Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.6% → 98.0%

Time: 1.3min

Alternatives: 8

Speedup: 1.1×

• could not determine a ground truth

  1. z = -1.5787419126249072e+25

Specification

\[z \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\left(\left|\sqrt[3]{2 \cdot \pi}\right| \cdot {\left(2 \cdot \pi\right)}^{0.16666666666666666}\right) \cdot e^{\left(z + -1\right) - 6.5}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 + \left(-1 - z\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 + \left(-1 - z\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 + \left(-1 - z\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (pow (+ (- 1.0 z) 6.5) (- 1.0 (+ z 0.5)))
   (*
    (* (fabs (cbrt (* 2.0 PI))) (pow (* 2.0 PI) 0.16666666666666666))
    (exp (- (+ z -1.0) 6.5))))
  (*
   (+
    (+
     (+ 0.9999999999998099 (/ 676.5203681218851 (- 1.0 z)))
     (+
      (/ -1259.1392167224028 (+ 3.0 (- -1.0 z)))
      (+
       (/ 771.3234287776531 (- 1.0 (+ z -2.0)))
       (/ -176.6150291621406 (+ (- 1.0 z) 3.0)))))
    (+
     (+
      (/ 12.507343278686905 (+ (- 1.0 z) 4.0))
      (/ -0.13857109526572012 (- 1.0 (+ z -5.0))))
     (+
      (/ 9.984369578019572e-6 (+ 8.0 (- -1.0 z)))
      (/ 1.5056327351493116e-7 (+ 9.0 (- -1.0 z))))))
   (/ PI (sin (* z PI))))))

C

double code(double z) {
	return (pow(((1.0 - z) + 6.5), (1.0 - (z + 0.5))) * ((fabs(cbrt((2.0 * ((double) M_PI)))) * pow((2.0 * ((double) M_PI)), 0.16666666666666666)) * exp(((z + -1.0) - 6.5)))) * ((((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + ((-1259.1392167224028 / (3.0 + (-1.0 - z))) + ((771.3234287776531 / (1.0 - (z + -2.0))) + (-176.6150291621406 / ((1.0 - z) + 3.0))))) + (((12.507343278686905 / ((1.0 - z) + 4.0)) + (-0.13857109526572012 / (1.0 - (z + -5.0)))) + ((9.984369578019572e-6 / (8.0 + (-1.0 - z))) + (1.5056327351493116e-7 / (9.0 + (-1.0 - z)))))) * (((double) M_PI) / sin((z * ((double) M_PI)))));
}

Java

public static double code(double z) {
	return (Math.pow(((1.0 - z) + 6.5), (1.0 - (z + 0.5))) * ((Math.abs(Math.cbrt((2.0 * Math.PI))) * Math.pow((2.0 * Math.PI), 0.16666666666666666)) * Math.exp(((z + -1.0) - 6.5)))) * ((((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + ((-1259.1392167224028 / (3.0 + (-1.0 - z))) + ((771.3234287776531 / (1.0 - (z + -2.0))) + (-176.6150291621406 / ((1.0 - z) + 3.0))))) + (((12.507343278686905 / ((1.0 - z) + 4.0)) + (-0.13857109526572012 / (1.0 - (z + -5.0)))) + ((9.984369578019572e-6 / (8.0 + (-1.0 - z))) + (1.5056327351493116e-7 / (9.0 + (-1.0 - z)))))) * (Math.PI / Math.sin((z * Math.PI))));
}

Julia

function code(z)
	return Float64(Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(1.0 - Float64(z + 0.5))) * Float64(Float64(abs(cbrt(Float64(2.0 * pi))) * (Float64(2.0 * pi) ^ 0.16666666666666666)) * exp(Float64(Float64(z + -1.0) - 6.5)))) * Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 - z))) + Float64(Float64(-1259.1392167224028 / Float64(3.0 + Float64(-1.0 - z))) + Float64(Float64(771.3234287776531 / Float64(1.0 - Float64(z + -2.0))) + Float64(-176.6150291621406 / Float64(Float64(1.0 - z) + 3.0))))) + Float64(Float64(Float64(12.507343278686905 / Float64(Float64(1.0 - z) + 4.0)) + Float64(-0.13857109526572012 / Float64(1.0 - Float64(z + -5.0)))) + Float64(Float64(9.984369578019572e-6 / Float64(8.0 + Float64(-1.0 - z))) + Float64(1.5056327351493116e-7 / Float64(9.0 + Float64(-1.0 - z)))))) * Float64(pi / sin(Float64(z * pi)))))
end

Wolfram

code[z_] := N[(N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(1.0 - N[(z + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Abs[N[Power[N[(2.0 * Pi), $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision] * N[Power[N[(2.0 * Pi), $MachinePrecision], 0.16666666666666666], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(z + -1.0), $MachinePrecision] - 6.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1259.1392167224028 / N[(3.0 + N[(-1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(1.0 - N[(z + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(1.0 - N[(z + -5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(8.0 + N[(-1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(9.0 + N[(-1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.5%

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

3. Simplified 98.3%

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

   1. pow1/2 98.3%

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

   2. add-cube-cbrt 98.5%

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

   3. unpow-prod-down 98.5%

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

   4. pow2 98.5%

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

5. Applied egg-rr 98.5%

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

   1. unpow1/2 98.5%

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

   2. unpow2 98.5%

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

   3. rem-sqrt-square 98.5%

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

   4. *-commutative 98.5%

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

   5. unpow1/2 98.5%

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

   6. *-commutative 98.5%

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

7. Simplified 98.5%

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

   1. *-un-lft-identity 98.5%

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

9. Applied egg-rr 98.5%

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

    1. *-lft-identity 98.5%

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

    2. unpow1/3 98.5%

       \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\left(\left|\sqrt[3]{2 \cdot \pi}\right| \cdot \sqrt{\color{blue}{{\left(2 \cdot \pi\right)}^{0.3333333333333333}}}\right) \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

    3. sqr-pow 98.5%

       \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\left(\left|\sqrt[3]{2 \cdot \pi}\right| \cdot \sqrt{\color{blue}{{\left(2 \cdot \pi\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \cdot {\left(2 \cdot \pi\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}}\right) \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

    4. rem-sqrt-square 98.5%

       \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\left(\left|\sqrt[3]{2 \cdot \pi}\right| \cdot \color{blue}{\left|{\left(2 \cdot \pi\right)}^{\left(\frac{0.3333333333333333}{2}\right)}\right|}\right) \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

    5. metadata-eval 98.5%

       \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\left(\left|\sqrt[3]{2 \cdot \pi}\right| \cdot \left|{\left(2 \cdot \pi\right)}^{\color{blue}{0.16666666666666666}}\right|\right) \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

    6. sqr-pow 98.3%

       \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\left(\left|\sqrt[3]{2 \cdot \pi}\right| \cdot \left|\color{blue}{{\left(2 \cdot \pi\right)}^{\left(\frac{0.16666666666666666}{2}\right)} \cdot {\left(2 \cdot \pi\right)}^{\left(\frac{0.16666666666666666}{2}\right)}}\right|\right) \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

    7. fabs-sqr 98.3%

       \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\left(\left|\sqrt[3]{2 \cdot \pi}\right| \cdot \color{blue}{\left({\left(2 \cdot \pi\right)}^{\left(\frac{0.16666666666666666}{2}\right)} \cdot {\left(2 \cdot \pi\right)}^{\left(\frac{0.16666666666666666}{2}\right)}\right)}\right) \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

    8. sqr-pow 98.5%

       \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\left(\left|\sqrt[3]{2 \cdot \pi}\right| \cdot \color{blue}{{\left(2 \cdot \pi\right)}^{0.16666666666666666}}\right) \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

11. Simplified 98.5%

    \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\left(\left|\sqrt[3]{2 \cdot \pi}\right| \cdot \color{blue}{{\left(2 \cdot \pi\right)}^{0.16666666666666666}}\right) \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 - \left(z + 1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z + 1\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

12. Final simplification 98.5%

    \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\left(\left|\sqrt[3]{2 \cdot \pi}\right| \cdot {\left(2 \cdot \pi\right)}^{0.16666666666666666}\right) \cdot e^{\left(z + -1\right) - 6.5}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 + \left(-1 - z\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{8 + \left(-1 - z\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{9 + \left(-1 - z\right)}\right)\right)\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \]

Alternative 2: 97.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (+
    (+
     (+ 0.9999999999998099 (/ 676.5203681218851 (- 1.0 z)))
     (+
      (/ -1259.1392167224028 (+ 3.0 (- -1.0 z)))
      (+
       (/ 771.3234287776531 (- 1.0 (+ z -2.0)))
       (/ -176.6150291621406 (+ (- 1.0 z) 3.0)))))
    (+
     (+
      (/ 12.507343278686905 (+ (- 1.0 z) 4.0))
      (/ -0.13857109526572012 (- 1.0 (+ z -5.0))))
     (+
      (/ 9.984369578019572e-6 (+ 8.0 (- -1.0 z)))
      (/ 1.5056327351493116e-7 (+ 9.0 (- -1.0 z))))))
   (/ PI (sin (* z PI))))
  (*
   (pow (+ (- 1.0 z) 6.5) (- 1.0 (+ z 0.5)))
   (* (exp (- (+ z -1.0) 6.5)) (sqrt (* 2.0 PI))))))

C

double code(double z) {
	return ((((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + ((-1259.1392167224028 / (3.0 + (-1.0 - z))) + ((771.3234287776531 / (1.0 - (z + -2.0))) + (-176.6150291621406 / ((1.0 - z) + 3.0))))) + (((12.507343278686905 / ((1.0 - z) + 4.0)) + (-0.13857109526572012 / (1.0 - (z + -5.0)))) + ((9.984369578019572e-6 / (8.0 + (-1.0 - z))) + (1.5056327351493116e-7 / (9.0 + (-1.0 - z)))))) * (((double) M_PI) / sin((z * ((double) M_PI))))) * (pow(((1.0 - z) + 6.5), (1.0 - (z + 0.5))) * (exp(((z + -1.0) - 6.5)) * sqrt((2.0 * ((double) M_PI)))));
}

Java

public static double code(double z) {
	return ((((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + ((-1259.1392167224028 / (3.0 + (-1.0 - z))) + ((771.3234287776531 / (1.0 - (z + -2.0))) + (-176.6150291621406 / ((1.0 - z) + 3.0))))) + (((12.507343278686905 / ((1.0 - z) + 4.0)) + (-0.13857109526572012 / (1.0 - (z + -5.0)))) + ((9.984369578019572e-6 / (8.0 + (-1.0 - z))) + (1.5056327351493116e-7 / (9.0 + (-1.0 - z)))))) * (Math.PI / Math.sin((z * Math.PI)))) * (Math.pow(((1.0 - z) + 6.5), (1.0 - (z + 0.5))) * (Math.exp(((z + -1.0) - 6.5)) * Math.sqrt((2.0 * Math.PI))));
}

Python

def code(z):
	return ((((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + ((-1259.1392167224028 / (3.0 + (-1.0 - z))) + ((771.3234287776531 / (1.0 - (z + -2.0))) + (-176.6150291621406 / ((1.0 - z) + 3.0))))) + (((12.507343278686905 / ((1.0 - z) + 4.0)) + (-0.13857109526572012 / (1.0 - (z + -5.0)))) + ((9.984369578019572e-6 / (8.0 + (-1.0 - z))) + (1.5056327351493116e-7 / (9.0 + (-1.0 - z)))))) * (math.pi / math.sin((z * math.pi)))) * (math.pow(((1.0 - z) + 6.5), (1.0 - (z + 0.5))) * (math.exp(((z + -1.0) - 6.5)) * math.sqrt((2.0 * math.pi))))

Julia

function code(z)
	return Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 - z))) + Float64(Float64(-1259.1392167224028 / Float64(3.0 + Float64(-1.0 - z))) + Float64(Float64(771.3234287776531 / Float64(1.0 - Float64(z + -2.0))) + Float64(-176.6150291621406 / Float64(Float64(1.0 - z) + 3.0))))) + Float64(Float64(Float64(12.507343278686905 / Float64(Float64(1.0 - z) + 4.0)) + Float64(-0.13857109526572012 / Float64(1.0 - Float64(z + -5.0)))) + Float64(Float64(9.984369578019572e-6 / Float64(8.0 + Float64(-1.0 - z))) + Float64(1.5056327351493116e-7 / Float64(9.0 + Float64(-1.0 - z)))))) * Float64(pi / sin(Float64(z * pi)))) * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(1.0 - Float64(z + 0.5))) * Float64(exp(Float64(Float64(z + -1.0) - 6.5)) * sqrt(Float64(2.0 * pi)))))
end

MATLAB

function tmp = code(z)
	tmp = ((((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + ((-1259.1392167224028 / (3.0 + (-1.0 - z))) + ((771.3234287776531 / (1.0 - (z + -2.0))) + (-176.6150291621406 / ((1.0 - z) + 3.0))))) + (((12.507343278686905 / ((1.0 - z) + 4.0)) + (-0.13857109526572012 / (1.0 - (z + -5.0)))) + ((9.984369578019572e-6 / (8.0 + (-1.0 - z))) + (1.5056327351493116e-7 / (9.0 + (-1.0 - z)))))) * (pi / sin((z * pi)))) * ((((1.0 - z) + 6.5) ^ (1.0 - (z + 0.5))) * (exp(((z + -1.0) - 6.5)) * sqrt((2.0 * pi))));
end

Wolfram

code[z_] := N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1259.1392167224028 / N[(3.0 + N[(-1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(1.0 - N[(z + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(1.0 - N[(z + -5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(8.0 + N[(-1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(9.0 + N[(-1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(1.0 - N[(z + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(N[(z + -1.0), $MachinePrecision] - 6.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.5%

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

3. Simplified 98.3%

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

4. Final simplification 98.3%

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

Alternative 3: 98.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (/ PI (sin (* z PI)))
   (* (* (sqrt (* 2.0 PI)) (exp (+ z -7.5))) (pow (- 7.5 z) (- 0.5 z))))
  (+
   (+
    (+
     (+
      (+ 0.9999999999998099 (/ 676.5203681218851 (- 1.0 z)))
      (/ -1259.1392167224028 (- (- 1.0 z) -1.0)))
     (+
      (/ 771.3234287776531 (- (- 1.0 z) -2.0))
      (/ -176.6150291621406 (- (- 1.0 z) -3.0))))
    (+
     (/ 12.507343278686905 (- (- 1.0 z) -4.0))
     (/ -0.13857109526572012 (- (- 1.0 z) -5.0))))
   (+
    (/ 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((z * ((double) M_PI)))) * ((sqrt((2.0 * ((double) M_PI))) * exp((z + -7.5))) * pow((7.5 - z), (0.5 - z)))) * (((((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) + (-176.6150291621406 / ((1.0 - z) - -3.0)))) + ((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0)))) + ((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((z * Math.PI))) * ((Math.sqrt((2.0 * Math.PI)) * Math.exp((z + -7.5))) * Math.pow((7.5 - z), (0.5 - z)))) * (((((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) + (-176.6150291621406 / ((1.0 - z) - -3.0)))) + ((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0)))) + ((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((z * math.pi))) * ((math.sqrt((2.0 * math.pi)) * math.exp((z + -7.5))) * math.pow((7.5 - z), (0.5 - z)))) * (((((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) + (-176.6150291621406 / ((1.0 - z) - -3.0)))) + ((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0)))) + ((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(z * pi))) * Float64(Float64(sqrt(Float64(2.0 * pi)) * exp(Float64(z + -7.5))) * (Float64(7.5 - z) ^ Float64(0.5 - z)))) * Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 - z))) + Float64(-1259.1392167224028 / Float64(Float64(1.0 - z) - -1.0))) + Float64(Float64(771.3234287776531 / Float64(Float64(1.0 - z) - -2.0)) + Float64(-176.6150291621406 / Float64(Float64(1.0 - z) - -3.0)))) + Float64(Float64(12.507343278686905 / Float64(Float64(1.0 - z) - -4.0)) + Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) - -5.0)))) + 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((z * pi))) * ((sqrt((2.0 * pi)) * exp((z + -7.5))) * ((7.5 - z) ^ (0.5 - z)))) * (((((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) + (-176.6150291621406 / ((1.0 - z) - -3.0)))) + ((12.507343278686905 / ((1.0 - z) - -4.0)) + (-0.13857109526572012 / ((1.0 - z) - -5.0)))) + ((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[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(N[(1.0 - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(N[(1.0 - z), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] - -4.0), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] - -5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 96.5%

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

3. Simplified 98.2%

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

5. Applied egg-rr 89.2%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)} \cdot e^{-\mathsf{fma}\left(-1, z, 7.5\right)}\right)\right)} - 1\right)}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
6. Step-by-step derivation

   1. expm1-def 98.2%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(0.5 - z\right)} \cdot e^{-\mathsf{fma}\left(-1, z, 7.5\right)}\right)\right)\right)}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   2. expm1-log1p 98.2%

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

   3. associate-*r* 98.2%

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

   4. *-commutative 98.2%

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

   5. fma-udef 98.2%

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

   6. neg-mul-1 98.2%

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

   7. +-commutative 98.2%

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

   8. neg-mul-1 98.2%

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

   9. associate-*l* 98.2%

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

   10. fma-udef 98.2%

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

   11. neg-mul-1 98.2%

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

   12. +-commutative 98.2%

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

   13. sub-neg 98.2%

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

   14. *-commutative 98.2%

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

7. Simplified 98.2%

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

8. Final simplification 98.2%

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

Alternative 4: 96.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 + \left(-1 - z\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (pow (+ (- 1.0 z) 6.5) (- 1.0 (+ z 0.5)))
   (* (sqrt (* 2.0 PI)) (exp (+ z -7.5))))
  (*
   (/ PI (sin (* z PI)))
   (+
    (+
     (+ 0.9999999999998099 (/ 676.5203681218851 (- 1.0 z)))
     (+
      (/ -1259.1392167224028 (+ 3.0 (- -1.0 z)))
      (+
       (/ 771.3234287776531 (- 1.0 (+ z -2.0)))
       (/ -176.6150291621406 (+ (- 1.0 z) 3.0)))))
    (+ 2.4783749183520145 (* z 0.49644474017195733))))))

C

double code(double z) {
	return (pow(((1.0 - z) + 6.5), (1.0 - (z + 0.5))) * (sqrt((2.0 * ((double) M_PI))) * exp((z + -7.5)))) * ((((double) M_PI) / sin((z * ((double) M_PI)))) * (((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + ((-1259.1392167224028 / (3.0 + (-1.0 - z))) + ((771.3234287776531 / (1.0 - (z + -2.0))) + (-176.6150291621406 / ((1.0 - z) + 3.0))))) + (2.4783749183520145 + (z * 0.49644474017195733))));
}

Java

public static double code(double z) {
	return (Math.pow(((1.0 - z) + 6.5), (1.0 - (z + 0.5))) * (Math.sqrt((2.0 * Math.PI)) * Math.exp((z + -7.5)))) * ((Math.PI / Math.sin((z * Math.PI))) * (((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + ((-1259.1392167224028 / (3.0 + (-1.0 - z))) + ((771.3234287776531 / (1.0 - (z + -2.0))) + (-176.6150291621406 / ((1.0 - z) + 3.0))))) + (2.4783749183520145 + (z * 0.49644474017195733))));
}

Python

def code(z):
	return (math.pow(((1.0 - z) + 6.5), (1.0 - (z + 0.5))) * (math.sqrt((2.0 * math.pi)) * math.exp((z + -7.5)))) * ((math.pi / math.sin((z * math.pi))) * (((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + ((-1259.1392167224028 / (3.0 + (-1.0 - z))) + ((771.3234287776531 / (1.0 - (z + -2.0))) + (-176.6150291621406 / ((1.0 - z) + 3.0))))) + (2.4783749183520145 + (z * 0.49644474017195733))))

Julia

function code(z)
	return Float64(Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(1.0 - Float64(z + 0.5))) * Float64(sqrt(Float64(2.0 * pi)) * exp(Float64(z + -7.5)))) * Float64(Float64(pi / sin(Float64(z * pi))) * Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 - z))) + Float64(Float64(-1259.1392167224028 / Float64(3.0 + Float64(-1.0 - z))) + Float64(Float64(771.3234287776531 / Float64(1.0 - Float64(z + -2.0))) + Float64(-176.6150291621406 / Float64(Float64(1.0 - z) + 3.0))))) + Float64(2.4783749183520145 + Float64(z * 0.49644474017195733)))))
end

MATLAB

function tmp = code(z)
	tmp = ((((1.0 - z) + 6.5) ^ (1.0 - (z + 0.5))) * (sqrt((2.0 * pi)) * exp((z + -7.5)))) * ((pi / sin((z * pi))) * (((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + ((-1259.1392167224028 / (3.0 + (-1.0 - z))) + ((771.3234287776531 / (1.0 - (z + -2.0))) + (-176.6150291621406 / ((1.0 - z) + 3.0))))) + (2.4783749183520145 + (z * 0.49644474017195733))));
end

Wolfram

code[z_] := N[(N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(1.0 - N[(z + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1259.1392167224028 / N[(3.0 + N[(-1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(1.0 - N[(z + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.4783749183520145 + N[(z * 0.49644474017195733), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.5%

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

3. Simplified 98.3%

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

4. Taylor expanded in z around 0 97.9%

   \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \color{blue}{\left(2.4783749183520145 + 0.49644474017195733 \cdot z\right)}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
5. Step-by-step derivation

   1. *-commutative 97.9%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(2.4783749183520145 + \color{blue}{z \cdot 0.49644474017195733}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

6. Simplified 97.9%

   \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \color{blue}{\left(2.4783749183520145 + z \cdot 0.49644474017195733\right)}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
7. Step-by-step derivation

   1. pow1 97.9%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \color{blue}{{\left(\sqrt{\pi \cdot 2} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)}^{1}}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   2. *-commutative 97.9%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot {\left(\sqrt{\color{blue}{2 \cdot \pi}} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)}^{1}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   3. distribute-neg-in 97.9%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot {\left(\sqrt{2 \cdot \pi} \cdot e^{\color{blue}{\left(-\left(1 - z\right)\right) + \left(-6.5\right)}}\right)}^{1}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   4. metadata-eval 97.9%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot {\left(\sqrt{2 \cdot \pi} \cdot e^{\left(-\left(1 - z\right)\right) + \color{blue}{-6.5}}\right)}^{1}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

8. Applied egg-rr 97.9%

   \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \color{blue}{{\left(\sqrt{2 \cdot \pi} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)}^{1}}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
9. Step-by-step derivation

   1. unpow1 97.9%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \color{blue}{\left(\sqrt{2 \cdot \pi} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   2. metadata-eval 97.9%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\left(-\left(1 - z\right)\right) + \color{blue}{\left(-6.5\right)}}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   3. distribute-neg-in 97.9%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\color{blue}{-\left(\left(1 - z\right) + 6.5\right)}}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   4. +-commutative 97.9%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{-\color{blue}{\left(6.5 + \left(1 - z\right)\right)}}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   5. sub-neg 97.9%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{-\left(6.5 + \color{blue}{\left(1 + \left(-z\right)\right)}\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   6. associate-+r+ 97.9%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{-\color{blue}{\left(\left(6.5 + 1\right) + \left(-z\right)\right)}}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   7. metadata-eval 97.9%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{-\left(\color{blue}{7.5} + \left(-z\right)\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   8. +-commutative 97.9%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{-\color{blue}{\left(\left(-z\right) + 7.5\right)}}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   9. distribute-neg-in 97.9%

      \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\color{blue}{\left(-\left(-z\right)\right) + \left(-7.5\right)}}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   10. remove-double-neg 97.9%

       \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\color{blue}{z} + \left(-7.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

   11. metadata-eval 97.9%

       \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{z + \color{blue}{-7.5}}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

10. Simplified 97.9%

    \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \color{blue}{\left(\sqrt{2 \cdot \pi} \cdot e^{z + -7.5}\right)}\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 - \left(z + 1\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

11. Final simplification 97.9%

    \[\leadsto \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-1259.1392167224028}{3 + \left(-1 - z\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right) \]

Alternative 5: 97.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

3. Simplified 95.9%

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

4. Taylor expanded in z around 0 97.7%

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

   1. *-commutative 97.7%

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

6. Simplified 97.7%

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

7. Final simplification 97.7%

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

Alternative 6: 96.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

3. Simplified 95.9%

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

4. Taylor expanded in z around 0 94.5%

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

   1. *-commutative 94.5%

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

   2. associate-/l* 94.4%

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

6. Simplified 94.4%

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

7. Taylor expanded in z around 0 96.8%

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

   1. pow1 96.8%

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

   2. sqrt-unprod 96.8%

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

   3. metadata-eval 96.8%

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

9. Applied egg-rr 96.8%

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

    1. unpow1 96.8%

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

    2. *-commutative 96.8%

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

    3. associate-/r/ 96.8%

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

    4. *-commutative 96.8%

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

    5. associate-*l* 96.9%

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

11. Simplified 96.9%

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

12. Final simplification 96.9%

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

Alternative 7: 96.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

3. Simplified 95.9%

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

4. Taylor expanded in z around 0 94.5%

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

   1. *-commutative 94.5%

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

   2. associate-/l* 94.4%

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

6. Simplified 94.4%

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

7. Taylor expanded in z around 0 96.8%

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

   1. pow1 96.8%

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

   2. sqrt-unprod 96.8%

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

   3. metadata-eval 96.8%

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

9. Applied egg-rr 96.8%

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

    1. unpow1 96.8%

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

    2. *-commutative 96.8%

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

    3. associate-/r/ 96.8%

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

    4. *-commutative 96.8%

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

    5. associate-*l* 96.9%

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

11. Simplified 96.9%

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

12. Taylor expanded in z around 0 96.6%

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

    1. associate-*l/ 96.5%

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

    2. *-commutative 96.5%

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

    3. associate-/l* 96.6%

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

    4. associate-/r* 96.9%

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

14. Simplified 96.9%

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

15. Final simplification 96.9%

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

Alternative 8: 96.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

3. Simplified 95.9%

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

4. Taylor expanded in z around 0 94.5%

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

   1. *-commutative 94.5%

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

   2. associate-/l* 94.4%

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

6. Simplified 94.4%

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

7. Taylor expanded in z around 0 96.8%

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

   1. associate-*r/ 97.1%

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

   2. sqrt-unprod 97.1%

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

   3. metadata-eval 97.1%

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

9. Applied egg-rr 97.1%

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

10. Final simplification 97.1%

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

Reproduce

herbie shell --seed 2023291 
(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))))))