Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.6% → 98.5%

Time: 1.9min

Alternatives: 10

Speedup: 1.5×

• could not determine a ground truth

  1. z = -2.1547006010850844e+220

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: 96.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.4%

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

4. Applied egg-rr 99.3%

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

6. Simplified 99.3%

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

7. Final simplification 99.3%

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

Alternative 2: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7} + \left(\frac{12.507343278686905}{5 - z} + \left(z \cdot \left(z \cdot \left(z \cdot -0.6900093756475991 - 2.76025133439795\right) - 11.042288315961857\right) - 44.17685104674093\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ PI (sin (* PI z)))
  (*
   (exp (+ z -7.5))
   (*
    (* (pow (- 7.5 z) (- 0.5 z)) (sqrt (* PI 2.0)))
    (+
     (+
      0.9999999999998099
      (+
       (/ 676.5203681218851 (- 1.0 z))
       (+ (/ -1259.1392167224028 (- 2.0 z)) (/ 771.3234287776531 (- 3.0 z)))))
     (+
      (/ 1.5056327351493116e-7 (+ (- 1.0 z) 7.0))
      (+
       (/ 12.507343278686905 (- 5.0 z))
       (-
        (*
         z
         (-
          (* z (- (* z -0.6900093756475991) 2.76025133439795))
          11.042288315961857))
        44.17685104674093))))))))

C

double code(double z) {
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (exp((z + -7.5)) * ((pow((7.5 - z), (0.5 - z)) * sqrt((((double) M_PI) * 2.0))) * ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (3.0 - z))))) + ((1.5056327351493116e-7 / ((1.0 - z) + 7.0)) + ((12.507343278686905 / (5.0 - z)) + ((z * ((z * ((z * -0.6900093756475991) - 2.76025133439795)) - 11.042288315961857)) - 44.17685104674093))))));
}

Java

public static double code(double z) {
	return (Math.PI / Math.sin((Math.PI * z))) * (Math.exp((z + -7.5)) * ((Math.pow((7.5 - z), (0.5 - z)) * Math.sqrt((Math.PI * 2.0))) * ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (3.0 - z))))) + ((1.5056327351493116e-7 / ((1.0 - z) + 7.0)) + ((12.507343278686905 / (5.0 - z)) + ((z * ((z * ((z * -0.6900093756475991) - 2.76025133439795)) - 11.042288315961857)) - 44.17685104674093))))));
}

Python

def code(z):
	return (math.pi / math.sin((math.pi * z))) * (math.exp((z + -7.5)) * ((math.pow((7.5 - z), (0.5 - z)) * math.sqrt((math.pi * 2.0))) * ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (3.0 - z))))) + ((1.5056327351493116e-7 / ((1.0 - z) + 7.0)) + ((12.507343278686905 / (5.0 - z)) + ((z * ((z * ((z * -0.6900093756475991) - 2.76025133439795)) - 11.042288315961857)) - 44.17685104674093))))))

Julia

function code(z)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(exp(Float64(z + -7.5)) * Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * sqrt(Float64(pi * 2.0))) * 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(1.5056327351493116e-7 / Float64(Float64(1.0 - z) + 7.0)) + Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * -0.6900093756475991) - 2.76025133439795)) - 11.042288315961857)) - 44.17685104674093)))))))
end

MATLAB

function tmp = code(z)
	tmp = (pi / sin((pi * z))) * (exp((z + -7.5)) * ((((7.5 - z) ^ (0.5 - z)) * sqrt((pi * 2.0))) * ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (3.0 - z))))) + ((1.5056327351493116e-7 / ((1.0 - z) + 7.0)) + ((12.507343278686905 / (5.0 - z)) + ((z * ((z * ((z * -0.6900093756475991) - 2.76025133439795)) - 11.042288315961857)) - 44.17685104674093))))));
end

Wolfram

code[z_] := N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $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[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] + 7.0), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(z * N[(N[(z * N[(N[(z * -0.6900093756475991), $MachinePrecision] - 2.76025133439795), $MachinePrecision]), $MachinePrecision] - 11.042288315961857), $MachinePrecision]), $MachinePrecision] - 44.17685104674093), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.4%

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

4. Applied egg-rr 99.3%

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

6. Simplified 99.3%

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

7. Taylor expanded in z around 0 99.0%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \color{blue}{\left(z \cdot \left(z \cdot \left(-0.6900093756475991 \cdot z - 2.76025133439795\right) - 11.042288315961857\right) - 44.17685104674093\right)}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right)\right) \]

8. Final simplification 99.0%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7} + \left(\frac{12.507343278686905}{5 - z} + \left(z \cdot \left(z \cdot \left(z \cdot -0.6900093756475991 - 2.76025133439795\right) - 11.042288315961857\right) - 44.17685104674093\right)\right)\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 3: 98.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.4%

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

3. Simplified 97.7%

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

5. Taylor expanded in z around inf 97.7%

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

   1. exp-to-pow 97.7%

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

   2. sub-neg 97.7%

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

   3. metadata-eval 97.7%

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

7. Simplified 97.7%

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

8. Taylor expanded in z around 0 98.4%

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

   1. *-commutative 98.4%

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

10. Simplified 98.4%

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

    1. add-exp-log 98.9%

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

    2. *-commutative 98.9%

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

    3. log-prod 98.9%

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

    4. add-log-exp 98.9%

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

    5. log-pow 98.9%

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

12. Applied egg-rr 98.9%

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

13. Final simplification 98.9%

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

Alternative 4: 97.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{z} \cdot \left(e^{z + -7.5} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right) + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(z \cdot -229.08220098308368 - 372.46179876865034\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ 1.0 z)
  (*
   (exp (+ z -7.5))
   (*
    (* (pow (- 7.5 z) (- 0.5 z)) (sqrt (* PI 2.0)))
    (+
     (+
      (+
       (/ 12.507343278686905 (- 5.0 z))
       (+
        (/ -176.6150291621406 (+ (- 1.0 z) 3.0))
        (+
         (/ -0.13857109526572012 (- 6.0 z))
         (/ 9.984369578019572e-6 (- 7.0 z)))))
      (/ 1.5056327351493116e-7 (+ (- 1.0 z) 7.0)))
     (+
      0.9999999999998099
      (+
       (/ 676.5203681218851 (- 1.0 z))
       (- (* z -229.08220098308368) 372.46179876865034))))))))

C

double code(double z) {
	return (1.0 / z) * (exp((z + -7.5)) * ((pow((7.5 - z), (0.5 - z)) * sqrt((((double) M_PI) * 2.0))) * ((((12.507343278686905 / (5.0 - z)) + ((-176.6150291621406 / ((1.0 - z) + 3.0)) + ((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))))) + (1.5056327351493116e-7 / ((1.0 - z) + 7.0))) + (0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((z * -229.08220098308368) - 372.46179876865034))))));
}

Java

public static double code(double z) {
	return (1.0 / z) * (Math.exp((z + -7.5)) * ((Math.pow((7.5 - z), (0.5 - z)) * Math.sqrt((Math.PI * 2.0))) * ((((12.507343278686905 / (5.0 - z)) + ((-176.6150291621406 / ((1.0 - z) + 3.0)) + ((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))))) + (1.5056327351493116e-7 / ((1.0 - z) + 7.0))) + (0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((z * -229.08220098308368) - 372.46179876865034))))));
}

Python

def code(z):
	return (1.0 / z) * (math.exp((z + -7.5)) * ((math.pow((7.5 - z), (0.5 - z)) * math.sqrt((math.pi * 2.0))) * ((((12.507343278686905 / (5.0 - z)) + ((-176.6150291621406 / ((1.0 - z) + 3.0)) + ((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))))) + (1.5056327351493116e-7 / ((1.0 - z) + 7.0))) + (0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((z * -229.08220098308368) - 372.46179876865034))))))

Julia

function code(z)
	return Float64(Float64(1.0 / z) * Float64(exp(Float64(z + -7.5)) * Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * sqrt(Float64(pi * 2.0))) * Float64(Float64(Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(Float64(-176.6150291621406 / Float64(Float64(1.0 - z) + 3.0)) + Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(9.984369578019572e-6 / Float64(7.0 - z))))) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) + 7.0))) + Float64(0.9999999999998099 + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(z * -229.08220098308368) - 372.46179876865034)))))))
end

MATLAB

function tmp = code(z)
	tmp = (1.0 / z) * (exp((z + -7.5)) * ((((7.5 - z) ^ (0.5 - z)) * sqrt((pi * 2.0))) * ((((12.507343278686905 / (5.0 - z)) + ((-176.6150291621406 / ((1.0 - z) + 3.0)) + ((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))))) + (1.5056327351493116e-7 / ((1.0 - z) + 7.0))) + (0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((z * -229.08220098308368) - 372.46179876865034))))));
end

Wolfram

code[z_] := N[(N[(1.0 / z), $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] + 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(z * -229.08220098308368), $MachinePrecision] - 372.46179876865034), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.4%

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

4. Applied egg-rr 99.3%

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

6. Simplified 99.3%

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

7. Taylor expanded in z around 0 98.5%

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

8. Taylor expanded in z around 0 98.6%

   \[\leadsto \frac{1}{z} \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \color{blue}{\left(-229.08220098308368 \cdot z - 372.46179876865034\right)}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right)\right) \]

9. Final simplification 98.6%

   \[\leadsto \frac{1}{z} \cdot \left(e^{z + -7.5} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-176.6150291621406}{\left(1 - z\right) + 3} + \left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right) + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(z \cdot -229.08220098308368 - 372.46179876865034\right)\right)\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 5: 97.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{z} \cdot \left(e^{z + -7.5} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7} + \left(\frac{12.507343278686905}{5 - z} + \left(z \cdot -11.042288315961857 - 44.17685104674093\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ 1.0 z)
  (*
   (exp (+ z -7.5))
   (*
    (* (pow (- 7.5 z) (- 0.5 z)) (sqrt (* PI 2.0)))
    (+
     (+
      0.9999999999998099
      (+
       (/ 676.5203681218851 (- 1.0 z))
       (+ (/ -1259.1392167224028 (- 2.0 z)) (/ 771.3234287776531 (- 3.0 z)))))
     (+
      (/ 1.5056327351493116e-7 (+ (- 1.0 z) 7.0))
      (+
       (/ 12.507343278686905 (- 5.0 z))
       (- (* z -11.042288315961857) 44.17685104674093))))))))

C

double code(double z) {
	return (1.0 / z) * (exp((z + -7.5)) * ((pow((7.5 - z), (0.5 - z)) * sqrt((((double) M_PI) * 2.0))) * ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (3.0 - z))))) + ((1.5056327351493116e-7 / ((1.0 - z) + 7.0)) + ((12.507343278686905 / (5.0 - z)) + ((z * -11.042288315961857) - 44.17685104674093))))));
}

Java

public static double code(double z) {
	return (1.0 / z) * (Math.exp((z + -7.5)) * ((Math.pow((7.5 - z), (0.5 - z)) * Math.sqrt((Math.PI * 2.0))) * ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (3.0 - z))))) + ((1.5056327351493116e-7 / ((1.0 - z) + 7.0)) + ((12.507343278686905 / (5.0 - z)) + ((z * -11.042288315961857) - 44.17685104674093))))));
}

Python

def code(z):
	return (1.0 / z) * (math.exp((z + -7.5)) * ((math.pow((7.5 - z), (0.5 - z)) * math.sqrt((math.pi * 2.0))) * ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (3.0 - z))))) + ((1.5056327351493116e-7 / ((1.0 - z) + 7.0)) + ((12.507343278686905 / (5.0 - z)) + ((z * -11.042288315961857) - 44.17685104674093))))))

Julia

function code(z)
	return Float64(Float64(1.0 / z) * Float64(exp(Float64(z + -7.5)) * Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * sqrt(Float64(pi * 2.0))) * 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(1.5056327351493116e-7 / Float64(Float64(1.0 - z) + 7.0)) + Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(Float64(z * -11.042288315961857) - 44.17685104674093)))))))
end

MATLAB

function tmp = code(z)
	tmp = (1.0 / z) * (exp((z + -7.5)) * ((((7.5 - z) ^ (0.5 - z)) * sqrt((pi * 2.0))) * ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (3.0 - z))))) + ((1.5056327351493116e-7 / ((1.0 - z) + 7.0)) + ((12.507343278686905 / (5.0 - z)) + ((z * -11.042288315961857) - 44.17685104674093))))));
end

Wolfram

code[z_] := N[(N[(1.0 / z), $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $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[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] + 7.0), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(z * -11.042288315961857), $MachinePrecision] - 44.17685104674093), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.4%

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

4. Applied egg-rr 99.3%

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

6. Simplified 99.3%

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

7. Taylor expanded in z around 0 98.5%

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

8. Taylor expanded in z around 0 98.5%

   \[\leadsto \frac{1}{z} \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \color{blue}{\left(-11.042288315961857 \cdot z - 44.17685104674093\right)}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right)\right) \]

9. Final simplification 98.5%

   \[\leadsto \frac{1}{z} \cdot \left(e^{z + -7.5} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7} + \left(\frac{12.507343278686905}{5 - z} + \left(z \cdot -11.042288315961857 - 44.17685104674093\right)\right)\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 6: 96.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{z} \cdot \left(e^{z + -7.5} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7} + \left(\frac{12.507343278686905}{5 - z} + -44.17685104674093\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ 1.0 z)
  (*
   (exp (+ z -7.5))
   (*
    (* (pow (- 7.5 z) (- 0.5 z)) (sqrt (* PI 2.0)))
    (+
     (+
      0.9999999999998099
      (+
       (/ 676.5203681218851 (- 1.0 z))
       (+ (/ -1259.1392167224028 (- 2.0 z)) (/ 771.3234287776531 (- 3.0 z)))))
     (+
      (/ 1.5056327351493116e-7 (+ (- 1.0 z) 7.0))
      (+ (/ 12.507343278686905 (- 5.0 z)) -44.17685104674093)))))))

C

double code(double z) {
	return (1.0 / z) * (exp((z + -7.5)) * ((pow((7.5 - z), (0.5 - z)) * sqrt((((double) M_PI) * 2.0))) * ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (3.0 - z))))) + ((1.5056327351493116e-7 / ((1.0 - z) + 7.0)) + ((12.507343278686905 / (5.0 - z)) + -44.17685104674093)))));
}

Java

public static double code(double z) {
	return (1.0 / z) * (Math.exp((z + -7.5)) * ((Math.pow((7.5 - z), (0.5 - z)) * Math.sqrt((Math.PI * 2.0))) * ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (3.0 - z))))) + ((1.5056327351493116e-7 / ((1.0 - z) + 7.0)) + ((12.507343278686905 / (5.0 - z)) + -44.17685104674093)))));
}

Python

def code(z):
	return (1.0 / z) * (math.exp((z + -7.5)) * ((math.pow((7.5 - z), (0.5 - z)) * math.sqrt((math.pi * 2.0))) * ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (3.0 - z))))) + ((1.5056327351493116e-7 / ((1.0 - z) + 7.0)) + ((12.507343278686905 / (5.0 - z)) + -44.17685104674093)))))

Julia

function code(z)
	return Float64(Float64(1.0 / z) * Float64(exp(Float64(z + -7.5)) * Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * sqrt(Float64(pi * 2.0))) * 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(1.5056327351493116e-7 / Float64(Float64(1.0 - z) + 7.0)) + Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + -44.17685104674093))))))
end

MATLAB

function tmp = code(z)
	tmp = (1.0 / z) * (exp((z + -7.5)) * ((((7.5 - z) ^ (0.5 - z)) * sqrt((pi * 2.0))) * ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (3.0 - z))))) + ((1.5056327351493116e-7 / ((1.0 - z) + 7.0)) + ((12.507343278686905 / (5.0 - z)) + -44.17685104674093)))));
end

Wolfram

code[z_] := N[(N[(1.0 / z), $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $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[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] + 7.0), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + -44.17685104674093), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.4%

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

4. Applied egg-rr 99.3%

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

6. Simplified 99.3%

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

7. Taylor expanded in z around 0 98.5%

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

8. Taylor expanded in z around 0 98.3%

   \[\leadsto \frac{1}{z} \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \color{blue}{-44.17685104674093}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right)\right) \]

9. Final simplification 98.3%

   \[\leadsto \frac{1}{z} \cdot \left(e^{z + -7.5} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7} + \left(\frac{12.507343278686905}{5 - z} + -44.17685104674093\right)\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 7: 96.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.4%

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

3. Simplified 97.7%

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

5. Taylor expanded in z around inf 97.7%

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

   1. exp-to-pow 97.7%

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

   2. sub-neg 97.7%

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

   3. metadata-eval 97.7%

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

7. Simplified 97.7%

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

8. Taylor expanded in z around 0 98.4%

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

   1. *-commutative 98.4%

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

10. Simplified 98.4%

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

11. Taylor expanded in z around 0 98.2%

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

    1. *-commutative 98.2%

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

13. Simplified 98.2%

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

14. Final simplification 98.2%

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

Alternative 8: 96.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.4%

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

3. Taylor expanded in z around 0 96.7%

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

   1. *-commutative 96.7%

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

   2. *-commutative 96.7%

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

   3. associate-*l* 96.7%

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

   4. *-commutative 96.7%

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

   5. *-commutative 96.7%

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

   6. associate-*l* 96.7%

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

5. Simplified 96.7%

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

   1. associate-*l/ 96.7%

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

   2. associate-*r* 96.7%

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

   3. pow1/2 96.7%

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

   4. pow1/2 96.7%

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

   5. pow-prod-down 96.7%

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

   6. metadata-eval 96.7%

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

   7. *-commutative 96.7%

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

7. Applied egg-rr 96.7%

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

   1. *-commutative 96.7%

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

   2. associate-*r* 96.7%

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

   3. associate-*l* 96.7%

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

   4. *-commutative 96.7%

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

   5. associate-*l* 96.7%

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

   6. unpow1/2 96.7%

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

   7. unpow1/2 96.7%

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

   8. pow-plus 97.1%

      \[\leadsto \frac{\left(e^{-7.5} \cdot \left(\sqrt{15} \cdot 263.3831869810514\right)\right) \cdot \color{blue}{{\pi}^{\left(0.5 + 1\right)}}}{\sin \left(\pi \cdot z\right)} \]

   9. metadata-eval 97.1%

      \[\leadsto \frac{\left(e^{-7.5} \cdot \left(\sqrt{15} \cdot 263.3831869810514\right)\right) \cdot {\pi}^{\color{blue}{1.5}}}{\sin \left(\pi \cdot z\right)} \]

   10. *-commutative 97.1%

       \[\leadsto \frac{\left(e^{-7.5} \cdot \left(\sqrt{15} \cdot 263.3831869810514\right)\right) \cdot {\pi}^{1.5}}{\sin \color{blue}{\left(z \cdot \pi\right)}} \]

9. Simplified 97.1%

   \[\leadsto \color{blue}{\frac{\left(e^{-7.5} \cdot \left(\sqrt{15} \cdot 263.3831869810514\right)\right) \cdot {\pi}^{1.5}}{\sin \left(z \cdot \pi\right)}} \]

10. Taylor expanded in z around 0 97.2%

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

    1. associate-*l/ 97.0%

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

12. Applied egg-rr 97.0%

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

    1. associate-/l* 97.2%

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

14. Simplified 97.2%

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

15. Final simplification 97.2%

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

Alternative 9: 96.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.4%

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

3. Taylor expanded in z around 0 96.7%

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

   1. *-commutative 96.7%

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

   2. *-commutative 96.7%

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

   3. associate-*l* 96.7%

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

   4. *-commutative 96.7%

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

   5. *-commutative 96.7%

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

   6. associate-*l* 96.7%

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

5. Simplified 96.7%

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

   1. associate-*l/ 96.7%

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

   2. associate-*r* 96.7%

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

   3. pow1/2 96.7%

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

   4. pow1/2 96.7%

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

   5. pow-prod-down 96.7%

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

   6. metadata-eval 96.7%

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

   7. *-commutative 96.7%

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

7. Applied egg-rr 96.7%

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

   1. *-commutative 96.7%

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

   2. associate-*r* 96.7%

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

   3. associate-*l* 96.7%

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

   4. *-commutative 96.7%

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

   5. associate-*l* 96.7%

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

   6. unpow1/2 96.7%

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

   7. unpow1/2 96.7%

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

   8. pow-plus 97.1%

      \[\leadsto \frac{\left(e^{-7.5} \cdot \left(\sqrt{15} \cdot 263.3831869810514\right)\right) \cdot \color{blue}{{\pi}^{\left(0.5 + 1\right)}}}{\sin \left(\pi \cdot z\right)} \]

   9. metadata-eval 97.1%

      \[\leadsto \frac{\left(e^{-7.5} \cdot \left(\sqrt{15} \cdot 263.3831869810514\right)\right) \cdot {\pi}^{\color{blue}{1.5}}}{\sin \left(\pi \cdot z\right)} \]

   10. *-commutative 97.1%

       \[\leadsto \frac{\left(e^{-7.5} \cdot \left(\sqrt{15} \cdot 263.3831869810514\right)\right) \cdot {\pi}^{1.5}}{\sin \color{blue}{\left(z \cdot \pi\right)}} \]

9. Simplified 97.1%

   \[\leadsto \color{blue}{\frac{\left(e^{-7.5} \cdot \left(\sqrt{15} \cdot 263.3831869810514\right)\right) \cdot {\pi}^{1.5}}{\sin \left(z \cdot \pi\right)}} \]

10. Taylor expanded in z around 0 97.2%

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

    1. pow1 97.2%

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

    2. associate-/l* 97.3%

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

12. Applied egg-rr 97.3%

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

    1. unpow1 97.3%

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

14. Simplified 97.3%

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

15. Final simplification 97.3%

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

Alternative 10: 96.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.4%

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

3. Taylor expanded in z around 0 96.7%

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

   1. *-commutative 96.7%

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

   2. *-commutative 96.7%

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

   3. associate-*l* 96.7%

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

   4. *-commutative 96.7%

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

   5. *-commutative 96.7%

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

   6. associate-*l* 96.7%

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

5. Simplified 96.7%

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

   1. associate-*l/ 96.7%

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

   2. associate-*r* 96.7%

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

   3. pow1/2 96.7%

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

   4. pow1/2 96.7%

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

   5. pow-prod-down 96.7%

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

   6. metadata-eval 96.7%

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

   7. *-commutative 96.7%

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

7. Applied egg-rr 96.7%

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

   1. *-commutative 96.7%

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

   2. associate-*r* 96.7%

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

   3. associate-*l* 96.7%

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

   4. *-commutative 96.7%

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

   5. associate-*l* 96.7%

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

   6. unpow1/2 96.7%

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

   7. unpow1/2 96.7%

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

   8. pow-plus 97.1%

      \[\leadsto \frac{\left(e^{-7.5} \cdot \left(\sqrt{15} \cdot 263.3831869810514\right)\right) \cdot \color{blue}{{\pi}^{\left(0.5 + 1\right)}}}{\sin \left(\pi \cdot z\right)} \]

   9. metadata-eval 97.1%

      \[\leadsto \frac{\left(e^{-7.5} \cdot \left(\sqrt{15} \cdot 263.3831869810514\right)\right) \cdot {\pi}^{\color{blue}{1.5}}}{\sin \left(\pi \cdot z\right)} \]

   10. *-commutative 97.1%

       \[\leadsto \frac{\left(e^{-7.5} \cdot \left(\sqrt{15} \cdot 263.3831869810514\right)\right) \cdot {\pi}^{1.5}}{\sin \color{blue}{\left(z \cdot \pi\right)}} \]

9. Simplified 97.1%

   \[\leadsto \color{blue}{\frac{\left(e^{-7.5} \cdot \left(\sqrt{15} \cdot 263.3831869810514\right)\right) \cdot {\pi}^{1.5}}{\sin \left(z \cdot \pi\right)}} \]

10. Taylor expanded in z around 0 97.2%

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

    1. associate-*r* 97.2%

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

    2. associate-/l* 97.4%

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

12. Simplified 97.4%

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

13. Final simplification 97.4%

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

Reproduce

herbie shell --seed 2024061 
(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))))))