Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.5% → 98.0%

Time: 54.2s

Alternatives: 20

Speedup: 1.6×

• could not determine a ground truth

  1. z = -3.492534972068779e+267

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: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.0%

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

3. Simplified 97.7%

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

6. Applied egg-rr 99.1%

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

7. Final simplification 99.1%

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

Alternative 2: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.0%

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

3. Simplified 97.7%

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

6. Applied egg-rr 99.0%

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

7. Final simplification 99.0%

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

Alternative 3: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.0%

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

3. Simplified 97.7%

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

6. Applied egg-rr 99.1%

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

8. Applied egg-rr 99.0%

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

9. Final simplification 99.0%

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

Alternative 4: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.0%

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

3. Simplified 97.7%

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

6. Applied egg-rr 98.9%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{2501468655737381}{200000000000000}, \mathsf{\_.f64}\left(5, z\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3147848041806007}{2500000000000}, \mathsf{\_.f64}\left(2, z\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{9999999999998099}{10000000000000000}, \mathsf{/.f64}\left(\frac{6765203681218851}{10000000000000}, \mathsf{\_.f64}\left(1, z\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{7713234287776531}{10000000000000}, \mathsf{\_.f64}\left(3, z\right)\right), \mathsf{/.f64}\left(\frac{883075145810703}{5000000000000}, \mathsf{\_.f64}\left(4, z\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right)\right), \left(\frac{\frac{\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}}{e^{\frac{15}{2} - z}}}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

   2. associate-/l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{2501468655737381}{200000000000000}, \mathsf{\_.f64}\left(5, z\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3147848041806007}{2500000000000}, \mathsf{\_.f64}\left(2, z\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{9999999999998099}{10000000000000000}, \mathsf{/.f64}\left(\frac{6765203681218851}{10000000000000}, \mathsf{\_.f64}\left(1, z\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{7713234287776531}{10000000000000}, \mathsf{\_.f64}\left(3, z\right)\right), \mathsf{/.f64}\left(\frac{883075145810703}{5000000000000}, \mathsf{\_.f64}\left(4, z\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right) \cdot e^{\frac{15}{2} - z}} \cdot \mathsf{PI}\left(\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{2501468655737381}{200000000000000}, \mathsf{\_.f64}\left(5, z\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3147848041806007}{2500000000000}, \mathsf{\_.f64}\left(2, z\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{9999999999998099}{10000000000000000}, \mathsf{/.f64}\left(\frac{6765203681218851}{10000000000000}, \mathsf{\_.f64}\left(1, z\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{7713234287776531}{10000000000000}, \mathsf{\_.f64}\left(3, z\right)\right), \mathsf{/.f64}\left(\frac{883075145810703}{5000000000000}, \mathsf{\_.f64}\left(4, z\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right)\right), \left(\frac{\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right) \cdot e^{\frac{15}{2} - z}} \cdot \mathsf{PI}\left(\right)\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{2501468655737381}{200000000000000}, \mathsf{\_.f64}\left(5, z\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3147848041806007}{2500000000000}, \mathsf{\_.f64}\left(2, z\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{9999999999998099}{10000000000000000}, \mathsf{/.f64}\left(\frac{6765203681218851}{10000000000000}, \mathsf{\_.f64}\left(1, z\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{7713234287776531}{10000000000000}, \mathsf{\_.f64}\left(3, z\right)\right), \mathsf{/.f64}\left(\frac{883075145810703}{5000000000000}, \mathsf{\_.f64}\left(4, z\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right)\right), \left(\frac{\left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \mathsf{PI}\left(\right)}{\color{blue}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right) \cdot e^{\frac{15}{2} - z}}}\right)\right) \]

   5. times-frac N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{2501468655737381}{200000000000000}, \mathsf{\_.f64}\left(5, z\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3147848041806007}{2500000000000}, \mathsf{\_.f64}\left(2, z\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{9999999999998099}{10000000000000000}, \mathsf{/.f64}\left(\frac{6765203681218851}{10000000000000}, \mathsf{\_.f64}\left(1, z\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{7713234287776531}{10000000000000}, \mathsf{\_.f64}\left(3, z\right)\right), \mathsf{/.f64}\left(\frac{883075145810703}{5000000000000}, \mathsf{\_.f64}\left(4, z\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right)\right), \left(\frac{\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{e^{\frac{15}{2} - z}}}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{2501468655737381}{200000000000000}, \mathsf{\_.f64}\left(5, z\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3147848041806007}{2500000000000}, \mathsf{\_.f64}\left(2, z\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{9999999999998099}{10000000000000000}, \mathsf{/.f64}\left(\frac{6765203681218851}{10000000000000}, \mathsf{\_.f64}\left(1, z\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{7713234287776531}{10000000000000}, \mathsf{\_.f64}\left(3, z\right)\right), \mathsf{/.f64}\left(\frac{883075145810703}{5000000000000}, \mathsf{\_.f64}\left(4, z\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)}\right), \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{e^{\frac{15}{2} - z}}\right)}\right)\right) \]

8. Applied egg-rr 98.9%

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

9. Final simplification 98.9%

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

Alternative 5: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.0%

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

3. Simplified 97.7%

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

6. Applied egg-rr 98.9%

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

7. Final simplification 98.9%

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

Alternative 6: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{\sin \left(z \cdot \pi\right)} \cdot \frac{\pi \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \left(\frac{-771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{z - 2}\right)\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{-12.507343278686905}{z - 5}\right) + 1.8820409189366395 \cdot 10^{-8}\right)\right)\right)}{e^{7.5 - z}} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ (* (sqrt (* 2.0 PI)) (pow (- 7.5 z) (- 0.5 z))) (sin (* z PI)))
  (/
   (*
    PI
    (+
     (+ (/ -0.13857109526572012 (- 6.0 z)) (/ 9.984369578019572e-6 (- 7.0 z)))
     (+
      (-
       (+ 0.9999999999998099 (/ 676.5203681218851 (- 1.0 z)))
       (+ (/ -771.3234287776531 (- 3.0 z)) (/ -1259.1392167224028 (- z 2.0))))
      (+
       (+ (/ -176.6150291621406 (- 4.0 z)) (/ -12.507343278686905 (- z 5.0)))
       1.8820409189366395e-8))))
   (exp (- 7.5 z)))))

C

double code(double z) {
	return ((sqrt((2.0 * ((double) M_PI))) * pow((7.5 - z), (0.5 - z))) / sin((z * ((double) M_PI)))) * ((((double) M_PI) * (((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) - ((-771.3234287776531 / (3.0 - z)) + (-1259.1392167224028 / (z - 2.0)))) + (((-176.6150291621406 / (4.0 - z)) + (-12.507343278686905 / (z - 5.0))) + 1.8820409189366395e-8)))) / exp((7.5 - z)));
}

Java

public static double code(double z) {
	return ((Math.sqrt((2.0 * Math.PI)) * Math.pow((7.5 - z), (0.5 - z))) / Math.sin((z * Math.PI))) * ((Math.PI * (((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) - ((-771.3234287776531 / (3.0 - z)) + (-1259.1392167224028 / (z - 2.0)))) + (((-176.6150291621406 / (4.0 - z)) + (-12.507343278686905 / (z - 5.0))) + 1.8820409189366395e-8)))) / Math.exp((7.5 - z)));
}

Python

def code(z):
	return ((math.sqrt((2.0 * math.pi)) * math.pow((7.5 - z), (0.5 - z))) / math.sin((z * math.pi))) * ((math.pi * (((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) - ((-771.3234287776531 / (3.0 - z)) + (-1259.1392167224028 / (z - 2.0)))) + (((-176.6150291621406 / (4.0 - z)) + (-12.507343278686905 / (z - 5.0))) + 1.8820409189366395e-8)))) / math.exp((7.5 - z)))

Julia

function code(z)
	return Float64(Float64(Float64(sqrt(Float64(2.0 * pi)) * (Float64(7.5 - z) ^ Float64(0.5 - z))) / sin(Float64(z * pi))) * Float64(Float64(pi * Float64(Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(9.984369578019572e-6 / Float64(7.0 - z))) + Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 - z))) - Float64(Float64(-771.3234287776531 / Float64(3.0 - z)) + Float64(-1259.1392167224028 / Float64(z - 2.0)))) + Float64(Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) + Float64(-12.507343278686905 / Float64(z - 5.0))) + 1.8820409189366395e-8)))) / exp(Float64(7.5 - z))))
end

MATLAB

function tmp = code(z)
	tmp = ((sqrt((2.0 * pi)) * ((7.5 - z) ^ (0.5 - z))) / sin((z * pi))) * ((pi * (((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) - ((-771.3234287776531 / (3.0 - z)) + (-1259.1392167224028 / (z - 2.0)))) + (((-176.6150291621406 / (4.0 - z)) + (-12.507343278686905 / (z - 5.0))) + 1.8820409189366395e-8)))) / exp((7.5 - z)));
end

Wolfram

code[z_] := N[(N[(N[(N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * N[(N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(-771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(z - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-12.507343278686905 / N[(z - 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.8820409189366395e-8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Exp[N[(7.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.0%

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

3. Simplified 97.7%

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in z around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{9999999999998099}{10000000000000000}, \mathsf{/.f64}\left(\frac{6765203681218851}{10000000000000}, \mathsf{\_.f64}\left(1, z\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-3147848041806007}{2500000000000}, \mathsf{\_.f64}\left(2, z\right)\right), \mathsf{/.f64}\left(\frac{-7713234287776531}{10000000000000}, \mathsf{\_.f64}\left(3, z\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-883075145810703}{5000000000000}, \mathsf{\_.f64}\left(4, z\right)\right), \mathsf{/.f64}\left(\frac{-2501468655737381}{200000000000000}, \mathsf{\_.f64}\left(5, z\right)\right)\right), \color{blue}{\frac{3764081837873279}{200000000000000000000000}}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(z, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]
8. Step-by-step derivation

9. Simplified 98.8%

   \[\leadsto \frac{\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{2 - z} - \frac{-771.3234287776531}{3 - z}\right)\right) + \left(\left(\frac{-176.6150291621406}{4 - z} - \frac{-12.507343278686905}{5 - z}\right) + \color{blue}{1.8820409189366395 \cdot 10^{-8}}\right)\right)\right) \cdot \pi}{e^{7.5 - z}} \cdot \frac{\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{\sin \left(z \cdot \pi\right)} \]

10. Final simplification 98.8%

    \[\leadsto \frac{\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{\sin \left(z \cdot \pi\right)} \cdot \frac{\pi \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \left(\frac{-771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{z - 2}\right)\right) + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{-12.507343278686905}{z - 5}\right) + 1.8820409189366395 \cdot 10^{-8}\right)\right)\right)}{e^{7.5 - z}} \]
11. Add Preprocessing

Alternative 7: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{\sin \left(z \cdot \pi\right)} \cdot \left(\frac{\pi}{e^{7.5 - z}} \cdot \left(\frac{-0.13857109526572012}{6 - z} + \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + 1.8820409189366395 \cdot 10^{-8}\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ (* (sqrt (* 2.0 PI)) (pow (- 7.5 z) (- 0.5 z))) (sin (* z PI)))
  (*
   (/ PI (exp (- 7.5 z)))
   (+
    (/ -0.13857109526572012 (- 6.0 z))
    (+
     (+
      0.9999999999998099
      (+
       (/ 676.5203681218851 (- 1.0 z))
       (+ (/ -1259.1392167224028 (- 2.0 z)) (/ 771.3234287776531 (- 3.0 z)))))
     (+
      (/ 9.984369578019572e-6 (- 7.0 z))
      (+
       (+ (/ -176.6150291621406 (- 4.0 z)) (/ 12.507343278686905 (- 5.0 z)))
       1.8820409189366395e-8)))))))

C

double code(double z) {
	return ((sqrt((2.0 * ((double) M_PI))) * pow((7.5 - z), (0.5 - z))) / sin((z * ((double) M_PI)))) * ((((double) M_PI) / exp((7.5 - z))) * ((-0.13857109526572012 / (6.0 - z)) + ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (3.0 - z))))) + ((9.984369578019572e-6 / (7.0 - z)) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + 1.8820409189366395e-8)))));
}

Java

public static double code(double z) {
	return ((Math.sqrt((2.0 * Math.PI)) * Math.pow((7.5 - z), (0.5 - z))) / Math.sin((z * Math.PI))) * ((Math.PI / Math.exp((7.5 - z))) * ((-0.13857109526572012 / (6.0 - z)) + ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (3.0 - z))))) + ((9.984369578019572e-6 / (7.0 - z)) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + 1.8820409189366395e-8)))));
}

Python

def code(z):
	return ((math.sqrt((2.0 * math.pi)) * math.pow((7.5 - z), (0.5 - z))) / math.sin((z * math.pi))) * ((math.pi / math.exp((7.5 - z))) * ((-0.13857109526572012 / (6.0 - z)) + ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (3.0 - z))))) + ((9.984369578019572e-6 / (7.0 - z)) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + 1.8820409189366395e-8)))))

Julia

function code(z)
	return Float64(Float64(Float64(sqrt(Float64(2.0 * pi)) * (Float64(7.5 - z) ^ Float64(0.5 - z))) / sin(Float64(z * pi))) * Float64(Float64(pi / exp(Float64(7.5 - z))) * Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(Float64(0.9999999999998099 + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) + Float64(771.3234287776531 / Float64(3.0 - z))))) + Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) + Float64(12.507343278686905 / Float64(5.0 - z))) + 1.8820409189366395e-8))))))
end

MATLAB

function tmp = code(z)
	tmp = ((sqrt((2.0 * pi)) * ((7.5 - z) ^ (0.5 - z))) / sin((z * pi))) * ((pi / exp((7.5 - z))) * ((-0.13857109526572012 / (6.0 - z)) + ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (3.0 - z))))) + ((9.984369578019572e-6 / (7.0 - z)) + (((-176.6150291621406 / (4.0 - z)) + (12.507343278686905 / (5.0 - z))) + 1.8820409189366395e-8)))));
end

Wolfram

code[z_] := N[(N[(N[(N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / N[Exp[N[(7.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(0.9999999999998099 + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.8820409189366395e-8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.0%

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

3. Simplified 97.7%

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

6. Applied egg-rr 99.1%

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

8. Applied egg-rr 99.0%

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

9. Taylor expanded in z around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(z, \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-883075145810703}{5000000000000}, \mathsf{\_.f64}\left(4, z\right)\right), \mathsf{/.f64}\left(\frac{2501468655737381}{200000000000000}, \mathsf{\_.f64}\left(5, z\right)\right)\right), \color{blue}{\frac{3764081837873279}{200000000000000000000000}}\right)\right), \mathsf{+.f64}\left(\frac{9999999999998099}{10000000000000000}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{6765203681218851}{10000000000000}, \mathsf{\_.f64}\left(1, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3147848041806007}{2500000000000}, \mathsf{\_.f64}\left(2, z\right)\right), \mathsf{/.f64}\left(\frac{7713234287776531}{10000000000000}, \mathsf{\_.f64}\left(3, z\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right)\right)\right) \]
10. Step-by-step derivation

11. Simplified 98.8%

    \[\leadsto \frac{\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + \color{blue}{1.8820409189366395 \cdot 10^{-8}}\right)\right) + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right)\right)\right)\right) \cdot \frac{\pi}{e^{7.5 - z}}\right) \]

12. Final simplification 98.8%

    \[\leadsto \frac{\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{\sin \left(z \cdot \pi\right)} \cdot \left(\frac{\pi}{e^{7.5 - z}} \cdot \left(\frac{-0.13857109526572012}{6 - z} + \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{12.507343278686905}{5 - z}\right) + 1.8820409189366395 \cdot 10^{-8}\right)\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 8: 97.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{\sin \left(z \cdot \pi\right)} \cdot \frac{\pi \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) - \left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{z - 8} - \left(z \cdot -10.53814559148631 + -41.65228863479777\right)\right) - \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \left(\frac{-771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{z - 2}\right)\right)\right)\right)}{e^{7.5 - z}} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ (* (sqrt (* 2.0 PI)) (pow (- 7.5 z) (- 0.5 z))) (sin (* z PI)))
  (/
   (*
    PI
    (-
     (+ (/ -0.13857109526572012 (- 6.0 z)) (/ 9.984369578019572e-6 (- 7.0 z)))
     (-
      (-
       (/ 1.5056327351493116e-7 (- z 8.0))
       (+ (* z -10.53814559148631) -41.65228863479777))
      (-
       (+ 0.9999999999998099 (/ 676.5203681218851 (- 1.0 z)))
       (+
        (/ -771.3234287776531 (- 3.0 z))
        (/ -1259.1392167224028 (- z 2.0)))))))
   (exp (- 7.5 z)))))

C

double code(double z) {
	return ((sqrt((2.0 * ((double) M_PI))) * pow((7.5 - z), (0.5 - z))) / sin((z * ((double) M_PI)))) * ((((double) M_PI) * (((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) - (((1.5056327351493116e-7 / (z - 8.0)) - ((z * -10.53814559148631) + -41.65228863479777)) - ((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) - ((-771.3234287776531 / (3.0 - z)) + (-1259.1392167224028 / (z - 2.0))))))) / exp((7.5 - z)));
}

Java

public static double code(double z) {
	return ((Math.sqrt((2.0 * Math.PI)) * Math.pow((7.5 - z), (0.5 - z))) / Math.sin((z * Math.PI))) * ((Math.PI * (((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) - (((1.5056327351493116e-7 / (z - 8.0)) - ((z * -10.53814559148631) + -41.65228863479777)) - ((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) - ((-771.3234287776531 / (3.0 - z)) + (-1259.1392167224028 / (z - 2.0))))))) / Math.exp((7.5 - z)));
}

Python

def code(z):
	return ((math.sqrt((2.0 * math.pi)) * math.pow((7.5 - z), (0.5 - z))) / math.sin((z * math.pi))) * ((math.pi * (((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) - (((1.5056327351493116e-7 / (z - 8.0)) - ((z * -10.53814559148631) + -41.65228863479777)) - ((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) - ((-771.3234287776531 / (3.0 - z)) + (-1259.1392167224028 / (z - 2.0))))))) / math.exp((7.5 - z)))

Julia

function code(z)
	return Float64(Float64(Float64(sqrt(Float64(2.0 * pi)) * (Float64(7.5 - z) ^ Float64(0.5 - z))) / sin(Float64(z * pi))) * Float64(Float64(pi * Float64(Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(9.984369578019572e-6 / Float64(7.0 - z))) - Float64(Float64(Float64(1.5056327351493116e-7 / Float64(z - 8.0)) - Float64(Float64(z * -10.53814559148631) + -41.65228863479777)) - Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 - z))) - Float64(Float64(-771.3234287776531 / Float64(3.0 - z)) + Float64(-1259.1392167224028 / Float64(z - 2.0))))))) / exp(Float64(7.5 - z))))
end

MATLAB

function tmp = code(z)
	tmp = ((sqrt((2.0 * pi)) * ((7.5 - z) ^ (0.5 - z))) / sin((z * pi))) * ((pi * (((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) - (((1.5056327351493116e-7 / (z - 8.0)) - ((z * -10.53814559148631) + -41.65228863479777)) - ((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) - ((-771.3234287776531 / (3.0 - z)) + (-1259.1392167224028 / (z - 2.0))))))) / exp((7.5 - z)));
end

Wolfram

code[z_] := N[(N[(N[(N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * N[(N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.5056327351493116e-7 / N[(z - 8.0), $MachinePrecision]), $MachinePrecision] - N[(N[(z * -10.53814559148631), $MachinePrecision] + -41.65228863479777), $MachinePrecision]), $MachinePrecision] - N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(-771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(z - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Exp[N[(7.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.0%

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

3. Simplified 97.7%

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in z around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{9999999999998099}{10000000000000000}, \mathsf{/.f64}\left(\frac{6765203681218851}{10000000000000}, \mathsf{\_.f64}\left(1, z\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-3147848041806007}{2500000000000}, \mathsf{\_.f64}\left(2, z\right)\right), \mathsf{/.f64}\left(\frac{-7713234287776531}{10000000000000}, \mathsf{\_.f64}\left(3, z\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{-105381455914863113}{10000000000000000} \cdot z - \frac{41652288634797769}{1000000000000000}\right)}, \mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(z, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{9999999999998099}{10000000000000000}, \mathsf{/.f64}\left(\frac{6765203681218851}{10000000000000}, \mathsf{\_.f64}\left(1, z\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-3147848041806007}{2500000000000}, \mathsf{\_.f64}\left(2, z\right)\right), \mathsf{/.f64}\left(\frac{-7713234287776531}{10000000000000}, \mathsf{\_.f64}\left(3, z\right)\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{-105381455914863113}{10000000000000000} \cdot z + \left(\mathsf{neg}\left(\frac{41652288634797769}{1000000000000000}\right)\right)\right), \mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(z, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{9999999999998099}{10000000000000000}, \mathsf{/.f64}\left(\frac{6765203681218851}{10000000000000}, \mathsf{\_.f64}\left(1, z\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-3147848041806007}{2500000000000}, \mathsf{\_.f64}\left(2, z\right)\right), \mathsf{/.f64}\left(\frac{-7713234287776531}{10000000000000}, \mathsf{\_.f64}\left(3, z\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{-105381455914863113}{10000000000000000} \cdot z\right), \left(\mathsf{neg}\left(\frac{41652288634797769}{1000000000000000}\right)\right)\right), \mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(z, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{9999999999998099}{10000000000000000}, \mathsf{/.f64}\left(\frac{6765203681218851}{10000000000000}, \mathsf{\_.f64}\left(1, z\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-3147848041806007}{2500000000000}, \mathsf{\_.f64}\left(2, z\right)\right), \mathsf{/.f64}\left(\frac{-7713234287776531}{10000000000000}, \mathsf{\_.f64}\left(3, z\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(z \cdot \frac{-105381455914863113}{10000000000000000}\right), \left(\mathsf{neg}\left(\frac{41652288634797769}{1000000000000000}\right)\right)\right), \mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(z, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{9999999999998099}{10000000000000000}, \mathsf{/.f64}\left(\frac{6765203681218851}{10000000000000}, \mathsf{\_.f64}\left(1, z\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-3147848041806007}{2500000000000}, \mathsf{\_.f64}\left(2, z\right)\right), \mathsf{/.f64}\left(\frac{-7713234287776531}{10000000000000}, \mathsf{\_.f64}\left(3, z\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \frac{-105381455914863113}{10000000000000000}\right), \left(\mathsf{neg}\left(\frac{41652288634797769}{1000000000000000}\right)\right)\right), \mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(z, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

   5. metadata-eval 98.1%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{9999999999998099}{10000000000000000}, \mathsf{/.f64}\left(\frac{6765203681218851}{10000000000000}, \mathsf{\_.f64}\left(1, z\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-3147848041806007}{2500000000000}, \mathsf{\_.f64}\left(2, z\right)\right), \mathsf{/.f64}\left(\frac{-7713234287776531}{10000000000000}, \mathsf{\_.f64}\left(3, z\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \frac{-105381455914863113}{10000000000000000}\right), \frac{-41652288634797769}{1000000000000000}\right), \mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(z, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

9. Simplified 98.1%

   \[\leadsto \frac{\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{2 - z} - \frac{-771.3234287776531}{3 - z}\right)\right) + \left(\color{blue}{\left(z \cdot -10.53814559148631 + -41.65228863479777\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \cdot \pi}{e^{7.5 - z}} \cdot \frac{\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{\sin \left(z \cdot \pi\right)} \]

10. Final simplification 98.1%

    \[\leadsto \frac{\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{\sin \left(z \cdot \pi\right)} \cdot \frac{\pi \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) - \left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{z - 8} - \left(z \cdot -10.53814559148631 + -41.65228863479777\right)\right) - \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) - \left(\frac{-771.3234287776531}{3 - z} + \frac{-1259.1392167224028}{z - 2}\right)\right)\right)\right)}{e^{7.5 - z}} \]
11. Add Preprocessing

Alternative 9: 97.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{\sin \left(z \cdot \pi\right)} \cdot \left(\frac{\pi}{e^{7.5 - z}} \cdot \left(\frac{-0.13857109526572012}{6 - z} + \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(z \cdot -10.53814559148631 + -41.65228863479777\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ (* (sqrt (* 2.0 PI)) (pow (- 7.5 z) (- 0.5 z))) (sin (* z PI)))
  (*
   (/ PI (exp (- 7.5 z)))
   (+
    (/ -0.13857109526572012 (- 6.0 z))
    (+
     (+
      0.9999999999998099
      (+
       (/ 676.5203681218851 (- 1.0 z))
       (+ (/ -1259.1392167224028 (- 2.0 z)) (/ 771.3234287776531 (- 3.0 z)))))
     (+
      (/ 9.984369578019572e-6 (- 7.0 z))
      (+
       (/ 1.5056327351493116e-7 (- 8.0 z))
       (+ (* z -10.53814559148631) -41.65228863479777))))))))

C

double code(double z) {
	return ((sqrt((2.0 * ((double) M_PI))) * pow((7.5 - z), (0.5 - z))) / sin((z * ((double) M_PI)))) * ((((double) M_PI) / exp((7.5 - z))) * ((-0.13857109526572012 / (6.0 - z)) + ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (3.0 - z))))) + ((9.984369578019572e-6 / (7.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + ((z * -10.53814559148631) + -41.65228863479777))))));
}

Java

public static double code(double z) {
	return ((Math.sqrt((2.0 * Math.PI)) * Math.pow((7.5 - z), (0.5 - z))) / Math.sin((z * Math.PI))) * ((Math.PI / Math.exp((7.5 - z))) * ((-0.13857109526572012 / (6.0 - z)) + ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (3.0 - z))))) + ((9.984369578019572e-6 / (7.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + ((z * -10.53814559148631) + -41.65228863479777))))));
}

Python

def code(z):
	return ((math.sqrt((2.0 * math.pi)) * math.pow((7.5 - z), (0.5 - z))) / math.sin((z * math.pi))) * ((math.pi / math.exp((7.5 - z))) * ((-0.13857109526572012 / (6.0 - z)) + ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (3.0 - z))))) + ((9.984369578019572e-6 / (7.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + ((z * -10.53814559148631) + -41.65228863479777))))))

Julia

function code(z)
	return Float64(Float64(Float64(sqrt(Float64(2.0 * pi)) * (Float64(7.5 - z) ^ Float64(0.5 - z))) / sin(Float64(z * pi))) * Float64(Float64(pi / exp(Float64(7.5 - z))) * Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(Float64(0.9999999999998099 + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) + Float64(771.3234287776531 / Float64(3.0 - z))))) + Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) + Float64(Float64(z * -10.53814559148631) + -41.65228863479777)))))))
end

MATLAB

function tmp = code(z)
	tmp = ((sqrt((2.0 * pi)) * ((7.5 - z) ^ (0.5 - z))) / sin((z * pi))) * ((pi / exp((7.5 - z))) * ((-0.13857109526572012 / (6.0 - z)) + ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (3.0 - z))))) + ((9.984369578019572e-6 / (7.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + ((z * -10.53814559148631) + -41.65228863479777))))));
end

Wolfram

code[z_] := N[(N[(N[(N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / N[Exp[N[(7.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(0.9999999999998099 + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(z * -10.53814559148631), $MachinePrecision] + -41.65228863479777), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.0%

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

3. Simplified 97.7%

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

6. Applied egg-rr 99.1%

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

8. Applied egg-rr 99.0%

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

9. Taylor expanded in z around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(z, \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{-105381455914863113}{10000000000000000} \cdot z - \frac{41652288634797769}{1000000000000000}\right)}, \mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{9999999999998099}{10000000000000000}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{6765203681218851}{10000000000000}, \mathsf{\_.f64}\left(1, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3147848041806007}{2500000000000}, \mathsf{\_.f64}\left(2, z\right)\right), \mathsf{/.f64}\left(\frac{7713234287776531}{10000000000000}, \mathsf{\_.f64}\left(3, z\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right)\right)\right) \]
10. Step-by-step derivation

    1. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(z, \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right), \mathsf{+.f64}\left(\left(\frac{-105381455914863113}{10000000000000000} \cdot z + \left(\mathsf{neg}\left(\frac{41652288634797769}{1000000000000000}\right)\right)\right), \mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{9999999999998099}{10000000000000000}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{6765203681218851}{10000000000000}, \mathsf{\_.f64}\left(1, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3147848041806007}{2500000000000}, \mathsf{\_.f64}\left(2, z\right)\right), \mathsf{/.f64}\left(\frac{7713234287776531}{10000000000000}, \mathsf{\_.f64}\left(3, z\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right)\right)\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(z, \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{-105381455914863113}{10000000000000000} \cdot z\right), \left(\mathsf{neg}\left(\frac{41652288634797769}{1000000000000000}\right)\right)\right), \mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{9999999999998099}{10000000000000000}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{6765203681218851}{10000000000000}, \mathsf{\_.f64}\left(1, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3147848041806007}{2500000000000}, \mathsf{\_.f64}\left(2, z\right)\right), \mathsf{/.f64}\left(\frac{7713234287776531}{10000000000000}, \mathsf{\_.f64}\left(3, z\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right)\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(z, \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(z \cdot \frac{-105381455914863113}{10000000000000000}\right), \left(\mathsf{neg}\left(\frac{41652288634797769}{1000000000000000}\right)\right)\right), \mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{9999999999998099}{10000000000000000}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{6765203681218851}{10000000000000}, \mathsf{\_.f64}\left(1, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3147848041806007}{2500000000000}, \mathsf{\_.f64}\left(2, z\right)\right), \mathsf{/.f64}\left(\frac{7713234287776531}{10000000000000}, \mathsf{\_.f64}\left(3, z\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right)\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(z, \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \frac{-105381455914863113}{10000000000000000}\right), \left(\mathsf{neg}\left(\frac{41652288634797769}{1000000000000000}\right)\right)\right), \mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{9999999999998099}{10000000000000000}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{6765203681218851}{10000000000000}, \mathsf{\_.f64}\left(1, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3147848041806007}{2500000000000}, \mathsf{\_.f64}\left(2, z\right)\right), \mathsf{/.f64}\left(\frac{7713234287776531}{10000000000000}, \mathsf{\_.f64}\left(3, z\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right)\right)\right) \]

    5. metadata-eval 98.1%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(z, \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, \frac{-105381455914863113}{10000000000000000}\right), \frac{-41652288634797769}{1000000000000000}\right), \mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{9999999999998099}{10000000000000000}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{6765203681218851}{10000000000000}, \mathsf{\_.f64}\left(1, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3147848041806007}{2500000000000}, \mathsf{\_.f64}\left(2, z\right)\right), \mathsf{/.f64}\left(\frac{7713234287776531}{10000000000000}, \mathsf{\_.f64}\left(3, z\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right)\right)\right) \]

11. Simplified 98.1%

    \[\leadsto \frac{\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\color{blue}{\left(z \cdot -10.53814559148631 + -41.65228863479777\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right) + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right)\right)\right)\right) \cdot \frac{\pi}{e^{7.5 - z}}\right) \]

12. Final simplification 98.1%

    \[\leadsto \frac{\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{\sin \left(z \cdot \pi\right)} \cdot \left(\frac{\pi}{e^{7.5 - z}} \cdot \left(\frac{-0.13857109526572012}{6 - z} + \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(z \cdot -10.53814559148631 + -41.65228863479777\right)\right)\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 10: 97.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\pi \cdot \frac{\frac{\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{e^{7.5 - z}}}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{12.507343278686905}{5 - z} + \left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot 544.9358906000987\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   PI
   (/
    (/ (* (sqrt (* 2.0 PI)) (pow (- 7.5 z) (- 0.5 z))) (exp (- 7.5 z)))
    (sin (* z PI))))
  (+
   (+
    (+ (/ -0.13857109526572012 (- 6.0 z)) (/ 9.984369578019572e-6 (- 7.0 z)))
    (/ 1.5056327351493116e-7 (- 8.0 z)))
   (+
    (/ 12.507343278686905 (- 5.0 z))
    (+
     260.9048120626994
     (* z (+ 436.3997278161676 (* z 544.9358906000987))))))))

C

double code(double z) {
	return (((double) M_PI) * (((sqrt((2.0 * ((double) M_PI))) * pow((7.5 - z), (0.5 - z))) / exp((7.5 - z))) / sin((z * ((double) M_PI))))) * ((((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (1.5056327351493116e-7 / (8.0 - z))) + ((12.507343278686905 / (5.0 - z)) + (260.9048120626994 + (z * (436.3997278161676 + (z * 544.9358906000987))))));
}

Java

public static double code(double z) {
	return (Math.PI * (((Math.sqrt((2.0 * Math.PI)) * Math.pow((7.5 - z), (0.5 - z))) / Math.exp((7.5 - z))) / Math.sin((z * Math.PI)))) * ((((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (1.5056327351493116e-7 / (8.0 - z))) + ((12.507343278686905 / (5.0 - z)) + (260.9048120626994 + (z * (436.3997278161676 + (z * 544.9358906000987))))));
}

Python

def code(z):
	return (math.pi * (((math.sqrt((2.0 * math.pi)) * math.pow((7.5 - z), (0.5 - z))) / math.exp((7.5 - z))) / math.sin((z * math.pi)))) * ((((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (1.5056327351493116e-7 / (8.0 - z))) + ((12.507343278686905 / (5.0 - z)) + (260.9048120626994 + (z * (436.3997278161676 + (z * 544.9358906000987))))))

Julia

function code(z)
	return Float64(Float64(pi * Float64(Float64(Float64(sqrt(Float64(2.0 * pi)) * (Float64(7.5 - z) ^ Float64(0.5 - z))) / exp(Float64(7.5 - z))) / sin(Float64(z * pi)))) * Float64(Float64(Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(9.984369578019572e-6 / Float64(7.0 - z))) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))) + Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(260.9048120626994 + Float64(z * Float64(436.3997278161676 + Float64(z * 544.9358906000987)))))))
end

MATLAB

function tmp = code(z)
	tmp = (pi * (((sqrt((2.0 * pi)) * ((7.5 - z) ^ (0.5 - z))) / exp((7.5 - z))) / sin((z * pi)))) * ((((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (1.5056327351493116e-7 / (8.0 - z))) + ((12.507343278686905 / (5.0 - z)) + (260.9048120626994 + (z * (436.3997278161676 + (z * 544.9358906000987))))));
end

Wolfram

code[z_] := N[(N[(Pi * N[(N[(N[(N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Exp[N[(7.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(260.9048120626994 + N[(z * N[(436.3997278161676 + N[(z * 544.9358906000987), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.0%

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

3. Simplified 97.7%

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

5. Taylor expanded in z around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + \frac{4708246094784852251}{8640000000000000} \cdot z\right)\right)}, \mathsf{/.f64}\left(\frac{2501468655737381}{200000000000000}, \mathsf{\_.f64}\left(5, z\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{7827144361880981797}{30000000000000000}, \left(z \cdot \left(\frac{314207804027640689}{720000000000000} + \frac{4708246094784852251}{8640000000000000} \cdot z\right)\right)\right), \mathsf{/.f64}\left(\frac{2501468655737381}{200000000000000}, \mathsf{\_.f64}\left(5, z\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{7827144361880981797}{30000000000000000}, \mathsf{*.f64}\left(z, \left(\frac{314207804027640689}{720000000000000} + \frac{4708246094784852251}{8640000000000000} \cdot z\right)\right)\right), \mathsf{/.f64}\left(\frac{2501468655737381}{200000000000000}, \mathsf{\_.f64}\left(5, z\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{7827144361880981797}{30000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{314207804027640689}{720000000000000}, \left(\frac{4708246094784852251}{8640000000000000} \cdot z\right)\right)\right)\right), \mathsf{/.f64}\left(\frac{2501468655737381}{200000000000000}, \mathsf{\_.f64}\left(5, z\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{7827144361880981797}{30000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{314207804027640689}{720000000000000}, \left(z \cdot \frac{4708246094784852251}{8640000000000000}\right)\right)\right)\right), \mathsf{/.f64}\left(\frac{2501468655737381}{200000000000000}, \mathsf{\_.f64}\left(5, z\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f64 97.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{7827144361880981797}{30000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{314207804027640689}{720000000000000}, \mathsf{*.f64}\left(z, \frac{4708246094784852251}{8640000000000000}\right)\right)\right)\right), \mathsf{/.f64}\left(\frac{2501468655737381}{200000000000000}, \mathsf{\_.f64}\left(5, z\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

7. Simplified 97.9%

   \[\leadsto \left(\left(\color{blue}{\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot 544.9358906000987\right)\right)} + \frac{12.507343278686905}{5 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right)\right) \cdot \left(\pi \cdot \frac{\frac{\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{e^{7.5 - z}}}{\sin \left(\pi \cdot z\right)}\right) \]

8. Final simplification 97.9%

   \[\leadsto \left(\pi \cdot \frac{\frac{\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{e^{7.5 - z}}}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{12.507343278686905}{5 - z} + \left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot 544.9358906000987\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 11: 97.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\pi \cdot \frac{\frac{\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{e^{7.5 - z}}}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(263.4062807184368 + z \cdot \left(436.9000215473151 + z \cdot \left(545.0359493463282 + z \cdot 606.6767878347069\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   PI
   (/
    (/ (* (sqrt (* 2.0 PI)) (pow (- 7.5 z) (- 0.5 z))) (exp (- 7.5 z)))
    (sin (* z PI))))
  (+
   (+
    (+ (/ -0.13857109526572012 (- 6.0 z)) (/ 9.984369578019572e-6 (- 7.0 z)))
    (/ 1.5056327351493116e-7 (- 8.0 z)))
   (+
    263.4062807184368
    (*
     z
     (+
      436.9000215473151
      (* z (+ 545.0359493463282 (* z 606.6767878347069)))))))))

C

double code(double z) {
	return (((double) M_PI) * (((sqrt((2.0 * ((double) M_PI))) * pow((7.5 - z), (0.5 - z))) / exp((7.5 - z))) / sin((z * ((double) M_PI))))) * ((((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (1.5056327351493116e-7 / (8.0 - z))) + (263.4062807184368 + (z * (436.9000215473151 + (z * (545.0359493463282 + (z * 606.6767878347069)))))));
}

Java

public static double code(double z) {
	return (Math.PI * (((Math.sqrt((2.0 * Math.PI)) * Math.pow((7.5 - z), (0.5 - z))) / Math.exp((7.5 - z))) / Math.sin((z * Math.PI)))) * ((((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (1.5056327351493116e-7 / (8.0 - z))) + (263.4062807184368 + (z * (436.9000215473151 + (z * (545.0359493463282 + (z * 606.6767878347069)))))));
}

Python

def code(z):
	return (math.pi * (((math.sqrt((2.0 * math.pi)) * math.pow((7.5 - z), (0.5 - z))) / math.exp((7.5 - z))) / math.sin((z * math.pi)))) * ((((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (1.5056327351493116e-7 / (8.0 - z))) + (263.4062807184368 + (z * (436.9000215473151 + (z * (545.0359493463282 + (z * 606.6767878347069)))))))

Julia

function code(z)
	return Float64(Float64(pi * Float64(Float64(Float64(sqrt(Float64(2.0 * pi)) * (Float64(7.5 - z) ^ Float64(0.5 - z))) / exp(Float64(7.5 - z))) / sin(Float64(z * pi)))) * Float64(Float64(Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(9.984369578019572e-6 / Float64(7.0 - z))) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))) + Float64(263.4062807184368 + Float64(z * Float64(436.9000215473151 + Float64(z * Float64(545.0359493463282 + Float64(z * 606.6767878347069))))))))
end

MATLAB

function tmp = code(z)
	tmp = (pi * (((sqrt((2.0 * pi)) * ((7.5 - z) ^ (0.5 - z))) / exp((7.5 - z))) / sin((z * pi)))) * ((((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (1.5056327351493116e-7 / (8.0 - z))) + (263.4062807184368 + (z * (436.9000215473151 + (z * (545.0359493463282 + (z * 606.6767878347069)))))));
end

Wolfram

code[z_] := N[(N[(Pi * N[(N[(N[(N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Exp[N[(7.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(263.4062807184368 + N[(z * N[(436.9000215473151 + N[(z * N[(545.0359493463282 + N[(z * 606.6767878347069), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.0%

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

3. Simplified 97.7%

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

5. Taylor expanded in z around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \left(\frac{39321001939258358983}{90000000000000000} + z \cdot \left(\frac{2943194126470171931171}{5400000000000000000} + \frac{196563279258445065194677}{324000000000000000000} \cdot z\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{7902188421553103227}{30000000000000000}, \left(z \cdot \left(\frac{39321001939258358983}{90000000000000000} + z \cdot \left(\frac{2943194126470171931171}{5400000000000000000} + \frac{196563279258445065194677}{324000000000000000000} \cdot z\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{7902188421553103227}{30000000000000000}, \mathsf{*.f64}\left(z, \left(\frac{39321001939258358983}{90000000000000000} + z \cdot \left(\frac{2943194126470171931171}{5400000000000000000} + \frac{196563279258445065194677}{324000000000000000000} \cdot z\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{7902188421553103227}{30000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{39321001939258358983}{90000000000000000}, \left(z \cdot \left(\frac{2943194126470171931171}{5400000000000000000} + \frac{196563279258445065194677}{324000000000000000000} \cdot z\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{7902188421553103227}{30000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{39321001939258358983}{90000000000000000}, \mathsf{*.f64}\left(z, \left(\frac{2943194126470171931171}{5400000000000000000} + \frac{196563279258445065194677}{324000000000000000000} \cdot z\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{7902188421553103227}{30000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{39321001939258358983}{90000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{2943194126470171931171}{5400000000000000000}, \left(\frac{196563279258445065194677}{324000000000000000000} \cdot z\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{7902188421553103227}{30000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{39321001939258358983}{90000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{2943194126470171931171}{5400000000000000000}, \left(z \cdot \frac{196563279258445065194677}{324000000000000000000}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 97.7%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{7902188421553103227}{30000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{39321001939258358983}{90000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{2943194126470171931171}{5400000000000000000}, \mathsf{*.f64}\left(z, \frac{196563279258445065194677}{324000000000000000000}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3764081837873279}{25000000000000000000000}, \mathsf{\_.f64}\left(8, z\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-3464277381643003}{25000000000000000}, \mathsf{\_.f64}\left(6, z\right)\right), \mathsf{/.f64}\left(\frac{2496092394504893}{250000000000000000000}, \mathsf{\_.f64}\left(7, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), z\right)\right)\right)\right)\right) \]

7. Simplified 97.7%

   \[\leadsto \left(\color{blue}{\left(263.4062807184368 + z \cdot \left(436.9000215473151 + z \cdot \left(545.0359493463282 + z \cdot 606.6767878347069\right)\right)\right)} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right)\right) \cdot \left(\pi \cdot \frac{\frac{\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{e^{7.5 - z}}}{\sin \left(\pi \cdot z\right)}\right) \]

8. Final simplification 97.7%

   \[\leadsto \left(\pi \cdot \frac{\frac{\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{e^{7.5 - z}}}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(263.4062807184368 + z \cdot \left(436.9000215473151 + z \cdot \left(545.0359493463282 + z \cdot 606.6767878347069\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 12: 96.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + \left(\pi \cdot \pi\right) \cdot 43.89719783017524\right)\right)}{z} \cdot \left(e^{z + -7.5} \cdot \frac{{\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{{\left(2 \cdot \pi\right)}^{-0.5}}\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/
   (+
    263.3831869810514
    (*
     z
     (+
      436.8961725563396
      (* z (+ 545.0353078428827 (* (* PI PI) 43.89719783017524))))))
   z)
  (* (exp (+ z -7.5)) (/ (pow (- 7.5 z) (- 0.5 z)) (pow (* 2.0 PI) -0.5)))))

C

double code(double z) {
	return ((263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + ((((double) M_PI) * ((double) M_PI)) * 43.89719783017524)))))) / z) * (exp((z + -7.5)) * (pow((7.5 - z), (0.5 - z)) / pow((2.0 * ((double) M_PI)), -0.5)));
}

Java

public static double code(double z) {
	return ((263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + ((Math.PI * Math.PI) * 43.89719783017524)))))) / z) * (Math.exp((z + -7.5)) * (Math.pow((7.5 - z), (0.5 - z)) / Math.pow((2.0 * Math.PI), -0.5)));
}

Python

def code(z):
	return ((263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + ((math.pi * math.pi) * 43.89719783017524)))))) / z) * (math.exp((z + -7.5)) * (math.pow((7.5 - z), (0.5 - z)) / math.pow((2.0 * math.pi), -0.5)))

Julia

function code(z)
	return Float64(Float64(Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * Float64(545.0353078428827 + Float64(Float64(pi * pi) * 43.89719783017524)))))) / z) * Float64(exp(Float64(z + -7.5)) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) / (Float64(2.0 * pi) ^ -0.5))))
end

MATLAB

function tmp = code(z)
	tmp = ((263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + ((pi * pi) * 43.89719783017524)))))) / z) * (exp((z + -7.5)) * (((7.5 - z) ^ (0.5 - z)) / ((2.0 * pi) ^ -0.5)));
end

Wolfram

code[z_] := N[(N[(N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * N[(545.0353078428827 + N[(N[(Pi * Pi), $MachinePrecision] * 43.89719783017524), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] / N[Power[N[(2.0 * Pi), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.0%

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

3. Simplified 97.7%

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

6. Applied egg-rr 99.1%

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

7. Taylor expanded in z around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} - \frac{-1106209385320415913103082059}{25200000000000000000000000} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{z}\right)}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} - \frac{-1106209385320415913103082059}{25200000000000000000000000} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right), z\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right)}, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right)\right) \]

9. Simplified 96.2%

   \[\leadsto \color{blue}{\frac{263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + \left(\pi \cdot \pi\right) \cdot 43.89719783017524\right)\right)}{z}} \cdot \frac{\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{e^{7.5 - z}} \]
10. Step-by-step derivation

    1. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{102757979785251069442117317613}{235200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{1106209385320415913103082059}{25200000000000000000000000}\right)\right)\right)\right)\right)\right), z\right), \left(\frac{\sqrt{2 \cdot \mathsf{PI}\left(\right)}}{e^{\frac{15}{2} - z}} \cdot \color{blue}{{\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}}\right)\right) \]

    2. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{102757979785251069442117317613}{235200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{1106209385320415913103082059}{25200000000000000000000000}\right)\right)\right)\right)\right)\right), z\right), \left(\frac{1}{\frac{e^{\frac{15}{2} - z}}{\sqrt{2 \cdot \mathsf{PI}\left(\right)}}} \cdot {\color{blue}{\left(\frac{15}{2} - z\right)}}^{\left(\frac{1}{2} - z\right)}\right)\right) \]

    3. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{102757979785251069442117317613}{235200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{1106209385320415913103082059}{25200000000000000000000000}\right)\right)\right)\right)\right)\right), z\right), \left(\frac{1 \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}}{\color{blue}{\frac{e^{\frac{15}{2} - z}}{\sqrt{2 \cdot \mathsf{PI}\left(\right)}}}}\right)\right) \]

    4. div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{102757979785251069442117317613}{235200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{1106209385320415913103082059}{25200000000000000000000000}\right)\right)\right)\right)\right)\right), z\right), \left(\frac{1 \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}}{e^{\frac{15}{2} - z} \cdot \color{blue}{\frac{1}{\sqrt{2 \cdot \mathsf{PI}\left(\right)}}}}\right)\right) \]

    5. times-frac N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{102757979785251069442117317613}{235200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{1106209385320415913103082059}{25200000000000000000000000}\right)\right)\right)\right)\right)\right), z\right), \left(\frac{1}{e^{\frac{15}{2} - z}} \cdot \color{blue}{\frac{{\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}}{\frac{1}{\sqrt{2 \cdot \mathsf{PI}\left(\right)}}}}\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{102757979785251069442117317613}{235200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{1106209385320415913103082059}{25200000000000000000000000}\right)\right)\right)\right)\right)\right), z\right), \mathsf{*.f64}\left(\left(\frac{1}{e^{\frac{15}{2} - z}}\right), \color{blue}{\left(\frac{{\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}}{\frac{1}{\sqrt{2 \cdot \mathsf{PI}\left(\right)}}}\right)}\right)\right) \]

    7. exp-diff N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{102757979785251069442117317613}{235200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{1106209385320415913103082059}{25200000000000000000000000}\right)\right)\right)\right)\right)\right), z\right), \mathsf{*.f64}\left(\left(\frac{1}{\frac{e^{\frac{15}{2}}}{e^{z}}}\right), \left(\frac{{\left(\frac{15}{2} - z\right)}^{\color{blue}{\left(\frac{1}{2} - z\right)}}}{\frac{1}{\sqrt{2 \cdot \mathsf{PI}\left(\right)}}}\right)\right)\right) \]

    8. associate-/r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{102757979785251069442117317613}{235200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{1106209385320415913103082059}{25200000000000000000000000}\right)\right)\right)\right)\right)\right), z\right), \mathsf{*.f64}\left(\left(\frac{1}{e^{\frac{15}{2}}} \cdot e^{z}\right), \left(\frac{\color{blue}{{\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}}}{\frac{1}{\sqrt{2 \cdot \mathsf{PI}\left(\right)}}}\right)\right)\right) \]

    9. rec-exp N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{102757979785251069442117317613}{235200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{1106209385320415913103082059}{25200000000000000000000000}\right)\right)\right)\right)\right)\right), z\right), \mathsf{*.f64}\left(\left(e^{\mathsf{neg}\left(\frac{15}{2}\right)} \cdot e^{z}\right), \left(\frac{{\color{blue}{\left(\frac{15}{2} - z\right)}}^{\left(\frac{1}{2} - z\right)}}{\frac{1}{\sqrt{2 \cdot \mathsf{PI}\left(\right)}}}\right)\right)\right) \]

    10. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{102757979785251069442117317613}{235200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{1106209385320415913103082059}{25200000000000000000000000}\right)\right)\right)\right)\right)\right), z\right), \mathsf{*.f64}\left(\left(e^{\frac{-15}{2}} \cdot e^{z}\right), \left(\frac{{\left(\color{blue}{\frac{15}{2}} - z\right)}^{\left(\frac{1}{2} - z\right)}}{\frac{1}{\sqrt{2 \cdot \mathsf{PI}\left(\right)}}}\right)\right)\right) \]

    11. prod-exp N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{102757979785251069442117317613}{235200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{1106209385320415913103082059}{25200000000000000000000000}\right)\right)\right)\right)\right)\right), z\right), \mathsf{*.f64}\left(\left(e^{\frac{-15}{2} + z}\right), \left(\frac{\color{blue}{{\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}}}{\frac{1}{\sqrt{2 \cdot \mathsf{PI}\left(\right)}}}\right)\right)\right) \]

    12. exp-lowering-exp.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{102757979785251069442117317613}{235200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{1106209385320415913103082059}{25200000000000000000000000}\right)\right)\right)\right)\right)\right), z\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(\frac{-15}{2} + z\right)\right), \left(\frac{\color{blue}{{\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}}}{\frac{1}{\sqrt{2 \cdot \mathsf{PI}\left(\right)}}}\right)\right)\right) \]

    13. +-lowering-+.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{102757979785251069442117317613}{235200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{1106209385320415913103082059}{25200000000000000000000000}\right)\right)\right)\right)\right)\right), z\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{+.f64}\left(\frac{-15}{2}, z\right)\right), \left(\frac{{\color{blue}{\left(\frac{15}{2} - z\right)}}^{\left(\frac{1}{2} - z\right)}}{\frac{1}{\sqrt{2 \cdot \mathsf{PI}\left(\right)}}}\right)\right)\right) \]

    14. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{102757979785251069442117317613}{235200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{1106209385320415913103082059}{25200000000000000000000000}\right)\right)\right)\right)\right)\right), z\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{+.f64}\left(\frac{-15}{2}, z\right)\right), \mathsf{/.f64}\left(\left({\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right), \color{blue}{\left(\frac{1}{\sqrt{2 \cdot \mathsf{PI}\left(\right)}}\right)}\right)\right)\right) \]

    15. pow-lowering-pow.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{102757979785251069442117317613}{235200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{1106209385320415913103082059}{25200000000000000000000000}\right)\right)\right)\right)\right)\right), z\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{+.f64}\left(\frac{-15}{2}, z\right)\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{15}{2} - z\right), \left(\frac{1}{2} - z\right)\right), \left(\frac{\color{blue}{1}}{\sqrt{2 \cdot \mathsf{PI}\left(\right)}}\right)\right)\right)\right) \]

    16. --lowering--.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{102757979785251069442117317613}{235200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{1106209385320415913103082059}{25200000000000000000000000}\right)\right)\right)\right)\right)\right), z\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{+.f64}\left(\frac{-15}{2}, z\right)\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \left(\frac{1}{2} - z\right)\right), \left(\frac{1}{\sqrt{2 \cdot \mathsf{PI}\left(\right)}}\right)\right)\right)\right) \]

    17. --lowering--.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{102757979785251069442117317613}{235200000000000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{1106209385320415913103082059}{25200000000000000000000000}\right)\right)\right)\right)\right)\right), z\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{+.f64}\left(\frac{-15}{2}, z\right)\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right), \left(\frac{1}{\sqrt{2 \cdot \mathsf{PI}\left(\right)}}\right)\right)\right)\right) \]

11. Applied egg-rr 97.0%

    \[\leadsto \frac{263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + \left(\pi \cdot \pi\right) \cdot 43.89719783017524\right)\right)}{z} \cdot \color{blue}{\left(e^{-7.5 + z} \cdot \frac{{\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{{\left(2 \cdot \pi\right)}^{-0.5}}\right)} \]

12. Final simplification 97.0%

    \[\leadsto \frac{263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + \left(\pi \cdot \pi\right) \cdot 43.89719783017524\right)\right)}{z} \cdot \left(e^{z + -7.5} \cdot \frac{{\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{{\left(2 \cdot \pi\right)}^{-0.5}}\right) \]
13. Add Preprocessing

Alternative 13: 96.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + \left(\pi \cdot \pi\right) \cdot 43.89719783017524\right)\right)\right) \cdot \frac{\frac{{\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{e^{7.5 - z} \cdot {\left(2 \cdot \pi\right)}^{-0.5}}}{z} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (+
   263.3831869810514
   (*
    z
    (+
     436.8961725563396
     (* z (+ 545.0353078428827 (* (* PI PI) 43.89719783017524))))))
  (/
   (/ (pow (- 7.5 z) (- 0.5 z)) (* (exp (- 7.5 z)) (pow (* 2.0 PI) -0.5)))
   z)))

C

double code(double z) {
	return (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + ((((double) M_PI) * ((double) M_PI)) * 43.89719783017524)))))) * ((pow((7.5 - z), (0.5 - z)) / (exp((7.5 - z)) * pow((2.0 * ((double) M_PI)), -0.5))) / z);
}

Java

public static double code(double z) {
	return (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + ((Math.PI * Math.PI) * 43.89719783017524)))))) * ((Math.pow((7.5 - z), (0.5 - z)) / (Math.exp((7.5 - z)) * Math.pow((2.0 * Math.PI), -0.5))) / z);
}

Python

def code(z):
	return (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + ((math.pi * math.pi) * 43.89719783017524)))))) * ((math.pow((7.5 - z), (0.5 - z)) / (math.exp((7.5 - z)) * math.pow((2.0 * math.pi), -0.5))) / z)

Julia

function code(z)
	return Float64(Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * Float64(545.0353078428827 + Float64(Float64(pi * pi) * 43.89719783017524)))))) * Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) / Float64(exp(Float64(7.5 - z)) * (Float64(2.0 * pi) ^ -0.5))) / z))
end

MATLAB

function tmp = code(z)
	tmp = (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + ((pi * pi) * 43.89719783017524)))))) * ((((7.5 - z) ^ (0.5 - z)) / (exp((7.5 - z)) * ((2.0 * pi) ^ -0.5))) / z);
end

Wolfram

code[z_] := N[(N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * N[(545.0353078428827 + N[(N[(Pi * Pi), $MachinePrecision] * 43.89719783017524), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] / N[(N[Exp[N[(7.5 - z), $MachinePrecision]], $MachinePrecision] * N[Power[N[(2.0 * Pi), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.0%

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

3. Simplified 97.7%

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

6. Applied egg-rr 99.1%

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

7. Taylor expanded in z around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} - \frac{-1106209385320415913103082059}{25200000000000000000000000} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{z}\right)}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} - \frac{-1106209385320415913103082059}{25200000000000000000000000} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right), z\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right), \mathsf{\_.f64}\left(\frac{1}{2}, z\right)\right)\right)}, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\frac{15}{2}, z\right)\right)\right)\right) \]

9. Simplified 96.2%

   \[\leadsto \color{blue}{\frac{263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + \left(\pi \cdot \pi\right) \cdot 43.89719783017524\right)\right)}{z}} \cdot \frac{\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{e^{7.5 - z}} \]

11. Applied egg-rr 97.0%

    \[\leadsto \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + \left(\pi \cdot \pi\right) \cdot 43.89719783017524\right)\right)\right) \cdot \frac{\frac{{\left(7.5 - z\right)}^{\left(0.5 - z\right)}}{e^{7.5 - z} \cdot {\left(2 \cdot \pi\right)}^{-0.5}}}{z}} \]
12. Add Preprocessing

Alternative 14: 96.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.0%

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

3. Simplified 97.7%

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

5. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\frac{\sqrt{2} \cdot \sqrt{\frac{15}{2}}}{z \cdot e^{\frac{15}{2}}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \color{blue}{\left(\frac{\sqrt{2} \cdot \sqrt{\frac{15}{2}}}{z \cdot e^{\frac{15}{2}}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

   2. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\frac{\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}{\color{blue}{z \cdot e^{\frac{15}{2}}}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right), \color{blue}{\left(z \cdot e^{\frac{15}{2}}\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(\color{blue}{z} \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{2}\right), \left(\sqrt{\frac{15}{2}}\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\sqrt{\frac{15}{2}}\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   8. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   9. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(z, \color{blue}{\left(e^{\frac{15}{2}}\right)}\right)\right)\right) \]

   11. exp-lowering-exp.f64 95.7%

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{exp.f64}\left(\frac{15}{2}\right)\right)\right)\right) \]

7. Simplified 95.7%

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

   1. times-frac N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\frac{\sqrt{2} \cdot \sqrt{\frac{15}{2}}}{z} \cdot \color{blue}{\frac{\sqrt{\mathsf{PI}\left(\right)}}{e^{\frac{15}{2}}}}\right)\right) \]

   2. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\frac{\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \frac{\sqrt{\mathsf{PI}\left(\right)}}{e^{\frac{15}{2}}}}{\color{blue}{z}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \frac{\sqrt{\mathsf{PI}\left(\right)}}{e^{\frac{15}{2}}}\right), \color{blue}{z}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{e^{\frac{15}{2}}}\right)\right), z\right)\right) \]

   5. sqrt-unprod N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{2 \cdot \frac{15}{2}}\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{e^{\frac{15}{2}}}\right)\right), z\right)\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot \frac{15}{2}\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{e^{\frac{15}{2}}}\right)\right), z\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{e^{\frac{15}{2}}}\right)\right), z\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{/.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right), \left(e^{\frac{15}{2}}\right)\right)\right), z\right)\right) \]

   9. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right), \left(e^{\frac{15}{2}}\right)\right)\right), z\right)\right) \]

   10. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \left(e^{\frac{15}{2}}\right)\right)\right), z\right)\right) \]

   11. exp-lowering-exp.f64 96.9%

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(15\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{exp.f64}\left(\frac{15}{2}\right)\right)\right), z\right)\right) \]

9. Applied egg-rr 96.9%

   \[\leadsto 263.3831869810514 \cdot \color{blue}{\frac{\sqrt{15} \cdot \frac{\sqrt{\pi}}{e^{7.5}}}{z}} \]
10. Add Preprocessing

Alternative 15: 96.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.0%

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

3. Simplified 97.7%

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

5. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\frac{\sqrt{2} \cdot \sqrt{\frac{15}{2}}}{z \cdot e^{\frac{15}{2}}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \color{blue}{\left(\frac{\sqrt{2} \cdot \sqrt{\frac{15}{2}}}{z \cdot e^{\frac{15}{2}}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

   2. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\frac{\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}{\color{blue}{z \cdot e^{\frac{15}{2}}}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right), \color{blue}{\left(z \cdot e^{\frac{15}{2}}\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(\color{blue}{z} \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{2}\right), \left(\sqrt{\frac{15}{2}}\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\sqrt{\frac{15}{2}}\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   8. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   9. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(z, \color{blue}{\left(e^{\frac{15}{2}}\right)}\right)\right)\right) \]

   11. exp-lowering-exp.f64 95.7%

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{exp.f64}\left(\frac{15}{2}\right)\right)\right)\right) \]

7. Simplified 95.7%

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

   1. times-frac N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\frac{\sqrt{2} \cdot \sqrt{\frac{15}{2}}}{z} \cdot \color{blue}{\frac{\sqrt{\mathsf{PI}\left(\right)}}{e^{\frac{15}{2}}}}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{e^{\frac{15}{2}}} \cdot \color{blue}{\frac{\sqrt{2} \cdot \sqrt{\frac{15}{2}}}{z}}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{e^{\frac{15}{2}}}\right), \color{blue}{\left(\frac{\sqrt{2} \cdot \sqrt{\frac{15}{2}}}{z}\right)}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right), \left(e^{\frac{15}{2}}\right)\right), \left(\frac{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{15}{2}}}}{z}\right)\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right), \left(e^{\frac{15}{2}}\right)\right), \left(\frac{\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{15}{2}}}{z}\right)\right)\right) \]

   6. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \left(e^{\frac{15}{2}}\right)\right), \left(\frac{\sqrt{\color{blue}{2}} \cdot \sqrt{\frac{15}{2}}}{z}\right)\right)\right) \]

   7. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{exp.f64}\left(\frac{15}{2}\right)\right), \left(\frac{\sqrt{2} \cdot \color{blue}{\sqrt{\frac{15}{2}}}}{z}\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{exp.f64}\left(\frac{15}{2}\right)\right), \mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right), \color{blue}{z}\right)\right)\right) \]

   9. sqrt-unprod N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{exp.f64}\left(\frac{15}{2}\right)\right), \mathsf{/.f64}\left(\left(\sqrt{2 \cdot \frac{15}{2}}\right), z\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{exp.f64}\left(\frac{15}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot \frac{15}{2}\right)\right), z\right)\right)\right) \]

   11. metadata-eval 96.7%

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{exp.f64}\left(\frac{15}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(15\right), z\right)\right)\right) \]

9. Applied egg-rr 96.7%

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

Alternative 16: 94.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{263.3831869810514 \cdot {\left(\pi \cdot 15\right)}^{0.5}}{z} \cdot e^{-7.5} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (* (/ (* 263.3831869810514 (pow (* PI 15.0) 0.5)) z) (exp -7.5)))

C

double code(double z) {
	return ((263.3831869810514 * pow((((double) M_PI) * 15.0), 0.5)) / z) * exp(-7.5);
}

Java

public static double code(double z) {
	return ((263.3831869810514 * Math.pow((Math.PI * 15.0), 0.5)) / z) * Math.exp(-7.5);
}

Python

def code(z):
	return ((263.3831869810514 * math.pow((math.pi * 15.0), 0.5)) / z) * math.exp(-7.5)

Julia

function code(z)
	return Float64(Float64(Float64(263.3831869810514 * (Float64(pi * 15.0) ^ 0.5)) / z) * exp(-7.5))
end

MATLAB

function tmp = code(z)
	tmp = ((263.3831869810514 * ((pi * 15.0) ^ 0.5)) / z) * exp(-7.5);
end

Wolfram

code[z_] := N[(N[(N[(263.3831869810514 * N[Power[N[(Pi * 15.0), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.0%

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

3. Simplified 97.7%

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

5. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\frac{\sqrt{2} \cdot \sqrt{\frac{15}{2}}}{z \cdot e^{\frac{15}{2}}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \color{blue}{\left(\frac{\sqrt{2} \cdot \sqrt{\frac{15}{2}}}{z \cdot e^{\frac{15}{2}}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

   2. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\frac{\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}{\color{blue}{z \cdot e^{\frac{15}{2}}}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right), \color{blue}{\left(z \cdot e^{\frac{15}{2}}\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(\color{blue}{z} \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{2}\right), \left(\sqrt{\frac{15}{2}}\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\sqrt{\frac{15}{2}}\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   8. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   9. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(z, \color{blue}{\left(e^{\frac{15}{2}}\right)}\right)\right)\right) \]

   11. exp-lowering-exp.f64 95.7%

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{exp.f64}\left(\frac{15}{2}\right)\right)\right)\right) \]

7. Simplified 95.7%

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

   1. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\frac{1}{\color{blue}{\frac{z \cdot e^{\frac{15}{2}}}{\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}}}\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\frac{1}{z \cdot e^{\frac{15}{2}}} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\left(\frac{1}{z \cdot e^{\frac{15}{2}}}\right), \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\left(\frac{1}{e^{\frac{15}{2}} \cdot z}\right), \left(\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{15}{2}}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   5. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\left(\frac{\frac{1}{e^{\frac{15}{2}}}}{z}\right), \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{e^{\frac{15}{2}}}\right), z\right), \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   7. rec-exp N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{15}{2}\right)}\right), z\right), \left(\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   8. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\frac{15}{2}\right)\right)\right), z\right), \left(\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \left(\left(\sqrt{\color{blue}{2}} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   10. sqrt-unprod N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \left(\sqrt{2 \cdot \frac{15}{2}} \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

   11. sqrt-unprod N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \left(\sqrt{\left(2 \cdot \frac{15}{2}\right) \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

   12. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \frac{15}{2}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \frac{15}{2}\right), \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(15, \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   15. PI-lowering-PI.f64 96.2%

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(15, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

9. Applied egg-rr 96.2%

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

    1. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\sqrt{15 \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\frac{e^{\frac{-15}{2}}}{z}}\right)\right) \]

    2. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\sqrt{15 \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{\color{blue}{\frac{z}{e^{\frac{-15}{2}}}}}\right)\right) \]

    3. un-div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\frac{\sqrt{15 \cdot \mathsf{PI}\left(\right)}}{\color{blue}{\frac{z}{e^{\frac{-15}{2}}}}}\right)\right) \]

    4. sqrt-prod N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\frac{\sqrt{15} \cdot \sqrt{\mathsf{PI}\left(\right)}}{\frac{\color{blue}{z}}{e^{\frac{-15}{2}}}}\right)\right) \]

    5. div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\frac{\sqrt{15} \cdot \sqrt{\mathsf{PI}\left(\right)}}{z \cdot \color{blue}{\frac{1}{e^{\frac{-15}{2}}}}}\right)\right) \]

    6. rec-exp N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\frac{\sqrt{15} \cdot \sqrt{\mathsf{PI}\left(\right)}}{z \cdot e^{\mathsf{neg}\left(\frac{-15}{2}\right)}}\right)\right) \]

    7. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\frac{\sqrt{15} \cdot \sqrt{\mathsf{PI}\left(\right)}}{z \cdot e^{\frac{15}{2}}}\right)\right) \]

    8. frac-times N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\frac{\sqrt{15}}{z} \cdot \color{blue}{\frac{\sqrt{\mathsf{PI}\left(\right)}}{e^{\frac{15}{2}}}}\right)\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{e^{\frac{15}{2}}} \cdot \color{blue}{\frac{\sqrt{15}}{z}}\right)\right) \]

    10. associate-*l/ N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{\sqrt{15}}{z}}{\color{blue}{e^{\frac{15}{2}}}}\right)\right) \]

    11. div-inv N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{\sqrt{15}}{z}\right) \cdot \color{blue}{\frac{1}{e^{\frac{15}{2}}}}\right)\right) \]

    12. rec-exp N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{\sqrt{15}}{z}\right) \cdot e^{\mathsf{neg}\left(\frac{15}{2}\right)}\right)\right) \]

    13. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{\sqrt{15}}{z}\right) \cdot e^{\frac{-15}{2}}\right)\right) \]

    14. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{\sqrt{15}}{z}\right), \color{blue}{\left(e^{\frac{-15}{2}}\right)}\right)\right) \]

11. Applied egg-rr 96.1%

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

    1. associate-*r* N/A

       \[\leadsto \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{\sqrt{\mathsf{PI}\left(\right) \cdot 15}}{z}\right) \cdot \color{blue}{e^{\frac{-15}{2}}} \]

    2. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{\sqrt{\mathsf{PI}\left(\right) \cdot 15}}{z}\right), \color{blue}{\left(e^{\frac{-15}{2}}\right)}\right) \]

    3. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot 15}}{z}\right), \left(e^{\color{blue}{\frac{-15}{2}}}\right)\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot 15}\right), z\right), \left(e^{\color{blue}{\frac{-15}{2}}}\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\sqrt{\mathsf{PI}\left(\right) \cdot 15}\right)\right), z\right), \left(e^{\frac{-15}{2}}\right)\right) \]

    6. pow1/2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left({\left(\mathsf{PI}\left(\right) \cdot 15\right)}^{\frac{1}{2}}\right)\right), z\right), \left(e^{\frac{-15}{2}}\right)\right) \]

    7. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{pow.f64}\left(\left(\mathsf{PI}\left(\right) \cdot 15\right), \frac{1}{2}\right)\right), z\right), \left(e^{\frac{-15}{2}}\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), 15\right), \frac{1}{2}\right)\right), z\right), \left(e^{\frac{-15}{2}}\right)\right) \]

    9. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 15\right), \frac{1}{2}\right)\right), z\right), \left(e^{\frac{-15}{2}}\right)\right) \]

    10. exp-lowering-exp.f64 96.4%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 15\right), \frac{1}{2}\right)\right), z\right), \mathsf{exp.f64}\left(\frac{-15}{2}\right)\right) \]

13. Applied egg-rr 96.4%

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

Alternative 17: 95.6% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.0%

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

3. Simplified 97.7%

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

5. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\frac{\sqrt{2} \cdot \sqrt{\frac{15}{2}}}{z \cdot e^{\frac{15}{2}}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \color{blue}{\left(\frac{\sqrt{2} \cdot \sqrt{\frac{15}{2}}}{z \cdot e^{\frac{15}{2}}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

   2. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\frac{\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}{\color{blue}{z \cdot e^{\frac{15}{2}}}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right), \color{blue}{\left(z \cdot e^{\frac{15}{2}}\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(\color{blue}{z} \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{2}\right), \left(\sqrt{\frac{15}{2}}\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\sqrt{\frac{15}{2}}\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   8. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   9. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(z, \color{blue}{\left(e^{\frac{15}{2}}\right)}\right)\right)\right) \]

   11. exp-lowering-exp.f64 95.7%

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{exp.f64}\left(\frac{15}{2}\right)\right)\right)\right) \]

7. Simplified 95.7%

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

   1. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\frac{1}{\color{blue}{\frac{z \cdot e^{\frac{15}{2}}}{\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}}}\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\frac{1}{z \cdot e^{\frac{15}{2}}} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\left(\frac{1}{z \cdot e^{\frac{15}{2}}}\right), \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\left(\frac{1}{e^{\frac{15}{2}} \cdot z}\right), \left(\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{15}{2}}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   5. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\left(\frac{\frac{1}{e^{\frac{15}{2}}}}{z}\right), \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{e^{\frac{15}{2}}}\right), z\right), \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   7. rec-exp N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{15}{2}\right)}\right), z\right), \left(\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   8. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\frac{15}{2}\right)\right)\right), z\right), \left(\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \left(\left(\sqrt{\color{blue}{2}} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   10. sqrt-unprod N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \left(\sqrt{2 \cdot \frac{15}{2}} \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

   11. sqrt-unprod N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \left(\sqrt{\left(2 \cdot \frac{15}{2}\right) \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

   12. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \frac{15}{2}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \frac{15}{2}\right), \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(15, \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   15. PI-lowering-PI.f64 96.2%

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(15, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

9. Applied egg-rr 96.2%

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

    1. *-commutative N/A

       \[\leadsto \left(\frac{e^{\frac{-15}{2}}}{z} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}\right) \cdot \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000}} \]

    2. *-commutative N/A

       \[\leadsto \left(\sqrt{15 \cdot \mathsf{PI}\left(\right)} \cdot \frac{e^{\frac{-15}{2}}}{z}\right) \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

    3. clear-num N/A

       \[\leadsto \left(\sqrt{15 \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{\frac{z}{e^{\frac{-15}{2}}}}\right) \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

    4. un-div-inv N/A

       \[\leadsto \frac{\sqrt{15 \cdot \mathsf{PI}\left(\right)}}{\frac{z}{e^{\frac{-15}{2}}}} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

    5. sqrt-prod N/A

       \[\leadsto \frac{\sqrt{15} \cdot \sqrt{\mathsf{PI}\left(\right)}}{\frac{z}{e^{\frac{-15}{2}}}} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

    6. div-inv N/A

       \[\leadsto \frac{\sqrt{15} \cdot \sqrt{\mathsf{PI}\left(\right)}}{z \cdot \frac{1}{e^{\frac{-15}{2}}}} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

    7. rec-exp N/A

       \[\leadsto \frac{\sqrt{15} \cdot \sqrt{\mathsf{PI}\left(\right)}}{z \cdot e^{\mathsf{neg}\left(\frac{-15}{2}\right)}} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

    8. metadata-eval N/A

       \[\leadsto \frac{\sqrt{15} \cdot \sqrt{\mathsf{PI}\left(\right)}}{z \cdot e^{\frac{15}{2}}} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

    9. frac-times N/A

       \[\leadsto \left(\frac{\sqrt{15}}{z} \cdot \frac{\sqrt{\mathsf{PI}\left(\right)}}{e^{\frac{15}{2}}}\right) \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

    10. *-commutative N/A

        \[\leadsto \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{e^{\frac{15}{2}}} \cdot \frac{\sqrt{15}}{z}\right) \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

    11. *-commutative N/A

        \[\leadsto \left(\frac{\sqrt{15}}{z} \cdot \frac{\sqrt{\mathsf{PI}\left(\right)}}{e^{\frac{15}{2}}}\right) \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

    12. frac-times N/A

        \[\leadsto \frac{\sqrt{15} \cdot \sqrt{\mathsf{PI}\left(\right)}}{z \cdot e^{\frac{15}{2}}} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

    13. sqrt-prod N/A

        \[\leadsto \frac{\sqrt{15 \cdot \mathsf{PI}\left(\right)}}{z \cdot e^{\frac{15}{2}}} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

    14. *-commutative N/A

        \[\leadsto \frac{\sqrt{15 \cdot \mathsf{PI}\left(\right)}}{e^{\frac{15}{2}} \cdot z} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]

    15. associate-*l/ N/A

        \[\leadsto \frac{\sqrt{15 \cdot \mathsf{PI}\left(\right)} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000}}{\color{blue}{e^{\frac{15}{2}} \cdot z}} \]

    16. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{15 \cdot \mathsf{PI}\left(\right)} \cdot \frac{1106209385320415913103082059}{4200000000000000000000000}\right), \color{blue}{\left(e^{\frac{15}{2}} \cdot z\right)}\right) \]

11. Applied egg-rr 96.4%

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

12. Final simplification 96.4%

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

Alternative 18: 95.6% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.0%

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

3. Simplified 97.7%

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

5. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\frac{\sqrt{2} \cdot \sqrt{\frac{15}{2}}}{z \cdot e^{\frac{15}{2}}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \color{blue}{\left(\frac{\sqrt{2} \cdot \sqrt{\frac{15}{2}}}{z \cdot e^{\frac{15}{2}}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

   2. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\frac{\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}{\color{blue}{z \cdot e^{\frac{15}{2}}}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right), \color{blue}{\left(z \cdot e^{\frac{15}{2}}\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(\color{blue}{z} \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{2}\right), \left(\sqrt{\frac{15}{2}}\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\sqrt{\frac{15}{2}}\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   8. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   9. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(z, \color{blue}{\left(e^{\frac{15}{2}}\right)}\right)\right)\right) \]

   11. exp-lowering-exp.f64 95.7%

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{exp.f64}\left(\frac{15}{2}\right)\right)\right)\right) \]

7. Simplified 95.7%

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

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}{\color{blue}{z \cdot e^{\frac{15}{2}}}} \]

   2. div-inv N/A

      \[\leadsto \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \cdot \color{blue}{\frac{1}{z \cdot e^{\frac{15}{2}}}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right), \color{blue}{\left(\frac{1}{z \cdot e^{\frac{15}{2}}}\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(\frac{\color{blue}{1}}{z \cdot e^{\frac{15}{2}}}\right)\right) \]

   5. sqrt-unprod N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\sqrt{2 \cdot \frac{15}{2}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(\frac{1}{z \cdot e^{\frac{15}{2}}}\right)\right) \]

   6. sqrt-unprod N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\sqrt{\left(2 \cdot \frac{15}{2}\right) \cdot \mathsf{PI}\left(\right)}\right)\right), \left(\frac{1}{z \cdot e^{\frac{15}{2}}}\right)\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \frac{15}{2}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(\frac{1}{z \cdot e^{\frac{15}{2}}}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \frac{15}{2}\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(\frac{1}{z \cdot e^{\frac{15}{2}}}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(15, \mathsf{PI}\left(\right)\right)\right)\right), \left(\frac{1}{z \cdot e^{\frac{15}{2}}}\right)\right) \]

   10. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(15, \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(\frac{1}{z \cdot e^{\frac{15}{2}}}\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(15, \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(\frac{1}{e^{\frac{15}{2}} \cdot \color{blue}{z}}\right)\right) \]

   12. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(15, \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(\frac{\frac{1}{e^{\frac{15}{2}}}}{\color{blue}{z}}\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(15, \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{1}{e^{\frac{15}{2}}}\right), \color{blue}{z}\right)\right) \]

   14. rec-exp N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(15, \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{15}{2}\right)}\right), z\right)\right) \]

   15. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(15, \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\frac{15}{2}\right)\right)\right), z\right)\right) \]

   16. metadata-eval 96.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(15, \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right)\right) \]

9. Applied egg-rr 96.3%

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

10. Final simplification 96.3%

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

Alternative 19: 95.5% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.0%

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

3. Simplified 97.7%

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

5. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\frac{\sqrt{2} \cdot \sqrt{\frac{15}{2}}}{z \cdot e^{\frac{15}{2}}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \color{blue}{\left(\frac{\sqrt{2} \cdot \sqrt{\frac{15}{2}}}{z \cdot e^{\frac{15}{2}}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

   2. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\frac{\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}{\color{blue}{z \cdot e^{\frac{15}{2}}}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right), \color{blue}{\left(z \cdot e^{\frac{15}{2}}\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(\color{blue}{z} \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{2}\right), \left(\sqrt{\frac{15}{2}}\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\sqrt{\frac{15}{2}}\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   8. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   9. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(z, \color{blue}{\left(e^{\frac{15}{2}}\right)}\right)\right)\right) \]

   11. exp-lowering-exp.f64 95.7%

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{exp.f64}\left(\frac{15}{2}\right)\right)\right)\right) \]

7. Simplified 95.7%

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

   1. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\frac{1}{\color{blue}{\frac{z \cdot e^{\frac{15}{2}}}{\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}}}\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\frac{1}{z \cdot e^{\frac{15}{2}}} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\left(\frac{1}{z \cdot e^{\frac{15}{2}}}\right), \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\left(\frac{1}{e^{\frac{15}{2}} \cdot z}\right), \left(\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{15}{2}}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   5. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\left(\frac{\frac{1}{e^{\frac{15}{2}}}}{z}\right), \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{e^{\frac{15}{2}}}\right), z\right), \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   7. rec-exp N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{15}{2}\right)}\right), z\right), \left(\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   8. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\frac{15}{2}\right)\right)\right), z\right), \left(\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \left(\left(\sqrt{\color{blue}{2}} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   10. sqrt-unprod N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \left(\sqrt{2 \cdot \frac{15}{2}} \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

   11. sqrt-unprod N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \left(\sqrt{\left(2 \cdot \frac{15}{2}\right) \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

   12. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \frac{15}{2}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \frac{15}{2}\right), \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(15, \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   15. PI-lowering-PI.f64 96.2%

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(15, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

9. Applied egg-rr 96.2%

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

    1. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\frac{e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}}{\color{blue}{z}}\right)\right) \]

    2. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\left(e^{\frac{-15}{2}} \cdot \sqrt{15 \cdot \mathsf{PI}\left(\right)}\right), \color{blue}{z}\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(e^{\frac{-15}{2}}\right), \left(\sqrt{15 \cdot \mathsf{PI}\left(\right)}\right)\right), z\right)\right) \]

    4. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), \left(\sqrt{15 \cdot \mathsf{PI}\left(\right)}\right)\right), z\right)\right) \]

    5. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), \mathsf{sqrt.f64}\left(\left(15 \cdot \mathsf{PI}\left(\right)\right)\right)\right), z\right)\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), \mathsf{sqrt.f64}\left(\left(\mathsf{PI}\left(\right) \cdot 15\right)\right)\right), z\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), 15\right)\right)\right), z\right)\right) \]

    8. PI-lowering-PI.f64 96.2%

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 15\right)\right)\right), z\right)\right) \]

11. Applied egg-rr 96.2%

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

Alternative 20: 95.4% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.0%

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

3. Simplified 97.7%

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

5. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\frac{\sqrt{2} \cdot \sqrt{\frac{15}{2}}}{z \cdot e^{\frac{15}{2}}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \color{blue}{\left(\frac{\sqrt{2} \cdot \sqrt{\frac{15}{2}}}{z \cdot e^{\frac{15}{2}}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

   2. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\frac{\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}{\color{blue}{z \cdot e^{\frac{15}{2}}}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right), \color{blue}{\left(z \cdot e^{\frac{15}{2}}\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(\color{blue}{z} \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{2}\right), \left(\sqrt{\frac{15}{2}}\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\sqrt{\frac{15}{2}}\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   8. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   9. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \left(z \cdot e^{\frac{15}{2}}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(z, \color{blue}{\left(e^{\frac{15}{2}}\right)}\right)\right)\right) \]

   11. exp-lowering-exp.f64 95.7%

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(\frac{15}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(z, \mathsf{exp.f64}\left(\frac{15}{2}\right)\right)\right)\right) \]

7. Simplified 95.7%

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

   1. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\frac{1}{\color{blue}{\frac{z \cdot e^{\frac{15}{2}}}{\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}}}\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \left(\frac{1}{z \cdot e^{\frac{15}{2}}} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\left(\frac{1}{z \cdot e^{\frac{15}{2}}}\right), \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\left(\frac{1}{e^{\frac{15}{2}} \cdot z}\right), \left(\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{15}{2}}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   5. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\left(\frac{\frac{1}{e^{\frac{15}{2}}}}{z}\right), \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{e^{\frac{15}{2}}}\right), z\right), \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   7. rec-exp N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{15}{2}\right)}\right), z\right), \left(\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   8. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\frac{15}{2}\right)\right)\right), z\right), \left(\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \left(\left(\sqrt{\color{blue}{2}} \cdot \sqrt{\frac{15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   10. sqrt-unprod N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \left(\sqrt{2 \cdot \frac{15}{2}} \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

   11. sqrt-unprod N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \left(\sqrt{\left(2 \cdot \frac{15}{2}\right) \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

   12. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \frac{15}{2}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \frac{15}{2}\right), \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(15, \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   15. PI-lowering-PI.f64 96.2%

       \[\leadsto \mathsf{*.f64}\left(\frac{1106209385320415913103082059}{4200000000000000000000000}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{exp.f64}\left(\frac{-15}{2}\right), z\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(15, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

9. Applied egg-rr 96.2%

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

10. Final simplification 96.2%

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

Reproduce

herbie shell --seed 2024164 
(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))))))