Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.5% → 99.1%

Time: 1.4min

Alternatives: 17

Speedup: 1.6×

• could not determine a ground truth

  1. z = -1.1451592277510167e+236

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: 96.5% 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: 99.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

   1. *-commutative 96.2%

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

3. Simplified 97.9%

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

   1. add-exp-log 98.1%

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

   2. *-commutative 98.1%

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

   3. log-prod 98.1%

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

   4. add-log-exp 99.3%

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

   5. log-pow 99.3%

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

   6. neg-mul-1 99.3%

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

   7. fma-def 99.3%

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

5. Applied egg-rr 99.3%

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

6. Final simplification 99.3%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.9999999999998099 + \frac{676.5203681218851}{1 - z}\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z - 7.5}\right) \cdot \left(\frac{\frac{\frac{1585431.567088306}{2 - z}}{2 - z} - t_0 \cdot t_0}{\frac{-1259.1392167224028}{2 - z} - t_0} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{2 + \left(1 - z\right)}\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{12.507343278686905}{5 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ 0.9999999999998099 (/ 676.5203681218851 (- 1.0 z)))))
   (*
    (/ PI (sin (* PI z)))
    (*
     (sqrt (* PI 2.0))
     (*
      (* (pow (- 7.5 z) (- 0.5 z)) (exp (- z 7.5)))
      (+
       (/
        (- (/ (/ 1585431.567088306 (- 2.0 z)) (- 2.0 z)) (* t_0 t_0))
        (- (/ -1259.1392167224028 (- 2.0 z)) t_0))
       (+
        (+
         (/ -176.6150291621406 (- 4.0 z))
         (/ 771.3234287776531 (+ 2.0 (- 1.0 z))))
        (+
         (+
          (/ 9.984369578019572e-6 (- 7.0 z))
          (/ -0.13857109526572012 (+ (- 1.0 z) 5.0)))
         (+
          (/ 12.507343278686905 (- 5.0 z))
          (/ 1.5056327351493116e-7 (- 8.0 z)))))))))))

C

double code(double z) {
	double t_0 = 0.9999999999998099 + (676.5203681218851 / (1.0 - z));
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (sqrt((((double) M_PI) * 2.0)) * ((pow((7.5 - z), (0.5 - z)) * exp((z - 7.5))) * (((((1585431.567088306 / (2.0 - z)) / (2.0 - z)) - (t_0 * t_0)) / ((-1259.1392167224028 / (2.0 - z)) - t_0)) + (((-176.6150291621406 / (4.0 - z)) + (771.3234287776531 / (2.0 + (1.0 - z)))) + (((9.984369578019572e-6 / (7.0 - z)) + (-0.13857109526572012 / ((1.0 - z) + 5.0))) + ((12.507343278686905 / (5.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))))));
}

Java

public static double code(double z) {
	double t_0 = 0.9999999999998099 + (676.5203681218851 / (1.0 - z));
	return (Math.PI / Math.sin((Math.PI * z))) * (Math.sqrt((Math.PI * 2.0)) * ((Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z - 7.5))) * (((((1585431.567088306 / (2.0 - z)) / (2.0 - z)) - (t_0 * t_0)) / ((-1259.1392167224028 / (2.0 - z)) - t_0)) + (((-176.6150291621406 / (4.0 - z)) + (771.3234287776531 / (2.0 + (1.0 - z)))) + (((9.984369578019572e-6 / (7.0 - z)) + (-0.13857109526572012 / ((1.0 - z) + 5.0))) + ((12.507343278686905 / (5.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))))));
}

Python

def code(z):
	t_0 = 0.9999999999998099 + (676.5203681218851 / (1.0 - z))
	return (math.pi / math.sin((math.pi * z))) * (math.sqrt((math.pi * 2.0)) * ((math.pow((7.5 - z), (0.5 - z)) * math.exp((z - 7.5))) * (((((1585431.567088306 / (2.0 - z)) / (2.0 - z)) - (t_0 * t_0)) / ((-1259.1392167224028 / (2.0 - z)) - t_0)) + (((-176.6150291621406 / (4.0 - z)) + (771.3234287776531 / (2.0 + (1.0 - z)))) + (((9.984369578019572e-6 / (7.0 - z)) + (-0.13857109526572012 / ((1.0 - z) + 5.0))) + ((12.507343278686905 / (5.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))))))

Julia

function code(z)
	t_0 = Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 - z)))
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(sqrt(Float64(pi * 2.0)) * Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z - 7.5))) * Float64(Float64(Float64(Float64(Float64(1585431.567088306 / Float64(2.0 - z)) / Float64(2.0 - z)) - Float64(t_0 * t_0)) / Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) - t_0)) + Float64(Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) + Float64(771.3234287776531 / Float64(2.0 + Float64(1.0 - z)))) + Float64(Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) + 5.0))) + Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z)))))))))
end

MATLAB

function tmp = code(z)
	t_0 = 0.9999999999998099 + (676.5203681218851 / (1.0 - z));
	tmp = (pi / sin((pi * z))) * (sqrt((pi * 2.0)) * ((((7.5 - z) ^ (0.5 - z)) * exp((z - 7.5))) * (((((1585431.567088306 / (2.0 - z)) / (2.0 - z)) - (t_0 * t_0)) / ((-1259.1392167224028 / (2.0 - z)) - t_0)) + (((-176.6150291621406 / (4.0 - z)) + (771.3234287776531 / (2.0 + (1.0 - z)))) + (((9.984369578019572e-6 / (7.0 - z)) + (-0.13857109526572012 / ((1.0 - z) + 5.0))) + ((12.507343278686905 / (5.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z - 7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(1585431.567088306 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(2.0 + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 96.2%

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

   1. associate-*l* 96.3%

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

3. Simplified 96.7%

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

   1. expm1-log1p-u 95.8%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right)}\right)\right) \]

   2. expm1-udef 95.8%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)} - 1\right)}\right)\right) \]

5. Applied egg-rr 95.8%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \left(-z\right)}\right)\right) + \left(\frac{771.3234287776531}{1 - \left(z - 2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z - 5\right)}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)} - 1\right)}\right)\right) \]
6. Step-by-step derivation

   1. expm1-def 95.8%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \left(-z\right)}\right)\right) + \left(\frac{771.3234287776531}{1 - \left(z - 2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z - 5\right)}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)}\right)\right) \]

   2. expm1-log1p 98.2%

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

   3. associate-+l+ 98.2%

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

   4. associate-+r+ 98.2%

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

   5. associate-+l+ 98.2%

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

7. Simplified 97.3%

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

   1. flip-+ 98.1%

      \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\color{blue}{\frac{\frac{-1259.1392167224028}{2 - z} \cdot \frac{-1259.1392167224028}{2 - z} - \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)}{\frac{-1259.1392167224028}{2 - z} - \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)}} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

9. Applied egg-rr 98.2%

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

    1. associate-*l/ 98.1%

       \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\frac{\color{blue}{\frac{-1259.1392167224028 \cdot \frac{-1259.1392167224028}{2 - z}}{2 - z}} - \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)}{\frac{-1259.1392167224028}{2 - z} - \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

    2. associate-*r/ 98.1%

       \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\frac{\frac{\color{blue}{\frac{-1259.1392167224028 \cdot -1259.1392167224028}{2 - z}}}{2 - z} - \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)}{\frac{-1259.1392167224028}{2 - z} - \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

    3. metadata-eval 98.1%

       \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\frac{\frac{\frac{\color{blue}{1585431.567088306}}{2 - z}}{2 - z} - \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)}{\frac{-1259.1392167224028}{2 - z} - \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

11. Simplified 98.2%

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\color{blue}{\frac{\frac{\frac{1585431.567088306}{2 - z}}{2 - z} - \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)}{\frac{-1259.1392167224028}{2 - z} - \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)}} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

12. Taylor expanded in z around inf 98.2%

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z - 7.5}\right)} \cdot \left(\frac{\frac{\frac{1585431.567088306}{2 - z}}{2 - z} - \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)}{\frac{-1259.1392167224028}{2 - z} - \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

13. Final simplification 98.2%

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z - 7.5}\right) \cdot \left(\frac{\frac{\frac{1585431.567088306}{2 - z}}{2 - z} - \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)}{\frac{-1259.1392167224028}{2 - z} - \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{2 + \left(1 - z\right)}\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{12.507343278686905}{5 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)\right) \]

Alternative 3: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

   1. associate-*l* 96.3%

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

3. Simplified 96.7%

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

   1. expm1-log1p-u 95.8%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right)}\right)\right) \]

   2. expm1-udef 95.8%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)} - 1\right)}\right)\right) \]

5. Applied egg-rr 95.8%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \left(-z\right)}\right)\right) + \left(\frac{771.3234287776531}{1 - \left(z - 2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z - 5\right)}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)} - 1\right)}\right)\right) \]
6. Step-by-step derivation

   1. expm1-def 95.8%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \left(-z\right)}\right)\right) + \left(\frac{771.3234287776531}{1 - \left(z - 2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z - 5\right)}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)}\right)\right) \]

   2. expm1-log1p 98.2%

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

   3. associate-+l+ 98.2%

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

   4. associate-+r+ 98.2%

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

   5. associate-+l+ 98.2%

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

7. Simplified 97.3%

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

   1. *-un-lft-identity 97.1%

      \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\color{blue}{1 \cdot \left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right)} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

   2. +-commutative 97.1%

      \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(1 \cdot \color{blue}{\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

9. Applied egg-rr 97.3%

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

    1. *-lft-identity 97.1%

       \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\color{blue}{\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

    2. associate-+l+ 98.1%

       \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\color{blue}{\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

11. Simplified 98.2%

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

12. Final simplification 98.2%

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

Alternative 4: 97.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.9999999999998099 + \frac{676.5203681218851}{1 - z}\\ \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(z + -1\right) - 6.5}\right) \cdot \left(\frac{\frac{\frac{1585431.567088306}{2 - z}}{2 - z} - t_0 \cdot t_0}{\frac{-1259.1392167224028}{2 - z} - t_0} + \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{12.507343278686905}{5 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ 0.9999999999998099 (/ 676.5203681218851 (- 1.0 z)))))
   (*
    (+ (* 0.16666666666666666 (* z (pow PI 2.0))) (/ 1.0 z))
    (*
     (sqrt (* PI 2.0))
     (*
      (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp (- (+ z -1.0) 6.5)))
      (+
       (/
        (- (/ (/ 1585431.567088306 (- 2.0 z)) (- 2.0 z)) (* t_0 t_0))
        (- (/ -1259.1392167224028 (- 2.0 z)) t_0))
       (+
        (+
         (+
          (/ 9.984369578019572e-6 (- 7.0 z))
          (/ -0.13857109526572012 (+ (- 1.0 z) 5.0)))
         (+
          (/ 12.507343278686905 (- 5.0 z))
          (/ 1.5056327351493116e-7 (- 8.0 z))))
        (+
         (/ 771.3234287776531 (- 3.0 z))
         (/ -176.6150291621406 (- 4.0 z))))))))))

C

double code(double z) {
	double t_0 = 0.9999999999998099 + (676.5203681218851 / (1.0 - z));
	return ((0.16666666666666666 * (z * pow(((double) M_PI), 2.0))) + (1.0 / z)) * (sqrt((((double) M_PI) * 2.0)) * ((pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(((z + -1.0) - 6.5))) * (((((1585431.567088306 / (2.0 - z)) / (2.0 - z)) - (t_0 * t_0)) / ((-1259.1392167224028 / (2.0 - z)) - t_0)) + ((((9.984369578019572e-6 / (7.0 - z)) + (-0.13857109526572012 / ((1.0 - z) + 5.0))) + ((12.507343278686905 / (5.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + ((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z)))))));
}

Java

public static double code(double z) {
	double t_0 = 0.9999999999998099 + (676.5203681218851 / (1.0 - z));
	return ((0.16666666666666666 * (z * Math.pow(Math.PI, 2.0))) + (1.0 / z)) * (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))) * (((((1585431.567088306 / (2.0 - z)) / (2.0 - z)) - (t_0 * t_0)) / ((-1259.1392167224028 / (2.0 - z)) - t_0)) + ((((9.984369578019572e-6 / (7.0 - z)) + (-0.13857109526572012 / ((1.0 - z) + 5.0))) + ((12.507343278686905 / (5.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + ((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z)))))));
}

Python

def code(z):
	t_0 = 0.9999999999998099 + (676.5203681218851 / (1.0 - z))
	return ((0.16666666666666666 * (z * math.pow(math.pi, 2.0))) + (1.0 / z)) * (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))) * (((((1585431.567088306 / (2.0 - z)) / (2.0 - z)) - (t_0 * t_0)) / ((-1259.1392167224028 / (2.0 - z)) - t_0)) + ((((9.984369578019572e-6 / (7.0 - z)) + (-0.13857109526572012 / ((1.0 - z) + 5.0))) + ((12.507343278686905 / (5.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + ((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z)))))))

Julia

function code(z)
	t_0 = Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 - z)))
	return Float64(Float64(Float64(0.16666666666666666 * Float64(z * (pi ^ 2.0))) + Float64(1.0 / z)) * Float64(sqrt(Float64(pi * 2.0)) * Float64(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(Float64(Float64(Float64(1585431.567088306 / Float64(2.0 - z)) / Float64(2.0 - z)) - Float64(t_0 * t_0)) / Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) - t_0)) + Float64(Float64(Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) + 5.0))) + Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z)))) + Float64(Float64(771.3234287776531 / Float64(3.0 - z)) + Float64(-176.6150291621406 / Float64(4.0 - z))))))))
end

MATLAB

function tmp = code(z)
	t_0 = 0.9999999999998099 + (676.5203681218851 / (1.0 - z));
	tmp = ((0.16666666666666666 * (z * (pi ^ 2.0))) + (1.0 / z)) * (sqrt((pi * 2.0)) * (((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(((z + -1.0) - 6.5))) * (((((1585431.567088306 / (2.0 - z)) / (2.0 - z)) - (t_0 * t_0)) / ((-1259.1392167224028 / (2.0 - z)) - t_0)) + ((((9.984369578019572e-6 / (7.0 - z)) + (-0.13857109526572012 / ((1.0 - z) + 5.0))) + ((12.507343278686905 / (5.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + ((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z)))))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(0.16666666666666666 * N[(z * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / z), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(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] * N[(N[(N[(N[(N[(1585431.567088306 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 96.2%

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

   1. associate-*l* 96.3%

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

3. Simplified 96.7%

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

   1. expm1-log1p-u 95.8%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right)}\right)\right) \]

   2. expm1-udef 95.8%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)} - 1\right)}\right)\right) \]

5. Applied egg-rr 95.8%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \left(-z\right)}\right)\right) + \left(\frac{771.3234287776531}{1 - \left(z - 2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z - 5\right)}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)} - 1\right)}\right)\right) \]
6. Step-by-step derivation

   1. expm1-def 95.8%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \left(-z\right)}\right)\right) + \left(\frac{771.3234287776531}{1 - \left(z - 2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z - 5\right)}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)}\right)\right) \]

   2. expm1-log1p 98.2%

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

   3. associate-+l+ 98.2%

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

   4. associate-+r+ 98.2%

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

   5. associate-+l+ 98.2%

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

7. Simplified 97.3%

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

8. Taylor expanded in z around 0 97.1%

   \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. *-un-lft-identity 97.1%

      \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right) + \left(\color{blue}{1 \cdot \left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)} + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

10. Applied egg-rr 97.1%

    \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right) + \left(\color{blue}{1 \cdot \left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)} + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]
11. Step-by-step derivation

    1. *-lft-identity 97.1%

       \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right) + \left(\color{blue}{\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)} + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

    2. +-commutative 97.1%

       \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{\color{blue}{2 + \left(1 - z\right)}}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

    3. associate-+r- 97.1%

       \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{\color{blue}{\left(2 + 1\right) - z}}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

    4. metadata-eval 97.1%

       \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{\color{blue}{3} - z}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

12. Simplified 97.1%

    \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right) + \left(\color{blue}{\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right)} + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]
13. Step-by-step derivation

    1. flip-+ 98.1%

       \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\color{blue}{\frac{\frac{-1259.1392167224028}{2 - z} \cdot \frac{-1259.1392167224028}{2 - z} - \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)}{\frac{-1259.1392167224028}{2 - z} - \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)}} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

14. Applied egg-rr 98.1%

    \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\color{blue}{\frac{\frac{-1259.1392167224028}{2 - z} \cdot \frac{-1259.1392167224028}{2 - z} - \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)}{\frac{-1259.1392167224028}{2 - z} - \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)}} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]
15. Step-by-step derivation

    1. associate-*l/ 98.1%

       \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\frac{\color{blue}{\frac{-1259.1392167224028 \cdot \frac{-1259.1392167224028}{2 - z}}{2 - z}} - \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)}{\frac{-1259.1392167224028}{2 - z} - \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

    2. associate-*r/ 98.1%

       \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\frac{\frac{\color{blue}{\frac{-1259.1392167224028 \cdot -1259.1392167224028}{2 - z}}}{2 - z} - \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)}{\frac{-1259.1392167224028}{2 - z} - \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

    3. metadata-eval 98.1%

       \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\frac{\frac{\frac{\color{blue}{1585431.567088306}}{2 - z}}{2 - z} - \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)}{\frac{-1259.1392167224028}{2 - z} - \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

16. Simplified 98.1%

    \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\color{blue}{\frac{\frac{\frac{1585431.567088306}{2 - z}}{2 - z} - \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)}{\frac{-1259.1392167224028}{2 - z} - \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)}} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

17. Final simplification 98.1%

    \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(z + -1\right) - 6.5}\right) \cdot \left(\frac{\frac{\frac{1585431.567088306}{2 - z}}{2 - z} - \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) \cdot \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)}{\frac{-1259.1392167224028}{2 - z} - \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)} + \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{12.507343278686905}{5 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right)\right)\right) \]

Alternative 5: 97.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(z + -1\right) - 6.5}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{12.507343278686905}{5 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (+ (* 0.16666666666666666 (* z (pow PI 2.0))) (/ 1.0 z))
  (*
   (sqrt (* PI 2.0))
   (*
    (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp (- (+ z -1.0) 6.5)))
    (+
     (+
      0.9999999999998099
      (+ (/ 676.5203681218851 (- 1.0 z)) (/ -1259.1392167224028 (- 2.0 z))))
     (+
      (+
       (+
        (/ 9.984369578019572e-6 (- 7.0 z))
        (/ -0.13857109526572012 (+ (- 1.0 z) 5.0)))
       (+
        (/ 12.507343278686905 (- 5.0 z))
        (/ 1.5056327351493116e-7 (- 8.0 z))))
      (+
       (/ 771.3234287776531 (- 3.0 z))
       (/ -176.6150291621406 (- 4.0 z)))))))))

C

double code(double z) {
	return ((0.16666666666666666 * (z * pow(((double) M_PI), 2.0))) + (1.0 / z)) * (sqrt((((double) M_PI) * 2.0)) * ((pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(((z + -1.0) - 6.5))) * ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + ((((9.984369578019572e-6 / (7.0 - z)) + (-0.13857109526572012 / ((1.0 - z) + 5.0))) + ((12.507343278686905 / (5.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + ((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z)))))));
}

Java

public static double code(double z) {
	return ((0.16666666666666666 * (z * Math.pow(Math.PI, 2.0))) + (1.0 / z)) * (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))) * ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + ((((9.984369578019572e-6 / (7.0 - z)) + (-0.13857109526572012 / ((1.0 - z) + 5.0))) + ((12.507343278686905 / (5.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + ((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z)))))));
}

Python

def code(z):
	return ((0.16666666666666666 * (z * math.pow(math.pi, 2.0))) + (1.0 / z)) * (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))) * ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + ((((9.984369578019572e-6 / (7.0 - z)) + (-0.13857109526572012 / ((1.0 - z) + 5.0))) + ((12.507343278686905 / (5.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + ((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z)))))))

Julia

function code(z)
	return Float64(Float64(Float64(0.16666666666666666 * Float64(z * (pi ^ 2.0))) + Float64(1.0 / z)) * Float64(sqrt(Float64(pi * 2.0)) * Float64(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(0.9999999999998099 + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(-1259.1392167224028 / Float64(2.0 - z)))) + Float64(Float64(Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) + 5.0))) + Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z)))) + Float64(Float64(771.3234287776531 / Float64(3.0 - z)) + Float64(-176.6150291621406 / Float64(4.0 - z))))))))
end

MATLAB

function tmp = code(z)
	tmp = ((0.16666666666666666 * (z * (pi ^ 2.0))) + (1.0 / z)) * (sqrt((pi * 2.0)) * (((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(((z + -1.0) - 6.5))) * ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + ((((9.984369578019572e-6 / (7.0 - z)) + (-0.13857109526572012 / ((1.0 - z) + 5.0))) + ((12.507343278686905 / (5.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + ((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z)))))));
end

Wolfram

code[z_] := N[(N[(N[(0.16666666666666666 * N[(z * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / z), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(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] * N[(N[(0.9999999999998099 + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

   1. associate-*l* 96.3%

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

3. Simplified 96.7%

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

   1. expm1-log1p-u 95.8%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right)}\right)\right) \]

   2. expm1-udef 95.8%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)} - 1\right)}\right)\right) \]

5. Applied egg-rr 95.8%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \left(-z\right)}\right)\right) + \left(\frac{771.3234287776531}{1 - \left(z - 2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z - 5\right)}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)} - 1\right)}\right)\right) \]
6. Step-by-step derivation

   1. expm1-def 95.8%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \left(-z\right)}\right)\right) + \left(\frac{771.3234287776531}{1 - \left(z - 2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z - 5\right)}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)}\right)\right) \]

   2. expm1-log1p 98.2%

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

   3. associate-+l+ 98.2%

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

   4. associate-+r+ 98.2%

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

   5. associate-+l+ 98.2%

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

7. Simplified 97.3%

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

8. Taylor expanded in z around 0 97.1%

   \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. *-un-lft-identity 97.1%

      \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right) + \left(\color{blue}{1 \cdot \left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)} + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

10. Applied egg-rr 97.1%

    \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right) + \left(\color{blue}{1 \cdot \left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)} + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]
11. Step-by-step derivation

    1. *-lft-identity 97.1%

       \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right) + \left(\color{blue}{\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)} + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

    2. +-commutative 97.1%

       \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{\color{blue}{2 + \left(1 - z\right)}}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

    3. associate-+r- 97.1%

       \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{\color{blue}{\left(2 + 1\right) - z}}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

    4. metadata-eval 97.1%

       \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{\color{blue}{3} - z}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

12. Simplified 97.1%

    \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right) + \left(\color{blue}{\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right)} + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]
13. Step-by-step derivation

    1. *-un-lft-identity 97.1%

       \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\color{blue}{1 \cdot \left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right)} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

    2. +-commutative 97.1%

       \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(1 \cdot \color{blue}{\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

14. Applied egg-rr 97.1%

    \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\color{blue}{1 \cdot \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]
15. Step-by-step derivation

    1. *-lft-identity 97.1%

       \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\color{blue}{\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

    2. associate-+l+ 98.1%

       \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\color{blue}{\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

16. Simplified 98.1%

    \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\color{blue}{\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

17. Final simplification 98.1%

    \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(z + -1\right) - 6.5}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{12.507343278686905}{5 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right)\right)\right) \]

Alternative 6: 97.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(z + -1\right) - 6.5}\right) \cdot \left(\left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{12.507343278686905}{5 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right) + \left(z \cdot 361.7355639412844 + 47.95075976068351\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (+ (* 0.16666666666666666 (* z (pow PI 2.0))) (/ 1.0 z))
  (*
   (sqrt (* PI 2.0))
   (*
    (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp (- (+ z -1.0) 6.5)))
    (+
     (+
      (+
       (+
        (/ 9.984369578019572e-6 (- 7.0 z))
        (/ -0.13857109526572012 (+ (- 1.0 z) 5.0)))
       (+
        (/ 12.507343278686905 (- 5.0 z))
        (/ 1.5056327351493116e-7 (- 8.0 z))))
      (+ (/ 771.3234287776531 (- 3.0 z)) (/ -176.6150291621406 (- 4.0 z))))
     (+ (* z 361.7355639412844) 47.95075976068351))))))

C

double code(double z) {
	return ((0.16666666666666666 * (z * pow(((double) M_PI), 2.0))) + (1.0 / z)) * (sqrt((((double) M_PI) * 2.0)) * ((pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(((z + -1.0) - 6.5))) * (((((9.984369578019572e-6 / (7.0 - z)) + (-0.13857109526572012 / ((1.0 - z) + 5.0))) + ((12.507343278686905 / (5.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + ((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z)))) + ((z * 361.7355639412844) + 47.95075976068351))));
}

Java

public static double code(double z) {
	return ((0.16666666666666666 * (z * Math.pow(Math.PI, 2.0))) + (1.0 / z)) * (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))) * (((((9.984369578019572e-6 / (7.0 - z)) + (-0.13857109526572012 / ((1.0 - z) + 5.0))) + ((12.507343278686905 / (5.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + ((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z)))) + ((z * 361.7355639412844) + 47.95075976068351))));
}

Python

def code(z):
	return ((0.16666666666666666 * (z * math.pow(math.pi, 2.0))) + (1.0 / z)) * (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))) * (((((9.984369578019572e-6 / (7.0 - z)) + (-0.13857109526572012 / ((1.0 - z) + 5.0))) + ((12.507343278686905 / (5.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + ((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z)))) + ((z * 361.7355639412844) + 47.95075976068351))))

Julia

function code(z)
	return Float64(Float64(Float64(0.16666666666666666 * Float64(z * (pi ^ 2.0))) + Float64(1.0 / z)) * Float64(sqrt(Float64(pi * 2.0)) * Float64(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(Float64(Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) + 5.0))) + Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z)))) + Float64(Float64(771.3234287776531 / Float64(3.0 - z)) + Float64(-176.6150291621406 / Float64(4.0 - z)))) + Float64(Float64(z * 361.7355639412844) + 47.95075976068351)))))
end

MATLAB

function tmp = code(z)
	tmp = ((0.16666666666666666 * (z * (pi ^ 2.0))) + (1.0 / z)) * (sqrt((pi * 2.0)) * (((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(((z + -1.0) - 6.5))) * (((((9.984369578019572e-6 / (7.0 - z)) + (-0.13857109526572012 / ((1.0 - z) + 5.0))) + ((12.507343278686905 / (5.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + ((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z)))) + ((z * 361.7355639412844) + 47.95075976068351))));
end

Wolfram

code[z_] := N[(N[(N[(0.16666666666666666 * N[(z * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / z), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(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] * N[(N[(N[(N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z * 361.7355639412844), $MachinePrecision] + 47.95075976068351), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

   1. associate-*l* 96.3%

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

3. Simplified 96.7%

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

   1. expm1-log1p-u 95.8%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right)}\right)\right) \]

   2. expm1-udef 95.8%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)} - 1\right)}\right)\right) \]

5. Applied egg-rr 95.8%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \left(-z\right)}\right)\right) + \left(\frac{771.3234287776531}{1 - \left(z - 2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z - 5\right)}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)} - 1\right)}\right)\right) \]
6. Step-by-step derivation

   1. expm1-def 95.8%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \left(-z\right)}\right)\right) + \left(\frac{771.3234287776531}{1 - \left(z - 2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z - 5\right)}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)}\right)\right) \]

   2. expm1-log1p 98.2%

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

   3. associate-+l+ 98.2%

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

   4. associate-+r+ 98.2%

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

   5. associate-+l+ 98.2%

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

7. Simplified 97.3%

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

8. Taylor expanded in z around 0 97.1%

   \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. *-un-lft-identity 97.1%

      \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right) + \left(\color{blue}{1 \cdot \left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)} + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

10. Applied egg-rr 97.1%

    \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right) + \left(\color{blue}{1 \cdot \left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)} + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]
11. Step-by-step derivation

    1. *-lft-identity 97.1%

       \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right) + \left(\color{blue}{\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)} + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

    2. +-commutative 97.1%

       \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{\color{blue}{2 + \left(1 - z\right)}}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

    3. associate-+r- 97.1%

       \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{\color{blue}{\left(2 + 1\right) - z}}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

    4. metadata-eval 97.1%

       \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{\color{blue}{3} - z}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

12. Simplified 97.1%

    \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right) + \left(\color{blue}{\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right)} + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

13. Taylor expanded in z around 0 97.5%

    \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\color{blue}{\left(47.95075976068351 + 361.7355639412844 \cdot z\right)} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]
14. Step-by-step derivation

    1. +-commutative 97.5%

       \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\color{blue}{\left(361.7355639412844 \cdot z + 47.95075976068351\right)} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

    2. *-commutative 97.5%

       \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\left(\color{blue}{z \cdot 361.7355639412844} + 47.95075976068351\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

15. Simplified 97.5%

    \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\color{blue}{\left(z \cdot 361.7355639412844 + 47.95075976068351\right)} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)\right) \]

16. Final simplification 97.5%

    \[\leadsto \left(0.16666666666666666 \cdot \left(z \cdot {\pi}^{2}\right) + \frac{1}{z}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(z + -1\right) - 6.5}\right) \cdot \left(\left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{12.507343278686905}{5 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right) + \left(z \cdot 361.7355639412844 + 47.95075976068351\right)\right)\right)\right) \]

Alternative 7: 96.6% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ (* (sqrt 2.0) (* (- z -1.0) (exp -7.5))) (/ z (pow (- 7.5 z) (- 0.5 z))))
  (* (sqrt PI) 263.3831869810514)))

C

double code(double z) {
	return ((sqrt(2.0) * ((z - -1.0) * exp(-7.5))) / (z / pow((7.5 - z), (0.5 - z)))) * (sqrt(((double) M_PI)) * 263.3831869810514);
}

Java

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

Python

def code(z):
	return ((math.sqrt(2.0) * ((z - -1.0) * math.exp(-7.5))) / (z / math.pow((7.5 - z), (0.5 - z)))) * (math.sqrt(math.pi) * 263.3831869810514)

Julia

function code(z)
	return Float64(Float64(Float64(sqrt(2.0) * Float64(Float64(z - -1.0) * exp(-7.5))) / Float64(z / (Float64(7.5 - z) ^ Float64(0.5 - z)))) * Float64(sqrt(pi) * 263.3831869810514))
end

MATLAB

function tmp = code(z)
	tmp = ((sqrt(2.0) * ((z - -1.0) * exp(-7.5))) / (z / ((7.5 - z) ^ (0.5 - z)))) * (sqrt(pi) * 263.3831869810514);
end

Wolfram

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

Derivation

1. Initial program 96.2%

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

   1. associate-*l* 96.3%

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

3. Simplified 96.7%

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

4. Taylor expanded in z around 0 95.4%

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

5. Taylor expanded in z around 0 95.4%

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

6. Taylor expanded in z around inf 95.9%

   \[\leadsto \color{blue}{263.3831869810514 \cdot \left(\frac{\sqrt{2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z - 7.5}\right)}{z} \cdot \sqrt{\pi}\right)} \]
7. Step-by-step derivation

   1. *-commutative 95.9%

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

   2. associate-*l* 96.0%

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

8. Simplified 96.2%

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

9. Taylor expanded in z around 0 97.4%

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

    1. distribute-lft-out 97.4%

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

    2. *-commutative 97.4%

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

    3. distribute-lft1-in 97.4%

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

11. Simplified 97.4%

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

12. Final simplification 97.4%

    \[\leadsto \frac{\sqrt{2} \cdot \left(\left(z - -1\right) \cdot e^{-7.5}\right)}{\frac{z}{{\left(7.5 - z\right)}^{\left(0.5 - z\right)}}} \cdot \left(\sqrt{\pi} \cdot 263.3831869810514\right) \]

Alternative 8: 96.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

   1. associate-*l* 96.3%

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

3. Simplified 96.7%

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

4. Taylor expanded in z around 0 95.4%

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

5. Taylor expanded in z around 0 95.4%

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

6. Taylor expanded in z around 0 95.5%

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

   1. *-commutative 95.5%

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

8. Simplified 95.5%

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

   1. sqrt-prod 96.6%

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

10. Applied egg-rr 96.6%

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

11. Final simplification 96.6%

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

Alternative 9: 96.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

   1. associate-*l* 96.3%

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

3. Simplified 96.7%

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

4. Taylor expanded in z around 0 95.4%

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

5. Taylor expanded in z around 0 95.4%

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

6. Taylor expanded in z around 0 96.0%

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

   1. associate-*r* 96.0%

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

   2. associate-/l* 96.4%

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

8. Simplified 96.4%

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

9. Final simplification 96.4%

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

Alternative 10: 96.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

   1. associate-*l* 96.3%

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

3. Simplified 96.7%

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

4. Taylor expanded in z around 0 95.4%

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

5. Taylor expanded in z around 0 95.4%

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

6. Taylor expanded in z around inf 95.9%

   \[\leadsto \color{blue}{263.3831869810514 \cdot \left(\frac{\sqrt{2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z - 7.5}\right)}{z} \cdot \sqrt{\pi}\right)} \]
7. Step-by-step derivation

   1. *-commutative 95.9%

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

   2. associate-*l* 96.0%

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

8. Simplified 96.2%

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

9. Taylor expanded in z around 0 96.2%

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

    1. associate-/l* 96.5%

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

    2. associate-/r/ 96.5%

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

11. Simplified 96.5%

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

12. Final simplification 96.5%

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

Alternative 11: 96.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double z) {
	return (1.0 / z) * (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))) * ((((-176.6150291621406 / (4.0 - z)) + (771.3234287776531 / (2.0 + (1.0 - z)))) + (((9.984369578019572e-6 / (7.0 - z)) + (-0.13857109526572012 / ((1.0 - z) + 5.0))) + ((12.507343278686905 / (5.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))) + ((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + (-1259.1392167224028 / (2.0 - z))))));
}

Python

def code(z):
	return (1.0 / z) * (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))) * ((((-176.6150291621406 / (4.0 - z)) + (771.3234287776531 / (2.0 + (1.0 - z)))) + (((9.984369578019572e-6 / (7.0 - z)) + (-0.13857109526572012 / ((1.0 - z) + 5.0))) + ((12.507343278686905 / (5.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))) + ((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + (-1259.1392167224028 / (2.0 - z))))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

   1. associate-*l* 96.3%

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

3. Simplified 96.7%

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

   1. expm1-log1p-u 95.8%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right)}\right)\right) \]

   2. expm1-udef 95.8%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)} - 1\right)}\right)\right) \]

5. Applied egg-rr 95.8%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \left(-z\right)}\right)\right) + \left(\frac{771.3234287776531}{1 - \left(z - 2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z - 5\right)}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)} - 1\right)}\right)\right) \]
6. Step-by-step derivation

   1. expm1-def 95.8%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 + \left(-z\right)}\right)\right) + \left(\frac{771.3234287776531}{1 - \left(z - 2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z - 5\right)}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)}\right)\right) \]

   2. expm1-log1p 98.2%

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

   3. associate-+l+ 98.2%

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

   4. associate-+r+ 98.2%

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

   5. associate-+l+ 98.2%

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

7. Simplified 97.3%

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

8. Taylor expanded in z around 0 96.4%

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

9. Final simplification 96.4%

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

Alternative 12: 96.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\pi \cdot 2} \cdot \left(\frac{1}{z} \cdot \left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{2 - z} + 215.45552095775327\right)\right)\right) \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (sqrt (* PI 2.0))
  (*
   (/ 1.0 z)
   (*
    (+
     (+
      (/ -0.13857109526572012 (- 6.0 z))
      (+
       (/ 9.984369578019572e-6 (- 7.0 z))
       (/ 1.5056327351493116e-7 (- 8.0 z))))
     (+
      (+ 0.9999999999998099 (/ 676.5203681218851 (- 1.0 z)))
      (+ (/ -1259.1392167224028 (- 2.0 z)) 215.45552095775327)))
    (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5)))))))

C

double code(double z) {
	return sqrt((((double) M_PI) * 2.0)) * ((1.0 / z) * ((((-0.13857109526572012 / (6.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + ((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + ((-1259.1392167224028 / (2.0 - z)) + 215.45552095775327))) * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5)))));
}

Java

public static double code(double z) {
	return Math.sqrt((Math.PI * 2.0)) * ((1.0 / z) * ((((-0.13857109526572012 / (6.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + ((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + ((-1259.1392167224028 / (2.0 - z)) + 215.45552095775327))) * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5)))));
}

Python

def code(z):
	return math.sqrt((math.pi * 2.0)) * ((1.0 / z) * ((((-0.13857109526572012 / (6.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + ((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + ((-1259.1392167224028 / (2.0 - z)) + 215.45552095775327))) * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5)))))

Julia

function code(z)
	return Float64(sqrt(Float64(pi * 2.0)) * Float64(Float64(1.0 / z) * Float64(Float64(Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z)))) + Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 - z))) + Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) + 215.45552095775327))) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5))))))
end

MATLAB

function tmp = code(z)
	tmp = sqrt((pi * 2.0)) * ((1.0 / z) * ((((-0.13857109526572012 / (6.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + ((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + ((-1259.1392167224028 / (2.0 - z)) + 215.45552095775327))) * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5)))));
end

Wolfram

code[z_] := N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / z), $MachinePrecision] * N[(N[(N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] + 215.45552095775327), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

   1. *-commutative 96.2%

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

3. Simplified 97.9%

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

4. Taylor expanded in z around 0 96.2%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{2 - z} + \color{blue}{215.45552095775327}\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 - \left(-z\right)}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

5. Taylor expanded in z around 0 96.3%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{2 - z} + 215.45552095775327\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 - \left(-z\right)}\right)\right) \cdot \color{blue}{\frac{1}{z}}\right) \]

6. Final simplification 96.3%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\frac{1}{z} \cdot \left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{2 - z} + 215.45552095775327\right)\right)\right) \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right)\right) \]

Alternative 13: 96.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{\pi \cdot 2}}{z} \cdot \left(263.3831869810514 \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-6.5 + \left(z + -1\right)}\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ (sqrt (* PI 2.0)) z)
  (*
   263.3831869810514
   (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ -6.5 (+ z -1.0)))))))

C

double code(double z) {
	return (sqrt((((double) M_PI) * 2.0)) / z) * (263.3831869810514 * (pow((7.5 - z), (0.5 - z)) * exp((-6.5 + (z + -1.0)))));
}

Java

public static double code(double z) {
	return (Math.sqrt((Math.PI * 2.0)) / z) * (263.3831869810514 * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((-6.5 + (z + -1.0)))));
}

Python

def code(z):
	return (math.sqrt((math.pi * 2.0)) / z) * (263.3831869810514 * (math.pow((7.5 - z), (0.5 - z)) * math.exp((-6.5 + (z + -1.0)))))

Julia

function code(z)
	return Float64(Float64(sqrt(Float64(pi * 2.0)) / z) * Float64(263.3831869810514 * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(-6.5 + Float64(z + -1.0))))))
end

MATLAB

function tmp = code(z)
	tmp = (sqrt((pi * 2.0)) / z) * (263.3831869810514 * (((7.5 - z) ^ (0.5 - z)) * exp((-6.5 + (z + -1.0)))));
end

Wolfram

code[z_] := N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] / z), $MachinePrecision] * N[(263.3831869810514 * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-6.5 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

   1. associate-*l* 96.3%

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

3. Simplified 96.7%

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

4. Taylor expanded in z around 0 95.4%

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

5. Taylor expanded in z around 0 95.4%

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

   1. associate-*l/ 95.4%

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

   2. *-un-lft-identity 95.4%

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

   3. associate-*l* 95.4%

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

   4. distribute-neg-in 95.4%

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

   5. metadata-eval 95.4%

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

7. Applied egg-rr 95.4%

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

   1. associate-/l* 96.2%

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

   2. associate-/r/ 96.3%

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

   3. *-commutative 96.3%

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

   4. associate-*r* 96.3%

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

   5. *-commutative 96.3%

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

   6. +-commutative 96.3%

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

   7. associate-+r- 96.3%

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

   8. metadata-eval 96.3%

      \[\leadsto \frac{\sqrt{2 \cdot \pi}}{z} \cdot \left(263.3831869810514 \cdot \left({\left(\color{blue}{7.5} - z\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right) \]

   9. +-commutative 96.3%

      \[\leadsto \frac{\sqrt{2 \cdot \pi}}{z} \cdot \left(263.3831869810514 \cdot \left({\left(7.5 - z\right)}^{\color{blue}{\left(-0.5 + \left(1 - z\right)\right)}} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right) \]

   10. associate-+r- 96.3%

       \[\leadsto \frac{\sqrt{2 \cdot \pi}}{z} \cdot \left(263.3831869810514 \cdot \left({\left(7.5 - z\right)}^{\color{blue}{\left(\left(-0.5 + 1\right) - z\right)}} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right) \]

   11. metadata-eval 96.3%

       \[\leadsto \frac{\sqrt{2 \cdot \pi}}{z} \cdot \left(263.3831869810514 \cdot \left({\left(7.5 - z\right)}^{\left(\color{blue}{0.5} - z\right)} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right) \]

   12. neg-sub0 96.3%

       \[\leadsto \frac{\sqrt{2 \cdot \pi}}{z} \cdot \left(263.3831869810514 \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\color{blue}{\left(0 - \left(1 - z\right)\right)} + -6.5}\right)\right) \]

   13. associate--r- 96.3%

       \[\leadsto \frac{\sqrt{2 \cdot \pi}}{z} \cdot \left(263.3831869810514 \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\color{blue}{\left(\left(0 - 1\right) + z\right)} + -6.5}\right)\right) \]

   14. metadata-eval 96.3%

       \[\leadsto \frac{\sqrt{2 \cdot \pi}}{z} \cdot \left(263.3831869810514 \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(\color{blue}{-1} + z\right) + -6.5}\right)\right) \]

9. Simplified 96.3%

   \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \pi}}{z} \cdot \left(263.3831869810514 \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right)} \]

10. Final simplification 96.3%

    \[\leadsto \frac{\sqrt{\pi \cdot 2}}{z} \cdot \left(263.3831869810514 \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-6.5 + \left(z + -1\right)}\right)\right) \]

Alternative 14: 95.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

   1. associate-*l* 96.3%

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

3. Simplified 96.7%

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

4. Taylor expanded in z around 0 95.4%

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

5. Taylor expanded in z around 0 95.4%

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

   1. associate-*l/ 95.4%

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

   2. *-un-lft-identity 95.4%

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

   3. associate-*l* 95.4%

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

   4. distribute-neg-in 95.4%

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

   5. metadata-eval 95.4%

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

7. Applied egg-rr 95.4%

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

   1. associate-/l* 96.2%

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

   2. *-commutative 96.2%

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

   3. associate-/r* 95.7%

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

   4. +-commutative 95.7%

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

   5. associate-+r- 95.7%

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

   6. metadata-eval 95.7%

      \[\leadsto \frac{\sqrt{2 \cdot \pi}}{\frac{\frac{z}{{\left(\color{blue}{7.5} - z\right)}^{\left(\left(1 - z\right) + -0.5\right)}}}{e^{\left(-\left(1 - z\right)\right) + -6.5} \cdot 263.3831869810514}} \]

   7. +-commutative 95.7%

      \[\leadsto \frac{\sqrt{2 \cdot \pi}}{\frac{\frac{z}{{\left(7.5 - z\right)}^{\color{blue}{\left(-0.5 + \left(1 - z\right)\right)}}}}{e^{\left(-\left(1 - z\right)\right) + -6.5} \cdot 263.3831869810514}} \]

   8. associate-+r- 95.7%

      \[\leadsto \frac{\sqrt{2 \cdot \pi}}{\frac{\frac{z}{{\left(7.5 - z\right)}^{\color{blue}{\left(\left(-0.5 + 1\right) - z\right)}}}}{e^{\left(-\left(1 - z\right)\right) + -6.5} \cdot 263.3831869810514}} \]

   9. metadata-eval 95.7%

      \[\leadsto \frac{\sqrt{2 \cdot \pi}}{\frac{\frac{z}{{\left(7.5 - z\right)}^{\left(\color{blue}{0.5} - z\right)}}}{e^{\left(-\left(1 - z\right)\right) + -6.5} \cdot 263.3831869810514}} \]

   10. +-commutative 95.7%

       \[\leadsto \frac{\sqrt{2 \cdot \pi}}{\frac{\frac{z}{{\left(7.5 - z\right)}^{\left(0.5 - z\right)}}}{e^{\color{blue}{-6.5 + \left(-\left(1 - z\right)\right)}} \cdot 263.3831869810514}} \]

   11. neg-sub0 95.7%

       \[\leadsto \frac{\sqrt{2 \cdot \pi}}{\frac{\frac{z}{{\left(7.5 - z\right)}^{\left(0.5 - z\right)}}}{e^{-6.5 + \color{blue}{\left(0 - \left(1 - z\right)\right)}} \cdot 263.3831869810514}} \]

   12. associate--r- 95.7%

       \[\leadsto \frac{\sqrt{2 \cdot \pi}}{\frac{\frac{z}{{\left(7.5 - z\right)}^{\left(0.5 - z\right)}}}{e^{-6.5 + \color{blue}{\left(\left(0 - 1\right) + z\right)}} \cdot 263.3831869810514}} \]

   13. metadata-eval 95.7%

       \[\leadsto \frac{\sqrt{2 \cdot \pi}}{\frac{\frac{z}{{\left(7.5 - z\right)}^{\left(0.5 - z\right)}}}{e^{-6.5 + \left(\color{blue}{-1} + z\right)} \cdot 263.3831869810514}} \]

   14. associate-+r+ 95.7%

       \[\leadsto \frac{\sqrt{2 \cdot \pi}}{\frac{\frac{z}{{\left(7.5 - z\right)}^{\left(0.5 - z\right)}}}{e^{\color{blue}{\left(-6.5 + -1\right) + z}} \cdot 263.3831869810514}} \]

   15. metadata-eval 95.7%

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

9. Simplified 95.7%

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

10. Final simplification 95.7%

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

Alternative 15: 95.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

   1. associate-*l* 96.3%

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

3. Simplified 96.7%

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

4. Taylor expanded in z around 0 95.4%

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

5. Taylor expanded in z around 0 95.4%

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

6. Taylor expanded in z around 0 95.5%

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

   1. *-commutative 95.5%

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

8. Simplified 95.5%

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

   1. associate-*l/ 95.5%

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

   2. *-un-lft-identity 95.5%

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

   3. *-commutative 95.5%

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

   4. associate-*l* 95.5%

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

10. Applied egg-rr 95.5%

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

    1. associate-/l* 95.3%

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

    2. associate-/r/ 95.3%

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

    3. *-commutative 95.3%

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

12. Simplified 95.3%

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

13. Final simplification 95.3%

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

Alternative 16: 95.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

   1. associate-*l* 96.3%

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

3. Simplified 96.7%

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

4. Taylor expanded in z around 0 95.4%

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

5. Taylor expanded in z around 0 95.4%

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

6. Taylor expanded in z around 0 95.5%

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

   1. *-commutative 95.5%

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

8. Simplified 95.5%

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

   1. associate-*l/ 95.5%

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

   2. *-un-lft-identity 95.5%

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

   3. *-commutative 95.5%

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

   4. associate-*l* 95.5%

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

10. Applied egg-rr 95.5%

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

    1. associate-/l* 95.3%

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

    2. *-commutative 95.3%

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

12. Simplified 95.3%

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

13. Final simplification 95.3%

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

Alternative 17: 95.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

   1. associate-*l* 96.3%

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

3. Simplified 96.7%

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

4. Taylor expanded in z around 0 95.4%

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

5. Taylor expanded in z around 0 95.4%

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

6. Taylor expanded in z around 0 95.5%

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

   1. *-commutative 95.5%

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

8. Simplified 95.5%

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

   1. associate-*l/ 95.5%

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

   2. *-un-lft-identity 95.5%

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

   3. *-commutative 95.5%

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

   4. associate-*l* 95.5%

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

10. Applied egg-rr 95.5%

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

11. Final simplification 95.5%

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

Reproduce

herbie shell --seed 2023172 
(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))))))