Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.3% → 99.2%

Time: 1.0min

Alternatives: 20

Speedup: 1.2×

• could not determine a ground truth

  1. z = -1.9871447063767578e+99

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: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 - z\right) \cdot \left(z - 2\right)\\ t_1 := \frac{676.5203681218851}{z + -1}\\ t_2 := \sin \left(\pi \cdot z\right)\\ t_3 := {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\\ t_4 := \left(2 - z\right) \cdot 771.3234287776531\\ \mathbf{if}\;z \leq -7:\\ \;\;\;\;\pi \cdot \frac{\left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} - \left(\frac{-0.13857109526572012}{\left(z + -1\right) - 5} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{z - 8} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z + -1\right) - 6}\right)\right)\right) - \left(t\_1 + \left(\left(\frac{-1259.1392167224028 \cdot \left(3 - z\right) + t\_4}{t\_0} + \frac{-176.6150291621406}{\left(z + -1\right) - 3}\right) - 0.9999999999998099\right)\right)\right) \cdot e^{\log \left(t\_3 \cdot \left(\sqrt{\pi} \cdot \sqrt{2}\right)\right) + \left(z + -7.5\right)}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\sqrt{\pi \cdot 2} \cdot \left(\left(t\_3 \cdot e^{z + -7.5}\right) \cdot \left(\frac{12.507343278686905}{5 - z} + \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right) - \left(t\_1 + \left(\frac{\mathsf{fma}\left(-1259.1392167224028, 3 - z, t\_4\right)}{t\_0} + \left(\frac{-176.6150291621406}{z - 4} - 0.9999999999998099\right)\right)\right)\right)\right)\right)}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (* (- 3.0 z) (- z 2.0)))
        (t_1 (/ 676.5203681218851 (+ z -1.0)))
        (t_2 (sin (* PI z)))
        (t_3 (pow (- 7.5 z) (- 0.5 z)))
        (t_4 (* (- 2.0 z) 771.3234287776531)))
   (if (<= z -7.0)
     (*
      PI
      (/
       (*
        (-
         (-
          (/ 12.507343278686905 (+ (- 1.0 z) 4.0))
          (+
           (/ -0.13857109526572012 (- (+ z -1.0) 5.0))
           (+
            (/ 1.5056327351493116e-7 (- z 8.0))
            (/ 9.984369578019572e-6 (- (+ z -1.0) 6.0)))))
         (+
          t_1
          (-
           (+
            (/ (+ (* -1259.1392167224028 (- 3.0 z)) t_4) t_0)
            (/ -176.6150291621406 (- (+ z -1.0) 3.0)))
           0.9999999999998099)))
        (exp (+ (log (* t_3 (* (sqrt PI) (sqrt 2.0)))) (+ z -7.5))))
       t_2))
     (*
      PI
      (/
       (*
        (sqrt (* PI 2.0))
        (*
         (* t_3 (exp (+ z -7.5)))
         (+
          (/ 12.507343278686905 (- 5.0 z))
          (-
           (+
            (/ -0.13857109526572012 (- 6.0 z))
            (+
             (/ 1.5056327351493116e-7 (- 8.0 z))
             (/ 9.984369578019572e-6 (- 7.0 z))))
           (+
            t_1
            (+
             (/ (fma -1259.1392167224028 (- 3.0 z) t_4) t_0)
             (- (/ -176.6150291621406 (- z 4.0)) 0.9999999999998099)))))))
       t_2)))))

C

double code(double z) {
	double t_0 = (3.0 - z) * (z - 2.0);
	double t_1 = 676.5203681218851 / (z + -1.0);
	double t_2 = sin((((double) M_PI) * z));
	double t_3 = pow((7.5 - z), (0.5 - z));
	double t_4 = (2.0 - z) * 771.3234287776531;
	double tmp;
	if (z <= -7.0) {
		tmp = ((double) M_PI) * (((((12.507343278686905 / ((1.0 - z) + 4.0)) - ((-0.13857109526572012 / ((z + -1.0) - 5.0)) + ((1.5056327351493116e-7 / (z - 8.0)) + (9.984369578019572e-6 / ((z + -1.0) - 6.0))))) - (t_1 + (((((-1259.1392167224028 * (3.0 - z)) + t_4) / t_0) + (-176.6150291621406 / ((z + -1.0) - 3.0))) - 0.9999999999998099))) * exp((log((t_3 * (sqrt(((double) M_PI)) * sqrt(2.0)))) + (z + -7.5)))) / t_2);
	} else {
		tmp = ((double) M_PI) * ((sqrt((((double) M_PI) * 2.0)) * ((t_3 * exp((z + -7.5))) * ((12.507343278686905 / (5.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + (9.984369578019572e-6 / (7.0 - z)))) - (t_1 + ((fma(-1259.1392167224028, (3.0 - z), t_4) / t_0) + ((-176.6150291621406 / (z - 4.0)) - 0.9999999999998099))))))) / t_2);
	}
	return tmp;
}

Julia

function code(z)
	t_0 = Float64(Float64(3.0 - z) * Float64(z - 2.0))
	t_1 = Float64(676.5203681218851 / Float64(z + -1.0))
	t_2 = sin(Float64(pi * z))
	t_3 = Float64(7.5 - z) ^ Float64(0.5 - z)
	t_4 = Float64(Float64(2.0 - z) * 771.3234287776531)
	tmp = 0.0
	if (z <= -7.0)
		tmp = Float64(pi * Float64(Float64(Float64(Float64(Float64(12.507343278686905 / Float64(Float64(1.0 - z) + 4.0)) - Float64(Float64(-0.13857109526572012 / Float64(Float64(z + -1.0) - 5.0)) + Float64(Float64(1.5056327351493116e-7 / Float64(z - 8.0)) + Float64(9.984369578019572e-6 / Float64(Float64(z + -1.0) - 6.0))))) - Float64(t_1 + Float64(Float64(Float64(Float64(Float64(-1259.1392167224028 * Float64(3.0 - z)) + t_4) / t_0) + Float64(-176.6150291621406 / Float64(Float64(z + -1.0) - 3.0))) - 0.9999999999998099))) * exp(Float64(log(Float64(t_3 * Float64(sqrt(pi) * sqrt(2.0)))) + Float64(z + -7.5)))) / t_2));
	else
		tmp = Float64(pi * Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64(Float64(t_3 * exp(Float64(z + -7.5))) * Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) + Float64(9.984369578019572e-6 / Float64(7.0 - z)))) - Float64(t_1 + Float64(Float64(fma(-1259.1392167224028, Float64(3.0 - z), t_4) / t_0) + Float64(Float64(-176.6150291621406 / Float64(z - 4.0)) - 0.9999999999998099))))))) / t_2));
	end
	return tmp
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(3.0 - z), $MachinePrecision] * N[(z - 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(676.5203681218851 / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(2.0 - z), $MachinePrecision] * 771.3234287776531), $MachinePrecision]}, If[LessEqual[z, -7.0], N[(Pi * N[(N[(N[(N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.13857109526572012 / N[(N[(z + -1.0), $MachinePrecision] - 5.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.5056327351493116e-7 / N[(z - 8.0), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(N[(z + -1.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 + N[(N[(N[(N[(N[(-1259.1392167224028 * N[(3.0 - z), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(z + -1.0), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.9999999999998099), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Log[N[(t$95$3 * N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(z + -7.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$3 * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 + N[(N[(N[(-1259.1392167224028 * N[(3.0 - z), $MachinePrecision] + t$95$4), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(-176.6150291621406 / N[(z - 4.0), $MachinePrecision]), $MachinePrecision] - 0.9999999999998099), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if z < -7

   1. Initial program 13.3%

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

   3. Simplified 13.3%

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

   6. Applied egg-rr 13.2%

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

   8. Simplified 13.2%

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

      1. frac-add 13.7%

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

   10. Applied egg-rr 13.7%

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

       1. add-exp-log 13.7%

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

       2. associate-*r* 13.7%

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

       3. *-commutative 13.7%

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

       4. metadata-eval 13.7%

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

       5. distribute-neg-in 13.7%

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

       6. log-prod 14.3%

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

   12. Applied egg-rr 100.0%

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

   13. Taylor expanded in z around inf 100.0%

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

   15. Simplified 100.0%

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

   if -7 < z

   1. Initial program 97.4%

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

   3. Simplified 96.9%

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

   6. Applied egg-rr 98.9%

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

   8. Simplified 99.2%

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

      1. frac-add 99.2%

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

   10. Applied egg-rr 99.2%

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

   12. Applied egg-rr 99.2%

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

   14. Simplified 99.2%

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

4. Final simplification 99.3%

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

Alternative 2: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z + -1\right) - -1\\ t_1 := \sin \left(\pi \cdot z\right)\\ t_2 := \sqrt{\pi \cdot 2}\\ t_3 := 7 - t\_0\\ \mathbf{if}\;\left(\left(\left(t\_2 \cdot {\left(0.5 + t\_3\right)}^{\left(0.5 - t\_0\right)}\right) \cdot e^{\left(t\_0 - 7\right) - 0.5}\right) \cdot \left(\left(\left(\frac{-0.13857109526572012}{6 + \left(-1 + \left(1 - z\right)\right)} - \left(\left(\left(\left(\left(\frac{676.5203681218851}{-1 + t\_0} - 0.9999999999998099\right) + \frac{-1259.1392167224028}{t\_0 - 2}\right) + \frac{771.3234287776531}{t\_0 - 3}\right) + \frac{-176.6150291621406}{t\_0 - 4}\right) + \frac{12.507343278686905}{t\_0 - 5}\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_3}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - t\_0}\right)\right) \cdot \frac{\pi}{t\_1} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\pi \cdot \frac{\left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} - \left(\frac{-0.13857109526572012}{\left(z + -1\right) - 5} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{z - 8} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z + -1\right) - 6}\right)\right)\right) - \left(\frac{676.5203681218851}{z + -1} + \left(\left(\frac{-1259.1392167224028 \cdot \left(3 - z\right) + \left(2 - z\right) \cdot 771.3234287776531}{\left(3 - z\right) \cdot \left(z - 2\right)} + \frac{-176.6150291621406}{\left(z + -1\right) - 3}\right) - 0.9999999999998099\right)\right)\right) \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(t\_2 \cdot e^{z + -7.5}\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \frac{\pi \cdot 0.9999999999998099}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (+ z -1.0) -1.0))
        (t_1 (sin (* PI z)))
        (t_2 (sqrt (* PI 2.0)))
        (t_3 (- 7.0 t_0)))
   (if (<=
        (*
         (*
          (* (* t_2 (pow (+ 0.5 t_3) (- 0.5 t_0))) (exp (- (- t_0 7.0) 0.5)))
          (+
           (+
            (-
             (/ -0.13857109526572012 (+ 6.0 (+ -1.0 (- 1.0 z))))
             (+
              (+
               (+
                (+
                 (- (/ 676.5203681218851 (+ -1.0 t_0)) 0.9999999999998099)
                 (/ -1259.1392167224028 (- t_0 2.0)))
                (/ 771.3234287776531 (- t_0 3.0)))
               (/ -176.6150291621406 (- t_0 4.0)))
              (/ 12.507343278686905 (- t_0 5.0))))
            (/ 9.984369578019572e-6 t_3))
           (/ 1.5056327351493116e-7 (- 8.0 t_0))))
         (/ PI t_1))
        2e+307)
     (*
      PI
      (/
       (*
        (-
         (-
          (/ 12.507343278686905 (+ (- 1.0 z) 4.0))
          (+
           (/ -0.13857109526572012 (- (+ z -1.0) 5.0))
           (+
            (/ 1.5056327351493116e-7 (- z 8.0))
            (/ 9.984369578019572e-6 (- (+ z -1.0) 6.0)))))
         (+
          (/ 676.5203681218851 (+ z -1.0))
          (-
           (+
            (/
             (+
              (* -1259.1392167224028 (- 3.0 z))
              (* (- 2.0 z) 771.3234287776531))
             (* (- 3.0 z) (- z 2.0)))
            (/ -176.6150291621406 (- (+ z -1.0) 3.0)))
           0.9999999999998099)))
        (* (pow (- 7.5 z) (- 0.5 z)) (* t_2 (exp (+ z -7.5)))))
       t_1))
     (*
      (* t_2 (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp -7.5)))
      (/ (* PI 0.9999999999998099) t_1)))))

C

double code(double z) {
	double t_0 = (z + -1.0) - -1.0;
	double t_1 = sin((((double) M_PI) * z));
	double t_2 = sqrt((((double) M_PI) * 2.0));
	double t_3 = 7.0 - t_0;
	double tmp;
	if (((((t_2 * pow((0.5 + t_3), (0.5 - t_0))) * exp(((t_0 - 7.0) - 0.5))) * ((((-0.13857109526572012 / (6.0 + (-1.0 + (1.0 - z)))) - ((((((676.5203681218851 / (-1.0 + t_0)) - 0.9999999999998099) + (-1259.1392167224028 / (t_0 - 2.0))) + (771.3234287776531 / (t_0 - 3.0))) + (-176.6150291621406 / (t_0 - 4.0))) + (12.507343278686905 / (t_0 - 5.0)))) + (9.984369578019572e-6 / t_3)) + (1.5056327351493116e-7 / (8.0 - t_0)))) * (((double) M_PI) / t_1)) <= 2e+307) {
		tmp = ((double) M_PI) * (((((12.507343278686905 / ((1.0 - z) + 4.0)) - ((-0.13857109526572012 / ((z + -1.0) - 5.0)) + ((1.5056327351493116e-7 / (z - 8.0)) + (9.984369578019572e-6 / ((z + -1.0) - 6.0))))) - ((676.5203681218851 / (z + -1.0)) + (((((-1259.1392167224028 * (3.0 - z)) + ((2.0 - z) * 771.3234287776531)) / ((3.0 - z) * (z - 2.0))) + (-176.6150291621406 / ((z + -1.0) - 3.0))) - 0.9999999999998099))) * (pow((7.5 - z), (0.5 - z)) * (t_2 * exp((z + -7.5))))) / t_1);
	} else {
		tmp = (t_2 * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(-7.5))) * ((((double) M_PI) * 0.9999999999998099) / t_1);
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = (z + -1.0) - -1.0;
	double t_1 = Math.sin((Math.PI * z));
	double t_2 = Math.sqrt((Math.PI * 2.0));
	double t_3 = 7.0 - t_0;
	double tmp;
	if (((((t_2 * Math.pow((0.5 + t_3), (0.5 - t_0))) * Math.exp(((t_0 - 7.0) - 0.5))) * ((((-0.13857109526572012 / (6.0 + (-1.0 + (1.0 - z)))) - ((((((676.5203681218851 / (-1.0 + t_0)) - 0.9999999999998099) + (-1259.1392167224028 / (t_0 - 2.0))) + (771.3234287776531 / (t_0 - 3.0))) + (-176.6150291621406 / (t_0 - 4.0))) + (12.507343278686905 / (t_0 - 5.0)))) + (9.984369578019572e-6 / t_3)) + (1.5056327351493116e-7 / (8.0 - t_0)))) * (Math.PI / t_1)) <= 2e+307) {
		tmp = Math.PI * (((((12.507343278686905 / ((1.0 - z) + 4.0)) - ((-0.13857109526572012 / ((z + -1.0) - 5.0)) + ((1.5056327351493116e-7 / (z - 8.0)) + (9.984369578019572e-6 / ((z + -1.0) - 6.0))))) - ((676.5203681218851 / (z + -1.0)) + (((((-1259.1392167224028 * (3.0 - z)) + ((2.0 - z) * 771.3234287776531)) / ((3.0 - z) * (z - 2.0))) + (-176.6150291621406 / ((z + -1.0) - 3.0))) - 0.9999999999998099))) * (Math.pow((7.5 - z), (0.5 - z)) * (t_2 * Math.exp((z + -7.5))))) / t_1);
	} else {
		tmp = (t_2 * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp(-7.5))) * ((Math.PI * 0.9999999999998099) / t_1);
	}
	return tmp;
}

Python

def code(z):
	t_0 = (z + -1.0) - -1.0
	t_1 = math.sin((math.pi * z))
	t_2 = math.sqrt((math.pi * 2.0))
	t_3 = 7.0 - t_0
	tmp = 0
	if ((((t_2 * math.pow((0.5 + t_3), (0.5 - t_0))) * math.exp(((t_0 - 7.0) - 0.5))) * ((((-0.13857109526572012 / (6.0 + (-1.0 + (1.0 - z)))) - ((((((676.5203681218851 / (-1.0 + t_0)) - 0.9999999999998099) + (-1259.1392167224028 / (t_0 - 2.0))) + (771.3234287776531 / (t_0 - 3.0))) + (-176.6150291621406 / (t_0 - 4.0))) + (12.507343278686905 / (t_0 - 5.0)))) + (9.984369578019572e-6 / t_3)) + (1.5056327351493116e-7 / (8.0 - t_0)))) * (math.pi / t_1)) <= 2e+307:
		tmp = math.pi * (((((12.507343278686905 / ((1.0 - z) + 4.0)) - ((-0.13857109526572012 / ((z + -1.0) - 5.0)) + ((1.5056327351493116e-7 / (z - 8.0)) + (9.984369578019572e-6 / ((z + -1.0) - 6.0))))) - ((676.5203681218851 / (z + -1.0)) + (((((-1259.1392167224028 * (3.0 - z)) + ((2.0 - z) * 771.3234287776531)) / ((3.0 - z) * (z - 2.0))) + (-176.6150291621406 / ((z + -1.0) - 3.0))) - 0.9999999999998099))) * (math.pow((7.5 - z), (0.5 - z)) * (t_2 * math.exp((z + -7.5))))) / t_1)
	else:
		tmp = (t_2 * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp(-7.5))) * ((math.pi * 0.9999999999998099) / t_1)
	return tmp

Julia

function code(z)
	t_0 = Float64(Float64(z + -1.0) - -1.0)
	t_1 = sin(Float64(pi * z))
	t_2 = sqrt(Float64(pi * 2.0))
	t_3 = Float64(7.0 - t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(t_2 * (Float64(0.5 + t_3) ^ Float64(0.5 - t_0))) * exp(Float64(Float64(t_0 - 7.0) - 0.5))) * Float64(Float64(Float64(Float64(-0.13857109526572012 / Float64(6.0 + Float64(-1.0 + Float64(1.0 - z)))) - Float64(Float64(Float64(Float64(Float64(Float64(676.5203681218851 / Float64(-1.0 + t_0)) - 0.9999999999998099) + 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(9.984369578019572e-6 / t_3)) + Float64(1.5056327351493116e-7 / Float64(8.0 - t_0)))) * Float64(pi / t_1)) <= 2e+307)
		tmp = Float64(pi * Float64(Float64(Float64(Float64(Float64(12.507343278686905 / Float64(Float64(1.0 - z) + 4.0)) - Float64(Float64(-0.13857109526572012 / Float64(Float64(z + -1.0) - 5.0)) + Float64(Float64(1.5056327351493116e-7 / Float64(z - 8.0)) + Float64(9.984369578019572e-6 / Float64(Float64(z + -1.0) - 6.0))))) - Float64(Float64(676.5203681218851 / Float64(z + -1.0)) + Float64(Float64(Float64(Float64(Float64(-1259.1392167224028 * Float64(3.0 - z)) + Float64(Float64(2.0 - z) * 771.3234287776531)) / Float64(Float64(3.0 - z) * Float64(z - 2.0))) + Float64(-176.6150291621406 / Float64(Float64(z + -1.0) - 3.0))) - 0.9999999999998099))) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * Float64(t_2 * exp(Float64(z + -7.5))))) / t_1));
	else
		tmp = Float64(Float64(t_2 * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(-7.5))) * Float64(Float64(pi * 0.9999999999998099) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = (z + -1.0) - -1.0;
	t_1 = sin((pi * z));
	t_2 = sqrt((pi * 2.0));
	t_3 = 7.0 - t_0;
	tmp = 0.0;
	if (((((t_2 * ((0.5 + t_3) ^ (0.5 - t_0))) * exp(((t_0 - 7.0) - 0.5))) * ((((-0.13857109526572012 / (6.0 + (-1.0 + (1.0 - z)))) - ((((((676.5203681218851 / (-1.0 + t_0)) - 0.9999999999998099) + (-1259.1392167224028 / (t_0 - 2.0))) + (771.3234287776531 / (t_0 - 3.0))) + (-176.6150291621406 / (t_0 - 4.0))) + (12.507343278686905 / (t_0 - 5.0)))) + (9.984369578019572e-6 / t_3)) + (1.5056327351493116e-7 / (8.0 - t_0)))) * (pi / t_1)) <= 2e+307)
		tmp = pi * (((((12.507343278686905 / ((1.0 - z) + 4.0)) - ((-0.13857109526572012 / ((z + -1.0) - 5.0)) + ((1.5056327351493116e-7 / (z - 8.0)) + (9.984369578019572e-6 / ((z + -1.0) - 6.0))))) - ((676.5203681218851 / (z + -1.0)) + (((((-1259.1392167224028 * (3.0 - z)) + ((2.0 - z) * 771.3234287776531)) / ((3.0 - z) * (z - 2.0))) + (-176.6150291621406 / ((z + -1.0) - 3.0))) - 0.9999999999998099))) * (((7.5 - z) ^ (0.5 - z)) * (t_2 * exp((z + -7.5))))) / t_1);
	else
		tmp = (t_2 * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(-7.5))) * ((pi * 0.9999999999998099) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(z + -1.0), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(7.0 - t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(t$95$2 * N[Power[N[(0.5 + t$95$3), $MachinePrecision], N[(0.5 - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t$95$0 - 7.0), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(-0.13857109526572012 / N[(6.0 + N[(-1.0 + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(N[(N[(676.5203681218851 / N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] - 0.9999999999998099), $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]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$3), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi / t$95$1), $MachinePrecision]), $MachinePrecision], 2e+307], N[(Pi * N[(N[(N[(N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.13857109526572012 / N[(N[(z + -1.0), $MachinePrecision] - 5.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.5056327351493116e-7 / N[(z - 8.0), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(N[(z + -1.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(676.5203681218851 / N[(z + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(-1259.1392167224028 * N[(3.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 - z), $MachinePrecision] * 771.3234287776531), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 - z), $MachinePrecision] * N[(z - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(z + -1.0), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.9999999999998099), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[(t$95$2 * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * 0.9999999999998099), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (PI.f64) (sin.f64 (*.f64 (PI.f64) z))) (*.f64 (*.f64 (*.f64 (sqrt.f64 (*.f64 (PI.f64) #s(literal 2 binary64))) (pow.f64 (+.f64 (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)) #s(literal 1/2 binary64)) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 1/2 binary64)))) (exp.f64 (neg.f64 (+.f64 (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)) #s(literal 1/2 binary64))))) (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 #s(literal 9999999999998099/10000000000000000 binary64) (/.f64 #s(literal 6765203681218851/10000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 1 binary64)))) (/.f64 #s(literal -3147848041806007/2500000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 2 binary64)))) (/.f64 #s(literal 7713234287776531/10000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 3 binary64)))) (/.f64 #s(literal -883075145810703/5000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 4 binary64)))) (/.f64 #s(literal 2501468655737381/200000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 5 binary64)))) (/.f64 #s(literal -3464277381643003/25000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 6 binary64)))) (/.f64 #s(literal 2496092394504893/250000000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)))) (/.f64 #s(literal 3764081837873279/25000000000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 8 binary64)))))) < 1.99999999999999997e307

   1. Initial program 97.3%

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

   3. Simplified 96.9%

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

   6. Applied egg-rr 98.9%

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

   8. Simplified 99.2%

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

      1. frac-add 99.2%

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

   10. Applied egg-rr 99.2%

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

       1. add-exp-log 97.8%

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

       2. associate-*r* 97.8%

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

       3. *-commutative 97.8%

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

       4. metadata-eval 97.8%

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

       5. distribute-neg-in 97.8%

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

       6. log-prod 97.8%

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

   12. Applied egg-rr 97.8%

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

       1. *-un-lft-identity 97.8%

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

       2. exp-sum 99.2%

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

       3. add-exp-log 99.2%

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

       4. *-commutative 99.2%

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

   14. Applied egg-rr 99.2%

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

       1. *-lft-identity 99.2%

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

       2. associate-*l* 99.2%

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

       3. fma-undefine 99.2%

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

       4. neg-mul-1 99.2%

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

       5. +-commutative 99.2%

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

       6. associate-+r- 99.2%

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

       7. metadata-eval 99.2%

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

       8. +-commutative 99.2%

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

       9. associate-+r- 99.2%

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

       10. metadata-eval 99.2%

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

       11. *-commutative 99.2%

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

       12. +-commutative 99.2%

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

       13. +-commutative 99.2%

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

       14. metadata-eval 99.2%

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

       15. sub-neg 99.2%

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

   16. Simplified 99.2%

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

   if 1.99999999999999997e307 < (*.f64 (/.f64 (PI.f64) (sin.f64 (*.f64 (PI.f64) z))) (*.f64 (*.f64 (*.f64 (sqrt.f64 (*.f64 (PI.f64) #s(literal 2 binary64))) (pow.f64 (+.f64 (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)) #s(literal 1/2 binary64)) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 1/2 binary64)))) (exp.f64 (neg.f64 (+.f64 (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)) #s(literal 1/2 binary64))))) (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 #s(literal 9999999999998099/10000000000000000 binary64) (/.f64 #s(literal 6765203681218851/10000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 1 binary64)))) (/.f64 #s(literal -3147848041806007/2500000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 2 binary64)))) (/.f64 #s(literal 7713234287776531/10000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 3 binary64)))) (/.f64 #s(literal -883075145810703/5000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 4 binary64)))) (/.f64 #s(literal 2501468655737381/200000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 5 binary64)))) (/.f64 #s(literal -3464277381643003/25000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 6 binary64)))) (/.f64 #s(literal 2496092394504893/250000000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)))) (/.f64 #s(literal 3764081837873279/25000000000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 8 binary64))))))

   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 inf 0.0%

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

      1. associate-*r/ 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in z around 0 100.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{e^{-7.5}}\right)\right) \cdot \frac{0.9999999999998099 \cdot \pi}{\sin \left(z \cdot \pi\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(0.5 + \left(7 - \left(\left(z + -1\right) - -1\right)\right)\right)}^{\left(0.5 - \left(\left(z + -1\right) - -1\right)\right)}\right) \cdot e^{\left(\left(\left(z + -1\right) - -1\right) - 7\right) - 0.5}\right) \cdot \left(\left(\left(\frac{-0.13857109526572012}{6 + \left(-1 + \left(1 - z\right)\right)} - \left(\left(\left(\left(\left(\frac{676.5203681218851}{-1 + \left(\left(z + -1\right) - -1\right)} - 0.9999999999998099\right) + \frac{-1259.1392167224028}{\left(\left(z + -1\right) - -1\right) - 2}\right) + \frac{771.3234287776531}{\left(\left(z + -1\right) - -1\right) - 3}\right) + \frac{-176.6150291621406}{\left(\left(z + -1\right) - -1\right) - 4}\right) + \frac{12.507343278686905}{\left(\left(z + -1\right) - -1\right) - 5}\right)\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - \left(\left(z + -1\right) - -1\right)}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - \left(\left(z + -1\right) - -1\right)}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\pi \cdot \frac{\left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} - \left(\frac{-0.13857109526572012}{\left(z + -1\right) - 5} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{z - 8} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z + -1\right) - 6}\right)\right)\right) - \left(\frac{676.5203681218851}{z + -1} + \left(\left(\frac{-1259.1392167224028 \cdot \left(3 - z\right) + \left(2 - z\right) \cdot 771.3234287776531}{\left(3 - z\right) \cdot \left(z - 2\right)} + \frac{-176.6150291621406}{\left(z + -1\right) - 3}\right) - 0.9999999999998099\right)\right)\right) \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot e^{z + -7.5}\right)\right)}{\sin \left(\pi \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\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^{-7.5}\right)\right) \cdot \frac{\pi \cdot 0.9999999999998099}{\sin \left(\pi \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double z) {
	return ((double) M_PI) * ((exp(((log((sqrt(((double) M_PI)) * (sqrt(2.0) * pow((7.5 - z), (0.5 - z))))) + (z + -1.0)) - 6.5)) * (((12.507343278686905 / ((1.0 - z) + 4.0)) - ((-0.13857109526572012 / ((z + -1.0) - 5.0)) + ((1.5056327351493116e-7 / (z - 8.0)) + (9.984369578019572e-6 / ((z + -1.0) - 6.0))))) - ((676.5203681218851 / (z + -1.0)) + (((((-1259.1392167224028 * (3.0 - z)) + ((2.0 - z) * 771.3234287776531)) / ((3.0 - z) * (z - 2.0))) + (-176.6150291621406 / ((z + -1.0) - 3.0))) - 0.9999999999998099)))) / sin((((double) M_PI) * z)));
}

Java

public static double code(double z) {
	return Math.PI * ((Math.exp(((Math.log((Math.sqrt(Math.PI) * (Math.sqrt(2.0) * Math.pow((7.5 - z), (0.5 - z))))) + (z + -1.0)) - 6.5)) * (((12.507343278686905 / ((1.0 - z) + 4.0)) - ((-0.13857109526572012 / ((z + -1.0) - 5.0)) + ((1.5056327351493116e-7 / (z - 8.0)) + (9.984369578019572e-6 / ((z + -1.0) - 6.0))))) - ((676.5203681218851 / (z + -1.0)) + (((((-1259.1392167224028 * (3.0 - z)) + ((2.0 - z) * 771.3234287776531)) / ((3.0 - z) * (z - 2.0))) + (-176.6150291621406 / ((z + -1.0) - 3.0))) - 0.9999999999998099)))) / Math.sin((Math.PI * z)));
}

Python

def code(z):
	return math.pi * ((math.exp(((math.log((math.sqrt(math.pi) * (math.sqrt(2.0) * math.pow((7.5 - z), (0.5 - z))))) + (z + -1.0)) - 6.5)) * (((12.507343278686905 / ((1.0 - z) + 4.0)) - ((-0.13857109526572012 / ((z + -1.0) - 5.0)) + ((1.5056327351493116e-7 / (z - 8.0)) + (9.984369578019572e-6 / ((z + -1.0) - 6.0))))) - ((676.5203681218851 / (z + -1.0)) + (((((-1259.1392167224028 * (3.0 - z)) + ((2.0 - z) * 771.3234287776531)) / ((3.0 - z) * (z - 2.0))) + (-176.6150291621406 / ((z + -1.0) - 3.0))) - 0.9999999999998099)))) / math.sin((math.pi * z)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.1%

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

3. Simplified 94.6%

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

6. Applied egg-rr 96.5%

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

8. Simplified 96.9%

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

   1. frac-add 96.9%

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

10. Applied egg-rr 96.9%

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

    1. add-exp-log 95.5%

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

    2. associate-*r* 95.5%

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

    3. *-commutative 95.5%

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

    4. metadata-eval 95.5%

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

    5. distribute-neg-in 95.5%

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

    6. log-prod 95.5%

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

12. Applied egg-rr 97.9%

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

13. Taylor expanded in z around inf 97.9%

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

14. Final simplification 97.9%

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

Alternative 4: 98.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ t_1 := \sin \left(\pi \cdot z\right)\\ \mathbf{if}\;z \leq -20000:\\ \;\;\;\;\left(t\_0 \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \frac{\pi \cdot 0.9999999999998099}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{t\_0 \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \left(\frac{12.507343278686905}{5 - z} + \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right) - \left(\frac{676.5203681218851}{z + -1} + \left(\frac{\mathsf{fma}\left(-1259.1392167224028, 3 - z, \left(2 - z\right) \cdot 771.3234287776531\right)}{\left(3 - z\right) \cdot \left(z - 2\right)} + \left(\frac{-176.6150291621406}{z - 4} - 0.9999999999998099\right)\right)\right)\right)\right)\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 2.0))) (t_1 (sin (* PI z))))
   (if (<= z -20000.0)
     (*
      (* t_0 (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp -7.5)))
      (/ (* PI 0.9999999999998099) t_1))
     (*
      PI
      (/
       (*
        t_0
        (*
         (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5)))
         (+
          (/ 12.507343278686905 (- 5.0 z))
          (-
           (+
            (/ -0.13857109526572012 (- 6.0 z))
            (+
             (/ 1.5056327351493116e-7 (- 8.0 z))
             (/ 9.984369578019572e-6 (- 7.0 z))))
           (+
            (/ 676.5203681218851 (+ z -1.0))
            (+
             (/
              (fma
               -1259.1392167224028
               (- 3.0 z)
               (* (- 2.0 z) 771.3234287776531))
              (* (- 3.0 z) (- z 2.0)))
             (- (/ -176.6150291621406 (- z 4.0)) 0.9999999999998099)))))))
       t_1)))))

C

double code(double z) {
	double t_0 = sqrt((((double) M_PI) * 2.0));
	double t_1 = sin((((double) M_PI) * z));
	double tmp;
	if (z <= -20000.0) {
		tmp = (t_0 * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(-7.5))) * ((((double) M_PI) * 0.9999999999998099) / t_1);
	} else {
		tmp = ((double) M_PI) * ((t_0 * ((pow((7.5 - z), (0.5 - z)) * exp((z + -7.5))) * ((12.507343278686905 / (5.0 - z)) + (((-0.13857109526572012 / (6.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + (9.984369578019572e-6 / (7.0 - z)))) - ((676.5203681218851 / (z + -1.0)) + ((fma(-1259.1392167224028, (3.0 - z), ((2.0 - z) * 771.3234287776531)) / ((3.0 - z) * (z - 2.0))) + ((-176.6150291621406 / (z - 4.0)) - 0.9999999999998099))))))) / t_1);
	}
	return tmp;
}

Julia

function code(z)
	t_0 = sqrt(Float64(pi * 2.0))
	t_1 = sin(Float64(pi * z))
	tmp = 0.0
	if (z <= -20000.0)
		tmp = Float64(Float64(t_0 * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(-7.5))) * Float64(Float64(pi * 0.9999999999998099) / t_1));
	else
		tmp = Float64(pi * Float64(Float64(t_0 * Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5))) * Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) + Float64(9.984369578019572e-6 / Float64(7.0 - z)))) - Float64(Float64(676.5203681218851 / Float64(z + -1.0)) + Float64(Float64(fma(-1259.1392167224028, Float64(3.0 - z), Float64(Float64(2.0 - z) * 771.3234287776531)) / Float64(Float64(3.0 - z) * Float64(z - 2.0))) + Float64(Float64(-176.6150291621406 / Float64(z - 4.0)) - 0.9999999999998099))))))) / t_1));
	end
	return tmp
end

Wolfram

code[z_] := Block[{t$95$0 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -20000.0], N[(N[(t$95$0 * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * 0.9999999999998099), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(N[(t$95$0 * N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(676.5203681218851 / N[(z + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-1259.1392167224028 * N[(3.0 - z), $MachinePrecision] + N[(N[(2.0 - z), $MachinePrecision] * 771.3234287776531), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 - z), $MachinePrecision] * N[(z - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-176.6150291621406 / N[(z - 4.0), $MachinePrecision]), $MachinePrecision] - 0.9999999999998099), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < -2e4

   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 inf 0.0%

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

      1. associate-*r/ 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in z around 0 100.0%

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

   if -2e4 < 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 96.9%

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

   6. Applied egg-rr 98.9%

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

   8. Simplified 99.2%

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

      1. frac-add 99.2%

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

   10. Applied egg-rr 99.2%

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

   12. Applied egg-rr 99.2%

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

   14. Simplified 99.2%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -20000:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \frac{\pi \cdot 0.9999999999998099}{\sin \left(\pi \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\sqrt{\pi \cdot 2} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \left(\frac{12.507343278686905}{5 - z} + \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right) - \left(\frac{676.5203681218851}{z + -1} + \left(\frac{\mathsf{fma}\left(-1259.1392167224028, 3 - z, \left(2 - z\right) \cdot 771.3234287776531\right)}{\left(3 - z\right) \cdot \left(z - 2\right)} + \left(\frac{-176.6150291621406}{z - 4} - 0.9999999999998099\right)\right)\right)\right)\right)\right)}{\sin \left(\pi \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ t_1 := \sin \left(\pi \cdot z\right)\\ \mathbf{if}\;z \leq -20000:\\ \;\;\;\;\left(t\_0 \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \frac{\pi \cdot 0.9999999999998099}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} - \left(\frac{-0.13857109526572012}{\left(z + -1\right) - 5} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{z - 8} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z + -1\right) - 6}\right)\right)\right) - \left(\frac{676.5203681218851}{z + -1} + \left(\left(\frac{-1259.1392167224028 \cdot \left(3 - z\right) + \left(2 - z\right) \cdot 771.3234287776531}{\left(3 - z\right) \cdot \left(z - 2\right)} + \frac{-176.6150291621406}{\left(z + -1\right) - 3}\right) - 0.9999999999998099\right)\right)\right) \cdot \left(t\_0 \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 2.0))) (t_1 (sin (* PI z))))
   (if (<= z -20000.0)
     (*
      (* t_0 (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp -7.5)))
      (/ (* PI 0.9999999999998099) t_1))
     (*
      PI
      (/
       (*
        (-
         (-
          (/ 12.507343278686905 (+ (- 1.0 z) 4.0))
          (+
           (/ -0.13857109526572012 (- (+ z -1.0) 5.0))
           (+
            (/ 1.5056327351493116e-7 (- z 8.0))
            (/ 9.984369578019572e-6 (- (+ z -1.0) 6.0)))))
         (+
          (/ 676.5203681218851 (+ z -1.0))
          (-
           (+
            (/
             (+
              (* -1259.1392167224028 (- 3.0 z))
              (* (- 2.0 z) 771.3234287776531))
             (* (- 3.0 z) (- z 2.0)))
            (/ -176.6150291621406 (- (+ z -1.0) 3.0)))
           0.9999999999998099)))
        (* t_0 (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5)))))
       t_1)))))

C

double code(double z) {
	double t_0 = sqrt((((double) M_PI) * 2.0));
	double t_1 = sin((((double) M_PI) * z));
	double tmp;
	if (z <= -20000.0) {
		tmp = (t_0 * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(-7.5))) * ((((double) M_PI) * 0.9999999999998099) / t_1);
	} else {
		tmp = ((double) M_PI) * (((((12.507343278686905 / ((1.0 - z) + 4.0)) - ((-0.13857109526572012 / ((z + -1.0) - 5.0)) + ((1.5056327351493116e-7 / (z - 8.0)) + (9.984369578019572e-6 / ((z + -1.0) - 6.0))))) - ((676.5203681218851 / (z + -1.0)) + (((((-1259.1392167224028 * (3.0 - z)) + ((2.0 - z) * 771.3234287776531)) / ((3.0 - z) * (z - 2.0))) + (-176.6150291621406 / ((z + -1.0) - 3.0))) - 0.9999999999998099))) * (t_0 * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5))))) / t_1);
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.sqrt((Math.PI * 2.0));
	double t_1 = Math.sin((Math.PI * z));
	double tmp;
	if (z <= -20000.0) {
		tmp = (t_0 * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp(-7.5))) * ((Math.PI * 0.9999999999998099) / t_1);
	} else {
		tmp = Math.PI * (((((12.507343278686905 / ((1.0 - z) + 4.0)) - ((-0.13857109526572012 / ((z + -1.0) - 5.0)) + ((1.5056327351493116e-7 / (z - 8.0)) + (9.984369578019572e-6 / ((z + -1.0) - 6.0))))) - ((676.5203681218851 / (z + -1.0)) + (((((-1259.1392167224028 * (3.0 - z)) + ((2.0 - z) * 771.3234287776531)) / ((3.0 - z) * (z - 2.0))) + (-176.6150291621406 / ((z + -1.0) - 3.0))) - 0.9999999999998099))) * (t_0 * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5))))) / t_1);
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.sqrt((math.pi * 2.0))
	t_1 = math.sin((math.pi * z))
	tmp = 0
	if z <= -20000.0:
		tmp = (t_0 * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp(-7.5))) * ((math.pi * 0.9999999999998099) / t_1)
	else:
		tmp = math.pi * (((((12.507343278686905 / ((1.0 - z) + 4.0)) - ((-0.13857109526572012 / ((z + -1.0) - 5.0)) + ((1.5056327351493116e-7 / (z - 8.0)) + (9.984369578019572e-6 / ((z + -1.0) - 6.0))))) - ((676.5203681218851 / (z + -1.0)) + (((((-1259.1392167224028 * (3.0 - z)) + ((2.0 - z) * 771.3234287776531)) / ((3.0 - z) * (z - 2.0))) + (-176.6150291621406 / ((z + -1.0) - 3.0))) - 0.9999999999998099))) * (t_0 * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5))))) / t_1)
	return tmp

Julia

function code(z)
	t_0 = sqrt(Float64(pi * 2.0))
	t_1 = sin(Float64(pi * z))
	tmp = 0.0
	if (z <= -20000.0)
		tmp = Float64(Float64(t_0 * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(-7.5))) * Float64(Float64(pi * 0.9999999999998099) / t_1));
	else
		tmp = Float64(pi * Float64(Float64(Float64(Float64(Float64(12.507343278686905 / Float64(Float64(1.0 - z) + 4.0)) - Float64(Float64(-0.13857109526572012 / Float64(Float64(z + -1.0) - 5.0)) + Float64(Float64(1.5056327351493116e-7 / Float64(z - 8.0)) + Float64(9.984369578019572e-6 / Float64(Float64(z + -1.0) - 6.0))))) - Float64(Float64(676.5203681218851 / Float64(z + -1.0)) + Float64(Float64(Float64(Float64(Float64(-1259.1392167224028 * Float64(3.0 - z)) + Float64(Float64(2.0 - z) * 771.3234287776531)) / Float64(Float64(3.0 - z) * Float64(z - 2.0))) + Float64(-176.6150291621406 / Float64(Float64(z + -1.0) - 3.0))) - 0.9999999999998099))) * Float64(t_0 * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5))))) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = sqrt((pi * 2.0));
	t_1 = sin((pi * z));
	tmp = 0.0;
	if (z <= -20000.0)
		tmp = (t_0 * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(-7.5))) * ((pi * 0.9999999999998099) / t_1);
	else
		tmp = pi * (((((12.507343278686905 / ((1.0 - z) + 4.0)) - ((-0.13857109526572012 / ((z + -1.0) - 5.0)) + ((1.5056327351493116e-7 / (z - 8.0)) + (9.984369578019572e-6 / ((z + -1.0) - 6.0))))) - ((676.5203681218851 / (z + -1.0)) + (((((-1259.1392167224028 * (3.0 - z)) + ((2.0 - z) * 771.3234287776531)) / ((3.0 - z) * (z - 2.0))) + (-176.6150291621406 / ((z + -1.0) - 3.0))) - 0.9999999999998099))) * (t_0 * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5))))) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -20000.0], N[(N[(t$95$0 * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * 0.9999999999998099), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(N[(N[(N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.13857109526572012 / N[(N[(z + -1.0), $MachinePrecision] - 5.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.5056327351493116e-7 / N[(z - 8.0), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(N[(z + -1.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(676.5203681218851 / N[(z + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(-1259.1392167224028 * N[(3.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 - z), $MachinePrecision] * 771.3234287776531), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 - z), $MachinePrecision] * N[(z - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(z + -1.0), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.9999999999998099), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * 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] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < -2e4

   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 inf 0.0%

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

      1. associate-*r/ 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in z around 0 100.0%

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

   if -2e4 < 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 96.9%

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

   6. Applied egg-rr 98.9%

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

   8. Simplified 99.2%

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

      1. frac-add 99.2%

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

   10. Applied egg-rr 99.2%

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

       1. pow1 99.2%

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

       2. neg-mul-1 99.2%

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

       3. fma-define 99.2%

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

   12. Applied egg-rr 99.2%

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

       1. unpow1 99.2%

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

       2. *-commutative 99.2%

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

       3. +-commutative 99.2%

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

       4. sub-neg 99.2%

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

       5. associate-+r+ 99.2%

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

       6. metadata-eval 99.2%

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

       7. sub-neg 99.2%

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

       8. +-commutative 99.2%

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

       9. sub-neg 99.2%

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

       10. associate-+r+ 99.2%

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

       11. metadata-eval 99.2%

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

       12. sub-neg 99.2%

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

       13. fma-undefine 99.2%

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

       14. neg-mul-1 99.2%

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

       15. metadata-eval 99.2%

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

       16. distribute-neg-in 99.2%

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

       17. +-commutative 99.2%

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

       18. sub-neg 99.2%

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

       19. associate-+r+ 99.2%

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

       20. metadata-eval 99.2%

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

       21. distribute-neg-in 99.2%

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

   14. Simplified 99.2%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -20000:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \frac{\pi \cdot 0.9999999999998099}{\sin \left(\pi \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} - \left(\frac{-0.13857109526572012}{\left(z + -1\right) - 5} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{z - 8} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z + -1\right) - 6}\right)\right)\right) - \left(\frac{676.5203681218851}{z + -1} + \left(\left(\frac{-1259.1392167224028 \cdot \left(3 - z\right) + \left(2 - z\right) \cdot 771.3234287776531}{\left(3 - z\right) \cdot \left(z - 2\right)} + \frac{-176.6150291621406}{\left(z + -1\right) - 3}\right) - 0.9999999999998099\right)\right)\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right)}{\sin \left(\pi \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ t_1 := \sin \left(\pi \cdot z\right)\\ t_2 := {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\\ \mathbf{if}\;z \leq -20000:\\ \;\;\;\;\left(t\_0 \cdot \left(t\_2 \cdot e^{-7.5}\right)\right) \cdot \frac{\pi \cdot 0.9999999999998099}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\left(t\_0 \cdot \left(t\_2 \cdot e^{\left(z + -1\right) + -6.5}\right)\right) \cdot \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} - \left(\frac{-0.13857109526572012}{\left(z + -1\right) - 5} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{z - 8} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z + -1\right) - 6}\right)\right)\right) + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 - \left(\left(\frac{771.3234287776531}{z - 3} - \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{\left(z + -1\right) - 3}\right)\right)\right)\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 2.0)))
        (t_1 (sin (* PI z)))
        (t_2 (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5))))
   (if (<= z -20000.0)
     (* (* t_0 (* t_2 (exp -7.5))) (/ (* PI 0.9999999999998099) t_1))
     (*
      PI
      (/
       (*
        (* t_0 (* t_2 (exp (+ (+ z -1.0) -6.5))))
        (+
         (-
          (/ 12.507343278686905 (+ (- 1.0 z) 4.0))
          (+
           (/ -0.13857109526572012 (- (+ z -1.0) 5.0))
           (+
            (/ 1.5056327351493116e-7 (- z 8.0))
            (/ 9.984369578019572e-6 (- (+ z -1.0) 6.0)))))
         (+
          (/ 676.5203681218851 (- 1.0 z))
          (-
           0.9999999999998099
           (+
            (-
             (/ 771.3234287776531 (- z 3.0))
             (/ -1259.1392167224028 (- 2.0 z)))
            (/ -176.6150291621406 (- (+ z -1.0) 3.0)))))))
       t_1)))))

C

double code(double z) {
	double t_0 = sqrt((((double) M_PI) * 2.0));
	double t_1 = sin((((double) M_PI) * z));
	double t_2 = pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5));
	double tmp;
	if (z <= -20000.0) {
		tmp = (t_0 * (t_2 * exp(-7.5))) * ((((double) M_PI) * 0.9999999999998099) / t_1);
	} else {
		tmp = ((double) M_PI) * (((t_0 * (t_2 * exp(((z + -1.0) + -6.5)))) * (((12.507343278686905 / ((1.0 - z) + 4.0)) - ((-0.13857109526572012 / ((z + -1.0) - 5.0)) + ((1.5056327351493116e-7 / (z - 8.0)) + (9.984369578019572e-6 / ((z + -1.0) - 6.0))))) + ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 - (((771.3234287776531 / (z - 3.0)) - (-1259.1392167224028 / (2.0 - z))) + (-176.6150291621406 / ((z + -1.0) - 3.0))))))) / t_1);
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.sqrt((Math.PI * 2.0));
	double t_1 = Math.sin((Math.PI * z));
	double t_2 = Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5));
	double tmp;
	if (z <= -20000.0) {
		tmp = (t_0 * (t_2 * Math.exp(-7.5))) * ((Math.PI * 0.9999999999998099) / t_1);
	} else {
		tmp = Math.PI * (((t_0 * (t_2 * Math.exp(((z + -1.0) + -6.5)))) * (((12.507343278686905 / ((1.0 - z) + 4.0)) - ((-0.13857109526572012 / ((z + -1.0) - 5.0)) + ((1.5056327351493116e-7 / (z - 8.0)) + (9.984369578019572e-6 / ((z + -1.0) - 6.0))))) + ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 - (((771.3234287776531 / (z - 3.0)) - (-1259.1392167224028 / (2.0 - z))) + (-176.6150291621406 / ((z + -1.0) - 3.0))))))) / t_1);
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.sqrt((math.pi * 2.0))
	t_1 = math.sin((math.pi * z))
	t_2 = math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5))
	tmp = 0
	if z <= -20000.0:
		tmp = (t_0 * (t_2 * math.exp(-7.5))) * ((math.pi * 0.9999999999998099) / t_1)
	else:
		tmp = math.pi * (((t_0 * (t_2 * math.exp(((z + -1.0) + -6.5)))) * (((12.507343278686905 / ((1.0 - z) + 4.0)) - ((-0.13857109526572012 / ((z + -1.0) - 5.0)) + ((1.5056327351493116e-7 / (z - 8.0)) + (9.984369578019572e-6 / ((z + -1.0) - 6.0))))) + ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 - (((771.3234287776531 / (z - 3.0)) - (-1259.1392167224028 / (2.0 - z))) + (-176.6150291621406 / ((z + -1.0) - 3.0))))))) / t_1)
	return tmp

Julia

function code(z)
	t_0 = sqrt(Float64(pi * 2.0))
	t_1 = sin(Float64(pi * z))
	t_2 = Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)
	tmp = 0.0
	if (z <= -20000.0)
		tmp = Float64(Float64(t_0 * Float64(t_2 * exp(-7.5))) * Float64(Float64(pi * 0.9999999999998099) / t_1));
	else
		tmp = Float64(pi * Float64(Float64(Float64(t_0 * Float64(t_2 * exp(Float64(Float64(z + -1.0) + -6.5)))) * Float64(Float64(Float64(12.507343278686905 / Float64(Float64(1.0 - z) + 4.0)) - Float64(Float64(-0.13857109526572012 / Float64(Float64(z + -1.0) - 5.0)) + Float64(Float64(1.5056327351493116e-7 / Float64(z - 8.0)) + Float64(9.984369578019572e-6 / Float64(Float64(z + -1.0) - 6.0))))) + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(0.9999999999998099 - Float64(Float64(Float64(771.3234287776531 / Float64(z - 3.0)) - Float64(-1259.1392167224028 / Float64(2.0 - z))) + Float64(-176.6150291621406 / Float64(Float64(z + -1.0) - 3.0))))))) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = sqrt((pi * 2.0));
	t_1 = sin((pi * z));
	t_2 = ((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5);
	tmp = 0.0;
	if (z <= -20000.0)
		tmp = (t_0 * (t_2 * exp(-7.5))) * ((pi * 0.9999999999998099) / t_1);
	else
		tmp = pi * (((t_0 * (t_2 * exp(((z + -1.0) + -6.5)))) * (((12.507343278686905 / ((1.0 - z) + 4.0)) - ((-0.13857109526572012 / ((z + -1.0) - 5.0)) + ((1.5056327351493116e-7 / (z - 8.0)) + (9.984369578019572e-6 / ((z + -1.0) - 6.0))))) + ((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 - (((771.3234287776531 / (z - 3.0)) - (-1259.1392167224028 / (2.0 - z))) + (-176.6150291621406 / ((z + -1.0) - 3.0))))))) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -20000.0], N[(N[(t$95$0 * N[(t$95$2 * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * 0.9999999999998099), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(N[(N[(t$95$0 * N[(t$95$2 * N[Exp[N[(N[(z + -1.0), $MachinePrecision] + -6.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.13857109526572012 / N[(N[(z + -1.0), $MachinePrecision] - 5.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.5056327351493116e-7 / N[(z - 8.0), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(N[(z + -1.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 - N[(N[(N[(771.3234287776531 / N[(z - 3.0), $MachinePrecision]), $MachinePrecision] - N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(z + -1.0), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if z < -2e4

   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 inf 0.0%

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

      1. associate-*r/ 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in z around 0 100.0%

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

   if -2e4 < 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 96.9%

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

   6. Applied egg-rr 98.9%

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

   8. Simplified 99.2%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -20000:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \frac{\pi \cdot 0.9999999999998099}{\sin \left(\pi \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(z + -1\right) + -6.5}\right)\right) \cdot \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} - \left(\frac{-0.13857109526572012}{\left(z + -1\right) - 5} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{z - 8} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z + -1\right) - 6}\right)\right)\right) + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 - \left(\left(\frac{771.3234287776531}{z - 3} - \frac{-1259.1392167224028}{2 - z}\right) + \frac{-176.6150291621406}{\left(z + -1\right) - 3}\right)\right)\right)\right)}{\sin \left(\pi \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ t_1 := \sin \left(\pi \cdot z\right)\\ \mathbf{if}\;z \leq -20000:\\ \;\;\;\;\left(t\_0 \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \frac{\pi \cdot 0.9999999999998099}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_0 \cdot \frac{\pi}{t\_1}\right) \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(z + -1\right) + -6.5}\right)\right) \cdot \left(\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) - \left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{z - 8} + \frac{9.984369578019572 \cdot 10^{-6}}{z - 7}\right) + \left(\left(\frac{771.3234287776531}{z - 3} - \left(\frac{-176.6150291621406}{4 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(\frac{-0.13857109526572012}{z - 6} + \frac{12.507343278686905}{z - 5}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 2.0))) (t_1 (sin (* PI z))))
   (if (<= z -20000.0)
     (*
      (* t_0 (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp -7.5)))
      (/ (* PI 0.9999999999998099) t_1))
     (*
      (*
       (* t_0 (/ PI t_1))
       (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ (+ z -1.0) -6.5))))
      (-
       (+ (/ 676.5203681218851 (- 1.0 z)) 0.9999999999998099)
       (+
        (+
         (/ 1.5056327351493116e-7 (- z 8.0))
         (/ 9.984369578019572e-6 (- z 7.0)))
        (+
         (-
          (/ 771.3234287776531 (- z 3.0))
          (+
           (/ -176.6150291621406 (- 4.0 z))
           (/ -1259.1392167224028 (- 2.0 z))))
         (+
          (/ -0.13857109526572012 (- z 6.0))
          (/ 12.507343278686905 (- z 5.0))))))))))

C

double code(double z) {
	double t_0 = sqrt((((double) M_PI) * 2.0));
	double t_1 = sin((((double) M_PI) * z));
	double tmp;
	if (z <= -20000.0) {
		tmp = (t_0 * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(-7.5))) * ((((double) M_PI) * 0.9999999999998099) / t_1);
	} else {
		tmp = ((t_0 * (((double) M_PI) / t_1)) * (pow((7.5 - z), (0.5 - z)) * exp(((z + -1.0) + -6.5)))) * (((676.5203681218851 / (1.0 - z)) + 0.9999999999998099) - (((1.5056327351493116e-7 / (z - 8.0)) + (9.984369578019572e-6 / (z - 7.0))) + (((771.3234287776531 / (z - 3.0)) - ((-176.6150291621406 / (4.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + ((-0.13857109526572012 / (z - 6.0)) + (12.507343278686905 / (z - 5.0))))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.sqrt((Math.PI * 2.0));
	double t_1 = Math.sin((Math.PI * z));
	double tmp;
	if (z <= -20000.0) {
		tmp = (t_0 * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp(-7.5))) * ((Math.PI * 0.9999999999998099) / t_1);
	} else {
		tmp = ((t_0 * (Math.PI / t_1)) * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp(((z + -1.0) + -6.5)))) * (((676.5203681218851 / (1.0 - z)) + 0.9999999999998099) - (((1.5056327351493116e-7 / (z - 8.0)) + (9.984369578019572e-6 / (z - 7.0))) + (((771.3234287776531 / (z - 3.0)) - ((-176.6150291621406 / (4.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + ((-0.13857109526572012 / (z - 6.0)) + (12.507343278686905 / (z - 5.0))))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.sqrt((math.pi * 2.0))
	t_1 = math.sin((math.pi * z))
	tmp = 0
	if z <= -20000.0:
		tmp = (t_0 * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp(-7.5))) * ((math.pi * 0.9999999999998099) / t_1)
	else:
		tmp = ((t_0 * (math.pi / t_1)) * (math.pow((7.5 - z), (0.5 - z)) * math.exp(((z + -1.0) + -6.5)))) * (((676.5203681218851 / (1.0 - z)) + 0.9999999999998099) - (((1.5056327351493116e-7 / (z - 8.0)) + (9.984369578019572e-6 / (z - 7.0))) + (((771.3234287776531 / (z - 3.0)) - ((-176.6150291621406 / (4.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + ((-0.13857109526572012 / (z - 6.0)) + (12.507343278686905 / (z - 5.0))))))
	return tmp

Julia

function code(z)
	t_0 = sqrt(Float64(pi * 2.0))
	t_1 = sin(Float64(pi * z))
	tmp = 0.0
	if (z <= -20000.0)
		tmp = Float64(Float64(t_0 * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(-7.5))) * Float64(Float64(pi * 0.9999999999998099) / t_1));
	else
		tmp = Float64(Float64(Float64(t_0 * Float64(pi / t_1)) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(Float64(z + -1.0) + -6.5)))) * Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + 0.9999999999998099) - Float64(Float64(Float64(1.5056327351493116e-7 / Float64(z - 8.0)) + Float64(9.984369578019572e-6 / Float64(z - 7.0))) + Float64(Float64(Float64(771.3234287776531 / Float64(z - 3.0)) - Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) + Float64(-1259.1392167224028 / Float64(2.0 - z)))) + Float64(Float64(-0.13857109526572012 / Float64(z - 6.0)) + Float64(12.507343278686905 / Float64(z - 5.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = sqrt((pi * 2.0));
	t_1 = sin((pi * z));
	tmp = 0.0;
	if (z <= -20000.0)
		tmp = (t_0 * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(-7.5))) * ((pi * 0.9999999999998099) / t_1);
	else
		tmp = ((t_0 * (pi / t_1)) * (((7.5 - z) ^ (0.5 - z)) * exp(((z + -1.0) + -6.5)))) * (((676.5203681218851 / (1.0 - z)) + 0.9999999999998099) - (((1.5056327351493116e-7 / (z - 8.0)) + (9.984369578019572e-6 / (z - 7.0))) + (((771.3234287776531 / (z - 3.0)) - ((-176.6150291621406 / (4.0 - z)) + (-1259.1392167224028 / (2.0 - z)))) + ((-0.13857109526572012 / (z - 6.0)) + (12.507343278686905 / (z - 5.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -20000.0], N[(N[(t$95$0 * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * 0.9999999999998099), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 * N[(Pi / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(z + -1.0), $MachinePrecision] + -6.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + 0.9999999999998099), $MachinePrecision] - N[(N[(N[(1.5056327351493116e-7 / N[(z - 8.0), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(z - 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(771.3234287776531 / N[(z - 3.0), $MachinePrecision]), $MachinePrecision] - N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(z - 6.0), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(z - 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < -2e4

   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 inf 0.0%

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

      1. associate-*r/ 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in z around 0 100.0%

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

   if -2e4 < 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 96.9%

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

   6. Applied egg-rr 99.1%

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

   8. Simplified 98.8%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -20000:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \frac{\pi \cdot 0.9999999999998099}{\sin \left(\pi \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{\pi \cdot 2} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{\left(z + -1\right) + -6.5}\right)\right) \cdot \left(\left(\frac{676.5203681218851}{1 - z} + 0.9999999999998099\right) - \left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{z - 8} + \frac{9.984369578019572 \cdot 10^{-6}}{z - 7}\right) + \left(\left(\frac{771.3234287776531}{z - 3} - \left(\frac{-176.6150291621406}{4 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \left(\frac{-0.13857109526572012}{z - 6} + \frac{12.507343278686905}{z - 5}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ t_1 := \sin \left(\pi \cdot z\right)\\ \mathbf{if}\;z \leq -20000:\\ \;\;\;\;\left(t\_0 \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \frac{\pi \cdot 0.9999999999998099}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-1259.1392167224028}{2 - z} - \frac{771.3234287776531}{z - 3}\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right)\right)\right)\right)\right) \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \frac{\pi}{t\_1}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 2.0))) (t_1 (sin (* PI z))))
   (if (<= z -20000.0)
     (*
      (* t_0 (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp -7.5)))
      (/ (* PI 0.9999999999998099) t_1))
     (*
      t_0
      (*
       (+
        0.9999999999998099
        (+
         (/ 676.5203681218851 (- 1.0 z))
         (+
          (- (/ -1259.1392167224028 (- 2.0 z)) (/ 771.3234287776531 (- z 3.0)))
          (+
           (+
            (/ 12.507343278686905 (- 5.0 z))
            (/ -176.6150291621406 (- 4.0 z)))
           (+
            (/ -0.13857109526572012 (- 6.0 z))
            (+
             (/ 1.5056327351493116e-7 (- 8.0 z))
             (/ 9.984369578019572e-6 (- 7.0 z))))))))
       (* (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5))) (/ PI t_1)))))))

C

double code(double z) {
	double t_0 = sqrt((((double) M_PI) * 2.0));
	double t_1 = sin((((double) M_PI) * z));
	double tmp;
	if (z <= -20000.0) {
		tmp = (t_0 * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(-7.5))) * ((((double) M_PI) * 0.9999999999998099) / t_1);
	} else {
		tmp = t_0 * ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (((-1259.1392167224028 / (2.0 - z)) - (771.3234287776531 / (z - 3.0))) + (((12.507343278686905 / (5.0 - z)) + (-176.6150291621406 / (4.0 - z))) + ((-0.13857109526572012 / (6.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + (9.984369578019572e-6 / (7.0 - z)))))))) * ((pow((7.5 - z), (0.5 - z)) * exp((z + -7.5))) * (((double) M_PI) / t_1)));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.sqrt((Math.PI * 2.0));
	double t_1 = Math.sin((Math.PI * z));
	double tmp;
	if (z <= -20000.0) {
		tmp = (t_0 * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp(-7.5))) * ((Math.PI * 0.9999999999998099) / t_1);
	} else {
		tmp = t_0 * ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (((-1259.1392167224028 / (2.0 - z)) - (771.3234287776531 / (z - 3.0))) + (((12.507343278686905 / (5.0 - z)) + (-176.6150291621406 / (4.0 - z))) + ((-0.13857109526572012 / (6.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + (9.984369578019572e-6 / (7.0 - z)))))))) * ((Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5))) * (Math.PI / t_1)));
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.sqrt((math.pi * 2.0))
	t_1 = math.sin((math.pi * z))
	tmp = 0
	if z <= -20000.0:
		tmp = (t_0 * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp(-7.5))) * ((math.pi * 0.9999999999998099) / t_1)
	else:
		tmp = t_0 * ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (((-1259.1392167224028 / (2.0 - z)) - (771.3234287776531 / (z - 3.0))) + (((12.507343278686905 / (5.0 - z)) + (-176.6150291621406 / (4.0 - z))) + ((-0.13857109526572012 / (6.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + (9.984369578019572e-6 / (7.0 - z)))))))) * ((math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5))) * (math.pi / t_1)))
	return tmp

Julia

function code(z)
	t_0 = sqrt(Float64(pi * 2.0))
	t_1 = sin(Float64(pi * z))
	tmp = 0.0
	if (z <= -20000.0)
		tmp = Float64(Float64(t_0 * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(-7.5))) * Float64(Float64(pi * 0.9999999999998099) / t_1));
	else
		tmp = Float64(t_0 * Float64(Float64(0.9999999999998099 + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) - Float64(771.3234287776531 / Float64(z - 3.0))) + Float64(Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(-176.6150291621406 / Float64(4.0 - z))) + Float64(Float64(-0.13857109526572012 / Float64(6.0 - z)) + Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) + Float64(9.984369578019572e-6 / Float64(7.0 - z)))))))) * Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5))) * Float64(pi / t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = sqrt((pi * 2.0));
	t_1 = sin((pi * z));
	tmp = 0.0;
	if (z <= -20000.0)
		tmp = (t_0 * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(-7.5))) * ((pi * 0.9999999999998099) / t_1);
	else
		tmp = t_0 * ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) + (((-1259.1392167224028 / (2.0 - z)) - (771.3234287776531 / (z - 3.0))) + (((12.507343278686905 / (5.0 - z)) + (-176.6150291621406 / (4.0 - z))) + ((-0.13857109526572012 / (6.0 - z)) + ((1.5056327351493116e-7 / (8.0 - z)) + (9.984369578019572e-6 / (7.0 - z)))))))) * ((((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5))) * (pi / t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -20000.0], N[(N[(t$95$0 * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * 0.9999999999998099), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(0.9999999999998099 + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] - N[(771.3234287776531 / N[(z - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(Pi / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < -2e4

   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 inf 0.0%

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

      1. associate-*r/ 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in z around 0 100.0%

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

   if -2e4 < 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 97.0%

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

      1. pow1 97.0%

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

   6. Applied egg-rr 96.9%

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

   8. Simplified 98.8%

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

   10. Applied egg-rr 98.8%

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

   12. Simplified 98.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -20000:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \frac{\pi \cdot 0.9999999999998099}{\sin \left(\pi \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{-1259.1392167224028}{2 - z} - \frac{771.3234287776531}{z - 3}\right) + \left(\left(\frac{12.507343278686905}{5 - z} + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right)\right)\right)\right)\right)\right) \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ t_1 := \sin \left(\pi \cdot z\right)\\ \mathbf{if}\;z \leq -20000:\\ \;\;\;\;\left(t\_0 \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \frac{\pi \cdot 0.9999999999998099}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \frac{\pi}{t\_1}\right) \cdot \left(\left(0.9999999999998099 - \left(\frac{771.3234287776531}{z - 3} - \left(\frac{-1259.1392167224028}{2 - z} - \frac{676.5203681218851}{z + -1}\right)\right)\right) + \left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) - \left(\left(\frac{12.507343278686905}{z - 5} + \frac{-176.6150291621406}{z - 4}\right) - \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 2.0))) (t_1 (sin (* PI z))))
   (if (<= z -20000.0)
     (*
      (* t_0 (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp -7.5)))
      (/ (* PI 0.9999999999998099) t_1))
     (*
      t_0
      (*
       (* (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5))) (/ PI t_1))
       (+
        (-
         0.9999999999998099
         (-
          (/ 771.3234287776531 (- z 3.0))
          (-
           (/ -1259.1392167224028 (- 2.0 z))
           (/ 676.5203681218851 (+ z -1.0)))))
        (-
         (+
          (/ 1.5056327351493116e-7 (- 8.0 z))
          (/ 9.984369578019572e-6 (- 7.0 z)))
         (-
          (+ (/ 12.507343278686905 (- z 5.0)) (/ -176.6150291621406 (- z 4.0)))
          (/ -0.13857109526572012 (- 6.0 z))))))))))

C

double code(double z) {
	double t_0 = sqrt((((double) M_PI) * 2.0));
	double t_1 = sin((((double) M_PI) * z));
	double tmp;
	if (z <= -20000.0) {
		tmp = (t_0 * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(-7.5))) * ((((double) M_PI) * 0.9999999999998099) / t_1);
	} else {
		tmp = t_0 * (((pow((7.5 - z), (0.5 - z)) * exp((z + -7.5))) * (((double) M_PI) / t_1)) * ((0.9999999999998099 - ((771.3234287776531 / (z - 3.0)) - ((-1259.1392167224028 / (2.0 - z)) - (676.5203681218851 / (z + -1.0))))) + (((1.5056327351493116e-7 / (8.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) - (((12.507343278686905 / (z - 5.0)) + (-176.6150291621406 / (z - 4.0))) - (-0.13857109526572012 / (6.0 - z))))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.sqrt((Math.PI * 2.0));
	double t_1 = Math.sin((Math.PI * z));
	double tmp;
	if (z <= -20000.0) {
		tmp = (t_0 * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp(-7.5))) * ((Math.PI * 0.9999999999998099) / t_1);
	} else {
		tmp = t_0 * (((Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5))) * (Math.PI / t_1)) * ((0.9999999999998099 - ((771.3234287776531 / (z - 3.0)) - ((-1259.1392167224028 / (2.0 - z)) - (676.5203681218851 / (z + -1.0))))) + (((1.5056327351493116e-7 / (8.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) - (((12.507343278686905 / (z - 5.0)) + (-176.6150291621406 / (z - 4.0))) - (-0.13857109526572012 / (6.0 - z))))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.sqrt((math.pi * 2.0))
	t_1 = math.sin((math.pi * z))
	tmp = 0
	if z <= -20000.0:
		tmp = (t_0 * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp(-7.5))) * ((math.pi * 0.9999999999998099) / t_1)
	else:
		tmp = t_0 * (((math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5))) * (math.pi / t_1)) * ((0.9999999999998099 - ((771.3234287776531 / (z - 3.0)) - ((-1259.1392167224028 / (2.0 - z)) - (676.5203681218851 / (z + -1.0))))) + (((1.5056327351493116e-7 / (8.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) - (((12.507343278686905 / (z - 5.0)) + (-176.6150291621406 / (z - 4.0))) - (-0.13857109526572012 / (6.0 - z))))))
	return tmp

Julia

function code(z)
	t_0 = sqrt(Float64(pi * 2.0))
	t_1 = sin(Float64(pi * z))
	tmp = 0.0
	if (z <= -20000.0)
		tmp = Float64(Float64(t_0 * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(-7.5))) * Float64(Float64(pi * 0.9999999999998099) / t_1));
	else
		tmp = Float64(t_0 * Float64(Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5))) * Float64(pi / t_1)) * Float64(Float64(0.9999999999998099 - Float64(Float64(771.3234287776531 / Float64(z - 3.0)) - Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) - Float64(676.5203681218851 / Float64(z + -1.0))))) + Float64(Float64(Float64(1.5056327351493116e-7 / Float64(8.0 - z)) + Float64(9.984369578019572e-6 / Float64(7.0 - z))) - Float64(Float64(Float64(12.507343278686905 / Float64(z - 5.0)) + Float64(-176.6150291621406 / Float64(z - 4.0))) - Float64(-0.13857109526572012 / Float64(6.0 - z)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = sqrt((pi * 2.0));
	t_1 = sin((pi * z));
	tmp = 0.0;
	if (z <= -20000.0)
		tmp = (t_0 * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(-7.5))) * ((pi * 0.9999999999998099) / t_1);
	else
		tmp = t_0 * (((((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5))) * (pi / t_1)) * ((0.9999999999998099 - ((771.3234287776531 / (z - 3.0)) - ((-1259.1392167224028 / (2.0 - z)) - (676.5203681218851 / (z + -1.0))))) + (((1.5056327351493116e-7 / (8.0 - z)) + (9.984369578019572e-6 / (7.0 - z))) - (((12.507343278686905 / (z - 5.0)) + (-176.6150291621406 / (z - 4.0))) - (-0.13857109526572012 / (6.0 - z))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -20000.0], N[(N[(t$95$0 * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * 0.9999999999998099), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(Pi / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(0.9999999999998099 - N[(N[(771.3234287776531 / N[(z - 3.0), $MachinePrecision]), $MachinePrecision] - N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] - N[(676.5203681218851 / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(12.507343278686905 / N[(z - 5.0), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(z - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < -2e4

   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 inf 0.0%

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

      1. associate-*r/ 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in z around 0 100.0%

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

   if -2e4 < 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 97.0%

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

      1. pow1 97.0%

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

   6. Applied egg-rr 96.9%

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

   8. Simplified 98.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -20000:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \frac{\pi \cdot 0.9999999999998099}{\sin \left(\pi \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(\left(0.9999999999998099 - \left(\frac{771.3234287776531}{z - 3} - \left(\frac{-1259.1392167224028}{2 - z} - \frac{676.5203681218851}{z + -1}\right)\right)\right) + \left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) - \left(\left(\frac{12.507343278686905}{z - 5} + \frac{-176.6150291621406}{z - 4}\right) - \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ t_1 := \sin \left(\pi \cdot z\right)\\ t_2 := \left(z + -1\right) + -6\\ \mathbf{if}\;z \leq -0.74:\\ \;\;\;\;\left(t\_0 \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \frac{\pi \cdot 0.9999999999998099}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\pi}{t\_1} \cdot \left(t\_0 \cdot \left({\left(7.5 - \left(\left(z + -1\right) - -1\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{-0.5 + t\_2}\right)\right)\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + z \cdot 606.6766809125655\right)\right)\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7} - \frac{9.984369578019572 \cdot 10^{-6}}{t\_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 2.0))) (t_1 (sin (* PI z))) (t_2 (+ (+ z -1.0) -6.0)))
   (if (<= z -0.74)
     (*
      (* t_0 (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp -7.5)))
      (/ (* PI 0.9999999999998099) t_1))
     (*
      (*
       (/ PI t_1)
       (*
        t_0
        (*
         (pow (- 7.5 (- (+ z -1.0) -1.0)) (- (- 1.0 z) 0.5))
         (exp (+ -0.5 t_2)))))
      (+
       (+
        263.3831855358925
        (*
         z
         (+
          436.8961723502244
          (* z (+ 545.0353078134797 (* z 606.6766809125655))))))
       (-
        (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0))
        (/ 9.984369578019572e-6 t_2)))))))

C

double code(double z) {
	double t_0 = sqrt((((double) M_PI) * 2.0));
	double t_1 = sin((((double) M_PI) * z));
	double t_2 = (z + -1.0) + -6.0;
	double tmp;
	if (z <= -0.74) {
		tmp = (t_0 * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(-7.5))) * ((((double) M_PI) * 0.9999999999998099) / t_1);
	} else {
		tmp = ((((double) M_PI) / t_1) * (t_0 * (pow((7.5 - ((z + -1.0) - -1.0)), ((1.0 - z) - 0.5)) * exp((-0.5 + t_2))))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))) + ((1.5056327351493116e-7 / ((1.0 - z) - -7.0)) - (9.984369578019572e-6 / t_2)));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.sqrt((Math.PI * 2.0));
	double t_1 = Math.sin((Math.PI * z));
	double t_2 = (z + -1.0) + -6.0;
	double tmp;
	if (z <= -0.74) {
		tmp = (t_0 * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp(-7.5))) * ((Math.PI * 0.9999999999998099) / t_1);
	} else {
		tmp = ((Math.PI / t_1) * (t_0 * (Math.pow((7.5 - ((z + -1.0) - -1.0)), ((1.0 - z) - 0.5)) * Math.exp((-0.5 + t_2))))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))) + ((1.5056327351493116e-7 / ((1.0 - z) - -7.0)) - (9.984369578019572e-6 / t_2)));
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.sqrt((math.pi * 2.0))
	t_1 = math.sin((math.pi * z))
	t_2 = (z + -1.0) + -6.0
	tmp = 0
	if z <= -0.74:
		tmp = (t_0 * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp(-7.5))) * ((math.pi * 0.9999999999998099) / t_1)
	else:
		tmp = ((math.pi / t_1) * (t_0 * (math.pow((7.5 - ((z + -1.0) - -1.0)), ((1.0 - z) - 0.5)) * math.exp((-0.5 + t_2))))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))) + ((1.5056327351493116e-7 / ((1.0 - z) - -7.0)) - (9.984369578019572e-6 / t_2)))
	return tmp

Julia

function code(z)
	t_0 = sqrt(Float64(pi * 2.0))
	t_1 = sin(Float64(pi * z))
	t_2 = Float64(Float64(z + -1.0) + -6.0)
	tmp = 0.0
	if (z <= -0.74)
		tmp = Float64(Float64(t_0 * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(-7.5))) * Float64(Float64(pi * 0.9999999999998099) / t_1));
	else
		tmp = Float64(Float64(Float64(pi / t_1) * Float64(t_0 * Float64((Float64(7.5 - Float64(Float64(z + -1.0) - -1.0)) ^ Float64(Float64(1.0 - z) - 0.5)) * exp(Float64(-0.5 + t_2))))) * Float64(Float64(263.3831855358925 + Float64(z * Float64(436.8961723502244 + Float64(z * Float64(545.0353078134797 + Float64(z * 606.6766809125655)))))) + Float64(Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0)) - Float64(9.984369578019572e-6 / t_2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = sqrt((pi * 2.0));
	t_1 = sin((pi * z));
	t_2 = (z + -1.0) + -6.0;
	tmp = 0.0;
	if (z <= -0.74)
		tmp = (t_0 * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(-7.5))) * ((pi * 0.9999999999998099) / t_1);
	else
		tmp = ((pi / t_1) * (t_0 * (((7.5 - ((z + -1.0) - -1.0)) ^ ((1.0 - z) - 0.5)) * exp((-0.5 + t_2))))) * ((263.3831855358925 + (z * (436.8961723502244 + (z * (545.0353078134797 + (z * 606.6766809125655)))))) + ((1.5056327351493116e-7 / ((1.0 - z) - -7.0)) - (9.984369578019572e-6 / t_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(z + -1.0), $MachinePrecision] + -6.0), $MachinePrecision]}, If[LessEqual[z, -0.74], N[(N[(t$95$0 * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * 0.9999999999998099), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / t$95$1), $MachinePrecision] * N[(t$95$0 * N[(N[Power[N[(7.5 - N[(N[(z + -1.0), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-0.5 + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(263.3831855358925 + N[(z * N[(436.8961723502244 + N[(z * N[(545.0353078134797 + N[(z * 606.6766809125655), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision] - N[(9.984369578019572e-6 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.73999999999999999

   1. Initial program 13.3%

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

   3. Simplified 13.3%

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

   5. Taylor expanded in z around inf 2.3%

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

      1. associate-*r/ 2.3%

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

   7. Simplified 2.3%

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

   8. Taylor expanded in z around 0 87.7%

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

   if -0.73999999999999999 < z

   1. Initial program 97.4%

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

   3. Simplified 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

   5. Taylor expanded in z around 0 98.6%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + 606.6766809125655 \cdot z\right)\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 98.6%

         \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + \color{blue}{z \cdot 606.6766809125655}\right)\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]

   7. Simplified 98.6%

      \[\leadsto \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{\left(-\left(\left(1 - z\right) - -6\right)\right) + -0.5}\right)\right)\right) \cdot \left(\color{blue}{\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + z \cdot 606.6766809125655\right)\right)\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.74:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \frac{\pi \cdot 0.9999999999998099}{\sin \left(\pi \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - \left(\left(z + -1\right) - -1\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{-0.5 + \left(\left(z + -1\right) + -6\right)}\right)\right)\right) \cdot \left(\left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot \left(545.0353078134797 + z \cdot 606.6766809125655\right)\right)\right) + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7} - \frac{9.984369578019572 \cdot 10^{-6}}{\left(z + -1\right) + -6}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ t_1 := \sin \left(\pi \cdot z\right)\\ t_2 := \left(z + -1\right) + -6\\ \mathbf{if}\;z \leq -1.45:\\ \;\;\;\;\left(t\_0 \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \frac{\pi \cdot 0.9999999999998099}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\pi}{t\_1} \cdot \left(t\_0 \cdot \left({\left(7.5 - \left(\left(z + -1\right) - -1\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{-0.5 + t\_2}\right)\right)\right) \cdot \left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7} - \frac{9.984369578019572 \cdot 10^{-6}}{t\_2}\right) + \left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 2.0))) (t_1 (sin (* PI z))) (t_2 (+ (+ z -1.0) -6.0)))
   (if (<= z -1.45)
     (*
      (* t_0 (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp -7.5)))
      (/ (* PI 0.9999999999998099) t_1))
     (*
      (*
       (/ PI t_1)
       (*
        t_0
        (*
         (pow (- 7.5 (- (+ z -1.0) -1.0)) (- (- 1.0 z) 0.5))
         (exp (+ -0.5 t_2)))))
      (+
       (-
        (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0))
        (/ 9.984369578019572e-6 t_2))
       (+
        263.3831855358925
        (* z (+ 436.8961723502244 (* z 545.0353078134797)))))))))

C

double code(double z) {
	double t_0 = sqrt((((double) M_PI) * 2.0));
	double t_1 = sin((((double) M_PI) * z));
	double t_2 = (z + -1.0) + -6.0;
	double tmp;
	if (z <= -1.45) {
		tmp = (t_0 * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(-7.5))) * ((((double) M_PI) * 0.9999999999998099) / t_1);
	} else {
		tmp = ((((double) M_PI) / t_1) * (t_0 * (pow((7.5 - ((z + -1.0) - -1.0)), ((1.0 - z) - 0.5)) * exp((-0.5 + t_2))))) * (((1.5056327351493116e-7 / ((1.0 - z) - -7.0)) - (9.984369578019572e-6 / t_2)) + (263.3831855358925 + (z * (436.8961723502244 + (z * 545.0353078134797)))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.sqrt((Math.PI * 2.0));
	double t_1 = Math.sin((Math.PI * z));
	double t_2 = (z + -1.0) + -6.0;
	double tmp;
	if (z <= -1.45) {
		tmp = (t_0 * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp(-7.5))) * ((Math.PI * 0.9999999999998099) / t_1);
	} else {
		tmp = ((Math.PI / t_1) * (t_0 * (Math.pow((7.5 - ((z + -1.0) - -1.0)), ((1.0 - z) - 0.5)) * Math.exp((-0.5 + t_2))))) * (((1.5056327351493116e-7 / ((1.0 - z) - -7.0)) - (9.984369578019572e-6 / t_2)) + (263.3831855358925 + (z * (436.8961723502244 + (z * 545.0353078134797)))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.sqrt((math.pi * 2.0))
	t_1 = math.sin((math.pi * z))
	t_2 = (z + -1.0) + -6.0
	tmp = 0
	if z <= -1.45:
		tmp = (t_0 * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp(-7.5))) * ((math.pi * 0.9999999999998099) / t_1)
	else:
		tmp = ((math.pi / t_1) * (t_0 * (math.pow((7.5 - ((z + -1.0) - -1.0)), ((1.0 - z) - 0.5)) * math.exp((-0.5 + t_2))))) * (((1.5056327351493116e-7 / ((1.0 - z) - -7.0)) - (9.984369578019572e-6 / t_2)) + (263.3831855358925 + (z * (436.8961723502244 + (z * 545.0353078134797)))))
	return tmp

Julia

function code(z)
	t_0 = sqrt(Float64(pi * 2.0))
	t_1 = sin(Float64(pi * z))
	t_2 = Float64(Float64(z + -1.0) + -6.0)
	tmp = 0.0
	if (z <= -1.45)
		tmp = Float64(Float64(t_0 * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(-7.5))) * Float64(Float64(pi * 0.9999999999998099) / t_1));
	else
		tmp = Float64(Float64(Float64(pi / t_1) * Float64(t_0 * Float64((Float64(7.5 - Float64(Float64(z + -1.0) - -1.0)) ^ Float64(Float64(1.0 - z) - 0.5)) * exp(Float64(-0.5 + t_2))))) * Float64(Float64(Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0)) - Float64(9.984369578019572e-6 / t_2)) + Float64(263.3831855358925 + Float64(z * Float64(436.8961723502244 + Float64(z * 545.0353078134797))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = sqrt((pi * 2.0));
	t_1 = sin((pi * z));
	t_2 = (z + -1.0) + -6.0;
	tmp = 0.0;
	if (z <= -1.45)
		tmp = (t_0 * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(-7.5))) * ((pi * 0.9999999999998099) / t_1);
	else
		tmp = ((pi / t_1) * (t_0 * (((7.5 - ((z + -1.0) - -1.0)) ^ ((1.0 - z) - 0.5)) * exp((-0.5 + t_2))))) * (((1.5056327351493116e-7 / ((1.0 - z) - -7.0)) - (9.984369578019572e-6 / t_2)) + (263.3831855358925 + (z * (436.8961723502244 + (z * 545.0353078134797)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(z + -1.0), $MachinePrecision] + -6.0), $MachinePrecision]}, If[LessEqual[z, -1.45], N[(N[(t$95$0 * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * 0.9999999999998099), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / t$95$1), $MachinePrecision] * N[(t$95$0 * N[(N[Power[N[(7.5 - N[(N[(z + -1.0), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-0.5 + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision] - N[(9.984369578019572e-6 / t$95$2), $MachinePrecision]), $MachinePrecision] + N[(263.3831855358925 + N[(z * N[(436.8961723502244 + N[(z * 545.0353078134797), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.44999999999999996

   1. Initial program 13.3%

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

   3. Simplified 13.3%

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

   5. Taylor expanded in z around inf 2.3%

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

      1. associate-*r/ 2.3%

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

   7. Simplified 2.3%

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

   8. Taylor expanded in z around 0 87.7%

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

   if -1.44999999999999996 < z

   1. Initial program 97.4%

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

   3. Simplified 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

   5. Taylor expanded in z around 0 98.5%

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

      1. *-commutative 98.5%

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

   7. Simplified 98.5%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \frac{\pi \cdot 0.9999999999998099}{\sin \left(\pi \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - \left(\left(z + -1\right) - -1\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{-0.5 + \left(\left(z + -1\right) + -6\right)}\right)\right)\right) \cdot \left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7} - \frac{9.984369578019572 \cdot 10^{-6}}{\left(z + -1\right) + -6}\right) + \left(263.3831855358925 + z \cdot \left(436.8961723502244 + z \cdot 545.0353078134797\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ t_1 := \sin \left(\pi \cdot z\right)\\ t_2 := \left(z + -1\right) + -6\\ \mathbf{if}\;z \leq -0.6:\\ \;\;\;\;\left(t\_0 \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \frac{\pi \cdot 0.9999999999998099}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\pi}{t\_1} \cdot \left(t\_0 \cdot \left({\left(7.5 - \left(\left(z + -1\right) - -1\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{-0.5 + t\_2}\right)\right)\right) \cdot \left(\left(263.3831855358925 + z \cdot 436.8961723502244\right) - \left(\frac{9.984369578019572 \cdot 10^{-6}}{t\_2} + \frac{1.5056327351493116 \cdot 10^{-7}}{-7 + \left(z + -1\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 2.0))) (t_1 (sin (* PI z))) (t_2 (+ (+ z -1.0) -6.0)))
   (if (<= z -0.6)
     (*
      (* t_0 (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp -7.5)))
      (/ (* PI 0.9999999999998099) t_1))
     (*
      (*
       (/ PI t_1)
       (*
        t_0
        (*
         (pow (- 7.5 (- (+ z -1.0) -1.0)) (- (- 1.0 z) 0.5))
         (exp (+ -0.5 t_2)))))
      (-
       (+ 263.3831855358925 (* z 436.8961723502244))
       (+
        (/ 9.984369578019572e-6 t_2)
        (/ 1.5056327351493116e-7 (+ -7.0 (+ z -1.0)))))))))

C

double code(double z) {
	double t_0 = sqrt((((double) M_PI) * 2.0));
	double t_1 = sin((((double) M_PI) * z));
	double t_2 = (z + -1.0) + -6.0;
	double tmp;
	if (z <= -0.6) {
		tmp = (t_0 * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(-7.5))) * ((((double) M_PI) * 0.9999999999998099) / t_1);
	} else {
		tmp = ((((double) M_PI) / t_1) * (t_0 * (pow((7.5 - ((z + -1.0) - -1.0)), ((1.0 - z) - 0.5)) * exp((-0.5 + t_2))))) * ((263.3831855358925 + (z * 436.8961723502244)) - ((9.984369578019572e-6 / t_2) + (1.5056327351493116e-7 / (-7.0 + (z + -1.0)))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.sqrt((Math.PI * 2.0));
	double t_1 = Math.sin((Math.PI * z));
	double t_2 = (z + -1.0) + -6.0;
	double tmp;
	if (z <= -0.6) {
		tmp = (t_0 * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp(-7.5))) * ((Math.PI * 0.9999999999998099) / t_1);
	} else {
		tmp = ((Math.PI / t_1) * (t_0 * (Math.pow((7.5 - ((z + -1.0) - -1.0)), ((1.0 - z) - 0.5)) * Math.exp((-0.5 + t_2))))) * ((263.3831855358925 + (z * 436.8961723502244)) - ((9.984369578019572e-6 / t_2) + (1.5056327351493116e-7 / (-7.0 + (z + -1.0)))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.sqrt((math.pi * 2.0))
	t_1 = math.sin((math.pi * z))
	t_2 = (z + -1.0) + -6.0
	tmp = 0
	if z <= -0.6:
		tmp = (t_0 * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp(-7.5))) * ((math.pi * 0.9999999999998099) / t_1)
	else:
		tmp = ((math.pi / t_1) * (t_0 * (math.pow((7.5 - ((z + -1.0) - -1.0)), ((1.0 - z) - 0.5)) * math.exp((-0.5 + t_2))))) * ((263.3831855358925 + (z * 436.8961723502244)) - ((9.984369578019572e-6 / t_2) + (1.5056327351493116e-7 / (-7.0 + (z + -1.0)))))
	return tmp

Julia

function code(z)
	t_0 = sqrt(Float64(pi * 2.0))
	t_1 = sin(Float64(pi * z))
	t_2 = Float64(Float64(z + -1.0) + -6.0)
	tmp = 0.0
	if (z <= -0.6)
		tmp = Float64(Float64(t_0 * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(-7.5))) * Float64(Float64(pi * 0.9999999999998099) / t_1));
	else
		tmp = Float64(Float64(Float64(pi / t_1) * Float64(t_0 * Float64((Float64(7.5 - Float64(Float64(z + -1.0) - -1.0)) ^ Float64(Float64(1.0 - z) - 0.5)) * exp(Float64(-0.5 + t_2))))) * Float64(Float64(263.3831855358925 + Float64(z * 436.8961723502244)) - Float64(Float64(9.984369578019572e-6 / t_2) + Float64(1.5056327351493116e-7 / Float64(-7.0 + Float64(z + -1.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = sqrt((pi * 2.0));
	t_1 = sin((pi * z));
	t_2 = (z + -1.0) + -6.0;
	tmp = 0.0;
	if (z <= -0.6)
		tmp = (t_0 * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(-7.5))) * ((pi * 0.9999999999998099) / t_1);
	else
		tmp = ((pi / t_1) * (t_0 * (((7.5 - ((z + -1.0) - -1.0)) ^ ((1.0 - z) - 0.5)) * exp((-0.5 + t_2))))) * ((263.3831855358925 + (z * 436.8961723502244)) - ((9.984369578019572e-6 / t_2) + (1.5056327351493116e-7 / (-7.0 + (z + -1.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(z + -1.0), $MachinePrecision] + -6.0), $MachinePrecision]}, If[LessEqual[z, -0.6], N[(N[(t$95$0 * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * 0.9999999999998099), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / t$95$1), $MachinePrecision] * N[(t$95$0 * N[(N[Power[N[(7.5 - N[(N[(z + -1.0), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-0.5 + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(263.3831855358925 + N[(z * 436.8961723502244), $MachinePrecision]), $MachinePrecision] - N[(N[(9.984369578019572e-6 / t$95$2), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(-7.0 + N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.599999999999999978

   1. Initial program 13.3%

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

   3. Simplified 13.3%

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

   5. Taylor expanded in z around inf 2.3%

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

      1. associate-*r/ 2.3%

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

   7. Simplified 2.3%

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

   8. Taylor expanded in z around 0 87.7%

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

   if -0.599999999999999978 < z

   1. Initial program 97.4%

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

   3. Simplified 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

   5. Taylor expanded in z around 0 98.1%

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

      1. *-commutative 98.1%

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

   7. Simplified 98.1%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.6:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \frac{\pi \cdot 0.9999999999998099}{\sin \left(\pi \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - \left(\left(z + -1\right) - -1\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{-0.5 + \left(\left(z + -1\right) + -6\right)}\right)\right)\right) \cdot \left(\left(263.3831855358925 + z \cdot 436.8961723502244\right) - \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(z + -1\right) + -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{-7 + \left(z + -1\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ t_1 := \sin \left(\pi \cdot z\right)\\ t_2 := {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\\ \mathbf{if}\;z \leq -0.74:\\ \;\;\;\;\left(t\_0 \cdot \left(t\_2 \cdot e^{-7.5}\right)\right) \cdot \frac{\pi \cdot 0.9999999999998099}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\left(t\_0 \cdot \left(t\_2 \cdot e^{\left(z + -1\right) + -6.5}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + z \cdot 606.6766809167608\right)\right)\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 2.0)))
        (t_1 (sin (* PI z)))
        (t_2 (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5))))
   (if (<= z -0.74)
     (* (* t_0 (* t_2 (exp -7.5))) (/ (* PI 0.9999999999998099) t_1))
     (*
      PI
      (/
       (*
        (* t_0 (* t_2 (exp (+ (+ z -1.0) -6.5))))
        (+
         263.3831869810514
         (*
          z
          (+
           436.8961725563396
           (* z (+ 545.0353078428827 (* z 606.6766809167608)))))))
       t_1)))))

C

double code(double z) {
	double t_0 = sqrt((((double) M_PI) * 2.0));
	double t_1 = sin((((double) M_PI) * z));
	double t_2 = pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5));
	double tmp;
	if (z <= -0.74) {
		tmp = (t_0 * (t_2 * exp(-7.5))) * ((((double) M_PI) * 0.9999999999998099) / t_1);
	} else {
		tmp = ((double) M_PI) * (((t_0 * (t_2 * exp(((z + -1.0) + -6.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (z * 606.6766809167608))))))) / t_1);
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.sqrt((Math.PI * 2.0));
	double t_1 = Math.sin((Math.PI * z));
	double t_2 = Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5));
	double tmp;
	if (z <= -0.74) {
		tmp = (t_0 * (t_2 * Math.exp(-7.5))) * ((Math.PI * 0.9999999999998099) / t_1);
	} else {
		tmp = Math.PI * (((t_0 * (t_2 * Math.exp(((z + -1.0) + -6.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (z * 606.6766809167608))))))) / t_1);
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.sqrt((math.pi * 2.0))
	t_1 = math.sin((math.pi * z))
	t_2 = math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5))
	tmp = 0
	if z <= -0.74:
		tmp = (t_0 * (t_2 * math.exp(-7.5))) * ((math.pi * 0.9999999999998099) / t_1)
	else:
		tmp = math.pi * (((t_0 * (t_2 * math.exp(((z + -1.0) + -6.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (z * 606.6766809167608))))))) / t_1)
	return tmp

Julia

function code(z)
	t_0 = sqrt(Float64(pi * 2.0))
	t_1 = sin(Float64(pi * z))
	t_2 = Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)
	tmp = 0.0
	if (z <= -0.74)
		tmp = Float64(Float64(t_0 * Float64(t_2 * exp(-7.5))) * Float64(Float64(pi * 0.9999999999998099) / t_1));
	else
		tmp = Float64(pi * Float64(Float64(Float64(t_0 * Float64(t_2 * exp(Float64(Float64(z + -1.0) + -6.5)))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * Float64(545.0353078428827 + Float64(z * 606.6766809167608))))))) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = sqrt((pi * 2.0));
	t_1 = sin((pi * z));
	t_2 = ((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5);
	tmp = 0.0;
	if (z <= -0.74)
		tmp = (t_0 * (t_2 * exp(-7.5))) * ((pi * 0.9999999999998099) / t_1);
	else
		tmp = pi * (((t_0 * (t_2 * exp(((z + -1.0) + -6.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (z * 606.6766809167608))))))) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -0.74], N[(N[(t$95$0 * N[(t$95$2 * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * 0.9999999999998099), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(N[(N[(t$95$0 * N[(t$95$2 * N[Exp[N[(N[(z + -1.0), $MachinePrecision] + -6.5), $MachinePrecision]], $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] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.73999999999999999

   1. Initial program 13.3%

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

   3. Simplified 13.3%

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

   5. Taylor expanded in z around inf 2.3%

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

      1. associate-*r/ 2.3%

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

   7. Simplified 2.3%

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

   8. Taylor expanded in z around 0 87.7%

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

   if -0.73999999999999999 < z

   1. Initial program 97.4%

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

   3. Simplified 96.9%

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

   6. Applied egg-rr 98.9%

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

   8. Simplified 99.2%

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

   9. Taylor expanded in z around 0 98.0%

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

       1. *-commutative 98.0%

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

   11. Simplified 98.0%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.74:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \frac{\pi \cdot 0.9999999999998099}{\sin \left(\pi \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(z + -1\right) + -6.5}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + z \cdot 606.6766809167608\right)\right)\right)}{\sin \left(\pi \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 97.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ t_1 := \sin \left(\pi \cdot z\right)\\ t_2 := {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\\ \mathbf{if}\;z \leq -1.45:\\ \;\;\;\;\left(t\_0 \cdot \left(t\_2 \cdot e^{-7.5}\right)\right) \cdot \frac{\pi \cdot 0.9999999999998099}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\left(t\_0 \cdot \left(t\_2 \cdot e^{\left(z + -1\right) + -6.5}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 2.0)))
        (t_1 (sin (* PI z)))
        (t_2 (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5))))
   (if (<= z -1.45)
     (* (* t_0 (* t_2 (exp -7.5))) (/ (* PI 0.9999999999998099) t_1))
     (*
      PI
      (/
       (*
        (* t_0 (* t_2 (exp (+ (+ z -1.0) -6.5))))
        (+
         263.3831869810514
         (* z (+ 436.8961725563396 (* z 545.0353078428827)))))
       t_1)))))

C

double code(double z) {
	double t_0 = sqrt((((double) M_PI) * 2.0));
	double t_1 = sin((((double) M_PI) * z));
	double t_2 = pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5));
	double tmp;
	if (z <= -1.45) {
		tmp = (t_0 * (t_2 * exp(-7.5))) * ((((double) M_PI) * 0.9999999999998099) / t_1);
	} else {
		tmp = ((double) M_PI) * (((t_0 * (t_2 * exp(((z + -1.0) + -6.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827))))) / t_1);
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.sqrt((Math.PI * 2.0));
	double t_1 = Math.sin((Math.PI * z));
	double t_2 = Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5));
	double tmp;
	if (z <= -1.45) {
		tmp = (t_0 * (t_2 * Math.exp(-7.5))) * ((Math.PI * 0.9999999999998099) / t_1);
	} else {
		tmp = Math.PI * (((t_0 * (t_2 * Math.exp(((z + -1.0) + -6.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827))))) / t_1);
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.sqrt((math.pi * 2.0))
	t_1 = math.sin((math.pi * z))
	t_2 = math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5))
	tmp = 0
	if z <= -1.45:
		tmp = (t_0 * (t_2 * math.exp(-7.5))) * ((math.pi * 0.9999999999998099) / t_1)
	else:
		tmp = math.pi * (((t_0 * (t_2 * math.exp(((z + -1.0) + -6.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827))))) / t_1)
	return tmp

Julia

function code(z)
	t_0 = sqrt(Float64(pi * 2.0))
	t_1 = sin(Float64(pi * z))
	t_2 = Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)
	tmp = 0.0
	if (z <= -1.45)
		tmp = Float64(Float64(t_0 * Float64(t_2 * exp(-7.5))) * Float64(Float64(pi * 0.9999999999998099) / t_1));
	else
		tmp = Float64(pi * Float64(Float64(Float64(t_0 * Float64(t_2 * exp(Float64(Float64(z + -1.0) + -6.5)))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * 545.0353078428827))))) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = sqrt((pi * 2.0));
	t_1 = sin((pi * z));
	t_2 = ((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5);
	tmp = 0.0;
	if (z <= -1.45)
		tmp = (t_0 * (t_2 * exp(-7.5))) * ((pi * 0.9999999999998099) / t_1);
	else
		tmp = pi * (((t_0 * (t_2 * exp(((z + -1.0) + -6.5)))) * (263.3831869810514 + (z * (436.8961725563396 + (z * 545.0353078428827))))) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1.45], N[(N[(t$95$0 * N[(t$95$2 * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * 0.9999999999998099), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(N[(N[(t$95$0 * N[(t$95$2 * N[Exp[N[(N[(z + -1.0), $MachinePrecision] + -6.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * 545.0353078428827), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.44999999999999996

   1. Initial program 13.3%

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

   3. Simplified 13.3%

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

   5. Taylor expanded in z around inf 2.3%

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

      1. associate-*r/ 2.3%

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

   7. Simplified 2.3%

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

   8. Taylor expanded in z around 0 87.7%

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

   if -1.44999999999999996 < z

   1. Initial program 97.4%

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

   3. Simplified 96.9%

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

   6. Applied egg-rr 98.9%

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

   8. Simplified 99.2%

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

   9. Taylor expanded in z around 0 98.0%

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

       1. *-commutative 98.0%

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

   11. Simplified 98.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \frac{\pi \cdot 0.9999999999998099}{\sin \left(\pi \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(z + -1\right) + -6.5}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot 545.0353078428827\right)\right)}{\sin \left(\pi \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 97.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.96:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \frac{\pi \cdot 0.9999999999998099}{\sin \left(\pi \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;263.3831869810514 \cdot \frac{\left(\sqrt{\pi} \cdot e^{-7.5}\right) \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (if (<= z -0.96)
   (*
    (*
     (sqrt (* PI 2.0))
     (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (exp -7.5)))
    (/ (* PI 0.9999999999998099) (sin (* PI z))))
   (*
    263.3831869810514
    (/ (* (* (sqrt PI) (exp -7.5)) (* (sqrt 2.0) (sqrt 7.5))) z))))

C

double code(double z) {
	double tmp;
	if (z <= -0.96) {
		tmp = (sqrt((((double) M_PI) * 2.0)) * (pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * exp(-7.5))) * ((((double) M_PI) * 0.9999999999998099) / sin((((double) M_PI) * z)));
	} else {
		tmp = 263.3831869810514 * (((sqrt(((double) M_PI)) * exp(-7.5)) * (sqrt(2.0) * sqrt(7.5))) / z);
	}
	return tmp;
}

Java

public static double code(double z) {
	double tmp;
	if (z <= -0.96) {
		tmp = (Math.sqrt((Math.PI * 2.0)) * (Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * Math.exp(-7.5))) * ((Math.PI * 0.9999999999998099) / Math.sin((Math.PI * z)));
	} else {
		tmp = 263.3831869810514 * (((Math.sqrt(Math.PI) * Math.exp(-7.5)) * (Math.sqrt(2.0) * Math.sqrt(7.5))) / z);
	}
	return tmp;
}

Python

def code(z):
	tmp = 0
	if z <= -0.96:
		tmp = (math.sqrt((math.pi * 2.0)) * (math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * math.exp(-7.5))) * ((math.pi * 0.9999999999998099) / math.sin((math.pi * z)))
	else:
		tmp = 263.3831869810514 * (((math.sqrt(math.pi) * math.exp(-7.5)) * (math.sqrt(2.0) * math.sqrt(7.5))) / z)
	return tmp

Julia

function code(z)
	tmp = 0.0
	if (z <= -0.96)
		tmp = Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64((Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * exp(-7.5))) * Float64(Float64(pi * 0.9999999999998099) / sin(Float64(pi * z))));
	else
		tmp = Float64(263.3831869810514 * Float64(Float64(Float64(sqrt(pi) * exp(-7.5)) * Float64(sqrt(2.0) * sqrt(7.5))) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	tmp = 0.0;
	if (z <= -0.96)
		tmp = (sqrt((pi * 2.0)) * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * exp(-7.5))) * ((pi * 0.9999999999998099) / sin((pi * z)));
	else
		tmp = 263.3831869810514 * (((sqrt(pi) * exp(-7.5)) * (sqrt(2.0) * sqrt(7.5))) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := If[LessEqual[z, -0.96], N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * 0.9999999999998099), $MachinePrecision] / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(263.3831869810514 * N[(N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.95999999999999996

   1. Initial program 13.3%

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

   3. Simplified 13.3%

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

   5. Taylor expanded in z around inf 2.3%

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

      1. associate-*r/ 2.3%

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

   7. Simplified 2.3%

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

   8. Taylor expanded in z around 0 87.7%

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

   if -0.95999999999999996 < z

   1. Initial program 97.4%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in z around 0 97.0%

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

      1. associate-*l/ 96.9%

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

      2. *-commutative 96.9%

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

      3. associate-*r* 97.6%

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

   7. Simplified 97.6%

      \[\leadsto \color{blue}{263.3831869810514 \cdot \frac{\left(\sqrt{\pi} \cdot e^{-7.5}\right) \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.96:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-7.5}\right)\right) \cdot \frac{\pi \cdot 0.9999999999998099}{\sin \left(\pi \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;263.3831869810514 \cdot \frac{\left(\sqrt{\pi} \cdot e^{-7.5}\right) \cdot \left(\sqrt{2} \cdot \sqrt{7.5}\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 96.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.1%

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

3. Simplified 94.7%

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

5. Taylor expanded in z around 0 94.4%

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

   1. associate-*l/ 94.3%

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

   2. *-commutative 94.3%

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

   3. associate-*r* 95.0%

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

7. Simplified 95.0%

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

Alternative 17: 96.0% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (sqrt (* PI 2.0))
   (* (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5)) (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(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * 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(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * 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(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)) * 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(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)) * 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)) * ((((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)) * 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[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $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 95.1%

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

3. Simplified 94.6%

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

5. Taylor expanded in z around 0 94.6%

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

   1. *-commutative 94.6%

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

7. Simplified 94.6%

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

8. Final simplification 94.6%

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

Alternative 18: 95.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.1%

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

3. Simplified 94.6%

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

5. Taylor expanded in z around 0 93.5%

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

6. Taylor expanded in z around 0 93.7%

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

7. Taylor expanded in z around 0 93.6%

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

   1. associate-*r/ 93.9%

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

   2. *-commutative 93.9%

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

9. Applied egg-rr 93.9%

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

    1. associate-/l* 93.6%

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

    2. associate-*r* 93.8%

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

    3. *-commutative 93.8%

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

    4. associate-*l* 93.7%

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

    5. associate-*r/ 94.3%

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

11. Simplified 94.3%

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

Alternative 19: 95.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.1%

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

3. Simplified 94.6%

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

5. Taylor expanded in z around 0 93.5%

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

6. Taylor expanded in z around 0 93.6%

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

   1. associate-*r* 94.4%

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

8. Simplified 94.4%

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

   1. associate-*r/ 94.9%

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

   2. associate-*l* 93.9%

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

   3. sqrt-unprod 93.9%

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

   4. metadata-eval 93.9%

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

10. Applied egg-rr 93.9%

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

11. Final simplification 93.9%

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

Alternative 20: 95.0% 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 95.1%

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

3. Simplified 94.6%

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

5. Taylor expanded in z around 0 93.5%

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

6. Taylor expanded in z around 0 93.6%

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

   1. associate-*r* 94.4%

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

8. Simplified 94.4%

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

   1. associate-*r/ 94.9%

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

   2. associate-*l* 93.9%

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

   3. sqrt-unprod 93.9%

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

   4. metadata-eval 93.9%

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

10. Applied egg-rr 93.9%

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

    1. associate-/l* 93.6%

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

    2. *-commutative 93.6%

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

12. Simplified 93.6%

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

13. Final simplification 93.6%

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

Reproduce

herbie shell --seed 2024129 
(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))))))