathsf{fma}\left(-1, \left(1 - z\right) + 6, -0.5\right)} \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right)}}\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
11. Step-by-step derivation

    1. associate-*l* 98.4%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \sqrt[3]{\color{blue}{\left(e^{\mathsf{fma}\left(-1, \left(1 - z\right) + 6, -0.5\right)} \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(\left(e^{\mathsf{fma}\left(-1, \left(1 - z\right) + 6, -0.5\right)} \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot \left(e^{\mathsf{fma}\left(-1, \left(1 - z\right) + 6, -0.5\right)} \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right)\right)}}\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    2. cube-unmult 98.4%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \sqrt[3]{\color{blue}{{\left(e^{\mathsf{fma}\left(-1, \left(1 - z\right) + 6, -0.5\right)} \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right)}^{3}}}\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

12. Simplified 98.4%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \color{blue}{\sqrt[3]{{\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\mathsf{fma}\left(-1, 7 - z, -0.5\right)}\right)}^{3}}}\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

13. Final simplification 98.4%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \sqrt[3]{{\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\mathsf{fma}\left(-1, 7 - z, -0.5\right)}\right)}^{3}}\right)\right) \cdot \left(\left(\left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right) - \left(\left(\frac{771.3234287776531}{-2 - \left(1 - z\right)} + \frac{-176.6150291621406}{-3 - \left(1 - z\right)}\right) - \frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{\frac{1585431.567088306}{2 - z}}{z - 2}}{0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}\right)}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
14. Add Preprocessing

Alternative 2: 97.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{676.5203681218851}{1 - z}\\ t_1 := 0.9999999999998099 + t\_0\\ \left(\left(\left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right) - \left(\left(\frac{771.3234287776531}{-2 - \left(1 - z\right)} + \frac{-176.6150291621406}{-3 - \left(1 - z\right)}\right) - \frac{t\_1 \cdot t\_1 + \frac{\frac{1585431.567088306}{2 - z}}{z - 2}}{0.9999999999998099 + \left(t\_0 + \frac{1259.1392167224028}{2 - z}\right)}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 + \sqrt[3]{\left(-z\right) \cdot \left(z \cdot z\right)}\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{-0.5 + \left(-6 - \left(1 - z\right)\right)}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (/ 676.5203681218851 (- 1.0 z))) (t_1 (+ 0.9999999999998099 t_0)))
   (*
    (+
     (-
      (+
       (/ 12.507343278686905 (- (- 1.0 z) -4.0))
       (/ -0.13857109526572012 (- (- 1.0 z) -5.0)))
      (-
       (+
        (/ 771.3234287776531 (- -2.0 (- 1.0 z)))
        (/ -176.6150291621406 (- -3.0 (- 1.0 z))))
       (/
        (+ (* t_1 t_1) (/ (/ 1585431.567088306 (- 2.0 z)) (- z 2.0)))
        (+ 0.9999999999998099 (+ t_0 (/ 1259.1392167224028 (- 2.0 z)))))))
     (+
      (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0))
      (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0))))
    (*
     (/ PI (sin (* PI z)))
     (*
      (sqrt (* PI 2.0))
      (*
       (pow (+ 7.5 (cbrt (* (- z) (* z z)))) (- (- 1.0 z) 0.5))
       (exp (+ -0.5 (- -6.0 (- 1.0 z))))))))))

C

double code(double z) {
	double t_0 = 676.5203681218851 / (1.0 - z);
	double t_1 = 0.9999999999998099 + t_0;
	return ((((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0))) - (((771.3234287776531 / (-2.0 - (1.0 - z))) + (-176.6150291621406 / (-3.0 - (1.0 - z)))) - (((t_1 * t_1) + ((1585431.567088306 / (2.0 - z)) / (z - 2.0))) / (0.9999999999998099 + (t_0 + (1259.1392167224028 / (2.0 - z))))))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)))) * ((((double) M_PI) / sin((((double) M_PI) * z))) * (sqrt((((double) M_PI) * 2.0)) * (pow((7.5 + cbrt((-z * (z * z)))), ((1.0 - z) - 0.5)) * exp((-0.5 + (-6.0 - (1.0 - z)))))));
}

Java

public static double code(double z) {
	double t_0 = 676.5203681218851 / (1.0 - z);
	double t_1 = 0.9999999999998099 + t_0;
	return ((((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0))) - (((771.3234287776531 / (-2.0 - (1.0 - z))) + (-176.6150291621406 / (-3.0 - (1.0 - z)))) - (((t_1 * t_1) + ((1585431.567088306 / (2.0 - z)) / (z - 2.0))) / (0.9999999999998099 + (t_0 + (1259.1392167224028 / (2.0 - z))))))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)))) * ((Math.PI / Math.sin((Math.PI * z))) * (Math.sqrt((Math.PI * 2.0)) * (Math.pow((7.5 + Math.cbrt((-z * (z * z)))), ((1.0 - z) - 0.5)) * Math.exp((-0.5 + (-6.0 - (1.0 - z)))))));
}

Julia

function code(z)
	t_0 = Float64(676.5203681218851 / Float64(1.0 - z))
	t_1 = Float64(0.9999999999998099 + t_0)
	return Float64(Float64(Float64(Float64(Float64(12.507343278686905 / Float64(Float64(1.0 - z) - -4.0)) + Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) - -5.0))) - Float64(Float64(Float64(771.3234287776531 / Float64(-2.0 - Float64(1.0 - z))) + Float64(-176.6150291621406 / Float64(-3.0 - Float64(1.0 - z)))) - Float64(Float64(Float64(t_1 * t_1) + Float64(Float64(1585431.567088306 / Float64(2.0 - z)) / Float64(z - 2.0))) / Float64(0.9999999999998099 + Float64(t_0 + Float64(1259.1392167224028 / Float64(2.0 - z))))))) + Float64(Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0)) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0)))) * Float64(Float64(pi / sin(Float64(pi * z))) * Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(7.5 + cbrt(Float64(Float64(-z) * Float64(z * z)))) ^ Float64(Float64(1.0 - z) - 0.5)) * exp(Float64(-0.5 + Float64(-6.0 - Float64(1.0 - z))))))))
end

Wolfram

code[z_] := Block[{t$95$0 = N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.9999999999998099 + t$95$0), $MachinePrecision]}, N[(N[(N[(N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] - -4.0), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] - -5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(771.3234287776531 / N[(-2.0 - N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(-3.0 - N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] + N[(N[(1585431.567088306 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] / N[(z - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.9999999999998099 + N[(t$95$0 + N[(1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(7.5 + N[Power[N[((-z) * N[(z * z), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-0.5 + N[(-6.0 - N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 96.6%

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

3. Simplified 98.2%

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

   1. flip-+ 98.2%

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

6. Applied egg-rr 98.2%

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

   1. +-commutative 98.2%

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

   2. +-commutative 98.2%

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

   3. associate-*l/ 98.2%

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

   4. associate-*r/ 98.2%

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

   5. metadata-eval 98.2%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{\color{blue}{1585431.567088306}}{1 - \left(z - 1\right)}}{1 - \left(z - 1\right)}}{\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   6. associate--r- 98.2%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{\color{blue}{\left(1 - z\right) + 1}}}{1 - \left(z - 1\right)}}{\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   7. +-commutative 98.2%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{\color{blue}{1 + \left(1 - z\right)}}}{1 - \left(z - 1\right)}}{\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   8. associate-+r- 98.2%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{\color{blue}{\left(1 + 1\right) - z}}}{1 - \left(z - 1\right)}}{\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   9. metadata-eval 98.2%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{\color{blue}{2} - z}}{1 - \left(z - 1\right)}}{\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   10. associate--r- 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{\color{blue}{\left(1 - z\right) + 1}}}{\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   11. +-commutative 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{\color{blue}{1 + \left(1 - z\right)}}}{\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   12. associate-+r- 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{\color{blue}{\left(1 + 1\right) - z}}}{\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   13. metadata-eval 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{\color{blue}{2} - z}}{\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   14. +-commutative 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{\color{blue}{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)} - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

8. Simplified 98.2%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\color{blue}{\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
9. Step-by-step derivation

   1. add-cbrt-cube 98.2%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\color{blue}{\sqrt[3]{\left(\left(\left(1 - z\right) + -1\right) \cdot \left(\left(1 - z\right) + -1\right)\right) \cdot \left(\left(1 - z\right) + -1\right)}} + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

10. Applied egg-rr 98.2%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\color{blue}{\sqrt[3]{\left(\left(\left(1 - z\right) + -1\right) \cdot \left(\left(1 - z\right) + -1\right)\right) \cdot \left(\left(1 - z\right) + -1\right)}} + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
11. Step-by-step derivation

    1. associate-*l* 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\sqrt[3]{\color{blue}{\left(\left(1 - z\right) + -1\right) \cdot \left(\left(\left(1 - z\right) + -1\right) \cdot \left(\left(1 - z\right) + -1\right)\right)}} + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    2. +-commutative 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\sqrt[3]{\color{blue}{\left(-1 + \left(1 - z\right)\right)} \cdot \left(\left(\left(1 - z\right) + -1\right) \cdot \left(\left(1 - z\right) + -1\right)\right)} + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    3. associate-+r- 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\sqrt[3]{\color{blue}{\left(\left(-1 + 1\right) - z\right)} \cdot \left(\left(\left(1 - z\right) + -1\right) \cdot \left(\left(1 - z\right) + -1\right)\right)} + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    4. metadata-eval 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\sqrt[3]{\left(\color{blue}{0} - z\right) \cdot \left(\left(\left(1 - z\right) + -1\right) \cdot \left(\left(1 - z\right) + -1\right)\right)} + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    5. neg-sub0 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\sqrt[3]{\color{blue}{\left(-z\right)} \cdot \left(\left(\left(1 - z\right) + -1\right) \cdot \left(\left(1 - z\right) + -1\right)\right)} + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    6. +-commutative 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\sqrt[3]{\left(-z\right) \cdot \left(\color{blue}{\left(-1 + \left(1 - z\right)\right)} \cdot \left(\left(1 - z\right) + -1\right)\right)} + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    7. associate-+r- 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\sqrt[3]{\left(-z\right) \cdot \left(\color{blue}{\left(\left(-1 + 1\right) - z\right)} \cdot \left(\left(1 - z\right) + -1\right)\right)} + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    8. metadata-eval 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\sqrt[3]{\left(-z\right) \cdot \left(\left(\color{blue}{0} - z\right) \cdot \left(\left(1 - z\right) + -1\right)\right)} + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    9. neg-sub0 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\sqrt[3]{\left(-z\right) \cdot \left(\color{blue}{\left(-z\right)} \cdot \left(\left(1 - z\right) + -1\right)\right)} + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    10. +-commutative 98.2%

        \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\sqrt[3]{\left(-z\right) \cdot \left(\left(-z\right) \cdot \color{blue}{\left(-1 + \left(1 - z\right)\right)}\right)} + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    11. associate-+r- 98.2%

        \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\sqrt[3]{\left(-z\right) \cdot \left(\left(-z\right) \cdot \color{blue}{\left(\left(-1 + 1\right) - z\right)}\right)} + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    12. metadata-eval 98.2%

        \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\sqrt[3]{\left(-z\right) \cdot \left(\left(-z\right) \cdot \left(\color{blue}{0} - z\right)\right)} + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    13. neg-sub0 98.2%

        \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\sqrt[3]{\left(-z\right) \cdot \left(\left(-z\right) \cdot \color{blue}{\left(-z\right)}\right)} + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

12. Simplified 98.2%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\color{blue}{\sqrt[3]{\left(-z\right) \cdot \left(\left(-z\right) \cdot \left(-z\right)\right)}} + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

13. Final simplification 98.2%

    \[\leadsto \left(\left(\left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right) - \left(\left(\frac{771.3234287776531}{-2 - \left(1 - z\right)} + \frac{-176.6150291621406}{-3 - \left(1 - z\right)}\right) - \frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{\frac{1585431.567088306}{2 - z}}{z - 2}}{0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}\right)}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 + \sqrt[3]{\left(-z\right) \cdot \left(z \cdot z\right)}\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{-0.5 + \left(-6 - \left(1 - z\right)\right)}\right)\right)\right) \]
14. Add Preprocessing

Alternative 3: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{676.5203681218851}{1 - z}\\ t_1 := 0.9999999999998099 + t\_0\\ \left(\left(\left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right) - \left(\left(\frac{771.3234287776531}{-2 - \left(1 - z\right)} + \frac{-176.6150291621406}{-3 - \left(1 - z\right)}\right) - \frac{t\_1 \cdot t\_1 + \frac{\frac{1585431.567088306}{2 - z}}{z - 2}}{0.9999999999998099 + \left(t\_0 + \frac{1259.1392167224028}{2 - z}\right)}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(e^{-0.5 + \left(-6 - \left(1 - z\right)\right)} \cdot {\left(7.5 + \left(-1 + \left(1 - z\right)\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (/ 676.5203681218851 (- 1.0 z))) (t_1 (+ 0.9999999999998099 t_0)))
   (*
    (+
     (-
      (+
       (/ 12.507343278686905 (- (- 1.0 z) -4.0))
       (/ -0.13857109526572012 (- (- 1.0 z) -5.0)))
      (-
       (+
        (/ 771.3234287776531 (- -2.0 (- 1.0 z)))
        (/ -176.6150291621406 (- -3.0 (- 1.0 z))))
       (/
        (+ (* t_1 t_1) (/ (/ 1585431.567088306 (- 2.0 z)) (- z 2.0)))
        (+ 0.9999999999998099 (+ t_0 (/ 1259.1392167224028 (- 2.0 z)))))))
     (+
      (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0))
      (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0))))
    (*
     (/ PI (sin (* PI z)))
     (*
      (sqrt (* PI 2.0))
      (*
       (exp (+ -0.5 (- -6.0 (- 1.0 z))))
       (pow (+ 7.5 (+ -1.0 (- 1.0 z))) (- (- 1.0 z) 0.5))))))))

C

double code(double z) {
	double t_0 = 676.5203681218851 / (1.0 - z);
	double t_1 = 0.9999999999998099 + t_0;
	return ((((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0))) - (((771.3234287776531 / (-2.0 - (1.0 - z))) + (-176.6150291621406 / (-3.0 - (1.0 - z)))) - (((t_1 * t_1) + ((1585431.567088306 / (2.0 - z)) / (z - 2.0))) / (0.9999999999998099 + (t_0 + (1259.1392167224028 / (2.0 - z))))))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)))) * ((((double) M_PI) / sin((((double) M_PI) * z))) * (sqrt((((double) M_PI) * 2.0)) * (exp((-0.5 + (-6.0 - (1.0 - z)))) * pow((7.5 + (-1.0 + (1.0 - z))), ((1.0 - z) - 0.5)))));
}

Java

public static double code(double z) {
	double t_0 = 676.5203681218851 / (1.0 - z);
	double t_1 = 0.9999999999998099 + t_0;
	return ((((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0))) - (((771.3234287776531 / (-2.0 - (1.0 - z))) + (-176.6150291621406 / (-3.0 - (1.0 - z)))) - (((t_1 * t_1) + ((1585431.567088306 / (2.0 - z)) / (z - 2.0))) / (0.9999999999998099 + (t_0 + (1259.1392167224028 / (2.0 - z))))))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)))) * ((Math.PI / Math.sin((Math.PI * z))) * (Math.sqrt((Math.PI * 2.0)) * (Math.exp((-0.5 + (-6.0 - (1.0 - z)))) * Math.pow((7.5 + (-1.0 + (1.0 - z))), ((1.0 - z) - 0.5)))));
}

Python

def code(z):
	t_0 = 676.5203681218851 / (1.0 - z)
	t_1 = 0.9999999999998099 + t_0
	return ((((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0))) - (((771.3234287776531 / (-2.0 - (1.0 - z))) + (-176.6150291621406 / (-3.0 - (1.0 - z)))) - (((t_1 * t_1) + ((1585431.567088306 / (2.0 - z)) / (z - 2.0))) / (0.9999999999998099 + (t_0 + (1259.1392167224028 / (2.0 - z))))))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)))) * ((math.pi / math.sin((math.pi * z))) * (math.sqrt((math.pi * 2.0)) * (math.exp((-0.5 + (-6.0 - (1.0 - z)))) * math.pow((7.5 + (-1.0 + (1.0 - z))), ((1.0 - z) - 0.5)))))

Julia

function code(z)
	t_0 = Float64(676.5203681218851 / Float64(1.0 - z))
	t_1 = Float64(0.9999999999998099 + t_0)
	return Float64(Float64(Float64(Float64(Float64(12.507343278686905 / Float64(Float64(1.0 - z) - -4.0)) + Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) - -5.0))) - Float64(Float64(Float64(771.3234287776531 / Float64(-2.0 - Float64(1.0 - z))) + Float64(-176.6150291621406 / Float64(-3.0 - Float64(1.0 - z)))) - Float64(Float64(Float64(t_1 * t_1) + Float64(Float64(1585431.567088306 / Float64(2.0 - z)) / Float64(z - 2.0))) / Float64(0.9999999999998099 + Float64(t_0 + Float64(1259.1392167224028 / Float64(2.0 - z))))))) + Float64(Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0)) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0)))) * Float64(Float64(pi / sin(Float64(pi * z))) * Float64(sqrt(Float64(pi * 2.0)) * Float64(exp(Float64(-0.5 + Float64(-6.0 - Float64(1.0 - z)))) * (Float64(7.5 + Float64(-1.0 + Float64(1.0 - z))) ^ Float64(Float64(1.0 - z) - 0.5))))))
end

MATLAB

function tmp = code(z)
	t_0 = 676.5203681218851 / (1.0 - z);
	t_1 = 0.9999999999998099 + t_0;
	tmp = ((((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0))) - (((771.3234287776531 / (-2.0 - (1.0 - z))) + (-176.6150291621406 / (-3.0 - (1.0 - z)))) - (((t_1 * t_1) + ((1585431.567088306 / (2.0 - z)) / (z - 2.0))) / (0.9999999999998099 + (t_0 + (1259.1392167224028 / (2.0 - z))))))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)))) * ((pi / sin((pi * z))) * (sqrt((pi * 2.0)) * (exp((-0.5 + (-6.0 - (1.0 - z)))) * ((7.5 + (-1.0 + (1.0 - z))) ^ ((1.0 - z) - 0.5)))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.9999999999998099 + t$95$0), $MachinePrecision]}, N[(N[(N[(N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] - -4.0), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] - -5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(771.3234287776531 / N[(-2.0 - N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(-3.0 - N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] + N[(N[(1585431.567088306 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] / N[(z - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.9999999999998099 + N[(t$95$0 + N[(1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(-0.5 + N[(-6.0 - N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 + N[(-1.0 + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 96.6%

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

3. Simplified 98.2%

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

   1. flip-+ 98.2%

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

6. Applied egg-rr 98.2%

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

   1. +-commutative 98.2%

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

   2. +-commutative 98.2%

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

   3. associate-*l/ 98.2%

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

   4. associate-*r/ 98.2%

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

   5. metadata-eval 98.2%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{\color{blue}{1585431.567088306}}{1 - \left(z - 1\right)}}{1 - \left(z - 1\right)}}{\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   6. associate--r- 98.2%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{\color{blue}{\left(1 - z\right) + 1}}}{1 - \left(z - 1\right)}}{\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   7. +-commutative 98.2%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{\color{blue}{1 + \left(1 - z\right)}}}{1 - \left(z - 1\right)}}{\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   8. associate-+r- 98.2%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{\color{blue}{\left(1 + 1\right) - z}}}{1 - \left(z - 1\right)}}{\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   9. metadata-eval 98.2%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{\color{blue}{2} - z}}{1 - \left(z - 1\right)}}{\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   10. associate--r- 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{\color{blue}{\left(1 - z\right) + 1}}}{\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   11. +-commutative 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{\color{blue}{1 + \left(1 - z\right)}}}{\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   12. associate-+r- 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{\color{blue}{\left(1 + 1\right) - z}}}{\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   13. metadata-eval 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{\color{blue}{2} - z}}{\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   14. +-commutative 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{\color{blue}{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)} - \frac{-1259.1392167224028}{1 - \left(z - 1\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

8. Simplified 98.2%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\color{blue}{\frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \frac{\frac{1585431.567088306}{2 - z}}{2 - z}}{0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}\right)}} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

9. Final simplification 98.2%

   \[\leadsto \left(\left(\left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right) - \left(\left(\frac{771.3234287776531}{-2 - \left(1 - z\right)} + \frac{-176.6150291621406}{-3 - \left(1 - z\right)}\right) - \frac{\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{\frac{1585431.567088306}{2 - z}}{z - 2}}{0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{1259.1392167224028}{2 - z}\right)}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(e^{-0.5 + \left(-6 - \left(1 - z\right)\right)} \cdot {\left(7.5 + \left(-1 + \left(1 - z\right)\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right)\right)\right) \]
10. Add Preprocessing

Alternative 4: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

3. Simplified 98.2%

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

   1. *-un-lft-identity 98.2%

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

   2. +-commutative 98.2%

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

   3. --rgt-identity 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. associate-+l- 98.2%

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

6. Applied egg-rr 98.2%

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

   1. *-lft-identity 98.2%

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

   2. associate-+l+ 98.2%

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

   3. associate--r- 98.2%

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

   4. +-commutative 98.2%

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

   5. associate-+r- 98.2%

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

   6. metadata-eval 98.2%

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

8. Simplified 98.2%

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

9. Final simplification 98.2%

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

Alternative 5: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

3. Simplified 98.2%

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

   1. associate-+l- 98.2%

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

6. Applied egg-rr 98.2%

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

   1. *-un-lft-identity 98.2%

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

   2. +-commutative 98.2%

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

   3. --rgt-identity 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. associate-+l- 98.2%

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

8. Applied egg-rr 98.2%

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

   1. *-lft-identity 98.2%

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

   2. associate-+l+ 98.2%

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

   3. associate--r- 98.2%

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

   4. +-commutative 98.2%

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

   5. associate-+r- 98.2%

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

   6. metadata-eval 98.2%

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

10. Simplified 98.2%

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

11. Final simplification 98.2%

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

Alternative 6: 96.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(e^{-0.5 + \left(-6 - \left(1 - z\right)\right)} \cdot {\left(7.5 - z\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right)\right)\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right) + \left(\left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \left(z \cdot 361.7355639412844 + 47.95075976068351\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (/ PI (sin (* PI z)))
   (*
    (sqrt (* PI 2.0))
    (* (exp (+ -0.5 (- -6.0 (- 1.0 z)))) (pow (- 7.5 z) (- (- 1.0 z) 0.5)))))
  (+
   (+
    (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0))
    (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0)))
   (+
    (+
     (/ 12.507343278686905 (- (- 1.0 z) -4.0))
     (/ -0.13857109526572012 (- (- 1.0 z) -5.0)))
    (+
     (+
      (/ 771.3234287776531 (- (- 1.0 z) -2.0))
      (/ -176.6150291621406 (- (- 1.0 z) -3.0)))
     (+ (* z 361.7355639412844) 47.95075976068351))))))

C

double code(double z) {
	return ((((double) M_PI) / sin((((double) M_PI) * z))) * (sqrt((((double) M_PI) * 2.0)) * (exp((-0.5 + (-6.0 - (1.0 - z)))) * pow((7.5 - z), ((1.0 - z) - 0.5))))) * (((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))) + (((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0))) + (((771.3234287776531 / ((1.0 - z) - -2.0)) + (-176.6150291621406 / ((1.0 - z) - -3.0))) + ((z * 361.7355639412844) + 47.95075976068351))));
}

Java

public static double code(double z) {
	return ((Math.PI / Math.sin((Math.PI * z))) * (Math.sqrt((Math.PI * 2.0)) * (Math.exp((-0.5 + (-6.0 - (1.0 - z)))) * Math.pow((7.5 - z), ((1.0 - z) - 0.5))))) * (((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))) + (((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0))) + (((771.3234287776531 / ((1.0 - z) - -2.0)) + (-176.6150291621406 / ((1.0 - z) - -3.0))) + ((z * 361.7355639412844) + 47.95075976068351))));
}

Python

def code(z):
	return ((math.pi / math.sin((math.pi * z))) * (math.sqrt((math.pi * 2.0)) * (math.exp((-0.5 + (-6.0 - (1.0 - z)))) * math.pow((7.5 - z), ((1.0 - z) - 0.5))))) * (((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))) + (((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0))) + (((771.3234287776531 / ((1.0 - z) - -2.0)) + (-176.6150291621406 / ((1.0 - z) - -3.0))) + ((z * 361.7355639412844) + 47.95075976068351))))

Julia

function code(z)
	return Float64(Float64(Float64(pi / sin(Float64(pi * z))) * Float64(sqrt(Float64(pi * 2.0)) * Float64(exp(Float64(-0.5 + Float64(-6.0 - Float64(1.0 - z)))) * (Float64(7.5 - z) ^ Float64(Float64(1.0 - z) - 0.5))))) * Float64(Float64(Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0)) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0))) + Float64(Float64(Float64(12.507343278686905 / Float64(Float64(1.0 - z) - -4.0)) + Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) - -5.0))) + Float64(Float64(Float64(771.3234287776531 / Float64(Float64(1.0 - z) - -2.0)) + Float64(-176.6150291621406 / Float64(Float64(1.0 - z) - -3.0))) + Float64(Float64(z * 361.7355639412844) + 47.95075976068351)))))
end

MATLAB

function tmp = code(z)
	tmp = ((pi / sin((pi * z))) * (sqrt((pi * 2.0)) * (exp((-0.5 + (-6.0 - (1.0 - z)))) * ((7.5 - z) ^ ((1.0 - z) - 0.5))))) * (((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))) + (((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0))) + (((771.3234287776531 / ((1.0 - z) - -2.0)) + (-176.6150291621406 / ((1.0 - z) - -3.0))) + ((z * 361.7355639412844) + 47.95075976068351))));
end

Wolfram

code[z_] := N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(-0.5 + N[(-6.0 - N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] - -4.0), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] - -5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(771.3234287776531 / N[(N[(1.0 - z), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z * 361.7355639412844), $MachinePrecision] + 47.95075976068351), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

3. Simplified 98.2%

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

   1. associate-+l- 98.2%

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

6. Applied egg-rr 98.2%

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

7. Taylor expanded in z around 0 97.1%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - \left(z - -1\right)\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\color{blue}{\left(47.95075976068351 + 361.7355639412844 \cdot z\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
8. Step-by-step derivation

   1. +-commutative 97.1%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - \left(z - -1\right)\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\color{blue}{\left(361.7355639412844 \cdot z + 47.95075976068351\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   2. *-commutative 97.1%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - \left(z - -1\right)\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\left(\color{blue}{z \cdot 361.7355639412844} + 47.95075976068351\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

9. Simplified 97.1%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - \left(z - -1\right)\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\color{blue}{\left(z \cdot 361.7355639412844 + 47.95075976068351\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

10. Taylor expanded in z around 0 97.1%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\color{blue}{-1 \cdot z} + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\left(z \cdot 361.7355639412844 + 47.95075976068351\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
11. Step-by-step derivation

    1. neg-mul-1 97.1%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\color{blue}{\left(-z\right)} + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\left(z \cdot 361.7355639412844 + 47.95075976068351\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

12. Simplified 97.1%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\color{blue}{\left(-z\right)} + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\left(z \cdot 361.7355639412844 + 47.95075976068351\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

13. Final simplification 97.1%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(e^{-0.5 + \left(-6 - \left(1 - z\right)\right)} \cdot {\left(7.5 - z\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right)\right)\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right) + \left(\left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \left(z \cdot 361.7355639412844 + 47.95075976068351\right)\right)\right)\right) \]
14. Add Preprocessing

Alternative 7: 96.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

3. Simplified 95.3%

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

5. Taylor expanded in z around 0 96.3%

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

   1. associate-*l/ 96.2%

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

   2. *-commutative 96.2%

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

   3. associate-*r* 96.9%

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

   4. *-commutative 96.9%

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

7. Simplified 96.9%

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

8. Final simplification 96.9%

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

Alternative 8: 95.5% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (* 263.3831869810514 (* (exp -7.5) (* (sqrt 15.0) (/ (sqrt PI) z)))))

C

double code(double z) {
	return 263.3831869810514 * (exp(-7.5) * (sqrt(15.0) * (sqrt(((double) M_PI)) / z)));
}

Java

public static double code(double z) {
	return 263.3831869810514 * (Math.exp(-7.5) * (Math.sqrt(15.0) * (Math.sqrt(Math.PI) / z)));
}

Python

def code(z):
	return 263.3831869810514 * (math.exp(-7.5) * (math.sqrt(15.0) * (math.sqrt(math.pi) / z)))

Julia

function code(z)
	return Float64(263.3831869810514 * Float64(exp(-7.5) * Float64(sqrt(15.0) * Float64(sqrt(pi) / z))))
end

MATLAB

function tmp = code(z)
	tmp = 263.3831869810514 * (exp(-7.5) * (sqrt(15.0) * (sqrt(pi) / z)));
end

Wolfram

code[z_] := N[(263.3831869810514 * N[(N[Exp[-7.5], $MachinePrecision] * N[(N[Sqrt[15.0], $MachinePrecision] * N[(N[Sqrt[Pi], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

3. Simplified 95.3%

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

5. Taylor expanded in z around 0 96.3%

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

   1. pow1 96.3%

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

   2. *-commutative 96.3%

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

   3. associate-/l* 96.5%

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

   4. sqrt-unprod 96.5%

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

   5. metadata-eval 96.5%

      \[\leadsto {\left(263.3831869810514 \cdot \left(\sqrt{\pi} \cdot \left(e^{-7.5} \cdot \frac{\sqrt{\color{blue}{15}}}{z}\right)\right)\right)}^{1} \]

7. Applied egg-rr 96.5%

   \[\leadsto \color{blue}{{\left(263.3831869810514 \cdot \left(\sqrt{\pi} \cdot \left(e^{-7.5} \cdot \frac{\sqrt{15}}{z}\right)\right)\right)}^{1}} \]
8. Step-by-step derivation

   1. unpow1 96.5%

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

   2. *-commutative 96.5%

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

   3. associate-*l* 96.2%

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

9. Simplified 96.2%

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

10. Taylor expanded in z around 0 96.3%

    \[\leadsto 263.3831869810514 \cdot \color{blue}{\left(\frac{e^{-7.5} \cdot \sqrt{15}}{z} \cdot \sqrt{\pi}\right)} \]
11. Step-by-step derivation

    1. associate-/l* 96.5%

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

    2. associate-*r* 96.2%

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

    3. associate-*l/ 95.7%

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

    4. associate-/l* 96.1%

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

12. Simplified 96.1%

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

13. Final simplification 96.1%

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

Alternative 9: 95.5% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (* 263.3831869810514 (* (exp -7.5) (* (sqrt PI) (/ (sqrt 15.0) z)))))

C

double code(double z) {
	return 263.3831869810514 * (exp(-7.5) * (sqrt(((double) M_PI)) * (sqrt(15.0) / z)));
}

Java

public static double code(double z) {
	return 263.3831869810514 * (Math.exp(-7.5) * (Math.sqrt(Math.PI) * (Math.sqrt(15.0) / z)));
}

Python

def code(z):
	return 263.3831869810514 * (math.exp(-7.5) * (math.sqrt(math.pi) * (math.sqrt(15.0) / z)))

Julia

function code(z)
	return Float64(263.3831869810514 * Float64(exp(-7.5) * Float64(sqrt(pi) * Float64(sqrt(15.0) / z))))
end

MATLAB

function tmp = code(z)
	tmp = 263.3831869810514 * (exp(-7.5) * (sqrt(pi) * (sqrt(15.0) / z)));
end

Wolfram

code[z_] := N[(263.3831869810514 * N[(N[Exp[-7.5], $MachinePrecision] * N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[Sqrt[15.0], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

3. Simplified 95.3%

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

5. Taylor expanded in z around 0 96.3%

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

   1. pow1 96.3%

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

   2. *-commutative 96.3%

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

   3. associate-/l* 96.5%

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

   4. sqrt-unprod 96.5%

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

   5. metadata-eval 96.5%

      \[\leadsto {\left(263.3831869810514 \cdot \left(\sqrt{\pi} \cdot \left(e^{-7.5} \cdot \frac{\sqrt{\color{blue}{15}}}{z}\right)\right)\right)}^{1} \]

7. Applied egg-rr 96.5%

   \[\leadsto \color{blue}{{\left(263.3831869810514 \cdot \left(\sqrt{\pi} \cdot \left(e^{-7.5} \cdot \frac{\sqrt{15}}{z}\right)\right)\right)}^{1}} \]
8. Step-by-step derivation

   1. unpow1 96.5%

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

   2. *-commutative 96.5%

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

   3. associate-*l* 96.2%

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

9. Simplified 96.2%

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

10. Final simplification 96.2%

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

Alternative 10: 95.6% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (* 263.3831869810514 (* (sqrt PI) (* (exp -7.5) (/ (sqrt 15.0) z)))))

C

double code(double z) {
	return 263.3831869810514 * (sqrt(((double) M_PI)) * (exp(-7.5) * (sqrt(15.0) / z)));
}

Java

public static double code(double z) {
	return 263.3831869810514 * (Math.sqrt(Math.PI) * (Math.exp(-7.5) * (Math.sqrt(15.0) / z)));
}

Python

def code(z):
	return 263.3831869810514 * (math.sqrt(math.pi) * (math.exp(-7.5) * (math.sqrt(15.0) / z)))

Julia

function code(z)
	return Float64(263.3831869810514 * Float64(sqrt(pi) * Float64(exp(-7.5) * Float64(sqrt(15.0) / z))))
end

MATLAB

function tmp = code(z)
	tmp = 263.3831869810514 * (sqrt(pi) * (exp(-7.5) * (sqrt(15.0) / z)));
end

Wolfram

code[z_] := N[(263.3831869810514 * N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] * N[(N[Sqrt[15.0], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

3. Simplified 95.3%

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

5. Taylor expanded in z around 0 96.3%

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

   1. *-un-lft-identity 96.3%

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

   2. associate-/l* 96.5%

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

   3. sqrt-unprod 96.5%

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

   4. metadata-eval 96.5%

      \[\leadsto 263.3831869810514 \cdot \left(\left(1 \cdot \left(e^{-7.5} \cdot \frac{\sqrt{\color{blue}{15}}}{z}\right)\right) \cdot \sqrt{\pi}\right) \]

7. Applied egg-rr 96.5%

   \[\leadsto 263.3831869810514 \cdot \left(\color{blue}{\left(1 \cdot \left(e^{-7.5} \cdot \frac{\sqrt{15}}{z}\right)\right)} \cdot \sqrt{\pi}\right) \]
8. Step-by-step derivation

   1. *-lft-identity 96.5%

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

9. Simplified 96.5%

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

10. Final simplification 96.5%

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

Alternative 11: 95.7% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (* (* 263.3831869810514 (sqrt PI)) (* (exp -7.5) (/ (sqrt 15.0) z))))

C

double code(double z) {
	return (263.3831869810514 * sqrt(((double) M_PI))) * (exp(-7.5) * (sqrt(15.0) / z));
}

Java

public static double code(double z) {
	return (263.3831869810514 * Math.sqrt(Math.PI)) * (Math.exp(-7.5) * (Math.sqrt(15.0) / z));
}

Python

def code(z):
	return (263.3831869810514 * math.sqrt(math.pi)) * (math.exp(-7.5) * (math.sqrt(15.0) / z))

Julia

function code(z)
	return Float64(Float64(263.3831869810514 * sqrt(pi)) * Float64(exp(-7.5) * Float64(sqrt(15.0) / z)))
end

MATLAB

function tmp = code(z)
	tmp = (263.3831869810514 * sqrt(pi)) * (exp(-7.5) * (sqrt(15.0) / z));
end

Wolfram

code[z_] := N[(N[(263.3831869810514 * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] * N[(N[Sqrt[15.0], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

3. Simplified 95.3%

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

5. Taylor expanded in z around 0 94.6%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\color{blue}{\left(46.9507597606837 + z \cdot \left(361.7355639412844 + 519.1279660315847 \cdot z\right)\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. *-commutative 94.6%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(46.9507597606837 + z \cdot \left(361.7355639412844 + \color{blue}{z \cdot 519.1279660315847}\right)\right) + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]

7. Simplified 94.6%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\color{blue}{\left(46.9507597606837 + z \cdot \left(361.7355639412844 + z \cdot 519.1279660315847\right)\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]

8. Taylor expanded in z around 0 96.3%

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

   1. *-commutative 96.3%

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

   2. associate-*l* 96.4%

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

   3. *-commutative 96.4%

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

   4. associate-/l* 96.6%

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

   5. *-commutative 96.6%

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

10. Simplified 96.6%

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

    1. associate-*r/ 96.4%

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

    2. sqrt-unprod 96.4%

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

    3. metadata-eval 96.4%

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

12. Applied egg-rr 96.4%

    \[\leadsto \color{blue}{\frac{\sqrt{15} \cdot e^{-7.5}}{z}} \cdot \left(\sqrt{\pi} \cdot 263.3831869810514\right) \]
13. Step-by-step derivation

    1. *-commutative 96.4%

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

    2. associate-/l* 96.6%

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

14. Simplified 96.6%

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

15. Final simplification 96.6%

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

Alternative 12: 95.8% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (* (* 263.3831869810514 (sqrt PI)) (* (sqrt 15.0) (/ (exp -7.5) z))))

C

double code(double z) {
	return (263.3831869810514 * sqrt(((double) M_PI))) * (sqrt(15.0) * (exp(-7.5) / z));
}

Java

public static double code(double z) {
	return (263.3831869810514 * Math.sqrt(Math.PI)) * (Math.sqrt(15.0) * (Math.exp(-7.5) / z));
}

Python

def code(z):
	return (263.3831869810514 * math.sqrt(math.pi)) * (math.sqrt(15.0) * (math.exp(-7.5) / z))

Julia

function code(z)
	return Float64(Float64(263.3831869810514 * sqrt(pi)) * Float64(sqrt(15.0) * Float64(exp(-7.5) / z)))
end

MATLAB

function tmp = code(z)
	tmp = (263.3831869810514 * sqrt(pi)) * (sqrt(15.0) * (exp(-7.5) / z));
end

Wolfram

code[z_] := N[(N[(263.3831869810514 * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[15.0], $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

3. Simplified 95.3%

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

5. Taylor expanded in z around 0 94.6%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\color{blue}{\left(46.9507597606837 + z \cdot \left(361.7355639412844 + 519.1279660315847 \cdot z\right)\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. *-commutative 94.6%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(46.9507597606837 + z \cdot \left(361.7355639412844 + \color{blue}{z \cdot 519.1279660315847}\right)\right) + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]

7. Simplified 94.6%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\color{blue}{\left(46.9507597606837 + z \cdot \left(361.7355639412844 + z \cdot 519.1279660315847\right)\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]

8. Taylor expanded in z around 0 96.3%

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

   1. *-commutative 96.3%

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

   2. associate-*l* 96.4%

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

   3. *-commutative 96.4%

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

   4. associate-/l* 96.6%

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

   5. *-commutative 96.6%

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

10. Simplified 96.6%

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

    1. pow1 96.6%

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

    2. *-commutative 96.6%

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

    3. sqrt-unprod 96.6%

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

    4. metadata-eval 96.6%

       \[\leadsto {\left(\frac{e^{-7.5}}{z} \cdot \sqrt{\color{blue}{15}}\right)}^{1} \cdot \left(\sqrt{\pi} \cdot 263.3831869810514\right) \]

12. Applied egg-rr 96.6%

    \[\leadsto \color{blue}{{\left(\frac{e^{-7.5}}{z} \cdot \sqrt{15}\right)}^{1}} \cdot \left(\sqrt{\pi} \cdot 263.3831869810514\right) \]

13. Final simplification 96.6%

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

Alternative 13: 95.8% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (* (sqrt 15.0) (* (/ (exp -7.5) z) (* 263.3831869810514 (sqrt PI)))))

C

double code(double z) {
	return sqrt(15.0) * ((exp(-7.5) / z) * (263.3831869810514 * sqrt(((double) M_PI))));
}

Java

public static double code(double z) {
	return Math.sqrt(15.0) * ((Math.exp(-7.5) / z) * (263.3831869810514 * Math.sqrt(Math.PI)));
}

Python

def code(z):
	return math.sqrt(15.0) * ((math.exp(-7.5) / z) * (263.3831869810514 * math.sqrt(math.pi)))

Julia

function code(z)
	return Float64(sqrt(15.0) * Float64(Float64(exp(-7.5) / z) * Float64(263.3831869810514 * sqrt(pi))))
end

MATLAB

function tmp = code(z)
	tmp = sqrt(15.0) * ((exp(-7.5) / z) * (263.3831869810514 * sqrt(pi)));
end

Wolfram

code[z_] := N[(N[Sqrt[15.0], $MachinePrecision] * N[(N[(N[Exp[-7.5], $MachinePrecision] / z), $MachinePrecision] * N[(263.3831869810514 * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

3. Simplified 95.3%

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

5. Taylor expanded in z around 0 94.6%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\color{blue}{\left(46.9507597606837 + z \cdot \left(361.7355639412844 + 519.1279660315847 \cdot z\right)\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. *-commutative 94.6%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(46.9507597606837 + z \cdot \left(361.7355639412844 + \color{blue}{z \cdot 519.1279660315847}\right)\right) + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]

7. Simplified 94.6%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\color{blue}{\left(46.9507597606837 + z \cdot \left(361.7355639412844 + z \cdot 519.1279660315847\right)\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]

8. Taylor expanded in z around 0 96.3%

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

   1. *-commutative 96.3%

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

   2. associate-*l* 96.4%

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

   3. *-commutative 96.4%

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

   4. associate-/l* 96.6%

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

   5. *-commutative 96.6%

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

10. Simplified 96.6%

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

    1. pow1 96.6%

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

    2. associate-*l* 96.6%

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

    3. sqrt-unprod 96.6%

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

    4. metadata-eval 96.6%

       \[\leadsto {\left(\sqrt{\color{blue}{15}} \cdot \left(\frac{e^{-7.5}}{z} \cdot \left(\sqrt{\pi} \cdot 263.3831869810514\right)\right)\right)}^{1} \]

12. Applied egg-rr 96.6%

    \[\leadsto \color{blue}{{\left(\sqrt{15} \cdot \left(\frac{e^{-7.5}}{z} \cdot \left(\sqrt{\pi} \cdot 263.3831869810514\right)\right)\right)}^{1}} \]

13. Final simplification 96.6%

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

Alternative 14: 95.2% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (* 263.3831869810514 (* (exp -7.5) (/ (sqrt (* PI 15.0)) z))))

C

double code(double z) {
	return 263.3831869810514 * (exp(-7.5) * (sqrt((((double) M_PI) * 15.0)) / z));
}

Java

public static double code(double z) {
	return 263.3831869810514 * (Math.exp(-7.5) * (Math.sqrt((Math.PI * 15.0)) / z));
}

Python

def code(z):
	return 263.3831869810514 * (math.exp(-7.5) * (math.sqrt((math.pi * 15.0)) / z))

Julia

function code(z)
	return Float64(263.3831869810514 * Float64(exp(-7.5) * Float64(sqrt(Float64(pi * 15.0)) / z)))
end

MATLAB

function tmp = code(z)
	tmp = 263.3831869810514 * (exp(-7.5) * (sqrt((pi * 15.0)) / z));
end

Wolfram

code[z_] := N[(263.3831869810514 * N[(N[Exp[-7.5], $MachinePrecision] * N[(N[Sqrt[N[(Pi * 15.0), $MachinePrecision]], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

3. Simplified 95.3%

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

5. Taylor expanded in z around 0 96.3%

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

   1. pow1 96.3%

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

   2. *-commutative 96.3%

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

   3. associate-/l* 96.5%

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

   4. sqrt-unprod 96.5%

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

   5. metadata-eval 96.5%

      \[\leadsto {\left(263.3831869810514 \cdot \left(\sqrt{\pi} \cdot \left(e^{-7.5} \cdot \frac{\sqrt{\color{blue}{15}}}{z}\right)\right)\right)}^{1} \]

7. Applied egg-rr 96.5%

   \[\leadsto \color{blue}{{\left(263.3831869810514 \cdot \left(\sqrt{\pi} \cdot \left(e^{-7.5} \cdot \frac{\sqrt{15}}{z}\right)\right)\right)}^{1}} \]
8. Step-by-step derivation

   1. unpow1 96.5%

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

   2. *-commutative 96.5%

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

   3. associate-*l* 96.2%

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

9. Simplified 96.2%

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

10. Taylor expanded in z around 0 96.3%

    \[\leadsto 263.3831869810514 \cdot \color{blue}{\left(\frac{e^{-7.5} \cdot \sqrt{15}}{z} \cdot \sqrt{\pi}\right)} \]
11. Step-by-step derivation

    1. associate-/l* 96.5%

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

    2. associate-*r* 96.2%

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

    3. associate-*l/ 95.7%

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

    4. associate-/l* 96.1%

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

12. Simplified 96.1%

    \[\leadsto 263.3831869810514 \cdot \color{blue}{\left(e^{-7.5} \cdot \left(\sqrt{15} \cdot \frac{\sqrt{\pi}}{z}\right)\right)} \]
13. Step-by-step derivation

    1. pow1 96.1%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{{\left(e^{-7.5} \cdot \left(\sqrt{15} \cdot \frac{\sqrt{\pi}}{z}\right)\right)}^{1}} \]

    2. metadata-eval 96.1%

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

    3. sqrt-unprod 96.1%

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

    4. associate-*r/ 95.7%

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

    5. sqrt-unprod 95.7%

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

    6. metadata-eval 95.7%

       \[\leadsto 263.3831869810514 \cdot {\left(e^{-7.5} \cdot \frac{\sqrt{\color{blue}{15}} \cdot \sqrt{\pi}}{z}\right)}^{1} \]

    7. pow1/2 95.7%

       \[\leadsto 263.3831869810514 \cdot {\left(e^{-7.5} \cdot \frac{\color{blue}{{15}^{0.5}} \cdot \sqrt{\pi}}{z}\right)}^{1} \]

    8. pow1/2 95.7%

       \[\leadsto 263.3831869810514 \cdot {\left(e^{-7.5} \cdot \frac{{15}^{0.5} \cdot \color{blue}{{\pi}^{0.5}}}{z}\right)}^{1} \]

    9. pow-prod-down 95.7%

       \[\leadsto 263.3831869810514 \cdot {\left(e^{-7.5} \cdot \frac{\color{blue}{{\left(15 \cdot \pi\right)}^{0.5}}}{z}\right)}^{1} \]

14. Applied egg-rr 95.7%

    \[\leadsto 263.3831869810514 \cdot \color{blue}{{\left(e^{-7.5} \cdot \frac{{\left(15 \cdot \pi\right)}^{0.5}}{z}\right)}^{1}} \]
15. Step-by-step derivation

    1. unpow1 95.7%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{\left(e^{-7.5} \cdot \frac{{\left(15 \cdot \pi\right)}^{0.5}}{z}\right)} \]

    2. unpow1/2 95.7%

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

    3. *-commutative 95.7%

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

16. Simplified 95.7%

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

17. Final simplification 95.7%

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

Reproduce

herbie shell --seed 2024080 
(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))))))