Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.5% → 98.4%

Time: 1.2min

Alternatives: 12

Speedup: 1.2×

• could not determine a ground truth

  1. z = -4.3863889525100276e+161

Specification

\[z \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

4. Applied egg-rr 97.5%

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

6. Simplified 99.2%

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

7. Final simplification 99.2%

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

Alternative 2: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

4. Applied egg-rr 97.5%

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

6. Simplified 99.2%

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

   1. *-un-lft-identity 99.2%

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

   2. +-commutative 99.2%

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

   3. associate-+l+ 98.5%

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

8. Applied egg-rr 98.5%

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

   1. *-lft-identity 98.5%

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

   2. associate-+l+ 99.2%

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

   3. associate-+l+ 99.2%

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

10. Simplified 99.2%

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

11. Final simplification 99.2%

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

Alternative 3: 97.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z - 7.5}\right) \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(\frac{771.3234287776531}{1 - \left(z - 2\right)} + \frac{-176.6150291621406}{4 - z}\right)\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7} + \left(2.4783748995316053 + z \cdot \left(0.4964447378194062 + z \cdot 0.09941724248999204\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ PI (sin (* PI z)))
  (*
   (* (* (pow (- 7.5 z) (- 0.5 z)) (exp (- z 7.5))) (sqrt (* PI 2.0)))
   (+
    (+ 0.9999999999998099 (/ 676.5203681218851 (- 1.0 z)))
    (+
     (+
      (/ -1259.1392167224028 (- 2.0 z))
      (+
       (/ 771.3234287776531 (- 1.0 (- z 2.0)))
       (/ -176.6150291621406 (- 4.0 z))))
     (+
      (/ 1.5056327351493116e-7 (+ (- 1.0 z) 7.0))
      (+
       2.4783748995316053
       (* z (+ 0.4964447378194062 (* z 0.09941724248999204))))))))))

C

double code(double z) {
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (((pow((7.5 - z), (0.5 - z)) * exp((z - 7.5))) * sqrt((((double) M_PI) * 2.0))) * ((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + (((-1259.1392167224028 / (2.0 - z)) + ((771.3234287776531 / (1.0 - (z - 2.0))) + (-176.6150291621406 / (4.0 - z)))) + ((1.5056327351493116e-7 / ((1.0 - z) + 7.0)) + (2.4783748995316053 + (z * (0.4964447378194062 + (z * 0.09941724248999204))))))));
}

Java

public static double code(double z) {
	return (Math.PI / Math.sin((Math.PI * z))) * (((Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z - 7.5))) * Math.sqrt((Math.PI * 2.0))) * ((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + (((-1259.1392167224028 / (2.0 - z)) + ((771.3234287776531 / (1.0 - (z - 2.0))) + (-176.6150291621406 / (4.0 - z)))) + ((1.5056327351493116e-7 / ((1.0 - z) + 7.0)) + (2.4783748995316053 + (z * (0.4964447378194062 + (z * 0.09941724248999204))))))));
}

Python

def code(z):
	return (math.pi / math.sin((math.pi * z))) * (((math.pow((7.5 - z), (0.5 - z)) * math.exp((z - 7.5))) * math.sqrt((math.pi * 2.0))) * ((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + (((-1259.1392167224028 / (2.0 - z)) + ((771.3234287776531 / (1.0 - (z - 2.0))) + (-176.6150291621406 / (4.0 - z)))) + ((1.5056327351493116e-7 / ((1.0 - z) + 7.0)) + (2.4783748995316053 + (z * (0.4964447378194062 + (z * 0.09941724248999204))))))))

Julia

function code(z)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z - 7.5))) * sqrt(Float64(pi * 2.0))) * Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 - z))) + Float64(Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) + Float64(Float64(771.3234287776531 / Float64(1.0 - Float64(z - 2.0))) + Float64(-176.6150291621406 / Float64(4.0 - z)))) + Float64(Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) + 7.0)) + Float64(2.4783748995316053 + Float64(z * Float64(0.4964447378194062 + Float64(z * 0.09941724248999204)))))))))
end

MATLAB

function tmp = code(z)
	tmp = (pi / sin((pi * z))) * (((((7.5 - z) ^ (0.5 - z)) * exp((z - 7.5))) * sqrt((pi * 2.0))) * ((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + (((-1259.1392167224028 / (2.0 - z)) + ((771.3234287776531 / (1.0 - (z - 2.0))) + (-176.6150291621406 / (4.0 - z)))) + ((1.5056327351493116e-7 / ((1.0 - z) + 7.0)) + (2.4783748995316053 + (z * (0.4964447378194062 + (z * 0.09941724248999204))))))));
end

Wolfram

code[z_] := N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z - 7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(1.0 - N[(z - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] + 7.0), $MachinePrecision]), $MachinePrecision] + N[(2.4783748995316053 + N[(z * N[(0.4964447378194062 + N[(z * 0.09941724248999204), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.3%

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

4. Applied egg-rr 97.5%

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

6. Simplified 99.2%

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

7. Taylor expanded in z around 0 99.0%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right) \cdot \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(\frac{771.3234287776531}{1 + \left(2 - z\right)} + \frac{-176.6150291621406}{4 - z}\right)\right) + \left(\color{blue}{\left(2.4783748995316053 + z \cdot \left(0.4964447378194062 + 0.09941724248999204 \cdot z\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. *-commutative 99.0%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right) \cdot \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(\frac{771.3234287776531}{1 + \left(2 - z\right)} + \frac{-176.6150291621406}{4 - z}\right)\right) + \left(\left(2.4783748995316053 + z \cdot \left(0.4964447378194062 + \color{blue}{z \cdot 0.09941724248999204}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]

9. Simplified 99.0%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right) \cdot \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(\frac{771.3234287776531}{1 + \left(2 - z\right)} + \frac{-176.6150291621406}{4 - z}\right)\right) + \left(\color{blue}{\left(2.4783748995316053 + z \cdot \left(0.4964447378194062 + z \cdot 0.09941724248999204\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]

10. Taylor expanded in z around inf 99.0%

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \color{blue}{e^{z - 7.5}}\right)\right) \cdot \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(\frac{771.3234287776531}{1 + \left(2 - z\right)} + \frac{-176.6150291621406}{4 - z}\right)\right) + \left(\left(2.4783748995316053 + z \cdot \left(0.4964447378194062 + z \cdot 0.09941724248999204\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)\right) \]

11. Final simplification 99.0%

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z - 7.5}\right) \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\left(\frac{-1259.1392167224028}{2 - z} + \left(\frac{771.3234287776531}{1 - \left(z - 2\right)} + \frac{-176.6150291621406}{4 - z}\right)\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7} + \left(2.4783748995316053 + z \cdot \left(0.4964447378194062 + z \cdot 0.09941724248999204\right)\right)\right)\right)\right)\right) \]
12. Add Preprocessing

Alternative 4: 97.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

3. Simplified 99.0%

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

5. Taylor expanded in z around 0 98.7%

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

   1. *-commutative 98.7%

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

7. Simplified 98.7%

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

8. Final simplification 98.7%

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

Alternative 5: 96.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-6.5 + \left(z + -1\right)}\right) \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ PI (sin (* PI z)))
  (*
   (*
    (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ -6.5 (+ z -1.0))))
    (sqrt (* PI 2.0)))
   (+ 263.3831869810514 (* z (+ 436.8961725563396 (* z 545.0353078428827)))))))

C

double code(double z) {
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (((pow((7.5 - z), (0.5 - z)) * exp((-6.5 + (z + -1.0)))) * sqrt((((double) M_PI) * 2.0))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))));
}

Java

public static double code(double z) {
	return (Math.PI / Math.sin((Math.PI * z))) * (((Math.pow((7.5 - z), (0.5 - z)) * Math.exp((-6.5 + (z + -1.0)))) * Math.sqrt((Math.PI * 2.0))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))));
}

Python

def code(z):
	return (math.pi / math.sin((math.pi * z))) * (((math.pow((7.5 - z), (0.5 - z)) * math.exp((-6.5 + (z + -1.0)))) * math.sqrt((math.pi * 2.0))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))))

Julia

function code(z)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(-6.5 + Float64(z + -1.0)))) * sqrt(Float64(pi * 2.0))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * 545.0353078428827))))))
end

MATLAB

function tmp = code(z)
	tmp = (pi / sin((pi * z))) * (((((7.5 - z) ^ (0.5 - z)) * exp((-6.5 + (z + -1.0)))) * sqrt((pi * 2.0))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827)))));
end

Wolfram

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

Derivation

1. Initial program 97.3%

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

4. Applied egg-rr 97.5%

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

6. Simplified 99.2%

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

7. Taylor expanded in z around 0 98.1%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + 545.0353078428827 \cdot z\right)\right)}\right) \]
8. Step-by-step derivation

   1. *-commutative 98.1%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + \color{blue}{z \cdot 545.0353078428827}\right)\right)\right) \]

9. Simplified 98.1%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(-1 + z\right) + -6.5}\right)\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)}\right) \]

10. Final simplification 98.1%

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-6.5 + \left(z + -1\right)}\right) \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)\right) \]
11. Add Preprocessing

Alternative 6: 96.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 547.6948589273117\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (* (sqrt (* PI 2.0)) (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5))))
  (*
   (/ PI (sin (* PI z)))
   (+ 263.3831869810514 (* z (+ 436.8961725563396 (* z 547.6948589273117)))))))

C

double code(double z) {
	return (sqrt((((double) M_PI) * 2.0)) * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5)))) * ((((double) M_PI) / sin((((double) M_PI) * z))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 547.6948589273117)))));
}

Java

public static double code(double z) {
	return (Math.sqrt((Math.PI * 2.0)) * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5)))) * ((Math.PI / Math.sin((Math.PI * z))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 547.6948589273117)))));
}

Python

def code(z):
	return (math.sqrt((math.pi * 2.0)) * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5)))) * ((math.pi / math.sin((math.pi * z))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 547.6948589273117)))))

Julia

function code(z)
	return Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5)))) * Float64(Float64(pi / sin(Float64(pi * z))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * 547.6948589273117))))))
end

MATLAB

function tmp = code(z)
	tmp = (sqrt((pi * 2.0)) * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5)))) * ((pi / sin((pi * z))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 547.6948589273117)))));
end

Wolfram

code[z_] := N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * 547.6948589273117), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.3%

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

3. Simplified 97.1%

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

5. Taylor expanded in z around 0 96.6%

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

6. Taylor expanded in z around 0 97.6%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + 547.6948589273117 \cdot z\right)\right)}\right) \]
7. Step-by-step derivation

   1. *-commutative 97.6%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + \color{blue}{z \cdot 547.6948589273117}\right)\right)\right) \]

8. Simplified 97.6%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 547.6948589273117\right)\right)}\right) \]

9. Final simplification 97.6%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 547.6948589273117\right)\right)\right) \]
10. Add Preprocessing

Alternative 7: 96.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{\pi \cdot 2} \cdot \left(e^{\left(z + -1\right) - 6.5} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)}\right)}{z \cdot 0.0037967495627271876} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (/
  (*
   (sqrt (* PI 2.0))
   (* (exp (- (+ z -1.0) 6.5)) (pow (+ (- 1.0 z) 6.5) (- 1.0 (+ z 0.5)))))
  (* z 0.0037967495627271876)))

C

double code(double z) {
	return (sqrt((((double) M_PI) * 2.0)) * (exp(((z + -1.0) - 6.5)) * pow(((1.0 - z) + 6.5), (1.0 - (z + 0.5))))) / (z * 0.0037967495627271876);
}

Java

public static double code(double z) {
	return (Math.sqrt((Math.PI * 2.0)) * (Math.exp(((z + -1.0) - 6.5)) * Math.pow(((1.0 - z) + 6.5), (1.0 - (z + 0.5))))) / (z * 0.0037967495627271876);
}

Python

def code(z):
	return (math.sqrt((math.pi * 2.0)) * (math.exp(((z + -1.0) - 6.5)) * math.pow(((1.0 - z) + 6.5), (1.0 - (z + 0.5))))) / (z * 0.0037967495627271876)

Julia

function code(z)
	return Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64(exp(Float64(Float64(z + -1.0) - 6.5)) * (Float64(Float64(1.0 - z) + 6.5) ^ Float64(1.0 - Float64(z + 0.5))))) / Float64(z * 0.0037967495627271876))
end

MATLAB

function tmp = code(z)
	tmp = (sqrt((pi * 2.0)) * (exp(((z + -1.0) - 6.5)) * (((1.0 - z) + 6.5) ^ (1.0 - (z + 0.5))))) / (z * 0.0037967495627271876);
end

Wolfram

code[z_] := N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(N[(z + -1.0), $MachinePrecision] - 6.5), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(1.0 - N[(z + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * 0.0037967495627271876), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.3%

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

3. Simplified 96.9%

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

5. Taylor expanded in z around 0 96.3%

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

   1. add-cbrt-cube 38.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \color{blue}{\sqrt[3]{\left(\frac{263.3831869810514}{z} \cdot \frac{263.3831869810514}{z}\right) \cdot \frac{263.3831869810514}{z}}} \]

7. Applied egg-rr 38.0%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \color{blue}{\sqrt[3]{\left(\frac{263.3831869810514}{z} \cdot \frac{263.3831869810514}{z}\right) \cdot \frac{263.3831869810514}{z}}} \]
8. Step-by-step derivation

   1. associate-*l* 38.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \sqrt[3]{\color{blue}{\frac{263.3831869810514}{z} \cdot \left(\frac{263.3831869810514}{z} \cdot \frac{263.3831869810514}{z}\right)}} \]

9. Simplified 38.0%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \color{blue}{\sqrt[3]{\frac{263.3831869810514}{z} \cdot \left(\frac{263.3831869810514}{z} \cdot \frac{263.3831869810514}{z}\right)}} \]
10. Step-by-step derivation

    1. associate-*r* 38.0%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \sqrt[3]{\color{blue}{\left(\frac{263.3831869810514}{z} \cdot \frac{263.3831869810514}{z}\right) \cdot \frac{263.3831869810514}{z}}} \]

    2. add-cbrt-cube 96.3%

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

    3. clear-num 96.4%

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

11. Applied egg-rr 96.4%

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

    1. un-div-inv 97.1%

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

    2. *-commutative 97.1%

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

    3. *-commutative 97.1%

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

    4. metadata-eval 97.1%

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

    5. sub-neg 97.1%

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

    6. associate--l- 97.1%

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

    7. div-inv 96.8%

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

    8. metadata-eval 97.6%

       \[\leadsto \frac{\sqrt{2 \cdot \pi} \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)}\right)}{z \cdot \color{blue}{0.0037967495627271876}} \]

13. Applied egg-rr 97.6%

    \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \pi} \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)}\right)}{z \cdot 0.0037967495627271876}} \]

14. Final simplification 97.6%

    \[\leadsto \frac{\sqrt{\pi \cdot 2} \cdot \left(e^{\left(z + -1\right) - 6.5} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)}\right)}{z \cdot 0.0037967495627271876} \]
15. Add Preprocessing

Alternative 8: 96.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{\pi \cdot 2}}{z} \cdot \frac{{\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot e^{-6.5 + \left(z + -1\right)}}{0.0037967495627271876} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ (sqrt (* PI 2.0)) z)
  (/
   (* (pow (+ (- 1.0 z) 6.5) (- 1.0 (+ z 0.5))) (exp (+ -6.5 (+ z -1.0))))
   0.0037967495627271876)))

C

double code(double z) {
	return (sqrt((((double) M_PI) * 2.0)) / z) * ((pow(((1.0 - z) + 6.5), (1.0 - (z + 0.5))) * exp((-6.5 + (z + -1.0)))) / 0.0037967495627271876);
}

Java

public static double code(double z) {
	return (Math.sqrt((Math.PI * 2.0)) / z) * ((Math.pow(((1.0 - z) + 6.5), (1.0 - (z + 0.5))) * Math.exp((-6.5 + (z + -1.0)))) / 0.0037967495627271876);
}

Python

def code(z):
	return (math.sqrt((math.pi * 2.0)) / z) * ((math.pow(((1.0 - z) + 6.5), (1.0 - (z + 0.5))) * math.exp((-6.5 + (z + -1.0)))) / 0.0037967495627271876)

Julia

function code(z)
	return Float64(Float64(sqrt(Float64(pi * 2.0)) / z) * Float64(Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(1.0 - Float64(z + 0.5))) * exp(Float64(-6.5 + Float64(z + -1.0)))) / 0.0037967495627271876))
end

MATLAB

function tmp = code(z)
	tmp = (sqrt((pi * 2.0)) / z) * (((((1.0 - z) + 6.5) ^ (1.0 - (z + 0.5))) * exp((-6.5 + (z + -1.0)))) / 0.0037967495627271876);
end

Wolfram

code[z_] := N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] / z), $MachinePrecision] * N[(N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(1.0 - N[(z + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-6.5 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 0.0037967495627271876), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.3%

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

3. Simplified 96.9%

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

5. Taylor expanded in z around 0 96.3%

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

   1. add-cbrt-cube 38.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \color{blue}{\sqrt[3]{\left(\frac{263.3831869810514}{z} \cdot \frac{263.3831869810514}{z}\right) \cdot \frac{263.3831869810514}{z}}} \]

7. Applied egg-rr 38.0%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \color{blue}{\sqrt[3]{\left(\frac{263.3831869810514}{z} \cdot \frac{263.3831869810514}{z}\right) \cdot \frac{263.3831869810514}{z}}} \]
8. Step-by-step derivation

   1. associate-*l* 38.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \sqrt[3]{\color{blue}{\frac{263.3831869810514}{z} \cdot \left(\frac{263.3831869810514}{z} \cdot \frac{263.3831869810514}{z}\right)}} \]

9. Simplified 38.0%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \color{blue}{\sqrt[3]{\frac{263.3831869810514}{z} \cdot \left(\frac{263.3831869810514}{z} \cdot \frac{263.3831869810514}{z}\right)}} \]
10. Step-by-step derivation

    1. associate-*r* 38.0%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \sqrt[3]{\color{blue}{\left(\frac{263.3831869810514}{z} \cdot \frac{263.3831869810514}{z}\right) \cdot \frac{263.3831869810514}{z}}} \]

    2. add-cbrt-cube 96.3%

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

    3. clear-num 96.4%

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

11. Applied egg-rr 96.4%

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

    1. un-div-inv 97.1%

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

    2. *-commutative 97.1%

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

    3. *-commutative 97.1%

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

    4. metadata-eval 97.1%

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

    5. sub-neg 97.1%

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

    6. associate--l- 97.1%

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

    7. div-inv 96.8%

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

    8. metadata-eval 97.6%

       \[\leadsto \frac{\sqrt{2 \cdot \pi} \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)}\right)}{z \cdot \color{blue}{0.0037967495627271876}} \]

13. Applied egg-rr 97.6%

    \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \pi} \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)}\right)}{z \cdot 0.0037967495627271876}} \]
14. Step-by-step derivation

    1. times-frac 97.5%

       \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \pi}}{z} \cdot \frac{e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)}}{0.0037967495627271876}} \]

    2. *-lft-identity 97.5%

       \[\leadsto \frac{\sqrt{2 \cdot \pi}}{\color{blue}{1 \cdot z}} \cdot \frac{e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)}}{0.0037967495627271876} \]

    3. *-commutative 97.5%

       \[\leadsto \frac{\sqrt{\color{blue}{\pi \cdot 2}}}{1 \cdot z} \cdot \frac{e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)}}{0.0037967495627271876} \]

    4. *-lft-identity 97.5%

       \[\leadsto \frac{\sqrt{\pi \cdot 2}}{\color{blue}{z}} \cdot \frac{e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)}}{0.0037967495627271876} \]

    5. *-commutative 97.5%

       \[\leadsto \frac{\sqrt{\pi \cdot 2}}{z} \cdot \frac{\color{blue}{{\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}}}{0.0037967495627271876} \]

    6. *-commutative 97.5%

       \[\leadsto \frac{\sqrt{\pi \cdot 2}}{z} \cdot \frac{\color{blue}{e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)}}}{0.0037967495627271876} \]

    7. distribute-neg-in 97.5%

       \[\leadsto \frac{\sqrt{\pi \cdot 2}}{z} \cdot \frac{e^{\color{blue}{\left(-\left(1 - z\right)\right) + \left(-6.5\right)}} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)}}{0.0037967495627271876} \]

    8. metadata-eval 97.5%

       \[\leadsto \frac{\sqrt{\pi \cdot 2}}{z} \cdot \frac{e^{\left(-\left(1 - z\right)\right) + \color{blue}{-6.5}} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)}}{0.0037967495627271876} \]

15. Simplified 97.5%

    \[\leadsto \color{blue}{\frac{\sqrt{\pi \cdot 2}}{z} \cdot \frac{e^{\left(-\left(1 - z\right)\right) + -6.5} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)}}{0.0037967495627271876}} \]

16. Final simplification 97.5%

    \[\leadsto \frac{\sqrt{\pi \cdot 2}}{z} \cdot \frac{{\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot e^{-6.5 + \left(z + -1\right)}}{0.0037967495627271876} \]
17. Add Preprocessing

Alternative 9: 95.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

3. Simplified 96.9%

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

5. Taylor expanded in z around 0 96.3%

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

   1. add-cbrt-cube 38.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \color{blue}{\sqrt[3]{\left(\frac{263.3831869810514}{z} \cdot \frac{263.3831869810514}{z}\right) \cdot \frac{263.3831869810514}{z}}} \]

7. Applied egg-rr 38.0%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \color{blue}{\sqrt[3]{\left(\frac{263.3831869810514}{z} \cdot \frac{263.3831869810514}{z}\right) \cdot \frac{263.3831869810514}{z}}} \]
8. Step-by-step derivation

   1. associate-*l* 38.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \sqrt[3]{\color{blue}{\frac{263.3831869810514}{z} \cdot \left(\frac{263.3831869810514}{z} \cdot \frac{263.3831869810514}{z}\right)}} \]

9. Simplified 38.0%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \color{blue}{\sqrt[3]{\frac{263.3831869810514}{z} \cdot \left(\frac{263.3831869810514}{z} \cdot \frac{263.3831869810514}{z}\right)}} \]
10. Step-by-step derivation

    1. associate-*r* 38.0%

       \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \sqrt[3]{\color{blue}{\left(\frac{263.3831869810514}{z} \cdot \frac{263.3831869810514}{z}\right) \cdot \frac{263.3831869810514}{z}}} \]

    2. add-cbrt-cube 96.3%

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

    3. clear-num 96.4%

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

11. Applied egg-rr 96.4%

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

    1. un-div-inv 97.1%

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

    2. *-commutative 97.1%

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

    3. *-commutative 97.1%

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

    4. metadata-eval 97.1%

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

    5. sub-neg 97.1%

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

    6. associate--l- 97.1%

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

    7. div-inv 96.8%

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

    8. metadata-eval 97.6%

       \[\leadsto \frac{\sqrt{2 \cdot \pi} \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)}\right)}{z \cdot \color{blue}{0.0037967495627271876}} \]

13. Applied egg-rr 97.6%

    \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \pi} \cdot \left(e^{-\left(\left(1 - z\right) + 6.5\right)} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)}\right)}{z \cdot 0.0037967495627271876}} \]

15. Simplified 97.2%

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

16. Final simplification 97.2%

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

Alternative 10: 95.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

3. Simplified 96.9%

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

5. Taylor expanded in z around 0 96.3%

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

6. Taylor expanded in z around 0 96.1%

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

   1. associate-*r/ 96.5%

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

   2. *-commutative 96.5%

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

   3. distribute-neg-in 96.5%

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

   4. metadata-eval 96.5%

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

8. Applied egg-rr 96.5%

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

9. Final simplification 96.5%

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

Alternative 11: 95.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

3. Simplified 96.9%

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

5. Taylor expanded in z around 0 96.3%

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

6. Taylor expanded in z around 0 96.1%

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

7. Taylor expanded in z around 0 96.1%

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

   1. associate-*r/ 96.4%

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

   2. associate-*r* 96.4%

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

   3. *-commutative 96.4%

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

9. Applied egg-rr 96.4%

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

10. Final simplification 96.4%

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

Alternative 12: 95.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

3. Simplified 96.9%

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

5. Taylor expanded in z around 0 96.3%

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

6. Taylor expanded in z around 0 96.1%

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

7. Taylor expanded in z around 0 96.1%

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

   1. associate-*r/ 96.4%

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

   2. associate-*r* 96.4%

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

   3. *-commutative 96.4%

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

9. Applied egg-rr 96.4%

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

    1. associate-/l* 96.1%

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

    2. associate-*l* 96.1%

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

    3. associate-*r* 96.2%

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

    4. *-commutative 96.2%

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

    5. associate-*l* 96.1%

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

11. Simplified 96.1%

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

Reproduce

herbie shell --seed 2024110 
(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))))))