Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.5% → 99.3%

Time: 1.1min

Alternatives: 9

Speedup: 1.1×

• could not determine a ground truth

  1. z = -2.4979715976058496e+224

Specification

\[z \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{\sin \left(\pi \cdot z\right)}\\ t_1 := \left(z + -1\right) - -1\\ t_2 := \left(1 - z\right) + -1\\ t_3 := t\_2 + 7\\ t_4 := {\left(t\_3 + 0.5\right)}^{\left(t\_2 + 0.5\right)} \cdot \sqrt{\pi \cdot 2}\\ t_5 := t\_4 \cdot e^{\left(t\_1 - 7\right) - 0.5}\\ t_6 := \frac{1.5056327351493116 \cdot 10^{-7}}{t\_2 + 8}\\ \mathbf{if}\;t\_0 \cdot \left(t\_5 \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{t\_3} - \left(\frac{-0.13857109526572012}{t\_1 - 6} - \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 + t\_2}\right) + \frac{-1259.1392167224028}{2 + t\_2}\right) + \frac{771.3234287776531}{t\_2 + 3}\right) + \frac{-176.6150291621406}{t\_2 + 4}\right) + \frac{12.507343278686905}{t\_2 + 5}\right)\right)\right) + t\_6\right)\right) \leq 2 \cdot 10^{+307}:\\ \;\;\;\;t\_0 \cdot \left(t\_5 \cdot \left(t\_6 + \left(\left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \left(\left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 + \left(5 - z\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(t\_4 \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right) \cdot \left(t\_6 + \left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot \left(545.0353078425886 + z \cdot 606.676680916724\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (/ PI (sin (* PI z))))
        (t_1 (- (+ z -1.0) -1.0))
        (t_2 (+ (- 1.0 z) -1.0))
        (t_3 (+ t_2 7.0))
        (t_4 (* (pow (+ t_3 0.5) (+ t_2 0.5)) (sqrt (* PI 2.0))))
        (t_5 (* t_4 (exp (- (- t_1 7.0) 0.5))))
        (t_6 (/ 1.5056327351493116e-7 (+ t_2 8.0))))
   (if (<=
        (*
         t_0
         (*
          t_5
          (+
           (-
            (/ 9.984369578019572e-6 t_3)
            (-
             (/ -0.13857109526572012 (- t_1 6.0))
             (+
              (+
               (+
                (+
                 (+ 0.9999999999998099 (/ 676.5203681218851 (+ 1.0 t_2)))
                 (/ -1259.1392167224028 (+ 2.0 t_2)))
                (/ 771.3234287776531 (+ t_2 3.0)))
               (/ -176.6150291621406 (+ t_2 4.0)))
              (/ 12.507343278686905 (+ t_2 5.0)))))
           t_6)))
        2e+307)
     (*
      t_0
      (*
       t_5
       (+
        t_6
        (+
         (+
          (/ 676.5203681218851 (- 1.0 z))
          (+
           0.9999999999998099
           (+
            (+
             (/ -1259.1392167224028 (- 2.0 z))
             (/ 771.3234287776531 (- 3.0 z)))
            (/ -176.6150291621406 (+ (- 1.0 z) 3.0)))))
         (+
          (+
           (/ 12.507343278686905 (+ (- 1.0 z) 4.0))
           (/ -0.13857109526572012 (+ 1.0 (- 5.0 z))))
          (/ 9.984369578019572e-6 (+ (- 1.0 z) 6.0)))))))
     (*
      t_0
      (*
       (* t_4 (* (+ z 1.0) (exp -7.5)))
       (+
        t_6
        (+
         263.383186962231
         (*
          z
          (+
           436.896172553987
           (* z (+ 545.0353078425886 (* z 606.676680916724))))))))))))

C

double code(double z) {
	double t_0 = ((double) M_PI) / sin((((double) M_PI) * z));
	double t_1 = (z + -1.0) - -1.0;
	double t_2 = (1.0 - z) + -1.0;
	double t_3 = t_2 + 7.0;
	double t_4 = pow((t_3 + 0.5), (t_2 + 0.5)) * sqrt((((double) M_PI) * 2.0));
	double t_5 = t_4 * exp(((t_1 - 7.0) - 0.5));
	double t_6 = 1.5056327351493116e-7 / (t_2 + 8.0);
	double tmp;
	if ((t_0 * (t_5 * (((9.984369578019572e-6 / t_3) - ((-0.13857109526572012 / (t_1 - 6.0)) - (((((0.9999999999998099 + (676.5203681218851 / (1.0 + t_2))) + (-1259.1392167224028 / (2.0 + t_2))) + (771.3234287776531 / (t_2 + 3.0))) + (-176.6150291621406 / (t_2 + 4.0))) + (12.507343278686905 / (t_2 + 5.0))))) + t_6))) <= 2e+307) {
		tmp = t_0 * (t_5 * (t_6 + (((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + (((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (3.0 - z))) + (-176.6150291621406 / ((1.0 - z) + 3.0))))) + (((12.507343278686905 / ((1.0 - z) + 4.0)) + (-0.13857109526572012 / (1.0 + (5.0 - z)))) + (9.984369578019572e-6 / ((1.0 - z) + 6.0))))));
	} else {
		tmp = t_0 * ((t_4 * ((z + 1.0) * exp(-7.5))) * (t_6 + (263.383186962231 + (z * (436.896172553987 + (z * (545.0353078425886 + (z * 606.676680916724))))))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.PI / Math.sin((Math.PI * z));
	double t_1 = (z + -1.0) - -1.0;
	double t_2 = (1.0 - z) + -1.0;
	double t_3 = t_2 + 7.0;
	double t_4 = Math.pow((t_3 + 0.5), (t_2 + 0.5)) * Math.sqrt((Math.PI * 2.0));
	double t_5 = t_4 * Math.exp(((t_1 - 7.0) - 0.5));
	double t_6 = 1.5056327351493116e-7 / (t_2 + 8.0);
	double tmp;
	if ((t_0 * (t_5 * (((9.984369578019572e-6 / t_3) - ((-0.13857109526572012 / (t_1 - 6.0)) - (((((0.9999999999998099 + (676.5203681218851 / (1.0 + t_2))) + (-1259.1392167224028 / (2.0 + t_2))) + (771.3234287776531 / (t_2 + 3.0))) + (-176.6150291621406 / (t_2 + 4.0))) + (12.507343278686905 / (t_2 + 5.0))))) + t_6))) <= 2e+307) {
		tmp = t_0 * (t_5 * (t_6 + (((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + (((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (3.0 - z))) + (-176.6150291621406 / ((1.0 - z) + 3.0))))) + (((12.507343278686905 / ((1.0 - z) + 4.0)) + (-0.13857109526572012 / (1.0 + (5.0 - z)))) + (9.984369578019572e-6 / ((1.0 - z) + 6.0))))));
	} else {
		tmp = t_0 * ((t_4 * ((z + 1.0) * Math.exp(-7.5))) * (t_6 + (263.383186962231 + (z * (436.896172553987 + (z * (545.0353078425886 + (z * 606.676680916724))))))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.pi / math.sin((math.pi * z))
	t_1 = (z + -1.0) - -1.0
	t_2 = (1.0 - z) + -1.0
	t_3 = t_2 + 7.0
	t_4 = math.pow((t_3 + 0.5), (t_2 + 0.5)) * math.sqrt((math.pi * 2.0))
	t_5 = t_4 * math.exp(((t_1 - 7.0) - 0.5))
	t_6 = 1.5056327351493116e-7 / (t_2 + 8.0)
	tmp = 0
	if (t_0 * (t_5 * (((9.984369578019572e-6 / t_3) - ((-0.13857109526572012 / (t_1 - 6.0)) - (((((0.9999999999998099 + (676.5203681218851 / (1.0 + t_2))) + (-1259.1392167224028 / (2.0 + t_2))) + (771.3234287776531 / (t_2 + 3.0))) + (-176.6150291621406 / (t_2 + 4.0))) + (12.507343278686905 / (t_2 + 5.0))))) + t_6))) <= 2e+307:
		tmp = t_0 * (t_5 * (t_6 + (((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + (((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (3.0 - z))) + (-176.6150291621406 / ((1.0 - z) + 3.0))))) + (((12.507343278686905 / ((1.0 - z) + 4.0)) + (-0.13857109526572012 / (1.0 + (5.0 - z)))) + (9.984369578019572e-6 / ((1.0 - z) + 6.0))))))
	else:
		tmp = t_0 * ((t_4 * ((z + 1.0) * math.exp(-7.5))) * (t_6 + (263.383186962231 + (z * (436.896172553987 + (z * (545.0353078425886 + (z * 606.676680916724))))))))
	return tmp

Julia

function code(z)
	t_0 = Float64(pi / sin(Float64(pi * z)))
	t_1 = Float64(Float64(z + -1.0) - -1.0)
	t_2 = Float64(Float64(1.0 - z) + -1.0)
	t_3 = Float64(t_2 + 7.0)
	t_4 = Float64((Float64(t_3 + 0.5) ^ Float64(t_2 + 0.5)) * sqrt(Float64(pi * 2.0)))
	t_5 = Float64(t_4 * exp(Float64(Float64(t_1 - 7.0) - 0.5)))
	t_6 = Float64(1.5056327351493116e-7 / Float64(t_2 + 8.0))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_5 * Float64(Float64(Float64(9.984369578019572e-6 / t_3) - Float64(Float64(-0.13857109526572012 / Float64(t_1 - 6.0)) - Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 + t_2))) + Float64(-1259.1392167224028 / Float64(2.0 + t_2))) + Float64(771.3234287776531 / Float64(t_2 + 3.0))) + Float64(-176.6150291621406 / Float64(t_2 + 4.0))) + Float64(12.507343278686905 / Float64(t_2 + 5.0))))) + t_6))) <= 2e+307)
		tmp = Float64(t_0 * Float64(t_5 * Float64(t_6 + Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(0.9999999999998099 + Float64(Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) + Float64(771.3234287776531 / Float64(3.0 - z))) + Float64(-176.6150291621406 / Float64(Float64(1.0 - z) + 3.0))))) + Float64(Float64(Float64(12.507343278686905 / Float64(Float64(1.0 - z) + 4.0)) + Float64(-0.13857109526572012 / Float64(1.0 + Float64(5.0 - z)))) + Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) + 6.0)))))));
	else
		tmp = Float64(t_0 * Float64(Float64(t_4 * Float64(Float64(z + 1.0) * exp(-7.5))) * Float64(t_6 + Float64(263.383186962231 + Float64(z * Float64(436.896172553987 + Float64(z * Float64(545.0353078425886 + Float64(z * 606.676680916724)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = pi / sin((pi * z));
	t_1 = (z + -1.0) - -1.0;
	t_2 = (1.0 - z) + -1.0;
	t_3 = t_2 + 7.0;
	t_4 = ((t_3 + 0.5) ^ (t_2 + 0.5)) * sqrt((pi * 2.0));
	t_5 = t_4 * exp(((t_1 - 7.0) - 0.5));
	t_6 = 1.5056327351493116e-7 / (t_2 + 8.0);
	tmp = 0.0;
	if ((t_0 * (t_5 * (((9.984369578019572e-6 / t_3) - ((-0.13857109526572012 / (t_1 - 6.0)) - (((((0.9999999999998099 + (676.5203681218851 / (1.0 + t_2))) + (-1259.1392167224028 / (2.0 + t_2))) + (771.3234287776531 / (t_2 + 3.0))) + (-176.6150291621406 / (t_2 + 4.0))) + (12.507343278686905 / (t_2 + 5.0))))) + t_6))) <= 2e+307)
		tmp = t_0 * (t_5 * (t_6 + (((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + (((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (3.0 - z))) + (-176.6150291621406 / ((1.0 - z) + 3.0))))) + (((12.507343278686905 / ((1.0 - z) + 4.0)) + (-0.13857109526572012 / (1.0 + (5.0 - z)))) + (9.984369578019572e-6 / ((1.0 - z) + 6.0))))));
	else
		tmp = t_0 * ((t_4 * ((z + 1.0) * exp(-7.5))) * (t_6 + (263.383186962231 + (z * (436.896172553987 + (z * (545.0353078425886 + (z * 606.676680916724))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z + -1.0), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 - z), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + 7.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[N[(t$95$3 + 0.5), $MachinePrecision], N[(t$95$2 + 0.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * N[Exp[N[(N[(t$95$1 - 7.0), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(1.5056327351493116e-7 / N[(t$95$2 + 8.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$5 * N[(N[(N[(9.984369578019572e-6 / t$95$3), $MachinePrecision] - N[(N[(-0.13857109526572012 / N[(t$95$1 - 6.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(t$95$2 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(t$95$2 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(t$95$2 + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+307], N[(t$95$0 * N[(t$95$5 * N[(t$95$6 + N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(1.0 + N[(5.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(t$95$4 * N[(N[(z + 1.0), $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$6 + N[(263.383186962231 + N[(z * N[(436.896172553987 + N[(z * N[(545.0353078425886 + N[(z * 606.676680916724), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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%

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

   4. Applied egg-rr 98.4%

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

      1. *-lft-identity 98.4%

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

      2. associate-+l+ 99.3%

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

   6. Simplified 99.3%

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

   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%

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

   3. Taylor expanded in z around 0 0.0%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot \left(545.0353078425886 + 606.676680916724 \cdot z\right)\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot \left(545.0353078425886 + \color{blue}{z \cdot 606.676680916724}\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. Simplified 0.0%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot \left(545.0353078425886 + z \cdot 606.676680916724\right)\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. Taylor expanded in z around 0 100.0%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot \color{blue}{\left(e^{-7.5} + z \cdot e^{-7.5}\right)}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot \left(545.0353078425886 + z \cdot 606.676680916724\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   7. Step-by-step derivation

      1. distribute-rgt1-in 100.0%

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot \color{blue}{\left(\left(z + 1\right) \cdot e^{-7.5}\right)}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot \left(545.0353078425886 + z \cdot 606.676680916724\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot \color{blue}{\left(\left(z + 1\right) \cdot e^{-7.5}\right)}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot \left(545.0353078425886 + z \cdot 606.676680916724\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\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)} \cdot \sqrt{\pi \cdot 2}\right) \cdot e^{\left(\left(\left(z + -1\right) - -1\right) - 7\right) - 0.5}\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) + -1\right) + 7} - \left(\frac{-0.13857109526572012}{\left(\left(z + -1\right) - -1\right) - 6} - \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 + \left(\left(1 - z\right) + -1\right)}\right) + \frac{-1259.1392167224028}{2 + \left(\left(1 - z\right) + -1\right)}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) + -1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) + -1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) + -1\right) + 5}\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) + -1\right) + 8}\right)\right) \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\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)} \cdot \sqrt{\pi \cdot 2}\right) \cdot e^{\left(\left(\left(z + -1\right) - -1\right) - 7\right) - 0.5}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) + -1\right) + 8} + \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}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 + \left(5 - z\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\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)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) + -1\right) + 8} + \left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot \left(545.0353078425886 + z \cdot 606.676680916724\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) + -1\\ t_1 := \frac{\pi}{\sin \left(\pi \cdot z\right)}\\ t_2 := \sqrt{\pi \cdot 2}\\ t_3 := \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\\ \mathbf{if}\;z \leq -500:\\ \;\;\;\;t\_1 \cdot \left(\left(\left({\left(\left(t\_0 + 7\right) + 0.5\right)}^{\left(t\_0 + 0.5\right)} \cdot t\_2\right) \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right) \cdot \left(t\_3 + \left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot \left(545.0353078425886 + z \cdot 606.676680916724\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\left(t\_3 + \left(\left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \left(\left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 + \left(5 - z\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6}\right)\right)\right) \cdot \left(t\_2 \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-6.5 - \left(1 - z\right)}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ (- 1.0 z) -1.0))
        (t_1 (/ PI (sin (* PI z))))
        (t_2 (sqrt (* PI 2.0)))
        (t_3 (/ 1.5056327351493116e-7 (+ t_0 8.0))))
   (if (<= z -500.0)
     (*
      t_1
      (*
       (*
        (* (pow (+ (+ t_0 7.0) 0.5) (+ t_0 0.5)) t_2)
        (* (+ z 1.0) (exp -7.5)))
       (+
        t_3
        (+
         263.383186962231
         (*
          z
          (+
           436.896172553987
           (* z (+ 545.0353078425886 (* z 606.676680916724)))))))))
     (*
      t_1
      (*
       (+
        t_3
        (+
         (+
          (/ 676.5203681218851 (- 1.0 z))
          (+
           0.9999999999998099
           (+
            (+
             (/ -1259.1392167224028 (- 2.0 z))
             (/ 771.3234287776531 (- 3.0 z)))
            (/ -176.6150291621406 (+ (- 1.0 z) 3.0)))))
         (+
          (+
           (/ 12.507343278686905 (+ (- 1.0 z) 4.0))
           (/ -0.13857109526572012 (+ 1.0 (- 5.0 z))))
          (/ 9.984369578019572e-6 (+ (- 1.0 z) 6.0)))))
       (* t_2 (* (pow (- 7.5 z) (- 0.5 z)) (exp (- -6.5 (- 1.0 z))))))))))

C

double code(double z) {
	double t_0 = (1.0 - z) + -1.0;
	double t_1 = ((double) M_PI) / sin((((double) M_PI) * z));
	double t_2 = sqrt((((double) M_PI) * 2.0));
	double t_3 = 1.5056327351493116e-7 / (t_0 + 8.0);
	double tmp;
	if (z <= -500.0) {
		tmp = t_1 * (((pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5)) * t_2) * ((z + 1.0) * exp(-7.5))) * (t_3 + (263.383186962231 + (z * (436.896172553987 + (z * (545.0353078425886 + (z * 606.676680916724))))))));
	} else {
		tmp = t_1 * ((t_3 + (((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + (((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (3.0 - z))) + (-176.6150291621406 / ((1.0 - z) + 3.0))))) + (((12.507343278686905 / ((1.0 - z) + 4.0)) + (-0.13857109526572012 / (1.0 + (5.0 - z)))) + (9.984369578019572e-6 / ((1.0 - z) + 6.0))))) * (t_2 * (pow((7.5 - z), (0.5 - z)) * exp((-6.5 - (1.0 - z))))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) + -1.0;
	double t_1 = Math.PI / Math.sin((Math.PI * z));
	double t_2 = Math.sqrt((Math.PI * 2.0));
	double t_3 = 1.5056327351493116e-7 / (t_0 + 8.0);
	double tmp;
	if (z <= -500.0) {
		tmp = t_1 * (((Math.pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5)) * t_2) * ((z + 1.0) * Math.exp(-7.5))) * (t_3 + (263.383186962231 + (z * (436.896172553987 + (z * (545.0353078425886 + (z * 606.676680916724))))))));
	} else {
		tmp = t_1 * ((t_3 + (((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + (((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (3.0 - z))) + (-176.6150291621406 / ((1.0 - z) + 3.0))))) + (((12.507343278686905 / ((1.0 - z) + 4.0)) + (-0.13857109526572012 / (1.0 + (5.0 - z)))) + (9.984369578019572e-6 / ((1.0 - z) + 6.0))))) * (t_2 * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((-6.5 - (1.0 - z))))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = (1.0 - z) + -1.0
	t_1 = math.pi / math.sin((math.pi * z))
	t_2 = math.sqrt((math.pi * 2.0))
	t_3 = 1.5056327351493116e-7 / (t_0 + 8.0)
	tmp = 0
	if z <= -500.0:
		tmp = t_1 * (((math.pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5)) * t_2) * ((z + 1.0) * math.exp(-7.5))) * (t_3 + (263.383186962231 + (z * (436.896172553987 + (z * (545.0353078425886 + (z * 606.676680916724))))))))
	else:
		tmp = t_1 * ((t_3 + (((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + (((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (3.0 - z))) + (-176.6150291621406 / ((1.0 - z) + 3.0))))) + (((12.507343278686905 / ((1.0 - z) + 4.0)) + (-0.13857109526572012 / (1.0 + (5.0 - z)))) + (9.984369578019572e-6 / ((1.0 - z) + 6.0))))) * (t_2 * (math.pow((7.5 - z), (0.5 - z)) * math.exp((-6.5 - (1.0 - z))))))
	return tmp

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) + -1.0)
	t_1 = Float64(pi / sin(Float64(pi * z)))
	t_2 = sqrt(Float64(pi * 2.0))
	t_3 = Float64(1.5056327351493116e-7 / Float64(t_0 + 8.0))
	tmp = 0.0
	if (z <= -500.0)
		tmp = Float64(t_1 * Float64(Float64(Float64((Float64(Float64(t_0 + 7.0) + 0.5) ^ Float64(t_0 + 0.5)) * t_2) * Float64(Float64(z + 1.0) * exp(-7.5))) * Float64(t_3 + Float64(263.383186962231 + Float64(z * Float64(436.896172553987 + Float64(z * Float64(545.0353078425886 + Float64(z * 606.676680916724)))))))));
	else
		tmp = Float64(t_1 * Float64(Float64(t_3 + Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(0.9999999999998099 + Float64(Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) + Float64(771.3234287776531 / Float64(3.0 - z))) + Float64(-176.6150291621406 / Float64(Float64(1.0 - z) + 3.0))))) + Float64(Float64(Float64(12.507343278686905 / Float64(Float64(1.0 - z) + 4.0)) + Float64(-0.13857109526572012 / Float64(1.0 + Float64(5.0 - z)))) + Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) + 6.0))))) * Float64(t_2 * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(-6.5 - Float64(1.0 - z)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = (1.0 - z) + -1.0;
	t_1 = pi / sin((pi * z));
	t_2 = sqrt((pi * 2.0));
	t_3 = 1.5056327351493116e-7 / (t_0 + 8.0);
	tmp = 0.0;
	if (z <= -500.0)
		tmp = t_1 * ((((((t_0 + 7.0) + 0.5) ^ (t_0 + 0.5)) * t_2) * ((z + 1.0) * exp(-7.5))) * (t_3 + (263.383186962231 + (z * (436.896172553987 + (z * (545.0353078425886 + (z * 606.676680916724))))))));
	else
		tmp = t_1 * ((t_3 + (((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + (((-1259.1392167224028 / (2.0 - z)) + (771.3234287776531 / (3.0 - z))) + (-176.6150291621406 / ((1.0 - z) + 3.0))))) + (((12.507343278686905 / ((1.0 - z) + 4.0)) + (-0.13857109526572012 / (1.0 + (5.0 - z)))) + (9.984369578019572e-6 / ((1.0 - z) + 6.0))))) * (t_2 * (((7.5 - z) ^ (0.5 - z)) * exp((-6.5 - (1.0 - z))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(1.5056327351493116e-7 / N[(t$95$0 + 8.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -500.0], N[(t$95$1 * N[(N[(N[(N[Power[N[(N[(t$95$0 + 7.0), $MachinePrecision] + 0.5), $MachinePrecision], N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[(z + 1.0), $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 + N[(263.383186962231 + N[(z * N[(436.896172553987 + N[(z * N[(545.0353078425886 + N[(z * 606.676680916724), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(t$95$3 + N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(1.0 + N[(5.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-6.5 - N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if z < -500

   1. Initial program 0.0%

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

   3. Taylor expanded in z around 0 0.0%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot \left(545.0353078425886 + 606.676680916724 \cdot z\right)\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot \left(545.0353078425886 + \color{blue}{z \cdot 606.676680916724}\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. Simplified 0.0%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot \left(545.0353078425886 + z \cdot 606.676680916724\right)\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. Taylor expanded in z around 0 100.0%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot \color{blue}{\left(e^{-7.5} + z \cdot e^{-7.5}\right)}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot \left(545.0353078425886 + z \cdot 606.676680916724\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   7. Step-by-step derivation

      1. distribute-rgt1-in 100.0%

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot \color{blue}{\left(\left(z + 1\right) \cdot e^{-7.5}\right)}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot \left(545.0353078425886 + z \cdot 606.676680916724\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot \color{blue}{\left(\left(z + 1\right) \cdot e^{-7.5}\right)}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot \left(545.0353078425886 + z \cdot 606.676680916724\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   if -500 < z

   1. Initial program 97.3%

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

   4. Applied egg-rr 98.4%

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

      1. *-lft-identity 98.4%

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

      2. associate-+l+ 99.3%

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

   6. Simplified 99.3%

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

      1. associate-+l+ 99.3%

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

      2. metadata-eval 99.3%

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

      3. sub-neg 99.3%

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

      4. metadata-eval 99.3%

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

      5. associate-+l+ 99.3%

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

      6. metadata-eval 99.3%

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

      7. associate-+l- 99.3%

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

      8. metadata-eval 99.3%

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

      9. sub-neg 99.3%

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

      10. metadata-eval 99.3%

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

      11. associate-+l+ 99.3%

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

      12. metadata-eval 99.3%

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

      13. sub-neg 99.3%

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

      14. metadata-eval 99.3%

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

      15. associate-+l+ 99.3%

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

   8. Applied egg-rr 99.3%

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

      1. unpow1 99.3%

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

      2. *-commutative 99.3%

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

      3. +-commutative 99.3%

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

      4. metadata-eval 99.3%

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

      5. associate-+r+ 99.3%

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

      6. associate-+r- 99.3%

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

      7. metadata-eval 99.3%

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 + \left(\color{blue}{0} - z\right)\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right) \cdot \left(\left(\left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \left(\left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-176.6150291621406}{3 + \left(1 - z\right)}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z - 5\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      8. neg-sub0 99.3%

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

      9. sub-neg 99.3%

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

      10. +-commutative 99.3%

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

      11. metadata-eval 99.3%

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

      12. associate-+r+ 99.3%

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

      13. associate-+r- 99.3%

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

      14. metadata-eval 99.3%

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 + \left(\color{blue}{0} - z\right)\right)} \cdot e^{\left(-\left(1 - z\right)\right) + -6.5}\right)\right) \cdot \left(\left(\left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \left(\left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-176.6150291621406}{3 + \left(1 - z\right)}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 - \left(z - 5\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      15. neg-sub0 99.3%

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

      16. sub-neg 99.3%

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

   10. Simplified 99.3%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -500:\\ \;\;\;\;\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\left(\left(\left(\left(1 - z\right) + -1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) + -1\right) + 0.5\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) + -1\right) + 8} + \left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot \left(545.0353078425886 + z \cdot 606.676680916724\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) + -1\right) + 8} + \left(\left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \left(\left(\frac{-1259.1392167224028}{2 - z} + \frac{771.3234287776531}{3 - z}\right) + \frac{-176.6150291621406}{\left(1 - z\right) + 3}\right)\right)\right) + \left(\left(\frac{12.507343278686905}{\left(1 - z\right) + 4} + \frac{-0.13857109526572012}{1 + \left(5 - z\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) + 6}\right)\right)\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-6.5 - \left(1 - z\right)}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z \leq -720:\\ \;\;\;\;\left(t\_0 \cdot \left(e^{-7.5} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right)\right) \cdot \frac{263.3831869810514}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) - \left(413.1371825859926 + z \cdot \left(239.6241957716607 + z \cdot \left(131.48506030840542 - z \cdot -69.84368720931951\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 2.0))))
   (if (<= z -720.0)
     (*
      (* t_0 (* (exp -7.5) (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5))))
      (/ 263.3831869810514 z))
     (*
      t_0
      (*
       (* (/ PI (sin (* PI z))) (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5))))
       (+
        (/ 676.5203681218851 (- 1.0 z))
        (-
         (+
          (/ 9.984369578019572e-6 (- 7.0 z))
          (/ 1.5056327351493116e-7 (- 8.0 z)))
         (+
          413.1371825859926
          (*
           z
           (+
            239.6241957716607
            (* z (- 131.48506030840542 (* z -69.84368720931951)))))))))))))

C

double code(double z) {
	double t_0 = sqrt((((double) M_PI) * 2.0));
	double tmp;
	if (z <= -720.0) {
		tmp = (t_0 * (exp(-7.5) * pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)))) * (263.3831869810514 / z);
	} else {
		tmp = t_0 * (((((double) M_PI) / sin((((double) M_PI) * z))) * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5)))) * ((676.5203681218851 / (1.0 - z)) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) - (413.1371825859926 + (z * (239.6241957716607 + (z * (131.48506030840542 - (z * -69.84368720931951)))))))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.sqrt((Math.PI * 2.0));
	double tmp;
	if (z <= -720.0) {
		tmp = (t_0 * (Math.exp(-7.5) * Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)))) * (263.3831869810514 / z);
	} else {
		tmp = t_0 * (((Math.PI / Math.sin((Math.PI * z))) * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5)))) * ((676.5203681218851 / (1.0 - z)) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) - (413.1371825859926 + (z * (239.6241957716607 + (z * (131.48506030840542 - (z * -69.84368720931951)))))))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.sqrt((math.pi * 2.0))
	tmp = 0
	if z <= -720.0:
		tmp = (t_0 * (math.exp(-7.5) * math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)))) * (263.3831869810514 / z)
	else:
		tmp = t_0 * (((math.pi / math.sin((math.pi * z))) * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5)))) * ((676.5203681218851 / (1.0 - z)) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) - (413.1371825859926 + (z * (239.6241957716607 + (z * (131.48506030840542 - (z * -69.84368720931951)))))))))
	return tmp

Julia

function code(z)
	t_0 = sqrt(Float64(pi * 2.0))
	tmp = 0.0
	if (z <= -720.0)
		tmp = Float64(Float64(t_0 * Float64(exp(-7.5) * (Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)))) * Float64(263.3831869810514 / z));
	else
		tmp = Float64(t_0 * Float64(Float64(Float64(pi / sin(Float64(pi * z))) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5)))) * Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))) - Float64(413.1371825859926 + Float64(z * Float64(239.6241957716607 + Float64(z * Float64(131.48506030840542 - Float64(z * -69.84368720931951))))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = sqrt((pi * 2.0));
	tmp = 0.0;
	if (z <= -720.0)
		tmp = (t_0 * (exp(-7.5) * (((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)))) * (263.3831869810514 / z);
	else
		tmp = t_0 * (((pi / sin((pi * z))) * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5)))) * ((676.5203681218851 / (1.0 - z)) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) - (413.1371825859926 + (z * (239.6241957716607 + (z * (131.48506030840542 - (z * -69.84368720931951)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -720.0], N[(N[(t$95$0 * N[(N[Exp[-7.5], $MachinePrecision] * N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 / z), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(413.1371825859926 + N[(z * N[(239.6241957716607 + N[(z * N[(131.48506030840542 - N[(z * -69.84368720931951), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -720

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in z around 0 0.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \color{blue}{\frac{263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 - -43.89719783017524 \cdot {\pi}^{2}\right)\right)}{z}} \]

   6. Taylor expanded in z around 0 80.0%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{e^{-7.5}}\right)\right) \cdot \frac{263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 - -43.89719783017524 \cdot {\pi}^{2}\right)\right)}{z} \]

   7. Taylor expanded in z around 0 80.0%

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

   if -720 < 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 97.4%

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

      1. pow1 97.4%

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

   6. Applied egg-rr 97.3%

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

   8. Simplified 99.0%

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

   9. Taylor expanded in z around 0 98.8%

      \[\leadsto \sqrt{2 \cdot \pi} \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\color{blue}{\left(z \cdot \left(z \cdot \left(-69.84368720931951 \cdot z - 131.48506030840542\right) - 239.6241957716607\right) - 413.1371825859926\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -720:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left(e^{-7.5} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right)\right) \cdot \frac{263.3831869810514}{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) - \left(413.1371825859926 + z \cdot \left(239.6241957716607 + z \cdot \left(131.48506030840542 - z \cdot -69.84368720931951\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) + -1\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\left(\left(t\_0 + 7\right) + 0.5\right)}^{\left(t\_0 + 0.5\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8} + \left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot \left(545.0353078425886 + z \cdot 606.676680916724\right)\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ (- 1.0 z) -1.0)))
   (*
    (/ PI (sin (* PI z)))
    (*
     (*
      (* (pow (+ (+ t_0 7.0) 0.5) (+ t_0 0.5)) (sqrt (* PI 2.0)))
      (* (+ z 1.0) (exp -7.5)))
     (+
      (/ 1.5056327351493116e-7 (+ t_0 8.0))
      (+
       263.383186962231
       (*
        z
        (+
         436.896172553987
         (* z (+ 545.0353078425886 (* z 606.676680916724)))))))))))

C

double code(double z) {
	double t_0 = (1.0 - z) + -1.0;
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (((pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5)) * sqrt((((double) M_PI) * 2.0))) * ((z + 1.0) * exp(-7.5))) * ((1.5056327351493116e-7 / (t_0 + 8.0)) + (263.383186962231 + (z * (436.896172553987 + (z * (545.0353078425886 + (z * 606.676680916724))))))));
}

Java

public static double code(double z) {
	double t_0 = (1.0 - z) + -1.0;
	return (Math.PI / Math.sin((Math.PI * z))) * (((Math.pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5)) * Math.sqrt((Math.PI * 2.0))) * ((z + 1.0) * Math.exp(-7.5))) * ((1.5056327351493116e-7 / (t_0 + 8.0)) + (263.383186962231 + (z * (436.896172553987 + (z * (545.0353078425886 + (z * 606.676680916724))))))));
}

Python

def code(z):
	t_0 = (1.0 - z) + -1.0
	return (math.pi / math.sin((math.pi * z))) * (((math.pow(((t_0 + 7.0) + 0.5), (t_0 + 0.5)) * math.sqrt((math.pi * 2.0))) * ((z + 1.0) * math.exp(-7.5))) * ((1.5056327351493116e-7 / (t_0 + 8.0)) + (263.383186962231 + (z * (436.896172553987 + (z * (545.0353078425886 + (z * 606.676680916724))))))))

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) + -1.0)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64((Float64(Float64(t_0 + 7.0) + 0.5) ^ Float64(t_0 + 0.5)) * sqrt(Float64(pi * 2.0))) * Float64(Float64(z + 1.0) * exp(-7.5))) * Float64(Float64(1.5056327351493116e-7 / Float64(t_0 + 8.0)) + Float64(263.383186962231 + Float64(z * Float64(436.896172553987 + Float64(z * Float64(545.0353078425886 + Float64(z * 606.676680916724)))))))))
end

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) + -1.0;
	tmp = (pi / sin((pi * z))) * ((((((t_0 + 7.0) + 0.5) ^ (t_0 + 0.5)) * sqrt((pi * 2.0))) * ((z + 1.0) * exp(-7.5))) * ((1.5056327351493116e-7 / (t_0 + 8.0)) + (263.383186962231 + (z * (436.896172553987 + (z * (545.0353078425886 + (z * 606.676680916724))))))));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] + -1.0), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(N[(t$95$0 + 7.0), $MachinePrecision] + 0.5), $MachinePrecision], N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(z + 1.0), $MachinePrecision] * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.5056327351493116e-7 / N[(t$95$0 + 8.0), $MachinePrecision]), $MachinePrecision] + N[(263.383186962231 + N[(z * N[(436.896172553987 + N[(z * N[(545.0353078425886 + N[(z * 606.676680916724), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 95.4%

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

3. Taylor expanded in z around 0 95.8%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot \left(545.0353078425886 + 606.676680916724 \cdot z\right)\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
4. Step-by-step derivation

   1. *-commutative 95.8%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot \left(545.0353078425886 + \color{blue}{z \cdot 606.676680916724}\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

5. Simplified 95.8%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot \left(545.0353078425886 + z \cdot 606.676680916724\right)\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Taylor expanded in z around 0 97.5%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot \color{blue}{\left(e^{-7.5} + z \cdot e^{-7.5}\right)}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot \left(545.0353078425886 + z \cdot 606.676680916724\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
7. Step-by-step derivation

   1. distribute-rgt1-in 97.5%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot \color{blue}{\left(\left(z + 1\right) \cdot e^{-7.5}\right)}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot \left(545.0353078425886 + z \cdot 606.676680916724\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

8. Simplified 97.5%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot \color{blue}{\left(\left(z + 1\right) \cdot e^{-7.5}\right)}\right) \cdot \left(\left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot \left(545.0353078425886 + z \cdot 606.676680916724\right)\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

9. Final simplification 97.5%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left({\left(\left(\left(\left(1 - z\right) + -1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) + -1\right) + 0.5\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) + -1\right) + 8} + \left(263.383186962231 + z \cdot \left(436.896172553987 + z \cdot \left(545.0353078425886 + z \cdot 606.676680916724\right)\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 5: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot 2}\\ \mathbf{if}\;z \leq -116:\\ \;\;\;\;\left(t\_0 \cdot \left(e^{-7.5} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right)\right) \cdot \frac{263.3831869810514}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(z \cdot \left(z \cdot -131.48506030840542 - 239.6241957716607\right) - 413.1371825859926\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 2.0))))
   (if (<= z -116.0)
     (*
      (* t_0 (* (exp -7.5) (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5))))
      (/ 263.3831869810514 z))
     (*
      t_0
      (*
       (* (/ PI (sin (* PI z))) (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5))))
       (+
        (/ 676.5203681218851 (- 1.0 z))
        (+
         (+
          (/ 9.984369578019572e-6 (- 7.0 z))
          (/ 1.5056327351493116e-7 (- 8.0 z)))
         (-
          (* z (- (* z -131.48506030840542) 239.6241957716607))
          413.1371825859926))))))))

C

double code(double z) {
	double t_0 = sqrt((((double) M_PI) * 2.0));
	double tmp;
	if (z <= -116.0) {
		tmp = (t_0 * (exp(-7.5) * pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)))) * (263.3831869810514 / z);
	} else {
		tmp = t_0 * (((((double) M_PI) / sin((((double) M_PI) * z))) * (pow((7.5 - z), (0.5 - z)) * exp((z + -7.5)))) * ((676.5203681218851 / (1.0 - z)) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((z * ((z * -131.48506030840542) - 239.6241957716607)) - 413.1371825859926))));
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = Math.sqrt((Math.PI * 2.0));
	double tmp;
	if (z <= -116.0) {
		tmp = (t_0 * (Math.exp(-7.5) * Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)))) * (263.3831869810514 / z);
	} else {
		tmp = t_0 * (((Math.PI / Math.sin((Math.PI * z))) * (Math.pow((7.5 - z), (0.5 - z)) * Math.exp((z + -7.5)))) * ((676.5203681218851 / (1.0 - z)) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((z * ((z * -131.48506030840542) - 239.6241957716607)) - 413.1371825859926))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = math.sqrt((math.pi * 2.0))
	tmp = 0
	if z <= -116.0:
		tmp = (t_0 * (math.exp(-7.5) * math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)))) * (263.3831869810514 / z)
	else:
		tmp = t_0 * (((math.pi / math.sin((math.pi * z))) * (math.pow((7.5 - z), (0.5 - z)) * math.exp((z + -7.5)))) * ((676.5203681218851 / (1.0 - z)) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((z * ((z * -131.48506030840542) - 239.6241957716607)) - 413.1371825859926))))
	return tmp

Julia

function code(z)
	t_0 = sqrt(Float64(pi * 2.0))
	tmp = 0.0
	if (z <= -116.0)
		tmp = Float64(Float64(t_0 * Float64(exp(-7.5) * (Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)))) * Float64(263.3831869810514 / z));
	else
		tmp = Float64(t_0 * Float64(Float64(Float64(pi / sin(Float64(pi * z))) * Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5)))) * Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))) + Float64(Float64(z * Float64(Float64(z * -131.48506030840542) - 239.6241957716607)) - 413.1371825859926)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = sqrt((pi * 2.0));
	tmp = 0.0;
	if (z <= -116.0)
		tmp = (t_0 * (exp(-7.5) * (((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)))) * (263.3831869810514 / z);
	else
		tmp = t_0 * (((pi / sin((pi * z))) * (((7.5 - z) ^ (0.5 - z)) * exp((z + -7.5)))) * ((676.5203681218851 / (1.0 - z)) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((z * ((z * -131.48506030840542) - 239.6241957716607)) - 413.1371825859926))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -116.0], N[(N[(t$95$0 * N[(N[Exp[-7.5], $MachinePrecision] * N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 / z), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z * N[(N[(z * -131.48506030840542), $MachinePrecision] - 239.6241957716607), $MachinePrecision]), $MachinePrecision] - 413.1371825859926), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -116

   1. Initial program 16.7%

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

   3. Simplified 16.7%

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

   5. Taylor expanded in z around 0 16.7%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \color{blue}{\frac{263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 - -43.89719783017524 \cdot {\pi}^{2}\right)\right)}{z}} \]

   6. Taylor expanded in z around 0 83.3%

      \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{e^{-7.5}}\right)\right) \cdot \frac{263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 - -43.89719783017524 \cdot {\pi}^{2}\right)\right)}{z} \]

   7. Taylor expanded in z around 0 83.3%

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

   if -116 < 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 97.4%

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

      1. pow1 97.4%

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

   6. Applied egg-rr 97.3%

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

   8. Simplified 99.0%

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

   9. Taylor expanded in z around 0 98.5%

      \[\leadsto \sqrt{2 \cdot \pi} \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\color{blue}{\left(z \cdot \left(-131.48506030840542 \cdot z - 239.6241957716607\right) - 413.1371825859926\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -116:\\ \;\;\;\;\left(\sqrt{\pi \cdot 2} \cdot \left(e^{-7.5} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right)\right) \cdot \frac{263.3831869810514}{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot 2} \cdot \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(z \cdot \left(z \cdot -131.48506030840542 - 239.6241957716607\right) - 413.1371825859926\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{\pi \cdot 2} \cdot \left(e^{-7.5} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right)\right) \cdot \frac{263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 - -43.89719783017524 \cdot {\pi}^{2}\right)\right)}{z} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (sqrt (* PI 2.0))
   (* (exp -7.5) (pow (+ (- 1.0 z) 6.5) (+ (- 1.0 z) -0.5))))
  (/
   (+
    263.3831869810514
    (*
     z
     (+
      436.8961725563396
      (* z (- 545.0353078428827 (* -43.89719783017524 (pow PI 2.0)))))))
   z)))

C

double code(double z) {
	return (sqrt((((double) M_PI) * 2.0)) * (exp(-7.5) * pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)))) * ((263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 - (-43.89719783017524 * pow(((double) M_PI), 2.0))))))) / z);
}

Java

public static double code(double z) {
	return (Math.sqrt((Math.PI * 2.0)) * (Math.exp(-7.5) * Math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)))) * ((263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 - (-43.89719783017524 * Math.pow(Math.PI, 2.0))))))) / z);
}

Python

def code(z):
	return (math.sqrt((math.pi * 2.0)) * (math.exp(-7.5) * math.pow(((1.0 - z) + 6.5), ((1.0 - z) + -0.5)))) * ((263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 - (-43.89719783017524 * math.pow(math.pi, 2.0))))))) / z)

Julia

function code(z)
	return Float64(Float64(sqrt(Float64(pi * 2.0)) * Float64(exp(-7.5) * (Float64(Float64(1.0 - z) + 6.5) ^ Float64(Float64(1.0 - z) + -0.5)))) * Float64(Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * Float64(545.0353078428827 - Float64(-43.89719783017524 * (pi ^ 2.0))))))) / z))
end

MATLAB

function tmp = code(z)
	tmp = (sqrt((pi * 2.0)) * (exp(-7.5) * (((1.0 - z) + 6.5) ^ ((1.0 - z) + -0.5)))) * ((263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 - (-43.89719783017524 * (pi ^ 2.0))))))) / z);
end

Wolfram

code[z_] := N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] * N[Power[N[(N[(1.0 - z), $MachinePrecision] + 6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * N[(545.0353078428827 - N[(-43.89719783017524 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.4%

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

3. Simplified 95.2%

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

5. Taylor expanded in z around 0 96.0%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \color{blue}{\frac{263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 - -43.89719783017524 \cdot {\pi}^{2}\right)\right)}{z}} \]

6. Taylor expanded in z around 0 96.4%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{e^{-7.5}}\right)\right) \cdot \frac{263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 - -43.89719783017524 \cdot {\pi}^{2}\right)\right)}{z} \]

7. Final simplification 96.4%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left(e^{-7.5} \cdot {\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)}\right)\right) \cdot \frac{263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 - -43.89719783017524 \cdot {\pi}^{2}\right)\right)}{z} \]
8. Add Preprocessing

Alternative 7: 95.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 95.2%

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

5. Taylor expanded in z around 0 96.0%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot e^{-\left(\left(1 - z\right) + 6.5\right)}\right)\right) \cdot \color{blue}{\frac{263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 - -43.89719783017524 \cdot {\pi}^{2}\right)\right)}{z}} \]

6. Taylor expanded in z around 0 96.4%

   \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot \left({\left(\left(1 - z\right) + 6.5\right)}^{\left(\left(1 - z\right) + -0.5\right)} \cdot \color{blue}{e^{-7.5}}\right)\right) \cdot \frac{263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 - -43.89719783017524 \cdot {\pi}^{2}\right)\right)}{z} \]

7. Taylor expanded in z around 0 96.3%

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

8. Final simplification 96.3%

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

Alternative 8: 96.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Simplified 95.5%

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

   1. pow1 95.5%

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

6. Applied egg-rr 95.4%

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

8. Simplified 97.0%

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

9. Taylor expanded in z around 0 96.2%

   \[\leadsto \sqrt{2 \cdot \pi} \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{-7.5 + z}\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \cdot \left(\frac{676.5203681218851}{1 - z} + \left(\color{blue}{\left(z \cdot \left(-131.48506030840542 \cdot z - 239.6241957716607\right) - 413.1371825859926\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right)\right)\right)\right) \]

10. Taylor expanded in z around 0 95.0%

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

    1. associate-*r/ 95.2%

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

12. Simplified 95.2%

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

13. Final simplification 95.2%

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

Alternative 9: 96.0% accurate, 1.7× speedup

Math

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

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(263.3831869810514 * Float64(sqrt(pi) * Float64(exp(-7.5) * Float64(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[(263.3831869810514 * N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] * N[(N[Sqrt[15.0], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.4%

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

4. Applied egg-rr 96.5%

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

   1. *-lft-identity 96.5%

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

   2. associate-+l+ 97.4%

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

6. Simplified 97.4%

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

7. Taylor expanded in z around 0 94.9%

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

   1. associate-*r* 94.9%

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

   2. associate-/l* 95.0%

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

9. Simplified 95.0%

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

    1. pow1 95.0%

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

    2. associate-*l* 95.1%

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

    3. sqrt-unprod 95.1%

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

    4. metadata-eval 95.1%

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

11. Applied egg-rr 95.1%

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

    1. unpow1 95.1%

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

    2. *-commutative 95.1%

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

13. Simplified 95.1%

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

Reproduce

herbie shell --seed 2024151 
(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))))))