Jmat.Real.gamma, branch z less than 0.5

Average Error: 1.7 → 0.5

Time: 1.1min

Precision: binary64

\[z \leq 0.5\]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{676.5203681218851}{1 - z}\\ t_1 := 0.9999999999998099 - t_0\\ t_2 := \left(2 - z\right) \cdot t_1\\ t_3 := \left(3 - z\right) \cdot t_2\\ t_4 := \left(4 - z\right) \cdot t_3\\ t_5 := \left(5 - z\right) \cdot t_4\\ t_6 := \left(6 - z\right) \cdot t_5\\ t_7 := \left(7 - z\right) \cdot t_6\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \frac{\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999996197 - t_0 \cdot t_0\right) \cdot \left(2 - z\right) + t_1 \cdot -1259.1392167224028\right) \cdot \left(3 - z\right) + t_2 \cdot 771.3234287776531\right) \cdot \left(4 - z\right) + t_3 \cdot -176.6150291621406\right) \cdot \left(5 - z\right) + t_4 \cdot 12.507343278686905\right) \cdot \left(6 - z\right) + t_5 \cdot -0.13857109526572012\right) \cdot \left(7 - z\right) + t_6 \cdot 9.984369578019572 \cdot 10^{-6}\right) \cdot \left(8 - z\right) + t_7 \cdot 1.5056327351493116 \cdot 10^{-7}\right)}{\left(8 - z\right) \cdot t_7} \end{array} \]

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 (- 0.9999999999998099 t_0))
        (t_2 (* (- 2.0 z) t_1))
        (t_3 (* (- 3.0 z) t_2))
        (t_4 (* (- 4.0 z) t_3))
        (t_5 (* (- 5.0 z) t_4))
        (t_6 (* (- 6.0 z) t_5))
        (t_7 (* (- 7.0 z) t_6)))
   (*
    (/ PI (sin (* PI z)))
    (/
     (*
      (* (* (sqrt (* PI 2.0)) (pow (- 7.5 z) (- 0.5 z))) (exp (+ z -7.5)))
      (+
       (*
        (+
         (*
          (+
           (*
            (+
             (*
              (+
               (*
                (+
                 (*
                  (+
                   (* (- 0.9999999999996197 (* t_0 t_0)) (- 2.0 z))
                   (* t_1 -1259.1392167224028))
                  (- 3.0 z))
                 (* t_2 771.3234287776531))
                (- 4.0 z))
               (* t_3 -176.6150291621406))
              (- 5.0 z))
             (* t_4 12.507343278686905))
            (- 6.0 z))
           (* t_5 -0.13857109526572012))
          (- 7.0 z))
         (* t_6 9.984369578019572e-6))
        (- 8.0 z))
       (* t_7 1.5056327351493116e-7)))
     (* (- 8.0 z) t_7)))))

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 = 0.9999999999998099 - t_0;
	double t_2 = (2.0 - z) * t_1;
	double t_3 = (3.0 - z) * t_2;
	double t_4 = (4.0 - z) * t_3;
	double t_5 = (5.0 - z) * t_4;
	double t_6 = (6.0 - z) * t_5;
	double t_7 = (7.0 - z) * t_6;
	return (((double) M_PI) / sin(((double) M_PI) * z)) * ((((sqrt(((double) M_PI) * 2.0) * pow((7.5 - z), (0.5 - z))) * exp(z + -7.5)) * (((((((((((((((0.9999999999996197 - (t_0 * t_0)) * (2.0 - z)) + (t_1 * -1259.1392167224028)) * (3.0 - z)) + (t_2 * 771.3234287776531)) * (4.0 - z)) + (t_3 * -176.6150291621406)) * (5.0 - z)) + (t_4 * 12.507343278686905)) * (6.0 - z)) + (t_5 * -0.13857109526572012)) * (7.0 - z)) + (t_6 * 9.984369578019572e-6)) * (8.0 - z)) + (t_7 * 1.5056327351493116e-7))) / ((8.0 - z) * t_7));
}

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 associate-*r/_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)}^{\left(0.5 - z\right)}\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(\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)}} \]

12. Final simplification 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)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999996197 - \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(2 - z\right) \cdot \left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right)\right) \cdot 771.3234287776531\right) \cdot \left(4 - z\right) + \left(\left(3 - z\right) \cdot \left(\left(2 - z\right) \cdot \left(0.9999999999998099 - \frac{676.5203681218851}{1 - 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(2 - z\right) \cdot \left(0.9999999999998099 - \frac{676.5203681218851}{1 - 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(2 - z\right) \cdot \left(0.9999999999998099 - \frac{676.5203681218851}{1 - 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(2 - z\right) \cdot \left(0.9999999999998099 - \frac{676.5203681218851}{1 - 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(2 - z\right) \cdot \left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot 1.5056327351493116 \cdot 10^{-7}\right)}{\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(2 - z\right) \cdot \left(0.9999999999998099 - \frac{676.5203681218851}{1 - z}\right)\right)\right)\right)\right)\right)\right)} \]

Reproduce

herbie shell --seed 2021329 
(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))))))