Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.4% → 98.5%

Time: 59.0s

Alternatives: 14

Speedup: 1.2×

• could not determine a ground truth

  1. z = -7.527923997849419e+195

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: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.4%

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

3. Simplified 97.2%

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

6. Applied egg-rr 98.8%

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

8. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{\left(\pi \cdot \left(\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right) \cdot \left(e^{\left(-1 + z\right) + -6.5} \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(0.9999999999998099 + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\right)\right)\right)}{\sin \left(z \cdot \pi\right)}} \]

9. Taylor expanded in z around inf 99.5%

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

10. Final simplification 99.5%

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

Alternative 2: 97.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.4%

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

3. Simplified 97.3%

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

   1. pow1 97.3%

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

6. Applied egg-rr 97.3%

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

8. Simplified 98.8%

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

9. Final simplification 98.8%

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

Alternative 3: 97.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\pi \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right)\right) \cdot \left(e^{z - 7.5} \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(0.9999999999998099 + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{-1259.1392167224028}{2 - z} - \frac{771.3234287776531}{z - 3}\right)\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)\right)\right)\right)}{\sin \left(\pi \cdot z\right)} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (/
  (*
   (* PI (* (pow (- 7.5 z) (- 0.5 z)) (sqrt (* PI 2.0))))
   (*
    (exp (- z 7.5))
    (+
     (/ 676.5203681218851 (- 1.0 z))
     (+
      (+
       0.9999999999998099
       (+
        (/ -176.6150291621406 (- 4.0 z))
        (- (/ -1259.1392167224028 (- 2.0 z)) (/ 771.3234287776531 (- z 3.0)))))
      (+
       2.4783749183520145
       (*
        z
        (+
         0.49644474017195733
         (* z (+ 0.09941724278406093 (* z 0.01990483129967024))))))))))
  (sin (* PI z))))

C

double code(double z) {
	return ((((double) M_PI) * (pow((7.5 - z), (0.5 - z)) * sqrt((((double) M_PI) * 2.0)))) * (exp((z - 7.5)) * ((676.5203681218851 / (1.0 - z)) + ((0.9999999999998099 + ((-176.6150291621406 / (4.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) - (771.3234287776531 / (z - 3.0))))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * (0.09941724278406093 + (z * 0.01990483129967024)))))))))) / sin((((double) M_PI) * z));
}

Java

public static double code(double z) {
	return ((Math.PI * (Math.pow((7.5 - z), (0.5 - z)) * Math.sqrt((Math.PI * 2.0)))) * (Math.exp((z - 7.5)) * ((676.5203681218851 / (1.0 - z)) + ((0.9999999999998099 + ((-176.6150291621406 / (4.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) - (771.3234287776531 / (z - 3.0))))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * (0.09941724278406093 + (z * 0.01990483129967024)))))))))) / Math.sin((Math.PI * z));
}

Python

def code(z):
	return ((math.pi * (math.pow((7.5 - z), (0.5 - z)) * math.sqrt((math.pi * 2.0)))) * (math.exp((z - 7.5)) * ((676.5203681218851 / (1.0 - z)) + ((0.9999999999998099 + ((-176.6150291621406 / (4.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) - (771.3234287776531 / (z - 3.0))))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * (0.09941724278406093 + (z * 0.01990483129967024)))))))))) / math.sin((math.pi * z))

Julia

function code(z)
	return Float64(Float64(Float64(pi * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * sqrt(Float64(pi * 2.0)))) * Float64(exp(Float64(z - 7.5)) * Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(0.9999999999998099 + Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) + Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) - Float64(771.3234287776531 / Float64(z - 3.0))))) + Float64(2.4783749183520145 + Float64(z * Float64(0.49644474017195733 + Float64(z * Float64(0.09941724278406093 + Float64(z * 0.01990483129967024)))))))))) / sin(Float64(pi * z)))
end

MATLAB

function tmp = code(z)
	tmp = ((pi * (((7.5 - z) ^ (0.5 - z)) * sqrt((pi * 2.0)))) * (exp((z - 7.5)) * ((676.5203681218851 / (1.0 - z)) + ((0.9999999999998099 + ((-176.6150291621406 / (4.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) - (771.3234287776531 / (z - 3.0))))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * (0.09941724278406093 + (z * 0.01990483129967024)))))))))) / sin((pi * z));
end

Wolfram

code[z_] := N[(N[(N[(Pi * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(z - 7.5), $MachinePrecision]], $MachinePrecision] * N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(0.9999999999998099 + N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] - N[(771.3234287776531 / N[(z - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.4783749183520145 + N[(z * N[(0.49644474017195733 + N[(z * N[(0.09941724278406093 + N[(z * 0.01990483129967024), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.4%

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

3. Simplified 97.2%

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

6. Applied egg-rr 98.8%

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

8. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{\left(\pi \cdot \left(\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right) \cdot \left(e^{\left(-1 + z\right) + -6.5} \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(0.9999999999998099 + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\right)\right)\right)}{\sin \left(z \cdot \pi\right)}} \]

9. Taylor expanded in z around inf 99.5%

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

10. Taylor expanded in z around 0 98.7%

    \[\leadsto \frac{\left(\pi \cdot \left(\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right) \cdot \left(e^{z - 7.5} \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(0.9999999999998099 + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \color{blue}{\left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + 0.01990483129967024 \cdot z\right)\right)\right)}\right)\right)\right)}{\sin \left(z \cdot \pi\right)} \]
11. Step-by-step derivation

    1. *-commutative 98.7%

       \[\leadsto \frac{\left(\pi \cdot \left(\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right) \cdot \left(e^{z - 7.5} \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(0.9999999999998099 + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + \color{blue}{z \cdot 0.01990483129967024}\right)\right)\right)\right)\right)\right)}{\sin \left(z \cdot \pi\right)} \]

12. Simplified 98.7%

    \[\leadsto \frac{\left(\pi \cdot \left(\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right) \cdot \left(e^{z - 7.5} \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(0.9999999999998099 + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \color{blue}{\left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)}\right)\right)\right)}{\sin \left(z \cdot \pi\right)} \]

13. Final simplification 98.7%

    \[\leadsto \frac{\left(\pi \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right)\right) \cdot \left(e^{z - 7.5} \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(0.9999999999998099 + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{-1259.1392167224028}{2 - z} - \frac{771.3234287776531}{z - 3}\right)\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot \left(0.09941724278406093 + z \cdot 0.01990483129967024\right)\right)\right)\right)\right)\right)}{\sin \left(\pi \cdot z\right)} \]
14. Add Preprocessing

Alternative 4: 97.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\pi \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right)\right) \cdot \left(e^{z - 7.5} \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(0.9999999999998099 + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{-1259.1392167224028}{2 - z} - \frac{771.3234287776531}{z - 3}\right)\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)\right)\right)\right)}{\sin \left(\pi \cdot z\right)} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (/
  (*
   (* PI (* (pow (- 7.5 z) (- 0.5 z)) (sqrt (* PI 2.0))))
   (*
    (exp (- z 7.5))
    (+
     (/ 676.5203681218851 (- 1.0 z))
     (+
      (+
       0.9999999999998099
       (+
        (/ -176.6150291621406 (- 4.0 z))
        (- (/ -1259.1392167224028 (- 2.0 z)) (/ 771.3234287776531 (- z 3.0)))))
      (+
       2.4783749183520145
       (* z (+ 0.49644474017195733 (* z 0.09941724278406093))))))))
  (sin (* PI z))))

C

double code(double z) {
	return ((((double) M_PI) * (pow((7.5 - z), (0.5 - z)) * sqrt((((double) M_PI) * 2.0)))) * (exp((z - 7.5)) * ((676.5203681218851 / (1.0 - z)) + ((0.9999999999998099 + ((-176.6150291621406 / (4.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) - (771.3234287776531 / (z - 3.0))))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * 0.09941724278406093)))))))) / sin((((double) M_PI) * z));
}

Java

public static double code(double z) {
	return ((Math.PI * (Math.pow((7.5 - z), (0.5 - z)) * Math.sqrt((Math.PI * 2.0)))) * (Math.exp((z - 7.5)) * ((676.5203681218851 / (1.0 - z)) + ((0.9999999999998099 + ((-176.6150291621406 / (4.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) - (771.3234287776531 / (z - 3.0))))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * 0.09941724278406093)))))))) / Math.sin((Math.PI * z));
}

Python

def code(z):
	return ((math.pi * (math.pow((7.5 - z), (0.5 - z)) * math.sqrt((math.pi * 2.0)))) * (math.exp((z - 7.5)) * ((676.5203681218851 / (1.0 - z)) + ((0.9999999999998099 + ((-176.6150291621406 / (4.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) - (771.3234287776531 / (z - 3.0))))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * 0.09941724278406093)))))))) / math.sin((math.pi * z))

Julia

function code(z)
	return Float64(Float64(Float64(pi * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * sqrt(Float64(pi * 2.0)))) * Float64(exp(Float64(z - 7.5)) * Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(0.9999999999998099 + Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) + Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) - Float64(771.3234287776531 / Float64(z - 3.0))))) + Float64(2.4783749183520145 + Float64(z * Float64(0.49644474017195733 + Float64(z * 0.09941724278406093)))))))) / sin(Float64(pi * z)))
end

MATLAB

function tmp = code(z)
	tmp = ((pi * (((7.5 - z) ^ (0.5 - z)) * sqrt((pi * 2.0)))) * (exp((z - 7.5)) * ((676.5203681218851 / (1.0 - z)) + ((0.9999999999998099 + ((-176.6150291621406 / (4.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) - (771.3234287776531 / (z - 3.0))))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * 0.09941724278406093)))))))) / sin((pi * z));
end

Wolfram

code[z_] := N[(N[(N[(Pi * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(z - 7.5), $MachinePrecision]], $MachinePrecision] * N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(0.9999999999998099 + N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] - N[(771.3234287776531 / N[(z - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.4783749183520145 + N[(z * N[(0.49644474017195733 + N[(z * 0.09941724278406093), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.4%

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

3. Simplified 97.2%

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

6. Applied egg-rr 98.8%

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

8. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{\left(\pi \cdot \left(\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right) \cdot \left(e^{\left(-1 + z\right) + -6.5} \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(0.9999999999998099 + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\right)\right)\right)}{\sin \left(z \cdot \pi\right)}} \]

9. Taylor expanded in z around inf 99.5%

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

10. Taylor expanded in z around 0 98.4%

    \[\leadsto \frac{\left(\pi \cdot \left(\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right) \cdot \left(e^{z - 7.5} \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(0.9999999999998099 + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \color{blue}{\left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + 0.09941724278406093 \cdot z\right)\right)}\right)\right)\right)}{\sin \left(z \cdot \pi\right)} \]
11. Step-by-step derivation

    1. *-commutative 98.4%

       \[\leadsto \frac{\left(\pi \cdot \left(\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right) \cdot \left(e^{z - 7.5} \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(0.9999999999998099 + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + \color{blue}{z \cdot 0.09941724278406093}\right)\right)\right)\right)\right)}{\sin \left(z \cdot \pi\right)} \]

12. Simplified 98.4%

    \[\leadsto \frac{\left(\pi \cdot \left(\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right) \cdot \left(e^{z - 7.5} \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(0.9999999999998099 + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \color{blue}{\left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)}\right)\right)\right)}{\sin \left(z \cdot \pi\right)} \]

13. Final simplification 98.4%

    \[\leadsto \frac{\left(\pi \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right)\right) \cdot \left(e^{z - 7.5} \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(0.9999999999998099 + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{-1259.1392167224028}{2 - z} - \frac{771.3234287776531}{z - 3}\right)\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)\right)\right)\right)}{\sin \left(\pi \cdot z\right)} \]
14. Add Preprocessing

Alternative 5: 97.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.4%

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

3. Simplified 97.2%

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

6. Applied egg-rr 98.8%

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

8. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{\left(\pi \cdot \left(\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right) \cdot \left(e^{\left(-1 + z\right) + -6.5} \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(0.9999999999998099 + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\right)\right)\right)}{\sin \left(z \cdot \pi\right)}} \]

9. Taylor expanded in z around inf 99.5%

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

10. Taylor expanded in z around 0 97.9%

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

    1. *-commutative 97.9%

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

12. Simplified 97.9%

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

13. Final simplification 97.9%

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

Alternative 6: 96.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(z + -1\right) - 6.5}\right)\right) \cdot \left(\left(\left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right) + \left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + z \cdot 606.656776085461\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (sqrt (* PI 2.0))
   (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp (- (+ z -1.0) 6.5))))
  (*
   (+
    (+
     2.4783749183520145
     (* z (+ 0.49644474017195733 (* z 0.09941724278406093))))
    (+
     260.9048120626994
     (*
      z
      (+
       436.3997278161676
       (* z (+ 544.9358906000987 (* z 606.656776085461)))))))
   (/ PI (sin (* PI z))))))

C

double code(double z) {
	return (sqrt((((double) M_PI) * 2.0)) * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(((z + -1.0) - 6.5)))) * (((2.4783749183520145 + (z * (0.49644474017195733 + (z * 0.09941724278406093)))) + (260.9048120626994 + (z * (436.3997278161676 + (z * (544.9358906000987 + (z * 606.656776085461))))))) * (((double) M_PI) / sin((((double) M_PI) * z))));
}

Java

public static double code(double z) {
	return (Math.sqrt((Math.PI * 2.0)) * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp(((z + -1.0) - 6.5)))) * (((2.4783749183520145 + (z * (0.49644474017195733 + (z * 0.09941724278406093)))) + (260.9048120626994 + (z * (436.3997278161676 + (z * (544.9358906000987 + (z * 606.656776085461))))))) * (Math.PI / Math.sin((Math.PI * z))));
}

Python

def code(z):
	return (math.sqrt((math.pi * 2.0)) * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp(((z + -1.0) - 6.5)))) * (((2.4783749183520145 + (z * (0.49644474017195733 + (z * 0.09941724278406093)))) + (260.9048120626994 + (z * (436.3997278161676 + (z * (544.9358906000987 + (z * 606.656776085461))))))) * (math.pi / math.sin((math.pi * z))))

Julia

function code(z)
	return Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(Float64(Float64(z + -1.0) - 6.5)))) * Float64(Float64(Float64(2.4783749183520145 + Float64(z * Float64(0.49644474017195733 + Float64(z * 0.09941724278406093)))) + Float64(260.9048120626994 + Float64(z * Float64(436.3997278161676 + Float64(z * Float64(544.9358906000987 + Float64(z * 606.656776085461))))))) * Float64(pi / sin(Float64(pi * z)))))
end

MATLAB

function tmp = code(z)
	tmp = (sqrt((pi * 2.0)) * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(((z + -1.0) - 6.5)))) * (((2.4783749183520145 + (z * (0.49644474017195733 + (z * 0.09941724278406093)))) + (260.9048120626994 + (z * (436.3997278161676 + (z * (544.9358906000987 + (z * 606.656776085461))))))) * (pi / sin((pi * z))));
end

Wolfram

code[z_] := N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(z + -1.0), $MachinePrecision] - 6.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(2.4783749183520145 + N[(z * N[(0.49644474017195733 + N[(z * 0.09941724278406093), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(260.9048120626994 + N[(z * N[(436.3997278161676 + N[(z * N[(544.9358906000987 + N[(z * 606.656776085461), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.4%

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

3. Simplified 97.2%

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

5. Taylor expanded in z around 0 96.1%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \color{blue}{\left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + 0.09941724278406093 \cdot z\right)\right)}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
6. Step-by-step derivation

   1. *-commutative 96.1%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + \color{blue}{z \cdot 0.09941724278406093}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

7. Simplified 96.1%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \color{blue}{\left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

8. Taylor expanded in z around 0 97.0%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + 606.656776085461 \cdot z\right)\right)\right)} + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
9. Step-by-step derivation

   1. *-commutative 97.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + \color{blue}{z \cdot 606.656776085461}\right)\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

10. Simplified 97.0%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + z \cdot 606.656776085461\right)\right)\right)} + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

11. Final simplification 97.0%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(z + -1\right) - 6.5}\right)\right) \cdot \left(\left(\left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right) + \left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + z \cdot 606.656776085461\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
12. Add Preprocessing

Alternative 7: 96.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right) + \left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + z \cdot 606.656776085461\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (+
    (+
     2.4783749183520145
     (* z (+ 0.49644474017195733 (* z 0.09941724278406093))))
    (+
     260.9048120626994
     (*
      z
      (+
       436.3997278161676
       (* z (+ 544.9358906000987 (* z 606.656776085461)))))))
   (/ PI (sin (* PI z))))
  (* (sqrt (* PI 2.0)) (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5))))))

C

double code(double z) {
	return (((2.4783749183520145 + (z * (0.49644474017195733 + (z * 0.09941724278406093)))) + (260.9048120626994 + (z * (436.3997278161676 + (z * (544.9358906000987 + (z * 606.656776085461))))))) * (((double) M_PI) / sin((((double) M_PI) * z)))) * (sqrt((((double) M_PI) * 2.0)) * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5))));
}

Java

public static double code(double z) {
	return (((2.4783749183520145 + (z * (0.49644474017195733 + (z * 0.09941724278406093)))) + (260.9048120626994 + (z * (436.3997278161676 + (z * (544.9358906000987 + (z * 606.656776085461))))))) * (Math.PI / Math.sin((Math.PI * z)))) * (Math.sqrt((Math.PI * 2.0)) * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5))));
}

Python

def code(z):
	return (((2.4783749183520145 + (z * (0.49644474017195733 + (z * 0.09941724278406093)))) + (260.9048120626994 + (z * (436.3997278161676 + (z * (544.9358906000987 + (z * 606.656776085461))))))) * (math.pi / math.sin((math.pi * z)))) * (math.sqrt((math.pi * 2.0)) * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5))))

Julia

function code(z)
	return Float64(Float64(Float64(Float64(2.4783749183520145 + Float64(z * Float64(0.49644474017195733 + Float64(z * 0.09941724278406093)))) + Float64(260.9048120626994 + Float64(z * Float64(436.3997278161676 + Float64(z * Float64(544.9358906000987 + Float64(z * 606.656776085461))))))) * Float64(pi / sin(Float64(pi * z)))) * Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5)))))
end

MATLAB

function tmp = code(z)
	tmp = (((2.4783749183520145 + (z * (0.49644474017195733 + (z * 0.09941724278406093)))) + (260.9048120626994 + (z * (436.3997278161676 + (z * (544.9358906000987 + (z * 606.656776085461))))))) * (pi / sin((pi * z)))) * (sqrt((pi * 2.0)) * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5))));
end

Wolfram

code[z_] := N[(N[(N[(N[(2.4783749183520145 + N[(z * N[(0.49644474017195733 + N[(z * 0.09941724278406093), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(260.9048120626994 + N[(z * N[(436.3997278161676 + N[(z * N[(544.9358906000987 + N[(z * 606.656776085461), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.4%

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

3. Simplified 97.2%

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

5. Taylor expanded in z around 0 96.1%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \color{blue}{\left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + 0.09941724278406093 \cdot z\right)\right)}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
6. Step-by-step derivation

   1. *-commutative 96.1%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + \color{blue}{z \cdot 0.09941724278406093}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

7. Simplified 96.1%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \color{blue}{\left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

8. Taylor expanded in z around 0 97.0%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + 606.656776085461 \cdot z\right)\right)\right)} + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
9. Step-by-step derivation

   1. *-commutative 97.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + \color{blue}{z \cdot 606.656776085461}\right)\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

10. Simplified 97.0%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + z \cdot 606.656776085461\right)\right)\right)} + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

11. Taylor expanded in z around inf 96.9%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \color{blue}{\left(e^{\log \left(7.5 - z\right) \cdot \left(0.5 - z\right)} \cdot e^{z - 7.5}\right)}\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + z \cdot 606.656776085461\right)\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
12. Step-by-step derivation

    1. exp-to-pow 96.9%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left(\color{blue}{{\left(7.5 - z\right)}^{\left(0.5 - z\right)}} \cdot e^{z - 7.5}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + z \cdot 606.656776085461\right)\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

    2. sub-neg 96.9%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\color{blue}{z + \left(-7.5\right)}}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + z \cdot 606.656776085461\right)\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

    3. metadata-eval 96.9%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + \color{blue}{-7.5}}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + z \cdot 606.656776085461\right)\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

13. Simplified 96.9%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \color{blue}{\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)}\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + z \cdot 606.656776085461\right)\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

14. Final simplification 96.9%

    \[\leadsto \left(\left(\left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right) + \left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + z \cdot 606.656776085461\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \]
15. Add Preprocessing

Alternative 8: 96.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(z + -1\right) - 6.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(z \cdot 606.656776085461 + 545.0353078428827\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (sqrt (* PI 2.0))
   (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp (- (+ z -1.0) 6.5))))
  (*
   (/ PI (sin (* PI z)))
   (+
    263.3831869810514
    (*
     z
     (+
      436.8961725563396
      (* z (+ (* z 606.656776085461) 545.0353078428827))))))))

C

double code(double z) {
	return (sqrt((((double) M_PI) * 2.0)) * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(((z + -1.0) - 6.5)))) * ((((double) M_PI) / sin((((double) M_PI) * z))) * (263.3831869810514 + (z * (436.8961725563396 + (z * ((z * 606.656776085461) + 545.0353078428827))))));
}

Java

public static double code(double z) {
	return (Math.sqrt((Math.PI * 2.0)) * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp(((z + -1.0) - 6.5)))) * ((Math.PI / Math.sin((Math.PI * z))) * (263.3831869810514 + (z * (436.8961725563396 + (z * ((z * 606.656776085461) + 545.0353078428827))))));
}

Python

def code(z):
	return (math.sqrt((math.pi * 2.0)) * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp(((z + -1.0) - 6.5)))) * ((math.pi / math.sin((math.pi * z))) * (263.3831869810514 + (z * (436.8961725563396 + (z * ((z * 606.656776085461) + 545.0353078428827))))))

Julia

function code(z)
	return Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(Float64(Float64(z + -1.0) - 6.5)))) * Float64(Float64(pi / sin(Float64(pi * z))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * Float64(Float64(z * 606.656776085461) + 545.0353078428827)))))))
end

MATLAB

function tmp = code(z)
	tmp = (sqrt((pi * 2.0)) * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(((z + -1.0) - 6.5)))) * ((pi / sin((pi * z))) * (263.3831869810514 + (z * (436.8961725563396 + (z * ((z * 606.656776085461) + 545.0353078428827))))));
end

Wolfram

code[z_] := N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(z + -1.0), $MachinePrecision] - 6.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * N[(N[(z * 606.656776085461), $MachinePrecision] + 545.0353078428827), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.4%

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

3. Simplified 97.2%

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

5. Taylor expanded in z around 0 96.1%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \color{blue}{\left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + 0.09941724278406093 \cdot z\right)\right)}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
6. Step-by-step derivation

   1. *-commutative 96.1%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + \color{blue}{z \cdot 0.09941724278406093}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

7. Simplified 96.1%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \color{blue}{\left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

8. Taylor expanded in z around 0 97.0%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + 606.656776085461 \cdot z\right)\right)\right)} + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
9. Step-by-step derivation

   1. *-commutative 97.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + \color{blue}{z \cdot 606.656776085461}\right)\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

10. Simplified 97.0%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + z \cdot 606.656776085461\right)\right)\right)} + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

11. Taylor expanded in z around 0 96.8%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.656776085461 \cdot z\right)\right)\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
12. Step-by-step derivation

    1. *-commutative 96.8%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + \color{blue}{z \cdot 606.656776085461}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

13. Simplified 96.8%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + z \cdot 606.656776085461\right)\right)\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

14. Final simplification 96.8%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(z + -1\right) - 6.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(z \cdot 606.656776085461 + 545.0353078428827\right)\right)\right)\right) \]
15. Add Preprocessing

Alternative 9: 96.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(z + -1\right) - 6.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (sqrt (* PI 2.0))
   (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp (- (+ z -1.0) 6.5))))
  (*
   (/ PI (sin (* PI z)))
   (+ 263.3831869810514 (* z (+ 436.8961725563396 (* z 545.0353078428827)))))))

C

double code(double z) {
	return (sqrt((((double) M_PI) * 2.0)) * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(((z + -1.0) - 6.5)))) * ((((double) M_PI) / sin((((double) M_PI) * z))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))));
}

Java

public static double code(double z) {
	return (Math.sqrt((Math.PI * 2.0)) * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp(((z + -1.0) - 6.5)))) * ((Math.PI / Math.sin((Math.PI * z))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))));
}

Python

def code(z):
	return (math.sqrt((math.pi * 2.0)) * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp(((z + -1.0) - 6.5)))) * ((math.pi / math.sin((math.pi * z))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))))

Julia

function code(z)
	return Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(Float64(Float64(z + -1.0) - 6.5)))) * Float64(Float64(pi / sin(Float64(pi * z))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * 545.0353078428827))))))
end

MATLAB

function tmp = code(z)
	tmp = (sqrt((pi * 2.0)) * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(((z + -1.0) - 6.5)))) * ((pi / sin((pi * z))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))));
end

Wolfram

code[z_] := N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(z + -1.0), $MachinePrecision] - 6.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * 545.0353078428827), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.4%

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

3. Simplified 97.2%

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

5. Taylor expanded in z around 0 96.6%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
6. Step-by-step derivation

   1. *-commutative 96.6%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot \left(436.8961725563396 + \color{blue}{z \cdot 545.0353078428827}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

7. Simplified 96.6%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

8. Final simplification 96.6%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(z + -1\right) - 6.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)\right) \]
9. Add Preprocessing

Alternative 10: 96.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + z \cdot 9.52240454844455\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (* (sqrt (* PI 2.0)) (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5))))
  (*
   (/ PI (sin (* PI z)))
   (+
    263.3831869810514
    (*
     z
     (+
      436.8961725563396
      (* z (+ 545.0353078428827 (* z 9.52240454844455)))))))))

C

double code(double z) {
	return (sqrt((((double) M_PI) * 2.0)) * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5)))) * ((((double) M_PI) / sin((((double) M_PI) * z))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (z * 9.52240454844455)))))));
}

Java

public static double code(double z) {
	return (Math.sqrt((Math.PI * 2.0)) * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5)))) * ((Math.PI / Math.sin((Math.PI * z))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (z * 9.52240454844455)))))));
}

Python

def code(z):
	return (math.sqrt((math.pi * 2.0)) * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5)))) * ((math.pi / math.sin((math.pi * z))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (z * 9.52240454844455)))))))

Julia

function code(z)
	return Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5)))) * Float64(Float64(pi / sin(Float64(pi * z))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * Float64(545.0353078428827 + Float64(z * 9.52240454844455))))))))
end

MATLAB

function tmp = code(z)
	tmp = (sqrt((pi * 2.0)) * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5)))) * ((pi / sin((pi * z))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (z * 9.52240454844455)))))));
end

Wolfram

code[z_] := N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * N[(545.0353078428827 + N[(z * 9.52240454844455), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.4%

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

3. Simplified 97.3%

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

5. Taylor expanded in z around 0 95.5%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\color{blue}{\left(46.9507597606837 + z \cdot \left(361.7355639412844 + 519.1279660315847 \cdot z\right)\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. *-commutative 95.5%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(46.9507597606837 + z \cdot \left(361.7355639412844 + \color{blue}{z \cdot 519.1279660315847}\right)\right) + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]

7. Simplified 95.5%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\color{blue}{\left(46.9507597606837 + z \cdot \left(361.7355639412844 + z \cdot 519.1279660315847\right)\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]

8. Taylor expanded in z around 0 95.5%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(46.9507597606837 + z \cdot \left(361.7355639412844 + z \cdot 519.1279660315847\right)\right) + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \color{blue}{\left(z \cdot \left(-2.659551084428952 \cdot z - 10.53814559148631\right) - 41.65228863479777\right)}\right)\right)\right) \]

9. Taylor expanded in z around 0 96.6%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 9.52240454844455 \cdot z\right)\right)\right)}\right) \]
10. Step-by-step derivation

    1. *-commutative 96.6%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + \color{blue}{z \cdot 9.52240454844455}\right)\right)\right)\right) \]

11. Simplified 96.6%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + z \cdot 9.52240454844455\right)\right)\right)}\right) \]

12. Final simplification 96.6%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + z \cdot 9.52240454844455\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 11: 96.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (* (sqrt (* PI 2.0)) (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5))))
  (*
   (/ PI (sin (* PI z)))
   (+ 263.3831869810514 (* z (+ 436.8961725563396 (* z 545.0353078428827)))))))

C

double code(double z) {
	return (sqrt((((double) M_PI) * 2.0)) * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5)))) * ((((double) M_PI) / sin((((double) M_PI) * z))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))));
}

Java

public static double code(double z) {
	return (Math.sqrt((Math.PI * 2.0)) * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5)))) * ((Math.PI / Math.sin((Math.PI * z))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))));
}

Python

def code(z):
	return (math.sqrt((math.pi * 2.0)) * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5)))) * ((math.pi / math.sin((math.pi * z))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))))

Julia

function code(z)
	return Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5)))) * Float64(Float64(pi / sin(Float64(pi * z))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * 545.0353078428827))))))
end

MATLAB

function tmp = code(z)
	tmp = (sqrt((pi * 2.0)) * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5)))) * ((pi / sin((pi * z))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))));
end

Wolfram

code[z_] := N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * 545.0353078428827), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.4%

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

3. Simplified 97.3%

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

5. Taylor expanded in z around 0 95.5%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\color{blue}{\left(46.9507597606837 + z \cdot \left(361.7355639412844 + 519.1279660315847 \cdot z\right)\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. *-commutative 95.5%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(46.9507597606837 + z \cdot \left(361.7355639412844 + \color{blue}{z \cdot 519.1279660315847}\right)\right) + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]

7. Simplified 95.5%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\color{blue}{\left(46.9507597606837 + z \cdot \left(361.7355639412844 + z \cdot 519.1279660315847\right)\right)} + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right)\right)\right)\right) \]

8. Taylor expanded in z around 0 95.5%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(46.9507597606837 + z \cdot \left(361.7355639412844 + z \cdot 519.1279660315847\right)\right) + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{6 - z} + \color{blue}{\left(z \cdot \left(-2.659551084428952 \cdot z - 10.53814559148631\right) - 41.65228863479777\right)}\right)\right)\right) \]

9. Taylor expanded in z around 0 96.5%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\right)}\right) \]
10. Step-by-step derivation

    1. *-commutative 96.5%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + \color{blue}{z \cdot 545.0353078428827}\right)\right)\right) \]

11. Simplified 96.5%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)}\right) \]

12. Final simplification 96.5%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)\right) \]
13. Add Preprocessing

Alternative 12: 96.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.4%

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

3. Simplified 97.2%

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

5. Taylor expanded in z around 0 95.9%

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

   1. *-commutative 95.9%

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

7. Simplified 95.9%

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

8. Final simplification 95.9%

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

Alternative 13: 96.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.4%

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

3. Simplified 97.2%

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

5. Taylor expanded in z around 0 94.5%

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

6. Taylor expanded in z around 0 94.7%

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

   1. associate-*r* 95.4%

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

8. Simplified 95.4%

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

   1. associate-*r/ 95.6%

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

   2. associate-*l* 94.6%

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

   3. sqrt-unprod 94.6%

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

   4. metadata-eval 94.6%

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

10. Applied egg-rr 94.6%

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

    1. *-commutative 94.6%

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

    2. associate-/l* 95.0%

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

12. Simplified 95.0%

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

    1. pow1 95.0%

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

    2. associate-*r* 95.6%

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

14. Applied egg-rr 95.6%

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

    1. unpow1 95.6%

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

    2. *-commutative 95.6%

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

    3. *-commutative 95.6%

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

16. Simplified 95.6%

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

Alternative 14: 95.6% accurate, 1.7× speedup

Math

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

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(Float64(sqrt(15.0) * 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[(N[Sqrt[15.0], $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.4%

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

3. Simplified 97.2%

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

5. Taylor expanded in z around 0 94.5%

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

6. Taylor expanded in z around 0 94.7%

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

   1. associate-*r* 95.4%

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

8. Simplified 95.4%

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

   1. associate-*r/ 95.6%

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

   2. associate-*l* 94.6%

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

   3. sqrt-unprod 94.6%

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

   4. metadata-eval 94.6%

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

10. Applied egg-rr 94.6%

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

    1. *-commutative 94.6%

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

    2. associate-/l* 95.0%

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

12. Simplified 95.0%

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

13. Taylor expanded in z around 0 95.1%

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

14. Final simplification 95.1%

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

Reproduce

herbie shell --seed 2024152 
(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))))))