Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.4% → 99.3%

Time: 1.9min

Alternatives: 11

Speedup: 1.4×

• could not determine a ground truth

  1. z = -1.3890607944231348e+35

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: 96.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := t\_0 + 7\\ t_2 := t\_1 + 0.5\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {t\_2}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{-t\_2}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{t\_0 + 1}\right) + \frac{-1259.1392167224028}{t\_0 + 2}\right) + \frac{771.3234287776531}{t\_0 + 3}\right) + \frac{-176.6150291621406}{t\_0 + 4}\right) + \frac{12.507343278686905}{t\_0 + 5}\right) + \frac{-0.13857109526572012}{t\_0 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_1}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\right)\right) \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) + 3\\ t_1 := \left(1 - z\right) + 2\\ \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{1 + \left(1 - z\right)}\right)\right) + \frac{\mathsf{fma}\left(771.3234287776531, t\_0, t\_1 \cdot -176.6150291621406\right)}{t\_0 \cdot t\_1}\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right) \cdot \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\left(\left(z + -1\right) + -6.5\right) + \left(0.5 - z\right) \cdot \log \left(1 + \left(6.5 - z\right)\right)}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ (- 1.0 z) 3.0)) (t_1 (+ (- 1.0 z) 2.0)))
   (*
    (+
     (+
      (+
       0.9999999999998099
       (+
        (/ 676.5203681218851 (- 1.0 z))
        (/ -1259.1392167224028 (+ 1.0 (- 1.0 z)))))
      (/ (fma 771.3234287776531 t_0 (* t_1 -176.6150291621406)) (* t_0 t_1)))
     (+
      (+
       (/ 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)))))
    (*
     (/ PI (sin (* z PI)))
     (*
      (sqrt (* 2.0 PI))
      (exp (+ (+ (+ z -1.0) -6.5) (* (- 0.5 z) (log (+ 1.0 (- 6.5 z)))))))))))

C

double code(double z) {
	double t_0 = (1.0 - z) + 3.0;
	double t_1 = (1.0 - z) + 2.0;
	return (((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (1.0 + (1.0 - z))))) + (fma(771.3234287776531, t_0, (t_1 * -176.6150291621406)) / (t_0 * t_1))) + (((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))))) * ((((double) M_PI) / sin((z * ((double) M_PI)))) * (sqrt((2.0 * ((double) M_PI))) * exp((((z + -1.0) + -6.5) + ((0.5 - z) * log((1.0 + (6.5 - z))))))));
}

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) + 3.0)
	t_1 = Float64(Float64(1.0 - z) + 2.0)
	return Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(-1259.1392167224028 / Float64(1.0 + Float64(1.0 - z))))) + Float64(fma(771.3234287776531, t_0, Float64(t_1 * -176.6150291621406)) / Float64(t_0 * t_1))) + Float64(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))))) * Float64(Float64(pi / sin(Float64(z * pi))) * Float64(sqrt(Float64(2.0 * pi)) * exp(Float64(Float64(Float64(z + -1.0) + -6.5) + Float64(Float64(0.5 - z) * log(Float64(1.0 + Float64(6.5 - z)))))))))
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] + 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - z), $MachinePrecision] + 2.0), $MachinePrecision]}, N[(N[(N[(N[(0.9999999999998099 + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(1.0 + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 * t$95$0 + N[(t$95$1 * -176.6150291621406), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(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] + 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] * N[(N[(Pi / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[(z + -1.0), $MachinePrecision] + -6.5), $MachinePrecision] + N[(N[(0.5 - z), $MachinePrecision] * N[Log[N[(1.0 + N[(6.5 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 96.6%

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

3. Simplified 98.2%

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

   1. add-exp-log 97.4%

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

   2. *-commutative 97.4%

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

   3. log-prod 97.4%

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

   4. add-log-exp 98.2%

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

   5. distribute-neg-in 98.2%

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

   6. metadata-eval 98.2%

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

   7. log-pow 98.2%

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

6. Applied egg-rr 98.2%

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

   1. frac-add 99.4%

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

   2. fma-define 99.3%

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

8. Applied egg-rr 99.3%

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

9. Final simplification 99.3%

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

Alternative 2: 98.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.6%

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

3. Simplified 98.4%

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

   1. div-inv 98.3%

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

6. Applied egg-rr 98.3%

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

   1. *-un-lft-identity 98.3%

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

   2. associate-+l+ 98.3%

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

8. Applied egg-rr 98.4%

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

   1. *-lft-identity 98.4%

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

   2. associate-+l+ 98.4%

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

   3. +-commutative 98.4%

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

   4. associate-+r+ 98.4%

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

   5. +-commutative 98.4%

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

10. Simplified 98.4%

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

11. Final simplification 98.4%

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

Alternative 3: 98.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

3. Simplified 98.4%

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

5. Final simplification 98.4%

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

Alternative 4: 97.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(z)
	tmp = sqrt((2.0 * pi)) * ((pi / sin((z * pi))) * (((7.5 - z) ^ (0.5 - z)) * (exp((z + -7.5)) * (0.9999999999998099 + (((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (2.0 - z))) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z))) + (2.4783734731930944 + (z * 0.49644453405676175)))))))));
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[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(0.9999999999998099 + N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.4783734731930944 + N[(z * 0.49644453405676175), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

3. Simplified 98.4%

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

   1. div-inv 98.3%

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

6. Applied egg-rr 98.3%

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

   1. *-un-lft-identity 98.3%

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

   2. associate-+l+ 98.3%

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

8. Applied egg-rr 98.4%

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

   1. *-lft-identity 98.4%

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

   2. associate-+l+ 98.4%

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

   3. +-commutative 98.4%

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

   4. associate-+r+ 98.4%

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

   5. +-commutative 98.4%

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

10. Simplified 98.4%

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

11. Taylor expanded in z around 0 97.5%

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

12. Final simplification 97.5%

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

Alternative 5: 97.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

3. Simplified 98.4%

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

   1. div-inv 98.3%

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

6. Applied egg-rr 98.3%

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

   1. *-un-lft-identity 98.3%

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

   2. associate-+l+ 98.3%

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

8. Applied egg-rr 98.4%

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

9. Taylor expanded in z around 0 97.0%

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

10. Final simplification 97.0%

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

Alternative 6: 95.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ 263.3831869810514 \cdot \left(\sqrt{15} \cdot \left(\sqrt{\pi} \cdot \frac{e^{-7.5}}{z}\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (* 263.3831869810514 (* (sqrt 15.0) (* (sqrt PI) (/ (exp -7.5) z)))))

C

double code(double z) {
	return 263.3831869810514 * (sqrt(15.0) * (sqrt(((double) M_PI)) * (exp(-7.5) / z)));
}

Java

public static double code(double z) {
	return 263.3831869810514 * (Math.sqrt(15.0) * (Math.sqrt(Math.PI) * (Math.exp(-7.5) / z)));
}

Python

def code(z):
	return 263.3831869810514 * (math.sqrt(15.0) * (math.sqrt(math.pi) * (math.exp(-7.5) / z)))

Julia

function code(z)
	return Float64(263.3831869810514 * Float64(sqrt(15.0) * Float64(sqrt(pi) * Float64(exp(-7.5) / z))))
end

MATLAB

function tmp = code(z)
	tmp = 263.3831869810514 * (sqrt(15.0) * (sqrt(pi) * (exp(-7.5) / z)));
end

Wolfram

code[z_] := N[(263.3831869810514 * N[(N[Sqrt[15.0], $MachinePrecision] * N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

3. Simplified 98.4%

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

5. Taylor expanded in z around 0 96.2%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(\left(\color{blue}{212.9540523020159} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{12.507343278686905}{5 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

6. Taylor expanded in z around 0 95.9%

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

   1. associate-/l* 96.1%

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

   2. *-commutative 96.1%

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

8. Simplified 96.1%

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

   1. *-un-lft-identity 96.1%

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

   2. associate-/r/ 96.1%

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

   3. sqrt-unprod 96.1%

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

   4. metadata-eval 96.1%

      \[\leadsto 263.3831869810514 \cdot \left(\left(1 \cdot \left(\frac{e^{-7.5}}{z} \cdot \sqrt{\color{blue}{15}}\right)\right) \cdot \sqrt{\pi}\right) \]

10. Applied egg-rr 96.1%

    \[\leadsto 263.3831869810514 \cdot \left(\color{blue}{\left(1 \cdot \left(\frac{e^{-7.5}}{z} \cdot \sqrt{15}\right)\right)} \cdot \sqrt{\pi}\right) \]
11. Step-by-step derivation

    1. *-lft-identity 96.1%

       \[\leadsto 263.3831869810514 \cdot \left(\color{blue}{\left(\frac{e^{-7.5}}{z} \cdot \sqrt{15}\right)} \cdot \sqrt{\pi}\right) \]

12. Simplified 96.1%

    \[\leadsto 263.3831869810514 \cdot \left(\color{blue}{\left(\frac{e^{-7.5}}{z} \cdot \sqrt{15}\right)} \cdot \sqrt{\pi}\right) \]

13. Taylor expanded in z around 0 95.9%

    \[\leadsto 263.3831869810514 \cdot \color{blue}{\left(\frac{e^{-7.5} \cdot \sqrt{15}}{z} \cdot \sqrt{\pi}\right)} \]
14. Step-by-step derivation

    1. associate-*l/ 96.1%

       \[\leadsto 263.3831869810514 \cdot \left(\color{blue}{\left(\frac{e^{-7.5}}{z} \cdot \sqrt{15}\right)} \cdot \sqrt{\pi}\right) \]

    2. *-commutative 96.1%

       \[\leadsto 263.3831869810514 \cdot \left(\color{blue}{\left(\sqrt{15} \cdot \frac{e^{-7.5}}{z}\right)} \cdot \sqrt{\pi}\right) \]

    3. associate-*l* 96.0%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{\left(\sqrt{15} \cdot \left(\frac{e^{-7.5}}{z} \cdot \sqrt{\pi}\right)\right)} \]

15. Simplified 96.0%

    \[\leadsto 263.3831869810514 \cdot \color{blue}{\left(\sqrt{15} \cdot \left(\frac{e^{-7.5}}{z} \cdot \sqrt{\pi}\right)\right)} \]

16. Final simplification 96.0%

    \[\leadsto 263.3831869810514 \cdot \left(\sqrt{15} \cdot \left(\sqrt{\pi} \cdot \frac{e^{-7.5}}{z}\right)\right) \]
17. Add Preprocessing

Alternative 7: 95.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ 263.3831869810514 \cdot \left(\sqrt{\pi} \cdot \left(\sqrt{15} \cdot \frac{e^{-7.5}}{z}\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (* 263.3831869810514 (* (sqrt PI) (* (sqrt 15.0) (/ (exp -7.5) z)))))

C

double code(double z) {
	return 263.3831869810514 * (sqrt(((double) M_PI)) * (sqrt(15.0) * (exp(-7.5) / z)));
}

Java

public static double code(double z) {
	return 263.3831869810514 * (Math.sqrt(Math.PI) * (Math.sqrt(15.0) * (Math.exp(-7.5) / z)));
}

Python

def code(z):
	return 263.3831869810514 * (math.sqrt(math.pi) * (math.sqrt(15.0) * (math.exp(-7.5) / z)))

Julia

function code(z)
	return Float64(263.3831869810514 * Float64(sqrt(pi) * Float64(sqrt(15.0) * Float64(exp(-7.5) / z))))
end

MATLAB

function tmp = code(z)
	tmp = 263.3831869810514 * (sqrt(pi) * (sqrt(15.0) * (exp(-7.5) / z)));
end

Wolfram

code[z_] := N[(263.3831869810514 * N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[Sqrt[15.0], $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

3. Simplified 98.4%

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

5. Taylor expanded in z around 0 96.2%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(\left(\color{blue}{212.9540523020159} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{12.507343278686905}{5 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

6. Taylor expanded in z around 0 95.9%

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

   1. associate-/l* 96.1%

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

   2. *-commutative 96.1%

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

8. Simplified 96.1%

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

   1. *-un-lft-identity 96.1%

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

   2. associate-/r/ 96.1%

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

   3. sqrt-unprod 96.1%

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

   4. metadata-eval 96.1%

      \[\leadsto 263.3831869810514 \cdot \left(\left(1 \cdot \left(\frac{e^{-7.5}}{z} \cdot \sqrt{\color{blue}{15}}\right)\right) \cdot \sqrt{\pi}\right) \]

10. Applied egg-rr 96.1%

    \[\leadsto 263.3831869810514 \cdot \left(\color{blue}{\left(1 \cdot \left(\frac{e^{-7.5}}{z} \cdot \sqrt{15}\right)\right)} \cdot \sqrt{\pi}\right) \]
11. Step-by-step derivation

    1. *-lft-identity 96.1%

       \[\leadsto 263.3831869810514 \cdot \left(\color{blue}{\left(\frac{e^{-7.5}}{z} \cdot \sqrt{15}\right)} \cdot \sqrt{\pi}\right) \]

12. Simplified 96.1%

    \[\leadsto 263.3831869810514 \cdot \left(\color{blue}{\left(\frac{e^{-7.5}}{z} \cdot \sqrt{15}\right)} \cdot \sqrt{\pi}\right) \]

13. Final simplification 96.1%

    \[\leadsto 263.3831869810514 \cdot \left(\sqrt{\pi} \cdot \left(\sqrt{15} \cdot \frac{e^{-7.5}}{z}\right)\right) \]
14. Add Preprocessing

Alternative 8: 95.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ 263.3831869810514 \cdot \frac{e^{-7.5}}{\frac{\frac{z}{\sqrt{15}}}{\sqrt{\pi}}} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (* 263.3831869810514 (/ (exp -7.5) (/ (/ z (sqrt 15.0)) (sqrt PI)))))

C

double code(double z) {
	return 263.3831869810514 * (exp(-7.5) / ((z / sqrt(15.0)) / sqrt(((double) M_PI))));
}

Java

public static double code(double z) {
	return 263.3831869810514 * (Math.exp(-7.5) / ((z / Math.sqrt(15.0)) / Math.sqrt(Math.PI)));
}

Python

def code(z):
	return 263.3831869810514 * (math.exp(-7.5) / ((z / math.sqrt(15.0)) / math.sqrt(math.pi)))

Julia

function code(z)
	return Float64(263.3831869810514 * Float64(exp(-7.5) / Float64(Float64(z / sqrt(15.0)) / sqrt(pi))))
end

MATLAB

function tmp = code(z)
	tmp = 263.3831869810514 * (exp(-7.5) / ((z / sqrt(15.0)) / sqrt(pi)));
end

Wolfram

code[z_] := N[(263.3831869810514 * N[(N[Exp[-7.5], $MachinePrecision] / N[(N[(z / N[Sqrt[15.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

3. Simplified 98.4%

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

5. Taylor expanded in z around 0 96.2%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(\left(\color{blue}{212.9540523020159} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{12.507343278686905}{5 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

6. Taylor expanded in z around 0 95.9%

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

   1. associate-/l* 96.1%

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

   2. *-commutative 96.1%

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

8. Simplified 96.1%

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

   1. pow1 96.1%

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

   2. associate-*l/ 96.3%

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

   3. sqrt-unprod 96.3%

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

   4. metadata-eval 96.3%

      \[\leadsto {\left(263.3831869810514 \cdot \frac{e^{-7.5} \cdot \sqrt{\pi}}{\frac{z}{\sqrt{\color{blue}{15}}}}\right)}^{1} \]

10. Applied egg-rr 96.3%

    \[\leadsto \color{blue}{{\left(263.3831869810514 \cdot \frac{e^{-7.5} \cdot \sqrt{\pi}}{\frac{z}{\sqrt{15}}}\right)}^{1}} \]
11. Step-by-step derivation

    1. unpow1 96.3%

       \[\leadsto \color{blue}{263.3831869810514 \cdot \frac{e^{-7.5} \cdot \sqrt{\pi}}{\frac{z}{\sqrt{15}}}} \]

    2. associate-/l* 96.1%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{\frac{e^{-7.5}}{\frac{\frac{z}{\sqrt{15}}}{\sqrt{\pi}}}} \]

12. Simplified 96.1%

    \[\leadsto \color{blue}{263.3831869810514 \cdot \frac{e^{-7.5}}{\frac{\frac{z}{\sqrt{15}}}{\sqrt{\pi}}}} \]

13. Final simplification 96.1%

    \[\leadsto 263.3831869810514 \cdot \frac{e^{-7.5}}{\frac{\frac{z}{\sqrt{15}}}{\sqrt{\pi}}} \]
14. Add Preprocessing

Alternative 9: 96.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ 263.3831869810514 \cdot \frac{e^{-7.5} \cdot \sqrt{\pi}}{\frac{z}{\sqrt{15}}} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (* 263.3831869810514 (/ (* (exp -7.5) (sqrt PI)) (/ z (sqrt 15.0)))))

C

double code(double z) {
	return 263.3831869810514 * ((exp(-7.5) * sqrt(((double) M_PI))) / (z / sqrt(15.0)));
}

Java

public static double code(double z) {
	return 263.3831869810514 * ((Math.exp(-7.5) * Math.sqrt(Math.PI)) / (z / Math.sqrt(15.0)));
}

Python

def code(z):
	return 263.3831869810514 * ((math.exp(-7.5) * math.sqrt(math.pi)) / (z / math.sqrt(15.0)))

Julia

function code(z)
	return Float64(263.3831869810514 * Float64(Float64(exp(-7.5) * sqrt(pi)) / Float64(z / sqrt(15.0))))
end

MATLAB

function tmp = code(z)
	tmp = 263.3831869810514 * ((exp(-7.5) * sqrt(pi)) / (z / sqrt(15.0)));
end

Wolfram

code[z_] := N[(263.3831869810514 * N[(N[(N[Exp[-7.5], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] / N[(z / N[Sqrt[15.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

3. Simplified 98.4%

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

5. Taylor expanded in z around 0 96.2%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(\left(\color{blue}{212.9540523020159} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{12.507343278686905}{5 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

6. Taylor expanded in z around 0 95.9%

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

   1. associate-/l* 96.1%

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

   2. *-commutative 96.1%

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

8. Simplified 96.1%

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

   1. associate-*l/ 96.3%

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

   2. sqrt-unprod 96.3%

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

   3. metadata-eval 96.3%

      \[\leadsto 263.3831869810514 \cdot \frac{e^{-7.5} \cdot \sqrt{\pi}}{\frac{z}{\sqrt{\color{blue}{15}}}} \]

10. Applied egg-rr 96.3%

    \[\leadsto 263.3831869810514 \cdot \color{blue}{\frac{e^{-7.5} \cdot \sqrt{\pi}}{\frac{z}{\sqrt{15}}}} \]

11. Final simplification 96.3%

    \[\leadsto 263.3831869810514 \cdot \frac{e^{-7.5} \cdot \sqrt{\pi}}{\frac{z}{\sqrt{15}}} \]
12. Add Preprocessing

Alternative 10: 95.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ 263.3831869810514 \cdot \left(\frac{e^{-7.5}}{z} \cdot \sqrt{\pi \cdot 15}\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (* 263.3831869810514 (* (/ (exp -7.5) z) (sqrt (* PI 15.0)))))

C

double code(double z) {
	return 263.3831869810514 * ((exp(-7.5) / z) * sqrt((((double) M_PI) * 15.0)));
}

Java

public static double code(double z) {
	return 263.3831869810514 * ((Math.exp(-7.5) / z) * Math.sqrt((Math.PI * 15.0)));
}

Python

def code(z):
	return 263.3831869810514 * ((math.exp(-7.5) / z) * math.sqrt((math.pi * 15.0)))

Julia

function code(z)
	return Float64(263.3831869810514 * Float64(Float64(exp(-7.5) / z) * sqrt(Float64(pi * 15.0))))
end

MATLAB

function tmp = code(z)
	tmp = 263.3831869810514 * ((exp(-7.5) / z) * sqrt((pi * 15.0)));
end

Wolfram

code[z_] := N[(263.3831869810514 * N[(N[(N[Exp[-7.5], $MachinePrecision] / z), $MachinePrecision] * N[Sqrt[N[(Pi * 15.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

3. Simplified 98.4%

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

5. Taylor expanded in z around 0 96.2%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(\left(\color{blue}{212.9540523020159} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{12.507343278686905}{5 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

6. Taylor expanded in z around 0 95.9%

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

   1. associate-/l* 96.1%

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

   2. *-commutative 96.1%

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

8. Simplified 96.1%

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

   1. *-un-lft-identity 96.1%

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

   2. associate-/r/ 96.1%

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

   3. sqrt-unprod 96.1%

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

   4. metadata-eval 96.1%

      \[\leadsto 263.3831869810514 \cdot \left(\left(1 \cdot \left(\frac{e^{-7.5}}{z} \cdot \sqrt{\color{blue}{15}}\right)\right) \cdot \sqrt{\pi}\right) \]

10. Applied egg-rr 96.1%

    \[\leadsto 263.3831869810514 \cdot \left(\color{blue}{\left(1 \cdot \left(\frac{e^{-7.5}}{z} \cdot \sqrt{15}\right)\right)} \cdot \sqrt{\pi}\right) \]
11. Step-by-step derivation

    1. *-lft-identity 96.1%

       \[\leadsto 263.3831869810514 \cdot \left(\color{blue}{\left(\frac{e^{-7.5}}{z} \cdot \sqrt{15}\right)} \cdot \sqrt{\pi}\right) \]

12. Simplified 96.1%

    \[\leadsto 263.3831869810514 \cdot \left(\color{blue}{\left(\frac{e^{-7.5}}{z} \cdot \sqrt{15}\right)} \cdot \sqrt{\pi}\right) \]
13. Step-by-step derivation

    1. pow1 96.1%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{{\left(\left(\frac{e^{-7.5}}{z} \cdot \sqrt{15}\right) \cdot \sqrt{\pi}\right)}^{1}} \]

14. Applied egg-rr 95.7%

    \[\leadsto 263.3831869810514 \cdot \color{blue}{{\left(\frac{e^{-7.5}}{z} \cdot {\left(15 \cdot \pi\right)}^{0.5}\right)}^{1}} \]
15. Step-by-step derivation

    1. unpow1 95.7%

       \[\leadsto 263.3831869810514 \cdot \color{blue}{\left(\frac{e^{-7.5}}{z} \cdot {\left(15 \cdot \pi\right)}^{0.5}\right)} \]

    2. unpow1/2 95.7%

       \[\leadsto 263.3831869810514 \cdot \left(\frac{e^{-7.5}}{z} \cdot \color{blue}{\sqrt{15 \cdot \pi}}\right) \]

    3. *-commutative 95.7%

       \[\leadsto 263.3831869810514 \cdot \left(\frac{e^{-7.5}}{z} \cdot \sqrt{\color{blue}{\pi \cdot 15}}\right) \]

16. Simplified 95.7%

    \[\leadsto 263.3831869810514 \cdot \color{blue}{\left(\frac{e^{-7.5}}{z} \cdot \sqrt{\pi \cdot 15}\right)} \]

17. Final simplification 95.7%

    \[\leadsto 263.3831869810514 \cdot \left(\frac{e^{-7.5}}{z} \cdot \sqrt{\pi \cdot 15}\right) \]
18. Add Preprocessing

Alternative 11: 95.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \left(263.3831869810514 \cdot \frac{e^{-7.5}}{z}\right) \cdot \sqrt{\pi \cdot 15} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (* (* 263.3831869810514 (/ (exp -7.5) z)) (sqrt (* PI 15.0))))

C

double code(double z) {
	return (263.3831869810514 * (exp(-7.5) / z)) * sqrt((((double) M_PI) * 15.0));
}

Java

public static double code(double z) {
	return (263.3831869810514 * (Math.exp(-7.5) / z)) * Math.sqrt((Math.PI * 15.0));
}

Python

def code(z):
	return (263.3831869810514 * (math.exp(-7.5) / z)) * math.sqrt((math.pi * 15.0))

Julia

function code(z)
	return Float64(Float64(263.3831869810514 * Float64(exp(-7.5) / z)) * sqrt(Float64(pi * 15.0)))
end

MATLAB

function tmp = code(z)
	tmp = (263.3831869810514 * (exp(-7.5) / z)) * sqrt((pi * 15.0));
end

Wolfram

code[z_] := N[(N[(263.3831869810514 * N[(N[Exp[-7.5], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(Pi * 15.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

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

3. Simplified 98.4%

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

5. Taylor expanded in z around 0 96.2%

   \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(\left(\color{blue}{212.9540523020159} + \frac{-0.13857109526572012}{6 - z}\right) + \frac{12.507343278686905}{5 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

6. Taylor expanded in z around 0 95.9%

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

   1. associate-/l* 96.1%

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

   2. *-commutative 96.1%

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

8. Simplified 96.1%

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

   1. *-un-lft-identity 96.1%

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

   2. associate-/r/ 96.1%

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

   3. sqrt-unprod 96.1%

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

   4. metadata-eval 96.1%

      \[\leadsto 263.3831869810514 \cdot \left(\left(1 \cdot \left(\frac{e^{-7.5}}{z} \cdot \sqrt{\color{blue}{15}}\right)\right) \cdot \sqrt{\pi}\right) \]

10. Applied egg-rr 96.1%

    \[\leadsto 263.3831869810514 \cdot \left(\color{blue}{\left(1 \cdot \left(\frac{e^{-7.5}}{z} \cdot \sqrt{15}\right)\right)} \cdot \sqrt{\pi}\right) \]
11. Step-by-step derivation

    1. *-lft-identity 96.1%

       \[\leadsto 263.3831869810514 \cdot \left(\color{blue}{\left(\frac{e^{-7.5}}{z} \cdot \sqrt{15}\right)} \cdot \sqrt{\pi}\right) \]

12. Simplified 96.1%

    \[\leadsto 263.3831869810514 \cdot \left(\color{blue}{\left(\frac{e^{-7.5}}{z} \cdot \sqrt{15}\right)} \cdot \sqrt{\pi}\right) \]
13. Step-by-step derivation

    1. pow1 96.1%

       \[\leadsto \color{blue}{{\left(263.3831869810514 \cdot \left(\left(\frac{e^{-7.5}}{z} \cdot \sqrt{15}\right) \cdot \sqrt{\pi}\right)\right)}^{1}} \]

14. Applied egg-rr 95.7%

    \[\leadsto \color{blue}{{\left(263.3831869810514 \cdot \left(\frac{e^{-7.5}}{z} \cdot {\left(15 \cdot \pi\right)}^{0.5}\right)\right)}^{1}} \]
15. Step-by-step derivation

    1. unpow1 95.7%

       \[\leadsto \color{blue}{263.3831869810514 \cdot \left(\frac{e^{-7.5}}{z} \cdot {\left(15 \cdot \pi\right)}^{0.5}\right)} \]

    2. associate-*r* 95.7%

       \[\leadsto \color{blue}{\left(263.3831869810514 \cdot \frac{e^{-7.5}}{z}\right) \cdot {\left(15 \cdot \pi\right)}^{0.5}} \]

    3. unpow1/2 95.7%

       \[\leadsto \left(263.3831869810514 \cdot \frac{e^{-7.5}}{z}\right) \cdot \color{blue}{\sqrt{15 \cdot \pi}} \]

16. Simplified 95.7%

    \[\leadsto \color{blue}{\left(263.3831869810514 \cdot \frac{e^{-7.5}}{z}\right) \cdot \sqrt{15 \cdot \pi}} \]

17. Final simplification 95.7%

    \[\leadsto \left(263.3831869810514 \cdot \frac{e^{-7.5}}{z}\right) \cdot \sqrt{\pi \cdot 15} \]
18. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(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))))))