Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.3% → 99.2%

Time: 1.5min

Alternatives: 7

Speedup: 1.8×

• could not determine a ground truth

  1. z = -5.445032166697791e+201

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: 96.3% 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: 99.2% accurate, 0.9× speedup

Math

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

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 (* (- 0.5 z) (log (fma -1.0 z 7.5)))) 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 + ((0.5 - z) * log(fma(-1.0, z, 7.5)))) - 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 + Float64(Float64(0.5 - z) * log(fma(-1.0, z, 7.5)))) - 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 + N[(N[(0.5 - z), $MachinePrecision] * N[Log[N[(-1.0 * z + 7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.1%

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

3. Simplified 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 98.0%

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

   2. *-commutative 98.0%

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

   3. log-prod 98.0%

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

   4. add-log-exp 99.2%

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

   5. log-pow 99.2%

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

   6. neg-mul-1 99.2%

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

   7. fma-def 99.2%

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

5. Applied egg-rr 99.2%

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

6. Taylor expanded in z around inf 99.2%

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

7. Final simplification 99.2%

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

Alternative 2: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.1%

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

3. Simplified 98.1%

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

5. Applied egg-rr 98.1%

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

   1. expm1-def 98.1%

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

   2. expm1-log1p 98.1%

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

   3. +-commutative 98.1%

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

   4. associate-+l+ 98.1%

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

   5. unsub-neg 98.1%

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

7. Simplified 98.1%

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

8. Taylor expanded in z around 0 98.1%

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

   1. neg-mul-1 98.1%

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

10. Simplified 98.1%

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

11. Final simplification 98.1%

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

Alternative 3: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

def code(z):
	return math.sqrt((math.pi * 2.0)) * ((math.pi / math.sin((math.pi * z))) * ((((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))))) * (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(pi / sin(Float64(pi * z))) * 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))))) * 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)) * ((pi / sin((pi * z))) * ((((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))))) * (((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[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 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[(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 96.1%

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

3. Simplified 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 98.0%

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

   2. *-commutative 98.0%

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

   3. log-prod 98.0%

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

   4. add-log-exp 99.2%

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

   5. log-pow 99.2%

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

   6. neg-mul-1 99.2%

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

   7. fma-def 99.2%

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

5. Applied egg-rr 99.2%

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

6. Taylor expanded in z around inf 99.2%

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

   1. associate--l+ 99.2%

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

   2. prod-exp 97.9%

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

   3. *-commutative 97.9%

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

   4. exp-prod 97.9%

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

   5. fma-def 97.9%

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

   6. rem-exp-log 97.9%

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

   7. neg-mul-1 97.9%

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

   8. +-commutative 97.9%

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

   9. sub-neg 97.9%

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

   10. sub-neg 97.9%

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

   11. metadata-eval 97.9%

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

8. Simplified 97.9%

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

9. Final simplification 97.9%

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

Alternative 4: 97.4% accurate, 1.4× speedup

Math

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

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 (+ (* (- 0.5 z) (log (fma -1.0 z 7.5))) (+ z -7.5))))
   (/ 1.0 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((((0.5 - z) * log(fma(-1.0, z, 7.5))) + (z + -7.5)))) * (1.0 / 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(Float64(0.5 - z) * log(fma(-1.0, z, 7.5))) + Float64(z + -7.5)))) * Float64(1.0 / 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[(N[(0.5 - z), $MachinePrecision] * N[Log[N[(-1.0 * z + 7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(z + -7.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.1%

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

3. Simplified 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 98.0%

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

   2. *-commutative 98.0%

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

   3. log-prod 98.0%

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

   4. add-log-exp 99.2%

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

   5. log-pow 99.2%

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

   6. neg-mul-1 99.2%

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

   7. fma-def 99.2%

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

5. Applied egg-rr 99.2%

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

6. Taylor expanded in z around 0 97.5%

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

7. Final simplification 97.5%

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

Alternative 5: 96.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{676.5203681218851}{1 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(2.4783734731930944 + z \cdot 0.49644453405676175\right)\right)\right)\right)\right) \cdot \left(\frac{1}{z} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{z + -7.5}\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (+
   (+
    (/ 676.5203681218851 (- 1.0 z))
    (+
     (/ -1259.1392167224028 (- 2.0 z))
     (+ 0.9999999999998099 (/ 771.3234287776531 (- 3.0 z)))))
   (+
    (/ -176.6150291621406 (- 4.0 z))
    (+
     (/ 9.984369578019572e-6 (- 7.0 z))
     (+
      (/ 1.5056327351493116e-7 (- 8.0 z))
      (+ 2.4783734731930944 (* z 0.49644453405676175))))))
  (*
   (/ 1.0 z)
   (* (pow (- 7.5 z) (- 0.5 z)) (* (sqrt (* PI 2.0)) (exp (+ z -7.5)))))))

C

double code(double z) {
	return (((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z))))) + ((-176.6150291621406 / (4.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + (2.4783734731930944 + (z * 0.49644453405676175)))))) * ((1.0 / z) * (pow((7.5 - z), (0.5 - z)) * (sqrt((((double) M_PI) * 2.0)) * exp((z + -7.5)))));
}

Java

public static double code(double z) {
	return (((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z))))) + ((-176.6150291621406 / (4.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + (2.4783734731930944 + (z * 0.49644453405676175)))))) * ((1.0 / z) * (Math.pow((7.5 - z), (0.5 - z)) * (Math.sqrt((Math.PI * 2.0)) * Math.exp((z + -7.5)))));
}

Python

def code(z):
	return (((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z))))) + ((-176.6150291621406 / (4.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + (2.4783734731930944 + (z * 0.49644453405676175)))))) * ((1.0 / z) * (math.pow((7.5 - z), (0.5 - z)) * (math.sqrt((math.pi * 2.0)) * math.exp((z + -7.5)))))

Julia

function code(z)
	return Float64(Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) + Float64(0.9999999999998099 + Float64(771.3234287776531 / Float64(3.0 - z))))) + Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) + Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) + Float64(2.4783734731930944 + Float64(z * 0.49644453405676175)))))) * Float64(Float64(1.0 / z) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * Float64(sqrt(Float64(pi * 2.0)) * exp(Float64(z + -7.5))))))
end

MATLAB

function tmp = code(z)
	tmp = (((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z))))) + ((-176.6150291621406 / (4.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + (2.4783734731930944 + (z * 0.49644453405676175)))))) * ((1.0 / z) * (((7.5 - z) ^ (0.5 - z)) * (sqrt((pi * 2.0)) * exp((z + -7.5)))));
end

Wolfram

code[z_] := N[(N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] + N[(2.4783734731930944 + N[(z * 0.49644453405676175), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / z), $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.1%

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

3. Simplified 97.9%

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

4. Taylor expanded in z around 0 97.3%

   \[\leadsto \left(\left(\frac{676.5203681218851}{1 - z} + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(\frac{-176.6150291621406}{4 - z} + \left(\left(\color{blue}{\left(2.4783734731930944 + 0.49644453405676175 \cdot z\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{z + -7.5}\right)\right)\right) \]
5. Step-by-step derivation

   1. *-commutative 97.3%

      \[\leadsto \left(\left(\frac{676.5203681218851}{1 - z} + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(2.4783734731930944 + \color{blue}{z \cdot 0.49644453405676175}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{z + -7.5}\right)\right)\right) \]

6. Simplified 97.3%

   \[\leadsto \left(\left(\frac{676.5203681218851}{1 - z} + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(\frac{-176.6150291621406}{4 - z} + \left(\left(\color{blue}{\left(2.4783734731930944 + z \cdot 0.49644453405676175\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{z + -7.5}\right)\right)\right) \]

7. Taylor expanded in z around 0 97.0%

   \[\leadsto \left(\left(\frac{676.5203681218851}{1 - z} + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(\frac{-176.6150291621406}{4 - z} + \left(\left(\left(2.4783734731930944 + z \cdot 0.49644453405676175\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right)\right) \cdot \left(\color{blue}{\frac{1}{z}} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{z + -7.5}\right)\right)\right) \]

8. Final simplification 97.0%

   \[\leadsto \left(\left(\frac{676.5203681218851}{1 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(2.4783734731930944 + z \cdot 0.49644453405676175\right)\right)\right)\right)\right) \cdot \left(\frac{1}{z} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{z + -7.5}\right)\right)\right) \]

Alternative 6: 94.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.1%

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

3. Simplified 96.7%

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

4. Taylor expanded in z around 0 95.5%

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

   1. associate-*r* 95.5%

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

   2. *-commutative 95.5%

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

6. Simplified 95.5%

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

7. Taylor expanded in z around 0 95.4%

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

   1. add-log-exp 5.5%

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

   2. associate-*l/ 5.5%

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

   3. *-un-lft-identity 5.5%

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

9. Applied egg-rr 5.5%

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

    1. *-un-lft-identity 5.5%

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

    2. log-prod 5.5%

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

    3. metadata-eval 5.5%

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

    4. add-log-exp 95.4%

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

    5. associate-*r* 95.4%

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

    6. associate-/l* 95.2%

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

    7. *-commutative 95.2%

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

11. Applied egg-rr 95.2%

    \[\leadsto \color{blue}{0 + \frac{\sqrt{\pi \cdot 2}}{\frac{z}{\left(263.3831869810514 \cdot \sqrt{7.5}\right) \cdot e^{-7.5}}}} \]
12. Step-by-step derivation

    1. +-lft-identity 95.2%

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

    2. associate-/r/ 95.2%

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

    3. *-commutative 95.2%

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

    4. *-commutative 95.2%

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

    5. *-rgt-identity 95.2%

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

    6. associate-*r/ 95.3%

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

    7. associate-*l* 94.8%

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

    8. *-commutative 94.8%

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

    9. associate-*r/ 94.7%

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

    10. associate-/l* 94.7%

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

    11. *-commutative 94.7%

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

    12. /-rgt-identity 94.7%

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

13. Simplified 94.7%

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

14. Final simplification 94.7%

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

Alternative 7: 95.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{\pi \cdot 2}}{0.0037967495627271876 \cdot \frac{z}{e^{-7.5} \cdot \sqrt{7.5}}} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (/
  (sqrt (* PI 2.0))
  (* 0.0037967495627271876 (/ z (* (exp -7.5) (sqrt 7.5))))))

C

double code(double z) {
	return sqrt((((double) M_PI) * 2.0)) / (0.0037967495627271876 * (z / (exp(-7.5) * sqrt(7.5))));
}

Java

public static double code(double z) {
	return Math.sqrt((Math.PI * 2.0)) / (0.0037967495627271876 * (z / (Math.exp(-7.5) * Math.sqrt(7.5))));
}

Python

def code(z):
	return math.sqrt((math.pi * 2.0)) / (0.0037967495627271876 * (z / (math.exp(-7.5) * math.sqrt(7.5))))

Julia

function code(z)
	return Float64(sqrt(Float64(pi * 2.0)) / Float64(0.0037967495627271876 * Float64(z / Float64(exp(-7.5) * sqrt(7.5)))))
end

MATLAB

function tmp = code(z)
	tmp = sqrt((pi * 2.0)) / (0.0037967495627271876 * (z / (exp(-7.5) * sqrt(7.5))));
end

Wolfram

code[z_] := N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] / N[(0.0037967495627271876 * N[(z / N[(N[Exp[-7.5], $MachinePrecision] * N[Sqrt[7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.1%

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

3. Simplified 96.7%

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

4. Taylor expanded in z around 0 95.5%

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

   1. associate-*r* 95.5%

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

   2. *-commutative 95.5%

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

6. Simplified 95.5%

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

7. Taylor expanded in z around 0 95.4%

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

   1. associate-*l/ 95.4%

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

   2. *-un-lft-identity 95.4%

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

9. Applied egg-rr 95.4%

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

    1. associate-/l* 96.3%

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

    2. *-commutative 96.3%

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

    3. associate-*l* 95.2%

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

11. Simplified 95.2%

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

12. Taylor expanded in z around 0 96.5%

    \[\leadsto \frac{\sqrt{2 \cdot \pi}}{\color{blue}{0.0037967495627271876 \cdot \frac{z}{e^{-7.5} \cdot \sqrt{7.5}}}} \]

13. Final simplification 96.5%

    \[\leadsto \frac{\sqrt{\pi \cdot 2}}{0.0037967495627271876 \cdot \frac{z}{e^{-7.5} \cdot \sqrt{7.5}}} \]

Reproduce

herbie shell --seed 2023194 
(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))))))