Jmat.Real.gamma, branch z less than 0.5

Average Error: 1.7 → 0.5

Time: 1.2min

Precision: binary64

\[z \leq 0.5\]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{676.5203681218851}{1 - z}\\ t_1 := \sqrt{e^{z + -7.5}}\\ t_2 := \mathsf{fma}\left(t_0, t_0 + 0.9999999999998099, 0.9999999999996197\right)\\ t_3 := \left(0.9999999999998099 - t_0\right) \cdot \left(2 - z\right)\\ t_4 := \left(3 - z\right) \cdot t_3\\ t_5 := \left(4 - z\right) \cdot t_4\\ t_6 := \left(5 - z\right) \cdot t_5\\ t_7 := \left(6 - z\right) \cdot t_6\\ t_8 := \left(7 - z\right) \cdot t_7\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \frac{\left(t_1 \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot \sqrt{7.5 - z}\right) \cdot t_1\right)\right) \cdot \left(\left(\left(\left(\left(\left(\frac{\mathsf{fma}\left(0.9999999999996197 - \frac{\frac{457679.80848377093}{1 - z}}{1 - z}, \left(4 - z \cdot z\right) \cdot t_2, \left(-1259.1392167224028 \cdot \left(0.9999999999994297 - {t_0}^{3}\right)\right) \cdot \left(z + 2\right)\right)}{t_2 \cdot \left(z + 2\right)} \cdot \left(3 - z\right) + t_3 \cdot 771.3234287776531\right) \cdot \left(4 - z\right) + t_4 \cdot -176.6150291621406\right) \cdot \left(5 - z\right) + t_5 \cdot 12.507343278686905\right) \cdot \left(6 - z\right) + t_6 \cdot -0.13857109526572012\right) \cdot \left(7 - z\right) + t_7 \cdot 9.984369578019572 \cdot 10^{-6}\right) \cdot \left(8 - z\right) + t_8 \cdot 1.5056327351493116 \cdot 10^{-7}\right)}{{\left(7.5 - z\right)}^{z} \cdot \left(\left(8 - z\right) \cdot t_8\right)} \end{array} \]

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 (/ 676.5203681218851 (- 1.0 z)))
        (t_1 (sqrt (exp (+ z -7.5))))
        (t_2 (fma t_0 (+ t_0 0.9999999999998099) 0.9999999999996197))
        (t_3 (* (- 0.9999999999998099 t_0) (- 2.0 z)))
        (t_4 (* (- 3.0 z) t_3))
        (t_5 (* (- 4.0 z) t_4))
        (t_6 (* (- 5.0 z) t_5))
        (t_7 (* (- 6.0 z) t_6))
        (t_8 (* (- 7.0 z) t_7)))
   (*
    (/ PI (sin (* PI z)))
    (/
     (*
      (* t_1 (* (* (sqrt (* PI 2.0)) (sqrt (- 7.5 z))) t_1))
      (+
       (*
        (+
         (*
          (+
           (*
            (+
             (*
              (+
               (*
                (+
                 (*
                  (/
                   (fma
                    (-
                     0.9999999999996197
                     (/ (/ 457679.80848377093 (- 1.0 z)) (- 1.0 z)))
                    (* (- 4.0 (* z z)) t_2)
                    (*
                     (*
                      -1259.1392167224028
                      (- 0.9999999999994297 (pow t_0 3.0)))
                     (+ z 2.0)))
                   (* t_2 (+ z 2.0)))
                  (- 3.0 z))
                 (* t_3 771.3234287776531))
                (- 4.0 z))
               (* t_4 -176.6150291621406))
              (- 5.0 z))
             (* t_5 12.507343278686905))
            (- 6.0 z))
           (* t_6 -0.13857109526572012))
          (- 7.0 z))
         (* t_7 9.984369578019572e-6))
        (- 8.0 z))
       (* t_8 1.5056327351493116e-7)))
     (* (pow (- 7.5 z) z) (* (- 8.0 z) t_8))))))

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 = 676.5203681218851 / (1.0 - z);
	double t_1 = sqrt(exp(z + -7.5));
	double t_2 = fma(t_0, (t_0 + 0.9999999999998099), 0.9999999999996197);
	double t_3 = (0.9999999999998099 - t_0) * (2.0 - z);
	double t_4 = (3.0 - z) * t_3;
	double t_5 = (4.0 - z) * t_4;
	double t_6 = (5.0 - z) * t_5;
	double t_7 = (6.0 - z) * t_6;
	double t_8 = (7.0 - z) * t_7;
	return (((double) M_PI) / sin(((double) M_PI) * z)) * (((t_1 * ((sqrt(((double) M_PI) * 2.0) * sqrt(7.5 - z)) * t_1)) * (((((((((((((fma((0.9999999999996197 - ((457679.80848377093 / (1.0 - z)) / (1.0 - z))), ((4.0 - (z * z)) * t_2), ((-1259.1392167224028 * (0.9999999999994297 - pow(t_0, 3.0))) * (z + 2.0))) / (t_2 * (z + 2.0))) * (3.0 - z)) + (t_3 * 771.3234287776531)) * (4.0 - z)) + (t_4 * -176.6150291621406)) * (5.0 - z)) + (t_5 * 12.507343278686905)) * (6.0 - z)) + (t_6 * -0.13857109526572012)) * (7.0 - z)) + (t_7 * 9.984369578019572e-6)) * (8.0 - z)) + (t_8 * 1.5056327351493116e-7))) / (pow((7.5 - z), z) * ((8.0 - z) * t_8)));
}

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. Simplified 1.7

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

3. Applied flip-+_binary64 1.7

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

4. Applied frac-add_binary64 1.7

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

5. Applied frac-add_binary64 1.7

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

6. Applied frac-add_binary64 1.7

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

7. Applied frac-add_binary64 1.0

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

8. Applied frac-add_binary64 1.0

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

9. Applied frac-add_binary64 1.0

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

10. Applied frac-add_binary64 0.5

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

11. Applied pow-sub_binary64 0.5

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

12. Applied associate-*r/_binary64 0.5

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

13. Applied associate-*l/_binary64 0.5

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

14. Applied frac-times_binary64 0.5

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

15. Applied flip3--_binary64 0.5

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

16. Applied associate-*l/_binary64 0.5

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

17. Applied flip--_binary64 0.5

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

18. Applied associate-*r/_binary64 0.5

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

19. Applied frac-add_binary64 1.6

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

20. Simplified 0.5

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \frac{\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{0.5}\right) \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\left(\left(\left(\frac{\color{blue}{\mathsf{fma}\left(0.9999999999996197 - \frac{\frac{457679.80848377093}{1 - z}}{1 - z}, \left(4 - z \cdot z\right) \cdot \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, 0.9999999999998099 + \frac{676.5203681218851}{1 - z}, 0.9999999999996197\right), \left(-1259.1392167224028 \cdot \left(0.9999999999994297 - {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}\right)\right) \cdot \left(z + 2\right)\right)}}{\left(2 + z\right) \cdot \left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} + 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)\right)} \cdot \left(3 - z\right) + \left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot 771.3234287776531\right) \cdot \left(4 - z\right) + \left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot -176.6150291621406\right) \cdot \left(5 - z\right) + \left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot \left(4 - z\right)\right) \cdot 12.507343278686905\right) \cdot \left(6 - z\right) + \left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot \left(4 - z\right)\right) \cdot \left(5 - z\right)\right) \cdot -0.13857109526572012\right) \cdot \left(7 - z\right) + \left(\left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot \left(4 - z\right)\right) \cdot \left(5 - z\right)\right) \cdot \left(6 - z\right)\right) \cdot 9.984369578019572 \cdot 10^{-6}\right) \cdot \left(8 - z\right) + \left(\left(\left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot \left(4 - z\right)\right) \cdot \left(5 - z\right)\right) \cdot \left(6 - z\right)\right) \cdot \left(7 - z\right)\right) \cdot 1.5056327351493116 \cdot 10^{-7}\right)}{{\left(7.5 - z\right)}^{z} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot \left(4 - z\right)\right) \cdot \left(5 - z\right)\right) \cdot \left(6 - z\right)\right) \cdot \left(7 - z\right)\right) \cdot \left(8 - z\right)\right)} \]

21. Simplified 0.5

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \frac{\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{0.5}\right) \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\left(\left(\left(\frac{\mathsf{fma}\left(0.9999999999996197 - \frac{\frac{457679.80848377093}{1 - z}}{1 - z}, \left(4 - z \cdot z\right) \cdot \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, 0.9999999999998099 + \frac{676.5203681218851}{1 - z}, 0.9999999999996197\right), \left(-1259.1392167224028 \cdot \left(0.9999999999994297 - {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}\right)\right) \cdot \left(z + 2\right)\right)}{\color{blue}{\mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, 0.9999999999998099 + \frac{676.5203681218851}{1 - z}, 0.9999999999996197\right) \cdot \left(z + 2\right)}} \cdot \left(3 - z\right) + \left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot 771.3234287776531\right) \cdot \left(4 - z\right) + \left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot -176.6150291621406\right) \cdot \left(5 - z\right) + \left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot \left(4 - z\right)\right) \cdot 12.507343278686905\right) \cdot \left(6 - z\right) + \left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot \left(4 - z\right)\right) \cdot \left(5 - z\right)\right) \cdot -0.13857109526572012\right) \cdot \left(7 - z\right) + \left(\left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot \left(4 - z\right)\right) \cdot \left(5 - z\right)\right) \cdot \left(6 - z\right)\right) \cdot 9.984369578019572 \cdot 10^{-6}\right) \cdot \left(8 - z\right) + \left(\left(\left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot \left(4 - z\right)\right) \cdot \left(5 - z\right)\right) \cdot \left(6 - z\right)\right) \cdot \left(7 - z\right)\right) \cdot 1.5056327351493116 \cdot 10^{-7}\right)}{{\left(7.5 - z\right)}^{z} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot \left(4 - z\right)\right) \cdot \left(5 - z\right)\right) \cdot \left(6 - z\right)\right) \cdot \left(7 - z\right)\right) \cdot \left(8 - z\right)\right)} \]

22. Applied add-sqr-sqrt_binary64 1.4

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \frac{\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{0.5}\right) \cdot \color{blue}{\left(\sqrt{e^{z + -7.5}} \cdot \sqrt{e^{z + -7.5}}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\frac{\mathsf{fma}\left(0.9999999999996197 - \frac{\frac{457679.80848377093}{1 - z}}{1 - z}, \left(4 - z \cdot z\right) \cdot \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, 0.9999999999998099 + \frac{676.5203681218851}{1 - z}, 0.9999999999996197\right), \left(-1259.1392167224028 \cdot \left(0.9999999999994297 - {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}\right)\right) \cdot \left(z + 2\right)\right)}{\mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, 0.9999999999998099 + \frac{676.5203681218851}{1 - z}, 0.9999999999996197\right) \cdot \left(z + 2\right)} \cdot \left(3 - z\right) + \left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot 771.3234287776531\right) \cdot \left(4 - z\right) + \left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot -176.6150291621406\right) \cdot \left(5 - z\right) + \left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot \left(4 - z\right)\right) \cdot 12.507343278686905\right) \cdot \left(6 - z\right) + \left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot \left(4 - z\right)\right) \cdot \left(5 - z\right)\right) \cdot -0.13857109526572012\right) \cdot \left(7 - z\right) + \left(\left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot \left(4 - z\right)\right) \cdot \left(5 - z\right)\right) \cdot \left(6 - z\right)\right) \cdot 9.984369578019572 \cdot 10^{-6}\right) \cdot \left(8 - z\right) + \left(\left(\left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot \left(4 - z\right)\right) \cdot \left(5 - z\right)\right) \cdot \left(6 - z\right)\right) \cdot \left(7 - z\right)\right) \cdot 1.5056327351493116 \cdot 10^{-7}\right)}{{\left(7.5 - z\right)}^{z} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot \left(4 - z\right)\right) \cdot \left(5 - z\right)\right) \cdot \left(6 - z\right)\right) \cdot \left(7 - z\right)\right) \cdot \left(8 - z\right)\right)} \]

23. Applied associate-*r*_binary64 0.5

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \frac{\color{blue}{\left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{0.5}\right) \cdot \sqrt{e^{z + -7.5}}\right) \cdot \sqrt{e^{z + -7.5}}\right)} \cdot \left(\left(\left(\left(\left(\left(\frac{\mathsf{fma}\left(0.9999999999996197 - \frac{\frac{457679.80848377093}{1 - z}}{1 - z}, \left(4 - z \cdot z\right) \cdot \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, 0.9999999999998099 + \frac{676.5203681218851}{1 - z}, 0.9999999999996197\right), \left(-1259.1392167224028 \cdot \left(0.9999999999994297 - {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}\right)\right) \cdot \left(z + 2\right)\right)}{\mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, 0.9999999999998099 + \frac{676.5203681218851}{1 - z}, 0.9999999999996197\right) \cdot \left(z + 2\right)} \cdot \left(3 - z\right) + \left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot 771.3234287776531\right) \cdot \left(4 - z\right) + \left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot -176.6150291621406\right) \cdot \left(5 - z\right) + \left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot \left(4 - z\right)\right) \cdot 12.507343278686905\right) \cdot \left(6 - z\right) + \left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot \left(4 - z\right)\right) \cdot \left(5 - z\right)\right) \cdot -0.13857109526572012\right) \cdot \left(7 - z\right) + \left(\left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot \left(4 - z\right)\right) \cdot \left(5 - z\right)\right) \cdot \left(6 - z\right)\right) \cdot 9.984369578019572 \cdot 10^{-6}\right) \cdot \left(8 - z\right) + \left(\left(\left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot \left(4 - z\right)\right) \cdot \left(5 - z\right)\right) \cdot \left(6 - z\right)\right) \cdot \left(7 - z\right)\right) \cdot 1.5056327351493116 \cdot 10^{-7}\right)}{{\left(7.5 - z\right)}^{z} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot \left(4 - z\right)\right) \cdot \left(5 - z\right)\right) \cdot \left(6 - z\right)\right) \cdot \left(7 - z\right)\right) \cdot \left(8 - z\right)\right)} \]

24. Simplified 0.5

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \frac{\left(\color{blue}{\left(\left(\sqrt{2 \cdot \pi} \cdot \sqrt{7.5 - z}\right) \cdot \sqrt{e^{z + -7.5}}\right)} \cdot \sqrt{e^{z + -7.5}}\right) \cdot \left(\left(\left(\left(\left(\left(\frac{\mathsf{fma}\left(0.9999999999996197 - \frac{\frac{457679.80848377093}{1 - z}}{1 - z}, \left(4 - z \cdot z\right) \cdot \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, 0.9999999999998099 + \frac{676.5203681218851}{1 - z}, 0.9999999999996197\right), \left(-1259.1392167224028 \cdot \left(0.9999999999994297 - {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}\right)\right) \cdot \left(z + 2\right)\right)}{\mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, 0.9999999999998099 + \frac{676.5203681218851}{1 - z}, 0.9999999999996197\right) \cdot \left(z + 2\right)} \cdot \left(3 - z\right) + \left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot 771.3234287776531\right) \cdot \left(4 - z\right) + \left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot -176.6150291621406\right) \cdot \left(5 - z\right) + \left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot \left(4 - z\right)\right) \cdot 12.507343278686905\right) \cdot \left(6 - z\right) + \left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot \left(4 - z\right)\right) \cdot \left(5 - z\right)\right) \cdot -0.13857109526572012\right) \cdot \left(7 - z\right) + \left(\left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot \left(4 - z\right)\right) \cdot \left(5 - z\right)\right) \cdot \left(6 - z\right)\right) \cdot 9.984369578019572 \cdot 10^{-6}\right) \cdot \left(8 - z\right) + \left(\left(\left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot \left(4 - z\right)\right) \cdot \left(5 - z\right)\right) \cdot \left(6 - z\right)\right) \cdot \left(7 - z\right)\right) \cdot 1.5056327351493116 \cdot 10^{-7}\right)}{{\left(7.5 - z\right)}^{z} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot \left(4 - z\right)\right) \cdot \left(5 - z\right)\right) \cdot \left(6 - z\right)\right) \cdot \left(7 - z\right)\right) \cdot \left(8 - z\right)\right)} \]

25. Final simplification 0.5

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \frac{\left(\sqrt{e^{z + -7.5}} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot \sqrt{7.5 - z}\right) \cdot \sqrt{e^{z + -7.5}}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\frac{\mathsf{fma}\left(0.9999999999996197 - \frac{\frac{457679.80848377093}{1 - z}}{1 - z}, \left(4 - z \cdot z\right) \cdot \mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + 0.9999999999998099, 0.9999999999996197\right), \left(-1259.1392167224028 \cdot \left(0.9999999999994297 - {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}\right)\right) \cdot \left(z + 2\right)\right)}{\mathsf{fma}\left(\frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} + 0.9999999999998099, 0.9999999999996197\right) \cdot \left(z + 2\right)} \cdot \left(3 - z\right) + \left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right) \cdot 771.3234287776531\right) \cdot \left(4 - z\right) + \left(\left(3 - z\right) \cdot \left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right)\right) \cdot -176.6150291621406\right) \cdot \left(5 - z\right) + \left(\left(4 - z\right) \cdot \left(\left(3 - z\right) \cdot \left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right)\right)\right) \cdot 12.507343278686905\right) \cdot \left(6 - z\right) + \left(\left(5 - z\right) \cdot \left(\left(4 - z\right) \cdot \left(\left(3 - z\right) \cdot \left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right)\right)\right)\right) \cdot -0.13857109526572012\right) \cdot \left(7 - z\right) + \left(\left(6 - z\right) \cdot \left(\left(5 - z\right) \cdot \left(\left(4 - z\right) \cdot \left(\left(3 - z\right) \cdot \left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right)\right)\right)\right)\right) \cdot 9.984369578019572 \cdot 10^{-6}\right) \cdot \left(8 - z\right) + \left(\left(7 - z\right) \cdot \left(\left(6 - z\right) \cdot \left(\left(5 - z\right) \cdot \left(\left(4 - z\right) \cdot \left(\left(3 - z\right) \cdot \left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right)\right)\right)\right)\right)\right) \cdot 1.5056327351493116 \cdot 10^{-7}\right)}{{\left(7.5 - z\right)}^{z} \cdot \left(\left(8 - z\right) \cdot \left(\left(7 - z\right) \cdot \left(\left(6 - z\right) \cdot \left(\left(5 - z\right) \cdot \left(\left(4 - z\right) \cdot \left(\left(3 - z\right) \cdot \left(\left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right) \cdot \left(2 - z\right)\right)\right)\right)\right)\right)\right)\right)} \]

Reproduce

herbie shell --seed 2022077 
(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))))))