Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.5% → 98.3%

Time: 2.2min

Alternatives: 15

Speedup: 1.1×

• could not determine a ground truth

  1. z = -4.0962908790014025e+161

Specification

\[z \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (/ 771.3234287776531 (- 3.0 z))))
   (*
    (+
     (+
      (/ 676.5203681218851 (- 1.0 z))
      (+
       (+ 0.9999999999998099 (cbrt (* t_0 (* t_0 t_0))))
       (/ -1259.1392167224028 (- 2.0 z))))
     (+
      (/ -176.6150291621406 (- 4.0 z))
      (+
       (+
        (+ (/ 12.507343278686905 (- 5.0 z)) (/ -0.13857109526572012 (- 6.0 z)))
        (/ 1.5056327351493116e-7 (- 8.0 z)))
       (/ 9.984369578019572e-6 (- 7.0 z)))))
    (*
     (/ PI (sin (* z PI)))
     (*
      (pow (- 7.5 z) (- 0.5 z))
      (cbrt (pow (* (sqrt (* 2.0 PI)) (exp (+ z -7.5))) 3.0)))))))

C

double code(double z) {
	double t_0 = 771.3234287776531 / (3.0 - z);
	return (((676.5203681218851 / (1.0 - z)) + ((0.9999999999998099 + cbrt((t_0 * (t_0 * t_0)))) + (-1259.1392167224028 / (2.0 - z)))) + ((-176.6150291621406 / (4.0 - z)) + ((((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z))) + (1.5056327351493116e-7 / (8.0 - z))) + (9.984369578019572e-6 / (7.0 - z))))) * ((((double) M_PI) / sin((z * ((double) M_PI)))) * (pow((7.5 - z), (0.5 - z)) * cbrt(pow((sqrt((2.0 * ((double) M_PI))) * exp((z + -7.5))), 3.0))));
}

Java

public static double code(double z) {
	double t_0 = 771.3234287776531 / (3.0 - z);
	return (((676.5203681218851 / (1.0 - z)) + ((0.9999999999998099 + Math.cbrt((t_0 * (t_0 * t_0)))) + (-1259.1392167224028 / (2.0 - z)))) + ((-176.6150291621406 / (4.0 - z)) + ((((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z))) + (1.5056327351493116e-7 / (8.0 - z))) + (9.984369578019572e-6 / (7.0 - z))))) * ((Math.PI / Math.sin((z * Math.PI))) * (Math.pow((7.5 - z), (0.5 - z)) * Math.cbrt(Math.pow((Math.sqrt((2.0 * Math.PI)) * Math.exp((z + -7.5))), 3.0))));
}

Julia

function code(z)
	t_0 = Float64(771.3234287776531 / Float64(3.0 - z))
	return Float64(Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(0.9999999999998099 + cbrt(Float64(t_0 * Float64(t_0 * t_0)))) + Float64(-1259.1392167224028 / Float64(2.0 - z)))) + Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) + Float64(Float64(Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(-0.13857109526572012 / Float64(6.0 - z))) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))) + Float64(9.984369578019572e-6 / Float64(7.0 - z))))) * Float64(Float64(pi / sin(Float64(z * pi))) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * cbrt((Float64(sqrt(Float64(2.0 * pi)) * exp(Float64(z + -7.5))) ^ 3.0)))))
end

Wolfram

code[z_] := Block[{t$95$0 = N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(0.9999999999998099 + N[Power[N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Power[N[Power[N[(N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 97.3%

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

   1. associate-*r* 97.5%

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

   2. *-commutative 97.5%

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

3. Simplified 98.9%

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

   1. add-cbrt-cube 99.3%

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

   2. pow3 99.3%

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

5. Applied egg-rr 99.3%

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

   1. add-cbrt-cube 99.3%

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

7. Applied egg-rr 99.3%

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

8. Final simplification 99.3%

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

Alternative 2: 98.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[z_] := N[(N[(N[(Pi / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Power[N[Power[N[(N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.3%

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

   1. associate-*r* 97.5%

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

   2. *-commutative 97.5%

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

3. Simplified 98.9%

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

   1. add-cbrt-cube 99.3%

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

   2. pow3 99.3%

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

5. Applied egg-rr 99.3%

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

6. Final simplification 99.3%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

   1. *-commutative 97.3%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 4: 98.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

   1. *-commutative 97.3%

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

3. Simplified 99.0%

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

4. Taylor expanded in z around inf 99.0%

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

5. Final simplification 99.0%

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

Alternative 5: 97.1% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ PI (sin (* z PI)))
  (*
   (sqrt (* 2.0 PI))
   (*
    (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp (- (+ z -1.0) 6.5)))
    (+
     (+
      0.9999999999998099
      (+
       (+
        (/ 676.5203681218851 (- 1.0 z))
        (/ -1259.1392167224028 (+ 1.0 (- 1.0 z))))
       (+ 212.9540523020159 (* z 74.66416387488323))))
     (+ (* z 0.49644474017195733) 2.4783749183520145))))))

C

double code(double z) {
	return (((double) M_PI) / sin((z * ((double) M_PI)))) * (sqrt((2.0 * ((double) M_PI))) * ((pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(((z + -1.0) - 6.5))) * ((0.9999999999998099 + (((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (1.0 + (1.0 - z)))) + (212.9540523020159 + (z * 74.66416387488323)))) + ((z * 0.49644474017195733) + 2.4783749183520145))));
}

Java

public static double code(double z) {
	return (Math.PI / Math.sin((z * Math.PI))) * (Math.sqrt((2.0 * Math.PI)) * ((Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp(((z + -1.0) - 6.5))) * ((0.9999999999998099 + (((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (1.0 + (1.0 - z)))) + (212.9540523020159 + (z * 74.66416387488323)))) + ((z * 0.49644474017195733) + 2.4783749183520145))));
}

Python

def code(z):
	return (math.pi / math.sin((z * math.pi))) * (math.sqrt((2.0 * math.pi)) * ((math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp(((z + -1.0) - 6.5))) * ((0.9999999999998099 + (((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (1.0 + (1.0 - z)))) + (212.9540523020159 + (z * 74.66416387488323)))) + ((z * 0.49644474017195733) + 2.4783749183520145))))

Julia

function code(z)
	return Float64(Float64(pi / sin(Float64(z * pi))) * Float64(sqrt(Float64(2.0 * pi)) * Float64(Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(Float64(Float64(z + -1.0) - 6.5))) * Float64(Float64(0.9999999999998099 + Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(-1259.1392167224028 / Float64(1.0 + Float64(1.0 - z)))) + Float64(212.9540523020159 + Float64(z * 74.66416387488323)))) + Float64(Float64(z * 0.49644474017195733) + 2.4783749183520145)))))
end

MATLAB

function tmp = code(z)
	tmp = (pi / sin((z * pi))) * (sqrt((2.0 * pi)) * (((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(((z + -1.0) - 6.5))) * ((0.9999999999998099 + (((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (1.0 + (1.0 - z)))) + (212.9540523020159 + (z * 74.66416387488323)))) + ((z * 0.49644474017195733) + 2.4783749183520145))));
end

Wolfram

code[z_] := N[(N[(Pi / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(z + -1.0), $MachinePrecision] - 6.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.9999999999998099 + N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(1.0 + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(212.9540523020159 + N[(z * 74.66416387488323), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z * 0.49644474017195733), $MachinePrecision] + 2.4783749183520145), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.3%

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

   1. associate-*l* 97.3%

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

3. Simplified 97.8%

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

   1. expm1-log1p-u 97.8%

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

   2. expm1-udef 97.8%

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

5. Applied egg-rr 97.8%

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

   1. expm1-def 97.8%

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

   2. expm1-log1p 97.8%

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

   3. +-commutative 97.8%

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

   4. associate-+r- 97.8%

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

   5. metadata-eval 97.8%

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

7. Simplified 97.8%

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

8. Taylor expanded in z around 0 97.0%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \color{blue}{\left(0.49644474017195733 \cdot z + 2.4783749183520145\right)}\right)\right)\right) \]

9. Taylor expanded in z around 0 98.2%

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

    1. *-commutative 98.2%

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

11. Simplified 98.2%

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

12. Final simplification 98.2%

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

Alternative 6: 97.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{676.5203681218851}{1 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-176.6150291621406}{4 - z} + \left(z \cdot 0.49644474017195733 + 2.4783749183520145\right)\right)\right) \cdot \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{z + -7.5}\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (+
   (+
    (/ 676.5203681218851 (- 1.0 z))
    (+
     (/ -1259.1392167224028 (- 2.0 z))
     (+ 0.9999999999998099 (/ 771.3234287776531 (- 3.0 z)))))
   (+
    (/ -176.6150291621406 (- 4.0 z))
    (+ (* z 0.49644474017195733) 2.4783749183520145)))
  (*
   (/ PI (sin (* z PI)))
   (* (pow (- 7.5 z) (- 0.5 z)) (* (sqrt (* 2.0 PI)) (exp (+ z -7.5)))))))

C

double code(double z) {
	return (((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z))))) + ((-176.6150291621406 / (4.0 - z)) + ((z * 0.49644474017195733) + 2.4783749183520145))) * ((((double) M_PI) / sin((z * ((double) M_PI)))) * (pow((7.5 - z), (0.5 - z)) * (sqrt((2.0 * ((double) M_PI))) * exp((z + -7.5)))));
}

Java

public static double code(double z) {
	return (((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z))))) + ((-176.6150291621406 / (4.0 - z)) + ((z * 0.49644474017195733) + 2.4783749183520145))) * ((Math.PI / Math.sin((z * Math.PI))) * (Math.pow((7.5 - z), (0.5 - z)) * (Math.sqrt((2.0 * Math.PI)) * Math.exp((z + -7.5)))));
}

Python

def code(z):
	return (((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z))))) + ((-176.6150291621406 / (4.0 - z)) + ((z * 0.49644474017195733) + 2.4783749183520145))) * ((math.pi / math.sin((z * math.pi))) * (math.pow((7.5 - z), (0.5 - z)) * (math.sqrt((2.0 * math.pi)) * math.exp((z + -7.5)))))

Julia

function code(z)
	return Float64(Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) + Float64(0.9999999999998099 + Float64(771.3234287776531 / Float64(3.0 - z))))) + Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) + Float64(Float64(z * 0.49644474017195733) + 2.4783749183520145))) * Float64(Float64(pi / sin(Float64(z * pi))) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * Float64(sqrt(Float64(2.0 * pi)) * exp(Float64(z + -7.5))))))
end

MATLAB

function tmp = code(z)
	tmp = (((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z))))) + ((-176.6150291621406 / (4.0 - z)) + ((z * 0.49644474017195733) + 2.4783749183520145))) * ((pi / sin((z * pi))) * (((7.5 - z) ^ (0.5 - z)) * (sqrt((2.0 * pi)) * exp((z + -7.5)))));
end

Wolfram

code[z_] := N[(N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(z * 0.49644474017195733), $MachinePrecision] + 2.4783749183520145), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.3%

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

   1. associate-*r* 97.5%

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

   2. *-commutative 97.5%

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

3. Simplified 98.9%

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

4. Taylor expanded in z around 0 98.1%

   \[\leadsto \left(\left(\frac{676.5203681218851}{1 - z} + \left(\left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(\frac{-176.6150291621406}{4 - z} + \color{blue}{\left(0.49644474017195733 \cdot z + 2.4783749183520145\right)}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{z + -7.5}\right)\right)\right) \]

5. Final simplification 98.1%

   \[\leadsto \left(\left(\frac{676.5203681218851}{1 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-176.6150291621406}{4 - z} + \left(z \cdot 0.49644474017195733 + 2.4783749183520145\right)\right)\right) \cdot \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{z + -7.5}\right)\right)\right) \]

Alternative 7: 96.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[z_] := N[(N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(Pi / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(263.4062807184368 + N[(z * 436.9000215473151), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.3%

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

   1. *-commutative 97.3%

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

3. Simplified 99.0%

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

4. Taylor expanded in z around 0 98.0%

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

   1. *-commutative 98.0%

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

6. Simplified 98.0%

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

7. Final simplification 98.0%

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

Alternative 8: 96.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

   1. *-commutative 97.3%

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

3. Simplified 99.0%

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

4. Taylor expanded in z around 0 97.1%

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

5. Taylor expanded in z around 0 97.2%

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

   1. *-commutative 97.2%

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

   2. *-commutative 97.2%

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

   3. associate-*r* 97.2%

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

7. Simplified 97.2%

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

8. Taylor expanded in z around 0 97.0%

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

   1. associate-/l* 97.0%

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

10. Simplified 97.0%

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

11. Taylor expanded in z around 0 96.9%

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

    1. *-commutative 96.9%

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

    2. associate-*r* 97.1%

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

    3. associate-*r* 97.1%

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

    4. associate-/l* 97.4%

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

    5. associate-*l/ 97.3%

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

    6. associate-*l* 97.2%

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

    7. *-commutative 97.2%

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

    8. associate-*l/ 97.2%

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

    9. associate-/l* 96.9%

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

    10. associate-*r* 96.9%

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

    11. associate-/l* 97.3%

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

13. Simplified 97.3%

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

14. Final simplification 97.3%

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

Alternative 9: 96.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

   1. *-commutative 97.3%

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

3. Simplified 99.0%

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

4. Taylor expanded in z around 0 97.1%

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

5. Taylor expanded in z around 0 97.2%

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

   1. *-commutative 97.2%

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

   2. *-commutative 97.2%

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

   3. associate-*r* 97.2%

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

7. Simplified 97.2%

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

8. Taylor expanded in z around 0 96.9%

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

   1. *-commutative 96.9%

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

   2. associate-*l* 97.1%

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

   3. associate-/l* 97.4%

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

   4. *-commutative 97.4%

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

   5. associate-/r* 97.4%

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

10. Simplified 97.4%

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

11. Final simplification 97.4%

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

Alternative 10: 96.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

   1. *-commutative 97.3%

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

3. Simplified 99.0%

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

4. Taylor expanded in z around 0 97.1%

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

5. Taylor expanded in z around 0 97.2%

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

   1. *-commutative 97.2%

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

   2. *-commutative 97.2%

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

   3. associate-*r* 97.2%

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

7. Simplified 97.2%

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

8. Taylor expanded in z around 0 96.9%

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

   1. *-commutative 96.9%

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

   2. associate-*l* 97.1%

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

   3. associate-/l* 97.4%

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

   4. *-commutative 97.4%

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

   5. associate-/r* 97.4%

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

10. Simplified 97.4%

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

    1. associate-*l/ 97.5%

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

    2. *-commutative 97.5%

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

    3. associate-/l/ 97.6%

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

12. Applied egg-rr 97.6%

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

13. Final simplification 97.6%

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

Alternative 11: 95.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

   1. *-commutative 97.3%

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

3. Simplified 99.0%

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

4. Taylor expanded in z around 0 97.1%

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

5. Taylor expanded in z around 0 97.2%

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

   1. *-commutative 97.2%

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

   2. *-commutative 97.2%

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

   3. associate-*r* 97.2%

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

7. Simplified 97.2%

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

8. Taylor expanded in z around 0 97.3%

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

9. Final simplification 97.3%

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

Alternative 12: 95.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

   1. *-commutative 97.3%

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

3. Simplified 99.0%

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

4. Taylor expanded in z around 0 97.1%

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

5. Taylor expanded in z around 0 97.2%

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

   1. *-commutative 97.2%

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

   2. *-commutative 97.2%

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

   3. associate-*r* 97.2%

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

7. Simplified 97.2%

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

8. Taylor expanded in z around 0 97.0%

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

   1. associate-/l* 97.0%

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

10. Simplified 97.0%

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

    1. pow1 97.0%

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

    2. associate-*r* 96.4%

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

    3. associate-/r/ 96.4%

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

12. Applied egg-rr 96.4%

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

    1. unpow1 96.4%

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

    2. *-commutative 96.4%

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

    3. associate-*l* 96.9%

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

    4. *-commutative 96.9%

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

    5. associate-/r/ 97.0%

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

    6. *-rgt-identity 97.0%

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

    7. associate-*r/ 97.0%

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

    8. associate-/r/ 96.9%

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

    9. associate-*l/ 96.9%

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

    10. *-lft-identity 96.9%

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

14. Simplified 96.9%

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

15. Final simplification 96.9%

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

Alternative 13: 95.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

   1. *-commutative 97.3%

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

3. Simplified 99.0%

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

4. Taylor expanded in z around 0 97.1%

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

5. Taylor expanded in z around 0 97.2%

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

   1. *-commutative 97.2%

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

   2. *-commutative 97.2%

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

   3. associate-*r* 97.2%

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

7. Simplified 97.2%

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

8. Taylor expanded in z around 0 97.0%

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

   1. associate-/l* 97.0%

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

10. Simplified 97.0%

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

11. Taylor expanded in z around 0 97.0%

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

    1. associate-/l* 97.0%

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

    2. *-rgt-identity 97.0%

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

    3. associate-*r/ 97.0%

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

    4. associate-/r/ 96.9%

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

    5. associate-*l/ 97.0%

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

    6. *-lft-identity 97.0%

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

13. Simplified 97.0%

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

14. Final simplification 97.0%

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

Alternative 14: 95.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

   1. *-commutative 97.3%

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

3. Simplified 99.0%

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

4. Taylor expanded in z around 0 97.1%

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

5. Taylor expanded in z around 0 97.2%

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

   1. *-commutative 97.2%

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

   2. *-commutative 97.2%

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

   3. associate-*r* 97.2%

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

7. Simplified 97.2%

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

8. Taylor expanded in z around 0 97.0%

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

   1. associate-/l* 97.0%

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

10. Simplified 97.0%

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

11. Final simplification 97.0%

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

Alternative 15: 95.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

   1. *-commutative 97.3%

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

3. Simplified 99.0%

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

4. Taylor expanded in z around 0 97.1%

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

5. Taylor expanded in z around 0 97.2%

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

   1. *-commutative 97.2%

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

   2. *-commutative 97.2%

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

   3. associate-*r* 97.2%

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

7. Simplified 97.2%

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

8. Taylor expanded in z around 0 97.0%

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

9. Final simplification 97.0%

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

Reproduce

herbie shell --seed 2023187 
(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))))))