Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.6% → 99.1%

Time: 1.4min

Alternatives: 9

Speedup: 1.1×

• could not determine a ground truth

  1. z = -4.1142221152577396e+273

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: 96.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

3. Simplified 98.3%

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

   1. add-exp-log 98.4%

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

   2. *-commutative 98.4%

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

   3. log-prod 98.4%

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

   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. Final simplification 99.2%

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

Alternative 2: 99.1% accurate, 0.9× speedup

Math

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

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)))))
    (exp (+ -7.5 (fma (- 0.5 z) (log (- 7.5 z)) z)))))))

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))))) * exp((-7.5 + fma((0.5 - z), log((7.5 - z)), z)))));
}

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

Derivation

1. Initial program 96.6%

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

3. Simplified 98.3%

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

   1. add-exp-log 98.4%

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

   2. *-commutative 98.4%

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

   3. log-prod 98.4%

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

   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. exp-diff 99.1%

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

   2. sub-neg 99.1%

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

   3. +-commutative 99.1%

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

   4. neg-mul-1 99.1%

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

   5. *-commutative 99.1%

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

   6. remove-double-neg 99.1%

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

   7. sub-neg 99.1%

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

   8. exp-diff 99.2%

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

   9. sub-neg 99.2%

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

8. Simplified 99.2%

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

9. Final simplification 99.2%

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

Alternative 3: 98.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ PI (sin (* PI z)))
  (*
   (*
    (sqrt (* PI 2.0))
    (*
     (pow (+ 7.5 (+ (- 1.0 z) -1.0)) (- (- 1.0 z) 0.5))
     (exp (+ (+ -6.0 (+ z -1.0)) -0.5))))
   (+
    (+
     (+
      (+
       (+ 0.9999999999998099 (/ 676.5203681218851 (- 1.0 z)))
       (/ -1259.1392167224028 (- (- 1.0 z) -1.0)))
      (+
       (/ 771.3234287776531 (- (- 1.0 z) -2.0))
       (/ -176.6150291621406 (- (- 1.0 z) -3.0))))
     (+
      (/ 12.507343278686905 (- (- 1.0 z) -4.0))
      (/ -0.13857109526572012 (- (- 1.0 z) -5.0))))
    (+
     (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0))
     (/ 1.5056327351493116e-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 + ((1.0 - z) + -1.0)), ((1.0 - z) - 0.5)) * exp(((-6.0 + (z + -1.0)) + -0.5)))) * (((((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) + (-176.6150291621406 / ((1.0 - z) - -3.0)))) + ((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0)))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-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 + ((1.0 - z) + -1.0)), ((1.0 - z) - 0.5)) * Math.exp(((-6.0 + (z + -1.0)) + -0.5)))) * (((((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) + (-176.6150291621406 / ((1.0 - z) - -3.0)))) + ((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0)))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-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 + ((1.0 - z) + -1.0)), ((1.0 - z) - 0.5)) * math.exp(((-6.0 + (z + -1.0)) + -0.5)))) * (((((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) + (-176.6150291621406 / ((1.0 - z) - -3.0)))) + ((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0)))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-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 + Float64(Float64(1.0 - z) + -1.0)) ^ Float64(Float64(1.0 - z) - 0.5)) * exp(Float64(Float64(-6.0 + Float64(z + -1.0)) + -0.5)))) * Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 - z))) + Float64(-1259.1392167224028 / Float64(Float64(1.0 - z) - -1.0))) + Float64(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 + ((1.0 - z) + -1.0)) ^ ((1.0 - z) - 0.5)) * exp(((-6.0 + (z + -1.0)) + -0.5)))) * (((((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) + (-176.6150291621406 / ((1.0 - z) - -3.0)))) + ((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0)))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-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 + N[(N[(1.0 - z), $MachinePrecision] + -1.0), $MachinePrecision]), $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[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(N[(1.0 - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(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 96.6%

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

3. Simplified 98.5%

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

4. Final simplification 98.5%

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

Alternative 4: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(z)
	return Float64(sqrt(Float64(pi * 2.0)) * Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5))) * Float64(Float64(Float64(771.3234287776531 / Float64(3.0 - z)) + Float64(Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) + Float64(12.507343278686905 / Float64(5.0 - z))) + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(-1259.1392167224028 / Float64(2.0 - z))))) + Float64(0.9999999999998099 + 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)))))))))
end

MATLAB

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

Derivation

1. Initial program 96.6%

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

3. Simplified 98.3%

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

   1. expm1-log1p-u 98.4%

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{2 - z} + \left(\frac{771.3234287776531}{3 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)\right)} + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 - \left(-z\right)}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   2. expm1-udef 98.4%

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\color{blue}{\left(e^{\mathsf{log1p}\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{2 - z} + \left(\frac{771.3234287776531}{3 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right)} - 1\right)} + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 - \left(-z\right)}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

5. Applied egg-rr 98.4%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\color{blue}{\left(e^{\mathsf{log1p}\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right) + \left(\frac{12.507343278686905}{5 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right)\right)} - 1\right)} + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 - \left(-z\right)}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
6. Step-by-step derivation

   1. expm1-def 98.4%

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right) + \left(\frac{12.507343278686905}{5 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right)\right)\right)} + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 - \left(-z\right)}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   2. expm1-log1p 96.7%

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

   3. associate-+l+ 98.4%

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

   4. associate-+r+ 98.4%

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

7. Simplified 98.4%

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

   1. expm1-log1p-u 98.4%

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \left(\left(\frac{771.3234287776531}{3 - z} + \frac{12.507343278686905}{5 - z}\right) + \frac{-176.6150291621406}{4 - z}\right)\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 - \left(-z\right)}\right)\right)\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   2. expm1-udef 97.3%

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \left(\left(\frac{771.3234287776531}{3 - z} + \frac{12.507343278686905}{5 - z}\right) + \frac{-176.6150291621406}{4 - z}\right)\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 - \left(-z\right)}\right)\right)} - 1\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

9. Applied egg-rr 97.3%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\left(0.9999999999998099 + \left(\left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \left(\frac{12.507343278686905}{5 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 - \left(-z\right)}\right)\right)} - 1\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

11. Simplified 98.4%

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

12. Final simplification 98.4%

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

Alternative 5: 97.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\pi \cdot 2} \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) + \left(263.4062807184368 + z \cdot 436.9000215473151\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (sqrt (* PI 2.0))
  (*
   (/ PI (sin (* PI z)))
   (*
    (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5)))
    (+
     (+
      (/ -0.13857109526572012 (- 6.0 z))
      (+
       (/ 9.984369578019572e-6 (- 7.0 z))
       (/ 1.5056327351493116e-7 (- 8.0 z))))
     (+ 263.4062807184368 (* z 436.9000215473151)))))))

C

double code(double z) {
	return sqrt((((double) M_PI) * 2.0)) * ((((double) M_PI) / sin((((double) M_PI) * z))) * ((pow((7.5 - z), (0.5 - z)) * exp((z + -7.5))) * (((-0.13857109526572012 / (6.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + (263.4062807184368 + (z * 436.9000215473151)))));
}

Java

public static double code(double z) {
	return Math.sqrt((Math.PI * 2.0)) * ((Math.PI / Math.sin((Math.PI * z))) * ((Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5))) * (((-0.13857109526572012 / (6.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + (263.4062807184368 + (z * 436.9000215473151)))));
}

Python

def code(z):
	return math.sqrt((math.pi * 2.0)) * ((math.pi / math.sin((math.pi * z))) * ((math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5))) * (((-0.13857109526572012 / (6.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + (263.4062807184368 + (z * 436.9000215473151)))))

Julia

function code(z)
	return Float64(sqrt(Float64(pi * 2.0)) * Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5))) * Float64(Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z)))) + Float64(263.4062807184368 + Float64(z * 436.9000215473151))))))
end

MATLAB

function tmp = code(z)
	tmp = sqrt((pi * 2.0)) * ((pi / sin((pi * z))) * ((((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5))) * (((-0.13857109526572012 / (6.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z)))) + (263.4062807184368 + (z * 436.9000215473151)))));
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[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(263.4062807184368 + N[(z * 436.9000215473151), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

3. Simplified 98.3%

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

   1. add-exp-log 98.4%

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

   2. *-commutative 98.4%

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

   3. log-prod 98.4%

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

   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. sub-neg 99.2%

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

   3. +-commutative 99.2%

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

   4. neg-mul-1 99.2%

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

   5. *-commutative 99.2%

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

   6. prod-exp 98.3%

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

   7. fma-def 98.3%

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

   8. exp-to-pow 98.3%

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

   9. neg-mul-1 98.3%

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

   10. +-commutative 98.3%

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

   11. sub-neg 98.3%

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

   12. neg-mul-1 98.3%

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

   13. +-commutative 98.3%

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

   14. sub-neg 98.3%

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

   15. sub-neg 98.3%

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

   16. metadata-eval 98.3%

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

8. Simplified 98.3%

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

9. Taylor expanded in z around 0 97.9%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\color{blue}{\left(263.4062807184368 + 436.9000215473151 \cdot z\right)} + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
10. Step-by-step derivation

    1. *-commutative 97.9%

       \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\left(263.4062807184368 + \color{blue}{z \cdot 436.9000215473151}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

11. Simplified 97.9%

    \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\color{blue}{\left(263.4062807184368 + z \cdot 436.9000215473151\right)} + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

12. Final simplification 97.9%

    \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) + \left(263.4062807184368 + z \cdot 436.9000215473151\right)\right)\right)\right) \]

Alternative 6: 96.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

3. Simplified 98.3%

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

4. Taylor expanded in z around 0 97.1%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\color{blue}{263.4062807184368} + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 - \left(-z\right)}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

5. Taylor expanded in z around 0 97.1%

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

   1. expm1-log1p-u 48.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi \cdot 2} \cdot \left(\left(263.3831869810514 \cdot \left(e^{-7.5} \cdot \sqrt{7.5}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right)\right)} \]

   2. expm1-udef 48.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\pi \cdot 2} \cdot \left(\left(263.3831869810514 \cdot \left(e^{-7.5} \cdot \sqrt{7.5}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right)} - 1} \]

   3. associate-*r/ 48.9%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\pi \cdot 2} \cdot \color{blue}{\frac{\left(263.3831869810514 \cdot \left(e^{-7.5} \cdot \sqrt{7.5}\right)\right) \cdot \pi}{\sin \left(\pi \cdot z\right)}}\right)} - 1 \]

   4. associate-*r* 48.9%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\pi \cdot 2} \cdot \frac{\color{blue}{\left(\left(263.3831869810514 \cdot e^{-7.5}\right) \cdot \sqrt{7.5}\right)} \cdot \pi}{\sin \left(\pi \cdot z\right)}\right)} - 1 \]

   5. *-commutative 48.9%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\pi \cdot 2} \cdot \frac{\left(\left(263.3831869810514 \cdot e^{-7.5}\right) \cdot \sqrt{7.5}\right) \cdot \pi}{\sin \color{blue}{\left(z \cdot \pi\right)}}\right)} - 1 \]

7. Applied egg-rr 48.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\pi \cdot 2} \cdot \frac{\left(\left(263.3831869810514 \cdot e^{-7.5}\right) \cdot \sqrt{7.5}\right) \cdot \pi}{\sin \left(z \cdot \pi\right)}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 48.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi \cdot 2} \cdot \frac{\left(\left(263.3831869810514 \cdot e^{-7.5}\right) \cdot \sqrt{7.5}\right) \cdot \pi}{\sin \left(z \cdot \pi\right)}\right)\right)} \]

   2. expm1-log1p 97.5%

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

   3. *-commutative 97.5%

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

   4. *-commutative 97.5%

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

   5. associate-*r* 97.5%

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

9. Simplified 97.5%

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

10. Final simplification 97.5%

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

Alternative 7: 96.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

3. Simplified 98.3%

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

4. Taylor expanded in z around 0 97.1%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\color{blue}{263.4062807184368} + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 - \left(-z\right)}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

5. Taylor expanded in z around 0 97.1%

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

6. Taylor expanded in z around 0 97.0%

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

   1. associate-/l* 97.3%

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

8. Simplified 97.3%

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

9. Taylor expanded in z around 0 97.0%

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

    1. associate-*l/ 97.4%

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

    2. *-commutative 97.4%

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

    3. associate-*l* 97.3%

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

11. Simplified 97.3%

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

12. Final simplification 97.3%

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

Alternative 8: 96.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

3. Simplified 98.3%

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

4. Taylor expanded in z around 0 97.1%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\color{blue}{263.4062807184368} + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 - \left(-z\right)}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

5. Taylor expanded in z around 0 97.1%

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

6. Taylor expanded in z around 0 97.0%

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

   1. associate-/l* 97.3%

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

8. Simplified 97.3%

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

   1. associate-/r/ 97.4%

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

10. Applied egg-rr 97.4%

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

11. Final simplification 97.4%

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

Alternative 9: 96.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

3. Simplified 98.3%

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

4. Taylor expanded in z around 0 97.1%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(\left(\color{blue}{263.4062807184368} + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 - \left(-z\right)}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

5. Taylor expanded in z around 0 97.1%

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

6. Taylor expanded in z around 0 97.0%

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

   1. *-commutative 97.0%

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

   2. associate-*l* 97.1%

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

   3. associate-/l* 97.4%

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

8. Simplified 97.4%

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

9. Final simplification 97.4%

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

Reproduce

herbie shell --seed 2023193 
(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))))))