Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.5% → 98.4%

Time: 1.1min

Alternatives: 9

Speedup: 1.4×

• could not determine a ground truth

  1. z = -3.9733988770036165e+181

Specification

\[z \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  PI
  (/
   1.0
   (/
    (sin (* PI z))
    (*
     (*
      (pow (- 7.5 z) (- 0.5 z))
      (* (exp (+ -1.0 (+ z -6.5))) (sqrt (* PI 2.0))))
     (+
      (/ 676.5203681218851 (- 1.0 z))
      (-
       0.9999999999998099
       (-
        (+
         (/ -176.6150291621406 (- z 4.0))
         (+ (/ -1259.1392167224028 (- z 2.0)) (/ 771.3234287776531 (- z 3.0))))
        (-
         (/ 12.507343278686905 (- 5.0 z))
         (-
          (+
           (/ 9.984369578019572e-6 (- z 7.0))
           (/ 1.5056327351493116e-7 (- z 8.0)))
          (/ -0.13857109526572012 (- 6.0 z))))))))))))

C

double code(double z) {
	return ((double) M_PI) * (1.0 / (sin((((double) M_PI) * z)) / ((pow((7.5 - z), (0.5 - z)) * (exp((-1.0 + (z + -6.5))) * sqrt((((double) M_PI) * 2.0)))) * ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 - (((-176.6150291621406 / (z - 4.0)) + ((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0)))) - ((12.507343278686905 / (5.0 - z)) - (((9.984369578019572e-6 / (z - 7.0)) + (1.5056327351493116e-7 / (z - 8.0))) - (-0.13857109526572012 / (6.0 - z))))))))));
}

Java

public static double code(double z) {
	return Math.PI * (1.0 / (Math.sin((Math.PI * z)) / ((Math.pow((7.5 - z), (0.5 - z)) * (Math.exp((-1.0 + (z + -6.5))) * Math.sqrt((Math.PI * 2.0)))) * ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 - (((-176.6150291621406 / (z - 4.0)) + ((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0)))) - ((12.507343278686905 / (5.0 - z)) - (((9.984369578019572e-6 / (z - 7.0)) + (1.5056327351493116e-7 / (z - 8.0))) - (-0.13857109526572012 / (6.0 - z))))))))));
}

Python

def code(z):
	return math.pi * (1.0 / (math.sin((math.pi * z)) / ((math.pow((7.5 - z), (0.5 - z)) * (math.exp((-1.0 + (z + -6.5))) * math.sqrt((math.pi * 2.0)))) * ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 - (((-176.6150291621406 / (z - 4.0)) + ((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0)))) - ((12.507343278686905 / (5.0 - z)) - (((9.984369578019572e-6 / (z - 7.0)) + (1.5056327351493116e-7 / (z - 8.0))) - (-0.13857109526572012 / (6.0 - z))))))))))

Julia

function code(z)
	return Float64(pi * Float64(1.0 / Float64(sin(Float64(pi * z)) / Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * Float64(exp(Float64(-1.0 + Float64(z + -6.5))) * sqrt(Float64(pi * 2.0)))) * Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(0.9999999999998099 - Float64(Float64(Float64(-176.6150291621406 / Float64(z - 4.0)) + Float64(Float64(-1259.1392167224028 / Float64(z - 2.0)) + Float64(771.3234287776531 / Float64(z - 3.0)))) - Float64(Float64(12.507343278686905 / Float64(5.0 - z)) - Float64(Float64(Float64(9.984369578019572e-6 / Float64(z - 7.0)) + Float64(1.5056327351493116e-7 / Float64(z - 8.0))) - Float64(-0.13857109526572012 / Float64(6.0 - z)))))))))))
end

MATLAB

function tmp = code(z)
	tmp = pi * (1.0 / (sin((pi * z)) / ((((7.5 - z) ^ (0.5 - z)) * (exp((-1.0 + (z + -6.5))) * sqrt((pi * 2.0)))) * ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 - (((-176.6150291621406 / (z - 4.0)) + ((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0)))) - ((12.507343278686905 / (5.0 - z)) - (((9.984369578019572e-6 / (z - 7.0)) + (1.5056327351493116e-7 / (z - 8.0))) - (-0.13857109526572012 / (6.0 - z))))))))));
end

Wolfram

code[z_] := N[(Pi * N[(1.0 / N[(N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision] / N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(-1.0 + N[(z + -6.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 - 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] - N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(9.984369578019572e-6 / N[(z - 7.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(z - 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

3. Simplified 96.1%

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

6. Applied egg-rr 97.7%

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

8. Simplified 98.0%

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

   1. distribute-rgt-in 98.1%

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

10. Applied egg-rr 98.1%

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

    1. clear-num 98.1%

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

    2. inv-pow 98.1%

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

12. Applied egg-rr 98.1%

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

14. Simplified 98.1%

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

15. Final simplification 98.1%

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

Alternative 2: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

3. Simplified 96.1%

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

6. Applied egg-rr 97.7%

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

8. Simplified 98.0%

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

   1. distribute-lft-in 98.1%

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

10. Applied egg-rr 98.1%

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

12. Simplified 98.1%

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

13. Final simplification 98.1%

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

Alternative 3: 97.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 + \left(-1 + \left(1 - z\right)\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-6 + \left(z + -1\right)\right) + -0.5}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + z \cdot 606.6766809125655\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (*
    (sqrt (* PI 2.0))
    (*
     (pow (+ 7.5 (+ -1.0 (- 1.0 z))) (- (- 1.0 z) 0.5))
     (exp (+ (+ -6.0 (+ z -1.0)) -0.5))))
   (/ PI (sin (* PI z))))
  (+
   (+
    263.3831855358925
    (*
     z
     (+
      436.8961723502244
      (* z (+ 545.0353078134797 (* z 606.6766809125655))))))
   (+
    (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0))
    (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0))))))

C

double code(double z) {
	return ((sqrt((((double) M_PI) * 2.0)) * (pow((7.5 + (-1.0 + (1.0 - z))), ((1.0 - z) - 0.5)) * exp(((-6.0 + (z + -1.0)) + -0.5)))) * (((double) M_PI) / sin((((double) M_PI) * z)))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))));
}

Java

public static double code(double z) {
	return ((Math.sqrt((Math.PI * 2.0)) * (Math.pow((7.5 + (-1.0 + (1.0 - z))), ((1.0 - z) - 0.5)) * Math.exp(((-6.0 + (z + -1.0)) + -0.5)))) * (Math.PI / Math.sin((Math.PI * z)))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))));
}

Python

def code(z):
	return ((math.sqrt((math.pi * 2.0)) * (math.pow((7.5 + (-1.0 + (1.0 - z))), ((1.0 - z) - 0.5)) * math.exp(((-6.0 + (z + -1.0)) + -0.5)))) * (math.pi / math.sin((math.pi * z)))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))))

Julia

function code(z)
	return Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(7.5 + Float64(-1.0 + Float64(1.0 - z))) ^ Float64(Float64(1.0 - z) - 0.5)) * exp(Float64(Float64(-6.0 + Float64(z + -1.0)) + -0.5)))) * Float64(pi / sin(Float64(pi * z)))) * Float64(Float64(263.3831855358925 + Float64(z * Float64(436.8961723502244 + Float64(z * Float64(545.0353078134797 + Float64(z * 606.6766809125655)))))) + Float64(Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0)) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0)))))
end

MATLAB

function tmp = code(z)
	tmp = ((sqrt((pi * 2.0)) * (((7.5 + (-1.0 + (1.0 - z))) ^ ((1.0 - z) - 0.5)) * exp(((-6.0 + (z + -1.0)) + -0.5)))) * (pi / sin((pi * z)))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))));
end

Wolfram

code[z_] := N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(7.5 + N[(-1.0 + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(-6.0 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(263.3831855358925 + N[(z * N[(436.8961723502244 + N[(z * N[(545.0353078134797 + N[(z * 606.6766809125655), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

3. Simplified 97.9%

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

5. Taylor expanded in z around 0 97.2%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
6. Step-by-step derivation

   1. *-commutative 97.2%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + \color{blue}{z \cdot 606.6766809125655}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

7. Simplified 97.2%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + z \cdot 606.6766809125655\right)\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

8. Final simplification 97.2%

   \[\leadsto \left(\left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 + \left(-1 + \left(1 - z\right)\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-6 + \left(z + -1\right)\right) + -0.5}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + z \cdot 606.6766809125655\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
9. Add Preprocessing

Alternative 4: 97.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\left(\sqrt{\pi \cdot 2} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-6.5 - \left(1 - z\right)}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(z \cdot \left(z \cdot \left(z \cdot -69.84368720512434 - 131.4850602790024\right) - 239.6241955655455\right) - 413.1371811408337\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (* (sqrt (* PI 2.0)) (/ PI (sin (* PI z))))
   (* (pow (- 7.5 z) (- 0.5 z)) (exp (- -6.5 (- 1.0 z)))))
  (+
   (/ 676.5203681218851 (- 1.0 z))
   (-
    (*
     z
     (-
      (* z (- (* z -69.84368720512434) 131.4850602790024))
      239.6241955655455))
    413.1371811408337))))

C

double code(double z) {
	return ((sqrt((((double) M_PI) * 2.0)) * (((double) M_PI) / sin((((double) M_PI) * z)))) * (pow((7.5 - z), (0.5 - z)) * exp((-6.5 - (1.0 - z))))) * ((676.5203681218851 / (1.0 - z)) + ((z * ((z * ((z * -69.84368720512434) - 131.4850602790024)) - 239.6241955655455)) - 413.1371811408337));
}

Java

public static double code(double z) {
	return ((Math.sqrt((Math.PI * 2.0)) * (Math.PI / Math.sin((Math.PI * z)))) * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((-6.5 - (1.0 - z))))) * ((676.5203681218851 / (1.0 - z)) + ((z * ((z * ((z * -69.84368720512434) - 131.4850602790024)) - 239.6241955655455)) - 413.1371811408337));
}

Python

def code(z):
	return ((math.sqrt((math.pi * 2.0)) * (math.pi / math.sin((math.pi * z)))) * (math.pow((7.5 - z), (0.5 - z)) * math.exp((-6.5 - (1.0 - z))))) * ((676.5203681218851 / (1.0 - z)) + ((z * ((z * ((z * -69.84368720512434) - 131.4850602790024)) - 239.6241955655455)) - 413.1371811408337))

Julia

function code(z)
	return Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64(pi / sin(Float64(pi * z)))) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(-6.5 - Float64(1.0 - z))))) * Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * -69.84368720512434) - 131.4850602790024)) - 239.6241955655455)) - 413.1371811408337)))
end

MATLAB

function tmp = code(z)
	tmp = ((sqrt((pi * 2.0)) * (pi / sin((pi * z)))) * (((7.5 - z) ^ (0.5 - z)) * exp((-6.5 - (1.0 - z))))) * ((676.5203681218851 / (1.0 - z)) + ((z * ((z * ((z * -69.84368720512434) - 131.4850602790024)) - 239.6241955655455)) - 413.1371811408337));
end

Wolfram

code[z_] := N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-6.5 - N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(z * N[(N[(z * N[(N[(z * -69.84368720512434), $MachinePrecision] - 131.4850602790024), $MachinePrecision]), $MachinePrecision] - 239.6241955655455), $MachinePrecision]), $MachinePrecision] - 413.1371811408337), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

3. Simplified 96.1%

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

6. Applied egg-rr 98.0%

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

8. Simplified 97.7%

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

9. Taylor expanded in z around 0 97.1%

   \[\leadsto \left(\left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \color{blue}{\left(z \cdot \left(z \cdot \left(-69.84368720512434 \cdot z - 131.4850602790024\right) - 239.6241955655455\right) - 413.1371811408337\right)}\right) \]

10. Final simplification 97.1%

    \[\leadsto \left(\left(\sqrt{\pi \cdot 2} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-6.5 - \left(1 - z\right)}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(z \cdot \left(z \cdot \left(z \cdot -69.84368720512434 - 131.4850602790024\right) - 239.6241955655455\right) - 413.1371811408337\right)\right) \]
11. Add Preprocessing

Alternative 5: 96.9% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  PI
  (/
   1.0
   (/
    (* PI z)
    (*
     (*
      (pow (- 7.5 z) (- 0.5 z))
      (* (exp (+ -1.0 (+ z -6.5))) (sqrt (* PI 2.0))))
     (+
      (/ 676.5203681218851 (- 1.0 z))
      (-
       0.9999999999998099
       (-
        (+
         (/ -176.6150291621406 (- z 4.0))
         (+ (/ -1259.1392167224028 (- z 2.0)) (/ 771.3234287776531 (- z 3.0))))
        (-
         (/ 12.507343278686905 (- 5.0 z))
         (-
          (+
           (/ 9.984369578019572e-6 (- z 7.0))
           (/ 1.5056327351493116e-7 (- z 8.0)))
          (/ -0.13857109526572012 (- 6.0 z))))))))))))

C

double code(double z) {
	return ((double) M_PI) * (1.0 / ((((double) M_PI) * z) / ((pow((7.5 - z), (0.5 - z)) * (exp((-1.0 + (z + -6.5))) * sqrt((((double) M_PI) * 2.0)))) * ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 - (((-176.6150291621406 / (z - 4.0)) + ((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0)))) - ((12.507343278686905 / (5.0 - z)) - (((9.984369578019572e-6 / (z - 7.0)) + (1.5056327351493116e-7 / (z - 8.0))) - (-0.13857109526572012 / (6.0 - z))))))))));
}

Java

public static double code(double z) {
	return Math.PI * (1.0 / ((Math.PI * z) / ((Math.pow((7.5 - z), (0.5 - z)) * (Math.exp((-1.0 + (z + -6.5))) * Math.sqrt((Math.PI * 2.0)))) * ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 - (((-176.6150291621406 / (z - 4.0)) + ((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0)))) - ((12.507343278686905 / (5.0 - z)) - (((9.984369578019572e-6 / (z - 7.0)) + (1.5056327351493116e-7 / (z - 8.0))) - (-0.13857109526572012 / (6.0 - z))))))))));
}

Python

def code(z):
	return math.pi * (1.0 / ((math.pi * z) / ((math.pow((7.5 - z), (0.5 - z)) * (math.exp((-1.0 + (z + -6.5))) * math.sqrt((math.pi * 2.0)))) * ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 - (((-176.6150291621406 / (z - 4.0)) + ((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0)))) - ((12.507343278686905 / (5.0 - z)) - (((9.984369578019572e-6 / (z - 7.0)) + (1.5056327351493116e-7 / (z - 8.0))) - (-0.13857109526572012 / (6.0 - z))))))))))

Julia

function code(z)
	return Float64(pi * Float64(1.0 / Float64(Float64(pi * z) / Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * Float64(exp(Float64(-1.0 + Float64(z + -6.5))) * sqrt(Float64(pi * 2.0)))) * Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(0.9999999999998099 - Float64(Float64(Float64(-176.6150291621406 / Float64(z - 4.0)) + Float64(Float64(-1259.1392167224028 / Float64(z - 2.0)) + Float64(771.3234287776531 / Float64(z - 3.0)))) - Float64(Float64(12.507343278686905 / Float64(5.0 - z)) - Float64(Float64(Float64(9.984369578019572e-6 / Float64(z - 7.0)) + Float64(1.5056327351493116e-7 / Float64(z - 8.0))) - Float64(-0.13857109526572012 / Float64(6.0 - z)))))))))))
end

MATLAB

function tmp = code(z)
	tmp = pi * (1.0 / ((pi * z) / ((((7.5 - z) ^ (0.5 - z)) * (exp((-1.0 + (z + -6.5))) * sqrt((pi * 2.0)))) * ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 - (((-176.6150291621406 / (z - 4.0)) + ((-1259.1392167224028 / (z - 2.0)) + (771.3234287776531 / (z - 3.0)))) - ((12.507343278686905 / (5.0 - z)) - (((9.984369578019572e-6 / (z - 7.0)) + (1.5056327351493116e-7 / (z - 8.0))) - (-0.13857109526572012 / (6.0 - z))))))))));
end

Wolfram

code[z_] := N[(Pi * N[(1.0 / N[(N[(Pi * z), $MachinePrecision] / N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(-1.0 + N[(z + -6.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 - 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] - N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(9.984369578019572e-6 / N[(z - 7.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(z - 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

3. Simplified 96.1%

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

6. Applied egg-rr 97.7%

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

8. Simplified 98.0%

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

   1. distribute-rgt-in 98.1%

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

10. Applied egg-rr 98.1%

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

    1. clear-num 98.1%

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

    2. inv-pow 98.1%

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

12. Applied egg-rr 98.1%

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

14. Simplified 98.1%

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

15. Taylor expanded in z around 0 96.7%

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

16. Final simplification 96.7%

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

Alternative 6: 95.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

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)))) * ((263.3831869810514 + (z * 436.8961725563396)) / z);
}

Python

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

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(263.3831869810514 + Float64(z * 436.8961725563396)) / z))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

3. Simplified 96.2%

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

   1. add-cbrt-cube 96.2%

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

6. Applied egg-rr 96.2%

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

   1. associate-*l* 96.2%

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

8. Simplified 96.2%

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

9. Taylor expanded in z around 0 95.9%

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

    1. *-commutative 95.9%

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

11. Simplified 95.9%

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

12. Final simplification 95.9%

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

Alternative 7: 95.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\pi \cdot 2} \cdot \left(\frac{263.3831869810514}{z} \cdot \left(e^{-1 + \left(z + -6.5\right)} \cdot \sqrt{7.5}\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (sqrt (* PI 2.0))
  (* (/ 263.3831869810514 z) (* (exp (+ -1.0 (+ z -6.5))) (sqrt 7.5)))))

C

double code(double z) {
	return sqrt((((double) M_PI) * 2.0)) * ((263.3831869810514 / z) * (exp((-1.0 + (z + -6.5))) * sqrt(7.5)));
}

Java

public static double code(double z) {
	return Math.sqrt((Math.PI * 2.0)) * ((263.3831869810514 / z) * (Math.exp((-1.0 + (z + -6.5))) * Math.sqrt(7.5)));
}

Python

def code(z):
	return math.sqrt((math.pi * 2.0)) * ((263.3831869810514 / z) * (math.exp((-1.0 + (z + -6.5))) * math.sqrt(7.5)))

Julia

function code(z)
	return Float64(sqrt(Float64(pi * 2.0)) * Float64(Float64(263.3831869810514 / z) * Float64(exp(Float64(-1.0 + Float64(z + -6.5))) * sqrt(7.5))))
end

MATLAB

function tmp = code(z)
	tmp = sqrt((pi * 2.0)) * ((263.3831869810514 / z) * (exp((-1.0 + (z + -6.5))) * sqrt(7.5)));
end

Wolfram

code[z_] := N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(263.3831869810514 / z), $MachinePrecision] * N[(N[Exp[N[(-1.0 + N[(z + -6.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

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

3. Simplified 96.1%

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

5. Taylor expanded in z around 0 94.3%

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

6. Taylor expanded in z around 0 94.5%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\sqrt{7.5}} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \frac{263.3831869810514}{z} \]
7. Step-by-step derivation

   1. associate-*r/ 94.5%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\pi \cdot 2} \cdot \left(\sqrt{7.5} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot 263.3831869810514}{z}} \]

   2. distribute-neg-in 94.5%

      \[\leadsto \frac{\left(\sqrt{\pi \cdot 2} \cdot \left(\sqrt{7.5} \cdot e^{\color{blue}{\left(-\left(1 - z\right)\right) + \left(-6.5\right)}}\right)\right) \cdot 263.3831869810514}{z} \]

   3. metadata-eval 94.5%

      \[\leadsto \frac{\left(\sqrt{\pi \cdot 2} \cdot \left(\sqrt{7.5} \cdot e^{\left(-\left(1 - z\right)\right) + \color{blue}{-6.5}}\right)\right) \cdot 263.3831869810514}{z} \]

8. Applied egg-rr 94.5%

   \[\leadsto \color{blue}{\frac{\left(\sqrt{\pi \cdot 2} \cdot \left(\sqrt{7.5} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right) \cdot 263.3831869810514}{z}} \]
9. Step-by-step derivation

   1. associate-/l* 94.5%

      \[\leadsto \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left(\sqrt{7.5} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right) \cdot \frac{263.3831869810514}{z}} \]

   2. associate-*r* 94.6%

      \[\leadsto \color{blue}{\sqrt{\pi \cdot 2} \cdot \left(\left(\sqrt{7.5} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right) \cdot \frac{263.3831869810514}{z}\right)} \]

   3. *-commutative 94.6%

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \color{blue}{\left(\frac{263.3831869810514}{z} \cdot \left(\sqrt{7.5} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right)} \]

   4. sub-neg 94.6%

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\frac{263.3831869810514}{z} \cdot \left(\sqrt{7.5} \cdot e^{\left(-\color{blue}{\left(1 + \left(-z\right)\right)}\right) + -6.5}\right)\right) \]

   5. distribute-neg-in 94.6%

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\frac{263.3831869810514}{z} \cdot \left(\sqrt{7.5} \cdot e^{\color{blue}{\left(\left(-1\right) + \left(-\left(-z\right)\right)\right)} + -6.5}\right)\right) \]

   6. metadata-eval 94.6%

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\frac{263.3831869810514}{z} \cdot \left(\sqrt{7.5} \cdot e^{\left(\color{blue}{-1} + \left(-\left(-z\right)\right)\right) + -6.5}\right)\right) \]

   7. remove-double-neg 94.6%

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\frac{263.3831869810514}{z} \cdot \left(\sqrt{7.5} \cdot e^{\left(-1 + \color{blue}{z}\right) + -6.5}\right)\right) \]

   8. associate-+r+ 94.6%

      \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\frac{263.3831869810514}{z} \cdot \left(\sqrt{7.5} \cdot e^{\color{blue}{-1 + \left(z + -6.5\right)}}\right)\right) \]

10. Simplified 94.6%

    \[\leadsto \color{blue}{\sqrt{\pi \cdot 2} \cdot \left(\frac{263.3831869810514}{z} \cdot \left(\sqrt{7.5} \cdot e^{-1 + \left(z + -6.5\right)}\right)\right)} \]

11. Final simplification 94.6%

    \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\frac{263.3831869810514}{z} \cdot \left(e^{-1 + \left(z + -6.5\right)} \cdot \sqrt{7.5}\right)\right) \]
12. Add Preprocessing

Alternative 8: 95.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

3. Simplified 96.1%

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

5. Taylor expanded in z around 0 94.3%

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

6. Taylor expanded in z around 0 94.5%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\sqrt{7.5}} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \frac{263.3831869810514}{z} \]

7. Taylor expanded in z around 0 94.5%

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

   1. associate-*r/ 94.5%

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

9. Applied egg-rr 94.5%

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

    1. associate-/l* 94.5%

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

    2. associate-*r* 94.6%

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

    3. *-commutative 94.6%

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

    4. *-commutative 94.6%

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

11. Simplified 94.6%

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

12. Final simplification 94.6%

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

Alternative 9: 94.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

3. Simplified 96.1%

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

5. Taylor expanded in z around 0 94.3%

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

6. Taylor expanded in z around 0 94.5%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left(\color{blue}{\sqrt{7.5}} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \frac{263.3831869810514}{z} \]

7. Taylor expanded in z around 0 94.5%

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

   1. associate-*r/ 94.5%

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

9. Applied egg-rr 94.5%

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

    1. associate-/l* 94.5%

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

    2. associate-*r* 94.6%

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

    3. *-commutative 94.6%

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

    4. associate-*l* 94.5%

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

11. Simplified 94.5%

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

12. Final simplification 94.5%

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

Reproduce

herbie shell --seed 2024157 
(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))))))