Jmat.Real.gamma, branch z less than 0.5

Average Error: 1.7 → 0.5

Time: 2.1min

Precision: binary64

\[z \leq 0.5\]

Math

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

↓

\[\begin{array}{l} t_0 := \left(1 - z\right) + 2\\ t_1 := \frac{771.3234287776531}{t_0}\\ t_2 := \left(1 - z\right) + 4\\ t_3 := 7.5 + \left(\left(1 - z\right) + -1\right)\\ t_4 := 0.9999999999998099 + \frac{676.5203681218851}{1 - z}\\ t_5 := t_4 + \frac{-1259.1392167224028}{1 + \left(1 - z\right)}\\ t_6 := t_5 - t_1\\ t_7 := \left(1 - z\right) + 5\\ t_8 := \left(1 - z\right) + 3\\ t_9 := t_8 \cdot t_6\\ t_10 := \left(1 - z\right) + 7\\ t_11 := \left(1 - z\right) + 6\\ t_12 := t_2 \cdot t_9\\ t_13 := t_7 \cdot t_12\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \frac{\mathsf{fma}\left(1.5056327351493116 \cdot 10^{-7}, t_11 \cdot \left(t_7 \cdot \left(t_2 \cdot \left(t_8 \cdot \left(\left(t_4 + \frac{-1259.1392167224028}{2 - z}\right) - \frac{771.3234287776531}{3 - z}\right)\right)\right)\right), t_10 \cdot \mathsf{fma}\left(t_11, \mathsf{fma}\left(-0.13857109526572012, t_12, t_7 \cdot \mathsf{fma}\left(12.507343278686905, t_9, t_2 \cdot \mathsf{fma}\left(-176.6150291621406, t_6, t_8 \cdot \mathsf{fma}\left(t_5, t_5, t_1 \cdot \frac{-771.3234287776531}{t_0}\right)\right)\right)\right), 9.984369578019572 \cdot 10^{-6} \cdot t_13\right)\right) \cdot \left(e^{z + -7.5} \cdot \left(\sqrt{\pi \cdot 2} \cdot {t_3}^{\left(1 - z\right)}\right)\right)}{\sqrt{t_3} \cdot \left(t_10 \cdot \left(t_11 \cdot t_13\right)\right)} \end{array} \]

FPCore

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

↓

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ (- 1.0 z) 2.0))
        (t_1 (/ 771.3234287776531 t_0))
        (t_2 (+ (- 1.0 z) 4.0))
        (t_3 (+ 7.5 (+ (- 1.0 z) -1.0)))
        (t_4 (+ 0.9999999999998099 (/ 676.5203681218851 (- 1.0 z))))
        (t_5 (+ t_4 (/ -1259.1392167224028 (+ 1.0 (- 1.0 z)))))
        (t_6 (- t_5 t_1))
        (t_7 (+ (- 1.0 z) 5.0))
        (t_8 (+ (- 1.0 z) 3.0))
        (t_9 (* t_8 t_6))
        (t_10 (+ (- 1.0 z) 7.0))
        (t_11 (+ (- 1.0 z) 6.0))
        (t_12 (* t_2 t_9))
        (t_13 (* t_7 t_12)))
   (*
    (/ PI (sin (* PI z)))
    (/
     (*
      (fma
       1.5056327351493116e-7
       (*
        t_11
        (*
         t_7
         (*
          t_2
          (*
           t_8
           (-
            (+ t_4 (/ -1259.1392167224028 (- 2.0 z)))
            (/ 771.3234287776531 (- 3.0 z)))))))
       (*
        t_10
        (fma
         t_11
         (fma
          -0.13857109526572012
          t_12
          (*
           t_7
           (fma
            12.507343278686905
            t_9
            (*
             t_2
             (fma
              -176.6150291621406
              t_6
              (* t_8 (fma t_5 t_5 (* t_1 (/ -771.3234287776531 t_0)))))))))
         (* 9.984369578019572e-6 t_13))))
      (* (exp (+ z -7.5)) (* (sqrt (* PI 2.0)) (pow t_3 (- 1.0 z)))))
     (* (sqrt t_3) (* t_10 (* t_11 t_13)))))))

C

double code(double z) {
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (((sqrt((((double) M_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))));
}

↓

double code(double z) {
	double t_0 = (1.0 - z) + 2.0;
	double t_1 = 771.3234287776531 / t_0;
	double t_2 = (1.0 - z) + 4.0;
	double t_3 = 7.5 + ((1.0 - z) + -1.0);
	double t_4 = 0.9999999999998099 + (676.5203681218851 / (1.0 - z));
	double t_5 = t_4 + (-1259.1392167224028 / (1.0 + (1.0 - z)));
	double t_6 = t_5 - t_1;
	double t_7 = (1.0 - z) + 5.0;
	double t_8 = (1.0 - z) + 3.0;
	double t_9 = t_8 * t_6;
	double t_10 = (1.0 - z) + 7.0;
	double t_11 = (1.0 - z) + 6.0;
	double t_12 = t_2 * t_9;
	double t_13 = t_7 * t_12;
	return (((double) M_PI) / sin((((double) M_PI) * z))) * ((fma(1.5056327351493116e-7, (t_11 * (t_7 * (t_2 * (t_8 * ((t_4 + (-1259.1392167224028 / (2.0 - z))) - (771.3234287776531 / (3.0 - z))))))), (t_10 * fma(t_11, fma(-0.13857109526572012, t_12, (t_7 * fma(12.507343278686905, t_9, (t_2 * fma(-176.6150291621406, t_6, (t_8 * fma(t_5, t_5, (t_1 * (-771.3234287776531 / t_0))))))))), (9.984369578019572e-6 * t_13)))) * (exp((z + -7.5)) * (sqrt((((double) M_PI) * 2.0)) * pow(t_3, (1.0 - z))))) / (sqrt(t_3) * (t_10 * (t_11 * t_13))));
}

Julia

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

↓

function code(z)
	t_0 = Float64(Float64(1.0 - z) + 2.0)
	t_1 = Float64(771.3234287776531 / t_0)
	t_2 = Float64(Float64(1.0 - z) + 4.0)
	t_3 = Float64(7.5 + Float64(Float64(1.0 - z) + -1.0))
	t_4 = Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 - z)))
	t_5 = Float64(t_4 + Float64(-1259.1392167224028 / Float64(1.0 + Float64(1.0 - z))))
	t_6 = Float64(t_5 - t_1)
	t_7 = Float64(Float64(1.0 - z) + 5.0)
	t_8 = Float64(Float64(1.0 - z) + 3.0)
	t_9 = Float64(t_8 * t_6)
	t_10 = Float64(Float64(1.0 - z) + 7.0)
	t_11 = Float64(Float64(1.0 - z) + 6.0)
	t_12 = Float64(t_2 * t_9)
	t_13 = Float64(t_7 * t_12)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(fma(1.5056327351493116e-7, Float64(t_11 * Float64(t_7 * Float64(t_2 * Float64(t_8 * Float64(Float64(t_4 + Float64(-1259.1392167224028 / Float64(2.0 - z))) - Float64(771.3234287776531 / Float64(3.0 - z))))))), Float64(t_10 * fma(t_11, fma(-0.13857109526572012, t_12, Float64(t_7 * fma(12.507343278686905, t_9, Float64(t_2 * fma(-176.6150291621406, t_6, Float64(t_8 * fma(t_5, t_5, Float64(t_1 * Float64(-771.3234287776531 / t_0))))))))), Float64(9.984369578019572e-6 * t_13)))) * Float64(exp(Float64(z + -7.5)) * Float64(sqrt(Float64(pi * 2.0)) * (t_3 ^ Float64(1.0 - z))))) / Float64(sqrt(t_3) * Float64(t_10 * Float64(t_11 * t_13)))))
end

Wolfram

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

↓

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(771.3234287776531 / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 - z), $MachinePrecision] + 4.0), $MachinePrecision]}, Block[{t$95$3 = N[(7.5 + N[(N[(1.0 - z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 + N[(-1259.1392167224028 / N[(1.0 + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 - t$95$1), $MachinePrecision]}, Block[{t$95$7 = N[(N[(1.0 - z), $MachinePrecision] + 5.0), $MachinePrecision]}, Block[{t$95$8 = N[(N[(1.0 - z), $MachinePrecision] + 3.0), $MachinePrecision]}, Block[{t$95$9 = N[(t$95$8 * t$95$6), $MachinePrecision]}, Block[{t$95$10 = N[(N[(1.0 - z), $MachinePrecision] + 7.0), $MachinePrecision]}, Block[{t$95$11 = N[(N[(1.0 - z), $MachinePrecision] + 6.0), $MachinePrecision]}, Block[{t$95$12 = N[(t$95$2 * t$95$9), $MachinePrecision]}, Block[{t$95$13 = N[(t$95$7 * t$95$12), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.5056327351493116e-7 * N[(t$95$11 * N[(t$95$7 * N[(t$95$2 * N[(t$95$8 * N[(N[(t$95$4 + N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$10 * N[(t$95$11 * N[(-0.13857109526572012 * t$95$12 + N[(t$95$7 * N[(12.507343278686905 * t$95$9 + N[(t$95$2 * N[(-176.6150291621406 * t$95$6 + N[(t$95$8 * N[(t$95$5 * t$95$5 + N[(t$95$1 * N[(-771.3234287776531 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 * t$95$13), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$3, N[(1.0 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$3], $MachinePrecision] * N[(t$95$10 * N[(t$95$11 * t$95$13), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]

Error

Bits error versus z

Derivation

1. Initial program 1.7

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

2. Applied flip-+_binary64 1.7

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

3. Applied frac-add_binary64 1.7

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

4. Applied frac-add_binary64 1.7

   \[\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(\color{blue}{\frac{\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) \cdot \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} \cdot \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 4\right) + \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) \cdot -176.6150291621406\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 5\right) + \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) \cdot \left(\left(\left(1 - z\right) - 1\right) + 4\right)\right) \cdot 12.507343278686905}{\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) \cdot \left(\left(\left(1 - z\right) - 1\right) + 4\right)\right) \cdot \left(\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 frac-add_binary64 1.7

   \[\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(\color{blue}{\frac{\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) \cdot \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} \cdot \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 4\right) + \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) \cdot -176.6150291621406\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 5\right) + \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) \cdot \left(\left(\left(1 - z\right) - 1\right) + 4\right)\right) \cdot 12.507343278686905\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 6\right) + \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) \cdot \left(\left(\left(1 - z\right) - 1\right) + 4\right)\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 5\right)\right) \cdot -0.13857109526572012}{\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) \cdot \left(\left(\left(1 - z\right) - 1\right) + 4\right)\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 5\right)\right) \cdot \left(\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. Applied frac-add_binary64 1.7

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\frac{\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) \cdot \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} \cdot \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 4\right) + \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) \cdot -176.6150291621406\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 5\right) + \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) \cdot \left(\left(\left(1 - z\right) - 1\right) + 4\right)\right) \cdot 12.507343278686905\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 6\right) + \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) \cdot \left(\left(\left(1 - z\right) - 1\right) + 4\right)\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 5\right)\right) \cdot -0.13857109526572012\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 7\right) + \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) \cdot \left(\left(\left(1 - z\right) - 1\right) + 4\right)\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 5\right)\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 6\right)\right) \cdot 9.984369578019572 \cdot 10^{-6}}{\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) \cdot \left(\left(\left(1 - z\right) - 1\right) + 4\right)\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 5\right)\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 6\right)\right) \cdot \left(\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. Applied frac-add_binary64 1.7

   \[\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}{\frac{\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) \cdot \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} \cdot \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 4\right) + \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) \cdot -176.6150291621406\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 5\right) + \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) \cdot \left(\left(\left(1 - z\right) - 1\right) + 4\right)\right) \cdot 12.507343278686905\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 6\right) + \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) \cdot \left(\left(\left(1 - z\right) - 1\right) + 4\right)\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 5\right)\right) \cdot -0.13857109526572012\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 7\right) + \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) \cdot \left(\left(\left(1 - z\right) - 1\right) + 4\right)\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 5\right)\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 6\right)\right) \cdot 9.984369578019572 \cdot 10^{-6}\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 8\right) + \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) \cdot \left(\left(\left(1 - z\right) - 1\right) + 4\right)\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 5\right)\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 6\right)\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 7\right)\right) \cdot 1.5056327351493116 \cdot 10^{-7}}{\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) \cdot \left(\left(\left(1 - z\right) - 1\right) + 4\right)\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 5\right)\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 6\right)\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 7\right)\right) \cdot \left(\left(\left(1 - z\right) - 1\right) + 8\right)}}\right) \]

8. Applied associate-+l-_binary64 1.7

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

9. Applied pow-sub_binary64 1.7

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

10. Applied associate-*r/_binary64 1.7

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

11. Applied associate-*l/_binary64 1.6

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

12. Applied frac-times_binary64 1.0

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

13. Simplified 0.5

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

14. Simplified 0.5

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

15. Taylor expanded in z around 0 0.5

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

16. Simplified 0.5

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

17. Applied add-cube-cbrt_binary64 0.5

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

18. Applied add-cube-cbrt_binary64 0.5

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

19. Applied prod-diff_binary64 0.5

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

20. Simplified 0.5

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

21. Simplified 0.5

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

22. Final simplification 0.5

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

Reproduce

herbie shell --seed 2022137 
(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))))))