Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.4% → 98.2%

Time: 1.7min

Alternatives: 10

Speedup: 1.4×

• could not determine a ground truth

  1. z = -1.1595711017921683e+282

Specification

\[z \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (/ PI (sin (* PI z)))
   (*
    (pow (- 7.5 z) (- 0.5 z))
    (cbrt (pow (* (sqrt (* PI 2.0)) (exp (+ z -7.5))) 3.0))))
  (+
   (+
    (+
     (+
      0.9999999999998099
      (+
       (/ -1259.1392167224028 (- 1.0 (+ z -1.0)))
       (/ 676.5203681218851 (- 1.0 z))))
     (+
      (/ 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) -7.0))))))

C

double code(double z) {
	return ((((double) M_PI) / sin((((double) M_PI) * z))) * (pow((7.5 - z), (0.5 - z)) * cbrt(pow((sqrt((((double) M_PI) * 2.0)) * exp((z + -7.5))), 3.0)))) * ((((0.9999999999998099 + ((-1259.1392167224028 / (1.0 - (z + -1.0))) + (676.5203681218851 / (1.0 - z)))) + ((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) - -7.0))));
}

Java

public static double code(double z) {
	return ((Math.PI / Math.sin((Math.PI * z))) * (Math.pow((7.5 - z), (0.5 - z)) * Math.cbrt(Math.pow((Math.sqrt((Math.PI * 2.0)) * Math.exp((z + -7.5))), 3.0)))) * ((((0.9999999999998099 + ((-1259.1392167224028 / (1.0 - (z + -1.0))) + (676.5203681218851 / (1.0 - z)))) + ((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) - -7.0))));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

3. Simplified 98.2%

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

6. Applied egg-rr 38.9%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)} \cdot \mathsf{reciprocal}\left(\left(e^{7.5 + \left(-z\right)}\right)\right)\right)\right)} - 1\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
7. Step-by-step derivation

   1. expm1-def 38.9%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)} \cdot \mathsf{reciprocal}\left(\left(e^{7.5 + \left(-z\right)}\right)\right)\right)\right)\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   2. expm1-log1p 38.9%

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

   3. associate-*r* 38.9%

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

   4. *-commutative 38.9%

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

   5. sub-neg 38.9%

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

   6. sub-neg 38.9%

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

   7. exp-to-pow 38.9%

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

   8. associate-*l* 38.9%

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

   9. exp-to-pow 38.9%

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

   10. reciprocal-undefine 98.2%

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

   11. neg-mul-1 98.2%

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

   12. exp-neg 98.2%

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

8. Simplified 98.2%

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

   1. add-cbrt-cube 98.5%

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

   2. pow3 98.5%

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

   3. *-commutative 98.5%

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

10. Applied egg-rr 98.5%

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

11. Final simplification 98.5%

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

Alternative 2: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

3. Simplified 98.2%

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

6. Applied egg-rr 38.9%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)} \cdot \mathsf{reciprocal}\left(\left(e^{7.5 + \left(-z\right)}\right)\right)\right)\right)} - 1\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
7. Step-by-step derivation

   1. expm1-def 38.9%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 + \left(-z\right)\right)}^{\left(0.5 + \left(-z\right)\right)} \cdot \mathsf{reciprocal}\left(\left(e^{7.5 + \left(-z\right)}\right)\right)\right)\right)\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   2. expm1-log1p 38.9%

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

   3. associate-*r* 38.9%

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

   4. *-commutative 38.9%

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

   5. sub-neg 38.9%

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

   6. sub-neg 38.9%

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

   7. exp-to-pow 38.9%

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

   8. associate-*l* 38.9%

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

   9. exp-to-pow 38.9%

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

   10. reciprocal-undefine 98.2%

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

   11. neg-mul-1 98.2%

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

   12. exp-neg 98.2%

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

8. Simplified 98.2%

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

   1. expm1-log1p-u 96.9%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{z + -7.5}\right)\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right)}\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   2. expm1-udef 96.8%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{z + -7.5}\right)\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)} - 1\right)}\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   3. sub-neg 96.8%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{z + -7.5}\right)\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(e^{\mathsf{log1p}\left(\frac{771.3234287776531}{\color{blue}{\left(1 - z\right) + \left(--2\right)}} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)} - 1\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   4. metadata-eval 96.8%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{z + -7.5}\right)\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(e^{\mathsf{log1p}\left(\frac{771.3234287776531}{\left(1 - z\right) + \color{blue}{2}} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)} - 1\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   5. sub-neg 96.8%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{z + -7.5}\right)\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(e^{\mathsf{log1p}\left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\color{blue}{\left(1 - z\right) + \left(--3\right)}}\right)} - 1\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   6. metadata-eval 96.8%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{z + -7.5}\right)\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(e^{\mathsf{log1p}\left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\left(1 - z\right) + \color{blue}{3}}\right)} - 1\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

10. Applied egg-rr 96.8%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{z + -7.5}\right)\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)} - 1\right)}\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
11. Step-by-step derivation

    1. expm1-def 96.9%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{z + -7.5}\right)\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)}\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    2. expm1-log1p 98.2%

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

    3. +-commutative 98.2%

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

    4. associate-+r- 98.2%

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

    5. metadata-eval 98.2%

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

    6. +-commutative 98.2%

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

    7. associate-+r- 98.2%

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

    8. metadata-eval 98.2%

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

12. Simplified 98.2%

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

    1. expm1-log1p-u 97.0%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{z + -7.5}\right)\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right)}\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    2. expm1-udef 97.0%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{z + -7.5}\right)\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)} - 1\right)}\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    3. +-commutative 97.0%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{z + -7.5}\right)\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{676.5203681218851}{\left(1 - z\right) - 0}}\right)} - 1\right)\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    4. associate--l- 97.0%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{z + -7.5}\right)\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(e^{\mathsf{log1p}\left(\frac{-1259.1392167224028}{\color{blue}{1 - \left(z + -1\right)}} + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right)} - 1\right)\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    5. --rgt-identity 97.0%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{z + -7.5}\right)\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(e^{\mathsf{log1p}\left(\frac{-1259.1392167224028}{1 - \left(z + -1\right)} + \frac{676.5203681218851}{\color{blue}{1 - z}}\right)} - 1\right)\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

14. Applied egg-rr 97.0%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{z + -7.5}\right)\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-1259.1392167224028}{1 - \left(z + -1\right)} + \frac{676.5203681218851}{1 - z}\right)} - 1\right)}\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
15. Step-by-step derivation

    1. expm1-def 97.0%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{z + -7.5}\right)\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1259.1392167224028}{1 - \left(z + -1\right)} + \frac{676.5203681218851}{1 - z}\right)\right)}\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    2. expm1-log1p 98.2%

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

16. Simplified 98.2%

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

17. Final simplification 98.2%

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

Alternative 3: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

3. Simplified 98.2%

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

5. Final simplification 98.2%

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

Alternative 4: 97.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) + \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(212.9540523020159 + z \cdot 74.66416387488323\right)\right)\right) \cdot \left(e^{-7.5} + \frac{e^{-7.5}}{z}\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (* (pow (- 7.5 z) (- 0.5 z)) (sqrt (* PI 2.0)))
  (*
   (+
    (+
     (+ (/ 12.507343278686905 (- 5.0 z)) (/ -0.13857109526572012 (- 6.0 z)))
     (+
      (/ 9.984369578019572e-6 (- 7.0 z))
      (/ 1.5056327351493116e-7 (- 8.0 z))))
    (+
     (+
      0.9999999999998099
      (+ (/ 676.5203681218851 (- 1.0 z)) (/ -1259.1392167224028 (- 2.0 z))))
     (+ 212.9540523020159 (* z 74.66416387488323))))
   (+ (exp -7.5) (/ (exp -7.5) z)))))

C

double code(double z) {
	return (pow((7.5 - z), (0.5 - z)) * sqrt((((double) M_PI) * 2.0))) * (((((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z))) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (212.9540523020159 + (z * 74.66416387488323)))) * (exp(-7.5) + (exp(-7.5) / z)));
}

Java

public static double code(double z) {
	return (Math.pow((7.5 - z), (0.5 - z)) * Math.sqrt((Math.PI * 2.0))) * (((((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z))) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (212.9540523020159 + (z * 74.66416387488323)))) * (Math.exp(-7.5) + (Math.exp(-7.5) / z)));
}

Python

def code(z):
	return (math.pow((7.5 - z), (0.5 - z)) * math.sqrt((math.pi * 2.0))) * (((((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z))) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (212.9540523020159 + (z * 74.66416387488323)))) * (math.exp(-7.5) + (math.exp(-7.5) / z)))

Julia

function code(z)
	return Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * sqrt(Float64(pi * 2.0))) * Float64(Float64(Float64(Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(-0.13857109526572012 / Float64(6.0 - z))) + Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z)))) + Float64(Float64(0.9999999999998099 + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(-1259.1392167224028 / Float64(2.0 - z)))) + Float64(212.9540523020159 + Float64(z * 74.66416387488323)))) * Float64(exp(-7.5) + Float64(exp(-7.5) / z))))
end

MATLAB

function tmp = code(z)
	tmp = (((7.5 - z) ^ (0.5 - z)) * sqrt((pi * 2.0))) * (((((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z))) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (212.9540523020159 + (z * 74.66416387488323)))) * (exp(-7.5) + (exp(-7.5) / z)));
end

Wolfram

code[z_] := N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.9999999999998099 + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(212.9540523020159 + N[(z * 74.66416387488323), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] + N[(N[Exp[-7.5], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

3. Simplified 98.2%

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

5. Taylor expanded in z around 0 96.1%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \color{blue}{\left(212.9540523020159 + 74.66416387488323 \cdot z\right)}\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right) \]
6. Step-by-step derivation

   1. *-commutative 96.1%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(212.9540523020159 + \color{blue}{z \cdot 74.66416387488323}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right) \]

7. Simplified 96.1%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \color{blue}{\left(212.9540523020159 + z \cdot 74.66416387488323\right)}\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right) \]

8. Taylor expanded in z around 0 96.6%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(212.9540523020159 + z \cdot 74.66416387488323\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \color{blue}{\left(e^{-7.5} + \frac{e^{-7.5}}{z}\right)}\right) \]

9. Final simplification 96.6%

   \[\leadsto \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) + \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(212.9540523020159 + z \cdot 74.66416387488323\right)\right)\right) \cdot \left(e^{-7.5} + \frac{e^{-7.5}}{z}\right)\right) \]
10. Add Preprocessing

Alternative 5: 96.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(212.9540523020159 + z \cdot 74.66416387488323\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(2.4783734731930944 + z \cdot 0.49644453405676175\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (* (pow (- 7.5 z) (- 0.5 z)) (sqrt (* PI 2.0)))
  (*
   (* (/ PI (sin (* PI z))) (exp (+ z -7.5)))
   (+
    (+
     (+
      0.9999999999998099
      (+ (/ 676.5203681218851 (- 1.0 z)) (/ -1259.1392167224028 (- 2.0 z))))
     (+ 212.9540523020159 (* z 74.66416387488323)))
    (+
     (+ (/ 9.984369578019572e-6 (- 7.0 z)) (/ 1.5056327351493116e-7 (- 8.0 z)))
     (+ 2.4783734731930944 (* z 0.49644453405676175)))))))

C

double code(double z) {
	return (pow((7.5 - z), (0.5 - z)) * sqrt((((double) M_PI) * 2.0))) * (((((double) M_PI) / sin((((double) M_PI) * z))) * exp((z + -7.5))) * (((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (212.9540523020159 + (z * 74.66416387488323))) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (2.4783734731930944 + (z * 0.49644453405676175)))));
}

Java

public static double code(double z) {
	return (Math.pow((7.5 - z), (0.5 - z)) * Math.sqrt((Math.PI * 2.0))) * (((Math.PI / Math.sin((Math.PI * z))) * Math.exp((z + -7.5))) * (((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (212.9540523020159 + (z * 74.66416387488323))) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (2.4783734731930944 + (z * 0.49644453405676175)))));
}

Python

def code(z):
	return (math.pow((7.5 - z), (0.5 - z)) * math.sqrt((math.pi * 2.0))) * (((math.pi / math.sin((math.pi * z))) * math.exp((z + -7.5))) * (((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (212.9540523020159 + (z * 74.66416387488323))) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (2.4783734731930944 + (z * 0.49644453405676175)))))

Julia

function code(z)
	return Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * sqrt(Float64(pi * 2.0))) * Float64(Float64(Float64(pi / sin(Float64(pi * z))) * exp(Float64(z + -7.5))) * Float64(Float64(Float64(0.9999999999998099 + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(-1259.1392167224028 / Float64(2.0 - z)))) + Float64(212.9540523020159 + Float64(z * 74.66416387488323))) + Float64(Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))) + Float64(2.4783734731930944 + Float64(z * 0.49644453405676175))))))
end

MATLAB

function tmp = code(z)
	tmp = (((7.5 - z) ^ (0.5 - z)) * sqrt((pi * 2.0))) * (((pi / sin((pi * z))) * exp((z + -7.5))) * (((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (212.9540523020159 + (z * 74.66416387488323))) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (2.4783734731930944 + (z * 0.49644453405676175)))));
end

Wolfram

code[z_] := N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.9999999999998099 + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(212.9540523020159 + N[(z * 74.66416387488323), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.4783734731930944 + N[(z * 0.49644453405676175), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

3. Simplified 98.2%

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

5. Taylor expanded in z around 0 96.1%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \color{blue}{\left(212.9540523020159 + 74.66416387488323 \cdot z\right)}\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right) \]
6. Step-by-step derivation

   1. *-commutative 96.1%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(212.9540523020159 + \color{blue}{z \cdot 74.66416387488323}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right) \]

7. Simplified 96.1%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \color{blue}{\left(212.9540523020159 + z \cdot 74.66416387488323\right)}\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right) \]

8. Taylor expanded in z around 0 96.1%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(212.9540523020159 + z \cdot 74.66416387488323\right)\right) + \left(\color{blue}{\left(2.4783734731930944 + 0.49644453405676175 \cdot z\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right) \]
9. Step-by-step derivation

   1. *-commutative 96.1%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(212.9540523020159 + z \cdot 74.66416387488323\right)\right) + \left(\left(2.4783734731930944 + \color{blue}{z \cdot 0.49644453405676175}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right) \]

10. Simplified 96.1%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(212.9540523020159 + z \cdot 74.66416387488323\right)\right) + \left(\color{blue}{\left(2.4783734731930944 + z \cdot 0.49644453405676175\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right) \]

11. Final simplification 96.1%

    \[\leadsto \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(212.9540523020159 + z \cdot 74.66416387488323\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(2.4783734731930944 + z \cdot 0.49644453405676175\right)\right)\right)\right) \]
12. Add Preprocessing

Alternative 6: 96.6% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (+
   (+
    (+
     (+
      0.9999999999998099
      (+
       (/ -1259.1392167224028 (- 1.0 (+ z -1.0)))
       (/ 676.5203681218851 (- 1.0 z))))
     (+
      (/ 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) -7.0))))
  (*
   (/ 1.0 z)
   (*
    (* (sqrt (* PI 2.0)) (pow (- 7.5 z) (- (- 1.0 z) 0.5)))
    (exp (- z 7.5))))))

C

double code(double z) {
	return ((((0.9999999999998099 + ((-1259.1392167224028 / (1.0 - (z + -1.0))) + (676.5203681218851 / (1.0 - z)))) + ((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) - -7.0)))) * ((1.0 / z) * ((sqrt((((double) M_PI) * 2.0)) * pow((7.5 - z), ((1.0 - z) - 0.5))) * exp((z - 7.5))));
}

Java

public static double code(double z) {
	return ((((0.9999999999998099 + ((-1259.1392167224028 / (1.0 - (z + -1.0))) + (676.5203681218851 / (1.0 - z)))) + ((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) - -7.0)))) * ((1.0 / z) * ((Math.sqrt((Math.PI * 2.0)) * Math.pow((7.5 - z), ((1.0 - z) - 0.5))) * Math.exp((z - 7.5))));
}

Python

def code(z):
	return ((((0.9999999999998099 + ((-1259.1392167224028 / (1.0 - (z + -1.0))) + (676.5203681218851 / (1.0 - z)))) + ((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) - -7.0)))) * ((1.0 / z) * ((math.sqrt((math.pi * 2.0)) * math.pow((7.5 - z), ((1.0 - z) - 0.5))) * math.exp((z - 7.5))))

Julia

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

MATLAB

function tmp = code(z)
	tmp = ((((0.9999999999998099 + ((-1259.1392167224028 / (1.0 - (z + -1.0))) + (676.5203681218851 / (1.0 - z)))) + ((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) - -7.0)))) * ((1.0 / z) * ((sqrt((pi * 2.0)) * ((7.5 - z) ^ ((1.0 - z) - 0.5))) * exp((z - 7.5))));
end

Wolfram

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

Derivation

1. Initial program 96.6%

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

3. Simplified 98.2%

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

5. Taylor expanded in z around 0 98.2%

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

   1. neg-mul-1 98.2%

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

7. Simplified 98.2%

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

8. Taylor expanded in z around 0 98.2%

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

   1. neg-mul-1 98.2%

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

10. Simplified 98.2%

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

11. Taylor expanded in z around 0 95.8%

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

12. Final simplification 95.8%

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

Alternative 7: 96.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) + \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(212.9540523020159 + z \cdot 74.66416387488323\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{1}{z}\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (* (pow (- 7.5 z) (- 0.5 z)) (sqrt (* PI 2.0)))
  (*
   (+
    (+
     (+ (/ 12.507343278686905 (- 5.0 z)) (/ -0.13857109526572012 (- 6.0 z)))
     (+
      (/ 9.984369578019572e-6 (- 7.0 z))
      (/ 1.5056327351493116e-7 (- 8.0 z))))
    (+
     (+
      0.9999999999998099
      (+ (/ 676.5203681218851 (- 1.0 z)) (/ -1259.1392167224028 (- 2.0 z))))
     (+ 212.9540523020159 (* z 74.66416387488323))))
   (* (exp (+ z -7.5)) (/ 1.0 z)))))

C

double code(double z) {
	return (pow((7.5 - z), (0.5 - z)) * sqrt((((double) M_PI) * 2.0))) * (((((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z))) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (212.9540523020159 + (z * 74.66416387488323)))) * (exp((z + -7.5)) * (1.0 / z)));
}

Java

public static double code(double z) {
	return (Math.pow((7.5 - z), (0.5 - z)) * Math.sqrt((Math.PI * 2.0))) * (((((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z))) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (212.9540523020159 + (z * 74.66416387488323)))) * (Math.exp((z + -7.5)) * (1.0 / z)));
}

Python

def code(z):
	return (math.pow((7.5 - z), (0.5 - z)) * math.sqrt((math.pi * 2.0))) * (((((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z))) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (212.9540523020159 + (z * 74.66416387488323)))) * (math.exp((z + -7.5)) * (1.0 / z)))

Julia

function code(z)
	return Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * sqrt(Float64(pi * 2.0))) * Float64(Float64(Float64(Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(-0.13857109526572012 / Float64(6.0 - z))) + Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z)))) + Float64(Float64(0.9999999999998099 + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(-1259.1392167224028 / Float64(2.0 - z)))) + Float64(212.9540523020159 + Float64(z * 74.66416387488323)))) * Float64(exp(Float64(z + -7.5)) * Float64(1.0 / z))))
end

MATLAB

function tmp = code(z)
	tmp = (((7.5 - z) ^ (0.5 - z)) * sqrt((pi * 2.0))) * (((((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z))) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (212.9540523020159 + (z * 74.66416387488323)))) * (exp((z + -7.5)) * (1.0 / z)));
end

Wolfram

code[z_] := N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.9999999999998099 + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(212.9540523020159 + N[(z * 74.66416387488323), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

3. Simplified 98.2%

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

5. Taylor expanded in z around 0 96.1%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \color{blue}{\left(212.9540523020159 + 74.66416387488323 \cdot z\right)}\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right) \]
6. Step-by-step derivation

   1. *-commutative 96.1%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(212.9540523020159 + \color{blue}{z \cdot 74.66416387488323}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right) \]

7. Simplified 96.1%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \color{blue}{\left(212.9540523020159 + z \cdot 74.66416387488323\right)}\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right) \]

8. Taylor expanded in z around 0 95.8%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(212.9540523020159 + z \cdot 74.66416387488323\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \color{blue}{\frac{1}{z}}\right)\right) \]

9. Final simplification 95.8%

   \[\leadsto \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) + \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(212.9540523020159 + z \cdot 74.66416387488323\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{1}{z}\right)\right) \]
10. Add Preprocessing

Alternative 8: 96.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) + \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(212.9540523020159 + z \cdot 74.66416387488323\right)\right)\right) \cdot \frac{e^{-7.5}}{z}\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (* (pow (- 7.5 z) (- 0.5 z)) (sqrt (* PI 2.0)))
  (*
   (+
    (+
     (+ (/ 12.507343278686905 (- 5.0 z)) (/ -0.13857109526572012 (- 6.0 z)))
     (+
      (/ 9.984369578019572e-6 (- 7.0 z))
      (/ 1.5056327351493116e-7 (- 8.0 z))))
    (+
     (+
      0.9999999999998099
      (+ (/ 676.5203681218851 (- 1.0 z)) (/ -1259.1392167224028 (- 2.0 z))))
     (+ 212.9540523020159 (* z 74.66416387488323))))
   (/ (exp -7.5) z))))

C

double code(double z) {
	return (pow((7.5 - z), (0.5 - z)) * sqrt((((double) M_PI) * 2.0))) * (((((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z))) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (212.9540523020159 + (z * 74.66416387488323)))) * (exp(-7.5) / z));
}

Java

public static double code(double z) {
	return (Math.pow((7.5 - z), (0.5 - z)) * Math.sqrt((Math.PI * 2.0))) * (((((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z))) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (212.9540523020159 + (z * 74.66416387488323)))) * (Math.exp(-7.5) / z));
}

Python

def code(z):
	return (math.pow((7.5 - z), (0.5 - z)) * math.sqrt((math.pi * 2.0))) * (((((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z))) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (212.9540523020159 + (z * 74.66416387488323)))) * (math.exp(-7.5) / z))

Julia

function code(z)
	return Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * sqrt(Float64(pi * 2.0))) * Float64(Float64(Float64(Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(-0.13857109526572012 / Float64(6.0 - z))) + Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z)))) + Float64(Float64(0.9999999999998099 + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(-1259.1392167224028 / Float64(2.0 - z)))) + Float64(212.9540523020159 + Float64(z * 74.66416387488323)))) * Float64(exp(-7.5) / z)))
end

MATLAB

function tmp = code(z)
	tmp = (((7.5 - z) ^ (0.5 - z)) * sqrt((pi * 2.0))) * (((((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z))) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (212.9540523020159 + (z * 74.66416387488323)))) * (exp(-7.5) / z));
end

Wolfram

code[z_] := N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.9999999999998099 + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(212.9540523020159 + N[(z * 74.66416387488323), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

3. Simplified 98.2%

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

5. Taylor expanded in z around 0 96.1%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \color{blue}{\left(212.9540523020159 + 74.66416387488323 \cdot z\right)}\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right) \]
6. Step-by-step derivation

   1. *-commutative 96.1%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(212.9540523020159 + \color{blue}{z \cdot 74.66416387488323}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right) \]

7. Simplified 96.1%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \color{blue}{\left(212.9540523020159 + z \cdot 74.66416387488323\right)}\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right) \]

8. Taylor expanded in z around 0 94.1%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(212.9540523020159 + z \cdot 74.66416387488323\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \color{blue}{\frac{e^{-7.5}}{z}}\right) \]

9. Final simplification 94.1%

   \[\leadsto \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(\left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) + \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(212.9540523020159 + z \cdot 74.66416387488323\right)\right)\right) \cdot \frac{e^{-7.5}}{z}\right) \]
10. Add Preprocessing

Alternative 9: 21.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\pi \cdot 2} \cdot \left(\left(\frac{e^{-7.5}}{z} \cdot 213.9540523020157\right) \cdot {\left(z + 7.5\right)}^{\left(z + 0.5\right)}\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (sqrt (* PI 2.0))
  (* (* (/ (exp -7.5) z) 213.9540523020157) (pow (+ z 7.5) (+ z 0.5)))))

C

double code(double z) {
	return sqrt((((double) M_PI) * 2.0)) * (((exp(-7.5) / z) * 213.9540523020157) * pow((z + 7.5), (z + 0.5)));
}

Java

public static double code(double z) {
	return Math.sqrt((Math.PI * 2.0)) * (((Math.exp(-7.5) / z) * 213.9540523020157) * Math.pow((z + 7.5), (z + 0.5)));
}

Python

def code(z):
	return math.sqrt((math.pi * 2.0)) * (((math.exp(-7.5) / z) * 213.9540523020157) * math.pow((z + 7.5), (z + 0.5)))

Julia

function code(z)
	return Float64(sqrt(Float64(pi * 2.0)) * Float64(Float64(Float64(exp(-7.5) / z) * 213.9540523020157) * (Float64(z + 7.5) ^ Float64(z + 0.5))))
end

MATLAB

function tmp = code(z)
	tmp = sqrt((pi * 2.0)) * (((exp(-7.5) / z) * 213.9540523020157) * ((z + 7.5) ^ (z + 0.5)));
end

Wolfram

code[z_] := N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[Exp[-7.5], $MachinePrecision] / z), $MachinePrecision] * 213.9540523020157), $MachinePrecision] * N[Power[N[(z + 7.5), $MachinePrecision], N[(z + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

3. Simplified 98.2%

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

5. Taylor expanded in z around 0 94.4%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \color{blue}{212.9540523020159}\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right) \]

6. Taylor expanded in z around inf 21.3%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\color{blue}{213.9540523020157} \cdot \left(e^{z + -7.5} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right) \]

7. Taylor expanded in z around 0 21.9%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(213.9540523020157 \cdot \color{blue}{\frac{e^{-7.5}}{z}}\right) \]
8. Step-by-step derivation

   1. expm1-log1p-u 11.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(213.9540523020157 \cdot \frac{e^{-7.5}}{z}\right)\right)\right)} \]

   2. expm1-udef 11.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(213.9540523020157 \cdot \frac{e^{-7.5}}{z}\right)\right)} - 1} \]

9. Applied egg-rr 11.2%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\pi \cdot 2} \cdot \left({\left(z + 7.5\right)}^{\left(0.5 + z\right)} \cdot \left(\frac{e^{-7.5}}{z} \cdot 213.9540523020157\right)\right)\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 11.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi \cdot 2} \cdot \left({\left(z + 7.5\right)}^{\left(0.5 + z\right)} \cdot \left(\frac{e^{-7.5}}{z} \cdot 213.9540523020157\right)\right)\right)\right)} \]

    2. expm1-log1p 21.2%

       \[\leadsto \color{blue}{\sqrt{\pi \cdot 2} \cdot \left({\left(z + 7.5\right)}^{\left(0.5 + z\right)} \cdot \left(\frac{e^{-7.5}}{z} \cdot 213.9540523020157\right)\right)} \]

    3. rem-log-exp 21.2%

       \[\leadsto \sqrt{\pi \cdot 2} \cdot \left({\left(z + 7.5\right)}^{\left(0.5 + z\right)} \cdot \left(\frac{e^{-7.5}}{z} \cdot \color{blue}{\log \left(e^{213.9540523020157}\right)}\right)\right) \]

    4. log-pow 5.6%

       \[\leadsto \sqrt{\pi \cdot 2} \cdot \left({\left(z + 7.5\right)}^{\left(0.5 + z\right)} \cdot \color{blue}{\log \left({\left(e^{213.9540523020157}\right)}^{\left(\frac{e^{-7.5}}{z}\right)}\right)}\right) \]

    5. *-commutative 5.6%

       \[\leadsto \sqrt{\color{blue}{2 \cdot \pi}} \cdot \left({\left(z + 7.5\right)}^{\left(0.5 + z\right)} \cdot \log \left({\left(e^{213.9540523020157}\right)}^{\left(\frac{e^{-7.5}}{z}\right)}\right)\right) \]

    6. +-commutative 5.6%

       \[\leadsto \sqrt{2 \cdot \pi} \cdot \left({\color{blue}{\left(7.5 + z\right)}}^{\left(0.5 + z\right)} \cdot \log \left({\left(e^{213.9540523020157}\right)}^{\left(\frac{e^{-7.5}}{z}\right)}\right)\right) \]

    7. +-commutative 5.6%

       \[\leadsto \sqrt{2 \cdot \pi} \cdot \left({\left(7.5 + z\right)}^{\color{blue}{\left(z + 0.5\right)}} \cdot \log \left({\left(e^{213.9540523020157}\right)}^{\left(\frac{e^{-7.5}}{z}\right)}\right)\right) \]

    8. log-pow 21.2%

       \[\leadsto \sqrt{2 \cdot \pi} \cdot \left({\left(7.5 + z\right)}^{\left(z + 0.5\right)} \cdot \color{blue}{\left(\frac{e^{-7.5}}{z} \cdot \log \left(e^{213.9540523020157}\right)\right)}\right) \]

    9. rem-log-exp 21.2%

       \[\leadsto \sqrt{2 \cdot \pi} \cdot \left({\left(7.5 + z\right)}^{\left(z + 0.5\right)} \cdot \left(\frac{e^{-7.5}}{z} \cdot \color{blue}{213.9540523020157}\right)\right) \]

    10. *-commutative 21.2%

        \[\leadsto \sqrt{2 \cdot \pi} \cdot \left({\left(7.5 + z\right)}^{\left(z + 0.5\right)} \cdot \color{blue}{\left(213.9540523020157 \cdot \frac{e^{-7.5}}{z}\right)}\right) \]

11. Simplified 21.2%

    \[\leadsto \color{blue}{\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 + z\right)}^{\left(z + 0.5\right)} \cdot \left(213.9540523020157 \cdot \frac{e^{-7.5}}{z}\right)\right)} \]

12. Final simplification 21.2%

    \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(\frac{e^{-7.5}}{z} \cdot 213.9540523020157\right) \cdot {\left(z + 7.5\right)}^{\left(z + 0.5\right)}\right) \]
13. Add Preprocessing

Alternative 10: 21.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\frac{e^{-7.5}}{z} \cdot 213.9540523020157\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (* (pow (- 7.5 z) (- 0.5 z)) (sqrt (* PI 2.0)))
  (* (/ (exp -7.5) z) 213.9540523020157)))

C

double code(double z) {
	return (pow((7.5 - z), (0.5 - z)) * sqrt((((double) M_PI) * 2.0))) * ((exp(-7.5) / z) * 213.9540523020157);
}

Java

public static double code(double z) {
	return (Math.pow((7.5 - z), (0.5 - z)) * Math.sqrt((Math.PI * 2.0))) * ((Math.exp(-7.5) / z) * 213.9540523020157);
}

Python

def code(z):
	return (math.pow((7.5 - z), (0.5 - z)) * math.sqrt((math.pi * 2.0))) * ((math.exp(-7.5) / z) * 213.9540523020157)

Julia

function code(z)
	return Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * sqrt(Float64(pi * 2.0))) * Float64(Float64(exp(-7.5) / z) * 213.9540523020157))
end

MATLAB

function tmp = code(z)
	tmp = (((7.5 - z) ^ (0.5 - z)) * sqrt((pi * 2.0))) * ((exp(-7.5) / z) * 213.9540523020157);
end

Wolfram

code[z_] := N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Exp[-7.5], $MachinePrecision] / z), $MachinePrecision] * 213.9540523020157), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

3. Simplified 98.2%

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

5. Taylor expanded in z around 0 94.4%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \color{blue}{212.9540523020159}\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left(e^{z + -7.5} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right) \]

6. Taylor expanded in z around inf 21.3%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\color{blue}{213.9540523020157} \cdot \left(e^{z + -7.5} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right) \]

7. Taylor expanded in z around 0 21.9%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(213.9540523020157 \cdot \color{blue}{\frac{e^{-7.5}}{z}}\right) \]

8. Final simplification 21.9%

   \[\leadsto \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\frac{e^{-7.5}}{z} \cdot 213.9540523020157\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(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))))))