Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.3% → 98.9%

Time: 1.2min

Alternatives: 17

Speedup: 1.2×

• could not determine a ground truth

  1. z = -6.877728229032268e+258

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: 96.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.9999999999998099 - \frac{676.5203681218851}{z + -1}\\ t_1 := \sqrt{\pi \cdot 2}\\ t_2 := \frac{\pi}{\sin \left(\pi \cdot z\right)}\\ \mathbf{if}\;z \leq -2000:\\ \;\;\;\;\left(t\_1 \cdot \left({\left(6.5 - \left(z + -1\right)\right)}^{\left(-0.5 - \left(z + -1\right)\right)} \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right)\right) \cdot \left(t\_2 \cdot \left(\left(\left(\left(t\_0 - \frac{771.3234287776531}{\left(z + -1\right) - 2}\right) - \frac{-1259.1392167224028}{-1 + \left(z + -1\right)}\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 \cdot \left(t\_1 \cdot \left({\left(\left(-1 - \left(z + -1\right)\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{-0.5 + \left(-6 + \left(z + -1\right)\right)}\right)\right)\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) + \left(\left(\left(\frac{-1259.1392167224028}{1 + \left(1 - z\right)} + t\_0\right) - \left(\frac{-176.6150291621406}{-3 + \left(z + -1\right)} + \frac{771.3234287776531}{-2 + \left(z + -1\right)}\right)\right) - \left(\frac{12.507343278686905}{-4 + \left(z + -1\right)} + \frac{-0.13857109526572012}{-5 + \left(z + -1\right)}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- 0.9999999999998099 (/ 676.5203681218851 (+ z -1.0))))
        (t_1 (sqrt (* PI 2.0)))
        (t_2 (/ PI (sin (* PI z)))))
   (if (<= z -2000.0)
     (*
      (*
       t_1
       (*
        (pow (- 6.5 (+ z -1.0)) (- -0.5 (+ z -1.0)))
        (* (+ z 1.0) (exp -7.5))))
      (*
       t_2
       (+
        (+
         (-
          (- t_0 (/ 771.3234287776531 (- (+ z -1.0) 2.0)))
          (/ -1259.1392167224028 (+ -1.0 (+ z -1.0))))
         (/ -176.6150291621406 (+ (- 1.0 z) 3.0)))
        (+ 2.4783749183520145 (* z 0.49644474017195733)))))
     (*
      (*
       t_2
       (*
        t_1
        (*
         (pow (+ (- -1.0 (+ z -1.0)) 7.5) (- (- 1.0 z) 0.5))
         (exp (+ -0.5 (+ -6.0 (+ z -1.0)))))))
      (+
       (+
        (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0))
        (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0)))
       (-
        (-
         (+ (/ -1259.1392167224028 (+ 1.0 (- 1.0 z))) t_0)
         (+
          (/ -176.6150291621406 (+ -3.0 (+ z -1.0)))
          (/ 771.3234287776531 (+ -2.0 (+ z -1.0)))))
        (+
         (/ 12.507343278686905 (+ -4.0 (+ z -1.0)))
         (/ -0.13857109526572012 (+ -5.0 (+ z -1.0))))))))))

C

double code(double z) {
	double t_0 = 0.9999999999998099 - (676.5203681218851 / (z + -1.0));
	double t_1 = sqrt((((double) M_PI) * 2.0));
	double t_2 = ((double) M_PI) / sin((((double) M_PI) * z));
	double tmp;
	if (z <= -2000.0) {
		tmp = (t_1 * (pow((6.5 - (z + -1.0)), (-0.5 - (z + -1.0))) * ((z + 1.0) * exp(-7.5)))) * (t_2 * ((((t_0 - (771.3234287776531 / ((z + -1.0) - 2.0))) - (-1259.1392167224028 / (-1.0 + (z + -1.0)))) + (-176.6150291621406 / ((1.0 - z) + 3.0))) + (2.4783749183520145 + (z * 0.49644474017195733))));
	} else {
		tmp = (t_2 * (t_1 * (pow(((-1.0 - (z + -1.0)) + 7.5), ((1.0 - z) - 0.5)) * exp((-0.5 + (-6.0 + (z + -1.0))))))) * (((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))) + ((((-1259.1392167224028 / (1.0 + (1.0 - z))) + t_0) - ((-176.6150291621406 / (-3.0 + (z + -1.0))) + (771.3234287776531 / (-2.0 + (z + -1.0))))) - ((12.507343278686905 / (-4.0 + (z + -1.0))) + (-0.13857109526572012 / (-5.0 + (z + -1.0))))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = 0.9999999999998099 - (676.5203681218851 / (z + -1.0));
	double t_1 = Math.sqrt((Math.PI * 2.0));
	double t_2 = Math.PI / Math.sin((Math.PI * z));
	double tmp;
	if (z <= -2000.0) {
		tmp = (t_1 * (Math.pow((6.5 - (z + -1.0)), (-0.5 - (z + -1.0))) * ((z + 1.0) * Math.exp(-7.5)))) * (t_2 * ((((t_0 - (771.3234287776531 / ((z + -1.0) - 2.0))) - (-1259.1392167224028 / (-1.0 + (z + -1.0)))) + (-176.6150291621406 / ((1.0 - z) + 3.0))) + (2.4783749183520145 + (z * 0.49644474017195733))));
	} else {
		tmp = (t_2 * (t_1 * (Math.pow(((-1.0 - (z + -1.0)) + 7.5), ((1.0 - z) - 0.5)) * Math.exp((-0.5 + (-6.0 + (z + -1.0))))))) * (((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))) + ((((-1259.1392167224028 / (1.0 + (1.0 - z))) + t_0) - ((-176.6150291621406 / (-3.0 + (z + -1.0))) + (771.3234287776531 / (-2.0 + (z + -1.0))))) - ((12.507343278686905 / (-4.0 + (z + -1.0))) + (-0.13857109526572012 / (-5.0 + (z + -1.0))))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = 0.9999999999998099 - (676.5203681218851 / (z + -1.0))
	t_1 = math.sqrt((math.pi * 2.0))
	t_2 = math.pi / math.sin((math.pi * z))
	tmp = 0
	if z <= -2000.0:
		tmp = (t_1 * (math.pow((6.5 - (z + -1.0)), (-0.5 - (z + -1.0))) * ((z + 1.0) * math.exp(-7.5)))) * (t_2 * ((((t_0 - (771.3234287776531 / ((z + -1.0) - 2.0))) - (-1259.1392167224028 / (-1.0 + (z + -1.0)))) + (-176.6150291621406 / ((1.0 - z) + 3.0))) + (2.4783749183520145 + (z * 0.49644474017195733))))
	else:
		tmp = (t_2 * (t_1 * (math.pow(((-1.0 - (z + -1.0)) + 7.5), ((1.0 - z) - 0.5)) * math.exp((-0.5 + (-6.0 + (z + -1.0))))))) * (((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))) + ((((-1259.1392167224028 / (1.0 + (1.0 - z))) + t_0) - ((-176.6150291621406 / (-3.0 + (z + -1.0))) + (771.3234287776531 / (-2.0 + (z + -1.0))))) - ((12.507343278686905 / (-4.0 + (z + -1.0))) + (-0.13857109526572012 / (-5.0 + (z + -1.0))))))
	return tmp

Julia

function code(z)
	t_0 = Float64(0.9999999999998099 - Float64(676.5203681218851 / Float64(z + -1.0)))
	t_1 = sqrt(Float64(pi * 2.0))
	t_2 = Float64(pi / sin(Float64(pi * z)))
	tmp = 0.0
	if (z <= -2000.0)
		tmp = Float64(Float64(t_1 * Float64((Float64(6.5 - Float64(z + -1.0)) ^ Float64(-0.5 - Float64(z + -1.0))) * Float64(Float64(z + 1.0) * exp(-7.5)))) * Float64(t_2 * Float64(Float64(Float64(Float64(t_0 - Float64(771.3234287776531 / Float64(Float64(z + -1.0) - 2.0))) - Float64(-1259.1392167224028 / Float64(-1.0 + Float64(z + -1.0)))) + Float64(-176.6150291621406 / Float64(Float64(1.0 - z) + 3.0))) + Float64(2.4783749183520145 + Float64(z * 0.49644474017195733)))));
	else
		tmp = Float64(Float64(t_2 * Float64(t_1 * Float64((Float64(Float64(-1.0 - Float64(z + -1.0)) + 7.5) ^ Float64(Float64(1.0 - z) - 0.5)) * exp(Float64(-0.5 + Float64(-6.0 + Float64(z + -1.0))))))) * Float64(Float64(Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0)) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0))) + Float64(Float64(Float64(Float64(-1259.1392167224028 / Float64(1.0 + Float64(1.0 - z))) + t_0) - Float64(Float64(-176.6150291621406 / Float64(-3.0 + Float64(z + -1.0))) + Float64(771.3234287776531 / Float64(-2.0 + Float64(z + -1.0))))) - Float64(Float64(12.507343278686905 / Float64(-4.0 + Float64(z + -1.0))) + Float64(-0.13857109526572012 / Float64(-5.0 + Float64(z + -1.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = 0.9999999999998099 - (676.5203681218851 / (z + -1.0));
	t_1 = sqrt((pi * 2.0));
	t_2 = pi / sin((pi * z));
	tmp = 0.0;
	if (z <= -2000.0)
		tmp = (t_1 * (((6.5 - (z + -1.0)) ^ (-0.5 - (z + -1.0))) * ((z + 1.0) * exp(-7.5)))) * (t_2 * ((((t_0 - (771.3234287776531 / ((z + -1.0) - 2.0))) - (-1259.1392167224028 / (-1.0 + (z + -1.0)))) + (-176.6150291621406 / ((1.0 - z) + 3.0))) + (2.4783749183520145 + (z * 0.49644474017195733))));
	else
		tmp = (t_2 * (t_1 * ((((-1.0 - (z + -1.0)) + 7.5) ^ ((1.0 - z) - 0.5)) * exp((-0.5 + (-6.0 + (z + -1.0))))))) * (((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))) + ((((-1259.1392167224028 / (1.0 + (1.0 - z))) + t_0) - ((-176.6150291621406 / (-3.0 + (z + -1.0))) + (771.3234287776531 / (-2.0 + (z + -1.0))))) - ((12.507343278686905 / (-4.0 + (z + -1.0))) + (-0.13857109526572012 / (-5.0 + (z + -1.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[(0.9999999999998099 - N[(676.5203681218851 / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2000.0], N[(N[(t$95$1 * N[(N[Power[N[(6.5 - N[(z + -1.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 - N[(z + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(z + 1.0), $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(N[(N[(N[(t$95$0 - N[(771.3234287776531 / N[(N[(z + -1.0), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1259.1392167224028 / N[(-1.0 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.4783749183520145 + N[(z * 0.49644474017195733), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[(t$95$1 * N[(N[Power[N[(N[(-1.0 - N[(z + -1.0), $MachinePrecision]), $MachinePrecision] + 7.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-0.5 + N[(-6.0 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(-1259.1392167224028 / N[(1.0 + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] - N[(N[(-176.6150291621406 / N[(-3.0 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(-2.0 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(12.507343278686905 / N[(-4.0 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(-5.0 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if z < -2e3

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in z around 0 0.0%

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

      1. *-commutative 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in z around 0 100.0%

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

      1. distribute-rgt1-in 100.0%

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

   10. Simplified 100.0%

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

   if -2e3 < z

   1. Initial program 97.3%

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

   3. Simplified 98.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2000:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(6.5 - \left(z + -1\right)\right)}^{\left(-0.5 - \left(z + -1\right)\right)} \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{z + -1}\right) - \frac{771.3234287776531}{\left(z + -1\right) - 2}\right) - \frac{-1259.1392167224028}{-1 + \left(z + -1\right)}\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(-1 - \left(z + -1\right)\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{-0.5 + \left(-6 + \left(z + -1\right)\right)}\right)\right)\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) + \left(\left(\left(\frac{-1259.1392167224028}{1 + \left(1 - z\right)} + \left(0.9999999999998099 - \frac{676.5203681218851}{z + -1}\right)\right) - \left(\frac{-176.6150291621406}{-3 + \left(z + -1\right)} + \frac{771.3234287776531}{-2 + \left(z + -1\right)}\right)\right) - \left(\frac{12.507343278686905}{-4 + \left(z + -1\right)} + \frac{-0.13857109526572012}{-5 + \left(z + -1\right)}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{\mathsf{fma}\left(-0.5 - \left(z + -1\right), \mathsf{log1p}\left(6.5 - z\right), z + -7.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{1 + \left(1 - z\right)} + \left(0.9999999999998099 - \frac{676.5203681218851}{z + -1}\right)\right) - \left(\frac{-176.6150291621406}{-3 + \left(z + -1\right)} + \frac{771.3234287776531}{-2 + \left(z + -1\right)}\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (/ PI (sin (* PI z)))
   (*
    (sqrt (* PI 2.0))
    (exp (fma (- -0.5 (+ z -1.0)) (log1p (- 6.5 z)) (+ z -7.5)))))
  (+
   (+
    (-
     (+
      (/ -1259.1392167224028 (+ 1.0 (- 1.0 z)))
      (- 0.9999999999998099 (/ 676.5203681218851 (+ z -1.0))))
     (+
      (/ -176.6150291621406 (+ -3.0 (+ z -1.0)))
      (/ 771.3234287776531 (+ -2.0 (+ z -1.0)))))
    (+ (/ 12.507343278686905 (- 5.0 z)) (/ -0.13857109526572012 (- 6.0 z))))
   (+
    (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0))
    (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0))))))

C

double code(double z) {
	return ((((double) M_PI) / sin((((double) M_PI) * z))) * (sqrt((((double) M_PI) * 2.0)) * exp(fma((-0.5 - (z + -1.0)), log1p((6.5 - z)), (z + -7.5))))) * (((((-1259.1392167224028 / (1.0 + (1.0 - z))) + (0.9999999999998099 - (676.5203681218851 / (z + -1.0)))) - ((-176.6150291621406 / (-3.0 + (z + -1.0))) + (771.3234287776531 / (-2.0 + (z + -1.0))))) + ((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z)))) + ((9.984369578019572e-6 / ((1.0 - z) - -6.0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0))));
}

Julia

function code(z)
	return Float64(Float64(Float64(pi / sin(Float64(pi * z))) * Float64(sqrt(Float64(pi * 2.0)) * exp(fma(Float64(-0.5 - Float64(z + -1.0)), log1p(Float64(6.5 - z)), Float64(z + -7.5))))) * Float64(Float64(Float64(Float64(Float64(-1259.1392167224028 / Float64(1.0 + Float64(1.0 - z))) + Float64(0.9999999999998099 - Float64(676.5203681218851 / Float64(z + -1.0)))) - Float64(Float64(-176.6150291621406 / Float64(-3.0 + Float64(z + -1.0))) + Float64(771.3234287776531 / Float64(-2.0 + Float64(z + -1.0))))) + Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(-0.13857109526572012 / Float64(6.0 - z)))) + Float64(Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0)) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0)))))
end

Wolfram

code[z_] := N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(-0.5 - N[(z + -1.0), $MachinePrecision]), $MachinePrecision] * N[Log[1 + N[(6.5 - z), $MachinePrecision]], $MachinePrecision] + N[(z + -7.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(-1259.1392167224028 / N[(1.0 + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 - N[(676.5203681218851 / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(-176.6150291621406 / N[(-3.0 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(-2.0 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.4%

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

3. Simplified 97.0%

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

   1. add-cube-cbrt 97.0%

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

6. Applied egg-rr 97.0%

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

   1. expm1-log1p-u 97.0%

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

   2. *-commutative 97.0%

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

   3. associate--l- 97.0%

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

8. Applied egg-rr 97.0%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi \cdot 2} \cdot \left(e^{\left(-\left(1 - \left(z + -6\right)\right)\right) + -0.5} \cdot {\left(1 - \left(z - 6.5\right)\right)}^{\left(1 - \left(z - -0.5\right)\right)}\right)\right)\right)}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
9. Step-by-step derivation

   1. *-commutative 97.0%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\color{blue}{2 \cdot \pi}} \cdot \left(e^{\left(-\left(1 - \left(z + -6\right)\right)\right) + -0.5} \cdot {\left(1 - \left(z - 6.5\right)\right)}^{\left(1 - \left(z - -0.5\right)\right)}\right)\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   2. +-commutative 97.0%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \pi} \cdot \left(e^{\color{blue}{-0.5 + \left(-\left(1 - \left(z + -6\right)\right)\right)}} \cdot {\left(1 - \left(z - 6.5\right)\right)}^{\left(1 - \left(z - -0.5\right)\right)}\right)\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   3. unsub-neg 97.0%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \pi} \cdot \left(e^{\color{blue}{-0.5 - \left(1 - \left(z + -6\right)\right)}} \cdot {\left(1 - \left(z - 6.5\right)\right)}^{\left(1 - \left(z - -0.5\right)\right)}\right)\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   4. associate--r+ 97.0%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \pi} \cdot \left(e^{-0.5 - \color{blue}{\left(\left(1 - z\right) - -6\right)}} \cdot {\left(1 - \left(z - 6.5\right)\right)}^{\left(1 - \left(z - -0.5\right)\right)}\right)\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   5. sub-neg 97.0%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \pi} \cdot \left(e^{-0.5 - \color{blue}{\left(\left(1 - z\right) + \left(--6\right)\right)}} \cdot {\left(1 - \left(z - 6.5\right)\right)}^{\left(1 - \left(z - -0.5\right)\right)}\right)\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   6. metadata-eval 97.0%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \pi} \cdot \left(e^{-0.5 - \left(\left(1 - z\right) + \color{blue}{6}\right)} \cdot {\left(1 - \left(z - 6.5\right)\right)}^{\left(1 - \left(z - -0.5\right)\right)}\right)\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   7. associate--r- 97.0%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \pi} \cdot \left(e^{-0.5 - \left(\left(1 - z\right) + 6\right)} \cdot {\color{blue}{\left(\left(1 - z\right) + 6.5\right)}}^{\left(1 - \left(z - -0.5\right)\right)}\right)\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   8. associate--r- 97.0%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \pi} \cdot \left(e^{-0.5 - \left(\left(1 - z\right) + 6\right)} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\color{blue}{\left(\left(1 - z\right) + -0.5\right)}}\right)\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

10. Simplified 97.0%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \pi} \cdot \left(e^{-0.5 - \left(\left(1 - z\right) + 6\right)} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right)\right)\right)}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
11. Step-by-step derivation

    1. *-un-lft-identity 97.0%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \pi} \cdot \left(e^{-0.5 - \left(\left(1 - z\right) + 6\right)} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right)\right)\right)\right)}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    2. expm1-log1p-u 97.0%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(1 \cdot \color{blue}{\left(\sqrt{2 \cdot \pi} \cdot \left(e^{-0.5 - \left(\left(1 - z\right) + 6\right)} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right)\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    3. *-commutative 97.0%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(1 \cdot \left(\sqrt{\color{blue}{\pi \cdot 2}} \cdot \left(e^{-0.5 - \left(\left(1 - z\right) + 6\right)} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right)\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    4. *-commutative 97.0%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(1 \cdot \left(\sqrt{\pi \cdot 2} \cdot \color{blue}{\left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-0.5 - \left(\left(1 - z\right) + 6\right)}\right)}\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    5. associate-+l- 97.0%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(1 \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\color{blue}{\left(1 - \left(z - 6.5\right)\right)}}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-0.5 - \left(\left(1 - z\right) + 6\right)}\right)\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    6. metadata-eval 97.0%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(1 \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(1 - \left(z - 6.5\right)\right)}^{\left(\left(1 - z\right) + \color{blue}{\left(-0.5\right)}\right)} \cdot e^{-0.5 - \left(\left(1 - z\right) + 6\right)}\right)\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    7. sub-neg 97.0%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(1 \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(1 - \left(z - 6.5\right)\right)}^{\color{blue}{\left(\left(1 - z\right) - 0.5\right)}} \cdot e^{-0.5 - \left(\left(1 - z\right) + 6\right)}\right)\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    8. associate--l- 97.0%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(1 \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(1 - \left(z - 6.5\right)\right)}^{\color{blue}{\left(1 - \left(z + 0.5\right)\right)}} \cdot e^{-0.5 - \left(\left(1 - z\right) + 6\right)}\right)\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    9. metadata-eval 97.0%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(1 \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(1 - \left(z - 6.5\right)\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot e^{-0.5 - \left(\left(1 - z\right) + \color{blue}{\left(--6\right)}\right)}\right)\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    10. sub-neg 97.0%

        \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(1 \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(1 - \left(z - 6.5\right)\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot e^{-0.5 - \color{blue}{\left(\left(1 - z\right) - -6\right)}}\right)\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    11. associate--l- 97.0%

        \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(1 \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(1 - \left(z - 6.5\right)\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot e^{-0.5 - \color{blue}{\left(1 - \left(z + -6\right)\right)}}\right)\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

12. Applied egg-rr 97.0%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(1 \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(1 - \left(z - 6.5\right)\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot e^{-0.5 - \left(1 - \left(z + -6\right)\right)}\right)\right)\right)}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
13. Step-by-step derivation

    1. *-lft-identity 97.0%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(1 - \left(z - 6.5\right)\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot e^{-0.5 - \left(1 - \left(z + -6\right)\right)}\right)\right)}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    2. *-commutative 97.0%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\color{blue}{2 \cdot \pi}} \cdot \left({\left(1 - \left(z - 6.5\right)\right)}^{\left(1 - \left(z + 0.5\right)\right)} \cdot e^{-0.5 - \left(1 - \left(z + -6\right)\right)}\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    3. exp-to-pow 97.0%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot \left(\color{blue}{e^{\log \left(1 - \left(z - 6.5\right)\right) \cdot \left(1 - \left(z + 0.5\right)\right)}} \cdot e^{-0.5 - \left(1 - \left(z + -6\right)\right)}\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    4. *-commutative 97.0%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot \left(e^{\color{blue}{\left(1 - \left(z + 0.5\right)\right) \cdot \log \left(1 - \left(z - 6.5\right)\right)}} \cdot e^{-0.5 - \left(1 - \left(z + -6\right)\right)}\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    5. exp-sum 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot \color{blue}{e^{\left(1 - \left(z + 0.5\right)\right) \cdot \log \left(1 - \left(z - 6.5\right)\right) + \left(-0.5 - \left(1 - \left(z + -6\right)\right)\right)}}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    6. fma-define 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\color{blue}{\mathsf{fma}\left(1 - \left(z + 0.5\right), \log \left(1 - \left(z - 6.5\right)\right), -0.5 - \left(1 - \left(z + -6\right)\right)\right)}}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

14. Simplified 98.2%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(\sqrt{2 \cdot \pi} \cdot e^{\mathsf{fma}\left(\left(1 - z\right) + -0.5, \mathsf{log1p}\left(\left(-z\right) + 6.5\right), -7.5 + z\right)}\right)}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
15. Step-by-step derivation

    1. div-inv 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\mathsf{fma}\left(\left(1 - z\right) + -0.5, \mathsf{log1p}\left(\left(-z\right) + 6.5\right), -7.5 + z\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\color{blue}{12.507343278686905 \cdot \frac{1}{\left(1 - z\right) - -4}} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    2. sub-neg 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\mathsf{fma}\left(\left(1 - z\right) + -0.5, \mathsf{log1p}\left(\left(-z\right) + 6.5\right), -7.5 + z\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(12.507343278686905 \cdot \frac{1}{\color{blue}{\left(1 - z\right) + \left(--4\right)}} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    3. metadata-eval 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\mathsf{fma}\left(\left(1 - z\right) + -0.5, \mathsf{log1p}\left(\left(-z\right) + 6.5\right), -7.5 + z\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(12.507343278686905 \cdot \frac{1}{\left(1 - z\right) + \color{blue}{4}} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    4. div-inv 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\mathsf{fma}\left(\left(1 - z\right) + -0.5, \mathsf{log1p}\left(\left(-z\right) + 6.5\right), -7.5 + z\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\color{blue}{\frac{12.507343278686905}{\left(1 - z\right) + 4}} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    5. sub-neg 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\mathsf{fma}\left(\left(1 - z\right) + -0.5, \mathsf{log1p}\left(\left(-z\right) + 6.5\right), -7.5 + z\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\color{blue}{\left(1 - z\right) + \left(--5\right)}}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    6. metadata-eval 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\mathsf{fma}\left(\left(1 - z\right) + -0.5, \mathsf{log1p}\left(\left(-z\right) + 6.5\right), -7.5 + z\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + \color{blue}{5}}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    7. *-un-lft-identity 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\mathsf{fma}\left(\left(1 - z\right) + -0.5, \mathsf{log1p}\left(\left(-z\right) + 6.5\right), -7.5 + z\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \color{blue}{1 \cdot \left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    8. metadata-eval 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\mathsf{fma}\left(\left(1 - z\right) + -0.5, \mathsf{log1p}\left(\left(-z\right) + 6.5\right), -7.5 + z\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + 1 \cdot \left(\frac{12.507343278686905}{\left(1 - z\right) + \color{blue}{\left(--4\right)}} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    9. sub-neg 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\mathsf{fma}\left(\left(1 - z\right) + -0.5, \mathsf{log1p}\left(\left(-z\right) + 6.5\right), -7.5 + z\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + 1 \cdot \left(\frac{12.507343278686905}{\color{blue}{\left(1 - z\right) - -4}} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    10. associate--l- 98.2%

        \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\mathsf{fma}\left(\left(1 - z\right) + -0.5, \mathsf{log1p}\left(\left(-z\right) + 6.5\right), -7.5 + z\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + 1 \cdot \left(\frac{12.507343278686905}{\color{blue}{1 - \left(z + -4\right)}} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    11. metadata-eval 98.2%

        \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\mathsf{fma}\left(\left(1 - z\right) + -0.5, \mathsf{log1p}\left(\left(-z\right) + 6.5\right), -7.5 + z\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + 1 \cdot \left(\frac{12.507343278686905}{1 - \left(z + -4\right)} + \frac{-0.13857109526572012}{\left(1 - z\right) + \color{blue}{\left(--5\right)}}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    12. sub-neg 98.2%

        \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\mathsf{fma}\left(\left(1 - z\right) + -0.5, \mathsf{log1p}\left(\left(-z\right) + 6.5\right), -7.5 + z\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + 1 \cdot \left(\frac{12.507343278686905}{1 - \left(z + -4\right)} + \frac{-0.13857109526572012}{\color{blue}{\left(1 - z\right) - -5}}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    13. associate--l- 98.2%

        \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\mathsf{fma}\left(\left(1 - z\right) + -0.5, \mathsf{log1p}\left(\left(-z\right) + 6.5\right), -7.5 + z\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + 1 \cdot \left(\frac{12.507343278686905}{1 - \left(z + -4\right)} + \frac{-0.13857109526572012}{\color{blue}{1 - \left(z + -5\right)}}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

16. Applied egg-rr 98.2%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\mathsf{fma}\left(\left(1 - z\right) + -0.5, \mathsf{log1p}\left(\left(-z\right) + 6.5\right), -7.5 + z\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \color{blue}{1 \cdot \left(\frac{12.507343278686905}{1 - \left(z + -4\right)} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
17. Step-by-step derivation

    1. *-lft-identity 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\mathsf{fma}\left(\left(1 - z\right) + -0.5, \mathsf{log1p}\left(\left(-z\right) + 6.5\right), -7.5 + z\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \color{blue}{\left(\frac{12.507343278686905}{1 - \left(z + -4\right)} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    2. +-commutative 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\mathsf{fma}\left(\left(1 - z\right) + -0.5, \mathsf{log1p}\left(\left(-z\right) + 6.5\right), -7.5 + z\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{1 - \color{blue}{\left(-4 + z\right)}} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    3. associate--r+ 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\mathsf{fma}\left(\left(1 - z\right) + -0.5, \mathsf{log1p}\left(\left(-z\right) + 6.5\right), -7.5 + z\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\color{blue}{\left(1 - -4\right) - z}} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    4. metadata-eval 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\mathsf{fma}\left(\left(1 - z\right) + -0.5, \mathsf{log1p}\left(\left(-z\right) + 6.5\right), -7.5 + z\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\color{blue}{5} - z} + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    5. +-commutative 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\mathsf{fma}\left(\left(1 - z\right) + -0.5, \mathsf{log1p}\left(\left(-z\right) + 6.5\right), -7.5 + z\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{1 - \color{blue}{\left(-5 + z\right)}}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    6. associate--r+ 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\mathsf{fma}\left(\left(1 - z\right) + -0.5, \mathsf{log1p}\left(\left(-z\right) + 6.5\right), -7.5 + z\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{\color{blue}{\left(1 - -5\right) - z}}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

    7. metadata-eval 98.2%

       \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\mathsf{fma}\left(\left(1 - z\right) + -0.5, \mathsf{log1p}\left(\left(-z\right) + 6.5\right), -7.5 + z\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{\color{blue}{6} - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

18. Simplified 98.2%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{\mathsf{fma}\left(\left(1 - z\right) + -0.5, \mathsf{log1p}\left(\left(-z\right) + 6.5\right), -7.5 + z\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \color{blue}{\left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

19. Final simplification 98.2%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{\mathsf{fma}\left(-0.5 - \left(z + -1\right), \mathsf{log1p}\left(6.5 - z\right), z + -7.5\right)}\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{1 + \left(1 - z\right)} + \left(0.9999999999998099 - \frac{676.5203681218851}{z + -1}\right)\right) - \left(\frac{-176.6150291621406}{-3 + \left(z + -1\right)} + \frac{771.3234287776531}{-2 + \left(z + -1\right)}\right)\right) + \left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
20. Add Preprocessing

Alternative 3: 97.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(e^{\mathsf{fma}\left(-0.5 - \left(z + -1\right), \mathsf{log1p}\left(6.5 - z\right), z + -7.5\right)} \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{\pi \cdot 2}\right)\right) \cdot \left(\left(0.9999999999998099 - \left(\frac{-1259.1392167224028}{z + -2} + \frac{676.5203681218851}{z + -1}\right)\right) + \left(\left(\frac{-0.13857109526572012}{5 - \left(z + -1\right)} - \left(\frac{12.507343278686905}{z - 5} + \left(\frac{771.3234287776531}{\left(z + -1\right) - 2} + \frac{-176.6150291621406}{z - 4}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{6 - \left(z + -1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{7 - \left(z + -1\right)}\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (exp (fma (- -0.5 (+ z -1.0)) (log1p (- 6.5 z)) (+ z -7.5)))
   (* (/ PI (sin (* PI z))) (sqrt (* PI 2.0))))
  (+
   (-
    0.9999999999998099
    (+ (/ -1259.1392167224028 (+ z -2.0)) (/ 676.5203681218851 (+ z -1.0))))
   (+
    (-
     (/ -0.13857109526572012 (- 5.0 (+ z -1.0)))
     (+
      (/ 12.507343278686905 (- z 5.0))
      (+
       (/ 771.3234287776531 (- (+ z -1.0) 2.0))
       (/ -176.6150291621406 (- z 4.0)))))
    (+
     (/ 9.984369578019572e-6 (- 6.0 (+ z -1.0)))
     (/ 1.5056327351493116e-7 (- 7.0 (+ z -1.0))))))))

C

double code(double z) {
	return (exp(fma((-0.5 - (z + -1.0)), log1p((6.5 - z)), (z + -7.5))) * ((((double) M_PI) / sin((((double) M_PI) * z))) * sqrt((((double) M_PI) * 2.0)))) * ((0.9999999999998099 - ((-1259.1392167224028 / (z + -2.0)) + (676.5203681218851 / (z + -1.0)))) + (((-0.13857109526572012 / (5.0 - (z + -1.0))) - ((12.507343278686905 / (z - 5.0)) + ((771.3234287776531 / ((z + -1.0) - 2.0)) + (-176.6150291621406 / (z - 4.0))))) + ((9.984369578019572e-6 / (6.0 - (z + -1.0))) + (1.5056327351493116e-7 / (7.0 - (z + -1.0))))));
}

Julia

function code(z)
	return Float64(Float64(exp(fma(Float64(-0.5 - Float64(z + -1.0)), log1p(Float64(6.5 - z)), Float64(z + -7.5))) * Float64(Float64(pi / sin(Float64(pi * z))) * sqrt(Float64(pi * 2.0)))) * Float64(Float64(0.9999999999998099 - Float64(Float64(-1259.1392167224028 / Float64(z + -2.0)) + Float64(676.5203681218851 / Float64(z + -1.0)))) + Float64(Float64(Float64(-0.13857109526572012 / Float64(5.0 - Float64(z + -1.0))) - Float64(Float64(12.507343278686905 / Float64(z - 5.0)) + Float64(Float64(771.3234287776531 / Float64(Float64(z + -1.0) - 2.0)) + Float64(-176.6150291621406 / Float64(z - 4.0))))) + Float64(Float64(9.984369578019572e-6 / Float64(6.0 - Float64(z + -1.0))) + Float64(1.5056327351493116e-7 / Float64(7.0 - Float64(z + -1.0)))))))
end

Wolfram

code[z_] := N[(N[(N[Exp[N[(N[(-0.5 - N[(z + -1.0), $MachinePrecision]), $MachinePrecision] * N[Log[1 + N[(6.5 - z), $MachinePrecision]], $MachinePrecision] + N[(z + -7.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.9999999999998099 - N[(N[(-1259.1392167224028 / N[(z + -2.0), $MachinePrecision]), $MachinePrecision] + N[(676.5203681218851 / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.13857109526572012 / N[(5.0 - N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(12.507343278686905 / N[(z - 5.0), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(N[(z + -1.0), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(z - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(6.0 - N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(7.0 - N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.4%

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

3. Simplified 97.0%

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

   1. add-cube-cbrt 97.0%

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

6. Applied egg-rr 97.0%

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

   1. expm1-log1p-u 97.0%

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

   2. *-commutative 97.0%

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

   3. associate--l- 97.0%

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

8. Applied egg-rr 97.0%

   \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi \cdot 2} \cdot \left(e^{\left(-\left(1 - \left(z + -6\right)\right)\right) + -0.5} \cdot {\left(1 - \left(z - 6.5\right)\right)}^{\left(1 - \left(z - -0.5\right)\right)}\right)\right)\right)}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
9. Step-by-step derivation

   1. *-commutative 97.0%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\color{blue}{2 \cdot \pi}} \cdot \left(e^{\left(-\left(1 - \left(z + -6\right)\right)\right) + -0.5} \cdot {\left(1 - \left(z - 6.5\right)\right)}^{\left(1 - \left(z - -0.5\right)\right)}\right)\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   2. +-commutative 97.0%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \pi} \cdot \left(e^{\color{blue}{-0.5 + \left(-\left(1 - \left(z + -6\right)\right)\right)}} \cdot {\left(1 - \left(z - 6.5\right)\right)}^{\left(1 - \left(z - -0.5\right)\right)}\right)\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   3. unsub-neg 97.0%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \pi} \cdot \left(e^{\color{blue}{-0.5 - \left(1 - \left(z + -6\right)\right)}} \cdot {\left(1 - \left(z - 6.5\right)\right)}^{\left(1 - \left(z - -0.5\right)\right)}\right)\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   4. associate--r+ 97.0%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \pi} \cdot \left(e^{-0.5 - \color{blue}{\left(\left(1 - z\right) - -6\right)}} \cdot {\left(1 - \left(z - 6.5\right)\right)}^{\left(1 - \left(z - -0.5\right)\right)}\right)\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   5. sub-neg 97.0%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \pi} \cdot \left(e^{-0.5 - \color{blue}{\left(\left(1 - z\right) + \left(--6\right)\right)}} \cdot {\left(1 - \left(z - 6.5\right)\right)}^{\left(1 - \left(z - -0.5\right)\right)}\right)\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   6. metadata-eval 97.0%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \pi} \cdot \left(e^{-0.5 - \left(\left(1 - z\right) + \color{blue}{6}\right)} \cdot {\left(1 - \left(z - 6.5\right)\right)}^{\left(1 - \left(z - -0.5\right)\right)}\right)\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   7. associate--r- 97.0%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \pi} \cdot \left(e^{-0.5 - \left(\left(1 - z\right) + 6\right)} \cdot {\color{blue}{\left(\left(1 - z\right) + 6.5\right)}}^{\left(1 - \left(z - -0.5\right)\right)}\right)\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   8. associate--r- 97.0%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \pi} \cdot \left(e^{-0.5 - \left(\left(1 - z\right) + 6\right)} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\color{blue}{\left(\left(1 - z\right) + -0.5\right)}}\right)\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

10. Simplified 97.0%

    \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \pi} \cdot \left(e^{-0.5 - \left(\left(1 - z\right) + 6\right)} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right)\right)\right)}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

12. Applied egg-rr 97.2%

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

14. Simplified 98.0%

    \[\leadsto \color{blue}{\left(e^{\mathsf{fma}\left(\left(1 - z\right) + -0.5, \mathsf{log1p}\left(\left(-z\right) + 6.5\right), -7.5 + z\right)} \cdot \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \sqrt{2 \cdot \pi}\right)\right) \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{-\left(z + -2\right)}\right)\right) + \left(\left(\left(\left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{4 - z}\right) + \frac{12.507343278686905}{\left(-z\right) + 5}\right) + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right)} \]

15. Final simplification 98.0%

    \[\leadsto \left(e^{\mathsf{fma}\left(-0.5 - \left(z + -1\right), \mathsf{log1p}\left(6.5 - z\right), z + -7.5\right)} \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{\pi \cdot 2}\right)\right) \cdot \left(\left(0.9999999999998099 - \left(\frac{-1259.1392167224028}{z + -2} + \frac{676.5203681218851}{z + -1}\right)\right) + \left(\left(\frac{-0.13857109526572012}{5 - \left(z + -1\right)} - \left(\frac{12.507343278686905}{z - 5} + \left(\frac{771.3234287776531}{\left(z + -1\right) - 2} + \frac{-176.6150291621406}{z - 4}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{6 - \left(z + -1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{7 - \left(z + -1\right)}\right)\right)\right) \]
16. Add Preprocessing

Alternative 4: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{676.5203681218851}{z + -1}\\ t_1 := \sqrt{\pi \cdot 2}\\ t_2 := \frac{\pi}{\sin \left(\pi \cdot z\right)}\\ \mathbf{if}\;z \leq -2000:\\ \;\;\;\;\left(t\_1 \cdot \left({\left(6.5 - \left(z + -1\right)\right)}^{\left(-0.5 - \left(z + -1\right)\right)} \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right)\right) \cdot \left(t\_2 \cdot \left(\left(\left(\left(\left(0.9999999999998099 - t\_0\right) - \frac{771.3234287776531}{\left(z + -1\right) - 2}\right) - \frac{-1259.1392167224028}{-1 + \left(z + -1\right)}\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\left(\left(\frac{-1259.1392167224028}{2 - z} - t\_0\right) + \left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-0.13857109526572012}{6 - z} + \frac{-176.6150291621406}{4 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \cdot \left(t\_2 \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (/ 676.5203681218851 (+ z -1.0)))
        (t_1 (sqrt (* PI 2.0)))
        (t_2 (/ PI (sin (* PI z)))))
   (if (<= z -2000.0)
     (*
      (*
       t_1
       (*
        (pow (- 6.5 (+ z -1.0)) (- -0.5 (+ z -1.0)))
        (* (+ z 1.0) (exp -7.5))))
      (*
       t_2
       (+
        (+
         (-
          (-
           (- 0.9999999999998099 t_0)
           (/ 771.3234287776531 (- (+ z -1.0) 2.0)))
          (/ -1259.1392167224028 (+ -1.0 (+ z -1.0))))
         (/ -176.6150291621406 (+ (- 1.0 z) 3.0)))
        (+ 2.4783749183520145 (* z 0.49644474017195733)))))
     (*
      t_1
      (*
       (+
        (- (/ -1259.1392167224028 (- 2.0 z)) t_0)
        (+
         0.9999999999998099
         (+
          (/ 771.3234287776531 (- 3.0 z))
          (+
           (+
            (/ 12.507343278686905 (- 5.0 z))
            (+
             (/ -0.13857109526572012 (- 6.0 z))
             (/ -176.6150291621406 (- 4.0 z))))
           (+
            (/ 9.984369578019572e-6 (- 7.0 z))
            (/ 1.5056327351493116e-7 (- 8.0 z)))))))
       (* t_2 (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5)))))))))

C

double code(double z) {
	double t_0 = 676.5203681218851 / (z + -1.0);
	double t_1 = sqrt((((double) M_PI) * 2.0));
	double t_2 = ((double) M_PI) / sin((((double) M_PI) * z));
	double tmp;
	if (z <= -2000.0) {
		tmp = (t_1 * (pow((6.5 - (z + -1.0)), (-0.5 - (z + -1.0))) * ((z + 1.0) * exp(-7.5)))) * (t_2 * (((((0.9999999999998099 - t_0) - (771.3234287776531 / ((z + -1.0) - 2.0))) - (-1259.1392167224028 / (-1.0 + (z + -1.0)))) + (-176.6150291621406 / ((1.0 - z) + 3.0))) + (2.4783749183520145 + (z * 0.49644474017195733))));
	} else {
		tmp = t_1 * ((((-1259.1392167224028 / (2.0 - z)) - t_0) + (0.9999999999998099 + ((771.3234287776531 / (3.0 - z)) + (((12.507343278686905 / (5.0 - z)) + ((-0.13857109526572012 / (6.0 - z)) + (-176.6150291621406 / (4.0 - z)))) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))))) * (t_2 * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5)))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = 676.5203681218851 / (z + -1.0);
	double t_1 = Math.sqrt((Math.PI * 2.0));
	double t_2 = Math.PI / Math.sin((Math.PI * z));
	double tmp;
	if (z <= -2000.0) {
		tmp = (t_1 * (Math.pow((6.5 - (z + -1.0)), (-0.5 - (z + -1.0))) * ((z + 1.0) * Math.exp(-7.5)))) * (t_2 * (((((0.9999999999998099 - t_0) - (771.3234287776531 / ((z + -1.0) - 2.0))) - (-1259.1392167224028 / (-1.0 + (z + -1.0)))) + (-176.6150291621406 / ((1.0 - z) + 3.0))) + (2.4783749183520145 + (z * 0.49644474017195733))));
	} else {
		tmp = t_1 * ((((-1259.1392167224028 / (2.0 - z)) - t_0) + (0.9999999999998099 + ((771.3234287776531 / (3.0 - z)) + (((12.507343278686905 / (5.0 - z)) + ((-0.13857109526572012 / (6.0 - z)) + (-176.6150291621406 / (4.0 - z)))) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))))) * (t_2 * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5)))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = 676.5203681218851 / (z + -1.0)
	t_1 = math.sqrt((math.pi * 2.0))
	t_2 = math.pi / math.sin((math.pi * z))
	tmp = 0
	if z <= -2000.0:
		tmp = (t_1 * (math.pow((6.5 - (z + -1.0)), (-0.5 - (z + -1.0))) * ((z + 1.0) * math.exp(-7.5)))) * (t_2 * (((((0.9999999999998099 - t_0) - (771.3234287776531 / ((z + -1.0) - 2.0))) - (-1259.1392167224028 / (-1.0 + (z + -1.0)))) + (-176.6150291621406 / ((1.0 - z) + 3.0))) + (2.4783749183520145 + (z * 0.49644474017195733))))
	else:
		tmp = t_1 * ((((-1259.1392167224028 / (2.0 - z)) - t_0) + (0.9999999999998099 + ((771.3234287776531 / (3.0 - z)) + (((12.507343278686905 / (5.0 - z)) + ((-0.13857109526572012 / (6.0 - z)) + (-176.6150291621406 / (4.0 - z)))) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))))) * (t_2 * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5)))))
	return tmp

Julia

function code(z)
	t_0 = Float64(676.5203681218851 / Float64(z + -1.0))
	t_1 = sqrt(Float64(pi * 2.0))
	t_2 = Float64(pi / sin(Float64(pi * z)))
	tmp = 0.0
	if (z <= -2000.0)
		tmp = Float64(Float64(t_1 * Float64((Float64(6.5 - Float64(z + -1.0)) ^ Float64(-0.5 - Float64(z + -1.0))) * Float64(Float64(z + 1.0) * exp(-7.5)))) * Float64(t_2 * Float64(Float64(Float64(Float64(Float64(0.9999999999998099 - t_0) - Float64(771.3234287776531 / Float64(Float64(z + -1.0) - 2.0))) - Float64(-1259.1392167224028 / Float64(-1.0 + Float64(z + -1.0)))) + Float64(-176.6150291621406 / Float64(Float64(1.0 - z) + 3.0))) + Float64(2.4783749183520145 + Float64(z * 0.49644474017195733)))));
	else
		tmp = Float64(t_1 * Float64(Float64(Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) - t_0) + Float64(0.9999999999998099 + Float64(Float64(771.3234287776531 / Float64(3.0 - z)) + Float64(Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(-176.6150291621406 / Float64(4.0 - z)))) + Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))))))) * Float64(t_2 * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = 676.5203681218851 / (z + -1.0);
	t_1 = sqrt((pi * 2.0));
	t_2 = pi / sin((pi * z));
	tmp = 0.0;
	if (z <= -2000.0)
		tmp = (t_1 * (((6.5 - (z + -1.0)) ^ (-0.5 - (z + -1.0))) * ((z + 1.0) * exp(-7.5)))) * (t_2 * (((((0.9999999999998099 - t_0) - (771.3234287776531 / ((z + -1.0) - 2.0))) - (-1259.1392167224028 / (-1.0 + (z + -1.0)))) + (-176.6150291621406 / ((1.0 - z) + 3.0))) + (2.4783749183520145 + (z * 0.49644474017195733))));
	else
		tmp = t_1 * ((((-1259.1392167224028 / (2.0 - z)) - t_0) + (0.9999999999998099 + ((771.3234287776531 / (3.0 - z)) + (((12.507343278686905 / (5.0 - z)) + ((-0.13857109526572012 / (6.0 - z)) + (-176.6150291621406 / (4.0 - z)))) + ((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))))))) * (t_2 * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[(676.5203681218851 / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2000.0], N[(N[(t$95$1 * N[(N[Power[N[(6.5 - N[(z + -1.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 - N[(z + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(z + 1.0), $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(N[(N[(N[(N[(0.9999999999998099 - t$95$0), $MachinePrecision] - N[(771.3234287776531 / N[(N[(z + -1.0), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1259.1392167224028 / N[(-1.0 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.4783749183520145 + N[(z * 0.49644474017195733), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] + N[(0.9999999999998099 + N[(N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if z < -2e3

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in z around 0 0.0%

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

      1. *-commutative 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in z around 0 100.0%

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

      1. distribute-rgt1-in 100.0%

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

   10. Simplified 100.0%

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

   if -2e3 < z

   1. Initial program 97.3%

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

   3. Simplified 97.3%

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

      1. pow1 97.3%

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

   6. Applied egg-rr 97.2%

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

   8. Simplified 97.5%

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

   10. Applied egg-rr 97.5%

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

   12. Simplified 98.7%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2000:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(6.5 - \left(z + -1\right)\right)}^{\left(-0.5 - \left(z + -1\right)\right)} \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{z + -1}\right) - \frac{771.3234287776531}{\left(z + -1\right) - 2}\right) - \frac{-1259.1392167224028}{-1 + \left(z + -1\right)}\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(\left(\left(\frac{-1259.1392167224028}{2 - z} - \frac{676.5203681218851}{z + -1}\right) + \left(0.9999999999998099 + \left(\frac{771.3234287776531}{3 - z} + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-0.13857109526572012}{6 - z} + \frac{-176.6150291621406}{4 - z}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{676.5203681218851}{z + -1}\\ t_1 := \sqrt{\pi \cdot 2}\\ t_2 := \frac{\pi}{\sin \left(\pi \cdot z\right)}\\ \mathbf{if}\;z \leq -2000:\\ \;\;\;\;\left(t\_1 \cdot \left({\left(6.5 - \left(z + -1\right)\right)}^{\left(-0.5 - \left(z + -1\right)\right)} \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right)\right) \cdot \left(t\_2 \cdot \left(\left(\left(\left(\left(0.9999999999998099 - t\_0\right) - \frac{771.3234287776531}{\left(z + -1\right) - 2}\right) - \frac{-1259.1392167224028}{-1 + \left(z + -1\right)}\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\left(t\_2 \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\left(\frac{-1259.1392167224028}{2 - z} - t\_0\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \frac{-176.6150291621406}{4 - z}\right)\right) - \left(\frac{771.3234287776531}{z - 3} - 0.9999999999998099\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (/ 676.5203681218851 (+ z -1.0)))
        (t_1 (sqrt (* PI 2.0)))
        (t_2 (/ PI (sin (* PI z)))))
   (if (<= z -2000.0)
     (*
      (*
       t_1
       (*
        (pow (- 6.5 (+ z -1.0)) (- -0.5 (+ z -1.0)))
        (* (+ z 1.0) (exp -7.5))))
      (*
       t_2
       (+
        (+
         (-
          (-
           (- 0.9999999999998099 t_0)
           (/ 771.3234287776531 (- (+ z -1.0) 2.0)))
          (/ -1259.1392167224028 (+ -1.0 (+ z -1.0))))
         (/ -176.6150291621406 (+ (- 1.0 z) 3.0)))
        (+ 2.4783749183520145 (* z 0.49644474017195733)))))
     (*
      t_1
      (*
       (* t_2 (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5))))
       (+
        (/ 9.984369578019572e-6 (- 7.0 z))
        (+
         (/ 1.5056327351493116e-7 (- 8.0 z))
         (+
          (- (/ -1259.1392167224028 (- 2.0 z)) t_0)
          (-
           (+
            (/ -0.13857109526572012 (- 6.0 z))
            (+
             (/ 12.507343278686905 (- 5.0 z))
             (/ -176.6150291621406 (- 4.0 z))))
           (- (/ 771.3234287776531 (- z 3.0)) 0.9999999999998099))))))))))

C

double code(double z) {
	double t_0 = 676.5203681218851 / (z + -1.0);
	double t_1 = sqrt((((double) M_PI) * 2.0));
	double t_2 = ((double) M_PI) / sin((((double) M_PI) * z));
	double tmp;
	if (z <= -2000.0) {
		tmp = (t_1 * (pow((6.5 - (z + -1.0)), (-0.5 - (z + -1.0))) * ((z + 1.0) * exp(-7.5)))) * (t_2 * (((((0.9999999999998099 - t_0) - (771.3234287776531 / ((z + -1.0) - 2.0))) - (-1259.1392167224028 / (-1.0 + (z + -1.0)))) + (-176.6150291621406 / ((1.0 - z) + 3.0))) + (2.4783749183520145 + (z * 0.49644474017195733))));
	} else {
		tmp = t_1 * ((t_2 * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5)))) * ((9.984369578019572e-6 / (7.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + (((-1259.1392167224028 / (2.0 - z)) - t_0) + (((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) + (-176.6150291621406 / (4.0 - z)))) - ((771.3234287776531 / (z - 3.0)) - 0.9999999999998099))))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = 676.5203681218851 / (z + -1.0);
	double t_1 = Math.sqrt((Math.PI * 2.0));
	double t_2 = Math.PI / Math.sin((Math.PI * z));
	double tmp;
	if (z <= -2000.0) {
		tmp = (t_1 * (Math.pow((6.5 - (z + -1.0)), (-0.5 - (z + -1.0))) * ((z + 1.0) * Math.exp(-7.5)))) * (t_2 * (((((0.9999999999998099 - t_0) - (771.3234287776531 / ((z + -1.0) - 2.0))) - (-1259.1392167224028 / (-1.0 + (z + -1.0)))) + (-176.6150291621406 / ((1.0 - z) + 3.0))) + (2.4783749183520145 + (z * 0.49644474017195733))));
	} else {
		tmp = t_1 * ((t_2 * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5)))) * ((9.984369578019572e-6 / (7.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + (((-1259.1392167224028 / (2.0 - z)) - t_0) + (((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) + (-176.6150291621406 / (4.0 - z)))) - ((771.3234287776531 / (z - 3.0)) - 0.9999999999998099))))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = 676.5203681218851 / (z + -1.0)
	t_1 = math.sqrt((math.pi * 2.0))
	t_2 = math.pi / math.sin((math.pi * z))
	tmp = 0
	if z <= -2000.0:
		tmp = (t_1 * (math.pow((6.5 - (z + -1.0)), (-0.5 - (z + -1.0))) * ((z + 1.0) * math.exp(-7.5)))) * (t_2 * (((((0.9999999999998099 - t_0) - (771.3234287776531 / ((z + -1.0) - 2.0))) - (-1259.1392167224028 / (-1.0 + (z + -1.0)))) + (-176.6150291621406 / ((1.0 - z) + 3.0))) + (2.4783749183520145 + (z * 0.49644474017195733))))
	else:
		tmp = t_1 * ((t_2 * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5)))) * ((9.984369578019572e-6 / (7.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + (((-1259.1392167224028 / (2.0 - z)) - t_0) + (((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) + (-176.6150291621406 / (4.0 - z)))) - ((771.3234287776531 / (z - 3.0)) - 0.9999999999998099))))))
	return tmp

Julia

function code(z)
	t_0 = Float64(676.5203681218851 / Float64(z + -1.0))
	t_1 = sqrt(Float64(pi * 2.0))
	t_2 = Float64(pi / sin(Float64(pi * z)))
	tmp = 0.0
	if (z <= -2000.0)
		tmp = Float64(Float64(t_1 * Float64((Float64(6.5 - Float64(z + -1.0)) ^ Float64(-0.5 - Float64(z + -1.0))) * Float64(Float64(z + 1.0) * exp(-7.5)))) * Float64(t_2 * Float64(Float64(Float64(Float64(Float64(0.9999999999998099 - t_0) - Float64(771.3234287776531 / Float64(Float64(z + -1.0) - 2.0))) - Float64(-1259.1392167224028 / Float64(-1.0 + Float64(z + -1.0)))) + Float64(-176.6150291621406 / Float64(Float64(1.0 - z) + 3.0))) + Float64(2.4783749183520145 + Float64(z * 0.49644474017195733)))));
	else
		tmp = Float64(t_1 * Float64(Float64(t_2 * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5)))) * Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) + Float64(Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) - t_0) + Float64(Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(-176.6150291621406 / Float64(4.0 - z)))) - Float64(Float64(771.3234287776531 / Float64(z - 3.0)) - 0.9999999999998099)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = 676.5203681218851 / (z + -1.0);
	t_1 = sqrt((pi * 2.0));
	t_2 = pi / sin((pi * z));
	tmp = 0.0;
	if (z <= -2000.0)
		tmp = (t_1 * (((6.5 - (z + -1.0)) ^ (-0.5 - (z + -1.0))) * ((z + 1.0) * exp(-7.5)))) * (t_2 * (((((0.9999999999998099 - t_0) - (771.3234287776531 / ((z + -1.0) - 2.0))) - (-1259.1392167224028 / (-1.0 + (z + -1.0)))) + (-176.6150291621406 / ((1.0 - z) + 3.0))) + (2.4783749183520145 + (z * 0.49644474017195733))));
	else
		tmp = t_1 * ((t_2 * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5)))) * ((9.984369578019572e-6 / (7.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + (((-1259.1392167224028 / (2.0 - z)) - t_0) + (((-0.13857109526572012 / (6.0 - z)) + ((12.507343278686905 / (5.0 - z)) + (-176.6150291621406 / (4.0 - z)))) - ((771.3234287776531 / (z - 3.0)) - 0.9999999999998099))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[(676.5203681218851 / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2000.0], N[(N[(t$95$1 * N[(N[Power[N[(6.5 - N[(z + -1.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 - N[(z + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(z + 1.0), $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(N[(N[(N[(N[(0.9999999999998099 - t$95$0), $MachinePrecision] - N[(771.3234287776531 / N[(N[(z + -1.0), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1259.1392167224028 / N[(-1.0 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.4783749183520145 + N[(z * 0.49644474017195733), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(t$95$2 * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] + N[(N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(771.3234287776531 / N[(z - 3.0), $MachinePrecision]), $MachinePrecision] - 0.9999999999998099), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if z < -2e3

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in z around 0 0.0%

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

      1. *-commutative 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in z around 0 100.0%

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

      1. distribute-rgt1-in 100.0%

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

   10. Simplified 100.0%

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

   if -2e3 < z

   1. Initial program 97.3%

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

   3. Simplified 97.3%

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

      1. pow1 97.3%

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

   6. Applied egg-rr 97.2%

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

   8. Simplified 98.7%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2000:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(6.5 - \left(z + -1\right)\right)}^{\left(-0.5 - \left(z + -1\right)\right)} \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{z + -1}\right) - \frac{771.3234287776531}{\left(z + -1\right) - 2}\right) - \frac{-1259.1392167224028}{-1 + \left(z + -1\right)}\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \left(\left(\frac{-1259.1392167224028}{2 - z} - \frac{676.5203681218851}{z + -1}\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{12.507343278686905}{5 - z} + \frac{-176.6150291621406}{4 - z}\right)\right) - \left(\frac{771.3234287776531}{z - 3} - 0.9999999999998099\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{676.5203681218851}{z + -1}\\ t_1 := \sqrt{\pi \cdot 2}\\ t_2 := \frac{\pi}{\sin \left(\pi \cdot z\right)}\\ \mathbf{if}\;z \leq -2000:\\ \;\;\;\;\left(t\_1 \cdot \left({\left(6.5 - \left(z + -1\right)\right)}^{\left(-0.5 - \left(z + -1\right)\right)} \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right)\right) \cdot \left(t\_2 \cdot \left(\left(\left(\left(\left(0.9999999999998099 - t\_0\right) - \frac{771.3234287776531}{\left(z + -1\right) - 2}\right) - \frac{-1259.1392167224028}{-1 + \left(z + -1\right)}\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\left(t\_2 \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} - t\_0\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-0.13857109526572012}{6 - z} + \frac{-176.6150291621406}{4 - z}\right)\right) + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (/ 676.5203681218851 (+ z -1.0)))
        (t_1 (sqrt (* PI 2.0)))
        (t_2 (/ PI (sin (* PI z)))))
   (if (<= z -2000.0)
     (*
      (*
       t_1
       (*
        (pow (- 6.5 (+ z -1.0)) (- -0.5 (+ z -1.0)))
        (* (+ z 1.0) (exp -7.5))))
      (*
       t_2
       (+
        (+
         (-
          (-
           (- 0.9999999999998099 t_0)
           (/ 771.3234287776531 (- (+ z -1.0) 2.0)))
          (/ -1259.1392167224028 (+ -1.0 (+ z -1.0))))
         (/ -176.6150291621406 (+ (- 1.0 z) 3.0)))
        (+ 2.4783749183520145 (* z 0.49644474017195733)))))
     (*
      t_1
      (*
       (* t_2 (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5))))
       (+
        (- (/ -1259.1392167224028 (- 2.0 z)) t_0)
        (+
         (+
          (/ 9.984369578019572e-6 (- 7.0 z))
          (/ 1.5056327351493116e-7 (- 8.0 z)))
         (+
          (+
           (/ 12.507343278686905 (- 5.0 z))
           (+
            (/ -0.13857109526572012 (- 6.0 z))
            (/ -176.6150291621406 (- 4.0 z))))
          (+ 0.9999999999998099 (/ 771.3234287776531 (- 3.0 z)))))))))))

C

double code(double z) {
	double t_0 = 676.5203681218851 / (z + -1.0);
	double t_1 = sqrt((((double) M_PI) * 2.0));
	double t_2 = ((double) M_PI) / sin((((double) M_PI) * z));
	double tmp;
	if (z <= -2000.0) {
		tmp = (t_1 * (pow((6.5 - (z + -1.0)), (-0.5 - (z + -1.0))) * ((z + 1.0) * exp(-7.5)))) * (t_2 * (((((0.9999999999998099 - t_0) - (771.3234287776531 / ((z + -1.0) - 2.0))) - (-1259.1392167224028 / (-1.0 + (z + -1.0)))) + (-176.6150291621406 / ((1.0 - z) + 3.0))) + (2.4783749183520145 + (z * 0.49644474017195733))));
	} else {
		tmp = t_1 * ((t_2 * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5)))) * (((-1259.1392167224028 / (2.0 - z)) - t_0) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((12.507343278686905 / (5.0 - z)) + ((-0.13857109526572012 / (6.0 - z)) + (-176.6150291621406 / (4.0 - z)))) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z)))))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = 676.5203681218851 / (z + -1.0);
	double t_1 = Math.sqrt((Math.PI * 2.0));
	double t_2 = Math.PI / Math.sin((Math.PI * z));
	double tmp;
	if (z <= -2000.0) {
		tmp = (t_1 * (Math.pow((6.5 - (z + -1.0)), (-0.5 - (z + -1.0))) * ((z + 1.0) * Math.exp(-7.5)))) * (t_2 * (((((0.9999999999998099 - t_0) - (771.3234287776531 / ((z + -1.0) - 2.0))) - (-1259.1392167224028 / (-1.0 + (z + -1.0)))) + (-176.6150291621406 / ((1.0 - z) + 3.0))) + (2.4783749183520145 + (z * 0.49644474017195733))));
	} else {
		tmp = t_1 * ((t_2 * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5)))) * (((-1259.1392167224028 / (2.0 - z)) - t_0) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((12.507343278686905 / (5.0 - z)) + ((-0.13857109526572012 / (6.0 - z)) + (-176.6150291621406 / (4.0 - z)))) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z)))))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = 676.5203681218851 / (z + -1.0)
	t_1 = math.sqrt((math.pi * 2.0))
	t_2 = math.pi / math.sin((math.pi * z))
	tmp = 0
	if z <= -2000.0:
		tmp = (t_1 * (math.pow((6.5 - (z + -1.0)), (-0.5 - (z + -1.0))) * ((z + 1.0) * math.exp(-7.5)))) * (t_2 * (((((0.9999999999998099 - t_0) - (771.3234287776531 / ((z + -1.0) - 2.0))) - (-1259.1392167224028 / (-1.0 + (z + -1.0)))) + (-176.6150291621406 / ((1.0 - z) + 3.0))) + (2.4783749183520145 + (z * 0.49644474017195733))))
	else:
		tmp = t_1 * ((t_2 * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5)))) * (((-1259.1392167224028 / (2.0 - z)) - t_0) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((12.507343278686905 / (5.0 - z)) + ((-0.13857109526572012 / (6.0 - z)) + (-176.6150291621406 / (4.0 - z)))) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z)))))))
	return tmp

Julia

function code(z)
	t_0 = Float64(676.5203681218851 / Float64(z + -1.0))
	t_1 = sqrt(Float64(pi * 2.0))
	t_2 = Float64(pi / sin(Float64(pi * z)))
	tmp = 0.0
	if (z <= -2000.0)
		tmp = Float64(Float64(t_1 * Float64((Float64(6.5 - Float64(z + -1.0)) ^ Float64(-0.5 - Float64(z + -1.0))) * Float64(Float64(z + 1.0) * exp(-7.5)))) * Float64(t_2 * Float64(Float64(Float64(Float64(Float64(0.9999999999998099 - t_0) - Float64(771.3234287776531 / Float64(Float64(z + -1.0) - 2.0))) - Float64(-1259.1392167224028 / Float64(-1.0 + Float64(z + -1.0)))) + Float64(-176.6150291621406 / Float64(Float64(1.0 - z) + 3.0))) + Float64(2.4783749183520145 + Float64(z * 0.49644474017195733)))));
	else
		tmp = Float64(t_1 * Float64(Float64(t_2 * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5)))) * Float64(Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) - t_0) + Float64(Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))) + Float64(Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(-176.6150291621406 / Float64(4.0 - z)))) + Float64(0.9999999999998099 + Float64(771.3234287776531 / Float64(3.0 - z))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = 676.5203681218851 / (z + -1.0);
	t_1 = sqrt((pi * 2.0));
	t_2 = pi / sin((pi * z));
	tmp = 0.0;
	if (z <= -2000.0)
		tmp = (t_1 * (((6.5 - (z + -1.0)) ^ (-0.5 - (z + -1.0))) * ((z + 1.0) * exp(-7.5)))) * (t_2 * (((((0.9999999999998099 - t_0) - (771.3234287776531 / ((z + -1.0) - 2.0))) - (-1259.1392167224028 / (-1.0 + (z + -1.0)))) + (-176.6150291621406 / ((1.0 - z) + 3.0))) + (2.4783749183520145 + (z * 0.49644474017195733))));
	else
		tmp = t_1 * ((t_2 * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5)))) * (((-1259.1392167224028 / (2.0 - z)) - t_0) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (((12.507343278686905 / (5.0 - z)) + ((-0.13857109526572012 / (6.0 - z)) + (-176.6150291621406 / (4.0 - z)))) + (0.9999999999998099 + (771.3234287776531 / (3.0 - z)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[(676.5203681218851 / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2000.0], N[(N[(t$95$1 * N[(N[Power[N[(6.5 - N[(z + -1.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 - N[(z + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(z + 1.0), $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(N[(N[(N[(N[(0.9999999999998099 - t$95$0), $MachinePrecision] - N[(771.3234287776531 / N[(N[(z + -1.0), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1259.1392167224028 / N[(-1.0 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.4783749183520145 + N[(z * 0.49644474017195733), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(t$95$2 * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] + N[(N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if z < -2e3

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in z around 0 0.0%

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

      1. *-commutative 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in z around 0 100.0%

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

      1. distribute-rgt1-in 100.0%

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

   10. Simplified 100.0%

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

   if -2e3 < z

   1. Initial program 97.3%

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

   3. Simplified 97.3%

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

      1. pow1 97.3%

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

   6. Applied egg-rr 97.2%

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

   8. Simplified 97.5%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2000:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(6.5 - \left(z + -1\right)\right)}^{\left(-0.5 - \left(z + -1\right)\right)} \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{z + -1}\right) - \frac{771.3234287776531}{\left(z + -1\right) - 2}\right) - \frac{-1259.1392167224028}{-1 + \left(z + -1\right)}\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} - \frac{676.5203681218851}{z + -1}\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \left(\frac{-0.13857109526572012}{6 - z} + \frac{-176.6150291621406}{4 - z}\right)\right) + \left(0.9999999999998099 + \frac{771.3234287776531}{3 - z}\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{676.5203681218851}{z + -1}\\ t_1 := \sqrt{\pi \cdot 2}\\ t_2 := \frac{\pi}{\sin \left(\pi \cdot z\right)}\\ \mathbf{if}\;z \leq -12:\\ \;\;\;\;\left(t\_1 \cdot \left({\left(6.5 - \left(z + -1\right)\right)}^{\left(-0.5 - \left(z + -1\right)\right)} \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right)\right) \cdot \left(t\_2 \cdot \left(\left(\left(\left(\left(0.9999999999998099 - t\_0\right) - \frac{771.3234287776531}{\left(z + -1\right) - 2}\right) - \frac{-1259.1392167224028}{-1 + \left(z + -1\right)}\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\left(t\_2 \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} - t\_0\right) - \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z - 7} - \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) - \left(216.4324257752088 + z \cdot \left(75.16060840893998 + z \cdot \left(25.90734178189493 + z \cdot 8.852513835830658\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (/ 676.5203681218851 (+ z -1.0)))
        (t_1 (sqrt (* PI 2.0)))
        (t_2 (/ PI (sin (* PI z)))))
   (if (<= z -12.0)
     (*
      (*
       t_1
       (*
        (pow (- 6.5 (+ z -1.0)) (- -0.5 (+ z -1.0)))
        (* (+ z 1.0) (exp -7.5))))
      (*
       t_2
       (+
        (+
         (-
          (-
           (- 0.9999999999998099 t_0)
           (/ 771.3234287776531 (- (+ z -1.0) 2.0)))
          (/ -1259.1392167224028 (+ -1.0 (+ z -1.0))))
         (/ -176.6150291621406 (+ (- 1.0 z) 3.0)))
        (+ 2.4783749183520145 (* z 0.49644474017195733)))))
     (*
      t_1
      (*
       (* t_2 (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5))))
       (-
        (- (/ -1259.1392167224028 (- 2.0 z)) t_0)
        (-
         (-
          (/ 9.984369578019572e-6 (- z 7.0))
          (/ 1.5056327351493116e-7 (- 8.0 z)))
         (+
          216.4324257752088
          (*
           z
           (+
            75.16060840893998
            (* z (+ 25.90734178189493 (* z 8.852513835830658)))))))))))))

C

double code(double z) {
	double t_0 = 676.5203681218851 / (z + -1.0);
	double t_1 = sqrt((((double) M_PI) * 2.0));
	double t_2 = ((double) M_PI) / sin((((double) M_PI) * z));
	double tmp;
	if (z <= -12.0) {
		tmp = (t_1 * (pow((6.5 - (z + -1.0)), (-0.5 - (z + -1.0))) * ((z + 1.0) * exp(-7.5)))) * (t_2 * (((((0.9999999999998099 - t_0) - (771.3234287776531 / ((z + -1.0) - 2.0))) - (-1259.1392167224028 / (-1.0 + (z + -1.0)))) + (-176.6150291621406 / ((1.0 - z) + 3.0))) + (2.4783749183520145 + (z * 0.49644474017195733))));
	} else {
		tmp = t_1 * ((t_2 * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5)))) * (((-1259.1392167224028 / (2.0 - z)) - t_0) - (((9.984369578019572e-6 / (z - 7.0)) - (1.5056327351493116e-7 / (8.0 - z))) - (216.4324257752088 + (z * (75.16060840893998 + (z * (25.90734178189493 + (z * 8.852513835830658)))))))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = 676.5203681218851 / (z + -1.0);
	double t_1 = Math.sqrt((Math.PI * 2.0));
	double t_2 = Math.PI / Math.sin((Math.PI * z));
	double tmp;
	if (z <= -12.0) {
		tmp = (t_1 * (Math.pow((6.5 - (z + -1.0)), (-0.5 - (z + -1.0))) * ((z + 1.0) * Math.exp(-7.5)))) * (t_2 * (((((0.9999999999998099 - t_0) - (771.3234287776531 / ((z + -1.0) - 2.0))) - (-1259.1392167224028 / (-1.0 + (z + -1.0)))) + (-176.6150291621406 / ((1.0 - z) + 3.0))) + (2.4783749183520145 + (z * 0.49644474017195733))));
	} else {
		tmp = t_1 * ((t_2 * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5)))) * (((-1259.1392167224028 / (2.0 - z)) - t_0) - (((9.984369578019572e-6 / (z - 7.0)) - (1.5056327351493116e-7 / (8.0 - z))) - (216.4324257752088 + (z * (75.16060840893998 + (z * (25.90734178189493 + (z * 8.852513835830658)))))))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = 676.5203681218851 / (z + -1.0)
	t_1 = math.sqrt((math.pi * 2.0))
	t_2 = math.pi / math.sin((math.pi * z))
	tmp = 0
	if z <= -12.0:
		tmp = (t_1 * (math.pow((6.5 - (z + -1.0)), (-0.5 - (z + -1.0))) * ((z + 1.0) * math.exp(-7.5)))) * (t_2 * (((((0.9999999999998099 - t_0) - (771.3234287776531 / ((z + -1.0) - 2.0))) - (-1259.1392167224028 / (-1.0 + (z + -1.0)))) + (-176.6150291621406 / ((1.0 - z) + 3.0))) + (2.4783749183520145 + (z * 0.49644474017195733))))
	else:
		tmp = t_1 * ((t_2 * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5)))) * (((-1259.1392167224028 / (2.0 - z)) - t_0) - (((9.984369578019572e-6 / (z - 7.0)) - (1.5056327351493116e-7 / (8.0 - z))) - (216.4324257752088 + (z * (75.16060840893998 + (z * (25.90734178189493 + (z * 8.852513835830658)))))))))
	return tmp

Julia

function code(z)
	t_0 = Float64(676.5203681218851 / Float64(z + -1.0))
	t_1 = sqrt(Float64(pi * 2.0))
	t_2 = Float64(pi / sin(Float64(pi * z)))
	tmp = 0.0
	if (z <= -12.0)
		tmp = Float64(Float64(t_1 * Float64((Float64(6.5 - Float64(z + -1.0)) ^ Float64(-0.5 - Float64(z + -1.0))) * Float64(Float64(z + 1.0) * exp(-7.5)))) * Float64(t_2 * Float64(Float64(Float64(Float64(Float64(0.9999999999998099 - t_0) - Float64(771.3234287776531 / Float64(Float64(z + -1.0) - 2.0))) - Float64(-1259.1392167224028 / Float64(-1.0 + Float64(z + -1.0)))) + Float64(-176.6150291621406 / Float64(Float64(1.0 - z) + 3.0))) + Float64(2.4783749183520145 + Float64(z * 0.49644474017195733)))));
	else
		tmp = Float64(t_1 * Float64(Float64(t_2 * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5)))) * Float64(Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) - t_0) - Float64(Float64(Float64(9.984369578019572e-6 / Float64(z - 7.0)) - Float64(1.5056327351493116e-7 / Float64(8.0 - z))) - Float64(216.4324257752088 + Float64(z * Float64(75.16060840893998 + Float64(z * Float64(25.90734178189493 + Float64(z * 8.852513835830658))))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = 676.5203681218851 / (z + -1.0);
	t_1 = sqrt((pi * 2.0));
	t_2 = pi / sin((pi * z));
	tmp = 0.0;
	if (z <= -12.0)
		tmp = (t_1 * (((6.5 - (z + -1.0)) ^ (-0.5 - (z + -1.0))) * ((z + 1.0) * exp(-7.5)))) * (t_2 * (((((0.9999999999998099 - t_0) - (771.3234287776531 / ((z + -1.0) - 2.0))) - (-1259.1392167224028 / (-1.0 + (z + -1.0)))) + (-176.6150291621406 / ((1.0 - z) + 3.0))) + (2.4783749183520145 + (z * 0.49644474017195733))));
	else
		tmp = t_1 * ((t_2 * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5)))) * (((-1259.1392167224028 / (2.0 - z)) - t_0) - (((9.984369578019572e-6 / (z - 7.0)) - (1.5056327351493116e-7 / (8.0 - z))) - (216.4324257752088 + (z * (75.16060840893998 + (z * (25.90734178189493 + (z * 8.852513835830658)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[(676.5203681218851 / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -12.0], N[(N[(t$95$1 * N[(N[Power[N[(6.5 - N[(z + -1.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 - N[(z + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(z + 1.0), $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(N[(N[(N[(N[(0.9999999999998099 - t$95$0), $MachinePrecision] - N[(771.3234287776531 / N[(N[(z + -1.0), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1259.1392167224028 / N[(-1.0 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.4783749183520145 + N[(z * 0.49644474017195733), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(t$95$2 * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] - N[(N[(N[(9.984369578019572e-6 / N[(z - 7.0), $MachinePrecision]), $MachinePrecision] - N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(216.4324257752088 + N[(z * N[(75.16060840893998 + N[(z * N[(25.90734178189493 + N[(z * 8.852513835830658), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if z < -12

   1. Initial program 14.6%

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

   3. Simplified 14.6%

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

   5. Taylor expanded in z around 0 0.1%

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

      1. *-commutative 0.1%

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

   7. Simplified 0.1%

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

   8. Taylor expanded in z around 0 84.6%

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

      1. distribute-rgt1-in 84.6%

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

   10. Simplified 84.6%

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

   if -12 < z

   1. Initial program 97.3%

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

   3. Simplified 97.3%

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

      1. pow1 97.3%

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

   6. Applied egg-rr 97.2%

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

   8. Simplified 97.5%

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

   9. Taylor expanded in z around 0 97.8%

      \[\leadsto \sqrt{2 \cdot \pi} \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\color{blue}{\left(216.4324257752088 + z \cdot \left(75.16060840893998 + z \cdot \left(25.90734178189493 + 8.852513835830658 \cdot z\right)\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative 97.8%

          \[\leadsto \sqrt{2 \cdot \pi} \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\left(216.4324257752088 + z \cdot \left(75.16060840893998 + z \cdot \left(25.90734178189493 + \color{blue}{z \cdot 8.852513835830658}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right) \]

   11. Simplified 97.8%

       \[\leadsto \sqrt{2 \cdot \pi} \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\color{blue}{\left(216.4324257752088 + z \cdot \left(75.16060840893998 + z \cdot \left(25.90734178189493 + z \cdot 8.852513835830658\right)\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -12:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(6.5 - \left(z + -1\right)\right)}^{\left(-0.5 - \left(z + -1\right)\right)} \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{z + -1}\right) - \frac{771.3234287776531}{\left(z + -1\right) - 2}\right) - \frac{-1259.1392167224028}{-1 + \left(z + -1\right)}\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right) + \left(2.4783749183520145 + z \cdot 0.49644474017195733\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} - \frac{676.5203681218851}{z + -1}\right) - \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z - 7} - \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) - \left(216.4324257752088 + z \cdot \left(75.16060840893998 + z \cdot \left(25.90734178189493 + z \cdot 8.852513835830658\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\pi \cdot 2} \cdot \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} - \frac{676.5203681218851}{z + -1}\right) - \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z - 7} - \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) - \left(216.4324257752088 + z \cdot \left(75.16060840893998 + z \cdot \left(25.90734178189493 + z \cdot 8.852513835830658\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (sqrt (* PI 2.0))
  (*
   (* (/ PI (sin (* PI z))) (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5))))
   (-
    (- (/ -1259.1392167224028 (- 2.0 z)) (/ 676.5203681218851 (+ z -1.0)))
    (-
     (- (/ 9.984369578019572e-6 (- z 7.0)) (/ 1.5056327351493116e-7 (- 8.0 z)))
     (+
      216.4324257752088
      (*
       z
       (+
        75.16060840893998
        (* z (+ 25.90734178189493 (* z 8.852513835830658)))))))))))

C

double code(double z) {
	return sqrt((((double) M_PI) * 2.0)) * (((((double) M_PI) / sin((((double) M_PI) * z))) * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5)))) * (((-1259.1392167224028 / (2.0 - z)) - (676.5203681218851 / (z + -1.0))) - (((9.984369578019572e-6 / (z - 7.0)) - (1.5056327351493116e-7 / (8.0 - z))) - (216.4324257752088 + (z * (75.16060840893998 + (z * (25.90734178189493 + (z * 8.852513835830658)))))))));
}

Java

public static double code(double z) {
	return Math.sqrt((Math.PI * 2.0)) * (((Math.PI / Math.sin((Math.PI * z))) * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5)))) * (((-1259.1392167224028 / (2.0 - z)) - (676.5203681218851 / (z + -1.0))) - (((9.984369578019572e-6 / (z - 7.0)) - (1.5056327351493116e-7 / (8.0 - z))) - (216.4324257752088 + (z * (75.16060840893998 + (z * (25.90734178189493 + (z * 8.852513835830658)))))))));
}

Python

def code(z):
	return math.sqrt((math.pi * 2.0)) * (((math.pi / math.sin((math.pi * z))) * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5)))) * (((-1259.1392167224028 / (2.0 - z)) - (676.5203681218851 / (z + -1.0))) - (((9.984369578019572e-6 / (z - 7.0)) - (1.5056327351493116e-7 / (8.0 - z))) - (216.4324257752088 + (z * (75.16060840893998 + (z * (25.90734178189493 + (z * 8.852513835830658)))))))))

Julia

function code(z)
	return Float64(sqrt(Float64(pi * 2.0)) * Float64(Float64(Float64(pi / sin(Float64(pi * z))) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5)))) * Float64(Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) - Float64(676.5203681218851 / Float64(z + -1.0))) - Float64(Float64(Float64(9.984369578019572e-6 / Float64(z - 7.0)) - Float64(1.5056327351493116e-7 / Float64(8.0 - z))) - Float64(216.4324257752088 + Float64(z * Float64(75.16060840893998 + Float64(z * Float64(25.90734178189493 + Float64(z * 8.852513835830658))))))))))
end

MATLAB

function tmp = code(z)
	tmp = sqrt((pi * 2.0)) * (((pi / sin((pi * z))) * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5)))) * (((-1259.1392167224028 / (2.0 - z)) - (676.5203681218851 / (z + -1.0))) - (((9.984369578019572e-6 / (z - 7.0)) - (1.5056327351493116e-7 / (8.0 - z))) - (216.4324257752088 + (z * (75.16060840893998 + (z * (25.90734178189493 + (z * 8.852513835830658)))))))));
end

Wolfram

code[z_] := N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] - N[(676.5203681218851 / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(9.984369578019572e-6 / N[(z - 7.0), $MachinePrecision]), $MachinePrecision] - N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(216.4324257752088 + N[(z * N[(75.16060840893998 + N[(z * N[(25.90734178189493 + N[(z * 8.852513835830658), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.4%

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

3. Simplified 95.4%

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

   1. pow1 95.4%

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

6. Applied egg-rr 95.3%

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

8. Simplified 95.6%

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

9. Taylor expanded in z around 0 95.5%

   \[\leadsto \sqrt{2 \cdot \pi} \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\color{blue}{\left(216.4324257752088 + z \cdot \left(75.16060840893998 + z \cdot \left(25.90734178189493 + 8.852513835830658 \cdot z\right)\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right) \]
10. Step-by-step derivation

    1. *-commutative 95.5%

       \[\leadsto \sqrt{2 \cdot \pi} \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\left(216.4324257752088 + z \cdot \left(75.16060840893998 + z \cdot \left(25.90734178189493 + \color{blue}{z \cdot 8.852513835830658}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right) \]

11. Simplified 95.5%

    \[\leadsto \sqrt{2 \cdot \pi} \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\color{blue}{\left(216.4324257752088 + z \cdot \left(75.16060840893998 + z \cdot \left(25.90734178189493 + z \cdot 8.852513835830658\right)\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right) \]

12. Final simplification 95.5%

    \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} - \frac{676.5203681218851}{z + -1}\right) - \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z - 7} - \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) - \left(216.4324257752088 + z \cdot \left(75.16060840893998 + z \cdot \left(25.90734178189493 + z \cdot 8.852513835830658\right)\right)\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 9: 96.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\pi \cdot 2} \cdot \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} - \frac{676.5203681218851}{z + -1}\right) - \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z - 7} - \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) - \left(216.4324257752088 + z \cdot \left(75.16060840893998 + z \cdot 25.90734178189493\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (sqrt (* PI 2.0))
  (*
   (* (/ PI (sin (* PI z))) (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5))))
   (-
    (- (/ -1259.1392167224028 (- 2.0 z)) (/ 676.5203681218851 (+ z -1.0)))
    (-
     (- (/ 9.984369578019572e-6 (- z 7.0)) (/ 1.5056327351493116e-7 (- 8.0 z)))
     (+
      216.4324257752088
      (* z (+ 75.16060840893998 (* z 25.90734178189493)))))))))

C

double code(double z) {
	return sqrt((((double) M_PI) * 2.0)) * (((((double) M_PI) / sin((((double) M_PI) * z))) * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5)))) * (((-1259.1392167224028 / (2.0 - z)) - (676.5203681218851 / (z + -1.0))) - (((9.984369578019572e-6 / (z - 7.0)) - (1.5056327351493116e-7 / (8.0 - z))) - (216.4324257752088 + (z * (75.16060840893998 + (z * 25.90734178189493)))))));
}

Java

public static double code(double z) {
	return Math.sqrt((Math.PI * 2.0)) * (((Math.PI / Math.sin((Math.PI * z))) * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5)))) * (((-1259.1392167224028 / (2.0 - z)) - (676.5203681218851 / (z + -1.0))) - (((9.984369578019572e-6 / (z - 7.0)) - (1.5056327351493116e-7 / (8.0 - z))) - (216.4324257752088 + (z * (75.16060840893998 + (z * 25.90734178189493)))))));
}

Python

def code(z):
	return math.sqrt((math.pi * 2.0)) * (((math.pi / math.sin((math.pi * z))) * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5)))) * (((-1259.1392167224028 / (2.0 - z)) - (676.5203681218851 / (z + -1.0))) - (((9.984369578019572e-6 / (z - 7.0)) - (1.5056327351493116e-7 / (8.0 - z))) - (216.4324257752088 + (z * (75.16060840893998 + (z * 25.90734178189493)))))))

Julia

function code(z)
	return Float64(sqrt(Float64(pi * 2.0)) * Float64(Float64(Float64(pi / sin(Float64(pi * z))) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5)))) * Float64(Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) - Float64(676.5203681218851 / Float64(z + -1.0))) - Float64(Float64(Float64(9.984369578019572e-6 / Float64(z - 7.0)) - Float64(1.5056327351493116e-7 / Float64(8.0 - z))) - Float64(216.4324257752088 + Float64(z * Float64(75.16060840893998 + Float64(z * 25.90734178189493))))))))
end

MATLAB

function tmp = code(z)
	tmp = sqrt((pi * 2.0)) * (((pi / sin((pi * z))) * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5)))) * (((-1259.1392167224028 / (2.0 - z)) - (676.5203681218851 / (z + -1.0))) - (((9.984369578019572e-6 / (z - 7.0)) - (1.5056327351493116e-7 / (8.0 - z))) - (216.4324257752088 + (z * (75.16060840893998 + (z * 25.90734178189493)))))));
end

Wolfram

code[z_] := N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] - N[(676.5203681218851 / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(9.984369578019572e-6 / N[(z - 7.0), $MachinePrecision]), $MachinePrecision] - N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(216.4324257752088 + N[(z * N[(75.16060840893998 + N[(z * 25.90734178189493), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.4%

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

3. Simplified 95.4%

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

   1. pow1 95.4%

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

6. Applied egg-rr 95.3%

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

8. Simplified 95.6%

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

9. Taylor expanded in z around 0 95.4%

   \[\leadsto \sqrt{2 \cdot \pi} \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\color{blue}{\left(216.4324257752088 + z \cdot \left(75.16060840893998 + 25.90734178189493 \cdot z\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right) \]
10. Step-by-step derivation

    1. *-commutative 95.4%

       \[\leadsto \sqrt{2 \cdot \pi} \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\left(216.4324257752088 + z \cdot \left(75.16060840893998 + \color{blue}{z \cdot 25.90734178189493}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right) \]

11. Simplified 95.4%

    \[\leadsto \sqrt{2 \cdot \pi} \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\color{blue}{\left(216.4324257752088 + z \cdot \left(75.16060840893998 + z \cdot 25.90734178189493\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right) \]

12. Final simplification 95.4%

    \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} - \frac{676.5203681218851}{z + -1}\right) - \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{z - 7} - \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) - \left(216.4324257752088 + z \cdot \left(75.16060840893998 + z \cdot 25.90734178189493\right)\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 10: 96.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{\pi \cdot 2} \cdot \left({\left(6.5 - \left(z + -1\right)\right)}^{\left(-0.5 - \left(z + -1\right)\right)} \cdot e^{\left(z + -1\right) - 6.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + z \cdot 606.656776085461\right)\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (sqrt (* PI 2.0))
   (* (pow (- 6.5 (+ z -1.0)) (- -0.5 (+ z -1.0))) (exp (- (+ z -1.0) 6.5))))
  (*
   (/ PI (sin (* PI z)))
   (+
    (+
     260.9048120626994
     (*
      z
      (+
       436.3997278161676
       (* z (+ 544.9358906000987 (* z 606.656776085461))))))
    (+
     2.4783749183520145
     (* z (+ 0.49644474017195733 (* z 0.09941724278406093))))))))

C

double code(double z) {
	return (sqrt((((double) M_PI) * 2.0)) * (pow((6.5 - (z + -1.0)), (-0.5 - (z + -1.0))) * exp(((z + -1.0) - 6.5)))) * ((((double) M_PI) / sin((((double) M_PI) * z))) * ((260.9048120626994 + (z * (436.3997278161676 + (z * (544.9358906000987 + (z * 606.656776085461)))))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * 0.09941724278406093))))));
}

Java

public static double code(double z) {
	return (Math.sqrt((Math.PI * 2.0)) * (Math.pow((6.5 - (z + -1.0)), (-0.5 - (z + -1.0))) * Math.exp(((z + -1.0) - 6.5)))) * ((Math.PI / Math.sin((Math.PI * z))) * ((260.9048120626994 + (z * (436.3997278161676 + (z * (544.9358906000987 + (z * 606.656776085461)))))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * 0.09941724278406093))))));
}

Python

def code(z):
	return (math.sqrt((math.pi * 2.0)) * (math.pow((6.5 - (z + -1.0)), (-0.5 - (z + -1.0))) * math.exp(((z + -1.0) - 6.5)))) * ((math.pi / math.sin((math.pi * z))) * ((260.9048120626994 + (z * (436.3997278161676 + (z * (544.9358906000987 + (z * 606.656776085461)))))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * 0.09941724278406093))))))

Julia

function code(z)
	return Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(6.5 - Float64(z + -1.0)) ^ Float64(-0.5 - Float64(z + -1.0))) * exp(Float64(Float64(z + -1.0) - 6.5)))) * Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(260.9048120626994 + Float64(z * Float64(436.3997278161676 + Float64(z * Float64(544.9358906000987 + Float64(z * 606.656776085461)))))) + Float64(2.4783749183520145 + Float64(z * Float64(0.49644474017195733 + Float64(z * 0.09941724278406093)))))))
end

MATLAB

function tmp = code(z)
	tmp = (sqrt((pi * 2.0)) * (((6.5 - (z + -1.0)) ^ (-0.5 - (z + -1.0))) * exp(((z + -1.0) - 6.5)))) * ((pi / sin((pi * z))) * ((260.9048120626994 + (z * (436.3997278161676 + (z * (544.9358906000987 + (z * 606.656776085461)))))) + (2.4783749183520145 + (z * (0.49644474017195733 + (z * 0.09941724278406093))))));
end

Wolfram

code[z_] := N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(6.5 - N[(z + -1.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 - N[(z + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(z + -1.0), $MachinePrecision] - 6.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(260.9048120626994 + N[(z * N[(436.3997278161676 + N[(z * N[(544.9358906000987 + N[(z * 606.656776085461), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.4783749183520145 + N[(z * N[(0.49644474017195733 + N[(z * 0.09941724278406093), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.4%

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

3. Simplified 95.2%

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

5. Taylor expanded in z around 0 95.1%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + 606.656776085461 \cdot z\right)\right)\right)} + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
6. Step-by-step derivation

   1. *-commutative 95.1%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + \color{blue}{z \cdot 606.656776085461}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

7. Simplified 95.1%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + z \cdot 606.656776085461\right)\right)\right)} + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

8. Taylor expanded in z around 0 95.1%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + z \cdot 606.656776085461\right)\right)\right) + \color{blue}{\left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + 0.09941724278406093 \cdot z\right)\right)}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
9. Step-by-step derivation

   1. *-commutative 95.1%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + z \cdot 606.656776085461\right)\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + \color{blue}{z \cdot 0.09941724278406093}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

10. Simplified 95.1%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + z \cdot 606.656776085461\right)\right)\right) + \color{blue}{\left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

11. Final simplification 95.1%

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(6.5 - \left(z + -1\right)\right)}^{\left(-0.5 - \left(z + -1\right)\right)} \cdot e^{\left(z + -1\right) - 6.5}\right)\right) \cdot \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + z \cdot 606.656776085461\right)\right)\right) + \left(2.4783749183520145 + z \cdot \left(0.49644474017195733 + z \cdot 0.09941724278406093\right)\right)\right)\right) \]
12. Add Preprocessing

Alternative 11: 96.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{\pi \cdot 2} \cdot \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} - \frac{676.5203681218851}{z + -1}\right) + \left(\left(216.4324257752088 + z \cdot 75.16060840893998\right) - \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - 7} - \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (sqrt (* PI 2.0))
  (*
   (* (/ PI (sin (* PI z))) (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5))))
   (+
    (- (/ -1259.1392167224028 (- 2.0 z)) (/ 676.5203681218851 (+ z -1.0)))
    (-
     (+ 216.4324257752088 (* z 75.16060840893998))
     (-
      (/ 9.984369578019572e-6 (- z 7.0))
      (/ 1.5056327351493116e-7 (- 8.0 z))))))))

C

double code(double z) {
	return sqrt((((double) M_PI) * 2.0)) * (((((double) M_PI) / sin((((double) M_PI) * z))) * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5)))) * (((-1259.1392167224028 / (2.0 - z)) - (676.5203681218851 / (z + -1.0))) + ((216.4324257752088 + (z * 75.16060840893998)) - ((9.984369578019572e-6 / (z - 7.0)) - (1.5056327351493116e-7 / (8.0 - z))))));
}

Java

public static double code(double z) {
	return Math.sqrt((Math.PI * 2.0)) * (((Math.PI / Math.sin((Math.PI * z))) * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5)))) * (((-1259.1392167224028 / (2.0 - z)) - (676.5203681218851 / (z + -1.0))) + ((216.4324257752088 + (z * 75.16060840893998)) - ((9.984369578019572e-6 / (z - 7.0)) - (1.5056327351493116e-7 / (8.0 - z))))));
}

Python

def code(z):
	return math.sqrt((math.pi * 2.0)) * (((math.pi / math.sin((math.pi * z))) * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5)))) * (((-1259.1392167224028 / (2.0 - z)) - (676.5203681218851 / (z + -1.0))) + ((216.4324257752088 + (z * 75.16060840893998)) - ((9.984369578019572e-6 / (z - 7.0)) - (1.5056327351493116e-7 / (8.0 - z))))))

Julia

function code(z)
	return Float64(sqrt(Float64(pi * 2.0)) * Float64(Float64(Float64(pi / sin(Float64(pi * z))) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5)))) * Float64(Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) - Float64(676.5203681218851 / Float64(z + -1.0))) + Float64(Float64(216.4324257752088 + Float64(z * 75.16060840893998)) - Float64(Float64(9.984369578019572e-6 / Float64(z - 7.0)) - Float64(1.5056327351493116e-7 / Float64(8.0 - z)))))))
end

MATLAB

function tmp = code(z)
	tmp = sqrt((pi * 2.0)) * (((pi / sin((pi * z))) * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5)))) * (((-1259.1392167224028 / (2.0 - z)) - (676.5203681218851 / (z + -1.0))) + ((216.4324257752088 + (z * 75.16060840893998)) - ((9.984369578019572e-6 / (z - 7.0)) - (1.5056327351493116e-7 / (8.0 - z))))));
end

Wolfram

code[z_] := N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] - N[(676.5203681218851 / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(216.4324257752088 + N[(z * 75.16060840893998), $MachinePrecision]), $MachinePrecision] - N[(N[(9.984369578019572e-6 / N[(z - 7.0), $MachinePrecision]), $MachinePrecision] - N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.4%

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

3. Simplified 95.4%

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

   1. pow1 95.4%

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

6. Applied egg-rr 95.3%

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

8. Simplified 95.6%

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

9. Taylor expanded in z around 0 95.0%

   \[\leadsto \sqrt{2 \cdot \pi} \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\color{blue}{\left(216.4324257752088 + 75.16060840893998 \cdot z\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right) \]
10. Step-by-step derivation

    1. *-commutative 95.0%

       \[\leadsto \sqrt{2 \cdot \pi} \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\left(216.4324257752088 + \color{blue}{z \cdot 75.16060840893998}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right) \]

11. Simplified 95.0%

    \[\leadsto \sqrt{2 \cdot \pi} \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right) + \left(\color{blue}{\left(216.4324257752088 + z \cdot 75.16060840893998\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right) \]

12. Final simplification 95.0%

    \[\leadsto \sqrt{\pi \cdot 2} \cdot \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\left(\frac{-1259.1392167224028}{2 - z} - \frac{676.5203681218851}{z + -1}\right) + \left(\left(216.4324257752088 + z \cdot 75.16060840893998\right) - \left(\frac{9.984369578019572 \cdot 10^{-6}}{z - 7} - \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 12: 96.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 95.2%

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

5. Taylor expanded in z around 0 94.9%

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

   1. *-commutative 94.9%

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

7. Simplified 94.9%

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

8. Final simplification 94.9%

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

Alternative 13: 96.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 95.2%

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

5. Taylor expanded in z around 0 94.8%

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

   1. *-commutative 94.8%

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

7. Simplified 94.8%

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

8. Final simplification 94.8%

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

Alternative 14: 96.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{\pi \cdot 2} \cdot \left({\left(6.5 - \left(z + -1\right)\right)}^{\left(-0.5 - \left(z + -1\right)\right)} \cdot e^{\left(z + -1\right) - 6.5}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + z \cdot 606.656776085461\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{6 - \left(z + -1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{7 - \left(z + -1\right)}\right) + \left(\frac{-0.13857109526572012}{5 - \left(z + -1\right)} - \frac{12.507343278686905}{\left(z + -1\right) - 4}\right)\right)\right) \cdot \frac{\pi}{\pi \cdot z}\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (sqrt (* PI 2.0))
   (* (pow (- 6.5 (+ z -1.0)) (- -0.5 (+ z -1.0))) (exp (- (+ z -1.0) 6.5))))
  (*
   (+
    (+
     260.9048120626994
     (*
      z
      (+
       436.3997278161676
       (* z (+ 544.9358906000987 (* z 606.656776085461))))))
    (+
     (+
      (/ 9.984369578019572e-6 (- 6.0 (+ z -1.0)))
      (/ 1.5056327351493116e-7 (- 7.0 (+ z -1.0))))
     (-
      (/ -0.13857109526572012 (- 5.0 (+ z -1.0)))
      (/ 12.507343278686905 (- (+ z -1.0) 4.0)))))
   (/ PI (* PI z)))))

C

double code(double z) {
	return (sqrt((((double) M_PI) * 2.0)) * (pow((6.5 - (z + -1.0)), (-0.5 - (z + -1.0))) * exp(((z + -1.0) - 6.5)))) * (((260.9048120626994 + (z * (436.3997278161676 + (z * (544.9358906000987 + (z * 606.656776085461)))))) + (((9.984369578019572e-6 / (6.0 - (z + -1.0))) + (1.5056327351493116e-7 / (7.0 - (z + -1.0)))) + ((-0.13857109526572012 / (5.0 - (z + -1.0))) - (12.507343278686905 / ((z + -1.0) - 4.0))))) * (((double) M_PI) / (((double) M_PI) * z)));
}

Java

public static double code(double z) {
	return (Math.sqrt((Math.PI * 2.0)) * (Math.pow((6.5 - (z + -1.0)), (-0.5 - (z + -1.0))) * Math.exp(((z + -1.0) - 6.5)))) * (((260.9048120626994 + (z * (436.3997278161676 + (z * (544.9358906000987 + (z * 606.656776085461)))))) + (((9.984369578019572e-6 / (6.0 - (z + -1.0))) + (1.5056327351493116e-7 / (7.0 - (z + -1.0)))) + ((-0.13857109526572012 / (5.0 - (z + -1.0))) - (12.507343278686905 / ((z + -1.0) - 4.0))))) * (Math.PI / (Math.PI * z)));
}

Python

def code(z):
	return (math.sqrt((math.pi * 2.0)) * (math.pow((6.5 - (z + -1.0)), (-0.5 - (z + -1.0))) * math.exp(((z + -1.0) - 6.5)))) * (((260.9048120626994 + (z * (436.3997278161676 + (z * (544.9358906000987 + (z * 606.656776085461)))))) + (((9.984369578019572e-6 / (6.0 - (z + -1.0))) + (1.5056327351493116e-7 / (7.0 - (z + -1.0)))) + ((-0.13857109526572012 / (5.0 - (z + -1.0))) - (12.507343278686905 / ((z + -1.0) - 4.0))))) * (math.pi / (math.pi * z)))

Julia

function code(z)
	return Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(6.5 - Float64(z + -1.0)) ^ Float64(-0.5 - Float64(z + -1.0))) * exp(Float64(Float64(z + -1.0) - 6.5)))) * Float64(Float64(Float64(260.9048120626994 + Float64(z * Float64(436.3997278161676 + Float64(z * Float64(544.9358906000987 + Float64(z * 606.656776085461)))))) + Float64(Float64(Float64(9.984369578019572e-6 / Float64(6.0 - Float64(z + -1.0))) + Float64(1.5056327351493116e-7 / Float64(7.0 - Float64(z + -1.0)))) + Float64(Float64(-0.13857109526572012 / Float64(5.0 - Float64(z + -1.0))) - Float64(12.507343278686905 / Float64(Float64(z + -1.0) - 4.0))))) * Float64(pi / Float64(pi * z))))
end

MATLAB

function tmp = code(z)
	tmp = (sqrt((pi * 2.0)) * (((6.5 - (z + -1.0)) ^ (-0.5 - (z + -1.0))) * exp(((z + -1.0) - 6.5)))) * (((260.9048120626994 + (z * (436.3997278161676 + (z * (544.9358906000987 + (z * 606.656776085461)))))) + (((9.984369578019572e-6 / (6.0 - (z + -1.0))) + (1.5056327351493116e-7 / (7.0 - (z + -1.0)))) + ((-0.13857109526572012 / (5.0 - (z + -1.0))) - (12.507343278686905 / ((z + -1.0) - 4.0))))) * (pi / (pi * z)));
end

Wolfram

code[z_] := N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(6.5 - N[(z + -1.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 - N[(z + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(z + -1.0), $MachinePrecision] - 6.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(260.9048120626994 + N[(z * N[(436.3997278161676 + N[(z * N[(544.9358906000987 + N[(z * 606.656776085461), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(9.984369578019572e-6 / N[(6.0 - N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(7.0 - N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(5.0 - N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(12.507343278686905 / N[(N[(z + -1.0), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi / N[(Pi * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.4%

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

3. Simplified 95.2%

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

5. Taylor expanded in z around 0 95.1%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + 606.656776085461 \cdot z\right)\right)\right)} + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]
6. Step-by-step derivation

   1. *-commutative 95.1%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + \color{blue}{z \cdot 606.656776085461}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

7. Simplified 95.1%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + z \cdot 606.656776085461\right)\right)\right)} + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \]

8. Taylor expanded in z around 0 94.7%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + z \cdot 606.656776085461\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{\left(1 - z\right) + 5}\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) + 7}\right)\right)\right) \cdot \frac{\pi}{\color{blue}{z \cdot \pi}}\right) \]

9. Final simplification 94.7%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(6.5 - \left(z + -1\right)\right)}^{\left(-0.5 - \left(z + -1\right)\right)} \cdot e^{\left(z + -1\right) - 6.5}\right)\right) \cdot \left(\left(\left(260.9048120626994 + z \cdot \left(436.3997278161676 + z \cdot \left(544.9358906000987 + z \cdot 606.656776085461\right)\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{6 - \left(z + -1\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{7 - \left(z + -1\right)}\right) + \left(\frac{-0.13857109526572012}{5 - \left(z + -1\right)} - \frac{12.507343278686905}{\left(z + -1\right) - 4}\right)\right)\right) \cdot \frac{\pi}{\pi \cdot z}\right) \]
10. Add Preprocessing

Alternative 15: 95.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 95.2%

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

5. Taylor expanded in z around 0 94.5%

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

   1. *-commutative 94.5%

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

7. Simplified 94.5%

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

8. Final simplification 94.5%

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

Alternative 16: 96.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 97.0%

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

5. Taylor expanded in z around 0 94.0%

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

6. Taylor expanded in z around 0 93.7%

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

7. Taylor expanded in z around 0 93.9%

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

   1. associate-*l/ 93.7%

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

   2. *-commutative 93.7%

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

   3. associate-*r* 94.5%

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

9. Simplified 94.5%

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

    1. associate-/l* 94.3%

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

    2. *-commutative 94.3%

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

    3. sqrt-unprod 94.3%

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

    4. metadata-eval 94.3%

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

11. Applied egg-rr 94.3%

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

12. Final simplification 94.3%

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

Alternative 17: 94.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 97.0%

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

5. Taylor expanded in z around 0 94.0%

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

6. Taylor expanded in z around 0 93.7%

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

7. Taylor expanded in z around 0 93.9%

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

   1. associate-*l/ 93.7%

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

   2. *-commutative 93.7%

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

   3. associate-*r* 94.5%

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

9. Simplified 94.5%

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

    1. associate-*l/ 94.3%

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

    2. associate-*l* 93.4%

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

    3. sqrt-unprod 93.4%

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

    4. metadata-eval 93.4%

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

11. Applied egg-rr 93.4%

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

    1. associate-/l* 93.4%

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

13. Simplified 93.4%

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

Reproduce

herbie shell --seed 2024137 
(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))))))