Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.4% → 98.4%

Time: 59.3s

Alternatives: 17

Speedup: 1.2×

• could not determine a ground truth

  1. z = -7.45382320326033e+242

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: 98.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

4. Applied egg-rr 97.6%

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

6. Simplified 98.4%

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

   1. frac-add 98.4%

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

   2. fma-define 98.4%

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

8. Applied egg-rr 98.4%

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

9. Taylor expanded in z around inf 98.4%

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

    1. exp-to-pow 98.4%

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

    2. *-commutative 98.4%

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

    3. sub-neg 98.4%

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

    4. metadata-eval 98.4%

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

    5. +-commutative 98.4%

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

11. Simplified 98.4%

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

12. Final simplification 98.4%

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

Alternative 2: 98.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

4. Applied egg-rr 97.6%

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

6. Simplified 98.4%

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

7. Taylor expanded in z around inf 98.4%

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

   1. exp-to-pow 98.4%

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

   2. *-commutative 98.4%

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

   3. *-commutative 98.4%

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

   4. sub-neg 98.4%

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

   5. metadata-eval 98.4%

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

   6. +-commutative 98.4%

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

9. Simplified 98.4%

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

10. Final simplification 98.4%

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

Alternative 3: 98.3% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ PI (sin (* PI z)))
  (*
   (* (sqrt (* PI 2.0)) (* (pow (- 7.5 z) (- 0.5 z)) (exp (- -6.5 (- 1.0 z)))))
   (+
    (/ 676.5203681218851 (- 1.0 z))
    (+
     (+
      (/ 12.507343278686905 (- 5.0 z))
      (+
       (-
        (/ -0.13857109526572012 (- 6.0 z))
        (/ 1.5056327351493116e-7 (- z 8.0)))
       (/ 9.984369578019572e-6 (- 7.0 z))))
     (+
      0.9999999999998099
      (+
       (/ -176.6150291621406 (- 4.0 z))
       (/
        (- (* -1259.1392167224028 (- z 3.0)) (* (- 2.0 z) 771.3234287776531))
        (* (- 3.0 z) (- z 2.0))))))))))

C

double code(double z) {
	return (((double) M_PI) / sin((((double) M_PI) * z))) * ((sqrt((((double) M_PI) * 2.0)) * (pow((7.5 - z), (0.5 - z)) * exp((-6.5 - (1.0 - z))))) * ((676.5203681218851 / (1.0 - z)) + (((12.507343278686905 / (5.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) - (1.5056327351493116e-7 / (z - 8.0))) + (9.984369578019572e-6 / (7.0 - z)))) + (0.9999999999998099 + ((-176.6150291621406 / (4.0 - z)) + (((-1259.1392167224028 * (z - 3.0)) - ((2.0 - z) * 771.3234287776531)) / ((3.0 - z) * (z - 2.0))))))));
}

Java

public static double code(double z) {
	return (Math.PI / Math.sin((Math.PI * z))) * ((Math.sqrt((Math.PI * 2.0)) * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((-6.5 - (1.0 - z))))) * ((676.5203681218851 / (1.0 - z)) + (((12.507343278686905 / (5.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) - (1.5056327351493116e-7 / (z - 8.0))) + (9.984369578019572e-6 / (7.0 - z)))) + (0.9999999999998099 + ((-176.6150291621406 / (4.0 - z)) + (((-1259.1392167224028 * (z - 3.0)) - ((2.0 - z) * 771.3234287776531)) / ((3.0 - z) * (z - 2.0))))))));
}

Python

def code(z):
	return (math.pi / math.sin((math.pi * z))) * ((math.sqrt((math.pi * 2.0)) * (math.pow((7.5 - z), (0.5 - z)) * math.exp((-6.5 - (1.0 - z))))) * ((676.5203681218851 / (1.0 - z)) + (((12.507343278686905 / (5.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) - (1.5056327351493116e-7 / (z - 8.0))) + (9.984369578019572e-6 / (7.0 - z)))) + (0.9999999999998099 + ((-176.6150291621406 / (4.0 - z)) + (((-1259.1392167224028 * (z - 3.0)) - ((2.0 - z) * 771.3234287776531)) / ((3.0 - z) * (z - 2.0))))))))

Julia

function code(z)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(-6.5 - Float64(1.0 - z))))) * Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) - Float64(1.5056327351493116e-7 / Float64(z - 8.0))) + Float64(9.984369578019572e-6 / Float64(7.0 - z)))) + Float64(0.9999999999998099 + Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) + Float64(Float64(Float64(-1259.1392167224028 * Float64(z - 3.0)) - Float64(Float64(2.0 - z) * 771.3234287776531)) / Float64(Float64(3.0 - z) * Float64(z - 2.0)))))))))
end

MATLAB

function tmp = code(z)
	tmp = (pi / sin((pi * z))) * ((sqrt((pi * 2.0)) * (((7.5 - z) ^ (0.5 - z)) * exp((-6.5 - (1.0 - z))))) * ((676.5203681218851 / (1.0 - z)) + (((12.507343278686905 / (5.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) - (1.5056327351493116e-7 / (z - 8.0))) + (9.984369578019572e-6 / (7.0 - z)))) + (0.9999999999998099 + ((-176.6150291621406 / (4.0 - z)) + (((-1259.1392167224028 * (z - 3.0)) - ((2.0 - z) * 771.3234287776531)) / ((3.0 - z) * (z - 2.0))))))));
end

Wolfram

code[z_] := N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-6.5 - N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] - N[(1.5056327351493116e-7 / N[(z - 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-1259.1392167224028 * N[(z - 3.0), $MachinePrecision]), $MachinePrecision] - N[(N[(2.0 - z), $MachinePrecision] * 771.3234287776531), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 - z), $MachinePrecision] * N[(z - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

4. Applied egg-rr 97.6%

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

6. Simplified 98.4%

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

   1. frac-add 98.4%

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

   2. fma-define 98.4%

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

8. Applied egg-rr 98.4%

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

   1. fma-undefine 98.4%

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

10. Applied egg-rr 98.4%

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

11. Final simplification 98.4%

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

Alternative 4: 98.3% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ PI (sin (* PI z)))
  (*
   (+
    (/ 676.5203681218851 (- 1.0 z))
    (+
     (+
      (/ 12.507343278686905 (- 5.0 z))
      (+
       (-
        (/ -0.13857109526572012 (- 6.0 z))
        (/ 1.5056327351493116e-7 (- z 8.0)))
       (/ 9.984369578019572e-6 (- 7.0 z))))
     (+
      0.9999999999998099
      (+
       (/ -176.6150291621406 (- 4.0 z))
       (-
        (/ -1259.1392167224028 (- 2.0 z))
        (/ 771.3234287776531 (- z 3.0)))))))
   (*
    (sqrt (* PI 2.0))
    (* (pow (- 7.5 z) (- 0.5 z)) (exp (- -6.5 (- 1.0 z))))))))

C

double code(double z) {
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (((676.5203681218851 / (1.0 - z)) + (((12.507343278686905 / (5.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) - (1.5056327351493116e-7 / (z - 8.0))) + (9.984369578019572e-6 / (7.0 - z)))) + (0.9999999999998099 + ((-176.6150291621406 / (4.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) - (771.3234287776531 / (z - 3.0))))))) * (sqrt((((double) M_PI) * 2.0)) * (pow((7.5 - z), (0.5 - z)) * exp((-6.5 - (1.0 - z))))));
}

Java

public static double code(double z) {
	return (Math.PI / Math.sin((Math.PI * z))) * (((676.5203681218851 / (1.0 - z)) + (((12.507343278686905 / (5.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) - (1.5056327351493116e-7 / (z - 8.0))) + (9.984369578019572e-6 / (7.0 - z)))) + (0.9999999999998099 + ((-176.6150291621406 / (4.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) - (771.3234287776531 / (z - 3.0))))))) * (Math.sqrt((Math.PI * 2.0)) * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((-6.5 - (1.0 - z))))));
}

Python

def code(z):
	return (math.pi / math.sin((math.pi * z))) * (((676.5203681218851 / (1.0 - z)) + (((12.507343278686905 / (5.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) - (1.5056327351493116e-7 / (z - 8.0))) + (9.984369578019572e-6 / (7.0 - z)))) + (0.9999999999998099 + ((-176.6150291621406 / (4.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) - (771.3234287776531 / (z - 3.0))))))) * (math.sqrt((math.pi * 2.0)) * (math.pow((7.5 - z), (0.5 - z)) * math.exp((-6.5 - (1.0 - z))))))

Julia

function code(z)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) - Float64(1.5056327351493116e-7 / Float64(z - 8.0))) + Float64(9.984369578019572e-6 / Float64(7.0 - z)))) + Float64(0.9999999999998099 + Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) + Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) - Float64(771.3234287776531 / Float64(z - 3.0))))))) * Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(-6.5 - Float64(1.0 - z)))))))
end

MATLAB

function tmp = code(z)
	tmp = (pi / sin((pi * z))) * (((676.5203681218851 / (1.0 - z)) + (((12.507343278686905 / (5.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) - (1.5056327351493116e-7 / (z - 8.0))) + (9.984369578019572e-6 / (7.0 - z)))) + (0.9999999999998099 + ((-176.6150291621406 / (4.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) - (771.3234287776531 / (z - 3.0))))))) * (sqrt((pi * 2.0)) * (((7.5 - z) ^ (0.5 - z)) * exp((-6.5 - (1.0 - z))))));
end

Wolfram

code[z_] := N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] - N[(1.5056327351493116e-7 / N[(z - 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] - N[(771.3234287776531 / N[(z - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-6.5 - N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

4. Applied egg-rr 97.6%

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

6. Simplified 98.4%

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

7. Final simplification 98.4%

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

Alternative 5: 98.3% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  PI
  (/
   (*
    (+
     (/ 676.5203681218851 (- 1.0 z))
     (+
      (+
       (/ 12.507343278686905 (- 5.0 z))
       (+
        (-
         (/ -0.13857109526572012 (- 6.0 z))
         (/ 1.5056327351493116e-7 (- z 8.0)))
        (/ 9.984369578019572e-6 (- 7.0 z))))
      (+
       0.9999999999998099
       (+
        (/ -176.6150291621406 (- 4.0 z))
        (-
         (/ -1259.1392167224028 (- 2.0 z))
         (/ 771.3234287776531 (- z 3.0)))))))
    (*
     (sqrt (* PI 2.0))
     (* (pow (- 7.5 z) (- 0.5 z)) (exp (- -6.5 (- 1.0 z))))))
   (sin (* PI z)))))

C

double code(double z) {
	return ((double) M_PI) * ((((676.5203681218851 / (1.0 - z)) + (((12.507343278686905 / (5.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) - (1.5056327351493116e-7 / (z - 8.0))) + (9.984369578019572e-6 / (7.0 - z)))) + (0.9999999999998099 + ((-176.6150291621406 / (4.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) - (771.3234287776531 / (z - 3.0))))))) * (sqrt((((double) M_PI) * 2.0)) * (pow((7.5 - z), (0.5 - z)) * exp((-6.5 - (1.0 - z)))))) / sin((((double) M_PI) * z)));
}

Java

public static double code(double z) {
	return Math.PI * ((((676.5203681218851 / (1.0 - z)) + (((12.507343278686905 / (5.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) - (1.5056327351493116e-7 / (z - 8.0))) + (9.984369578019572e-6 / (7.0 - z)))) + (0.9999999999998099 + ((-176.6150291621406 / (4.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) - (771.3234287776531 / (z - 3.0))))))) * (Math.sqrt((Math.PI * 2.0)) * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((-6.5 - (1.0 - z)))))) / Math.sin((Math.PI * z)));
}

Python

def code(z):
	return math.pi * ((((676.5203681218851 / (1.0 - z)) + (((12.507343278686905 / (5.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) - (1.5056327351493116e-7 / (z - 8.0))) + (9.984369578019572e-6 / (7.0 - z)))) + (0.9999999999998099 + ((-176.6150291621406 / (4.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) - (771.3234287776531 / (z - 3.0))))))) * (math.sqrt((math.pi * 2.0)) * (math.pow((7.5 - z), (0.5 - z)) * math.exp((-6.5 - (1.0 - z)))))) / math.sin((math.pi * z)))

Julia

function code(z)
	return Float64(pi * Float64(Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) - Float64(1.5056327351493116e-7 / Float64(z - 8.0))) + Float64(9.984369578019572e-6 / Float64(7.0 - z)))) + Float64(0.9999999999998099 + Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) + Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) - Float64(771.3234287776531 / Float64(z - 3.0))))))) * Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(-6.5 - Float64(1.0 - z)))))) / sin(Float64(pi * z))))
end

MATLAB

function tmp = code(z)
	tmp = pi * ((((676.5203681218851 / (1.0 - z)) + (((12.507343278686905 / (5.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) - (1.5056327351493116e-7 / (z - 8.0))) + (9.984369578019572e-6 / (7.0 - z)))) + (0.9999999999998099 + ((-176.6150291621406 / (4.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) - (771.3234287776531 / (z - 3.0))))))) * (sqrt((pi * 2.0)) * (((7.5 - z) ^ (0.5 - z)) * exp((-6.5 - (1.0 - z)))))) / sin((pi * z)));
end

Wolfram

code[z_] := N[(Pi * N[(N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] - N[(1.5056327351493116e-7 / N[(z - 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] - N[(771.3234287776531 / N[(z - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-6.5 - N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[N[(Pi * 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 96.1%

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

6. Applied egg-rr 97.9%

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

8. Simplified 98.4%

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

9. Final simplification 98.4%

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

Alternative 6: 97.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[z_] := N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] - N[(N[(1.5056327351493116e-7 / N[(z - 8.0), $MachinePrecision]), $MachinePrecision] + N[(N[(676.5203681218851 / N[(z + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-1259.1392167224028 / N[(z - 2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(z - 3.0), $MachinePrecision]), $MachinePrecision] - 0.9999999999998099), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(z - 6.0), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(z - 5.0), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(z - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 96.2%

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

   1. pow1 96.2%

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

6. Applied egg-rr 96.2%

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

8. Simplified 98.0%

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

9. Final simplification 98.0%

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

Alternative 7: 97.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[z_] := N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] - N[(N[(771.3234287776531 / N[(z - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1259.1392167224028 / N[(z - 2.0), $MachinePrecision]), $MachinePrecision] - 0.9999999999998099), $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 96.1%

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

6. Applied egg-rr 97.2%

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

8. Simplified 97.5%

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

10. Applied egg-rr 97.4%

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

12. Simplified 97.9%

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

13. Final simplification 97.9%

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

Alternative 8: 97.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-6.5 - \left(1 - z\right)}\right)\right)}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} - \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + -7}\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/
   (*
    PI
    (*
     (sqrt (* PI 2.0))
     (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp (- -6.5 (- 1.0 z))))))
   (sin (* PI z)))
  (+
   (+ 263.3831855358925 (* z (+ 436.8961723502244 (* z 545.0353078134797))))
   (-
    (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0))
    (/ 1.5056327351493116e-7 (+ (+ z -1.0) -7.0))))))

C

double code(double z) {
	return ((((double) M_PI) * (sqrt((((double) M_PI) * 2.0)) * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp((-6.5 - (1.0 - z)))))) / sin((((double) M_PI) * z))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * 545.0353078134797)))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) - (1.5056327351493116e-7 / ((z + -1.0) + -7.0))));
}

Java

public static double code(double z) {
	return ((Math.PI * (Math.sqrt((Math.PI * 2.0)) * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp((-6.5 - (1.0 - z)))))) / Math.sin((Math.PI * z))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * 545.0353078134797)))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) - (1.5056327351493116e-7 / ((z + -1.0) + -7.0))));
}

Python

def code(z):
	return ((math.pi * (math.sqrt((math.pi * 2.0)) * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp((-6.5 - (1.0 - z)))))) / math.sin((math.pi * z))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * 545.0353078134797)))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) - (1.5056327351493116e-7 / ((z + -1.0) + -7.0))))

Julia

function code(z)
	return Float64(Float64(Float64(pi * Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(Float64(-6.5 - Float64(1.0 - z)))))) / sin(Float64(pi * z))) * Float64(Float64(263.3831855358925 + Float64(z * Float64(436.8961723502244 + Float64(z * 545.0353078134797)))) + Float64(Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0)) - Float64(1.5056327351493116e-7 / Float64(Float64(z + -1.0) + -7.0)))))
end

MATLAB

function tmp = code(z)
	tmp = ((pi * (sqrt((pi * 2.0)) * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp((-6.5 - (1.0 - z)))))) / sin((pi * z))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * 545.0353078134797)))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) - (1.5056327351493116e-7 / ((z + -1.0) + -7.0))));
end

Wolfram

code[z_] := N[(N[(N[(Pi * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-6.5 - N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(263.3831855358925 + N[(z * N[(436.8961723502244 + N[(z * 545.0353078134797), $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[(z + -1.0), $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 98.1%

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

5. Taylor expanded in z around 0 96.5%

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

   1. *-commutative 96.5%

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

7. Simplified 96.5%

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

9. Applied egg-rr 96.7%

   \[\leadsto \color{blue}{\frac{\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right)}{\sin \left(\pi \cdot z\right)}} \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

10. Final simplification 96.7%

    \[\leadsto \frac{\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-6.5 - \left(1 - z\right)}\right)\right)}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} - \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + -7}\right)\right) \]
11. Add Preprocessing

Alternative 9: 96.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\pi \cdot \frac{e^{-6.5 + \left(z + -1\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right)}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + z \cdot 606.6766809167608\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   PI
   (/
    (*
     (exp (+ -6.5 (+ z -1.0)))
     (* (sqrt (* PI 2.0)) (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5))))
    (sin (* PI z))))
  (+
   263.3831869810514
   (*
    z
    (+
     436.8961725563396
     (* z (+ 545.0353078428827 (* z 606.6766809167608))))))))

C

double code(double z) {
	return (((double) M_PI) * ((exp((-6.5 + (z + -1.0))) * (sqrt((((double) M_PI) * 2.0)) * pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)))) / sin((((double) M_PI) * z)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (z * 606.6766809167608))))));
}

Java

public static double code(double z) {
	return (Math.PI * ((Math.exp((-6.5 + (z + -1.0))) * (Math.sqrt((Math.PI * 2.0)) * Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)))) / Math.sin((Math.PI * z)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (z * 606.6766809167608))))));
}

Python

def code(z):
	return (math.pi * ((math.exp((-6.5 + (z + -1.0))) * (math.sqrt((math.pi * 2.0)) * math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)))) / math.sin((math.pi * z)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (z * 606.6766809167608))))))

Julia

function code(z)
	return Float64(Float64(pi * Float64(Float64(exp(Float64(-6.5 + Float64(z + -1.0))) * Float64(sqrt(Float64(pi * 2.0)) * (Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)))) / sin(Float64(pi * z)))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * Float64(545.0353078428827 + Float64(z * 606.6766809167608)))))))
end

MATLAB

function tmp = code(z)
	tmp = (pi * ((exp((-6.5 + (z + -1.0))) * (sqrt((pi * 2.0)) * (((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)))) / sin((pi * z)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (z * 606.6766809167608))))));
end

Wolfram

code[z_] := N[(N[(Pi * N[(N[(N[Exp[N[(-6.5 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * N[(545.0353078428827 + N[(z * 606.6766809167608), $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 96.1%

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

6. Applied egg-rr 98.4%

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

8. Simplified 98.1%

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

9. Taylor expanded in z around 0 96.4%

   \[\leadsto \left(\pi \cdot \frac{\left(\sqrt{2 \cdot \pi} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}}{\sin \left(z \cdot \pi\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)} \]
10. Step-by-step derivation

    1. *-commutative 96.4%

       \[\leadsto \left(\pi \cdot \frac{\left(\sqrt{2 \cdot \pi} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + \color{blue}{z \cdot 606.6766809167608}\right)\right)\right) \]

11. Simplified 96.4%

    \[\leadsto \left(\pi \cdot \frac{\left(\sqrt{2 \cdot \pi} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}}{\sin \left(z \cdot \pi\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + z \cdot 606.6766809167608\right)\right)\right)} \]

12. Final simplification 96.4%

    \[\leadsto \left(\pi \cdot \frac{e^{-6.5 + \left(z + -1\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right)}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + z \cdot 606.6766809167608\right)\right)\right) \]
13. Add Preprocessing

Alternative 10: 97.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\pi \cdot \frac{e^{-6.5 + \left(z + -1\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right)}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   PI
   (/
    (*
     (exp (+ -6.5 (+ z -1.0)))
     (* (sqrt (* PI 2.0)) (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5))))
    (sin (* PI z))))
  (+ 263.3831869810514 (* z (+ 436.8961725563396 (* z 545.0353078428827))))))

C

double code(double z) {
	return (((double) M_PI) * ((exp((-6.5 + (z + -1.0))) * (sqrt((((double) M_PI) * 2.0)) * pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)))) / sin((((double) M_PI) * z)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827))));
}

Java

public static double code(double z) {
	return (Math.PI * ((Math.exp((-6.5 + (z + -1.0))) * (Math.sqrt((Math.PI * 2.0)) * Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)))) / Math.sin((Math.PI * z)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827))));
}

Python

def code(z):
	return (math.pi * ((math.exp((-6.5 + (z + -1.0))) * (math.sqrt((math.pi * 2.0)) * math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)))) / math.sin((math.pi * z)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827))))

Julia

function code(z)
	return Float64(Float64(pi * Float64(Float64(exp(Float64(-6.5 + Float64(z + -1.0))) * Float64(sqrt(Float64(pi * 2.0)) * (Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)))) / sin(Float64(pi * z)))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * 545.0353078428827)))))
end

MATLAB

function tmp = code(z)
	tmp = (pi * ((exp((-6.5 + (z + -1.0))) * (sqrt((pi * 2.0)) * (((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)))) / sin((pi * z)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827))));
end

Wolfram

code[z_] := N[(N[(Pi * N[(N[(N[Exp[N[(-6.5 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * 545.0353078428827), $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 96.1%

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

6. Applied egg-rr 98.4%

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

8. Simplified 98.1%

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

9. Taylor expanded in z around 0 96.4%

   \[\leadsto \left(\pi \cdot \frac{\left(\sqrt{2 \cdot \pi} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}}{\sin \left(z \cdot \pi\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\right)} \]
10. Step-by-step derivation

    1. *-commutative 96.4%

       \[\leadsto \left(\pi \cdot \frac{\left(\sqrt{2 \cdot \pi} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + \color{blue}{z \cdot 545.0353078428827}\right)\right) \]

11. Simplified 96.4%

    \[\leadsto \left(\pi \cdot \frac{\left(\sqrt{2 \cdot \pi} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}}{\sin \left(z \cdot \pi\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)} \]

12. Final simplification 96.4%

    \[\leadsto \left(\pi \cdot \frac{e^{-6.5 + \left(z + -1\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right)}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right) \]
13. Add Preprocessing

Alternative 11: 96.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\pi \cdot \frac{e^{-6.5 + \left(z + -1\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right)}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(263.3831869810514 + z \cdot 436.8961725563396\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   PI
   (/
    (*
     (exp (+ -6.5 (+ z -1.0)))
     (* (sqrt (* PI 2.0)) (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5))))
    (sin (* PI z))))
  (+ 263.3831869810514 (* z 436.8961725563396))))

C

double code(double z) {
	return (((double) M_PI) * ((exp((-6.5 + (z + -1.0))) * (sqrt((((double) M_PI) * 2.0)) * pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)))) / sin((((double) M_PI) * z)))) * (263.3831869810514 + (z * 436.8961725563396));
}

Java

public static double code(double z) {
	return (Math.PI * ((Math.exp((-6.5 + (z + -1.0))) * (Math.sqrt((Math.PI * 2.0)) * Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)))) / Math.sin((Math.PI * z)))) * (263.3831869810514 + (z * 436.8961725563396));
}

Python

def code(z):
	return (math.pi * ((math.exp((-6.5 + (z + -1.0))) * (math.sqrt((math.pi * 2.0)) * math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)))) / math.sin((math.pi * z)))) * (263.3831869810514 + (z * 436.8961725563396))

Julia

function code(z)
	return Float64(Float64(pi * Float64(Float64(exp(Float64(-6.5 + Float64(z + -1.0))) * Float64(sqrt(Float64(pi * 2.0)) * (Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)))) / sin(Float64(pi * z)))) * Float64(263.3831869810514 + Float64(z * 436.8961725563396)))
end

MATLAB

function tmp = code(z)
	tmp = (pi * ((exp((-6.5 + (z + -1.0))) * (sqrt((pi * 2.0)) * (((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)))) / sin((pi * z)))) * (263.3831869810514 + (z * 436.8961725563396));
end

Wolfram

code[z_] := N[(N[(Pi * N[(N[(N[Exp[N[(-6.5 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * 436.8961725563396), $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 96.1%

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

6. Applied egg-rr 98.4%

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

8. Simplified 98.1%

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

9. Taylor expanded in z around 0 95.8%

   \[\leadsto \left(\pi \cdot \frac{\left(\sqrt{2 \cdot \pi} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}}{\sin \left(z \cdot \pi\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + 436.8961725563396 \cdot z\right)} \]
10. Step-by-step derivation

    1. *-commutative 95.8%

       \[\leadsto \left(\pi \cdot \frac{\left(\sqrt{2 \cdot \pi} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(263.3831869810514 + \color{blue}{z \cdot 436.8961725563396}\right) \]

11. Simplified 95.8%

    \[\leadsto \left(\pi \cdot \frac{\left(\sqrt{2 \cdot \pi} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right) \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}}{\sin \left(z \cdot \pi\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot 436.8961725563396\right)} \]

12. Final simplification 95.8%

    \[\leadsto \left(\pi \cdot \frac{e^{-6.5 + \left(z + -1\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right)}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(263.3831869810514 + z \cdot 436.8961725563396\right) \]
13. Add Preprocessing

Alternative 12: 96.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

3. Simplified 96.2%

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

5. Taylor expanded in z around 0 94.5%

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

   1. associate-*l/ 94.4%

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

   2. *-commutative 94.4%

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

   3. associate-*r* 95.1%

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

7. Simplified 95.1%

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

Alternative 13: 96.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(z + -1\right) - 6.5}\right)\right) \cdot \frac{263.3831869810514 + z \cdot 436.8961725563396}{z} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (sqrt (* PI 2.0))
   (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp (- (+ z -1.0) 6.5))))
  (/ (+ 263.3831869810514 (* z 436.8961725563396)) z)))

C

double code(double z) {
	return (sqrt((((double) M_PI) * 2.0)) * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(((z + -1.0) - 6.5)))) * ((263.3831869810514 + (z * 436.8961725563396)) / z);
}

Java

public static double code(double z) {
	return (Math.sqrt((Math.PI * 2.0)) * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp(((z + -1.0) - 6.5)))) * ((263.3831869810514 + (z * 436.8961725563396)) / z);
}

Python

def code(z):
	return (math.sqrt((math.pi * 2.0)) * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp(((z + -1.0) - 6.5)))) * ((263.3831869810514 + (z * 436.8961725563396)) / z)

Julia

function code(z)
	return Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(Float64(Float64(z + -1.0) - 6.5)))) * Float64(Float64(263.3831869810514 + Float64(z * 436.8961725563396)) / z))
end

MATLAB

function tmp = code(z)
	tmp = (sqrt((pi * 2.0)) * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(((z + -1.0) - 6.5)))) * ((263.3831869810514 + (z * 436.8961725563396)) / z);
end

Wolfram

code[z_] := N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(z + -1.0), $MachinePrecision] - 6.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(263.3831869810514 + N[(z * 436.8961725563396), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

3. Simplified 96.1%

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

5. Taylor expanded in z around 0 95.0%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \color{blue}{\frac{263.3831869810514 + 436.8961725563396 \cdot z}{z}} \]
6. Step-by-step derivation

   1. *-commutative 95.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \frac{263.3831869810514 + \color{blue}{z \cdot 436.8961725563396}}{z} \]

7. Simplified 95.0%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \color{blue}{\frac{263.3831869810514 + z \cdot 436.8961725563396}{z}} \]

8. Final simplification 95.0%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(z + -1\right) - 6.5}\right)\right) \cdot \frac{263.3831869810514 + z \cdot 436.8961725563396}{z} \]
9. Add Preprocessing

Alternative 14: 95.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

3. Simplified 96.1%

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

5. Taylor expanded in z around 0 93.5%

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

   1. associate-*r/ 93.8%

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

   2. *-commutative 93.8%

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

   3. metadata-eval 93.8%

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

   4. sub-neg 93.8%

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

   5. associate--l- 93.8%

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

7. Applied egg-rr 93.8%

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

9. Simplified 94.2%

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

10. Final simplification 94.2%

    \[\leadsto 263.3831869810514 \cdot \frac{e^{z + -7.5} \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)}{z} \]
11. Add Preprocessing

Alternative 15: 95.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{263.3831869810514 \cdot \left(e^{-6.5 + \left(z + -1\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \sqrt{7.5}\right)\right)}{z} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (/
  (*
   263.3831869810514
   (* (exp (+ -6.5 (+ z -1.0))) (* (sqrt (* PI 2.0)) (sqrt 7.5))))
  z))

C

double code(double z) {
	return (263.3831869810514 * (exp((-6.5 + (z + -1.0))) * (sqrt((((double) M_PI) * 2.0)) * sqrt(7.5)))) / z;
}

Java

public static double code(double z) {
	return (263.3831869810514 * (Math.exp((-6.5 + (z + -1.0))) * (Math.sqrt((Math.PI * 2.0)) * Math.sqrt(7.5)))) / z;
}

Python

def code(z):
	return (263.3831869810514 * (math.exp((-6.5 + (z + -1.0))) * (math.sqrt((math.pi * 2.0)) * math.sqrt(7.5)))) / z

Julia

function code(z)
	return Float64(Float64(263.3831869810514 * Float64(exp(Float64(-6.5 + Float64(z + -1.0))) * Float64(sqrt(Float64(pi * 2.0)) * sqrt(7.5)))) / z)
end

MATLAB

function tmp = code(z)
	tmp = (263.3831869810514 * (exp((-6.5 + (z + -1.0))) * (sqrt((pi * 2.0)) * sqrt(7.5)))) / z;
end

Wolfram

code[z_] := N[(N[(263.3831869810514 * N[(N[Exp[N[(-6.5 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $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 96.1%

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

5. Taylor expanded in z around 0 93.5%

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

6. Taylor expanded in z around 0 93.7%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\sqrt{7.5}} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \frac{263.3831869810514}{z} \]
7. Step-by-step derivation

   1. associate-*r/ 94.0%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\pi \cdot 2} \cdot \left(\sqrt{7.5} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot 263.3831869810514}{z}} \]

   2. associate-*r* 94.0%

      \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{\pi \cdot 2} \cdot \sqrt{7.5}\right) \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)} \cdot 263.3831869810514}{z} \]

   3. distribute-neg-in 94.0%

      \[\leadsto \frac{\left(\left(\sqrt{\pi \cdot 2} \cdot \sqrt{7.5}\right) \cdot e^{\color{blue}{\left(-\left(1 - z\right)\right) + \left(-6.5\right)}}\right) \cdot 263.3831869810514}{z} \]

   4. metadata-eval 94.0%

      \[\leadsto \frac{\left(\left(\sqrt{\pi \cdot 2} \cdot \sqrt{7.5}\right) \cdot e^{\left(-\left(1 - z\right)\right) + \color{blue}{-6.5}}\right) \cdot 263.3831869810514}{z} \]

   5. +-commutative 94.0%

      \[\leadsto \frac{\left(\left(\sqrt{\pi \cdot 2} \cdot \sqrt{7.5}\right) \cdot e^{\color{blue}{-6.5 + \left(-\left(1 - z\right)\right)}}\right) \cdot 263.3831869810514}{z} \]

8. Applied egg-rr 94.0%

   \[\leadsto \color{blue}{\frac{\left(\left(\sqrt{\pi \cdot 2} \cdot \sqrt{7.5}\right) \cdot e^{-6.5 + \left(-\left(1 - z\right)\right)}\right) \cdot 263.3831869810514}{z}} \]

9. Final simplification 94.0%

   \[\leadsto \frac{263.3831869810514 \cdot \left(e^{-6.5 + \left(z + -1\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \sqrt{7.5}\right)\right)}{z} \]
10. Add Preprocessing

Alternative 16: 95.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[z_] := N[(N[(263.3831869810514 * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] * N[Sqrt[7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $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 96.1%

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

5. Taylor expanded in z around 0 93.5%

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

6. Taylor expanded in z around 0 93.7%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\sqrt{7.5}} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \frac{263.3831869810514}{z} \]

7. Taylor expanded in z around 0 93.7%

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

   1. associate-*r/ 94.0%

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

9. Applied egg-rr 94.0%

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

10. Final simplification 94.0%

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

Alternative 17: 95.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[z_] := N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[7.5], $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] * N[(263.3831869810514 / 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 96.1%

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

5. Taylor expanded in z around 0 93.5%

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

6. Taylor expanded in z around 0 93.7%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\sqrt{7.5}} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \frac{263.3831869810514}{z} \]

7. Taylor expanded in z around 0 93.7%

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

   1. associate-*r/ 94.0%

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

9. Applied egg-rr 94.0%

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

    1. associate-/l* 93.7%

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

    2. associate-*r* 93.8%

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

    3. *-commutative 93.8%

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

    4. associate-*l* 93.7%

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

11. Simplified 93.7%

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

12. Final simplification 93.7%

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

Reproduce

herbie shell --seed 2024182 
(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))))))