Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.5% → 99.2%

Time: 2.0min

Alternatives: 14

Speedup: 1.1×

• could not determine a ground truth

  1. z = -1.3777278562754414e+138

Specification

\[z \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(z + -1\right)\\ t_1 := \sin \left(z \cdot \pi\right)\\ t_2 := \left(1 - z\right) + -1\\ t_3 := \sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\\ \mathbf{if}\;z \leq -13.8:\\ \;\;\;\;\frac{\pi}{t\_1} \cdot \left(e^{\log t\_3 + \left(z + -7.5\right)} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{t\_2 + 8} - \left(\frac{9.984369578019572 \cdot 10^{-6}}{t\_0 - 7} - \left(\frac{-0.13857109526572012}{t\_2 + 6} - \left(\left(\left(\left(\frac{-1259.1392167224028}{z - 2} + \left(\frac{676.5203681218851}{z + -1} - 0.9999999999998099\right)\right) + \frac{771.3234287776531}{t\_0 - 3}\right) + \frac{-176.6150291621406}{t\_0 - 4}\right) + \frac{12.507343278686905}{t\_0 - 5}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \frac{1}{t\_1}\right) \cdot \left(t\_3 \cdot \left(e^{z + -7.5} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} - \left(\left(\frac{12.507343278686905}{z - 5} + \left(\frac{771.3234287776531}{z - 3} + \frac{-176.6150291621406}{z - 4}\right)\right) + \left(\left(\frac{-0.13857109526572012}{z - 6} + \frac{9.984369578019572 \cdot 10^{-6}}{z - 7}\right) - \left(0.9999999999998099 + {\left(\sqrt[3]{\frac{-1259.1392167224028}{2 - z} + \frac{676.5203681218851}{1 - z}}\right)}^{3}\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (+ z -1.0)))
        (t_1 (sin (* z PI)))
        (t_2 (+ (- 1.0 z) -1.0))
        (t_3 (* (sqrt (* PI 2.0)) (pow (- 7.5 z) (- 0.5 z)))))
   (if (<= z -13.8)
     (*
      (/ PI t_1)
      (*
       (exp (+ (log t_3) (+ z -7.5)))
       (-
        (/ 1.5056327351493116e-7 (+ t_2 8.0))
        (-
         (/ 9.984369578019572e-6 (- t_0 7.0))
         (-
          (/ -0.13857109526572012 (+ t_2 6.0))
          (+
           (+
            (+
             (+
              (/ -1259.1392167224028 (- z 2.0))
              (- (/ 676.5203681218851 (+ z -1.0)) 0.9999999999998099))
             (/ 771.3234287776531 (- t_0 3.0)))
            (/ -176.6150291621406 (- t_0 4.0)))
           (/ 12.507343278686905 (- t_0 5.0))))))))
     (*
      (* PI (/ 1.0 t_1))
      (*
       t_3
       (*
        (exp (+ z -7.5))
        (-
         (/ 1.5056327351493116e-7 (- 8.0 z))
         (+
          (+
           (/ 12.507343278686905 (- z 5.0))
           (+
            (/ 771.3234287776531 (- z 3.0))
            (/ -176.6150291621406 (- z 4.0))))
          (-
           (+
            (/ -0.13857109526572012 (- z 6.0))
            (/ 9.984369578019572e-6 (- z 7.0)))
           (+
            0.9999999999998099
            (pow
             (cbrt
              (+
               (/ -1259.1392167224028 (- 2.0 z))
               (/ 676.5203681218851 (- 1.0 z))))
             3.0)))))))))))

C

double code(double z) {
	double t_0 = 1.0 + (z + -1.0);
	double t_1 = sin((z * ((double) M_PI)));
	double t_2 = (1.0 - z) + -1.0;
	double t_3 = sqrt((((double) M_PI) * 2.0)) * pow((7.5 - z), (0.5 - z));
	double tmp;
	if (z <= -13.8) {
		tmp = (((double) M_PI) / t_1) * (exp((log(t_3) + (z + -7.5))) * ((1.5056327351493116e-7 / (t_2 + 8.0)) - ((9.984369578019572e-6 / (t_0 - 7.0)) - ((-0.13857109526572012 / (t_2 + 6.0)) - (((((-1259.1392167224028 / (z - 2.0)) + ((676.5203681218851 / (z + -1.0)) - 0.9999999999998099)) + (771.3234287776531 / (t_0 - 3.0))) + (-176.6150291621406 / (t_0 - 4.0))) + (12.507343278686905 / (t_0 - 5.0)))))));
	} else {
		tmp = (((double) M_PI) * (1.0 / t_1)) * (t_3 * (exp((z + -7.5)) * ((1.5056327351493116e-7 / (8.0 - z)) - (((12.507343278686905 / (z - 5.0)) + ((771.3234287776531 / (z - 3.0)) + (-176.6150291621406 / (z - 4.0)))) + (((-0.13857109526572012 / (z - 6.0)) + (9.984369578019572e-6 / (z - 7.0))) - (0.9999999999998099 + pow(cbrt(((-1259.1392167224028 / (2.0 - z)) + (676.5203681218851 / (1.0 - z)))), 3.0)))))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = 1.0 + (z + -1.0);
	double t_1 = Math.sin((z * Math.PI));
	double t_2 = (1.0 - z) + -1.0;
	double t_3 = Math.sqrt((Math.PI * 2.0)) * Math.pow((7.5 - z), (0.5 - z));
	double tmp;
	if (z <= -13.8) {
		tmp = (Math.PI / t_1) * (Math.exp((Math.log(t_3) + (z + -7.5))) * ((1.5056327351493116e-7 / (t_2 + 8.0)) - ((9.984369578019572e-6 / (t_0 - 7.0)) - ((-0.13857109526572012 / (t_2 + 6.0)) - (((((-1259.1392167224028 / (z - 2.0)) + ((676.5203681218851 / (z + -1.0)) - 0.9999999999998099)) + (771.3234287776531 / (t_0 - 3.0))) + (-176.6150291621406 / (t_0 - 4.0))) + (12.507343278686905 / (t_0 - 5.0)))))));
	} else {
		tmp = (Math.PI * (1.0 / t_1)) * (t_3 * (Math.exp((z + -7.5)) * ((1.5056327351493116e-7 / (8.0 - z)) - (((12.507343278686905 / (z - 5.0)) + ((771.3234287776531 / (z - 3.0)) + (-176.6150291621406 / (z - 4.0)))) + (((-0.13857109526572012 / (z - 6.0)) + (9.984369578019572e-6 / (z - 7.0))) - (0.9999999999998099 + Math.pow(Math.cbrt(((-1259.1392167224028 / (2.0 - z)) + (676.5203681218851 / (1.0 - z)))), 3.0)))))));
	}
	return tmp;
}

Julia

function code(z)
	t_0 = Float64(1.0 + Float64(z + -1.0))
	t_1 = sin(Float64(z * pi))
	t_2 = Float64(Float64(1.0 - z) + -1.0)
	t_3 = Float64(sqrt(Float64(pi * 2.0)) * (Float64(7.5 - z) ^ Float64(0.5 - z)))
	tmp = 0.0
	if (z <= -13.8)
		tmp = Float64(Float64(pi / t_1) * Float64(exp(Float64(log(t_3) + Float64(z + -7.5))) * Float64(Float64(1.5056327351493116e-7 / Float64(t_2 + 8.0)) - Float64(Float64(9.984369578019572e-6 / Float64(t_0 - 7.0)) - Float64(Float64(-0.13857109526572012 / Float64(t_2 + 6.0)) - Float64(Float64(Float64(Float64(Float64(-1259.1392167224028 / Float64(z - 2.0)) + Float64(Float64(676.5203681218851 / Float64(z + -1.0)) - 0.9999999999998099)) + Float64(771.3234287776531 / Float64(t_0 - 3.0))) + Float64(-176.6150291621406 / Float64(t_0 - 4.0))) + Float64(12.507343278686905 / Float64(t_0 - 5.0))))))));
	else
		tmp = Float64(Float64(pi * Float64(1.0 / t_1)) * Float64(t_3 * Float64(exp(Float64(z + -7.5)) * Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) - Float64(Float64(Float64(12.507343278686905 / Float64(z - 5.0)) + Float64(Float64(771.3234287776531 / Float64(z - 3.0)) + Float64(-176.6150291621406 / Float64(z - 4.0)))) + Float64(Float64(Float64(-0.13857109526572012 / Float64(z - 6.0)) + Float64(9.984369578019572e-6 / Float64(z - 7.0))) - Float64(0.9999999999998099 + (cbrt(Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) + Float64(676.5203681218851 / Float64(1.0 - z)))) ^ 3.0))))))));
	end
	return tmp
end

Wolfram

code[z_] := Block[{t$95$0 = N[(1.0 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 - z), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -13.8], N[(N[(Pi / t$95$1), $MachinePrecision] * N[(N[Exp[N[(N[Log[t$95$3], $MachinePrecision] + N[(z + -7.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(1.5056327351493116e-7 / N[(t$95$2 + 8.0), $MachinePrecision]), $MachinePrecision] - N[(N[(9.984369578019572e-6 / N[(t$95$0 - 7.0), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.13857109526572012 / N[(t$95$2 + 6.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(N[(-1259.1392167224028 / N[(z - 2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(676.5203681218851 / N[(z + -1.0), $MachinePrecision]), $MachinePrecision] - 0.9999999999998099), $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]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(12.507343278686905 / N[(z - 5.0), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(z - 3.0), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(z - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.13857109526572012 / N[(z - 6.0), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(z - 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.9999999999998099 + N[Power[N[Power[N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] + N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if z < -13.800000000000001

   1. Initial program 19.3%

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

      1. +-commutative 19.3%

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

      2. +-commutative 19.3%

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

      3. add-exp-log 19.3%

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

      4. expm1-define 19.3%

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{-1259.1392167224028}{2 + \color{blue}{\mathsf{expm1}\left(\log \left(1 - z\right)\right)}} + \left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      5. sub-neg 19.3%

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{-1259.1392167224028}{2 + \mathsf{expm1}\left(\log \color{blue}{\left(1 + \left(-z\right)\right)}\right)} + \left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      6. log1p-define 19.3%

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{-1259.1392167224028}{2 + \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(-z\right)}\right)} + \left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      7. expm1-log1p-u 19.3%

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

      8. sub-neg 19.3%

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

      9. +-commutative 19.3%

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

      10. add-exp-log 19.3%

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

      11. +-commutative 19.3%

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

      12. log1p-define 19.3%

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{-1259.1392167224028}{2 - z} + \left(\frac{676.5203681218851}{e^{\color{blue}{\mathsf{log1p}\left(\left(1 - z\right) - 1\right)}}} + 0.9999999999998099\right)\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      13. add-exp-log 19.3%

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{-1259.1392167224028}{2 - z} + \left(\frac{676.5203681218851}{e^{\mathsf{log1p}\left(\color{blue}{e^{\log \left(1 - z\right)}} - 1\right)}} + 0.9999999999998099\right)\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      14. expm1-define 19.3%

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{-1259.1392167224028}{2 - z} + \left(\frac{676.5203681218851}{e^{\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left(1 - z\right)\right)}\right)}} + 0.9999999999998099\right)\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      15. log1p-expm1-u 19.3%

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

      16. add-exp-log 19.3%

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

   4. Applied egg-rr 19.3%

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

   6. Applied egg-rr 19.3%

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

      1. add-exp-log 20.0%

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

      2. *-commutative 20.0%

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

      3. +-commutative 20.0%

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

      4. sub-neg 20.0%

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

      5. *-commutative 20.0%

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

      6. log-prod 20.0%

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

      7. add-log-exp 100.0%

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

      8. distribute-neg-in 100.0%

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

      9. metadata-eval 100.0%

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

   8. Applied egg-rr 100.0%

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

   if -13.800000000000001 < z

   1. Initial program 97.4%

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

   3. Simplified 99.2%

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

      1. add-cube-cbrt 99.2%

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

      2. pow3 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. div-inv 99.3%

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

      2. *-commutative 99.3%

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

   8. Applied egg-rr 99.3%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -13.8:\\ \;\;\;\;\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(e^{\log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) + \left(z + -7.5\right)} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) + -1\right) + 8} - \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 + \left(z + -1\right)\right) - 7} - \left(\frac{-0.13857109526572012}{\left(\left(1 - z\right) + -1\right) + 6} - \left(\left(\left(\left(\frac{-1259.1392167224028}{z - 2} + \left(\frac{676.5203681218851}{z + -1} - 0.9999999999998099\right)\right) + \frac{771.3234287776531}{\left(1 + \left(z + -1\right)\right) - 3}\right) + \frac{-176.6150291621406}{\left(1 + \left(z + -1\right)\right) - 4}\right) + \frac{12.507343278686905}{\left(1 + \left(z + -1\right)\right) - 5}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \frac{1}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} - \left(\left(\frac{12.507343278686905}{z - 5} + \left(\frac{771.3234287776531}{z - 3} + \frac{-176.6150291621406}{z - 4}\right)\right) + \left(\left(\frac{-0.13857109526572012}{z - 6} + \frac{9.984369578019572 \cdot 10^{-6}}{z - 7}\right) - \left(0.9999999999998099 + {\left(\sqrt[3]{\frac{-1259.1392167224028}{2 - z} + \frac{676.5203681218851}{1 - z}}\right)}^{3}\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(z + -1\right)\\ t_1 := \sin \left(z \cdot \pi\right)\\ t_2 := \left(1 - z\right) + -1\\ t_3 := \sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\\ \mathbf{if}\;z \leq -0.0085:\\ \;\;\;\;\frac{\pi}{t\_1} \cdot \left(e^{\log t\_3 + \left(z + -7.5\right)} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{t\_2 + 8} - \left(\frac{9.984369578019572 \cdot 10^{-6}}{t\_0 - 7} - \left(\frac{-0.13857109526572012}{t\_2 + 6} - \left(\left(\left(\left(\frac{-1259.1392167224028}{z - 2} + \left(\frac{676.5203681218851}{z + -1} - 0.9999999999998099\right)\right) + \frac{771.3234287776531}{t\_0 - 3}\right) + \frac{-176.6150291621406}{t\_0 - 4}\right) + \frac{12.507343278686905}{t\_0 - 5}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \frac{1}{t\_1}\right) \cdot \left(t\_3 \cdot \left(e^{z + -7.5} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} - \left(\left(\frac{12.507343278686905}{z - 5} + \left(\frac{771.3234287776531}{z - 3} + \frac{-176.6150291621406}{z - 4}\right)\right) + \left(\left(\frac{-0.13857109526572012}{z - 6} + \frac{9.984369578019572 \cdot 10^{-6}}{z - 7}\right) - \left(0.9999999999998099 + \mathsf{fma}\left(676.5203681218851, \frac{1}{1 - z}, \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (+ z -1.0)))
        (t_1 (sin (* z PI)))
        (t_2 (+ (- 1.0 z) -1.0))
        (t_3 (* (sqrt (* PI 2.0)) (pow (- 7.5 z) (- 0.5 z)))))
   (if (<= z -0.0085)
     (*
      (/ PI t_1)
      (*
       (exp (+ (log t_3) (+ z -7.5)))
       (-
        (/ 1.5056327351493116e-7 (+ t_2 8.0))
        (-
         (/ 9.984369578019572e-6 (- t_0 7.0))
         (-
          (/ -0.13857109526572012 (+ t_2 6.0))
          (+
           (+
            (+
             (+
              (/ -1259.1392167224028 (- z 2.0))
              (- (/ 676.5203681218851 (+ z -1.0)) 0.9999999999998099))
             (/ 771.3234287776531 (- t_0 3.0)))
            (/ -176.6150291621406 (- t_0 4.0)))
           (/ 12.507343278686905 (- t_0 5.0))))))))
     (*
      (* PI (/ 1.0 t_1))
      (*
       t_3
       (*
        (exp (+ z -7.5))
        (-
         (/ 1.5056327351493116e-7 (- 8.0 z))
         (+
          (+
           (/ 12.507343278686905 (- z 5.0))
           (+
            (/ 771.3234287776531 (- z 3.0))
            (/ -176.6150291621406 (- z 4.0))))
          (-
           (+
            (/ -0.13857109526572012 (- z 6.0))
            (/ 9.984369578019572e-6 (- z 7.0)))
           (+
            0.9999999999998099
            (fma
             676.5203681218851
             (/ 1.0 (- 1.0 z))
             (/ -1259.1392167224028 (- 2.0 z)))))))))))))

C

double code(double z) {
	double t_0 = 1.0 + (z + -1.0);
	double t_1 = sin((z * ((double) M_PI)));
	double t_2 = (1.0 - z) + -1.0;
	double t_3 = sqrt((((double) M_PI) * 2.0)) * pow((7.5 - z), (0.5 - z));
	double tmp;
	if (z <= -0.0085) {
		tmp = (((double) M_PI) / t_1) * (exp((log(t_3) + (z + -7.5))) * ((1.5056327351493116e-7 / (t_2 + 8.0)) - ((9.984369578019572e-6 / (t_0 - 7.0)) - ((-0.13857109526572012 / (t_2 + 6.0)) - (((((-1259.1392167224028 / (z - 2.0)) + ((676.5203681218851 / (z + -1.0)) - 0.9999999999998099)) + (771.3234287776531 / (t_0 - 3.0))) + (-176.6150291621406 / (t_0 - 4.0))) + (12.507343278686905 / (t_0 - 5.0)))))));
	} else {
		tmp = (((double) M_PI) * (1.0 / t_1)) * (t_3 * (exp((z + -7.5)) * ((1.5056327351493116e-7 / (8.0 - z)) - (((12.507343278686905 / (z - 5.0)) + ((771.3234287776531 / (z - 3.0)) + (-176.6150291621406 / (z - 4.0)))) + (((-0.13857109526572012 / (z - 6.0)) + (9.984369578019572e-6 / (z - 7.0))) - (0.9999999999998099 + fma(676.5203681218851, (1.0 / (1.0 - z)), (-1259.1392167224028 / (2.0 - z)))))))));
	}
	return tmp;
}

Julia

function code(z)
	t_0 = Float64(1.0 + Float64(z + -1.0))
	t_1 = sin(Float64(z * pi))
	t_2 = Float64(Float64(1.0 - z) + -1.0)
	t_3 = Float64(sqrt(Float64(pi * 2.0)) * (Float64(7.5 - z) ^ Float64(0.5 - z)))
	tmp = 0.0
	if (z <= -0.0085)
		tmp = Float64(Float64(pi / t_1) * Float64(exp(Float64(log(t_3) + Float64(z + -7.5))) * Float64(Float64(1.5056327351493116e-7 / Float64(t_2 + 8.0)) - Float64(Float64(9.984369578019572e-6 / Float64(t_0 - 7.0)) - Float64(Float64(-0.13857109526572012 / Float64(t_2 + 6.0)) - Float64(Float64(Float64(Float64(Float64(-1259.1392167224028 / Float64(z - 2.0)) + Float64(Float64(676.5203681218851 / Float64(z + -1.0)) - 0.9999999999998099)) + Float64(771.3234287776531 / Float64(t_0 - 3.0))) + Float64(-176.6150291621406 / Float64(t_0 - 4.0))) + Float64(12.507343278686905 / Float64(t_0 - 5.0))))))));
	else
		tmp = Float64(Float64(pi * Float64(1.0 / t_1)) * Float64(t_3 * Float64(exp(Float64(z + -7.5)) * Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) - Float64(Float64(Float64(12.507343278686905 / Float64(z - 5.0)) + Float64(Float64(771.3234287776531 / Float64(z - 3.0)) + Float64(-176.6150291621406 / Float64(z - 4.0)))) + Float64(Float64(Float64(-0.13857109526572012 / Float64(z - 6.0)) + Float64(9.984369578019572e-6 / Float64(z - 7.0))) - Float64(0.9999999999998099 + fma(676.5203681218851, Float64(1.0 / Float64(1.0 - z)), Float64(-1259.1392167224028 / Float64(2.0 - z))))))))));
	end
	return tmp
end

Wolfram

code[z_] := Block[{t$95$0 = N[(1.0 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 - z), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.0085], N[(N[(Pi / t$95$1), $MachinePrecision] * N[(N[Exp[N[(N[Log[t$95$3], $MachinePrecision] + N[(z + -7.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(1.5056327351493116e-7 / N[(t$95$2 + 8.0), $MachinePrecision]), $MachinePrecision] - N[(N[(9.984369578019572e-6 / N[(t$95$0 - 7.0), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.13857109526572012 / N[(t$95$2 + 6.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(N[(-1259.1392167224028 / N[(z - 2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(676.5203681218851 / N[(z + -1.0), $MachinePrecision]), $MachinePrecision] - 0.9999999999998099), $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]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(12.507343278686905 / N[(z - 5.0), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(z - 3.0), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(z - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.13857109526572012 / N[(z - 6.0), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(z - 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.9999999999998099 + N[(676.5203681218851 * N[(1.0 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.0085000000000000006

   1. Initial program 48.8%

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

      1. +-commutative 48.8%

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

      2. +-commutative 48.8%

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

      3. add-exp-log 48.8%

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

      4. expm1-define 48.8%

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{-1259.1392167224028}{2 + \color{blue}{\mathsf{expm1}\left(\log \left(1 - z\right)\right)}} + \left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      5. sub-neg 48.8%

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{-1259.1392167224028}{2 + \mathsf{expm1}\left(\log \color{blue}{\left(1 + \left(-z\right)\right)}\right)} + \left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      6. log1p-define 48.8%

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{-1259.1392167224028}{2 + \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(-z\right)}\right)} + \left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right)\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      7. expm1-log1p-u 48.8%

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

      8. sub-neg 48.8%

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

      9. +-commutative 48.8%

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

      10. add-exp-log 48.8%

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

      11. +-commutative 48.8%

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

      12. log1p-define 48.8%

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{-1259.1392167224028}{2 - z} + \left(\frac{676.5203681218851}{e^{\color{blue}{\mathsf{log1p}\left(\left(1 - z\right) - 1\right)}}} + 0.9999999999998099\right)\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      13. add-exp-log 48.8%

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{-1259.1392167224028}{2 - z} + \left(\frac{676.5203681218851}{e^{\mathsf{log1p}\left(\color{blue}{e^{\log \left(1 - z\right)}} - 1\right)}} + 0.9999999999998099\right)\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      14. expm1-define 48.8%

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\frac{-1259.1392167224028}{2 - z} + \left(\frac{676.5203681218851}{e^{\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left(1 - z\right)\right)}\right)}} + 0.9999999999998099\right)\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      15. log1p-expm1-u 48.8%

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

      16. add-exp-log 48.8%

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

   4. Applied egg-rr 48.8%

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

   6. Applied egg-rr 48.3%

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

      1. add-exp-log 48.9%

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

      2. *-commutative 48.9%

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

      3. +-commutative 48.9%

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

      4. sub-neg 48.9%

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

      5. *-commutative 48.9%

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

      6. log-prod 48.9%

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

      7. add-log-exp 98.9%

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

      8. distribute-neg-in 98.9%

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

      9. metadata-eval 98.9%

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

   8. Applied egg-rr 98.9%

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

   if -0.0085000000000000006 < z

   1. Initial program 97.4%

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

   3. Simplified 99.2%

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

      1. add-cube-cbrt 99.3%

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

      2. pow3 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. div-inv 99.3%

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

      2. *-commutative 99.3%

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

   8. Applied egg-rr 99.3%

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

      1. rem-cube-cbrt 99.3%

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

      2. div-inv 99.3%

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

      3. fma-define 99.3%

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

   10. Applied egg-rr 99.3%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.0085:\\ \;\;\;\;\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(e^{\log \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) + \left(z + -7.5\right)} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) + -1\right) + 8} - \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 + \left(z + -1\right)\right) - 7} - \left(\frac{-0.13857109526572012}{\left(\left(1 - z\right) + -1\right) + 6} - \left(\left(\left(\left(\frac{-1259.1392167224028}{z - 2} + \left(\frac{676.5203681218851}{z + -1} - 0.9999999999998099\right)\right) + \frac{771.3234287776531}{\left(1 + \left(z + -1\right)\right) - 3}\right) + \frac{-176.6150291621406}{\left(1 + \left(z + -1\right)\right) - 4}\right) + \frac{12.507343278686905}{\left(1 + \left(z + -1\right)\right) - 5}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \frac{1}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} - \left(\left(\frac{12.507343278686905}{z - 5} + \left(\frac{771.3234287776531}{z - 3} + \frac{-176.6150291621406}{z - 4}\right)\right) + \left(\left(\frac{-0.13857109526572012}{z - 6} + \frac{9.984369578019572 \cdot 10^{-6}}{z - 7}\right) - \left(0.9999999999998099 + \mathsf{fma}\left(676.5203681218851, \frac{1}{1 - z}, \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.9%

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

3. Simplified 97.7%

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

   1. add-cube-cbrt 97.7%

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

   2. pow3 97.7%

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

6. Applied egg-rr 97.7%

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

   1. div-inv 97.7%

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

   2. *-commutative 97.7%

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

8. Applied egg-rr 97.7%

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

   1. rem-cube-cbrt 97.7%

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

   2. div-inv 97.7%

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

   3. fma-define 97.7%

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

10. Applied egg-rr 97.7%

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

11. Final simplification 97.7%

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

Alternative 4: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.9%

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

3. Simplified 97.7%

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

   1. add-cube-cbrt 97.7%

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

   2. pow3 97.7%

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

6. Applied egg-rr 97.7%

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

   1. div-inv 97.7%

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

   2. *-commutative 97.7%

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

8. Applied egg-rr 97.7%

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

   1. rem-cube-cbrt 97.7%

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

10. Applied egg-rr 97.7%

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

11. Final simplification 97.7%

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

Alternative 5: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.9%

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

3. Simplified 97.7%

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

5. Final simplification 97.7%

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

Alternative 6: 97.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \left(46.9507597606837 + z \cdot \left(361.7355639412844 + z \cdot 519.1279660315847\right)\right)\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ PI (sin (* z PI)))
  (*
   (* (sqrt (* PI 2.0)) (pow (- 7.5 z) (- 0.5 z)))
   (*
    (exp (+ z -7.5))
    (+
     (/ 1.5056327351493116e-7 (- 8.0 z))
     (+
      (+
       (/ 12.507343278686905 (- 5.0 z))
       (+ (/ 771.3234287776531 (- 3.0 z)) (/ -176.6150291621406 (- 4.0 z))))
      (+
       (+
        (/ -0.13857109526572012 (- 6.0 z))
        (/ 9.984369578019572e-6 (- 7.0 z)))
       (+
        0.9999999999998099
        (+
         46.9507597606837
         (* z (+ 361.7355639412844 (* z 519.1279660315847))))))))))))

C

double code(double z) {
	return (((double) M_PI) / sin((z * ((double) M_PI)))) * ((sqrt((((double) M_PI) * 2.0)) * pow((7.5 - z), (0.5 - z))) * (exp((z + -7.5)) * ((1.5056327351493116e-7 / (8.0 - z)) + (((12.507343278686905 / (5.0 - z)) + ((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z)))) + (((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (0.9999999999998099 + (46.9507597606837 + (z * (361.7355639412844 + (z * 519.1279660315847))))))))));
}

Java

public static double code(double z) {
	return (Math.PI / Math.sin((z * Math.PI))) * ((Math.sqrt((Math.PI * 2.0)) * Math.pow((7.5 - z), (0.5 - z))) * (Math.exp((z + -7.5)) * ((1.5056327351493116e-7 / (8.0 - z)) + (((12.507343278686905 / (5.0 - z)) + ((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z)))) + (((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (0.9999999999998099 + (46.9507597606837 + (z * (361.7355639412844 + (z * 519.1279660315847))))))))));
}

Python

def code(z):
	return (math.pi / math.sin((z * math.pi))) * ((math.sqrt((math.pi * 2.0)) * math.pow((7.5 - z), (0.5 - z))) * (math.exp((z + -7.5)) * ((1.5056327351493116e-7 / (8.0 - z)) + (((12.507343278686905 / (5.0 - z)) + ((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z)))) + (((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (0.9999999999998099 + (46.9507597606837 + (z * (361.7355639412844 + (z * 519.1279660315847))))))))))

Julia

function code(z)
	return Float64(Float64(pi / sin(Float64(z * pi))) * Float64(Float64(sqrt(Float64(pi * 2.0)) * (Float64(7.5 - z) ^ Float64(0.5 - z))) * Float64(exp(Float64(z + -7.5)) * Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) + Float64(Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(Float64(771.3234287776531 / Float64(3.0 - z)) + Float64(-176.6150291621406 / Float64(4.0 - z)))) + Float64(Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(9.984369578019572e-6 / Float64(7.0 - z))) + Float64(0.9999999999998099 + Float64(46.9507597606837 + Float64(z * Float64(361.7355639412844 + Float64(z * 519.1279660315847)))))))))))
end

MATLAB

function tmp = code(z)
	tmp = (pi / sin((z * pi))) * ((sqrt((pi * 2.0)) * ((7.5 - z) ^ (0.5 - z))) * (exp((z + -7.5)) * ((1.5056327351493116e-7 / (8.0 - z)) + (((12.507343278686905 / (5.0 - z)) + ((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z)))) + (((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (0.9999999999998099 + (46.9507597606837 + (z * (361.7355639412844 + (z * 519.1279660315847))))))))));
end

Wolfram

code[z_] := N[(N[(Pi / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(46.9507597606837 + N[(z * N[(361.7355639412844 + N[(z * 519.1279660315847), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.9%

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

3. Simplified 97.7%

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

5. Taylor expanded in z around 0 96.1%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \color{blue}{\left(46.9507597606837 + z \cdot \left(361.7355639412844 + 519.1279660315847 \cdot z\right)\right)}\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \]

6. Final simplification 96.1%

   \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \left(46.9507597606837 + z \cdot \left(361.7355639412844 + z \cdot 519.1279660315847\right)\right)\right)\right)\right)\right)\right)\right) \]
7. Add Preprocessing

Alternative 7: 97.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \left(46.9507597606837 + z \cdot 361.7355639412844\right)\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ PI (sin (* z PI)))
  (*
   (* (sqrt (* PI 2.0)) (pow (- 7.5 z) (- 0.5 z)))
   (*
    (exp (+ z -7.5))
    (+
     (/ 1.5056327351493116e-7 (- 8.0 z))
     (+
      (+
       (/ 12.507343278686905 (- 5.0 z))
       (+ (/ 771.3234287776531 (- 3.0 z)) (/ -176.6150291621406 (- 4.0 z))))
      (+
       (+
        (/ -0.13857109526572012 (- 6.0 z))
        (/ 9.984369578019572e-6 (- 7.0 z)))
       (+
        0.9999999999998099
        (+ 46.9507597606837 (* z 361.7355639412844))))))))))

C

double code(double z) {
	return (((double) M_PI) / sin((z * ((double) M_PI)))) * ((sqrt((((double) M_PI) * 2.0)) * pow((7.5 - z), (0.5 - z))) * (exp((z + -7.5)) * ((1.5056327351493116e-7 / (8.0 - z)) + (((12.507343278686905 / (5.0 - z)) + ((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z)))) + (((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (0.9999999999998099 + (46.9507597606837 + (z * 361.7355639412844))))))));
}

Java

public static double code(double z) {
	return (Math.PI / Math.sin((z * Math.PI))) * ((Math.sqrt((Math.PI * 2.0)) * Math.pow((7.5 - z), (0.5 - z))) * (Math.exp((z + -7.5)) * ((1.5056327351493116e-7 / (8.0 - z)) + (((12.507343278686905 / (5.0 - z)) + ((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z)))) + (((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (0.9999999999998099 + (46.9507597606837 + (z * 361.7355639412844))))))));
}

Python

def code(z):
	return (math.pi / math.sin((z * math.pi))) * ((math.sqrt((math.pi * 2.0)) * math.pow((7.5 - z), (0.5 - z))) * (math.exp((z + -7.5)) * ((1.5056327351493116e-7 / (8.0 - z)) + (((12.507343278686905 / (5.0 - z)) + ((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z)))) + (((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (0.9999999999998099 + (46.9507597606837 + (z * 361.7355639412844))))))))

Julia

function code(z)
	return Float64(Float64(pi / sin(Float64(z * pi))) * Float64(Float64(sqrt(Float64(pi * 2.0)) * (Float64(7.5 - z) ^ Float64(0.5 - z))) * Float64(exp(Float64(z + -7.5)) * Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) + Float64(Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(Float64(771.3234287776531 / Float64(3.0 - z)) + Float64(-176.6150291621406 / Float64(4.0 - z)))) + Float64(Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(9.984369578019572e-6 / Float64(7.0 - z))) + Float64(0.9999999999998099 + Float64(46.9507597606837 + Float64(z * 361.7355639412844)))))))))
end

MATLAB

function tmp = code(z)
	tmp = (pi / sin((z * pi))) * ((sqrt((pi * 2.0)) * ((7.5 - z) ^ (0.5 - z))) * (exp((z + -7.5)) * ((1.5056327351493116e-7 / (8.0 - z)) + (((12.507343278686905 / (5.0 - z)) + ((771.3234287776531 / (3.0 - z)) + (-176.6150291621406 / (4.0 - z)))) + (((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (0.9999999999998099 + (46.9507597606837 + (z * 361.7355639412844))))))));
end

Wolfram

code[z_] := N[(N[(Pi / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(46.9507597606837 + N[(z * 361.7355639412844), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.9%

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

3. Simplified 97.7%

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

5. Taylor expanded in z around 0 95.6%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \color{blue}{\left(46.9507597606837 + 361.7355639412844 \cdot z\right)}\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \]

6. Final simplification 95.6%

   \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \left(46.9507597606837 + z \cdot 361.7355639412844\right)\right)\right)\right)\right)\right)\right) \]
7. Add Preprocessing

Alternative 8: 96.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\pi \cdot \frac{1}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \left(46.9507597606837 + z \cdot \left(361.7355639412844 + z \cdot 519.1279660315847\right)\right)\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \left(212.9540523020159 + z \cdot 74.66416387488323\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (* PI (/ 1.0 (sin (* z PI))))
  (*
   (* (sqrt (* PI 2.0)) (pow (- 7.5 z) (- 0.5 z)))
   (*
    (exp (+ z -7.5))
    (+
     (/ 1.5056327351493116e-7 (- 8.0 z))
     (+
      (+
       (+
        (/ -0.13857109526572012 (- 6.0 z))
        (/ 9.984369578019572e-6 (- 7.0 z)))
       (+
        0.9999999999998099
        (+
         46.9507597606837
         (* z (+ 361.7355639412844 (* z 519.1279660315847))))))
      (+
       (/ 12.507343278686905 (- 5.0 z))
       (+ 212.9540523020159 (* z 74.66416387488323)))))))))

C

double code(double z) {
	return (((double) M_PI) * (1.0 / sin((z * ((double) M_PI))))) * ((sqrt((((double) M_PI) * 2.0)) * pow((7.5 - z), (0.5 - z))) * (exp((z + -7.5)) * ((1.5056327351493116e-7 / (8.0 - z)) + ((((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (0.9999999999998099 + (46.9507597606837 + (z * (361.7355639412844 + (z * 519.1279660315847)))))) + ((12.507343278686905 / (5.0 - z)) + (212.9540523020159 + (z * 74.66416387488323)))))));
}

Java

public static double code(double z) {
	return (Math.PI * (1.0 / Math.sin((z * Math.PI)))) * ((Math.sqrt((Math.PI * 2.0)) * Math.pow((7.5 - z), (0.5 - z))) * (Math.exp((z + -7.5)) * ((1.5056327351493116e-7 / (8.0 - z)) + ((((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (0.9999999999998099 + (46.9507597606837 + (z * (361.7355639412844 + (z * 519.1279660315847)))))) + ((12.507343278686905 / (5.0 - z)) + (212.9540523020159 + (z * 74.66416387488323)))))));
}

Python

def code(z):
	return (math.pi * (1.0 / math.sin((z * math.pi)))) * ((math.sqrt((math.pi * 2.0)) * math.pow((7.5 - z), (0.5 - z))) * (math.exp((z + -7.5)) * ((1.5056327351493116e-7 / (8.0 - z)) + ((((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (0.9999999999998099 + (46.9507597606837 + (z * (361.7355639412844 + (z * 519.1279660315847)))))) + ((12.507343278686905 / (5.0 - z)) + (212.9540523020159 + (z * 74.66416387488323)))))))

Julia

function code(z)
	return Float64(Float64(pi * Float64(1.0 / sin(Float64(z * pi)))) * Float64(Float64(sqrt(Float64(pi * 2.0)) * (Float64(7.5 - z) ^ Float64(0.5 - z))) * Float64(exp(Float64(z + -7.5)) * Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) + Float64(Float64(Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(9.984369578019572e-6 / Float64(7.0 - z))) + Float64(0.9999999999998099 + Float64(46.9507597606837 + Float64(z * Float64(361.7355639412844 + Float64(z * 519.1279660315847)))))) + Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(212.9540523020159 + Float64(z * 74.66416387488323))))))))
end

MATLAB

function tmp = code(z)
	tmp = (pi * (1.0 / sin((z * pi)))) * ((sqrt((pi * 2.0)) * ((7.5 - z) ^ (0.5 - z))) * (exp((z + -7.5)) * ((1.5056327351493116e-7 / (8.0 - z)) + ((((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (0.9999999999998099 + (46.9507597606837 + (z * (361.7355639412844 + (z * 519.1279660315847)))))) + ((12.507343278686905 / (5.0 - z)) + (212.9540523020159 + (z * 74.66416387488323)))))));
end

Wolfram

code[z_] := N[(N[(Pi * N[(1.0 / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(46.9507597606837 + N[(z * N[(361.7355639412844 + N[(z * 519.1279660315847), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(212.9540523020159 + N[(z * 74.66416387488323), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.9%

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

3. Simplified 97.7%

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

5. Taylor expanded in z around 0 96.1%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \color{blue}{\left(46.9507597606837 + z \cdot \left(361.7355639412844 + 519.1279660315847 \cdot z\right)\right)}\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \]

6. Taylor expanded in z around 0 95.1%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \left(46.9507597606837 + z \cdot \left(361.7355639412844 + 519.1279660315847 \cdot z\right)\right)\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \color{blue}{\left(212.9540523020159 + 74.66416387488323 \cdot z\right)}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \]
7. Step-by-step derivation

   1. div-inv 97.7%

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

   2. *-commutative 97.7%

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

8. Applied egg-rr 95.1%

   \[\leadsto \color{blue}{\left(\pi \cdot \frac{1}{\sin \left(z \cdot \pi\right)}\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \left(46.9507597606837 + z \cdot \left(361.7355639412844 + 519.1279660315847 \cdot z\right)\right)\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \left(212.9540523020159 + 74.66416387488323 \cdot z\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \]

9. Final simplification 95.1%

   \[\leadsto \left(\pi \cdot \frac{1}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \left(46.9507597606837 + z \cdot \left(361.7355639412844 + z \cdot 519.1279660315847\right)\right)\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \left(212.9540523020159 + z \cdot 74.66416387488323\right)\right)\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 9: 96.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \left(46.9507597606837 + z \cdot \left(361.7355639412844 + z \cdot 519.1279660315847\right)\right)\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \left(212.9540523020159 + z \cdot 74.66416387488323\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ PI (sin (* z PI)))
  (*
   (* (sqrt (* PI 2.0)) (pow (- 7.5 z) (- 0.5 z)))
   (*
    (exp (+ z -7.5))
    (+
     (/ 1.5056327351493116e-7 (- 8.0 z))
     (+
      (+
       (+
        (/ -0.13857109526572012 (- 6.0 z))
        (/ 9.984369578019572e-6 (- 7.0 z)))
       (+
        0.9999999999998099
        (+
         46.9507597606837
         (* z (+ 361.7355639412844 (* z 519.1279660315847))))))
      (+
       (/ 12.507343278686905 (- 5.0 z))
       (+ 212.9540523020159 (* z 74.66416387488323)))))))))

C

double code(double z) {
	return (((double) M_PI) / sin((z * ((double) M_PI)))) * ((sqrt((((double) M_PI) * 2.0)) * pow((7.5 - z), (0.5 - z))) * (exp((z + -7.5)) * ((1.5056327351493116e-7 / (8.0 - z)) + ((((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (0.9999999999998099 + (46.9507597606837 + (z * (361.7355639412844 + (z * 519.1279660315847)))))) + ((12.507343278686905 / (5.0 - z)) + (212.9540523020159 + (z * 74.66416387488323)))))));
}

Java

public static double code(double z) {
	return (Math.PI / Math.sin((z * Math.PI))) * ((Math.sqrt((Math.PI * 2.0)) * Math.pow((7.5 - z), (0.5 - z))) * (Math.exp((z + -7.5)) * ((1.5056327351493116e-7 / (8.0 - z)) + ((((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (0.9999999999998099 + (46.9507597606837 + (z * (361.7355639412844 + (z * 519.1279660315847)))))) + ((12.507343278686905 / (5.0 - z)) + (212.9540523020159 + (z * 74.66416387488323)))))));
}

Python

def code(z):
	return (math.pi / math.sin((z * math.pi))) * ((math.sqrt((math.pi * 2.0)) * math.pow((7.5 - z), (0.5 - z))) * (math.exp((z + -7.5)) * ((1.5056327351493116e-7 / (8.0 - z)) + ((((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (0.9999999999998099 + (46.9507597606837 + (z * (361.7355639412844 + (z * 519.1279660315847)))))) + ((12.507343278686905 / (5.0 - z)) + (212.9540523020159 + (z * 74.66416387488323)))))))

Julia

function code(z)
	return Float64(Float64(pi / sin(Float64(z * pi))) * Float64(Float64(sqrt(Float64(pi * 2.0)) * (Float64(7.5 - z) ^ Float64(0.5 - z))) * Float64(exp(Float64(z + -7.5)) * Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) + Float64(Float64(Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(9.984369578019572e-6 / Float64(7.0 - z))) + Float64(0.9999999999998099 + Float64(46.9507597606837 + Float64(z * Float64(361.7355639412844 + Float64(z * 519.1279660315847)))))) + Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(212.9540523020159 + Float64(z * 74.66416387488323))))))))
end

MATLAB

function tmp = code(z)
	tmp = (pi / sin((z * pi))) * ((sqrt((pi * 2.0)) * ((7.5 - z) ^ (0.5 - z))) * (exp((z + -7.5)) * ((1.5056327351493116e-7 / (8.0 - z)) + ((((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (0.9999999999998099 + (46.9507597606837 + (z * (361.7355639412844 + (z * 519.1279660315847)))))) + ((12.507343278686905 / (5.0 - z)) + (212.9540523020159 + (z * 74.66416387488323)))))));
end

Wolfram

code[z_] := N[(N[(Pi / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(46.9507597606837 + N[(z * N[(361.7355639412844 + N[(z * 519.1279660315847), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(212.9540523020159 + N[(z * 74.66416387488323), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.9%

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

3. Simplified 97.7%

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

5. Taylor expanded in z around 0 96.1%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \color{blue}{\left(46.9507597606837 + z \cdot \left(361.7355639412844 + 519.1279660315847 \cdot z\right)\right)}\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \]

6. Taylor expanded in z around 0 95.1%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \left(46.9507597606837 + z \cdot \left(361.7355639412844 + 519.1279660315847 \cdot z\right)\right)\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \color{blue}{\left(212.9540523020159 + 74.66416387488323 \cdot z\right)}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \]

7. Final simplification 95.1%

   \[\leadsto \frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \left(46.9507597606837 + z \cdot \left(361.7355639412844 + z \cdot 519.1279660315847\right)\right)\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \left(212.9540523020159 + z \cdot 74.66416387488323\right)\right)\right)\right)\right)\right) \]
8. Add Preprocessing

Alternative 10: 96.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \left(46.9507597606837 + z \cdot \left(361.7355639412844 + z \cdot 519.1279660315847\right)\right)\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \left(212.9540523020159 + z \cdot 74.66416387488323\right)\right)\right)\right)\right)\right) \cdot \frac{1}{z} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (* (sqrt (* PI 2.0)) (pow (- 7.5 z) (- 0.5 z)))
   (*
    (exp (+ z -7.5))
    (+
     (/ 1.5056327351493116e-7 (- 8.0 z))
     (+
      (+
       (+
        (/ -0.13857109526572012 (- 6.0 z))
        (/ 9.984369578019572e-6 (- 7.0 z)))
       (+
        0.9999999999998099
        (+
         46.9507597606837
         (* z (+ 361.7355639412844 (* z 519.1279660315847))))))
      (+
       (/ 12.507343278686905 (- 5.0 z))
       (+ 212.9540523020159 (* z 74.66416387488323)))))))
  (/ 1.0 z)))

C

double code(double z) {
	return ((sqrt((((double) M_PI) * 2.0)) * pow((7.5 - z), (0.5 - z))) * (exp((z + -7.5)) * ((1.5056327351493116e-7 / (8.0 - z)) + ((((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (0.9999999999998099 + (46.9507597606837 + (z * (361.7355639412844 + (z * 519.1279660315847)))))) + ((12.507343278686905 / (5.0 - z)) + (212.9540523020159 + (z * 74.66416387488323))))))) * (1.0 / z);
}

Java

public static double code(double z) {
	return ((Math.sqrt((Math.PI * 2.0)) * Math.pow((7.5 - z), (0.5 - z))) * (Math.exp((z + -7.5)) * ((1.5056327351493116e-7 / (8.0 - z)) + ((((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (0.9999999999998099 + (46.9507597606837 + (z * (361.7355639412844 + (z * 519.1279660315847)))))) + ((12.507343278686905 / (5.0 - z)) + (212.9540523020159 + (z * 74.66416387488323))))))) * (1.0 / z);
}

Python

def code(z):
	return ((math.sqrt((math.pi * 2.0)) * math.pow((7.5 - z), (0.5 - z))) * (math.exp((z + -7.5)) * ((1.5056327351493116e-7 / (8.0 - z)) + ((((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (0.9999999999998099 + (46.9507597606837 + (z * (361.7355639412844 + (z * 519.1279660315847)))))) + ((12.507343278686905 / (5.0 - z)) + (212.9540523020159 + (z * 74.66416387488323))))))) * (1.0 / z)

Julia

function code(z)
	return Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (Float64(7.5 - z) ^ Float64(0.5 - z))) * Float64(exp(Float64(z + -7.5)) * Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) + Float64(Float64(Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(9.984369578019572e-6 / Float64(7.0 - z))) + Float64(0.9999999999998099 + Float64(46.9507597606837 + Float64(z * Float64(361.7355639412844 + Float64(z * 519.1279660315847)))))) + Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(212.9540523020159 + Float64(z * 74.66416387488323))))))) * Float64(1.0 / z))
end

MATLAB

function tmp = code(z)
	tmp = ((sqrt((pi * 2.0)) * ((7.5 - z) ^ (0.5 - z))) * (exp((z + -7.5)) * ((1.5056327351493116e-7 / (8.0 - z)) + ((((-0.13857109526572012 / (6.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) + (0.9999999999998099 + (46.9507597606837 + (z * (361.7355639412844 + (z * 519.1279660315847)))))) + ((12.507343278686905 / (5.0 - z)) + (212.9540523020159 + (z * 74.66416387488323))))))) * (1.0 / z);
end

Wolfram

code[z_] := N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(46.9507597606837 + N[(z * N[(361.7355639412844 + N[(z * 519.1279660315847), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(212.9540523020159 + N[(z * 74.66416387488323), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.9%

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

3. Simplified 97.7%

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

5. Taylor expanded in z around 0 96.1%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \color{blue}{\left(46.9507597606837 + z \cdot \left(361.7355639412844 + 519.1279660315847 \cdot z\right)\right)}\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \]

6. Taylor expanded in z around 0 95.1%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \left(46.9507597606837 + z \cdot \left(361.7355639412844 + 519.1279660315847 \cdot z\right)\right)\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \color{blue}{\left(212.9540523020159 + 74.66416387488323 \cdot z\right)}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \]

7. Taylor expanded in z around 0 94.9%

   \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \left(46.9507597606837 + z \cdot \left(361.7355639412844 + 519.1279660315847 \cdot z\right)\right)\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \left(212.9540523020159 + 74.66416387488323 \cdot z\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \]

8. Final simplification 94.9%

   \[\leadsto \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \left(46.9507597606837 + z \cdot \left(361.7355639412844 + z \cdot 519.1279660315847\right)\right)\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \left(212.9540523020159 + z \cdot 74.66416387488323\right)\right)\right)\right)\right)\right) \cdot \frac{1}{z} \]
9. Add Preprocessing

Alternative 11: 95.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \frac{1}{z}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + 47.95075976068351\right)\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\frac{12.507343278686905}{5 - z} + 212.9540523020159\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (* (* (sqrt (* PI 2.0)) (pow (- 7.5 z) (- 0.5 z))) (/ 1.0 z))
  (*
   (exp (+ z -7.5))
   (+
    (+
     (/ -0.13857109526572012 (- 6.0 z))
     (+ (/ 9.984369578019572e-6 (- 7.0 z)) 47.95075976068351))
    (+
     (/ 1.5056327351493116e-7 (- 8.0 z))
     (+ (/ 12.507343278686905 (- 5.0 z)) 212.9540523020159))))))

C

double code(double z) {
	return ((sqrt((((double) M_PI) * 2.0)) * pow((7.5 - z), (0.5 - z))) * (1.0 / z)) * (exp((z + -7.5)) * (((-0.13857109526572012 / (6.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + 47.95075976068351)) + ((1.5056327351493116e-7 / (8.0 - z)) + ((12.507343278686905 / (5.0 - z)) + 212.9540523020159))));
}

Java

public static double code(double z) {
	return ((Math.sqrt((Math.PI * 2.0)) * Math.pow((7.5 - z), (0.5 - z))) * (1.0 / z)) * (Math.exp((z + -7.5)) * (((-0.13857109526572012 / (6.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + 47.95075976068351)) + ((1.5056327351493116e-7 / (8.0 - z)) + ((12.507343278686905 / (5.0 - z)) + 212.9540523020159))));
}

Python

def code(z):
	return ((math.sqrt((math.pi * 2.0)) * math.pow((7.5 - z), (0.5 - z))) * (1.0 / z)) * (math.exp((z + -7.5)) * (((-0.13857109526572012 / (6.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + 47.95075976068351)) + ((1.5056327351493116e-7 / (8.0 - z)) + ((12.507343278686905 / (5.0 - z)) + 212.9540523020159))))

Julia

function code(z)
	return Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (Float64(7.5 - z) ^ Float64(0.5 - z))) * Float64(1.0 / z)) * Float64(exp(Float64(z + -7.5)) * Float64(Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + 47.95075976068351)) + Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) + Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + 212.9540523020159)))))
end

MATLAB

function tmp = code(z)
	tmp = ((sqrt((pi * 2.0)) * ((7.5 - z) ^ (0.5 - z))) * (1.0 / z)) * (exp((z + -7.5)) * (((-0.13857109526572012 / (6.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + 47.95075976068351)) + ((1.5056327351493116e-7 / (8.0 - z)) + ((12.507343278686905 / (5.0 - z)) + 212.9540523020159))));
end

Wolfram

code[z_] := N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + 47.95075976068351), $MachinePrecision]), $MachinePrecision] + N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + 212.9540523020159), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.9%

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

3. Simplified 97.7%

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

5. Taylor expanded in z around 0 94.4%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \color{blue}{46.9507597606837}\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \]

6. Taylor expanded in z around 0 93.6%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + 46.9507597606837\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \color{blue}{212.9540523020159}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \]

7. Taylor expanded in z around 0 93.5%

   \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + 46.9507597606837\right)\right) + \left(\frac{12.507343278686905}{5 - z} + 212.9540523020159\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \]
8. Step-by-step derivation

   1. associate-*r* 93.7%

      \[\leadsto \color{blue}{\left(\frac{1}{z} \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right)\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + 46.9507597606837\right)\right) + \left(\frac{12.507343278686905}{5 - z} + 212.9540523020159\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)} \]

   2. associate-+l+ 93.7%

      \[\leadsto \left(\frac{1}{z} \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right)\right) \cdot \left(e^{z + -7.5} \cdot \color{blue}{\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + 46.9507597606837\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + 212.9540523020159\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)}\right) \]

   3. associate-+l+ 93.7%

      \[\leadsto \left(\frac{1}{z} \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right)\right) \cdot \left(e^{z + -7.5} \cdot \left(\color{blue}{\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(0.9999999999998099 + 46.9507597606837\right)\right)\right)} + \left(\left(\frac{12.507343278686905}{5 - z} + 212.9540523020159\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \]

   4. metadata-eval 93.7%

      \[\leadsto \left(\frac{1}{z} \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right)\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \color{blue}{47.95075976068351}\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + 212.9540523020159\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \]

9. Applied egg-rr 93.7%

   \[\leadsto \color{blue}{\left(\frac{1}{z} \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right)\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + 47.95075976068351\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + 212.9540523020159\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)} \]

10. Final simplification 93.7%

    \[\leadsto \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \frac{1}{z}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + 47.95075976068351\right)\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\frac{12.507343278686905}{5 - z} + 212.9540523020159\right)\right)\right)\right) \]
11. Add Preprocessing

Alternative 12: 95.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{z} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + 47.95075976068351\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + 212.9540523020159\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ 1.0 z)
  (*
   (sqrt (* PI 2.0))
   (*
    (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5)))
    (+
     (+
      (/ -0.13857109526572012 (- 6.0 z))
      (+ (/ 9.984369578019572e-6 (- 7.0 z)) 47.95075976068351))
     (+
      (/ 12.507343278686905 (- 5.0 z))
      (+ (/ 1.5056327351493116e-7 (- 8.0 z)) 212.9540523020159)))))))

C

double code(double z) {
	return (1.0 / z) * (sqrt((((double) M_PI) * 2.0)) * ((pow((7.5 - z), (0.5 - z)) * exp((z + -7.5))) * (((-0.13857109526572012 / (6.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + 47.95075976068351)) + ((12.507343278686905 / (5.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + 212.9540523020159)))));
}

Java

public static double code(double z) {
	return (1.0 / z) * (Math.sqrt((Math.PI * 2.0)) * ((Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5))) * (((-0.13857109526572012 / (6.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + 47.95075976068351)) + ((12.507343278686905 / (5.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + 212.9540523020159)))));
}

Python

def code(z):
	return (1.0 / z) * (math.sqrt((math.pi * 2.0)) * ((math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5))) * (((-0.13857109526572012 / (6.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + 47.95075976068351)) + ((12.507343278686905 / (5.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + 212.9540523020159)))))

Julia

function code(z)
	return Float64(Float64(1.0 / z) * Float64(sqrt(Float64(pi * 2.0)) * Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5))) * Float64(Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + 47.95075976068351)) + Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) + 212.9540523020159))))))
end

MATLAB

function tmp = code(z)
	tmp = (1.0 / z) * (sqrt((pi * 2.0)) * ((((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5))) * (((-0.13857109526572012 / (6.0 - z)) + ((9.984369578019572e-6 / (7.0 - z)) + 47.95075976068351)) + ((12.507343278686905 / (5.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + 212.9540523020159)))));
end

Wolfram

code[z_] := N[(N[(1.0 / z), $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + 47.95075976068351), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] + 212.9540523020159), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.9%

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

3. Simplified 97.7%

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

5. Taylor expanded in z around 0 94.4%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \color{blue}{46.9507597606837}\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \]

6. Taylor expanded in z around 0 93.6%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + 46.9507597606837\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \color{blue}{212.9540523020159}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \]

7. Taylor expanded in z around 0 93.5%

   \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + 46.9507597606837\right)\right) + \left(\frac{12.507343278686905}{5 - z} + 212.9540523020159\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \]
8. Step-by-step derivation

   1. *-un-lft-identity 93.5%

      \[\leadsto \frac{1}{z} \cdot \color{blue}{\left(1 \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + 46.9507597606837\right)\right) + \left(\frac{12.507343278686905}{5 - z} + 212.9540523020159\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)} \]

   2. associate-*r* 93.5%

      \[\leadsto \frac{1}{z} \cdot \left(1 \cdot \color{blue}{\left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + 46.9507597606837\right)\right) + \left(\frac{12.507343278686905}{5 - z} + 212.9540523020159\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)}\right) \]

9. Applied egg-rr 93.5%

   \[\leadsto \frac{1}{z} \cdot \color{blue}{\left(1 \cdot \left(\left(\left(\sqrt{2 \cdot \pi} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + 47.95075976068351\right) + \left(\left(\frac{12.507343278686905}{5 - z} + 212.9540523020159\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)} \]
10. Step-by-step derivation

    1. *-lft-identity 93.5%

       \[\leadsto \frac{1}{z} \cdot \color{blue}{\left(\left(\left(\sqrt{2 \cdot \pi} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + 47.95075976068351\right) + \left(\left(\frac{12.507343278686905}{5 - z} + 212.9540523020159\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)} \]

    2. associate-*r* 93.5%

       \[\leadsto \frac{1}{z} \cdot \color{blue}{\left(\left(\sqrt{2 \cdot \pi} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + 47.95075976068351\right) + \left(\left(\frac{12.507343278686905}{5 - z} + 212.9540523020159\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)} \]

    3. associate-*r* 93.5%

       \[\leadsto \frac{1}{z} \cdot \color{blue}{\left(\sqrt{2 \cdot \pi} \cdot \left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + 47.95075976068351\right) + \left(\left(\frac{12.507343278686905}{5 - z} + 212.9540523020159\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)} \]

    4. associate-*r* 93.5%

       \[\leadsto \frac{1}{z} \cdot \left(\sqrt{2 \cdot \pi} \cdot \color{blue}{\left(\left({\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + 47.95075976068351\right) + \left(\left(\frac{12.507343278686905}{5 - z} + 212.9540523020159\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)}\right) \]

    5. +-commutative 93.5%

       \[\leadsto \frac{1}{z} \cdot \left(\sqrt{2 \cdot \pi} \cdot \left(\left({\color{blue}{\left(7.5 + \left(-z\right)\right)}}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + 47.95075976068351\right) + \left(\left(\frac{12.507343278686905}{5 - z} + 212.9540523020159\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right) \]

    6. sub-neg 93.5%

       \[\leadsto \frac{1}{z} \cdot \left(\sqrt{2 \cdot \pi} \cdot \left(\left({\color{blue}{\left(7.5 - z\right)}}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + 47.95075976068351\right) + \left(\left(\frac{12.507343278686905}{5 - z} + 212.9540523020159\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right) \]

11. Simplified 93.5%

    \[\leadsto \frac{1}{z} \cdot \color{blue}{\left(\sqrt{2 \cdot \pi} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + 47.95075976068351\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \left(212.9540523020159 + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right)} \]

12. Final simplification 93.5%

    \[\leadsto \frac{1}{z} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + 47.95075976068351\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + 212.9540523020159\right)\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 13: 95.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{z} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{-7.5} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{z - 8} + \left(\left(\frac{12.507343278686905}{z - 5} - 212.9540523020159\right) - \left(47.95075976068351 - \left(\frac{-0.13857109526572012}{z - 6} + \frac{9.984369578019572 \cdot 10^{-6}}{z - 7}\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ -1.0 z)
  (*
   (* (sqrt (* PI 2.0)) (pow (- 7.5 z) (- 0.5 z)))
   (*
    (exp -7.5)
    (+
     (/ 1.5056327351493116e-7 (- z 8.0))
     (-
      (- (/ 12.507343278686905 (- z 5.0)) 212.9540523020159)
      (-
       47.95075976068351
       (+
        (/ -0.13857109526572012 (- z 6.0))
        (/ 9.984369578019572e-6 (- z 7.0))))))))))

C

double code(double z) {
	return (-1.0 / z) * ((sqrt((((double) M_PI) * 2.0)) * pow((7.5 - z), (0.5 - z))) * (exp(-7.5) * ((1.5056327351493116e-7 / (z - 8.0)) + (((12.507343278686905 / (z - 5.0)) - 212.9540523020159) - (47.95075976068351 - ((-0.13857109526572012 / (z - 6.0)) + (9.984369578019572e-6 / (z - 7.0))))))));
}

Java

public static double code(double z) {
	return (-1.0 / z) * ((Math.sqrt((Math.PI * 2.0)) * Math.pow((7.5 - z), (0.5 - z))) * (Math.exp(-7.5) * ((1.5056327351493116e-7 / (z - 8.0)) + (((12.507343278686905 / (z - 5.0)) - 212.9540523020159) - (47.95075976068351 - ((-0.13857109526572012 / (z - 6.0)) + (9.984369578019572e-6 / (z - 7.0))))))));
}

Python

def code(z):
	return (-1.0 / z) * ((math.sqrt((math.pi * 2.0)) * math.pow((7.5 - z), (0.5 - z))) * (math.exp(-7.5) * ((1.5056327351493116e-7 / (z - 8.0)) + (((12.507343278686905 / (z - 5.0)) - 212.9540523020159) - (47.95075976068351 - ((-0.13857109526572012 / (z - 6.0)) + (9.984369578019572e-6 / (z - 7.0))))))))

Julia

function code(z)
	return Float64(Float64(-1.0 / z) * Float64(Float64(sqrt(Float64(pi * 2.0)) * (Float64(7.5 - z) ^ Float64(0.5 - z))) * Float64(exp(-7.5) * Float64(Float64(1.5056327351493116e-7 / Float64(z - 8.0)) + Float64(Float64(Float64(12.507343278686905 / Float64(z - 5.0)) - 212.9540523020159) - Float64(47.95075976068351 - Float64(Float64(-0.13857109526572012 / Float64(z - 6.0)) + Float64(9.984369578019572e-6 / Float64(z - 7.0)))))))))
end

MATLAB

function tmp = code(z)
	tmp = (-1.0 / z) * ((sqrt((pi * 2.0)) * ((7.5 - z) ^ (0.5 - z))) * (exp(-7.5) * ((1.5056327351493116e-7 / (z - 8.0)) + (((12.507343278686905 / (z - 5.0)) - 212.9540523020159) - (47.95075976068351 - ((-0.13857109526572012 / (z - 6.0)) + (9.984369578019572e-6 / (z - 7.0))))))));
end

Wolfram

code[z_] := N[(N[(-1.0 / z), $MachinePrecision] * N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] * N[(N[(1.5056327351493116e-7 / N[(z - 8.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(12.507343278686905 / N[(z - 5.0), $MachinePrecision]), $MachinePrecision] - 212.9540523020159), $MachinePrecision] - N[(47.95075976068351 - N[(N[(-0.13857109526572012 / N[(z - 6.0), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(z - 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.9%

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

3. Simplified 97.7%

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

5. Taylor expanded in z around 0 94.4%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \color{blue}{46.9507597606837}\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \]

6. Taylor expanded in z around 0 93.6%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + 46.9507597606837\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \color{blue}{212.9540523020159}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \]

7. Taylor expanded in z around 0 93.5%

   \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + 46.9507597606837\right)\right) + \left(\frac{12.507343278686905}{5 - z} + 212.9540523020159\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \]

8. Taylor expanded in z around 0 93.8%

   \[\leadsto \frac{1}{z} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\color{blue}{e^{-7.5}} \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + 46.9507597606837\right)\right) + \left(\frac{12.507343278686905}{5 - z} + 212.9540523020159\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \]

9. Final simplification 93.8%

   \[\leadsto \frac{-1}{z} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{-7.5} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{z - 8} + \left(\left(\frac{12.507343278686905}{z - 5} - 212.9540523020159\right) - \left(47.95075976068351 - \left(\frac{-0.13857109526572012}{z - 6} + \frac{9.984369578019572 \cdot 10^{-6}}{z - 7}\right)\right)\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 14: 95.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(47.95075976068351 - \left(\frac{-0.13857109526572012}{z - 6} + \frac{9.984369578019572 \cdot 10^{-6}}{z - 7}\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + 212.9540523020159\right)\right)\right)\right)}{z} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (/
  (*
   (* (sqrt (* PI 2.0)) (pow (- 7.5 z) (- 0.5 z)))
   (*
    (exp (+ z -7.5))
    (+
     (-
      47.95075976068351
      (+
       (/ -0.13857109526572012 (- z 6.0))
       (/ 9.984369578019572e-6 (- z 7.0))))
     (+
      (/ 12.507343278686905 (- 5.0 z))
      (+ (/ 1.5056327351493116e-7 (- 8.0 z)) 212.9540523020159)))))
  z))

C

double code(double z) {
	return ((sqrt((((double) M_PI) * 2.0)) * pow((7.5 - z), (0.5 - z))) * (exp((z + -7.5)) * ((47.95075976068351 - ((-0.13857109526572012 / (z - 6.0)) + (9.984369578019572e-6 / (z - 7.0)))) + ((12.507343278686905 / (5.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + 212.9540523020159))))) / z;
}

Java

public static double code(double z) {
	return ((Math.sqrt((Math.PI * 2.0)) * Math.pow((7.5 - z), (0.5 - z))) * (Math.exp((z + -7.5)) * ((47.95075976068351 - ((-0.13857109526572012 / (z - 6.0)) + (9.984369578019572e-6 / (z - 7.0)))) + ((12.507343278686905 / (5.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + 212.9540523020159))))) / z;
}

Python

def code(z):
	return ((math.sqrt((math.pi * 2.0)) * math.pow((7.5 - z), (0.5 - z))) * (math.exp((z + -7.5)) * ((47.95075976068351 - ((-0.13857109526572012 / (z - 6.0)) + (9.984369578019572e-6 / (z - 7.0)))) + ((12.507343278686905 / (5.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + 212.9540523020159))))) / z

Julia

function code(z)
	return Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (Float64(7.5 - z) ^ Float64(0.5 - z))) * Float64(exp(Float64(z + -7.5)) * Float64(Float64(47.95075976068351 - Float64(Float64(-0.13857109526572012 / Float64(z - 6.0)) + Float64(9.984369578019572e-6 / Float64(z - 7.0)))) + Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) + 212.9540523020159))))) / z)
end

MATLAB

function tmp = code(z)
	tmp = ((sqrt((pi * 2.0)) * ((7.5 - z) ^ (0.5 - z))) * (exp((z + -7.5)) * ((47.95075976068351 - ((-0.13857109526572012 / (z - 6.0)) + (9.984369578019572e-6 / (z - 7.0)))) + ((12.507343278686905 / (5.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + 212.9540523020159))))) / z;
end

Wolfram

code[z_] := N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(N[(47.95075976068351 - N[(N[(-0.13857109526572012 / N[(z - 6.0), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(z - 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] + 212.9540523020159), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Derivation

1. Initial program 95.9%

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

3. Simplified 97.7%

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

5. Taylor expanded in z around 0 94.4%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + \color{blue}{46.9507597606837}\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{771.3234287776531}{3 - z} + \frac{-176.6150291621406}{4 - z}\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \]

6. Taylor expanded in z around 0 93.6%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + 46.9507597606837\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \color{blue}{212.9540523020159}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \]

7. Taylor expanded in z around 0 93.5%

   \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + 46.9507597606837\right)\right) + \left(\frac{12.507343278686905}{5 - z} + 212.9540523020159\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right) \]
8. Step-by-step derivation

   1. *-un-lft-identity 93.5%

      \[\leadsto \color{blue}{1 \cdot \left(\frac{1}{z} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + 46.9507597606837\right)\right) + \left(\frac{12.507343278686905}{5 - z} + 212.9540523020159\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)} \]

   2. associate-*l/ 93.5%

      \[\leadsto 1 \cdot \color{blue}{\frac{1 \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + \left(0.9999999999998099 + 46.9507597606837\right)\right) + \left(\frac{12.507343278686905}{5 - z} + 212.9540523020159\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)}{z}} \]

9. Applied egg-rr 93.5%

   \[\leadsto \color{blue}{1 \cdot \frac{\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}\right) \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + 47.95075976068351\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + 212.9540523020159\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)}{z}} \]
10. Step-by-step derivation

    1. *-lft-identity 93.5%

       \[\leadsto \color{blue}{\frac{\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-z\right) + 7.5\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}\right) \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + 47.95075976068351\right)\right) + \left(\left(\frac{12.507343278686905}{5 - z} + 212.9540523020159\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)}{z}} \]

11. Simplified 93.5%

    \[\leadsto \color{blue}{\frac{\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \left(e^{-7.5 + z} \cdot \left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + 47.95075976068351\right) + \left(\frac{12.507343278686905}{5 - z} + \left(212.9540523020159 + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)}{z}} \]

12. Final simplification 93.5%

    \[\leadsto \frac{\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(47.95075976068351 - \left(\frac{-0.13857109526572012}{z - 6} + \frac{9.984369578019572 \cdot 10^{-6}}{z - 7}\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + 212.9540523020159\right)\right)\right)\right)}{z} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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))))))