Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.3% → 98.2%

Time: 1.5min

Alternatives: 11

Speedup: 1.1×

• could not determine a ground truth

  1. z = -4.260335033897821e+264

Specification

\[z \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ (- 1.0 z) -1.0)) (t_1 (+ t_0 7.0)))
   (*
    (/ PI (sin (* PI z)))
    (*
     (*
      (* (* (sqrt PI) (sqrt 2.0)) (pow (+ t_1 0.5) (+ t_0 0.5)))
      (exp (- (- (- (+ z -1.0) -1.0) 7.0) 0.5)))
     (+
      (+
       (+
        (+
         (+
          (+
           (+
            0.9999999999998099
            (+
             (/ 676.5203681218851 (- 1.0 z))
             (/ -1259.1392167224028 (- 2.0 z))))
           (/ 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;
	return (((double) M_PI) / sin((((double) M_PI) * z))) * ((((sqrt(((double) M_PI)) * sqrt(2.0)) * pow((t_1 + 0.5), (t_0 + 0.5))) * exp(((((z + -1.0) - -1.0) - 7.0) - 0.5))) * (((((((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (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;
	return (Math.PI / Math.sin((Math.PI * z))) * ((((Math.sqrt(Math.PI) * Math.sqrt(2.0)) * Math.pow((t_1 + 0.5), (t_0 + 0.5))) * Math.exp(((((z + -1.0) - -1.0) - 7.0) - 0.5))) * (((((((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (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
	return (math.pi / math.sin((math.pi * z))) * ((((math.sqrt(math.pi) * math.sqrt(2.0)) * math.pow((t_1 + 0.5), (t_0 + 0.5))) * math.exp(((((z + -1.0) - -1.0) - 7.0) - 0.5))) * (((((((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (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)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(Float64(sqrt(pi) * sqrt(2.0)) * (Float64(t_1 + 0.5) ^ Float64(t_0 + 0.5))) * exp(Float64(Float64(Float64(Float64(z + -1.0) - -1.0) - 7.0) - 0.5))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(-1259.1392167224028 / Float64(2.0 - z)))) + 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;
	tmp = (pi / sin((pi * z))) * ((((sqrt(pi) * sqrt(2.0)) * ((t_1 + 0.5) ^ (t_0 + 0.5))) * exp(((((z + -1.0) - -1.0) - 7.0) - 0.5))) * (((((((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (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]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(t$95$1 + 0.5), $MachinePrecision], N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(N[(N[(z + -1.0), $MachinePrecision] - -1.0), $MachinePrecision] - 7.0), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(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]]]

Derivation

1. Initial program 95.9%

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

   1. expm1-log1p-u 94.3%

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

   2. expm1-udef 94.3%

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

3. Applied egg-rr 94.3%

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

   1. expm1-def 94.3%

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

   2. expm1-log1p 95.9%

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

   3. +-commutative 95.9%

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

   4. associate-+l+ 97.0%

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

5. Simplified 97.0%

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

   1. sqrt-prod 97.7%

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

7. Applied egg-rr 97.7%

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

8. Final simplification 97.7%

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

Alternative 2: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) + -1\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \frac{771.3234287776531}{t_0 + 3}\right) + \frac{-176.6150291621406}{t_0 + 4}\right) + \frac{12.507343278686905}{t_0 + 5}\right) + \frac{-0.13857109526572012}{t_0 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t_0 + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t_0 + 8}\right) \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot e^{z + -7.5}\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ (- 1.0 z) -1.0)))
   (*
    (/ PI (sin (* PI z)))
    (*
     (+
      (+
       (+
        (+
         (+
          (+
           (+
            0.9999999999998099
            (+
             (/ 676.5203681218851 (- 1.0 z))
             (/ -1259.1392167224028 (- 2.0 z))))
           (/ 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_0 7.0)))
      (/ 1.5056327351493116e-7 (+ t_0 8.0)))
     (*
      (pow (- 7.5 z) (- 0.5 z))
      (* (sqrt 2.0) (* (sqrt PI) (exp (+ z -7.5)))))))))

C

double code(double z) {
	double t_0 = (1.0 - z) + -1.0;
	return (((double) M_PI) / sin((((double) M_PI) * z))) * ((((((((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (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_0 + 7.0))) + (1.5056327351493116e-7 / (t_0 + 8.0))) * (pow((7.5 - z), (0.5 - z)) * (sqrt(2.0) * (sqrt(((double) M_PI)) * exp((z + -7.5))))));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) + -1.0;
	return (Math.PI / Math.sin((Math.PI * z))) * ((((((((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (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_0 + 7.0))) + (1.5056327351493116e-7 / (t_0 + 8.0))) * (Math.pow((7.5 - z), (0.5 - z)) * (Math.sqrt(2.0) * (Math.sqrt(Math.PI) * Math.exp((z + -7.5))))));
}

Python

def code(z):
	t_0 = (1.0 - z) + -1.0
	return (math.pi / math.sin((math.pi * z))) * ((((((((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (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_0 + 7.0))) + (1.5056327351493116e-7 / (t_0 + 8.0))) * (math.pow((7.5 - z), (0.5 - z)) * (math.sqrt(2.0) * (math.sqrt(math.pi) * math.exp((z + -7.5))))))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) + -1.0)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(-1259.1392167224028 / Float64(2.0 - z)))) + 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 / Float64(t_0 + 7.0))) + Float64(1.5056327351493116e-7 / Float64(t_0 + 8.0))) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * Float64(sqrt(2.0) * Float64(sqrt(pi) * exp(Float64(z + -7.5)))))))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) + -1.0;
	tmp = (pi / sin((pi * z))) * ((((((((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + (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_0 + 7.0))) + (1.5056327351493116e-7 / (t_0 + 8.0))) * (((7.5 - z) ^ (0.5 - z)) * (sqrt(2.0) * (sqrt(pi) * exp((z + -7.5))))));
end

Wolfram

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

Derivation

1. Initial program 95.9%

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

   1. expm1-log1p-u 94.3%

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

   2. expm1-udef 94.3%

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

3. Applied egg-rr 94.3%

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

   1. expm1-def 94.3%

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

   2. expm1-log1p 95.9%

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

   3. +-commutative 95.9%

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

   4. associate-+l+ 97.0%

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

5. Simplified 97.0%

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

   1. expm1-log1p-u 97.0%

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

   2. expm1-udef 88.5%

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

7. Applied egg-rr 88.5%

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

   1. expm1-def 97.0%

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

   2. expm1-log1p 97.0%

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

   3. *-commutative 97.0%

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

   4. sub-neg 97.0%

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

   5. exp-to-pow 97.0%

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

   6. associate-*l* 97.0%

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

   7. exp-to-pow 97.0%

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

   8. neg-sub0 97.0%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(e^{\color{blue}{0 - \left(7.5 - z\right)}} \cdot \sqrt{\pi \cdot 2}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   9. associate--r- 97.0%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(e^{\color{blue}{\left(0 - 7.5\right) + z}} \cdot \sqrt{\pi \cdot 2}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   10. metadata-eval 97.0%

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

9. Simplified 97.0%

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

10. Taylor expanded in z around inf 97.0%

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

    1. associate-*r* 97.7%

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

    2. sub-neg 97.7%

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

    3. metadata-eval 97.7%

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

    4. +-commutative 97.7%

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

12. Simplified 97.7%

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

13. Final simplification 97.7%

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

Alternative 3: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot {\left(\sqrt[3]{\pi \cdot 2}\right)}^{1.5}\right)\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (/ PI (sin (* PI z)))
   (*
    (pow (- 7.5 z) (- 0.5 z))
    (* (exp (+ z -7.5)) (pow (cbrt (* PI 2.0)) 1.5))))
  (+
   (+
    (+
     (+
      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))) * (pow((7.5 - z), (0.5 - z)) * (exp((z + -7.5)) * pow(cbrt((((double) M_PI) * 2.0)), 1.5)))) * ((((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.pow((7.5 - z), (0.5 - z)) * (Math.exp((z + -7.5)) * Math.pow(Math.cbrt((Math.PI * 2.0)), 1.5)))) * ((((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(Float64(pi / sin(Float64(pi * z))) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * Float64(exp(Float64(z + -7.5)) * (cbrt(Float64(pi * 2.0)) ^ 1.5)))) * Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(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

Wolfram

code[z_] := N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[Power[N[Power[N[(Pi * 2.0), $MachinePrecision], 1/3], $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.9999999999998099 + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(N[(1.0 - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(N[(1.0 - z), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] - -4.0), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] - -5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.9%

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

3. Simplified 97.4%

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

   1. pow1/2 97.4%

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

   2. add-cube-cbrt 97.6%

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

   3. unpow-prod-down 97.6%

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

5. Applied egg-rr 97.6%

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

   1. unpow1/2 97.6%

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

   2. rem-sqrt-square 97.6%

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

   3. unpow1/2 97.6%

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

7. Simplified 97.6%

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

9. Applied egg-rr 88.5%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\left|\sqrt[3]{\pi \cdot 2}\right| \cdot {\left(\pi \cdot 2\right)}^{0.16666666666666666}\right) \cdot \left({\left(7.5 - z\right)}^{\left(0.5 + \left(-z\right)\right)} \cdot e^{-\left(7.5 - z\right)}\right)\right)} - 1\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

11. Simplified 97.6%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(e^{-7.5 + z} \cdot {\left(\sqrt[3]{\pi \cdot 2}\right)}^{1.5}\right)\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{\left(1 - z\right) - 0} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

12. Final simplification 97.6%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot {\left(\sqrt[3]{\pi \cdot 2}\right)}^{1.5}\right)\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

Alternative 4: 98.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.9%

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

3. Simplified 97.6%

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

4. Final simplification 97.6%

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

Alternative 5: 96.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 519.1279660315847\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{\left(z + -1\right) - 6.5}\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (+ 263.3831869810514 (* z (+ 436.8961725563396 (* z 519.1279660315847))))
  (*
   (/ PI (sin (* PI z)))
   (*
    (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5))
    (* (sqrt (* PI 2.0)) (exp (- (+ z -1.0) 6.5)))))))

C

double code(double z) {
	return (263.3831869810514 + (z * (436.8961725563396 + (z * 519.1279660315847)))) * ((((double) M_PI) / sin((((double) M_PI) * z))) * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * (sqrt((((double) M_PI) * 2.0)) * exp(((z + -1.0) - 6.5)))));
}

Java

public static double code(double z) {
	return (263.3831869810514 + (z * (436.8961725563396 + (z * 519.1279660315847)))) * ((Math.PI / Math.sin((Math.PI * z))) * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * (Math.sqrt((Math.PI * 2.0)) * Math.exp(((z + -1.0) - 6.5)))));
}

Python

def code(z):
	return (263.3831869810514 + (z * (436.8961725563396 + (z * 519.1279660315847)))) * ((math.pi / math.sin((math.pi * z))) * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * (math.sqrt((math.pi * 2.0)) * math.exp(((z + -1.0) - 6.5)))))

Julia

function code(z)
	return Float64(Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * 519.1279660315847)))) * Float64(Float64(pi / sin(Float64(pi * z))) * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * Float64(sqrt(Float64(pi * 2.0)) * exp(Float64(Float64(z + -1.0) - 6.5))))))
end

MATLAB

function tmp = code(z)
	tmp = (263.3831869810514 + (z * (436.8961725563396 + (z * 519.1279660315847)))) * ((pi / sin((pi * z))) * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * (sqrt((pi * 2.0)) * exp(((z + -1.0) - 6.5)))));
end

Wolfram

code[z_] := N[(N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * 519.1279660315847), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(z + -1.0), $MachinePrecision] - 6.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.9%

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

3. Simplified 96.2%

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

4. Taylor expanded in z around 0 94.3%

   \[\leadsto \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \color{blue}{\left(2.4783749183520145 + 0.49644474017195733 \cdot z\right)}\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)\right) \]
5. Step-by-step derivation

   1. *-commutative 94.3%

      \[\leadsto \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(2.4783749183520145 + \color{blue}{z \cdot 0.49644474017195733}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)\right) \]

6. Simplified 94.3%

   \[\leadsto \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \color{blue}{\left(2.4783749183520145 + z \cdot 0.49644474017195733\right)}\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)\right) \]

7. Taylor expanded in z around 0 95.2%

   \[\leadsto \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \color{blue}{\left(212.9540523020159 + 74.66416387488323 \cdot z\right)}\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)\right) \]
8. Step-by-step derivation

   1. *-commutative 95.2%

      \[\leadsto \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \left(212.9540523020159 + \color{blue}{z \cdot 74.66416387488323}\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)\right) \]

9. Simplified 95.2%

   \[\leadsto \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \color{blue}{\left(212.9540523020159 + z \cdot 74.66416387488323\right)}\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)\right) \]

10. Taylor expanded in z around 0 95.4%

    \[\leadsto \color{blue}{\left(263.3831869810514 + \left(436.8961725563396 \cdot z + 519.1279660315847 \cdot {z}^{2}\right)\right)} \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)\right) \]
11. Step-by-step derivation

    1. *-commutative 95.4%

       \[\leadsto \left(263.3831869810514 + \left(\color{blue}{z \cdot 436.8961725563396} + 519.1279660315847 \cdot {z}^{2}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)\right) \]

    2. *-commutative 95.4%

       \[\leadsto \left(263.3831869810514 + \left(z \cdot 436.8961725563396 + \color{blue}{{z}^{2} \cdot 519.1279660315847}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)\right) \]

    3. unpow2 95.4%

       \[\leadsto \left(263.3831869810514 + \left(z \cdot 436.8961725563396 + \color{blue}{\left(z \cdot z\right)} \cdot 519.1279660315847\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)\right) \]

    4. associate-*l* 95.4%

       \[\leadsto \left(263.3831869810514 + \left(z \cdot 436.8961725563396 + \color{blue}{z \cdot \left(z \cdot 519.1279660315847\right)}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)\right) \]

    5. distribute-lft-out 95.4%

       \[\leadsto \left(263.3831869810514 + \color{blue}{z \cdot \left(436.8961725563396 + z \cdot 519.1279660315847\right)}\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)\right) \]

12. Simplified 95.4%

    \[\leadsto \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 519.1279660315847\right)\right)} \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)\right) \]

13. Final simplification 95.4%

    \[\leadsto \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 519.1279660315847\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{\left(z + -1\right) - 6.5}\right)\right)\right) \]

Alternative 6: 96.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.9%

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

3. Simplified 96.2%

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

4. Taylor expanded in z around 0 94.3%

   \[\leadsto \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \color{blue}{\left(2.4783749183520145 + 0.49644474017195733 \cdot z\right)}\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)\right) \]
5. Step-by-step derivation

   1. *-commutative 94.3%

      \[\leadsto \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(2.4783749183520145 + \color{blue}{z \cdot 0.49644474017195733}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)\right) \]

6. Simplified 94.3%

   \[\leadsto \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \color{blue}{\left(2.4783749183520145 + z \cdot 0.49644474017195733\right)}\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)\right) \]

7. Taylor expanded in z around 0 95.2%

   \[\leadsto \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \color{blue}{\left(212.9540523020159 + 74.66416387488323 \cdot z\right)}\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)\right) \]
8. Step-by-step derivation

   1. *-commutative 95.2%

      \[\leadsto \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \left(212.9540523020159 + \color{blue}{z \cdot 74.66416387488323}\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)\right) \]

9. Simplified 95.2%

   \[\leadsto \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \color{blue}{\left(212.9540523020159 + z \cdot 74.66416387488323\right)}\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)\right) \]

10. Taylor expanded in z around 0 94.7%

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

    1. *-commutative 94.7%

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

12. Simplified 94.7%

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

13. Final simplification 94.7%

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

Alternative 7: 95.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.9%

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

3. Simplified 96.2%

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

4. Taylor expanded in z around 0 94.3%

   \[\leadsto \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \color{blue}{\left(2.4783749183520145 + 0.49644474017195733 \cdot z\right)}\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)\right) \]
5. Step-by-step derivation

   1. *-commutative 94.3%

      \[\leadsto \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(2.4783749183520145 + \color{blue}{z \cdot 0.49644474017195733}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)\right) \]

6. Simplified 94.3%

   \[\leadsto \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \color{blue}{\left(2.4783749183520145 + z \cdot 0.49644474017195733\right)}\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)\right) \]

7. Taylor expanded in z around 0 95.2%

   \[\leadsto \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \color{blue}{\left(212.9540523020159 + 74.66416387488323 \cdot z\right)}\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)\right) \]
8. Step-by-step derivation

   1. *-commutative 95.2%

      \[\leadsto \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \left(212.9540523020159 + \color{blue}{z \cdot 74.66416387488323}\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)\right) \]

9. Simplified 95.2%

   \[\leadsto \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \color{blue}{\left(212.9540523020159 + z \cdot 74.66416387488323\right)}\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right)\right) \]

10. Taylor expanded in z around 0 94.3%

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

11. Final simplification 94.3%

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

Alternative 8: 14.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 3.4783734731929044 \cdot \left(e^{z + -7.5} \cdot \frac{\pi}{\frac{\sin \left(\pi \cdot z\right)}{{\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}}}\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  3.4783734731929044
  (*
   (exp (+ z -7.5))
   (/ PI (/ (sin (* PI z)) (* (pow (- 7.5 z) (- 0.5 z)) (sqrt (* PI 2.0))))))))

C

double code(double z) {
	return 3.4783734731929044 * (exp((z + -7.5)) * (((double) M_PI) / (sin((((double) M_PI) * z)) / (pow((7.5 - z), (0.5 - z)) * sqrt((((double) M_PI) * 2.0))))));
}

Java

public static double code(double z) {
	return 3.4783734731929044 * (Math.exp((z + -7.5)) * (Math.PI / (Math.sin((Math.PI * z)) / (Math.pow((7.5 - z), (0.5 - z)) * Math.sqrt((Math.PI * 2.0))))));
}

Python

def code(z):
	return 3.4783734731929044 * (math.exp((z + -7.5)) * (math.pi / (math.sin((math.pi * z)) / (math.pow((7.5 - z), (0.5 - z)) * math.sqrt((math.pi * 2.0))))))

Julia

function code(z)
	return Float64(3.4783734731929044 * Float64(exp(Float64(z + -7.5)) * Float64(pi / Float64(sin(Float64(pi * z)) / Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * sqrt(Float64(pi * 2.0)))))))
end

MATLAB

function tmp = code(z)
	tmp = 3.4783734731929044 * (exp((z + -7.5)) * (pi / (sin((pi * z)) / (((7.5 - z) ^ (0.5 - z)) * sqrt((pi * 2.0))))));
end

Wolfram

code[z_] := N[(3.4783734731929044 * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(Pi / N[(N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision] / N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.9%

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

3. Simplified 97.6%

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

4. Taylor expanded in z around inf 14.8%

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

5. Taylor expanded in z around 0 14.8%

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

6. Taylor expanded in z around inf 14.8%

   \[\leadsto \left(0.9999999999998099 + \color{blue}{2.4783734731930944}\right) \cdot \frac{\left(\left(\pi \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}}{\sin \left(\pi \cdot z\right)} \]
7. Step-by-step derivation

   1. expm1-log1p-u 8.1%

      \[\leadsto \left(0.9999999999998099 + 2.4783734731930944\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\left(\pi \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}}{\sin \left(\pi \cdot z\right)}\right)\right)} \]

   2. expm1-udef 8.1%

      \[\leadsto \left(0.9999999999998099 + 2.4783734731930944\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\left(\left(\pi \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}}{\sin \left(\pi \cdot z\right)}\right)} - 1\right)} \]

8. Applied egg-rr 8.1%

   \[\leadsto \left(0.9999999999998099 + 2.4783734731930944\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)}\right)}{\frac{\sin \left(\pi \cdot z\right)}{e^{-7.5 + z}}}\right)} - 1\right)} \]
9. Step-by-step derivation

   1. expm1-def 8.1%

      \[\leadsto \left(0.9999999999998099 + 2.4783734731930944\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)}\right)}{\frac{\sin \left(\pi \cdot z\right)}{e^{-7.5 + z}}}\right)\right)} \]

   2. expm1-log1p 14.8%

      \[\leadsto \left(0.9999999999998099 + 2.4783734731930944\right) \cdot \color{blue}{\frac{\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)}\right)}{\frac{\sin \left(\pi \cdot z\right)}{e^{-7.5 + z}}}} \]

   3. associate-/r/ 14.8%

      \[\leadsto \left(0.9999999999998099 + 2.4783734731930944\right) \cdot \color{blue}{\left(\frac{\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)}\right)}{\sin \left(\pi \cdot z\right)} \cdot e^{-7.5 + z}\right)} \]

   4. *-commutative 14.8%

      \[\leadsto \left(0.9999999999998099 + 2.4783734731930944\right) \cdot \color{blue}{\left(e^{-7.5 + z} \cdot \frac{\pi \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)}\right)}{\sin \left(\pi \cdot z\right)}\right)} \]

   5. associate-/l* 14.8%

      \[\leadsto \left(0.9999999999998099 + 2.4783734731930944\right) \cdot \left(e^{-7.5 + z} \cdot \color{blue}{\frac{\pi}{\frac{\sin \left(\pi \cdot z\right)}{\sqrt{\pi \cdot 2} \cdot {\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)}}}}\right) \]

   6. fma-udef 14.8%

      \[\leadsto \left(0.9999999999998099 + 2.4783734731930944\right) \cdot \left(e^{-7.5 + z} \cdot \frac{\pi}{\frac{\sin \left(\pi \cdot z\right)}{\sqrt{\pi \cdot 2} \cdot {\color{blue}{\left(-1 \cdot z + 7.5\right)}}^{\left(0.5 - z\right)}}}\right) \]

   7. +-commutative 14.8%

      \[\leadsto \left(0.9999999999998099 + 2.4783734731930944\right) \cdot \left(e^{-7.5 + z} \cdot \frac{\pi}{\frac{\sin \left(\pi \cdot z\right)}{\sqrt{\pi \cdot 2} \cdot {\color{blue}{\left(7.5 + -1 \cdot z\right)}}^{\left(0.5 - z\right)}}}\right) \]

   8. mul-1-neg 14.8%

      \[\leadsto \left(0.9999999999998099 + 2.4783734731930944\right) \cdot \left(e^{-7.5 + z} \cdot \frac{\pi}{\frac{\sin \left(\pi \cdot z\right)}{\sqrt{\pi \cdot 2} \cdot {\left(7.5 + \color{blue}{\left(-z\right)}\right)}^{\left(0.5 - z\right)}}}\right) \]

   9. sub-neg 14.8%

      \[\leadsto \left(0.9999999999998099 + 2.4783734731930944\right) \cdot \left(e^{-7.5 + z} \cdot \frac{\pi}{\frac{\sin \left(\pi \cdot z\right)}{\sqrt{\pi \cdot 2} \cdot {\color{blue}{\left(7.5 - z\right)}}^{\left(0.5 - z\right)}}}\right) \]

10. Simplified 14.8%

    \[\leadsto \left(0.9999999999998099 + 2.4783734731930944\right) \cdot \color{blue}{\left(e^{-7.5 + z} \cdot \frac{\pi}{\frac{\sin \left(\pi \cdot z\right)}{\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}}}\right)} \]

11. Final simplification 14.8%

    \[\leadsto 3.4783734731929044 \cdot \left(e^{z + -7.5} \cdot \frac{\pi}{\frac{\sin \left(\pi \cdot z\right)}{{\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}}}\right) \]

Alternative 9: 14.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{e^{z + -7.5} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\pi \cdot \sqrt{\pi \cdot 2}\right)\right)}{\sin \left(\pi \cdot z\right)} \cdot 3.4783749183518244 \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/
   (* (exp (+ z -7.5)) (* (pow (- 7.5 z) (- 0.5 z)) (* PI (sqrt (* PI 2.0)))))
   (sin (* PI z)))
  3.4783749183518244))

C

double code(double z) {
	return ((exp((z + -7.5)) * (pow((7.5 - z), (0.5 - z)) * (((double) M_PI) * sqrt((((double) M_PI) * 2.0))))) / sin((((double) M_PI) * z))) * 3.4783749183518244;
}

Java

public static double code(double z) {
	return ((Math.exp((z + -7.5)) * (Math.pow((7.5 - z), (0.5 - z)) * (Math.PI * Math.sqrt((Math.PI * 2.0))))) / Math.sin((Math.PI * z))) * 3.4783749183518244;
}

Python

def code(z):
	return ((math.exp((z + -7.5)) * (math.pow((7.5 - z), (0.5 - z)) * (math.pi * math.sqrt((math.pi * 2.0))))) / math.sin((math.pi * z))) * 3.4783749183518244

Julia

function code(z)
	return Float64(Float64(Float64(exp(Float64(z + -7.5)) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * Float64(pi * sqrt(Float64(pi * 2.0))))) / sin(Float64(pi * z))) * 3.4783749183518244)
end

MATLAB

function tmp = code(z)
	tmp = ((exp((z + -7.5)) * (((7.5 - z) ^ (0.5 - z)) * (pi * sqrt((pi * 2.0))))) / sin((pi * z))) * 3.4783749183518244;
end

Wolfram

code[z_] := N[(N[(N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[(Pi * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 3.4783749183518244), $MachinePrecision]

Derivation

1. Initial program 95.9%

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

3. Simplified 97.6%

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

4. Taylor expanded in z around inf 14.8%

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

5. Taylor expanded in z around 0 14.8%

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

6. Taylor expanded in z around 0 14.8%

   \[\leadsto \left(0.9999999999998099 + \color{blue}{2.4783749183520145}\right) \cdot \frac{\left(\left(\pi \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}}{\sin \left(\pi \cdot z\right)} \]

7. Final simplification 14.8%

   \[\leadsto \frac{e^{z + -7.5} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\pi \cdot \sqrt{\pi \cdot 2}\right)\right)}{\sin \left(\pi \cdot z\right)} \cdot 3.4783749183518244 \]

Alternative 10: 14.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 3.4783734731929044 \cdot \frac{\left(\sqrt{2} \cdot \sqrt{7.5}\right) \cdot \left(e^{-7.5} \cdot {\pi}^{1.5}\right)}{\sin \left(\pi \cdot z\right)} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  3.4783734731929044
  (/
   (* (* (sqrt 2.0) (sqrt 7.5)) (* (exp -7.5) (pow PI 1.5)))
   (sin (* PI z)))))

C

double code(double z) {
	return 3.4783734731929044 * (((sqrt(2.0) * sqrt(7.5)) * (exp(-7.5) * pow(((double) M_PI), 1.5))) / sin((((double) M_PI) * z)));
}

Java

public static double code(double z) {
	return 3.4783734731929044 * (((Math.sqrt(2.0) * Math.sqrt(7.5)) * (Math.exp(-7.5) * Math.pow(Math.PI, 1.5))) / Math.sin((Math.PI * z)));
}

Python

def code(z):
	return 3.4783734731929044 * (((math.sqrt(2.0) * math.sqrt(7.5)) * (math.exp(-7.5) * math.pow(math.pi, 1.5))) / math.sin((math.pi * z)))

Julia

function code(z)
	return Float64(3.4783734731929044 * Float64(Float64(Float64(sqrt(2.0) * sqrt(7.5)) * Float64(exp(-7.5) * (pi ^ 1.5))) / sin(Float64(pi * z))))
end

MATLAB

function tmp = code(z)
	tmp = 3.4783734731929044 * (((sqrt(2.0) * sqrt(7.5)) * (exp(-7.5) * (pi ^ 1.5))) / sin((pi * z)));
end

Wolfram

code[z_] := N[(3.4783734731929044 * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[7.5], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] * N[Power[Pi, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.9%

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

3. Simplified 97.6%

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

4. Taylor expanded in z around inf 14.8%

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

5. Taylor expanded in z around 0 14.8%

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

6. Taylor expanded in z around inf 14.8%

   \[\leadsto \left(0.9999999999998099 + \color{blue}{2.4783734731930944}\right) \cdot \frac{\left(\left(\pi \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}}{\sin \left(\pi \cdot z\right)} \]

7. Taylor expanded in z around 0 14.4%

   \[\leadsto \left(0.9999999999998099 + 2.4783734731930944\right) \cdot \frac{\color{blue}{\sqrt{{\pi}^{3}} \cdot \left(e^{-7.5} \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)\right)}}{\sin \left(\pi \cdot z\right)} \]
8. Step-by-step derivation

   1. *-commutative 14.4%

      \[\leadsto \left(0.9999999999998099 + 2.4783734731930944\right) \cdot \frac{\color{blue}{\left(e^{-7.5} \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)\right) \cdot \sqrt{{\pi}^{3}}}}{\sin \left(\pi \cdot z\right)} \]

   2. *-commutative 14.4%

      \[\leadsto \left(0.9999999999998099 + 2.4783734731930944\right) \cdot \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{7.5}\right) \cdot e^{-7.5}\right)} \cdot \sqrt{{\pi}^{3}}}{\sin \left(\pi \cdot z\right)} \]

   3. associate-*l* 14.4%

      \[\leadsto \left(0.9999999999998099 + 2.4783734731930944\right) \cdot \frac{\color{blue}{\left(\sqrt{2} \cdot \sqrt{7.5}\right) \cdot \left(e^{-7.5} \cdot \sqrt{{\pi}^{3}}\right)}}{\sin \left(\pi \cdot z\right)} \]

   4. metadata-eval 14.4%

      \[\leadsto \left(0.9999999999998099 + 2.4783734731930944\right) \cdot \frac{\left(\sqrt{2} \cdot \sqrt{7.5}\right) \cdot \left(e^{-7.5} \cdot \sqrt{{\pi}^{\color{blue}{\left(2 \cdot 1.5\right)}}}\right)}{\sin \left(\pi \cdot z\right)} \]

   5. pow-sqr 14.4%

      \[\leadsto \left(0.9999999999998099 + 2.4783734731930944\right) \cdot \frac{\left(\sqrt{2} \cdot \sqrt{7.5}\right) \cdot \left(e^{-7.5} \cdot \sqrt{\color{blue}{{\pi}^{1.5} \cdot {\pi}^{1.5}}}\right)}{\sin \left(\pi \cdot z\right)} \]

   6. rem-sqrt-square 14.4%

      \[\leadsto \left(0.9999999999998099 + 2.4783734731930944\right) \cdot \frac{\left(\sqrt{2} \cdot \sqrt{7.5}\right) \cdot \left(e^{-7.5} \cdot \color{blue}{\left|{\pi}^{1.5}\right|}\right)}{\sin \left(\pi \cdot z\right)} \]

   7. sqr-pow 14.4%

      \[\leadsto \left(0.9999999999998099 + 2.4783734731930944\right) \cdot \frac{\left(\sqrt{2} \cdot \sqrt{7.5}\right) \cdot \left(e^{-7.5} \cdot \left|\color{blue}{{\pi}^{\left(\frac{1.5}{2}\right)} \cdot {\pi}^{\left(\frac{1.5}{2}\right)}}\right|\right)}{\sin \left(\pi \cdot z\right)} \]

   8. fabs-sqr 14.4%

      \[\leadsto \left(0.9999999999998099 + 2.4783734731930944\right) \cdot \frac{\left(\sqrt{2} \cdot \sqrt{7.5}\right) \cdot \left(e^{-7.5} \cdot \color{blue}{\left({\pi}^{\left(\frac{1.5}{2}\right)} \cdot {\pi}^{\left(\frac{1.5}{2}\right)}\right)}\right)}{\sin \left(\pi \cdot z\right)} \]

   9. sqr-pow 14.4%

      \[\leadsto \left(0.9999999999998099 + 2.4783734731930944\right) \cdot \frac{\left(\sqrt{2} \cdot \sqrt{7.5}\right) \cdot \left(e^{-7.5} \cdot \color{blue}{{\pi}^{1.5}}\right)}{\sin \left(\pi \cdot z\right)} \]

9. Simplified 14.4%

   \[\leadsto \left(0.9999999999998099 + 2.4783734731930944\right) \cdot \frac{\color{blue}{\left(\sqrt{2} \cdot \sqrt{7.5}\right) \cdot \left(e^{-7.5} \cdot {\pi}^{1.5}\right)}}{\sin \left(\pi \cdot z\right)} \]

10. Final simplification 14.4%

    \[\leadsto 3.4783734731929044 \cdot \frac{\left(\sqrt{2} \cdot \sqrt{7.5}\right) \cdot \left(e^{-7.5} \cdot {\pi}^{1.5}\right)}{\sin \left(\pi \cdot z\right)} \]

Alternative 11: 14.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ 3.4783734731929044 \cdot \left(\sqrt{\pi} \cdot \left(\left(\sqrt{2} \cdot \sqrt{7.5}\right) \cdot \frac{e^{-7.5}}{z}\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  3.4783734731929044
  (* (sqrt PI) (* (* (sqrt 2.0) (sqrt 7.5)) (/ (exp -7.5) z)))))

C

double code(double z) {
	return 3.4783734731929044 * (sqrt(((double) M_PI)) * ((sqrt(2.0) * sqrt(7.5)) * (exp(-7.5) / z)));
}

Java

public static double code(double z) {
	return 3.4783734731929044 * (Math.sqrt(Math.PI) * ((Math.sqrt(2.0) * Math.sqrt(7.5)) * (Math.exp(-7.5) / z)));
}

Python

def code(z):
	return 3.4783734731929044 * (math.sqrt(math.pi) * ((math.sqrt(2.0) * math.sqrt(7.5)) * (math.exp(-7.5) / z)))

Julia

function code(z)
	return Float64(3.4783734731929044 * Float64(sqrt(pi) * Float64(Float64(sqrt(2.0) * sqrt(7.5)) * Float64(exp(-7.5) / z))))
end

MATLAB

function tmp = code(z)
	tmp = 3.4783734731929044 * (sqrt(pi) * ((sqrt(2.0) * sqrt(7.5)) * (exp(-7.5) / z)));
end

Wolfram

code[z_] := N[(3.4783734731929044 * N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[7.5], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.9%

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

3. Simplified 97.6%

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

4. Taylor expanded in z around inf 14.8%

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

5. Taylor expanded in z around 0 14.8%

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

6. Taylor expanded in z around inf 14.8%

   \[\leadsto \left(0.9999999999998099 + \color{blue}{2.4783734731930944}\right) \cdot \frac{\left(\left(\pi \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}}{\sin \left(\pi \cdot z\right)} \]

7. Taylor expanded in z around 0 14.3%

   \[\leadsto \left(0.9999999999998099 + 2.4783734731930944\right) \cdot \color{blue}{\left(\frac{e^{-7.5} \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)}{z} \cdot \sqrt{\pi}\right)} \]
8. Step-by-step derivation

   1. *-commutative 14.3%

      \[\leadsto \left(0.9999999999998099 + 2.4783734731930944\right) \cdot \color{blue}{\left(\sqrt{\pi} \cdot \frac{e^{-7.5} \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)}{z}\right)} \]

   2. associate-/l* 14.3%

      \[\leadsto \left(0.9999999999998099 + 2.4783734731930944\right) \cdot \left(\sqrt{\pi} \cdot \color{blue}{\frac{e^{-7.5}}{\frac{z}{\sqrt{2} \cdot \sqrt{7.5}}}}\right) \]

   3. associate-/r/ 14.3%

      \[\leadsto \left(0.9999999999998099 + 2.4783734731930944\right) \cdot \left(\sqrt{\pi} \cdot \color{blue}{\left(\frac{e^{-7.5}}{z} \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)\right)}\right) \]

9. Simplified 14.3%

   \[\leadsto \left(0.9999999999998099 + 2.4783734731930944\right) \cdot \color{blue}{\left(\sqrt{\pi} \cdot \left(\frac{e^{-7.5}}{z} \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)\right)\right)} \]

10. Final simplification 14.3%

    \[\leadsto 3.4783734731929044 \cdot \left(\sqrt{\pi} \cdot \left(\left(\sqrt{2} \cdot \sqrt{7.5}\right) \cdot \frac{e^{-7.5}}{z}\right)\right) \]

Reproduce

herbie shell --seed 2023314 
(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))))))