Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.6% → 98.5%

Time: 2.0min

Alternatives: 11

Speedup: 1.4×

• could not determine a ground truth

  1. z = -1.3152941140288807e+267

Specification

\[z \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.6% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[z_] := N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(1.0 + N[(6.5 - z), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(N[(z - 6.5), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(-176.6150291621406 / N[(1.0 - N[(z + -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + 0.9999999999998099), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.0%

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

4. Applied egg-rr 98.9%

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

6. Applied egg-rr 98.9%

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

7. Final simplification 98.9%

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

Alternative 2: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[z_] := N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(6.5 + N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(z + -1.0), $MachinePrecision] - 6.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] + 5.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(1.0 + N[(6.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-176.6150291621406 / N[(1.0 - N[(z + -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] + N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.0%

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

4. Applied egg-rr 98.9%

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

6. Applied egg-rr 98.5%

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

8. Simplified 98.6%

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

   1. *-un-lft-identity 98.6%

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

   2. associate-+r+ 98.6%

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

10. Applied egg-rr 98.6%

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

    1. *-lft-identity 98.6%

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

    2. +-commutative 98.6%

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

    3. associate-+l+ 98.6%

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

12. Simplified 98.6%

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

13. Final simplification 98.6%

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

Alternative 3: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

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((-6.5 + (z + -1.0))))) * (((676.5203681218851 / (1.0 - z)) + 0.9999999999998099) + ((((771.3234287776531 / (3.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (-176.6150291621406 / (4.0 - z)))) + ((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z)))) + ((1.5056327351493116e-7 / (8.0 - z)) + (9.984369578019572e-6 / (7.0 - z)))));
}

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((-6.5 + (z + -1.0))))) * (((676.5203681218851 / (1.0 - z)) + 0.9999999999998099) + ((((771.3234287776531 / (3.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (-176.6150291621406 / (4.0 - z)))) + ((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z)))) + ((1.5056327351493116e-7 / (8.0 - z)) + (9.984369578019572e-6 / (7.0 - z)))));
}

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((-6.5 + (z + -1.0))))) * (((676.5203681218851 / (1.0 - z)) + 0.9999999999998099) + ((((771.3234287776531 / (3.0 - z)) + ((-1259.1392167224028 / (2.0 - z)) + (-176.6150291621406 / (4.0 - z)))) + ((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z)))) + ((1.5056327351493116e-7 / (8.0 - z)) + (9.984369578019572e-6 / (7.0 - z)))))

Julia

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

MATLAB

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

Wolfram

code[z_] := N[(N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-6.5 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + 0.9999999999998099), $MachinePrecision] + N[(N[(N[(N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.0%

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

3. Simplified 97.2%

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

6. Applied egg-rr 98.8%

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

8. Simplified 98.5%

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

9. Final simplification 98.5%

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

Alternative 4: 97.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left({\left(6.5 + \left(1 - z\right)\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(z + -1\right) - 6.5}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + z \cdot 606.6766809167608\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (* (/ PI (sin (* PI z))) (sqrt (* PI 2.0)))
  (*
   (* (pow (+ 6.5 (- 1.0 z)) (+ (- 1.0 z) -0.5)) (exp (- (+ z -1.0) 6.5)))
   (+
    263.3831869810514
    (*
     z
     (+
      436.8961725563396
      (* z (+ 545.0353078428827 (* z 606.6766809167608)))))))))

C

double code(double z) {
	return ((((double) M_PI) / sin((((double) M_PI) * z))) * sqrt((((double) M_PI) * 2.0))) * ((pow((6.5 + (1.0 - z)), ((1.0 - z) + -0.5)) * exp(((z + -1.0) - 6.5))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (z * 606.6766809167608)))))));
}

Java

public static double code(double z) {
	return ((Math.PI / Math.sin((Math.PI * z))) * Math.sqrt((Math.PI * 2.0))) * ((Math.pow((6.5 + (1.0 - z)), ((1.0 - z) + -0.5)) * Math.exp(((z + -1.0) - 6.5))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (z * 606.6766809167608)))))));
}

Python

def code(z):
	return ((math.pi / math.sin((math.pi * z))) * math.sqrt((math.pi * 2.0))) * ((math.pow((6.5 + (1.0 - z)), ((1.0 - z) + -0.5)) * math.exp(((z + -1.0) - 6.5))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (z * 606.6766809167608)))))))

Julia

function code(z)
	return Float64(Float64(Float64(pi / sin(Float64(pi * z))) * sqrt(Float64(pi * 2.0))) * Float64(Float64((Float64(6.5 + Float64(1.0 - z)) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(Float64(Float64(z + -1.0) - 6.5))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * Float64(545.0353078428827 + Float64(z * 606.6766809167608))))))))
end

MATLAB

function tmp = code(z)
	tmp = ((pi / sin((pi * z))) * sqrt((pi * 2.0))) * ((((6.5 + (1.0 - z)) ^ ((1.0 - z) + -0.5)) * exp(((z + -1.0) - 6.5))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (z * 606.6766809167608)))))));
end

Wolfram

code[z_] := N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(6.5 + N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(z + -1.0), $MachinePrecision] - 6.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * N[(545.0353078428827 + N[(z * 606.6766809167608), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.0%

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

4. Applied egg-rr 98.9%

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

6. Applied egg-rr 98.5%

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

8. Simplified 98.6%

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

9. Taylor expanded in z around 0 98.0%

   \[\leadsto \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)}\right) \]
10. Step-by-step derivation

    1. *-commutative 98.0%

       \[\leadsto \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + \color{blue}{z \cdot 606.6766809167608}\right)\right)\right)\right) \]

11. Simplified 98.0%

    \[\leadsto \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \left(\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + z \cdot 606.6766809167608\right)\right)\right)}\right) \]

12. Final simplification 98.0%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left({\left(6.5 + \left(1 - z\right)\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(z + -1\right) - 6.5}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + z \cdot 606.6766809167608\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 5: 97.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[z_] := N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(1.0 + N[(6.5 - z), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(N[(z - 6.5), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(-176.6150291621406 / N[(1.0 - N[(z + -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + 0.9999999999998099), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.0%

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

4. Applied egg-rr 98.9%

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

6. Applied egg-rr 98.9%

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

7. Taylor expanded in z around 0 97.8%

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

8. Final simplification 97.8%

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

Alternative 6: 96.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{\pi \cdot 2} \cdot \left({\left(6.5 + \left(1 - z\right)\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right)\right) \cdot \frac{263.3831869810514 + z \cdot 436.8961725563396}{z} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (sqrt (* PI 2.0))
   (* (pow (+ 6.5 (- 1.0 z)) (+ (- 1.0 z) -0.5)) (* (+ z 1.0) (exp -7.5))))
  (/ (+ 263.3831869810514 (* z 436.8961725563396)) z)))

C

double code(double z) {
	return (sqrt((((double) M_PI) * 2.0)) * (pow((6.5 + (1.0 - z)), ((1.0 - z) + -0.5)) * ((z + 1.0) * exp(-7.5)))) * ((263.3831869810514 + (z * 436.8961725563396)) / z);
}

Java

public static double code(double z) {
	return (Math.sqrt((Math.PI * 2.0)) * (Math.pow((6.5 + (1.0 - z)), ((1.0 - z) + -0.5)) * ((z + 1.0) * Math.exp(-7.5)))) * ((263.3831869810514 + (z * 436.8961725563396)) / z);
}

Python

def code(z):
	return (math.sqrt((math.pi * 2.0)) * (math.pow((6.5 + (1.0 - z)), ((1.0 - z) + -0.5)) * ((z + 1.0) * math.exp(-7.5)))) * ((263.3831869810514 + (z * 436.8961725563396)) / z)

Julia

function code(z)
	return Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(6.5 + Float64(1.0 - z)) ^ Float64(Float64(1.0 - z) + -0.5)) * Float64(Float64(z + 1.0) * exp(-7.5)))) * Float64(Float64(263.3831869810514 + Float64(z * 436.8961725563396)) / z))
end

MATLAB

function tmp = code(z)
	tmp = (sqrt((pi * 2.0)) * (((6.5 + (1.0 - z)) ^ ((1.0 - z) + -0.5)) * ((z + 1.0) * exp(-7.5)))) * ((263.3831869810514 + (z * 436.8961725563396)) / z);
end

Wolfram

code[z_] := N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(6.5 + N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[(z + 1.0), $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(263.3831869810514 + N[(z * 436.8961725563396), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.0%

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

3. Simplified 97.2%

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

5. Taylor expanded in z around 0 97.1%

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

   1. *-commutative 97.1%

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

7. Simplified 97.1%

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

8. Taylor expanded in z around 0 97.5%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(e^{-7.5} + z \cdot e^{-7.5}\right)}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
9. Step-by-step derivation

   1. distribute-rgt1-in 97.5%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(\left(z + 1\right) \cdot e^{-7.5}\right)}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

10. Simplified 97.5%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{\left(\left(z + 1\right) \cdot e^{-7.5}\right)}\right)\right) \cdot \left(\left(263.3831869810514 + z \cdot 436.8961725563396\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

11. Taylor expanded in z around 0 97.2%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right)\right) \cdot \color{blue}{\frac{263.3831869810514 + 436.8961725563396 \cdot z}{z}} \]

12. Final simplification 97.2%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(6.5 + \left(1 - z\right)\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right)\right) \cdot \frac{263.3831869810514 + z \cdot 436.8961725563396}{z} \]
13. Add Preprocessing

Alternative 7: 95.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-6.5 + \left(z + -1\right)}\right)\right) \cdot \frac{263.3831869810514 + z \cdot 436.3997278161676}{z} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (* (sqrt (* PI 2.0)) (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ -6.5 (+ z -1.0)))))
  (/ (+ 263.3831869810514 (* z 436.3997278161676)) z)))

C

double code(double z) {
	return (sqrt((((double) M_PI) * 2.0)) * (pow((7.5 - z), (0.5 - z)) * exp((-6.5 + (z + -1.0))))) * ((263.3831869810514 + (z * 436.3997278161676)) / z);
}

Java

public static double code(double z) {
	return (Math.sqrt((Math.PI * 2.0)) * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((-6.5 + (z + -1.0))))) * ((263.3831869810514 + (z * 436.3997278161676)) / z);
}

Python

def code(z):
	return (math.sqrt((math.pi * 2.0)) * (math.pow((7.5 - z), (0.5 - z)) * math.exp((-6.5 + (z + -1.0))))) * ((263.3831869810514 + (z * 436.3997278161676)) / z)

Julia

function code(z)
	return Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(-6.5 + Float64(z + -1.0))))) * Float64(Float64(263.3831869810514 + Float64(z * 436.3997278161676)) / z))
end

MATLAB

function tmp = code(z)
	tmp = (sqrt((pi * 2.0)) * (((7.5 - z) ^ (0.5 - z)) * exp((-6.5 + (z + -1.0))))) * ((263.3831869810514 + (z * 436.3997278161676)) / z);
end

Wolfram

code[z_] := N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-6.5 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(263.3831869810514 + N[(z * 436.3997278161676), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.0%

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

3. Simplified 97.2%

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

5. Taylor expanded in z around 0 95.6%

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

   1. pow1 95.6%

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

   2. distribute-neg-in 95.6%

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

   3. metadata-eval 95.6%

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

7. Applied egg-rr 95.6%

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

   1. unpow1 95.6%

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

   2. +-commutative 95.6%

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

   3. metadata-eval 95.6%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\color{blue}{\left(7.5 + -1\right)} + \left(1 - z\right)\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + 2.4783749183520145\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   4. associate-+r+ 95.6%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\color{blue}{\left(7.5 + \left(-1 + \left(1 - z\right)\right)\right)}}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + 2.4783749183520145\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   5. associate-+r- 95.6%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 + \color{blue}{\left(\left(-1 + 1\right) - z\right)}\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + 2.4783749183520145\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   6. metadata-eval 95.6%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 + \left(\color{blue}{0} - z\right)\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + 2.4783749183520145\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   7. neg-sub0 95.6%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 + \color{blue}{\left(-z\right)}\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + 2.4783749183520145\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. sub-neg 95.6%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\color{blue}{\left(7.5 - z\right)}}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + 2.4783749183520145\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   9. +-commutative 95.6%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\color{blue}{\left(-0.5 + \left(1 - z\right)\right)}} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + 2.4783749183520145\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   10. metadata-eval 95.6%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(\color{blue}{\left(0.5 + -1\right)} + \left(1 - z\right)\right)} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + 2.4783749183520145\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   11. associate-+r+ 95.6%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\color{blue}{\left(0.5 + \left(-1 + \left(1 - z\right)\right)\right)}} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + 2.4783749183520145\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   12. associate-+r- 95.6%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 + \color{blue}{\left(\left(-1 + 1\right) - z\right)}\right)} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + 2.4783749183520145\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   13. metadata-eval 95.6%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 + \left(\color{blue}{0} - z\right)\right)} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + 2.4783749183520145\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   14. neg-sub0 95.6%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 + \color{blue}{\left(-z\right)}\right)} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + 2.4783749183520145\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   15. sub-neg 95.6%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\color{blue}{\left(0.5 - z\right)}} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + 2.4783749183520145\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

9. Simplified 95.6%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \color{blue}{\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)}\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) + 1} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + 2.4783749183520145\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

10. Taylor expanded in z around 0 96.8%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right) \cdot \color{blue}{\frac{263.3831869810514 + 436.3997278161676 \cdot z}{z}} \]
11. Step-by-step derivation

    1. *-commutative 96.8%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right) \cdot \frac{263.3831869810514 + \color{blue}{z \cdot 436.3997278161676}}{z} \]

12. Simplified 96.8%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right) \cdot \color{blue}{\frac{263.3831869810514 + z \cdot 436.3997278161676}{z}} \]

13. Final simplification 96.8%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-6.5 + \left(z + -1\right)}\right)\right) \cdot \frac{263.3831869810514 + z \cdot 436.3997278161676}{z} \]
14. Add Preprocessing

Alternative 8: 95.9% 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 97.0%

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

3. Simplified 97.3%

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

5. Taylor expanded in z around 0 96.4%

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

   1. *-un-lft-identity 96.4%

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

   2. associate-/l* 96.6%

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

   3. sqrt-unprod 96.6%

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

   4. metadata-eval 96.6%

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

7. Applied egg-rr 96.6%

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

   1. *-lft-identity 96.6%

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

9. Simplified 96.6%

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

Alternative 9: 95.8% accurate, 1.7× speedup

Math

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

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

Derivation

1. Initial program 97.0%

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

3. Simplified 97.3%

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

5. Taylor expanded in z around 0 96.4%

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

   1. pow1 96.4%

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

   2. associate-/l* 96.6%

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

   3. sqrt-unprod 96.6%

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

   4. metadata-eval 96.6%

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

7. Applied egg-rr 96.6%

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

   1. unpow1 96.6%

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

   2. associate-*l* 96.6%

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

9. Simplified 96.6%

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

Alternative 10: 95.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.0%

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

3. Simplified 97.2%

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

5. Taylor expanded in z around 0 96.2%

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

6. Taylor expanded in z around 0 96.4%

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

7. Taylor expanded in z around 0 96.3%

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

   1. pow1 96.3%

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

   2. associate-*r* 96.3%

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

   3. pow1/2 96.3%

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

   4. *-commutative 96.3%

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

   5. pow1/2 96.3%

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

   6. pow-prod-down 96.3%

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

   7. *-commutative 96.3%

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

9. Applied egg-rr 96.3%

   \[\leadsto \color{blue}{{\left({\left(\left(\pi \cdot 2\right) \cdot 7.5\right)}^{0.5} \cdot e^{-7.5}\right)}^{1}} \cdot \frac{263.3831869810514}{z} \]
10. Step-by-step derivation

    1. unpow1 96.3%

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

    2. *-commutative 96.3%

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

    3. unpow1/2 96.3%

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

    4. associate-*l* 96.3%

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

    5. metadata-eval 96.3%

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

11. Simplified 96.3%

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

Alternative 11: 94.7% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.0%

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

3. Simplified 97.2%

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

5. Taylor expanded in z around 0 96.2%

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

6. Taylor expanded in z around 0 96.4%

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

7. Taylor expanded in z around 0 96.3%

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

   1. associate-*r/ 95.9%

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

   2. associate-*r* 95.9%

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

   3. pow1/2 95.9%

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

   4. *-commutative 95.9%

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

   5. pow1/2 95.9%

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

   6. pow-prod-down 95.9%

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

   7. *-commutative 95.9%

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

9. Applied egg-rr 95.9%

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

    1. associate-*r/ 96.3%

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

    2. *-commutative 96.3%

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

    3. associate-*l* 96.2%

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

    4. unpow1/2 96.2%

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

    5. associate-*l* 96.2%

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

    6. metadata-eval 96.2%

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

11. Simplified 96.2%

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

Reproduce

herbie shell --seed 2024143 
(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))))))