Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.4% → 99.2%

Time: 11.1s

Alternatives: 10

Speedup: 1.7×

• could not determine a ground truth

  1. z = -9783211722922.025

Specification

\[z \leq 0.5\]

Math

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

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: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := -1 \cdot z + 7\\ t_1 := t\_0 + 0.5\\ \mathbf{if}\;z \leq -500:\\ \;\;\;\;\left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{\pi + \pi}\right)\right) \cdot 0.1456731240789439\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {t\_1}^{\left(-1 \cdot z + 0.5\right)}\right) \cdot e^{-t\_1}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} - \frac{1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, z + -2, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{176.6150291621406}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{1}{\frac{\left(-z\right) - -5}{12.507343278686905}}\right) + \frac{-0.13857109526572012}{-1 \cdot z + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_0}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{-1 \cdot z + 8}\right)\right)\\ \end{array} \]

FPCore

(FPCore (z)
  :precision binary64
  (let* ((t_0 (+ (* -1.0 z) 7.0)) (t_1 (+ t_0 0.5)))
  (if (<= z -500.0)
    (*
     (*
      (/ PI (sin (* z PI)))
      (* (pow (- (- 1.0 z) -6.5) (- (- 1.0 z) 0.5)) (sqrt (+ PI PI))))
     0.1456731240789439)
    (*
     (/ PI (sin (* PI z)))
     (*
      (*
       (* (sqrt (* PI 2.0)) (pow t_1 (+ (* -1.0 z) 0.5)))
       (exp (- t_1)))
      (+
       (+
        (+
         (+
          (+
           0.9999999999998099
           (+
            (-
             (/ 676.5203681218851 (- 1.0 z))
             (/ 1259.1392167224028 (- (- 1.0 z) -1.0)))
            (-
             (/
              771.3234287776531
              (/
               (- (pow 1.0 3.0) (pow (+ z -2.0) 3.0))
               (fma
                1.0
                1.0
                (fma (+ z -2.0) (+ z -2.0) (* 1.0 (+ z -2.0))))))
             (/ 176.6150291621406 (- (- 1.0 z) -3.0)))))
          (/ 1.0 (/ (- (- z) -5.0) 12.507343278686905)))
         (/ -0.13857109526572012 (+ (* -1.0 z) 6.0)))
        (/ 9.984369578019572e-6 t_0))
       (/ 1.5056327351493116e-7 (+ (* -1.0 z) 8.0))))))))

C

double code(double z) {
	double t_0 = (-1.0 * z) + 7.0;
	double t_1 = t_0 + 0.5;
	double tmp;
	if (z <= -500.0) {
		tmp = ((((double) M_PI) / sin((z * ((double) M_PI)))) * (pow(((1.0 - z) - -6.5), ((1.0 - z) - 0.5)) * sqrt((((double) M_PI) + ((double) M_PI))))) * 0.1456731240789439;
	} else {
		tmp = (((double) M_PI) / sin((((double) M_PI) * z))) * (((sqrt((((double) M_PI) * 2.0)) * pow(t_1, ((-1.0 * z) + 0.5))) * exp(-t_1)) * (((((0.9999999999998099 + (((676.5203681218851 / (1.0 - z)) - (1259.1392167224028 / ((1.0 - z) - -1.0))) + ((771.3234287776531 / ((pow(1.0, 3.0) - pow((z + -2.0), 3.0)) / fma(1.0, 1.0, fma((z + -2.0), (z + -2.0), (1.0 * (z + -2.0)))))) - (176.6150291621406 / ((1.0 - z) - -3.0))))) + (1.0 / ((-z - -5.0) / 12.507343278686905))) + (-0.13857109526572012 / ((-1.0 * z) + 6.0))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / ((-1.0 * z) + 8.0))));
	}
	return tmp;
}

Julia

function code(z)
	t_0 = Float64(Float64(-1.0 * z) + 7.0)
	t_1 = Float64(t_0 + 0.5)
	tmp = 0.0
	if (z <= -500.0)
		tmp = Float64(Float64(Float64(pi / sin(Float64(z * pi))) * Float64((Float64(Float64(1.0 - z) - -6.5) ^ Float64(Float64(1.0 - z) - 0.5)) * sqrt(Float64(pi + pi)))) * 0.1456731240789439);
	else
		tmp = Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (t_1 ^ Float64(Float64(-1.0 * z) + 0.5))) * exp(Float64(-t_1))) * Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) - Float64(1259.1392167224028 / Float64(Float64(1.0 - z) - -1.0))) + Float64(Float64(771.3234287776531 / Float64(Float64((1.0 ^ 3.0) - (Float64(z + -2.0) ^ 3.0)) / fma(1.0, 1.0, fma(Float64(z + -2.0), Float64(z + -2.0), Float64(1.0 * Float64(z + -2.0)))))) - Float64(176.6150291621406 / Float64(Float64(1.0 - z) - -3.0))))) + Float64(1.0 / Float64(Float64(Float64(-z) - -5.0) / 12.507343278686905))) + Float64(-0.13857109526572012 / Float64(Float64(-1.0 * z) + 6.0))) + Float64(9.984369578019572e-6 / t_0)) + Float64(1.5056327351493116e-7 / Float64(Float64(-1.0 * z) + 8.0)))));
	end
	return tmp
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(-1.0 * z), $MachinePrecision] + 7.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 0.5), $MachinePrecision]}, If[LessEqual[z, -500.0], N[(N[(N[(Pi / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] - -6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi + Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.1456731240789439), $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$1, N[(N[(-1.0 * z), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$1)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(0.9999999999998099 + N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[(1259.1392167224028 / N[(N[(1.0 - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(N[(N[Power[1.0, 3.0], $MachinePrecision] - N[Power[N[(z + -2.0), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 * 1.0 + N[(N[(z + -2.0), $MachinePrecision] * N[(z + -2.0), $MachinePrecision] + N[(1.0 * N[(z + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[((-z) - -5.0), $MachinePrecision] / 12.507343278686905), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(-1.0 * z), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(-1.0 * z), $MachinePrecision] + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < -500

   1. Initial program 96.4%

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

   3. Applied rewrites 96.4%

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

   5. Applied rewrites 96.4%

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

   6. Taylor expanded in z around 0

      \[\leadsto \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{\pi + \pi}\right)\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot e^{\frac{-15}{2}}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{\pi + \pi}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \color{blue}{e^{\frac{-15}{2}}}\right) \]

      2. lower-exp.f64 96.0%

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

   8. Applied rewrites 96.0%

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

   9. Evaluated real constant 96.0%

      \[\leadsto \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{\pi + \pi}\right)\right) \cdot 0.1456731240789439 \]

   if -500 < z

   1. Initial program 96.4%

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

   3. Applied rewrites 98.3%

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

      1. lift--.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\color{blue}{\left(1 - z\right) - -2}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\color{blue}{\left(1 - z\right)} - -2} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      3. associate--l- N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\color{blue}{1 - \left(z + -2\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      4. flip3-- N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\color{blue}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{1 \cdot 1 + \left(\left(z + -2\right) \cdot \left(z + -2\right) + 1 \cdot \left(z + -2\right)\right)}}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      5. lower-unsound-/.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\color{blue}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{1 \cdot 1 + \left(\left(z + -2\right) \cdot \left(z + -2\right) + 1 \cdot \left(z + -2\right)\right)}}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      6. lower-unsound--.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{\color{blue}{{1}^{3} - {\left(z + -2\right)}^{3}}}{1 \cdot 1 + \left(\left(z + -2\right) \cdot \left(z + -2\right) + 1 \cdot \left(z + -2\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      7. lower-unsound-pow.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{\color{blue}{{1}^{3}} - {\left(z + -2\right)}^{3}}{1 \cdot 1 + \left(\left(z + -2\right) \cdot \left(z + -2\right) + 1 \cdot \left(z + -2\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      8. lower-unsound-pow.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - \color{blue}{{\left(z + -2\right)}^{3}}}{1 \cdot 1 + \left(\left(z + -2\right) \cdot \left(z + -2\right) + 1 \cdot \left(z + -2\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\color{blue}{\left(z + -2\right)}}^{3}}{1 \cdot 1 + \left(\left(z + -2\right) \cdot \left(z + -2\right) + 1 \cdot \left(z + -2\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      10. lower-unsound-fma.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\color{blue}{\mathsf{fma}\left(1, 1, \left(z + -2\right) \cdot \left(z + -2\right) + 1 \cdot \left(z + -2\right)\right)}}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      11. lower-unsound-fma.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \color{blue}{\mathsf{fma}\left(z + -2, z + -2, 1 \cdot \left(z + -2\right)\right)}\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(\color{blue}{z + -2}, z + -2, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, \color{blue}{z + -2}, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      14. lower-unsound-*.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, z + -2, \color{blue}{1 \cdot \left(z + -2\right)}\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      15. lower-+.f64 98.3%

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in z around 0

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

      1. lower-*.f64 98.3%

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in z around 0

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

       1. lower-*.f64 98.3%

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

   11. Applied rewrites 98.3%

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

   12. Taylor expanded in z around 0

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

       1. lower-*.f64 98.3%

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

   14. Applied rewrites 98.3%

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

   15. Taylor expanded in z around 0

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

       1. lower-*.f64 98.3%

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

   17. Applied rewrites 98.3%

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

   18. Taylor expanded in z around 0

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

       1. lower-*.f64 98.3%

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

   20. Applied rewrites 98.3%

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

   21. Taylor expanded in z around 0

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

       1. lower-*.f64 98.3%

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

   23. Applied rewrites 98.3%

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

   24. Taylor expanded in z around 0

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

       1. lower-*.f64 98.3%

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

   26. Applied rewrites 98.3%

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

       1. lift-/.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}^{\left(-1 \cdot z + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, z + -2, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \color{blue}{\frac{\frac{2501468655737381}{200000000000000}}{-1 \cdot z + 5}}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{-1 \cdot z + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{-1 \cdot z + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{-1 \cdot z + 8}\right)\right) \]

       2. div-flip N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}^{\left(-1 \cdot z + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, z + -2, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \color{blue}{\frac{1}{\frac{-1 \cdot z + 5}{\frac{2501468655737381}{200000000000000}}}}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{-1 \cdot z + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{-1 \cdot z + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{-1 \cdot z + 8}\right)\right) \]

       3. lower-unsound-/.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}^{\left(-1 \cdot z + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, z + -2, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \color{blue}{\frac{1}{\frac{-1 \cdot z + 5}{\frac{2501468655737381}{200000000000000}}}}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{-1 \cdot z + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{-1 \cdot z + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{-1 \cdot z + 8}\right)\right) \]

       4. lower-unsound-/.f64 98.3%

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

       5. lift-+.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}^{\left(-1 \cdot z + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, z + -2, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{1}{\frac{\color{blue}{-1 \cdot z + 5}}{\frac{2501468655737381}{200000000000000}}}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{-1 \cdot z + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{-1 \cdot z + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{-1 \cdot z + 8}\right)\right) \]

       6. add-flip N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}^{\left(-1 \cdot z + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, z + -2, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{1}{\frac{\color{blue}{-1 \cdot z - \left(\mathsf{neg}\left(5\right)\right)}}{\frac{2501468655737381}{200000000000000}}}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{-1 \cdot z + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{-1 \cdot z + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{-1 \cdot z + 8}\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}^{\left(-1 \cdot z + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, z + -2, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{1}{\frac{-1 \cdot z - \color{blue}{-5}}{\frac{2501468655737381}{200000000000000}}}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{-1 \cdot z + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{-1 \cdot z + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{-1 \cdot z + 8}\right)\right) \]

       8. lower--.f64 98.3%

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

       9. lift-*.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}^{\left(-1 \cdot z + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, z + -2, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{1}{\frac{-1 \cdot \color{blue}{z} - -5}{\frac{2501468655737381}{200000000000000}}}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{-1 \cdot z + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{-1 \cdot z + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{-1 \cdot z + 8}\right)\right) \]

       10. mul-1-neg N/A

           \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}^{\left(-1 \cdot z + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, z + -2, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{1}{\frac{\left(\mathsf{neg}\left(z\right)\right) - -5}{\frac{2501468655737381}{200000000000000}}}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{-1 \cdot z + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{-1 \cdot z + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{-1 \cdot z + 8}\right)\right) \]

       11. lower-neg.f64 98.3%

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

   28. Applied rewrites 98.3%

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

Alternative 2: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \left(-z\right) + 7\\ t_1 := t\_0 + 0.5\\ \mathbf{if}\;z \leq -500:\\ \;\;\;\;\left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{\pi + \pi}\right)\right) \cdot 0.1456731240789439\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {t\_1}^{\left(\left(-z\right) + 0.5\right)}\right) \cdot e^{-t\_1}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} - \frac{1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, z + -2, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{176.6150291621406}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{12.507343278686905}{\left(-z\right) + 5}\right) + \frac{-0.13857109526572012}{\left(-z\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_0}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(-z\right) + 8}\right)\right)\\ \end{array} \]

FPCore

(FPCore (z)
  :precision binary64
  (let* ((t_0 (+ (- z) 7.0)) (t_1 (+ t_0 0.5)))
  (if (<= z -500.0)
    (*
     (*
      (/ PI (sin (* z PI)))
      (* (pow (- (- 1.0 z) -6.5) (- (- 1.0 z) 0.5)) (sqrt (+ PI PI))))
     0.1456731240789439)
    (*
     (/ PI (sin (* PI z)))
     (*
      (* (* (sqrt (* PI 2.0)) (pow t_1 (+ (- z) 0.5))) (exp (- t_1)))
      (+
       (+
        (+
         (+
          (+
           0.9999999999998099
           (+
            (-
             (/ 676.5203681218851 (- 1.0 z))
             (/ 1259.1392167224028 (- (- 1.0 z) -1.0)))
            (-
             (/
              771.3234287776531
              (/
               (- (pow 1.0 3.0) (pow (+ z -2.0) 3.0))
               (fma
                1.0
                1.0
                (fma (+ z -2.0) (+ z -2.0) (* 1.0 (+ z -2.0))))))
             (/ 176.6150291621406 (- (- 1.0 z) -3.0)))))
          (/ 12.507343278686905 (+ (- z) 5.0)))
         (/ -0.13857109526572012 (+ (- z) 6.0)))
        (/ 9.984369578019572e-6 t_0))
       (/ 1.5056327351493116e-7 (+ (- z) 8.0))))))))

C

double code(double z) {
	double t_0 = -z + 7.0;
	double t_1 = t_0 + 0.5;
	double tmp;
	if (z <= -500.0) {
		tmp = ((((double) M_PI) / sin((z * ((double) M_PI)))) * (pow(((1.0 - z) - -6.5), ((1.0 - z) - 0.5)) * sqrt((((double) M_PI) + ((double) M_PI))))) * 0.1456731240789439;
	} else {
		tmp = (((double) M_PI) / sin((((double) M_PI) * z))) * (((sqrt((((double) M_PI) * 2.0)) * pow(t_1, (-z + 0.5))) * exp(-t_1)) * (((((0.9999999999998099 + (((676.5203681218851 / (1.0 - z)) - (1259.1392167224028 / ((1.0 - z) - -1.0))) + ((771.3234287776531 / ((pow(1.0, 3.0) - pow((z + -2.0), 3.0)) / fma(1.0, 1.0, fma((z + -2.0), (z + -2.0), (1.0 * (z + -2.0)))))) - (176.6150291621406 / ((1.0 - z) - -3.0))))) + (12.507343278686905 / (-z + 5.0))) + (-0.13857109526572012 / (-z + 6.0))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / (-z + 8.0))));
	}
	return tmp;
}

Julia

function code(z)
	t_0 = Float64(Float64(-z) + 7.0)
	t_1 = Float64(t_0 + 0.5)
	tmp = 0.0
	if (z <= -500.0)
		tmp = Float64(Float64(Float64(pi / sin(Float64(z * pi))) * Float64((Float64(Float64(1.0 - z) - -6.5) ^ Float64(Float64(1.0 - z) - 0.5)) * sqrt(Float64(pi + pi)))) * 0.1456731240789439);
	else
		tmp = Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (t_1 ^ Float64(Float64(-z) + 0.5))) * exp(Float64(-t_1))) * Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) - Float64(1259.1392167224028 / Float64(Float64(1.0 - z) - -1.0))) + Float64(Float64(771.3234287776531 / Float64(Float64((1.0 ^ 3.0) - (Float64(z + -2.0) ^ 3.0)) / fma(1.0, 1.0, fma(Float64(z + -2.0), Float64(z + -2.0), Float64(1.0 * Float64(z + -2.0)))))) - Float64(176.6150291621406 / Float64(Float64(1.0 - z) - -3.0))))) + Float64(12.507343278686905 / Float64(Float64(-z) + 5.0))) + Float64(-0.13857109526572012 / Float64(Float64(-z) + 6.0))) + Float64(9.984369578019572e-6 / t_0)) + Float64(1.5056327351493116e-7 / Float64(Float64(-z) + 8.0)))));
	end
	return tmp
end

Wolfram

code[z_] := Block[{t$95$0 = N[((-z) + 7.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 0.5), $MachinePrecision]}, If[LessEqual[z, -500.0], N[(N[(N[(Pi / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] - -6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi + Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.1456731240789439), $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$1, N[((-z) + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$1)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(0.9999999999998099 + N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[(1259.1392167224028 / N[(N[(1.0 - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(N[(N[Power[1.0, 3.0], $MachinePrecision] - N[Power[N[(z + -2.0), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 * 1.0 + N[(N[(z + -2.0), $MachinePrecision] * N[(z + -2.0), $MachinePrecision] + N[(1.0 * N[(z + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[((-z) + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[((-z) + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[((-z) + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < -500

   1. Initial program 96.4%

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

   3. Applied rewrites 96.4%

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

   5. Applied rewrites 96.4%

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

   6. Taylor expanded in z around 0

      \[\leadsto \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{\pi + \pi}\right)\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot e^{\frac{-15}{2}}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{\pi + \pi}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \color{blue}{e^{\frac{-15}{2}}}\right) \]

      2. lower-exp.f64 96.0%

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

   8. Applied rewrites 96.0%

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

   9. Evaluated real constant 96.0%

      \[\leadsto \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{\pi + \pi}\right)\right) \cdot 0.1456731240789439 \]

   if -500 < z

   1. Initial program 96.4%

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

   3. Applied rewrites 98.3%

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

      1. lift--.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\color{blue}{\left(1 - z\right) - -2}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\color{blue}{\left(1 - z\right)} - -2} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      3. associate--l- N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\color{blue}{1 - \left(z + -2\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      4. flip3-- N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\color{blue}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{1 \cdot 1 + \left(\left(z + -2\right) \cdot \left(z + -2\right) + 1 \cdot \left(z + -2\right)\right)}}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      5. lower-unsound-/.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\color{blue}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{1 \cdot 1 + \left(\left(z + -2\right) \cdot \left(z + -2\right) + 1 \cdot \left(z + -2\right)\right)}}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      6. lower-unsound--.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{\color{blue}{{1}^{3} - {\left(z + -2\right)}^{3}}}{1 \cdot 1 + \left(\left(z + -2\right) \cdot \left(z + -2\right) + 1 \cdot \left(z + -2\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      7. lower-unsound-pow.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{\color{blue}{{1}^{3}} - {\left(z + -2\right)}^{3}}{1 \cdot 1 + \left(\left(z + -2\right) \cdot \left(z + -2\right) + 1 \cdot \left(z + -2\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      8. lower-unsound-pow.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - \color{blue}{{\left(z + -2\right)}^{3}}}{1 \cdot 1 + \left(\left(z + -2\right) \cdot \left(z + -2\right) + 1 \cdot \left(z + -2\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\color{blue}{\left(z + -2\right)}}^{3}}{1 \cdot 1 + \left(\left(z + -2\right) \cdot \left(z + -2\right) + 1 \cdot \left(z + -2\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      10. lower-unsound-fma.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\color{blue}{\mathsf{fma}\left(1, 1, \left(z + -2\right) \cdot \left(z + -2\right) + 1 \cdot \left(z + -2\right)\right)}}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      11. lower-unsound-fma.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \color{blue}{\mathsf{fma}\left(z + -2, z + -2, 1 \cdot \left(z + -2\right)\right)}\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(\color{blue}{z + -2}, z + -2, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, \color{blue}{z + -2}, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      14. lower-unsound-*.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, z + -2, \color{blue}{1 \cdot \left(z + -2\right)}\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      15. lower-+.f64 98.3%

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in z around 0

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

      1. lower-*.f64 98.3%

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in z around 0

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

       1. lower-*.f64 98.3%

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

   11. Applied rewrites 98.3%

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

   12. Taylor expanded in z around 0

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

       1. lower-*.f64 98.3%

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

   14. Applied rewrites 98.3%

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

   15. Taylor expanded in z around 0

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

       1. lower-*.f64 98.3%

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

   17. Applied rewrites 98.3%

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

   18. Taylor expanded in z around 0

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

       1. lower-*.f64 98.3%

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

   20. Applied rewrites 98.3%

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

   21. Taylor expanded in z around 0

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

       1. lower-*.f64 98.3%

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

   23. Applied rewrites 98.3%

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

   24. Taylor expanded in z around 0

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

       1. lower-*.f64 98.3%

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

   26. Applied rewrites 98.3%

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

       1. lift-*.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(-1 \cdot \color{blue}{z} + 7\right) + \frac{1}{2}\right)}^{\left(-1 \cdot z + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, z + -2, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{-1 \cdot z + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{-1 \cdot z + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{-1 \cdot z + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{-1 \cdot z + 8}\right)\right) \]

       2. mul-1-neg N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\mathsf{neg}\left(z\right)\right) + 7\right) + \frac{1}{2}\right)}^{\left(-1 \cdot z + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, z + -2, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{-1 \cdot z + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{-1 \cdot z + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{-1 \cdot z + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{-1 \cdot z + 8}\right)\right) \]

       3. lower-neg.f64 98.3%

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

   28. Applied rewrites 98.3%

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

       1. lift-*.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(-1 \cdot \color{blue}{z} + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, z + -2, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{-1 \cdot z + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{-1 \cdot z + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{-1 \cdot z + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{-1 \cdot z + 8}\right)\right) \]

       2. mul-1-neg N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\mathsf{neg}\left(z\right)\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(-1 \cdot z + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, z + -2, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{-1 \cdot z + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{-1 \cdot z + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{-1 \cdot z + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{-1 \cdot z + 8}\right)\right) \]

       3. lower-neg.f64 98.3%

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

   30. Applied rewrites 98.3%

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

       1. lift-*.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(-1 \cdot \color{blue}{z} + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, z + -2, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{-1 \cdot z + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{-1 \cdot z + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{-1 \cdot z + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{-1 \cdot z + 8}\right)\right) \]

       2. mul-1-neg N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\mathsf{neg}\left(z\right)\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, z + -2, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{-1 \cdot z + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{-1 \cdot z + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{-1 \cdot z + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{-1 \cdot z + 8}\right)\right) \]

       3. lower-neg.f64 98.3%

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

   32. Applied rewrites 98.3%

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

       1. lift-*.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, z + -2, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{-1 \cdot \color{blue}{z} + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{-1 \cdot z + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{-1 \cdot z + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{-1 \cdot z + 8}\right)\right) \]

       2. mul-1-neg N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, z + -2, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\mathsf{neg}\left(z\right)\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{-1 \cdot z + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{-1 \cdot z + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{-1 \cdot z + 8}\right)\right) \]

       3. lower-neg.f64 98.3%

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

   34. Applied rewrites 98.3%

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

       1. lift-*.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, z + -2, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(-z\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{-1 \cdot \color{blue}{z} + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{-1 \cdot z + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{-1 \cdot z + 8}\right)\right) \]

       2. mul-1-neg N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, z + -2, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(-z\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\mathsf{neg}\left(z\right)\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{-1 \cdot z + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{-1 \cdot z + 8}\right)\right) \]

       3. lower-neg.f64 98.3%

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

   36. Applied rewrites 98.3%

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

       1. lift-*.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, z + -2, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(-z\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(-z\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{-1 \cdot \color{blue}{z} + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{-1 \cdot z + 8}\right)\right) \]

       2. mul-1-neg N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, z + -2, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(-z\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(-z\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\mathsf{neg}\left(z\right)\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{-1 \cdot z + 8}\right)\right) \]

       3. lower-neg.f64 98.3%

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

   38. Applied rewrites 98.3%

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

       1. lift-*.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, z + -2, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(-z\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(-z\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(-z\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{-1 \cdot \color{blue}{z} + 8}\right)\right) \]

       2. mul-1-neg N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(-z\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(-z\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, z + -2, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(-z\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(-z\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(-z\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\mathsf{neg}\left(z\right)\right) + 8}\right)\right) \]

       3. lower-neg.f64 98.3%

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

   40. Applied rewrites 98.3%

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

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := t\_0 + 7\\ t_2 := t\_1 + 0.5\\ \mathbf{if}\;z \leq -500:\\ \;\;\;\;\left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{\pi + \pi}\right)\right) \cdot 0.1456731240789439\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {t\_2}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{-t\_2}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} - \frac{1259.1392167224028}{\left(1 - z\right) - -1}\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} - \frac{176.6150291621406}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{12.507343278686905}{t\_0 + 5}\right) + \frac{-0.13857109526572012}{t\_0 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_1}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\right)\right)\\ \end{array} \]

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)))
  (if (<= z -500.0)
    (*
     (*
      (/ PI (sin (* z PI)))
      (* (pow (- (- 1.0 z) -6.5) (- (- 1.0 z) 0.5)) (sqrt (+ PI PI))))
     0.1456731240789439)
    (*
     (/ PI (sin (* PI z)))
     (*
      (* (* (sqrt (* PI 2.0)) (pow t_2 (+ t_0 0.5))) (exp (- t_2)))
      (+
       (+
        (+
         (+
          (+
           0.9999999999998099
           (+
            (-
             (/ 676.5203681218851 (- 1.0 z))
             (/ 1259.1392167224028 (- (- 1.0 z) -1.0)))
            (-
             (/ 771.3234287776531 (- (- 1.0 z) -2.0))
             (/ 176.6150291621406 (- (- 1.0 z) -3.0)))))
          (/ 12.507343278686905 (+ 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;
	double tmp;
	if (z <= -500.0) {
		tmp = ((((double) M_PI) / sin((z * ((double) M_PI)))) * (pow(((1.0 - z) - -6.5), ((1.0 - z) - 0.5)) * sqrt((((double) M_PI) + ((double) M_PI))))) * 0.1456731240789439;
	} else {
		tmp = (((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 / (1.0 - z)) - (1259.1392167224028 / ((1.0 - z) - -1.0))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) - (176.6150291621406 / ((1.0 - z) - -3.0))))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
	}
	return tmp;
}

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;
	double tmp;
	if (z <= -500.0) {
		tmp = ((Math.PI / Math.sin((z * Math.PI))) * (Math.pow(((1.0 - z) - -6.5), ((1.0 - z) - 0.5)) * Math.sqrt((Math.PI + Math.PI)))) * 0.1456731240789439;
	} else {
		tmp = (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 / (1.0 - z)) - (1259.1392167224028 / ((1.0 - z) - -1.0))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) - (176.6150291621406 / ((1.0 - z) - -3.0))))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
	}
	return tmp;
}

Python

def code(z):
	t_0 = (1.0 - z) - 1.0
	t_1 = t_0 + 7.0
	t_2 = t_1 + 0.5
	tmp = 0
	if z <= -500.0:
		tmp = ((math.pi / math.sin((z * math.pi))) * (math.pow(((1.0 - z) - -6.5), ((1.0 - z) - 0.5)) * math.sqrt((math.pi + math.pi)))) * 0.1456731240789439
	else:
		tmp = (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 / (1.0 - z)) - (1259.1392167224028 / ((1.0 - z) - -1.0))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) - (176.6150291621406 / ((1.0 - z) - -3.0))))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))))
	return tmp

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)
	tmp = 0.0
	if (z <= -500.0)
		tmp = Float64(Float64(Float64(pi / sin(Float64(z * pi))) * Float64((Float64(Float64(1.0 - z) - -6.5) ^ Float64(Float64(1.0 - z) - 0.5)) * sqrt(Float64(pi + pi)))) * 0.1456731240789439);
	else
		tmp = 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(0.9999999999998099 + Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) - Float64(1259.1392167224028 / Float64(Float64(1.0 - z) - -1.0))) + Float64(Float64(771.3234287776531 / Float64(Float64(1.0 - z) - -2.0)) - Float64(176.6150291621406 / Float64(Float64(1.0 - z) - -3.0))))) + Float64(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
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = (1.0 - z) - 1.0;
	t_1 = t_0 + 7.0;
	t_2 = t_1 + 0.5;
	tmp = 0.0;
	if (z <= -500.0)
		tmp = ((pi / sin((z * pi))) * ((((1.0 - z) - -6.5) ^ ((1.0 - z) - 0.5)) * sqrt((pi + pi)))) * 0.1456731240789439;
	else
		tmp = (pi / sin((pi * z))) * (((sqrt((pi * 2.0)) * (t_2 ^ (t_0 + 0.5))) * exp(-t_2)) * (((((0.9999999999998099 + (((676.5203681218851 / (1.0 - z)) - (1259.1392167224028 / ((1.0 - z) - -1.0))) + ((771.3234287776531 / ((1.0 - z) - -2.0)) - (176.6150291621406 / ((1.0 - z) - -3.0))))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
	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[(t$95$0 + 7.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + 0.5), $MachinePrecision]}, If[LessEqual[z, -500.0], N[(N[(N[(Pi / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] - -6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi + Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.1456731240789439), $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[(0.9999999999998099 + N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[(1259.1392167224028 / N[(N[(1.0 - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(N[(1.0 - z), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] - N[(176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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]]]]]

Derivation

1. Split input into 2 regimes

2. if z < -500

   1. Initial program 96.4%

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

   3. Applied rewrites 96.4%

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

   5. Applied rewrites 96.4%

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

   6. Taylor expanded in z around 0

      \[\leadsto \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{\pi + \pi}\right)\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot e^{\frac{-15}{2}}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{\pi + \pi}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \color{blue}{e^{\frac{-15}{2}}}\right) \]

      2. lower-exp.f64 96.0%

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

   8. Applied rewrites 96.0%

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

   9. Evaluated real constant 96.0%

      \[\leadsto \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{\pi + \pi}\right)\right) \cdot 0.1456731240789439 \]

   if -500 < z

   1. Initial program 96.4%

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

   3. Applied rewrites 98.3%

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

Alternative 4: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \left(-z\right) + 7.5\\ t_1 := \sqrt{\pi + \pi}\\ t_2 := \left(1 - z\right) - 1\\ t_3 := t\_2 + 7\\ t_4 := t\_3 + 0.5\\ t_5 := \frac{\pi}{\sin \left(z \cdot \pi\right)}\\ \mathbf{if}\;\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {t\_4}^{\left(t\_2 + 0.5\right)}\right) \cdot e^{-t\_4}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{t\_2 + 1}\right) + \frac{-1259.1392167224028}{t\_2 + 2}\right) + \frac{771.3234287776531}{t\_2 + 3}\right) + \frac{-176.6150291621406}{t\_2 + 4}\right) + \frac{12.507343278686905}{t\_2 + 5}\right) + \frac{-0.13857109526572012}{t\_2 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_3}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_2 + 8}\right)\right) \leq 2 \cdot 10^{+200}:\\ \;\;\;\;\left(t\_5 \cdot \left(e^{-t\_0} \cdot \left({t\_0}^{\left(\left(-z\right) - -0.5\right)} \cdot t\_1\right)\right)\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(-z\right) - -8} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(-z\right) - -7} + \left(\frac{-0.13857109526572012}{\left(-z\right) - -6} + \left(\frac{12.507343278686905}{\left(-z\right) - -5} + \left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} - \frac{1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{-771.3234287776531}{-2 - \left(1 - z\right)} - \frac{176.6150291621406}{\left(1 - z\right) - -3}\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_5 \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot t\_1\right)\right) \cdot 0.1456731240789439\\ \end{array} \]

FPCore

(FPCore (z)
  :precision binary64
  (let* ((t_0 (+ (- z) 7.5))
       (t_1 (sqrt (+ PI PI)))
       (t_2 (- (- 1.0 z) 1.0))
       (t_3 (+ t_2 7.0))
       (t_4 (+ t_3 0.5))
       (t_5 (/ PI (sin (* z PI)))))
  (if (<=
       (*
        (/ PI (sin (* PI z)))
        (*
         (* (* (sqrt (* PI 2.0)) (pow t_4 (+ t_2 0.5))) (exp (- t_4)))
         (+
          (+
           (+
            (+
             (+
              (+
               (+
                (+
                 0.9999999999998099
                 (/ 676.5203681218851 (+ t_2 1.0)))
                (/ -1259.1392167224028 (+ t_2 2.0)))
               (/ 771.3234287776531 (+ t_2 3.0)))
              (/ -176.6150291621406 (+ t_2 4.0)))
             (/ 12.507343278686905 (+ t_2 5.0)))
            (/ -0.13857109526572012 (+ t_2 6.0)))
           (/ 9.984369578019572e-6 t_3))
          (/ 1.5056327351493116e-7 (+ t_2 8.0)))))
       2e+200)
    (*
     (* t_5 (* (exp (- t_0)) (* (pow t_0 (- (- z) -0.5)) t_1)))
     (+
      (/ 1.5056327351493116e-7 (- (- z) -8.0))
      (+
       (/ 9.984369578019572e-6 (- (- z) -7.0))
       (+
        (/ -0.13857109526572012 (- (- z) -6.0))
        (+
         (/ 12.507343278686905 (- (- z) -5.0))
         (+
          (+
           0.9999999999998099
           (-
            (/ 676.5203681218851 (- 1.0 z))
            (/ 1259.1392167224028 (- (- 1.0 z) -1.0))))
          (-
           (/ -771.3234287776531 (- -2.0 (- 1.0 z)))
           (/ 176.6150291621406 (- (- 1.0 z) -3.0)))))))))
    (*
     (* t_5 (* (pow (- (- 1.0 z) -6.5) (- (- 1.0 z) 0.5)) t_1))
     0.1456731240789439))))

C

double code(double z) {
	double t_0 = -z + 7.5;
	double t_1 = sqrt((((double) M_PI) + ((double) M_PI)));
	double t_2 = (1.0 - z) - 1.0;
	double t_3 = t_2 + 7.0;
	double t_4 = t_3 + 0.5;
	double t_5 = ((double) M_PI) / sin((z * ((double) M_PI)));
	double tmp;
	if (((((double) M_PI) / sin((((double) M_PI) * z))) * (((sqrt((((double) M_PI) * 2.0)) * pow(t_4, (t_2 + 0.5))) * exp(-t_4)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_2 + 1.0))) + (-1259.1392167224028 / (t_2 + 2.0))) + (771.3234287776531 / (t_2 + 3.0))) + (-176.6150291621406 / (t_2 + 4.0))) + (12.507343278686905 / (t_2 + 5.0))) + (-0.13857109526572012 / (t_2 + 6.0))) + (9.984369578019572e-6 / t_3)) + (1.5056327351493116e-7 / (t_2 + 8.0))))) <= 2e+200) {
		tmp = (t_5 * (exp(-t_0) * (pow(t_0, (-z - -0.5)) * t_1))) * ((1.5056327351493116e-7 / (-z - -8.0)) + ((9.984369578019572e-6 / (-z - -7.0)) + ((-0.13857109526572012 / (-z - -6.0)) + ((12.507343278686905 / (-z - -5.0)) + ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) - (1259.1392167224028 / ((1.0 - z) - -1.0)))) + ((-771.3234287776531 / (-2.0 - (1.0 - z))) - (176.6150291621406 / ((1.0 - z) - -3.0))))))));
	} else {
		tmp = (t_5 * (pow(((1.0 - z) - -6.5), ((1.0 - z) - 0.5)) * t_1)) * 0.1456731240789439;
	}
	return tmp;
}

Java

public static double code(double z) {
	double t_0 = -z + 7.5;
	double t_1 = Math.sqrt((Math.PI + Math.PI));
	double t_2 = (1.0 - z) - 1.0;
	double t_3 = t_2 + 7.0;
	double t_4 = t_3 + 0.5;
	double t_5 = Math.PI / Math.sin((z * Math.PI));
	double tmp;
	if (((Math.PI / Math.sin((Math.PI * z))) * (((Math.sqrt((Math.PI * 2.0)) * Math.pow(t_4, (t_2 + 0.5))) * Math.exp(-t_4)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_2 + 1.0))) + (-1259.1392167224028 / (t_2 + 2.0))) + (771.3234287776531 / (t_2 + 3.0))) + (-176.6150291621406 / (t_2 + 4.0))) + (12.507343278686905 / (t_2 + 5.0))) + (-0.13857109526572012 / (t_2 + 6.0))) + (9.984369578019572e-6 / t_3)) + (1.5056327351493116e-7 / (t_2 + 8.0))))) <= 2e+200) {
		tmp = (t_5 * (Math.exp(-t_0) * (Math.pow(t_0, (-z - -0.5)) * t_1))) * ((1.5056327351493116e-7 / (-z - -8.0)) + ((9.984369578019572e-6 / (-z - -7.0)) + ((-0.13857109526572012 / (-z - -6.0)) + ((12.507343278686905 / (-z - -5.0)) + ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) - (1259.1392167224028 / ((1.0 - z) - -1.0)))) + ((-771.3234287776531 / (-2.0 - (1.0 - z))) - (176.6150291621406 / ((1.0 - z) - -3.0))))))));
	} else {
		tmp = (t_5 * (Math.pow(((1.0 - z) - -6.5), ((1.0 - z) - 0.5)) * t_1)) * 0.1456731240789439;
	}
	return tmp;
}

Python

def code(z):
	t_0 = -z + 7.5
	t_1 = math.sqrt((math.pi + math.pi))
	t_2 = (1.0 - z) - 1.0
	t_3 = t_2 + 7.0
	t_4 = t_3 + 0.5
	t_5 = math.pi / math.sin((z * math.pi))
	tmp = 0
	if ((math.pi / math.sin((math.pi * z))) * (((math.sqrt((math.pi * 2.0)) * math.pow(t_4, (t_2 + 0.5))) * math.exp(-t_4)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_2 + 1.0))) + (-1259.1392167224028 / (t_2 + 2.0))) + (771.3234287776531 / (t_2 + 3.0))) + (-176.6150291621406 / (t_2 + 4.0))) + (12.507343278686905 / (t_2 + 5.0))) + (-0.13857109526572012 / (t_2 + 6.0))) + (9.984369578019572e-6 / t_3)) + (1.5056327351493116e-7 / (t_2 + 8.0))))) <= 2e+200:
		tmp = (t_5 * (math.exp(-t_0) * (math.pow(t_0, (-z - -0.5)) * t_1))) * ((1.5056327351493116e-7 / (-z - -8.0)) + ((9.984369578019572e-6 / (-z - -7.0)) + ((-0.13857109526572012 / (-z - -6.0)) + ((12.507343278686905 / (-z - -5.0)) + ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) - (1259.1392167224028 / ((1.0 - z) - -1.0)))) + ((-771.3234287776531 / (-2.0 - (1.0 - z))) - (176.6150291621406 / ((1.0 - z) - -3.0))))))))
	else:
		tmp = (t_5 * (math.pow(((1.0 - z) - -6.5), ((1.0 - z) - 0.5)) * t_1)) * 0.1456731240789439
	return tmp

Julia

function code(z)
	t_0 = Float64(Float64(-z) + 7.5)
	t_1 = sqrt(Float64(pi + pi))
	t_2 = Float64(Float64(1.0 - z) - 1.0)
	t_3 = Float64(t_2 + 7.0)
	t_4 = Float64(t_3 + 0.5)
	t_5 = Float64(pi / sin(Float64(z * pi)))
	tmp = 0.0
	if (Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (t_4 ^ Float64(t_2 + 0.5))) * exp(Float64(-t_4))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(t_2 + 1.0))) + Float64(-1259.1392167224028 / Float64(t_2 + 2.0))) + Float64(771.3234287776531 / Float64(t_2 + 3.0))) + Float64(-176.6150291621406 / Float64(t_2 + 4.0))) + Float64(12.507343278686905 / Float64(t_2 + 5.0))) + Float64(-0.13857109526572012 / Float64(t_2 + 6.0))) + Float64(9.984369578019572e-6 / t_3)) + Float64(1.5056327351493116e-7 / Float64(t_2 + 8.0))))) <= 2e+200)
		tmp = Float64(Float64(t_5 * Float64(exp(Float64(-t_0)) * Float64((t_0 ^ Float64(Float64(-z) - -0.5)) * t_1))) * Float64(Float64(1.5056327351493116e-7 / Float64(Float64(-z) - -8.0)) + Float64(Float64(9.984369578019572e-6 / Float64(Float64(-z) - -7.0)) + Float64(Float64(-0.13857109526572012 / Float64(Float64(-z) - -6.0)) + Float64(Float64(12.507343278686905 / Float64(Float64(-z) - -5.0)) + Float64(Float64(0.9999999999998099 + Float64(Float64(676.5203681218851 / Float64(1.0 - z)) - Float64(1259.1392167224028 / Float64(Float64(1.0 - z) - -1.0)))) + Float64(Float64(-771.3234287776531 / Float64(-2.0 - Float64(1.0 - z))) - Float64(176.6150291621406 / Float64(Float64(1.0 - z) - -3.0)))))))));
	else
		tmp = Float64(Float64(t_5 * Float64((Float64(Float64(1.0 - z) - -6.5) ^ Float64(Float64(1.0 - z) - 0.5)) * t_1)) * 0.1456731240789439);
	end
	return tmp
end

MATLAB

function tmp_2 = code(z)
	t_0 = -z + 7.5;
	t_1 = sqrt((pi + pi));
	t_2 = (1.0 - z) - 1.0;
	t_3 = t_2 + 7.0;
	t_4 = t_3 + 0.5;
	t_5 = pi / sin((z * pi));
	tmp = 0.0;
	if (((pi / sin((pi * z))) * (((sqrt((pi * 2.0)) * (t_4 ^ (t_2 + 0.5))) * exp(-t_4)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_2 + 1.0))) + (-1259.1392167224028 / (t_2 + 2.0))) + (771.3234287776531 / (t_2 + 3.0))) + (-176.6150291621406 / (t_2 + 4.0))) + (12.507343278686905 / (t_2 + 5.0))) + (-0.13857109526572012 / (t_2 + 6.0))) + (9.984369578019572e-6 / t_3)) + (1.5056327351493116e-7 / (t_2 + 8.0))))) <= 2e+200)
		tmp = (t_5 * (exp(-t_0) * ((t_0 ^ (-z - -0.5)) * t_1))) * ((1.5056327351493116e-7 / (-z - -8.0)) + ((9.984369578019572e-6 / (-z - -7.0)) + ((-0.13857109526572012 / (-z - -6.0)) + ((12.507343278686905 / (-z - -5.0)) + ((0.9999999999998099 + ((676.5203681218851 / (1.0 - z)) - (1259.1392167224028 / ((1.0 - z) - -1.0)))) + ((-771.3234287776531 / (-2.0 - (1.0 - z))) - (176.6150291621406 / ((1.0 - z) - -3.0))))))));
	else
		tmp = (t_5 * ((((1.0 - z) - -6.5) ^ ((1.0 - z) - 0.5)) * t_1)) * 0.1456731240789439;
	end
	tmp_2 = tmp;
end

Wolfram

code[z_] := Block[{t$95$0 = N[((-z) + 7.5), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(Pi + Pi), $MachinePrecision]], $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[(t$95$3 + 0.5), $MachinePrecision]}, Block[{t$95$5 = N[(Pi / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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$4, N[(t$95$2 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$4)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(t$95$2 + 2.0), $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] + N[(-0.13857109526572012 / N[(t$95$2 + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$3), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(t$95$2 + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+200], N[(N[(t$95$5 * N[(N[Exp[(-t$95$0)], $MachinePrecision] * N[(N[Power[t$95$0, N[((-z) - -0.5), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.5056327351493116e-7 / N[((-z) - -8.0), $MachinePrecision]), $MachinePrecision] + N[(N[(9.984369578019572e-6 / N[((-z) - -7.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[((-z) - -6.0), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[((-z) - -5.0), $MachinePrecision]), $MachinePrecision] + N[(N[(0.9999999999998099 + N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[(1259.1392167224028 / N[(N[(1.0 - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-771.3234287776531 / N[(-2.0 - N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$5 * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] - -6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * 0.1456731240789439), $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)))))) < 1.9999999999999999e200

   1. Initial program 96.4%

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

   3. Applied rewrites 98.3%

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

      1. lift--.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\color{blue}{\left(1 - z\right) - -2}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\color{blue}{\left(1 - z\right)} - -2} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      3. associate--l- N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\color{blue}{1 - \left(z + -2\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      4. flip3-- N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\color{blue}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{1 \cdot 1 + \left(\left(z + -2\right) \cdot \left(z + -2\right) + 1 \cdot \left(z + -2\right)\right)}}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      5. lower-unsound-/.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\color{blue}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{1 \cdot 1 + \left(\left(z + -2\right) \cdot \left(z + -2\right) + 1 \cdot \left(z + -2\right)\right)}}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      6. lower-unsound--.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{\color{blue}{{1}^{3} - {\left(z + -2\right)}^{3}}}{1 \cdot 1 + \left(\left(z + -2\right) \cdot \left(z + -2\right) + 1 \cdot \left(z + -2\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      7. lower-unsound-pow.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{\color{blue}{{1}^{3}} - {\left(z + -2\right)}^{3}}{1 \cdot 1 + \left(\left(z + -2\right) \cdot \left(z + -2\right) + 1 \cdot \left(z + -2\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      8. lower-unsound-pow.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - \color{blue}{{\left(z + -2\right)}^{3}}}{1 \cdot 1 + \left(\left(z + -2\right) \cdot \left(z + -2\right) + 1 \cdot \left(z + -2\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\color{blue}{\left(z + -2\right)}}^{3}}{1 \cdot 1 + \left(\left(z + -2\right) \cdot \left(z + -2\right) + 1 \cdot \left(z + -2\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      10. lower-unsound-fma.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\color{blue}{\mathsf{fma}\left(1, 1, \left(z + -2\right) \cdot \left(z + -2\right) + 1 \cdot \left(z + -2\right)\right)}}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      11. lower-unsound-fma.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \color{blue}{\mathsf{fma}\left(z + -2, z + -2, 1 \cdot \left(z + -2\right)\right)}\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(\color{blue}{z + -2}, z + -2, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, \color{blue}{z + -2}, 1 \cdot \left(z + -2\right)\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      14. lower-unsound-*.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} - \frac{\frac{3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\frac{{1}^{3} - {\left(z + -2\right)}^{3}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(z + -2, z + -2, \color{blue}{1 \cdot \left(z + -2\right)}\right)\right)}} - \frac{\frac{883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      15. lower-+.f64 98.3%

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in z around 0

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

      1. lower-*.f64 98.3%

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in z around 0

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

       1. lower-*.f64 98.3%

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

   11. Applied rewrites 98.3%

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

   12. Taylor expanded in z around 0

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

       1. lower-*.f64 98.3%

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

   14. Applied rewrites 98.3%

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

   15. Taylor expanded in z around 0

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

       1. lower-*.f64 98.3%

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

   17. Applied rewrites 98.3%

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

   18. Taylor expanded in z around 0

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

       1. lower-*.f64 98.3%

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

   20. Applied rewrites 98.3%

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

   21. Taylor expanded in z around 0

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

       1. lower-*.f64 98.3%

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

   23. Applied rewrites 98.3%

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

   24. Taylor expanded in z around 0

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

       1. lower-*.f64 98.3%

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

   26. Applied rewrites 98.3%

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

   28. Applied rewrites 97.7%

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

   if 1.9999999999999999e200 < (*.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 96.4%

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

   3. Applied rewrites 96.4%

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

   5. Applied rewrites 96.4%

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

   6. Taylor expanded in z around 0

      \[\leadsto \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{\pi + \pi}\right)\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot e^{\frac{-15}{2}}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{\pi + \pi}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \color{blue}{e^{\frac{-15}{2}}}\right) \]

      2. lower-exp.f64 96.0%

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

   8. Applied rewrites 96.0%

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

   9. Evaluated real constant 96.0%

      \[\leadsto \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{\pi + \pi}\right)\right) \cdot 0.1456731240789439 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := -\log \left(-z\right)\\ t_1 := \left(1 - z\right) - 1\\ t_2 := \left(t\_1 + 7\right) + 0.5\\ \mathbf{if}\;z \leq -0.46:\\ \;\;\;\;\frac{\pi \cdot \left(\left(e^{\left(-z\right) \cdot \left(\left(-t\_0\right) + \left(-\frac{\mathsf{fma}\left(-0.5, t\_0, 7.5\right)}{z}\right)\right) + \left(z - 7.5\right)} \cdot \sqrt{\pi + \pi}\right) \cdot 0.9999999999998099\right)}{\sin \left(z \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(2.5066282746310007 \cdot {t\_2}^{\left(t\_1 + 0.5\right)}\right) \cdot e^{-t\_2}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (z)
  :precision binary64
  (let* ((t_0 (- (log (- z))))
       (t_1 (- (- 1.0 z) 1.0))
       (t_2 (+ (+ t_1 7.0) 0.5)))
  (if (<= z -0.46)
    (/
     (*
      PI
      (*
       (*
        (exp
         (+
          (* (- z) (+ (- t_0) (- (/ (fma -0.5 t_0 7.5) z))))
          (- z 7.5)))
        (sqrt (+ PI PI)))
       0.9999999999998099))
     (sin (* z PI)))
    (*
     (/ PI (sin (* PI z)))
     (*
      (* (* 2.5066282746310007 (pow t_2 (+ t_1 0.5))) (exp (- t_2)))
      (+
       263.3831869810514
       (*
        z
        (+
         436.8961725563396
         (* z (+ 545.0353078428827 (* 606.6766809167608 z)))))))))))

C

double code(double z) {
	double t_0 = -log(-z);
	double t_1 = (1.0 - z) - 1.0;
	double t_2 = (t_1 + 7.0) + 0.5;
	double tmp;
	if (z <= -0.46) {
		tmp = (((double) M_PI) * ((exp(((-z * (-t_0 + -(fma(-0.5, t_0, 7.5) / z))) + (z - 7.5))) * sqrt((((double) M_PI) + ((double) M_PI)))) * 0.9999999999998099)) / sin((z * ((double) M_PI)));
	} else {
		tmp = (((double) M_PI) / sin((((double) M_PI) * z))) * (((2.5066282746310007 * pow(t_2, (t_1 + 0.5))) * exp(-t_2)) * (263.3831869810514 + (z * (436.8961725563396 + (z * (545.0353078428827 + (606.6766809167608 * z)))))));
	}
	return tmp;
}

Julia

function code(z)
	t_0 = Float64(-log(Float64(-z)))
	t_1 = Float64(Float64(1.0 - z) - 1.0)
	t_2 = Float64(Float64(t_1 + 7.0) + 0.5)
	tmp = 0.0
	if (z <= -0.46)
		tmp = Float64(Float64(pi * Float64(Float64(exp(Float64(Float64(Float64(-z) * Float64(Float64(-t_0) + Float64(-Float64(fma(-0.5, t_0, 7.5) / z)))) + Float64(z - 7.5))) * sqrt(Float64(pi + pi))) * 0.9999999999998099)) / sin(Float64(z * pi)));
	else
		tmp = Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(2.5066282746310007 * (t_2 ^ Float64(t_1 + 0.5))) * exp(Float64(-t_2))) * Float64(263.3831869810514 + Float64(z * Float64(436.8961725563396 + Float64(z * Float64(545.0353078428827 + Float64(606.6766809167608 * z))))))));
	end
	return tmp
end

Wolfram

code[z_] := Block[{t$95$0 = (-N[Log[(-z)], $MachinePrecision])}, Block[{t$95$1 = N[(N[(1.0 - z), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 + 7.0), $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[z, -0.46], N[(N[(Pi * N[(N[(N[Exp[N[(N[((-z) * N[((-t$95$0) + (-N[(N[(-0.5 * t$95$0 + 7.5), $MachinePrecision] / z), $MachinePrecision])), $MachinePrecision]), $MachinePrecision] + N[(z - 7.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi + Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.9999999999998099), $MachinePrecision]), $MachinePrecision] / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(2.5066282746310007 * N[Power[t$95$2, N[(t$95$1 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$2)], $MachinePrecision]), $MachinePrecision] * N[(263.3831869810514 + N[(z * N[(436.8961725563396 + N[(z * N[(545.0353078428827 + N[(606.6766809167608 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.46000000000000002

   1. Initial program 96.4%

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

   2. Taylor expanded in z around inf

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(\frac{9999999999998099}{10000000000000000} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2 \cdot \pi}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\frac{9999999999998099}{10000000000000000} \cdot \color{blue}{\left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\frac{9999999999998099}{10000000000000000} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \color{blue}{\left(e^{z - \frac{15}{2}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)}\right)\right) \]

   4. Applied rewrites 14.0%

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

   5. Taylor expanded in z around -inf

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(0.9999999999998099 \cdot \left(e^{-1 \cdot \left(z \cdot \left(-1 \cdot \log \left(\frac{-1}{z}\right) + -1 \cdot \frac{\frac{15}{2} + \frac{-1}{2} \cdot \log \left(\frac{-1}{z}\right)}{z}\right)\right)} \cdot \left(e^{\color{blue}{z - 7.5}} \cdot \sqrt{2 \cdot \pi}\right)\right)\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\frac{9999999999998099}{10000000000000000} \cdot \left(e^{-1 \cdot \left(z \cdot \left(-1 \cdot \log \left(\frac{-1}{z}\right) + -1 \cdot \frac{\frac{15}{2} + \frac{-1}{2} \cdot \log \left(\frac{-1}{z}\right)}{z}\right)\right)} \cdot \left(e^{z - \color{blue}{\frac{15}{2}}} \cdot \sqrt{2 \cdot \pi}\right)\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\frac{9999999999998099}{10000000000000000} \cdot \left(e^{-1 \cdot \left(z \cdot \left(-1 \cdot \log \left(\frac{-1}{z}\right) + -1 \cdot \frac{\frac{15}{2} + \frac{-1}{2} \cdot \log \left(\frac{-1}{z}\right)}{z}\right)\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2 \cdot \pi}\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\frac{9999999999998099}{10000000000000000} \cdot \left(e^{-1 \cdot \left(z \cdot \mathsf{fma}\left(-1, \log \left(\frac{-1}{z}\right), -1 \cdot \frac{\frac{15}{2} + \frac{-1}{2} \cdot \log \left(\frac{-1}{z}\right)}{z}\right)\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2 \cdot \pi}\right)\right)\right) \]

      4. lower-log.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\frac{9999999999998099}{10000000000000000} \cdot \left(e^{-1 \cdot \left(z \cdot \mathsf{fma}\left(-1, \log \left(\frac{-1}{z}\right), -1 \cdot \frac{\frac{15}{2} + \frac{-1}{2} \cdot \log \left(\frac{-1}{z}\right)}{z}\right)\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2 \cdot \pi}\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\frac{9999999999998099}{10000000000000000} \cdot \left(e^{-1 \cdot \left(z \cdot \mathsf{fma}\left(-1, \log \left(\frac{-1}{z}\right), -1 \cdot \frac{\frac{15}{2} + \frac{-1}{2} \cdot \log \left(\frac{-1}{z}\right)}{z}\right)\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2 \cdot \pi}\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\frac{9999999999998099}{10000000000000000} \cdot \left(e^{-1 \cdot \left(z \cdot \mathsf{fma}\left(-1, \log \left(\frac{-1}{z}\right), -1 \cdot \frac{\frac{15}{2} + \frac{-1}{2} \cdot \log \left(\frac{-1}{z}\right)}{z}\right)\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2 \cdot \pi}\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\frac{9999999999998099}{10000000000000000} \cdot \left(e^{-1 \cdot \left(z \cdot \mathsf{fma}\left(-1, \log \left(\frac{-1}{z}\right), -1 \cdot \frac{\frac{15}{2} + \frac{-1}{2} \cdot \log \left(\frac{-1}{z}\right)}{z}\right)\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2 \cdot \pi}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\frac{9999999999998099}{10000000000000000} \cdot \left(e^{-1 \cdot \left(z \cdot \mathsf{fma}\left(-1, \log \left(\frac{-1}{z}\right), -1 \cdot \frac{\frac{15}{2} + \frac{-1}{2} \cdot \log \left(\frac{-1}{z}\right)}{z}\right)\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2 \cdot \pi}\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\frac{9999999999998099}{10000000000000000} \cdot \left(e^{-1 \cdot \left(z \cdot \mathsf{fma}\left(-1, \log \left(\frac{-1}{z}\right), -1 \cdot \frac{\frac{15}{2} + \frac{-1}{2} \cdot \log \left(\frac{-1}{z}\right)}{z}\right)\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2 \cdot \pi}\right)\right)\right) \]

      10. lower-log.f64 N/A

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\frac{9999999999998099}{10000000000000000} \cdot \left(e^{-1 \cdot \left(z \cdot \mathsf{fma}\left(-1, \log \left(\frac{-1}{z}\right), -1 \cdot \frac{\frac{15}{2} + \frac{-1}{2} \cdot \log \left(\frac{-1}{z}\right)}{z}\right)\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2 \cdot \pi}\right)\right)\right) \]

      11. lower-/.f64 3.7%

          \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(0.9999999999998099 \cdot \left(e^{-1 \cdot \left(z \cdot \mathsf{fma}\left(-1, \log \left(\frac{-1}{z}\right), -1 \cdot \frac{7.5 + -0.5 \cdot \log \left(\frac{-1}{z}\right)}{z}\right)\right)} \cdot \left(e^{z - 7.5} \cdot \sqrt{2 \cdot \pi}\right)\right)\right) \]

   7. Applied rewrites 3.7%

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

   9. Applied rewrites 4.6%

      \[\leadsto \color{blue}{\frac{\pi \cdot \left(\left(e^{\left(-z\right) \cdot \left(\left(-\left(-\log \left(-z\right)\right)\right) + \left(-\frac{\mathsf{fma}\left(-0.5, -\log \left(-z\right), 7.5\right)}{z}\right)\right) + \left(z - 7.5\right)} \cdot \sqrt{\pi + \pi}\right) \cdot 0.9999999999998099\right)}{\sin \left(z \cdot \pi\right)}} \]

   if -0.46000000000000002 < z

   1. Initial program 96.4%

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

   2. Taylor expanded in z around 0

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)}\right) \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + \color{blue}{z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \color{blue}{\left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + \color{blue}{z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)}\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \color{blue}{\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \color{blue}{\frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z}\right)\right)\right)\right) \]

      6. lower-*.f64 96.5%

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot \color{blue}{z}\right)\right)\right)\right) \]

   4. Applied rewrites 96.5%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)}\right) \]

   5. Evaluated real constant 97.3%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{2.5066282746310007} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 96.0% accurate, 2.8× speedup

Math

\[\left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{\pi + \pi}\right)\right) \cdot 0.1456731240789439 \]

FPCore

(FPCore (z)
  :precision binary64
  (*
 (*
  (/ PI (sin (* z PI)))
  (* (pow (- (- 1.0 z) -6.5) (- (- 1.0 z) 0.5)) (sqrt (+ PI PI))))
 0.1456731240789439))

C

double code(double z) {
	return ((((double) M_PI) / sin((z * ((double) M_PI)))) * (pow(((1.0 - z) - -6.5), ((1.0 - z) - 0.5)) * sqrt((((double) M_PI) + ((double) M_PI))))) * 0.1456731240789439;
}

Java

public static double code(double z) {
	return ((Math.PI / Math.sin((z * Math.PI))) * (Math.pow(((1.0 - z) - -6.5), ((1.0 - z) - 0.5)) * Math.sqrt((Math.PI + Math.PI)))) * 0.1456731240789439;
}

Python

def code(z):
	return ((math.pi / math.sin((z * math.pi))) * (math.pow(((1.0 - z) - -6.5), ((1.0 - z) - 0.5)) * math.sqrt((math.pi + math.pi)))) * 0.1456731240789439

Julia

function code(z)
	return Float64(Float64(Float64(pi / sin(Float64(z * pi))) * Float64((Float64(Float64(1.0 - z) - -6.5) ^ Float64(Float64(1.0 - z) - 0.5)) * sqrt(Float64(pi + pi)))) * 0.1456731240789439)
end

MATLAB

function tmp = code(z)
	tmp = ((pi / sin((z * pi))) * ((((1.0 - z) - -6.5) ^ ((1.0 - z) - 0.5)) * sqrt((pi + pi)))) * 0.1456731240789439;
end

Wolfram

code[z_] := N[(N[(N[(Pi / N[Sin[N[(z * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(N[(1.0 - z), $MachinePrecision] - -6.5), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi + Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.1456731240789439), $MachinePrecision]

Derivation

1. Initial program 96.4%

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

3. Applied rewrites 96.4%

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

5. Applied rewrites 96.4%

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

6. Taylor expanded in z around 0

   \[\leadsto \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{\pi + \pi}\right)\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot e^{\frac{-15}{2}}\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{\pi + \pi}\right)\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \color{blue}{e^{\frac{-15}{2}}}\right) \]

   2. lower-exp.f64 96.0%

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

8. Applied rewrites 96.0%

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

9. Evaluated real constant 96.0%

   \[\leadsto \left(\frac{\pi}{\sin \left(z \cdot \pi\right)} \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{\pi + \pi}\right)\right) \cdot 0.1456731240789439 \]
10. Add Preprocessing

Alternative 7: 96.0% accurate, 10.5× speedup

Math

\[263.3831869810514 \cdot \frac{e^{-7.5} \cdot 6.864684246478268}{z} \]

FPCore

(FPCore (z)
  :precision binary64
  (* 263.3831869810514 (/ (* (exp -7.5) 6.864684246478268) z)))

C

double code(double z) {
	return 263.3831869810514 * ((exp(-7.5) * 6.864684246478268) / z);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(z)
use fmin_fmax_functions
    real(8), intent (in) :: z
    code = 263.3831869810514d0 * ((exp((-7.5d0)) * 6.864684246478268d0) / z)
end function

Java

public static double code(double z) {
	return 263.3831869810514 * ((Math.exp(-7.5) * 6.864684246478268) / z);
}

Python

def code(z):
	return 263.3831869810514 * ((math.exp(-7.5) * 6.864684246478268) / z)

Julia

function code(z)
	return Float64(263.3831869810514 * Float64(Float64(exp(-7.5) * 6.864684246478268) / z))
end

MATLAB

function tmp = code(z)
	tmp = 263.3831869810514 * ((exp(-7.5) * 6.864684246478268) / z);
end

Wolfram

code[z_] := N[(263.3831869810514 * N[(N[(N[Exp[-7.5], $MachinePrecision] * 6.864684246478268), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.4%

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

2. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \pi} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)}{z}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \color{blue}{\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)}{z}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)}{\color{blue}{z}} \]

4. Applied rewrites 95.3%

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

5. Evaluated real constant 96.0%

   \[\leadsto 263.3831869810514 \cdot \frac{e^{-7.5} \cdot 6.864684246478268}{z} \]
6. Add Preprocessing

Alternative 8: 96.0% accurate, 26.7× speedup

Math

\[\frac{1}{\frac{z}{1.0000000000000002}} \]

FPCore

(FPCore (z)
  :precision binary64
  (/ 1.0 (/ z 1.0000000000000002)))

C

double code(double z) {
	return 1.0 / (z / 1.0000000000000002);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(z)
use fmin_fmax_functions
    real(8), intent (in) :: z
    code = 1.0d0 / (z / 1.0000000000000002d0)
end function

Java

public static double code(double z) {
	return 1.0 / (z / 1.0000000000000002);
}

Python

def code(z):
	return 1.0 / (z / 1.0000000000000002)

Julia

function code(z)
	return Float64(1.0 / Float64(z / 1.0000000000000002))
end

MATLAB

function tmp = code(z)
	tmp = 1.0 / (z / 1.0000000000000002);
end

Wolfram

code[z_] := N[(1.0 / N[(z / 1.0000000000000002), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.4%

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

2. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \pi} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)}{z}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \color{blue}{\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)}{z}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)}{\color{blue}{z}} \]

4. Applied rewrites 95.3%

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

5. Evaluated real constant 95.3%

   \[\leadsto 263.3831869810514 \cdot \frac{0.003796749562727188}{z} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \color{blue}{\frac{\frac{2188677109237409}{576460752303423488}}{z}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{\frac{2188677109237409}{576460752303423488}}{\color{blue}{z}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{2188677109237409}{576460752303423488}}{\color{blue}{z}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{2188677109237409}{576460752303423488}}{\color{blue}{z}} \]

   5. metadata-eval 95.9%

      \[\leadsto \frac{1.0000000000000002}{z} \]

7. Applied rewrites 95.9%

   \[\leadsto \frac{1.0000000000000002}{\color{blue}{z}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\frac{345876451382054143332511925913682434220733}{345876451382054092800000000000000000000000}}{\color{blue}{z}} \]

   2. div-flip N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{z}{\frac{345876451382054143332511925913682434220733}{345876451382054092800000000000000000000000}}}} \]

   3. lower-unsound-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{z}{\frac{345876451382054143332511925913682434220733}{345876451382054092800000000000000000000000}}}} \]

   4. lower-unsound-/.f64 96.0%

      \[\leadsto \frac{1}{\frac{z}{\color{blue}{1.0000000000000002}}} \]

9. Applied rewrites 96.0%

   \[\leadsto \frac{1}{\color{blue}{\frac{z}{1.0000000000000002}}} \]
10. Add Preprocessing

Alternative 9: 95.9% accurate, 29.4× speedup

Math

\[1.0000000000000002 \cdot \frac{1}{z} \]

FPCore

(FPCore (z)
  :precision binary64
  (* 1.0000000000000002 (/ 1.0 z)))

C

double code(double z) {
	return 1.0000000000000002 * (1.0 / z);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(z)
use fmin_fmax_functions
    real(8), intent (in) :: z
    code = 1.0000000000000002d0 * (1.0d0 / z)
end function

Java

public static double code(double z) {
	return 1.0000000000000002 * (1.0 / z);
}

Python

def code(z):
	return 1.0000000000000002 * (1.0 / z)

Julia

function code(z)
	return Float64(1.0000000000000002 * Float64(1.0 / z))
end

MATLAB

function tmp = code(z)
	tmp = 1.0000000000000002 * (1.0 / z);
end

Wolfram

code[z_] := N[(1.0000000000000002 * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.4%

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

2. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \pi} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)}{z}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \color{blue}{\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)}{z}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)}{\color{blue}{z}} \]

4. Applied rewrites 95.3%

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

5. Evaluated real constant 95.3%

   \[\leadsto 263.3831869810514 \cdot \frac{0.003796749562727188}{z} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \color{blue}{\frac{\frac{2188677109237409}{576460752303423488}}{z}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{\frac{2188677109237409}{576460752303423488}}{\color{blue}{z}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{2188677109237409}{576460752303423488}}{\color{blue}{z}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{2188677109237409}{576460752303423488}}{\color{blue}{z}} \]

   5. metadata-eval 95.9%

      \[\leadsto \frac{1.0000000000000002}{z} \]

7. Applied rewrites 95.9%

   \[\leadsto \frac{1.0000000000000002}{\color{blue}{z}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\frac{345876451382054143332511925913682434220733}{345876451382054092800000000000000000000000}}{\color{blue}{z}} \]

   2. mult-flip N/A

      \[\leadsto \frac{345876451382054143332511925913682434220733}{345876451382054092800000000000000000000000} \cdot \color{blue}{\frac{1}{z}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{345876451382054143332511925913682434220733}{345876451382054092800000000000000000000000} \cdot \color{blue}{\frac{1}{z}} \]

   4. lower-/.f64 95.9%

      \[\leadsto 1.0000000000000002 \cdot \frac{1}{\color{blue}{z}} \]

9. Applied rewrites 95.9%

   \[\leadsto 1.0000000000000002 \cdot \color{blue}{\frac{1}{z}} \]
10. Add Preprocessing

Alternative 10: 95.9% accurate, 47.8× speedup

Math

\[\frac{1.0000000000000002}{z} \]

FPCore

(FPCore (z)
  :precision binary64
  (/ 1.0000000000000002 z))

C

double code(double z) {
	return 1.0000000000000002 / z;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(z)
use fmin_fmax_functions
    real(8), intent (in) :: z
    code = 1.0000000000000002d0 / z
end function

Java

public static double code(double z) {
	return 1.0000000000000002 / z;
}

Python

def code(z):
	return 1.0000000000000002 / z

Julia

function code(z)
	return Float64(1.0000000000000002 / z)
end

MATLAB

function tmp = code(z)
	tmp = 1.0000000000000002 / z;
end

Wolfram

code[z_] := N[(1.0000000000000002 / z), $MachinePrecision]

Derivation

1. Initial program 96.4%

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

2. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \pi} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)}{z}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \color{blue}{\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)}{z}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot {\frac{15}{2}}^{\frac{1}{2}}\right)}{\color{blue}{z}} \]

4. Applied rewrites 95.3%

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

5. Evaluated real constant 95.3%

   \[\leadsto 263.3831869810514 \cdot \frac{0.003796749562727188}{z} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \color{blue}{\frac{\frac{2188677109237409}{576460752303423488}}{z}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{\frac{2188677109237409}{576460752303423488}}{\color{blue}{z}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{2188677109237409}{576460752303423488}}{\color{blue}{z}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{2188677109237409}{576460752303423488}}{\color{blue}{z}} \]

   5. metadata-eval 95.9%

      \[\leadsto \frac{1.0000000000000002}{z} \]

7. Applied rewrites 95.9%

   \[\leadsto \frac{1.0000000000000002}{\color{blue}{z}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025210 
(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))))))