Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.4% → 98.1%

Time: 2.0min

Alternatives: 8

Speedup: 1.3×

• could not determine a ground truth

  1. z = -2.0978095140132458e+235

Specification

\[z \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.9%

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

3. Simplified 95.8%

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

   1. *-un-lft-identity 95.8%

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

   2. +-commutative 95.8%

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

   3. +-commutative 95.8%

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

   4. associate-+l- 95.8%

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

6. Applied egg-rr 95.8%

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

   1. *-lft-identity 95.8%

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

   2. associate-+l+ 97.0%

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

   3. +-commutative 97.0%

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

   4. associate-+l+ 97.3%

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

   5. +-commutative 97.3%

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

   6. associate-+r- 97.3%

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

   7. metadata-eval 97.3%

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

   8. associate--r- 97.3%

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

   9. +-commutative 97.3%

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

   10. associate-+r- 97.3%

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

   11. metadata-eval 97.3%

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

8. Simplified 97.3%

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

10. Applied egg-rr 97.2%

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

12. Simplified 98.7%

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

13. Final simplification 98.7%

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

Alternative 2: 96.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + z \cdot 8.852513840025837\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (/ PI (sin (* PI z)))
   (* (sqrt (* PI 2.0)) (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5)))))
  (+
   263.3831869810514
   (*
    z
    (+
     436.8961725563396
     (* z (+ 545.0353078428827 (* z 8.852513840025837))))))))

C

double code(double z) {
	return ((((double) M_PI) / sin((((double) M_PI) * z))) * (sqrt((((double) M_PI) * 2.0)) * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5))))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (z * 8.852513840025837))))));
}

Java

public static double code(double z) {
	return ((Math.PI / Math.sin((Math.PI * z))) * (Math.sqrt((Math.PI * 2.0)) * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5))))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (z * 8.852513840025837))))));
}

Python

def code(z):
	return ((math.pi / math.sin((math.pi * z))) * (math.sqrt((math.pi * 2.0)) * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5))))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (z * 8.852513840025837))))))

Julia

function code(z)
	return Float64(Float64(Float64(pi / sin(Float64(pi * z))) * Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5))))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * Float64(545.0353078428827 + Float64(z * 8.852513840025837)))))))
end

MATLAB

function tmp = code(z)
	tmp = ((pi / sin((pi * z))) * (sqrt((pi * 2.0)) * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5))))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (z * 8.852513840025837))))));
end

Wolfram

code[z_] := N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * N[(545.0353078428827 + N[(z * 8.852513840025837), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.9%

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

3. Simplified 98.6%

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

5. Taylor expanded in z around 0 97.9%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\color{blue}{\left(47.95075976068351 + z \cdot \left(361.7355639412844 + 519.1279660315847 \cdot z\right)\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
6. Step-by-step derivation

   1. +-commutative 97.9%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\color{blue}{\left(z \cdot \left(361.7355639412844 + 519.1279660315847 \cdot z\right) + 47.95075976068351\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   2. *-commutative 97.9%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\left(z \cdot \left(361.7355639412844 + \color{blue}{z \cdot 519.1279660315847}\right) + 47.95075976068351\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

7. Simplified 97.9%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(\left(\color{blue}{\left(z \cdot \left(361.7355639412844 + z \cdot 519.1279660315847\right) + 47.95075976068351\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

8. Taylor expanded in z around 0 97.5%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 8.852513840025837 \cdot z\right)\right)\right)} \]
9. Step-by-step derivation

   1. *-commutative 97.5%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + \color{blue}{z \cdot 8.852513840025837}\right)\right)\right) \]

10. Simplified 97.5%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + z \cdot 8.852513840025837\right)\right)\right)} \]

11. Taylor expanded in z around inf 97.5%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \color{blue}{\left(e^{\log \left(7.5 - z\right) \cdot \left(0.5 - z\right)} \cdot e^{z - 7.5}\right)}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + z \cdot 8.852513840025837\right)\right)\right) \]
12. Step-by-step derivation

    1. exp-to-pow 97.5%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\color{blue}{{\left(7.5 - z\right)}^{\left(0.5 - z\right)}} \cdot e^{z - 7.5}\right)\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + z \cdot 8.852513840025837\right)\right)\right) \]

    2. *-commutative 97.5%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \color{blue}{\left(e^{z - 7.5} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + z \cdot 8.852513840025837\right)\right)\right) \]

    3. sub-neg 97.5%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(e^{\color{blue}{z + \left(-7.5\right)}} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + z \cdot 8.852513840025837\right)\right)\right) \]

    4. metadata-eval 97.5%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(e^{z + \color{blue}{-7.5}} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + z \cdot 8.852513840025837\right)\right)\right) \]

    5. +-commutative 97.5%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(e^{\color{blue}{-7.5 + z}} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + z \cdot 8.852513840025837\right)\right)\right) \]

13. Simplified 97.5%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \color{blue}{\left(e^{-7.5 + z} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + z \cdot 8.852513840025837\right)\right)\right) \]

14. Final simplification 97.5%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + z \cdot 8.852513840025837\right)\right)\right) \]
15. Add Preprocessing

Alternative 3: 96.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.9%

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

3. Simplified 96.0%

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

5. Taylor expanded in z around 0 96.6%

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

   1. associate-*l/ 96.5%

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

   2. *-commutative 96.5%

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

   3. associate-*r* 97.2%

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

   4. *-commutative 97.2%

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

7. Simplified 97.2%

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

8. Final simplification 97.2%

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

Alternative 4: 96.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{\left(263.3831869810514 \cdot \sqrt[3]{{\pi}^{1.5}}\right) \cdot \left(e^{-7.5} \cdot \sqrt{15}\right)}{z} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (/
  (* (* 263.3831869810514 (cbrt (pow PI 1.5))) (* (exp -7.5) (sqrt 15.0)))
  z))

C

double code(double z) {
	return ((263.3831869810514 * cbrt(pow(((double) M_PI), 1.5))) * (exp(-7.5) * sqrt(15.0))) / z;
}

Java

public static double code(double z) {
	return ((263.3831869810514 * Math.cbrt(Math.pow(Math.PI, 1.5))) * (Math.exp(-7.5) * Math.sqrt(15.0))) / z;
}

Julia

function code(z)
	return Float64(Float64(Float64(263.3831869810514 * cbrt((pi ^ 1.5))) * Float64(exp(-7.5) * sqrt(15.0))) / z)
end

Wolfram

code[z_] := N[(N[(N[(263.3831869810514 * N[Power[N[Power[Pi, 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] * N[Sqrt[15.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Derivation

1. Initial program 96.9%

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

3. Simplified 96.0%

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

5. Taylor expanded in z around 0 96.6%

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

   1. add-cbrt-cube 96.6%

      \[\leadsto 263.3831869810514 \cdot \left(\frac{e^{-7.5} \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)}{z} \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}}}\right) \]

   2. add-sqr-sqrt 96.6%

      \[\leadsto 263.3831869810514 \cdot \left(\frac{e^{-7.5} \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)}{z} \cdot \sqrt[3]{\color{blue}{\pi} \cdot \sqrt{\pi}}\right) \]

7. Applied egg-rr 96.6%

   \[\leadsto 263.3831869810514 \cdot \left(\frac{e^{-7.5} \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)}{z} \cdot \color{blue}{\sqrt[3]{\pi \cdot \sqrt{\pi}}}\right) \]
8. Step-by-step derivation

   1. *-commutative 96.6%

      \[\leadsto 263.3831869810514 \cdot \left(\frac{e^{-7.5} \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)}{z} \cdot \sqrt[3]{\color{blue}{\sqrt{\pi} \cdot \pi}}\right) \]

   2. unpow1/2 96.6%

      \[\leadsto 263.3831869810514 \cdot \left(\frac{e^{-7.5} \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)}{z} \cdot \sqrt[3]{\color{blue}{{\pi}^{0.5}} \cdot \pi}\right) \]

   3. pow-plus 96.4%

      \[\leadsto 263.3831869810514 \cdot \left(\frac{e^{-7.5} \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)}{z} \cdot \sqrt[3]{\color{blue}{{\pi}^{\left(0.5 + 1\right)}}}\right) \]

   4. metadata-eval 96.4%

      \[\leadsto 263.3831869810514 \cdot \left(\frac{e^{-7.5} \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)}{z} \cdot \sqrt[3]{{\pi}^{\color{blue}{1.5}}}\right) \]

9. Simplified 96.4%

   \[\leadsto 263.3831869810514 \cdot \left(\frac{e^{-7.5} \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)}{z} \cdot \color{blue}{\sqrt[3]{{\pi}^{1.5}}}\right) \]
10. Step-by-step derivation

    1. pow1 96.4%

       \[\leadsto \color{blue}{{\left(263.3831869810514 \cdot \left(\frac{e^{-7.5} \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)}{z} \cdot \sqrt[3]{{\pi}^{1.5}}\right)\right)}^{1}} \]

    2. *-commutative 96.4%

       \[\leadsto {\left(263.3831869810514 \cdot \color{blue}{\left(\sqrt[3]{{\pi}^{1.5}} \cdot \frac{e^{-7.5} \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)}{z}\right)}\right)}^{1} \]

    3. associate-/l* 96.1%

       \[\leadsto {\left(263.3831869810514 \cdot \left(\sqrt[3]{{\pi}^{1.5}} \cdot \color{blue}{\left(e^{-7.5} \cdot \frac{\sqrt{2} \cdot \sqrt{7.5}}{z}\right)}\right)\right)}^{1} \]

    4. sqrt-unprod 96.1%

       \[\leadsto {\left(263.3831869810514 \cdot \left(\sqrt[3]{{\pi}^{1.5}} \cdot \left(e^{-7.5} \cdot \frac{\color{blue}{\sqrt{2 \cdot 7.5}}}{z}\right)\right)\right)}^{1} \]

    5. metadata-eval 96.1%

       \[\leadsto {\left(263.3831869810514 \cdot \left(\sqrt[3]{{\pi}^{1.5}} \cdot \left(e^{-7.5} \cdot \frac{\sqrt{\color{blue}{15}}}{z}\right)\right)\right)}^{1} \]

11. Applied egg-rr 96.1%

    \[\leadsto \color{blue}{{\left(263.3831869810514 \cdot \left(\sqrt[3]{{\pi}^{1.5}} \cdot \left(e^{-7.5} \cdot \frac{\sqrt{15}}{z}\right)\right)\right)}^{1}} \]
12. Step-by-step derivation

    1. unpow1 96.1%

       \[\leadsto \color{blue}{263.3831869810514 \cdot \left(\sqrt[3]{{\pi}^{1.5}} \cdot \left(e^{-7.5} \cdot \frac{\sqrt{15}}{z}\right)\right)} \]

    2. associate-*r* 96.0%

       \[\leadsto \color{blue}{\left(263.3831869810514 \cdot \sqrt[3]{{\pi}^{1.5}}\right) \cdot \left(e^{-7.5} \cdot \frac{\sqrt{15}}{z}\right)} \]

    3. associate-*r/ 96.3%

       \[\leadsto \left(263.3831869810514 \cdot \sqrt[3]{{\pi}^{1.5}}\right) \cdot \color{blue}{\frac{e^{-7.5} \cdot \sqrt{15}}{z}} \]

13. Simplified 96.3%

    \[\leadsto \color{blue}{\left(263.3831869810514 \cdot \sqrt[3]{{\pi}^{1.5}}\right) \cdot \frac{e^{-7.5} \cdot \sqrt{15}}{z}} \]
14. Step-by-step derivation

    1. associate-*r/ 97.1%

       \[\leadsto \color{blue}{\frac{\left(263.3831869810514 \cdot \sqrt[3]{{\pi}^{1.5}}\right) \cdot \left(e^{-7.5} \cdot \sqrt{15}\right)}{z}} \]

15. Applied egg-rr 97.1%

    \[\leadsto \color{blue}{\frac{\left(263.3831869810514 \cdot \sqrt[3]{{\pi}^{1.5}}\right) \cdot \left(e^{-7.5} \cdot \sqrt{15}\right)}{z}} \]

16. Final simplification 97.1%

    \[\leadsto \frac{\left(263.3831869810514 \cdot \sqrt[3]{{\pi}^{1.5}}\right) \cdot \left(e^{-7.5} \cdot \sqrt{15}\right)}{z} \]
17. Add Preprocessing

Alternative 5: 95.9% 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 96.9%

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

3. Simplified 96.0%

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

5. Taylor expanded in z around 0 96.6%

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

   1. associate-*l/ 96.5%

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

   2. sqrt-unprod 96.5%

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

   3. metadata-eval 96.5%

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

7. Applied egg-rr 96.5%

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

   1. associate-/l* 96.7%

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

9. Simplified 96.7%

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

10. Final simplification 96.7%

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

Alternative 6: 96.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.9%

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

3. Simplified 96.0%

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

5. Taylor expanded in z around 0 96.6%

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

   1. associate-/l* 96.8%

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

   2. sqrt-unprod 96.8%

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

   3. metadata-eval 96.8%

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

7. Applied egg-rr 96.8%

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

8. Final simplification 96.8%

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

Alternative 7: 96.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.9%

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

3. Simplified 96.0%

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

5. Taylor expanded in z around 0 96.6%

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

   1. associate-/l* 96.8%

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

   2. sqrt-unprod 96.8%

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

   3. metadata-eval 96.8%

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

7. Applied egg-rr 96.8%

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

8. Taylor expanded in z around 0 96.6%

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

   1. *-commutative 96.6%

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

   2. associate-/l* 96.8%

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

10. Simplified 96.8%

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

11. Final simplification 96.8%

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

Alternative 8: 96.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.9%

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

3. Simplified 96.0%

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

5. Taylor expanded in z around 0 96.6%

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

   1. pow1 96.6%

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

   2. *-commutative 96.6%

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

   3. associate-/l* 96.8%

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

   4. sqrt-unprod 96.8%

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

   5. metadata-eval 96.8%

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

7. Applied egg-rr 96.8%

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

   1. unpow1 96.8%

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

   2. *-commutative 96.8%

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

   3. associate-*r* 96.8%

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

9. Simplified 96.8%

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

10. Final simplification 96.8%

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

Reproduce

herbie shell --seed 2024077 
(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))))))