Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.4% → 97.4%

Time: 17.8s

Alternatives: 13

Speedup: 1.5×

• could not determine a ground truth

  1. z = -571883232514.6993

Specification

\[z \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(t\_0 + 7\right) + 0.5\right)}^{\left(\frac{{t\_0}^{3} + 0.125}{z \cdot z + \left(0.25 - t\_0 \cdot 0.5\right)}\right)}\right) \cdot e^{z - 7.5}\right) \cdot \left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)))
   (*
    (/ PI (sin (* PI z)))
    (*
     (*
      (*
       (sqrt (* PI 2.0))
       (pow
        (+ (+ t_0 7.0) 0.5)
        (/ (+ (pow t_0 3.0) 0.125) (+ (* z z) (- 0.25 (* t_0 0.5))))))
      (exp (- z 7.5)))
     (+
      (+
       (+
        (+
         (+
          (+
           (+ 0.9999999999998099 (/ 676.5203681218851 (- 1.0 z)))
           (/ -1259.1392167224028 (- (- 1.0 z) -1.0)))
          (/ 771.3234287776531 (- (- 1.0 z) -2.0)))
         (/ -176.6150291621406 (- (- 1.0 z) -3.0)))
        (/ 12.507343278686905 (- (- 1.0 z) -4.0)))
       (+
        (/ -0.13857109526572012 (- (- 1.0 z) -5.0))
        (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0))))
      (/ 1.5056327351493116e-7 (+ t_0 8.0)))))))

C

double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (((sqrt((((double) M_PI) * 2.0)) * pow(((t_0 + 7.0) + 0.5), ((pow(t_0, 3.0) + 0.125) / ((z * z) + (0.25 - (t_0 * 0.5)))))) * exp((z - 7.5))) * (((((((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + (771.3234287776531 / ((1.0 - z) - -2.0))) + (-176.6150291621406 / ((1.0 - z) - -3.0))) + (12.507343278686905 / ((1.0 - z) - -4.0))) + ((-0.13857109526572012 / ((1.0 - z) - -5.0)) + (9.984369578019572e-6 / ((1.0 - z) - -6.0)))) + (1.5056327351493116e-7 / (t_0 + 8.0))));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	return (Math.PI / Math.sin((Math.PI * z))) * (((Math.sqrt((Math.PI * 2.0)) * Math.pow(((t_0 + 7.0) + 0.5), ((Math.pow(t_0, 3.0) + 0.125) / ((z * z) + (0.25 - (t_0 * 0.5)))))) * Math.exp((z - 7.5))) * (((((((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + (771.3234287776531 / ((1.0 - z) - -2.0))) + (-176.6150291621406 / ((1.0 - z) - -3.0))) + (12.507343278686905 / ((1.0 - z) - -4.0))) + ((-0.13857109526572012 / ((1.0 - z) - -5.0)) + (9.984369578019572e-6 / ((1.0 - z) - -6.0)))) + (1.5056327351493116e-7 / (t_0 + 8.0))));
}

Python

def code(z):
	t_0 = (1.0 - z) - 1.0
	return (math.pi / math.sin((math.pi * z))) * (((math.sqrt((math.pi * 2.0)) * math.pow(((t_0 + 7.0) + 0.5), ((math.pow(t_0, 3.0) + 0.125) / ((z * z) + (0.25 - (t_0 * 0.5)))))) * math.exp((z - 7.5))) * (((((((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + (771.3234287776531 / ((1.0 - z) - -2.0))) + (-176.6150291621406 / ((1.0 - z) - -3.0))) + (12.507343278686905 / ((1.0 - z) - -4.0))) + ((-0.13857109526572012 / ((1.0 - z) - -5.0)) + (9.984369578019572e-6 / ((1.0 - z) - -6.0)))) + (1.5056327351493116e-7 / (t_0 + 8.0))))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - 1.0)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (Float64(Float64(t_0 + 7.0) + 0.5) ^ Float64(Float64((t_0 ^ 3.0) + 0.125) / Float64(Float64(z * z) + Float64(0.25 - Float64(t_0 * 0.5)))))) * exp(Float64(z - 7.5))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 - z))) + Float64(-1259.1392167224028 / Float64(Float64(1.0 - z) - -1.0))) + Float64(771.3234287776531 / Float64(Float64(1.0 - z) - -2.0))) + Float64(-176.6150291621406 / Float64(Float64(1.0 - z) - -3.0))) + Float64(12.507343278686905 / Float64(Float64(1.0 - z) - -4.0))) + Float64(Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) - -5.0)) + Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0)))) + Float64(1.5056327351493116e-7 / Float64(t_0 + 8.0)))))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - 1.0;
	tmp = (pi / sin((pi * z))) * (((sqrt((pi * 2.0)) * (((t_0 + 7.0) + 0.5) ^ (((t_0 ^ 3.0) + 0.125) / ((z * z) + (0.25 - (t_0 * 0.5)))))) * exp((z - 7.5))) * (((((((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + (771.3234287776531 / ((1.0 - z) - -2.0))) + (-176.6150291621406 / ((1.0 - z) - -3.0))) + (12.507343278686905 / ((1.0 - z) - -4.0))) + ((-0.13857109526572012 / ((1.0 - z) - -5.0)) + (9.984369578019572e-6 / ((1.0 - z) - -6.0)))) + (1.5056327351493116e-7 / (t_0 + 8.0))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - 1.0), $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[N[(N[(t$95$0 + 7.0), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] + 0.125), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] + N[(0.25 - N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(z - 7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(N[(1.0 - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(N[(1.0 - z), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] - -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] - -5.0), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(t$95$0 + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 96.6%

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

4. Applied rewrites 97.7%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\color{blue}{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. flip3-+ N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\color{blue}{\left(\frac{{\left(\left(1 - z\right) - 1\right)}^{3} + {\frac{1}{2}}^{3}}{\left(\left(1 - z\right) - 1\right) \cdot \left(\left(1 - z\right) - 1\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(\left(1 - z\right) - 1\right) \cdot \frac{1}{2}\right)}\right)}}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\color{blue}{\left(\frac{{\left(\left(1 - z\right) - 1\right)}^{3} + {\frac{1}{2}}^{3}}{\left(\left(1 - z\right) - 1\right) \cdot \left(\left(1 - z\right) - 1\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(\left(1 - z\right) - 1\right) \cdot \frac{1}{2}\right)}\right)}}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\frac{\color{blue}{{\left(\left(1 - z\right) - 1\right)}^{3} + {\frac{1}{2}}^{3}}}{\left(\left(1 - z\right) - 1\right) \cdot \left(\left(1 - z\right) - 1\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(\left(1 - z\right) - 1\right) \cdot \frac{1}{2}\right)}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\frac{\color{blue}{{\left(\left(1 - z\right) - 1\right)}^{3}} + {\frac{1}{2}}^{3}}{\left(\left(1 - z\right) - 1\right) \cdot \left(\left(1 - z\right) - 1\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(\left(1 - z\right) - 1\right) \cdot \frac{1}{2}\right)}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\frac{{\left(\left(1 - z\right) - 1\right)}^{3} + \color{blue}{\frac{1}{8}}}{\left(\left(1 - z\right) - 1\right) \cdot \left(\left(1 - z\right) - 1\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(\left(1 - z\right) - 1\right) \cdot \frac{1}{2}\right)}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\frac{{\left(\left(1 - z\right) - 1\right)}^{3} + \frac{1}{8}}{\color{blue}{\left(\left(1 - z\right) - 1\right) \cdot \left(\left(1 - z\right) - 1\right)} + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(\left(1 - z\right) - 1\right) \cdot \frac{1}{2}\right)}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\frac{{\left(\left(1 - z\right) - 1\right)}^{3} + \frac{1}{8}}{\color{blue}{\left(\left(1 - z\right) - 1\right) \cdot \left(\left(1 - z\right) - 1\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(\left(1 - z\right) - 1\right) \cdot \frac{1}{2}\right)}}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\frac{{\left(\left(1 - z\right) - 1\right)}^{3} + \frac{1}{8}}{\color{blue}{\left(\left(1 - z\right) - 1\right) \cdot \left(\left(1 - z\right) - 1\right)} + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(\left(1 - z\right) - 1\right) \cdot \frac{1}{2}\right)}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   10. pow2 N/A

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\frac{{\left(\left(1 - z\right) - 1\right)}^{3} + \frac{1}{8}}{\color{blue}{{\left(\left(1 - z\right) - 1\right)}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(\left(1 - z\right) - 1\right) \cdot \frac{1}{2}\right)}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   11. lower-pow.f64 N/A

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\frac{{\left(\left(1 - z\right) - 1\right)}^{3} + \frac{1}{8}}{\color{blue}{{\left(\left(1 - z\right) - 1\right)}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(\left(1 - z\right) - 1\right) \cdot \frac{1}{2}\right)}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   12. lower--.f64 N/A

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\frac{{\left(\left(1 - z\right) - 1\right)}^{3} + \frac{1}{8}}{{\left(\left(1 - z\right) - 1\right)}^{2} + \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{2} - \left(\left(1 - z\right) - 1\right) \cdot \frac{1}{2}\right)}}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\frac{{\left(\left(1 - z\right) - 1\right)}^{3} + \frac{1}{8}}{{\left(\left(1 - z\right) - 1\right)}^{2} + \left(\color{blue}{\frac{1}{4}} - \left(\left(1 - z\right) - 1\right) \cdot \frac{1}{2}\right)}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   14. lower-*.f64 97.7

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\frac{{\left(\left(1 - z\right) - 1\right)}^{3} + 0.125}{{\left(\left(1 - z\right) - 1\right)}^{2} + \left(0.25 - \color{blue}{\left(\left(1 - z\right) - 1\right) \cdot 0.5}\right)}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Applied rewrites 97.7%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\color{blue}{\left(\frac{{\left(\left(1 - z\right) - 1\right)}^{3} + 0.125}{{\left(\left(1 - z\right) - 1\right)}^{2} + \left(0.25 - \left(\left(1 - z\right) - 1\right) \cdot 0.5\right)}\right)}}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

7. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\frac{{\left(\left(1 - z\right) - 1\right)}^{3} + \frac{1}{8}}{\color{blue}{{z}^{2}} + \left(\frac{1}{4} - \left(\left(1 - z\right) - 1\right) \cdot \frac{1}{2}\right)}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
8. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\frac{{\left(\left(1 - z\right) - 1\right)}^{3} + \frac{1}{8}}{z \cdot \color{blue}{z} + \left(\frac{1}{4} - \left(\left(1 - z\right) - 1\right) \cdot \frac{1}{2}\right)}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-*.f64 97.7

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\frac{{\left(\left(1 - z\right) - 1\right)}^{3} + 0.125}{z \cdot \color{blue}{z} + \left(0.25 - \left(\left(1 - z\right) - 1\right) \cdot 0.5\right)}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

9. Applied rewrites 97.7%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\frac{{\left(\left(1 - z\right) - 1\right)}^{3} + 0.125}{\color{blue}{z \cdot z} + \left(0.25 - \left(\left(1 - z\right) - 1\right) \cdot 0.5\right)}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

10. Taylor expanded in z around 0

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\frac{{\left(\left(1 - z\right) - 1\right)}^{3} + \frac{1}{8}}{z \cdot z + \left(\frac{1}{4} - \left(\left(1 - z\right) - 1\right) \cdot \frac{1}{2}\right)}\right)}\right) \cdot e^{\color{blue}{z - \frac{15}{2}}}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
11. Step-by-step derivation

    1. lift--.f64 97.7

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\frac{{\left(\left(1 - z\right) - 1\right)}^{3} + 0.125}{z \cdot z + \left(0.25 - \left(\left(1 - z\right) - 1\right) \cdot 0.5\right)}\right)}\right) \cdot e^{z - \color{blue}{7.5}}\right) \cdot \left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

12. Applied rewrites 97.7%

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\frac{{\left(\left(1 - z\right) - 1\right)}^{3} + 0.125}{z \cdot z + \left(0.25 - \left(\left(1 - z\right) - 1\right) \cdot 0.5\right)}\right)}\right) \cdot e^{\color{blue}{z - 7.5}}\right) \cdot \left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

13. Final simplification 97.7%

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\frac{{\left(\left(1 - z\right) - 1\right)}^{3} + 0.125}{z \cdot z + \left(0.25 - \left(\left(1 - z\right) - 1\right) \cdot 0.5\right)}\right)}\right) \cdot e^{z - 7.5}\right) \cdot \left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
14. Add Preprocessing

Alternative 2: 97.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

4. Applied rewrites 97.7%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

5. Taylor expanded in z around inf

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lift-sqrt.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(\color{blue}{e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lift-PI.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left(e^{\color{blue}{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \color{blue}{\left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. exp-to-pow N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(\color{blue}{e^{z - \frac{15}{2}}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(\color{blue}{e^{z - \frac{15}{2}}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lower--.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{\color{blue}{z - \frac{15}{2}}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lower--.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \color{blue}{\sqrt{2}}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   10. lower-exp.f64 N/A

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{\color{blue}{2}}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   11. lower--.f64 N/A

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   12. lower-sqrt.f64 97.7

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z - 7.5} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

7. Applied rewrites 97.7%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\color{blue}{\left(\sqrt{\pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z - 7.5} \cdot \sqrt{2}\right)\right)\right)} \cdot \left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

8. Final simplification 97.7%

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

Alternative 3: 97.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

4. Applied rewrites 97.7%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

5. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(1 - z\right) - 0}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \color{blue}{\frac{3764081837873279}{200000000000000000000000}}\right)\right) \]
6. Step-by-step derivation

7. Applied rewrites 97.3%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right) + \color{blue}{1.8820409189366395 \cdot 10^{-8}}\right)\right) \]

8. Final simplification 97.3%

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

Alternative 4: 97.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(t\_0 + 7\right) + 0.5\right)}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{\left(\left(\left(-1 + z\right) - -1\right) - 7\right) - 0.5}\right) \cdot \left(\left(\left(263.4062807184368 + z \cdot \left(436.9000215473151 + z \cdot \left(545.0359493463282 + 606.6767878347069 \cdot z\right)\right)\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)))
   (*
    (/ PI (sin (* PI z)))
    (*
     (*
      (* (sqrt (* PI 2.0)) (pow (+ (+ t_0 7.0) 0.5) (+ t_0 0.5)))
      (exp (- (- (- (+ -1.0 z) -1.0) 7.0) 0.5)))
     (+
      (+
       (+
        263.4062807184368
        (*
         z
         (+
          436.9000215473151
          (* z (+ 545.0359493463282 (* 606.6767878347069 z))))))
       (+
        (/ -0.13857109526572012 (- (- 1.0 z) -5.0))
        (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0))))
      (/ 1.5056327351493116e-7 (+ t_0 8.0)))))))

C

double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (((sqrt((((double) M_PI) * 2.0)) * pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5))) * exp(((((-1.0 + z) - -1.0) - 7.0) - 0.5))) * (((263.4062807184368 + (z * (436.9000215473151 + (z * (545.0359493463282 + (606.6767878347069 * z)))))) + ((-0.13857109526572012 / ((1.0 - z) - -5.0)) + (9.984369578019572e-6 / ((1.0 - z) - -6.0)))) + (1.5056327351493116e-7 / (t_0 + 8.0))));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) - 1.0;
	return (Math.PI / Math.sin((Math.PI * z))) * (((Math.sqrt((Math.PI * 2.0)) * Math.pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5))) * Math.exp(((((-1.0 + z) - -1.0) - 7.0) - 0.5))) * (((263.4062807184368 + (z * (436.9000215473151 + (z * (545.0359493463282 + (606.6767878347069 * z)))))) + ((-0.13857109526572012 / ((1.0 - z) - -5.0)) + (9.984369578019572e-6 / ((1.0 - z) - -6.0)))) + (1.5056327351493116e-7 / (t_0 + 8.0))));
}

Python

def code(z):
	t_0 = (1.0 - z) - 1.0
	return (math.pi / math.sin((math.pi * z))) * (((math.sqrt((math.pi * 2.0)) * math.pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5))) * math.exp(((((-1.0 + z) - -1.0) - 7.0) - 0.5))) * (((263.4062807184368 + (z * (436.9000215473151 + (z * (545.0359493463282 + (606.6767878347069 * z)))))) + ((-0.13857109526572012 / ((1.0 - z) - -5.0)) + (9.984369578019572e-6 / ((1.0 - z) - -6.0)))) + (1.5056327351493116e-7 / (t_0 + 8.0))))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - 1.0)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (Float64(Float64(t_0 + 7.0) + 0.5) ^ Float64(t_0 + 0.5))) * exp(Float64(Float64(Float64(Float64(-1.0 + z) - -1.0) - 7.0) - 0.5))) * Float64(Float64(Float64(263.4062807184368 + Float64(z * Float64(436.9000215473151 + Float64(z * Float64(545.0359493463282 + Float64(606.6767878347069 * z)))))) + Float64(Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) - -5.0)) + Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0)))) + Float64(1.5056327351493116e-7 / Float64(t_0 + 8.0)))))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - 1.0;
	tmp = (pi / sin((pi * z))) * (((sqrt((pi * 2.0)) * (((t_0 + 7.0) + 0.5) ^ (t_0 + 0.5))) * exp(((((-1.0 + z) - -1.0) - 7.0) - 0.5))) * (((263.4062807184368 + (z * (436.9000215473151 + (z * (545.0359493463282 + (606.6767878347069 * z)))))) + ((-0.13857109526572012 / ((1.0 - z) - -5.0)) + (9.984369578019572e-6 / ((1.0 - z) - -6.0)))) + (1.5056327351493116e-7 / (t_0 + 8.0))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - 1.0), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[(t$95$0 + 7.0), $MachinePrecision] + 0.5), $MachinePrecision], N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(N[(N[(-1.0 + z), $MachinePrecision] - -1.0), $MachinePrecision] - 7.0), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(263.4062807184368 + N[(z * N[(436.9000215473151 + N[(z * N[(545.0359493463282 + N[(606.6767878347069 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] - -5.0), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(t$95$0 + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 96.6%

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

4. Applied rewrites 97.7%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

5. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\color{blue}{\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \left(\frac{39321001939258358983}{90000000000000000} + z \cdot \left(\frac{2943194126470171931171}{5400000000000000000} + \frac{196563279258445065194677}{324000000000000000000} \cdot z\right)\right)\right)} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{7902188421553103227}{30000000000000000} + \color{blue}{z \cdot \left(\frac{39321001939258358983}{90000000000000000} + z \cdot \left(\frac{2943194126470171931171}{5400000000000000000} + \frac{196563279258445065194677}{324000000000000000000} \cdot z\right)\right)}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \color{blue}{\left(\frac{39321001939258358983}{90000000000000000} + z \cdot \left(\frac{2943194126470171931171}{5400000000000000000} + \frac{196563279258445065194677}{324000000000000000000} \cdot z\right)\right)}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \left(\frac{39321001939258358983}{90000000000000000} + \color{blue}{z \cdot \left(\frac{2943194126470171931171}{5400000000000000000} + \frac{196563279258445065194677}{324000000000000000000} \cdot z\right)}\right)\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \left(\frac{39321001939258358983}{90000000000000000} + z \cdot \color{blue}{\left(\frac{2943194126470171931171}{5400000000000000000} + \frac{196563279258445065194677}{324000000000000000000} \cdot z\right)}\right)\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \left(\frac{39321001939258358983}{90000000000000000} + z \cdot \left(\frac{2943194126470171931171}{5400000000000000000} + \color{blue}{\frac{196563279258445065194677}{324000000000000000000} \cdot z}\right)\right)\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lower-*.f64 97.2

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(263.4062807184368 + z \cdot \left(436.9000215473151 + z \cdot \left(545.0359493463282 + 606.6767878347069 \cdot \color{blue}{z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

7. Applied rewrites 97.2%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\color{blue}{\left(263.4062807184368 + z \cdot \left(436.9000215473151 + z \cdot \left(545.0359493463282 + 606.6767878347069 \cdot z\right)\right)\right)} + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

8. Final simplification 97.2%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{\left(\left(\left(-1 + z\right) - -1\right) - 7\right) - 0.5}\right) \cdot \left(\left(\left(263.4062807184368 + z \cdot \left(436.9000215473151 + z \cdot \left(545.0359493463282 + 606.6767878347069 \cdot z\right)\right)\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
9. Add Preprocessing

Alternative 5: 97.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{\left(\left(-1 + z\right) + -6\right) - 0.5}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (/ PI (sin (* PI z)))
   (*
    (*
     (* (sqrt PI) (sqrt 2.0))
     (pow (+ (- (- 1.0 z) -6.0) 0.5) (- (- 1.0 z) 0.5)))
    (exp (- (+ (+ -1.0 z) -6.0) 0.5))))
  (+
   263.3831869810514
   (*
    z
    (+
     436.8961725563396
     (* z (+ 545.0353078428827 (* 606.6766809167608 z))))))))

C

double code(double z) {
	return ((((double) M_PI) / sin((((double) M_PI) * z))) * (((sqrt(((double) M_PI)) * sqrt(2.0)) * pow((((1.0 - z) - -6.0) + 0.5), ((1.0 - z) - 0.5))) * exp((((-1.0 + z) + -6.0) - 0.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))));
}

Java

public static double code(double z) {
	return ((Math.PI / Math.sin((Math.PI * z))) * (((Math.sqrt(Math.PI) * Math.sqrt(2.0)) * Math.pow((((1.0 - z) - -6.0) + 0.5), ((1.0 - z) - 0.5))) * Math.exp((((-1.0 + z) + -6.0) - 0.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))));
}

Python

def code(z):
	return ((math.pi / math.sin((math.pi * z))) * (((math.sqrt(math.pi) * math.sqrt(2.0)) * math.pow((((1.0 - z) - -6.0) + 0.5), ((1.0 - z) - 0.5))) * math.exp((((-1.0 + z) + -6.0) - 0.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))))

Julia

function code(z)
	return Float64(Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(sqrt(pi) * sqrt(2.0)) * (Float64(Float64(Float64(1.0 - z) - -6.0) + 0.5) ^ Float64(Float64(1.0 - z) - 0.5))) * exp(Float64(Float64(Float64(-1.0 + z) + -6.0) - 0.5)))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * Float64(545.0353078428827 + Float64(606.6766809167608 * z)))))))
end

MATLAB

function tmp = code(z)
	tmp = ((pi / sin((pi * z))) * (((sqrt(pi) * sqrt(2.0)) * ((((1.0 - z) - -6.0) + 0.5) ^ ((1.0 - z) - 0.5))) * exp((((-1.0 + z) + -6.0) - 0.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))));
end

Wolfram

code[z_] := N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(N[(-1.0 + z), $MachinePrecision] + -6.0), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * N[(545.0353078428827 + N[(606.6766809167608 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

4. Applied rewrites 96.8%

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

5. Taylor expanded in z around 0

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)} \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 96.7

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot \color{blue}{z}\right)\right)\right) \]

7. Applied rewrites 96.7%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)} \]
8. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sqrt-prod N/A

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

   5. lower-*.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-sqrt.f64 97.2

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \color{blue}{\sqrt{2}}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right) \]

9. Applied rewrites 97.2%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right) \]

10. Final simplification 97.2%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{\left(\left(-1 + z\right) + -6\right) - 0.5}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right) \]
11. Add Preprocessing

Alternative 6: 96.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{z - 7.5}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (/ PI (sin (* PI z)))
   (*
    (* (sqrt (* PI 2.0)) (pow (+ (- (- 1.0 z) -6.0) 0.5) (- (- 1.0 z) 0.5)))
    (exp (- z 7.5))))
  (+
   263.3831869810514
   (*
    z
    (+
     436.8961725563396
     (* z (+ 545.0353078428827 (* 606.6766809167608 z))))))))

C

double code(double z) {
	return ((((double) M_PI) / sin((((double) M_PI) * z))) * ((sqrt((((double) M_PI) * 2.0)) * pow((((1.0 - z) - -6.0) + 0.5), ((1.0 - z) - 0.5))) * exp((z - 7.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))));
}

Java

public static double code(double z) {
	return ((Math.PI / Math.sin((Math.PI * z))) * ((Math.sqrt((Math.PI * 2.0)) * Math.pow((((1.0 - z) - -6.0) + 0.5), ((1.0 - z) - 0.5))) * Math.exp((z - 7.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))));
}

Python

def code(z):
	return ((math.pi / math.sin((math.pi * z))) * ((math.sqrt((math.pi * 2.0)) * math.pow((((1.0 - z) - -6.0) + 0.5), ((1.0 - z) - 0.5))) * math.exp((z - 7.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))))

Julia

function code(z)
	return Float64(Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(sqrt(Float64(pi * 2.0)) * (Float64(Float64(Float64(1.0 - z) - -6.0) + 0.5) ^ Float64(Float64(1.0 - z) - 0.5))) * exp(Float64(z - 7.5)))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * Float64(545.0353078428827 + Float64(606.6766809167608 * z)))))))
end

MATLAB

function tmp = code(z)
	tmp = ((pi / sin((pi * z))) * ((sqrt((pi * 2.0)) * ((((1.0 - z) - -6.0) + 0.5) ^ ((1.0 - z) - 0.5))) * exp((z - 7.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))));
end

Wolfram

code[z_] := N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(z - 7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * N[(545.0353078428827 + N[(606.6766809167608 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

4. Applied rewrites 96.8%

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

5. Taylor expanded in z around 0

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)} \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 96.7

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot \color{blue}{z}\right)\right)\right) \]

7. Applied rewrites 96.7%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)} \]

8. Taylor expanded in z around 0

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{\color{blue}{z - \frac{15}{2}}}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right) \]
9. Step-by-step derivation

   1. lift--.f64 96.7

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{z - \color{blue}{7.5}}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right) \]

10. Applied rewrites 96.7%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{\color{blue}{z - 7.5}}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right) \]
11. Add Preprocessing

Alternative 7: 96.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{\left(\left(-1 + z\right) + -6\right) - 0.5}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (/ PI (sin (* PI z)))
   (*
    (* (sqrt (* PI 2.0)) (pow (- 7.5 z) (- 0.5 z)))
    (exp (- (+ (+ -1.0 z) -6.0) 0.5))))
  (+
   263.3831869810514
   (*
    z
    (+
     436.8961725563396
     (* z (+ 545.0353078428827 (* 606.6766809167608 z))))))))

C

double code(double z) {
	return ((((double) M_PI) / sin((((double) M_PI) * z))) * ((sqrt((((double) M_PI) * 2.0)) * pow((7.5 - z), (0.5 - z))) * exp((((-1.0 + z) + -6.0) - 0.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))));
}

Java

public static double code(double z) {
	return ((Math.PI / Math.sin((Math.PI * z))) * ((Math.sqrt((Math.PI * 2.0)) * Math.pow((7.5 - z), (0.5 - z))) * Math.exp((((-1.0 + z) + -6.0) - 0.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))));
}

Python

def code(z):
	return ((math.pi / math.sin((math.pi * z))) * ((math.sqrt((math.pi * 2.0)) * math.pow((7.5 - z), (0.5 - z))) * math.exp((((-1.0 + z) + -6.0) - 0.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))))

Julia

function code(z)
	return Float64(Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(sqrt(Float64(pi * 2.0)) * (Float64(7.5 - z) ^ Float64(0.5 - z))) * exp(Float64(Float64(Float64(-1.0 + z) + -6.0) - 0.5)))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * Float64(545.0353078428827 + Float64(606.6766809167608 * z)))))))
end

MATLAB

function tmp = code(z)
	tmp = ((pi / sin((pi * z))) * ((sqrt((pi * 2.0)) * ((7.5 - z) ^ (0.5 - z))) * exp((((-1.0 + z) + -6.0) - 0.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))));
end

Wolfram

code[z_] := N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(N[(-1.0 + z), $MachinePrecision] + -6.0), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * N[(545.0353078428827 + N[(606.6766809167608 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

4. Applied rewrites 96.8%

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

5. Taylor expanded in z around 0

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)} \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 96.7

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot \color{blue}{z}\right)\right)\right) \]

7. Applied rewrites 96.7%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)} \]

8. Taylor expanded in z around inf

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot \color{blue}{e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right) \]
9. Step-by-step derivation

   1. pow-to-exp N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\frac{15}{2} - z\right)}^{\color{blue}{\left(\frac{1}{2} - z\right)}}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right) \]

   2. lift-pow.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\frac{15}{2} - z\right)}^{\color{blue}{\left(\frac{1}{2} - z\right)}}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right) \]

   3. lift--.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\color{blue}{\frac{1}{2}} - z\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right) \]

   4. lift--.f64 96.7

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - \color{blue}{z}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right) \]

10. Applied rewrites 96.7%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot \color{blue}{{\left(7.5 - z\right)}^{\left(0.5 - z\right)}}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right) \]

11. Final simplification 96.7%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{\left(\left(-1 + z\right) + -6\right) - 0.5}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right) \]
12. Add Preprocessing

Alternative 8: 96.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double z) {
	return (((double) M_PI) / sin((((double) M_PI) * z))) * ((sqrt(((double) M_PI)) * (pow((7.5 - z), (0.5 - z)) * (exp((z - 7.5)) * sqrt(2.0)))) * (263.3831869810514 + (z * (436.8961725563396 + (545.0353078428827 * z)))));
}

Java

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

Python

def code(z):
	return (math.pi / math.sin((math.pi * z))) * ((math.sqrt(math.pi) * (math.pow((7.5 - z), (0.5 - z)) * (math.exp((z - 7.5)) * math.sqrt(2.0)))) * (263.3831869810514 + (z * (436.8961725563396 + (545.0353078428827 * z)))))

Julia

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

MATLAB

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

Wolfram

code[z_] := N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(z - 7.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(545.0353078428827 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

3. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\color{blue}{\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
4. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + \color{blue}{z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \color{blue}{\left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + \color{blue}{z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \color{blue}{\left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)}\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \color{blue}{\frac{65521081538557082921549}{108000000000000000000} \cdot z}\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lower-*.f64 96.5

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot \color{blue}{z}\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

5. Applied rewrites 96.5%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right)} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Taylor expanded in z around inf

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right)} \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)}\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lift-sqrt.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(\color{blue}{e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lift-PI.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left(e^{\color{blue}{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \color{blue}{\left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)}\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. exp-to-pow N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(\color{blue}{e^{z - \frac{15}{2}}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(\color{blue}{e^{z - \frac{15}{2}}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lower--.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{\color{blue}{z - \frac{15}{2}}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lower--.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \color{blue}{\sqrt{2}}\right)\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   10. lower-exp.f64 N/A

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{\color{blue}{2}}\right)\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   11. lower--.f64 N/A

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(\frac{9876869457595968283}{37500000000000000} + z \cdot \left(\frac{131068851705067315609}{300000000000000000} + z \cdot \left(\frac{367898832774098786021}{675000000000000000} + \frac{65521081538557082921549}{108000000000000000000} \cdot z\right)\right)\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   12. lower-sqrt.f64 96.5

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z - 7.5} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

8. Applied rewrites 96.5%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\color{blue}{\left(\sqrt{\pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z - 7.5} \cdot \sqrt{2}\right)\right)\right)} \cdot \left(\left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

9. Taylor expanded in z around 0

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

    1. lower-+.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. lower-+.f64 N/A

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

    4. lower-*.f64 96.3

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

11. Applied rewrites 96.3%

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

Alternative 9: 96.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left({z}^{-1} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{\left(\left(-1 + z\right) + -6\right) - 0.5}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (pow z -1.0)
   (*
    (* (sqrt (* PI 2.0)) (pow (+ (- (- 1.0 z) -6.0) 0.5) (- (- 1.0 z) 0.5)))
    (exp (- (+ (+ -1.0 z) -6.0) 0.5))))
  (+
   263.3831869810514
   (*
    z
    (+
     436.8961725563396
     (* z (+ 545.0353078428827 (* 606.6766809167608 z))))))))

C

double code(double z) {
	return (pow(z, -1.0) * ((sqrt((((double) M_PI) * 2.0)) * pow((((1.0 - z) - -6.0) + 0.5), ((1.0 - z) - 0.5))) * exp((((-1.0 + z) + -6.0) - 0.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))));
}

Java

public static double code(double z) {
	return (Math.pow(z, -1.0) * ((Math.sqrt((Math.PI * 2.0)) * Math.pow((((1.0 - z) - -6.0) + 0.5), ((1.0 - z) - 0.5))) * Math.exp((((-1.0 + z) + -6.0) - 0.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))));
}

Python

def code(z):
	return (math.pow(z, -1.0) * ((math.sqrt((math.pi * 2.0)) * math.pow((((1.0 - z) - -6.0) + 0.5), ((1.0 - z) - 0.5))) * math.exp((((-1.0 + z) + -6.0) - 0.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))))

Julia

function code(z)
	return Float64(Float64((z ^ -1.0) * Float64(Float64(sqrt(Float64(pi * 2.0)) * (Float64(Float64(Float64(1.0 - z) - -6.0) + 0.5) ^ Float64(Float64(1.0 - z) - 0.5))) * exp(Float64(Float64(Float64(-1.0 + z) + -6.0) - 0.5)))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * Float64(545.0353078428827 + Float64(606.6766809167608 * z)))))))
end

MATLAB

function tmp = code(z)
	tmp = ((z ^ -1.0) * ((sqrt((pi * 2.0)) * ((((1.0 - z) - -6.0) + 0.5) ^ ((1.0 - z) - 0.5))) * exp((((-1.0 + z) + -6.0) - 0.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))));
end

Wolfram

code[z_] := N[(N[(N[Power[z, -1.0], $MachinePrecision] * N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(N[(-1.0 + z), $MachinePrecision] + -6.0), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * N[(545.0353078428827 + N[(606.6766809167608 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

4. Applied rewrites 96.8%

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

5. Taylor expanded in z around 0

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)} \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 96.7

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot \color{blue}{z}\right)\right)\right) \]

7. Applied rewrites 96.7%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)} \]

8. Taylor expanded in z around 0

   \[\leadsto \left(\color{blue}{\frac{1}{z}} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right) \]
9. Step-by-step derivation

   1. inv-pow N/A

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

   2. lower-pow.f64 96.1

      \[\leadsto \left({z}^{\color{blue}{-1}} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right) \]

10. Applied rewrites 96.1%

    \[\leadsto \left(\color{blue}{{z}^{-1}} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right) \]

11. Final simplification 96.1%

    \[\leadsto \left({z}^{-1} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{\left(\left(-1 + z\right) + -6\right) - 0.5}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right) \]
12. Add Preprocessing

Alternative 10: 96.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\pi}{z \cdot \pi} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{\left(\left(-1 + z\right) + -6\right) - 0.5}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (/ PI (* z PI))
   (*
    (* (sqrt (* PI 2.0)) (pow (+ (- (- 1.0 z) -6.0) 0.5) (- (- 1.0 z) 0.5)))
    (exp (- (+ (+ -1.0 z) -6.0) 0.5))))
  (+
   263.3831869810514
   (*
    z
    (+
     436.8961725563396
     (* z (+ 545.0353078428827 (* 606.6766809167608 z))))))))

C

double code(double z) {
	return ((((double) M_PI) / (z * ((double) M_PI))) * ((sqrt((((double) M_PI) * 2.0)) * pow((((1.0 - z) - -6.0) + 0.5), ((1.0 - z) - 0.5))) * exp((((-1.0 + z) + -6.0) - 0.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))));
}

Java

public static double code(double z) {
	return ((Math.PI / (z * Math.PI)) * ((Math.sqrt((Math.PI * 2.0)) * Math.pow((((1.0 - z) - -6.0) + 0.5), ((1.0 - z) - 0.5))) * Math.exp((((-1.0 + z) + -6.0) - 0.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))));
}

Python

def code(z):
	return ((math.pi / (z * math.pi)) * ((math.sqrt((math.pi * 2.0)) * math.pow((((1.0 - z) - -6.0) + 0.5), ((1.0 - z) - 0.5))) * math.exp((((-1.0 + z) + -6.0) - 0.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))))

Julia

function code(z)
	return Float64(Float64(Float64(pi / Float64(z * pi)) * Float64(Float64(sqrt(Float64(pi * 2.0)) * (Float64(Float64(Float64(1.0 - z) - -6.0) + 0.5) ^ Float64(Float64(1.0 - z) - 0.5))) * exp(Float64(Float64(Float64(-1.0 + z) + -6.0) - 0.5)))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * Float64(545.0353078428827 + Float64(606.6766809167608 * z)))))))
end

MATLAB

function tmp = code(z)
	tmp = ((pi / (z * pi)) * ((sqrt((pi * 2.0)) * ((((1.0 - z) - -6.0) + 0.5) ^ ((1.0 - z) - 0.5))) * exp((((-1.0 + z) + -6.0) - 0.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))));
end

Wolfram

code[z_] := N[(N[(N[(Pi / N[(z * Pi), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(N[(-1.0 + z), $MachinePrecision] + -6.0), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * N[(545.0353078428827 + N[(606.6766809167608 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

4. Applied rewrites 96.8%

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

5. Taylor expanded in z around 0

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)} \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 96.7

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot \color{blue}{z}\right)\right)\right) \]

7. Applied rewrites 96.7%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)} \]

8. Taylor expanded in z around 0

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

   1. lower-*.f64 N/A

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

   2. lift-PI.f64 96.1

      \[\leadsto \left(\frac{\pi}{z \cdot \pi} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right) \]

10. Applied rewrites 96.1%

    \[\leadsto \left(\frac{\pi}{\color{blue}{z \cdot \pi}} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right) \]

11. Final simplification 96.1%

    \[\leadsto \left(\frac{\pi}{z \cdot \pi} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{\left(\left(-1 + z\right) + -6\right) - 0.5}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right) \]
12. Add Preprocessing

Alternative 11: 95.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \left(\frac{e^{-7.5} \cdot \sqrt{15}}{z} \cdot \sqrt{\pi}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (* (/ (* (exp -7.5) (sqrt 15.0)) z) (sqrt PI))
  (+
   263.3831869810514
   (*
    z
    (+
     436.8961725563396
     (* z (+ 545.0353078428827 (* 606.6766809167608 z))))))))

C

double code(double z) {
	return (((exp(-7.5) * sqrt(15.0)) / z) * sqrt(((double) M_PI))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))));
}

Java

public static double code(double z) {
	return (((Math.exp(-7.5) * Math.sqrt(15.0)) / z) * Math.sqrt(Math.PI)) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))));
}

Python

def code(z):
	return (((math.exp(-7.5) * math.sqrt(15.0)) / z) * math.sqrt(math.pi)) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))))

Julia

function code(z)
	return Float64(Float64(Float64(Float64(exp(-7.5) * sqrt(15.0)) / z) * sqrt(pi)) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * Float64(545.0353078428827 + Float64(606.6766809167608 * z)))))))
end

MATLAB

function tmp = code(z)
	tmp = (((exp(-7.5) * sqrt(15.0)) / z) * sqrt(pi)) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z))))));
end

Wolfram

code[z_] := N[(N[(N[(N[(N[Exp[-7.5], $MachinePrecision] * N[Sqrt[15.0], $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * N[(545.0353078428827 + N[(606.6766809167608 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

4. Applied rewrites 96.8%

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

5. Taylor expanded in z around 0

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + \frac{1}{2}\right)}\right)\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)} \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 96.7

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot \color{blue}{z}\right)\right)\right) \]

7. Applied rewrites 96.7%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}\right)\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)} \]

8. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right) \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z} \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right) \]

   2. sinh-+-cosh-rev N/A

      \[\leadsto \left(\frac{\left(\cosh \frac{-15}{2} + \sinh \frac{-15}{2}\right) \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right) \]

   3. sqrt-unprod N/A

      \[\leadsto \left(\frac{\left(\cosh \frac{-15}{2} + \sinh \frac{-15}{2}\right) \cdot \sqrt{2 \cdot \frac{15}{2}}}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \left(\frac{\left(\cosh \frac{-15}{2} + \sinh \frac{-15}{2}\right) \cdot \sqrt{15}}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right) \]

   5. lift-cosh.f64 N/A

      \[\leadsto \left(\frac{\left(\cosh \frac{-15}{2} + \sinh \frac{-15}{2}\right) \cdot \sqrt{15}}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right) \]

   6. lift-sinh.f64 N/A

      \[\leadsto \left(\frac{\left(\cosh \frac{-15}{2} + \sinh \frac{-15}{2}\right) \cdot \sqrt{15}}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right) \]

   7. lift-+.f64 N/A

      \[\leadsto \left(\frac{\left(\cosh \frac{-15}{2} + \sinh \frac{-15}{2}\right) \cdot \sqrt{15}}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right) \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{\left(\cosh \frac{-15}{2} + \sinh \frac{-15}{2}\right) \cdot \sqrt{15}}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \left(\frac{\left(\cosh \frac{-15}{2} + \sinh \frac{-15}{2}\right) \cdot \sqrt{15}}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\frac{\left(\cosh \frac{-15}{2} + \sinh \frac{-15}{2}\right) \cdot \sqrt{15}}{z} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right) \]

10. Applied rewrites 95.3%

    \[\leadsto \color{blue}{\left(\frac{e^{-7.5} \cdot \sqrt{15}}{z} \cdot \sqrt{\pi}\right)} \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right) \]
11. Add Preprocessing

Alternative 12: 95.5% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

3. Taylor expanded in z around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 95.2%

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

Alternative 13: 95.5% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

3. Taylor expanded in z around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 95.2%

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

6. Taylor expanded in z around 0

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

8. Applied rewrites 95.2%

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

Reproduce

herbie shell --seed 2025057 
(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))))))