Jmat.Real.gamma, branch z less than 0.5

Average Error: 1.8 → 0.4

Time: 1.1min

Precision: binary64

\[z \leq 0.5\]

Math

\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

↓

\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z}\right) \cdot e^{-7.5}\right) \cdot \left(\left(\left(\left(\frac{\left(4 - z\right) \cdot \left(\left(\left(2 - z\right) \cdot \left(0.9999999999994298 + {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}\right) + -1259.1392167224028 \cdot \left(0.9999999999996199 + \frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} - 0.9999999999998099\right)\right)\right) \cdot \left(3 - z\right) + \left(0.9999999999996199 + \frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} - 0.9999999999998099\right)\right) \cdot \left(\left(2 - z\right) \cdot 771.3234287776531\right)\right) + \left(0.9999999999996199 + \frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} - 0.9999999999998099\right)\right) \cdot \left(\left(2 - z\right) \cdot \left(\left(3 - z\right) \cdot -176.6150291621406\right)\right)}{\left(0.9999999999996199 + \frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} - 0.9999999999998099\right)\right) \cdot \left(\left(2 - z\right) \cdot \left(\left(4 - z\right) \cdot \left(3 - z\right)\right)\right)} + \frac{12.507343278686905}{5 - z}\right) + \frac{-0.13857109526572012}{6 - z}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{8 - z}\right)\right)\]

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-06 (+ (- (- 1.0 z) 1.0) 7.0)))
    (/ 1.5056327351493116e-07 (+ (- (- 1.0 z) 1.0) 8.0))))))

↓

(FPCore (z)
 :precision binary64
 (*
  (/ PI (sin (* PI z)))
  (*
   (* (* (* (sqrt (* PI 2.0)) (pow (- 7.5 z) (- 0.5 z))) (exp z)) (exp -7.5))
   (+
    (+
     (+
      (+
       (/
        (+
         (*
          (- 4.0 z)
          (+
           (*
            (+
             (*
              (- 2.0 z)
              (+ 0.9999999999994298 (pow (/ 676.5203681218851 (- 1.0 z)) 3.0)))
             (*
              -1259.1392167224028
              (+
               0.9999999999996199
               (*
                (/ 676.5203681218851 (- 1.0 z))
                (- (/ 676.5203681218851 (- 1.0 z)) 0.9999999999998099)))))
            (- 3.0 z))
           (*
            (+
             0.9999999999996199
             (*
              (/ 676.5203681218851 (- 1.0 z))
              (- (/ 676.5203681218851 (- 1.0 z)) 0.9999999999998099)))
            (* (- 2.0 z) 771.3234287776531))))
         (*
          (+
           0.9999999999996199
           (*
            (/ 676.5203681218851 (- 1.0 z))
            (- (/ 676.5203681218851 (- 1.0 z)) 0.9999999999998099)))
          (* (- 2.0 z) (* (- 3.0 z) -176.6150291621406))))
        (*
         (+
          0.9999999999996199
          (*
           (/ 676.5203681218851 (- 1.0 z))
           (- (/ 676.5203681218851 (- 1.0 z)) 0.9999999999998099)))
         (* (- 2.0 z) (* (- 4.0 z) (- 3.0 z)))))
       (/ 12.507343278686905 (- 5.0 z)))
      (/ -0.13857109526572012 (- 6.0 z)))
     (/ 9.984369578019572e-06 (- 7.0 z)))
    (/ 1.5056327351493116e-07 (- 8.0 z))))))

C

double code(double z) {
	return ((double) ((((double) M_PI) / ((double) sin(((double) (((double) M_PI) * z))))) * ((double) (((double) (((double) (((double) sqrt(((double) (((double) M_PI) * 2.0)))) * ((double) pow(((double) (((double) (((double) (((double) (1.0 - z)) - 1.0)) + 7.0)) + 0.5)), ((double) (((double) (((double) (1.0 - z)) - 1.0)) + 0.5)))))) * ((double) exp(((double) -(((double) (((double) (((double) (((double) (1.0 - z)) - 1.0)) + 7.0)) + 0.5)))))))) * ((double) (((double) (((double) (((double) (((double) (((double) (((double) (((double) (0.9999999999998099 + (676.5203681218851 / ((double) (((double) (((double) (1.0 - z)) - 1.0)) + 1.0))))) + (-1259.1392167224028 / ((double) (((double) (((double) (1.0 - z)) - 1.0)) + 2.0))))) + (771.3234287776531 / ((double) (((double) (((double) (1.0 - z)) - 1.0)) + 3.0))))) + (-176.6150291621406 / ((double) (((double) (((double) (1.0 - z)) - 1.0)) + 4.0))))) + (12.507343278686905 / ((double) (((double) (((double) (1.0 - z)) - 1.0)) + 5.0))))) + (-0.13857109526572012 / ((double) (((double) (((double) (1.0 - z)) - 1.0)) + 6.0))))) + (9.984369578019572e-06 / ((double) (((double) (((double) (1.0 - z)) - 1.0)) + 7.0))))) + (1.5056327351493116e-07 / ((double) (((double) (((double) (1.0 - z)) - 1.0)) + 8.0)))))))));
}

↓

double code(double z) {
	return ((double) ((((double) M_PI) / ((double) sin(((double) (((double) M_PI) * z))))) * ((double) (((double) (((double) (((double) (((double) sqrt(((double) (((double) M_PI) * 2.0)))) * ((double) pow(((double) (7.5 - z)), ((double) (0.5 - z)))))) * ((double) exp(z)))) * ((double) exp(-7.5)))) * ((double) (((double) (((double) (((double) ((((double) (((double) (((double) (4.0 - z)) * ((double) (((double) (((double) (((double) (((double) (2.0 - z)) * ((double) (0.9999999999994298 + ((double) pow((676.5203681218851 / ((double) (1.0 - z))), 3.0)))))) + ((double) (-1259.1392167224028 * ((double) (0.9999999999996199 + ((double) ((676.5203681218851 / ((double) (1.0 - z))) * ((double) ((676.5203681218851 / ((double) (1.0 - z))) - 0.9999999999998099)))))))))) * ((double) (3.0 - z)))) + ((double) (((double) (0.9999999999996199 + ((double) ((676.5203681218851 / ((double) (1.0 - z))) * ((double) ((676.5203681218851 / ((double) (1.0 - z))) - 0.9999999999998099)))))) * ((double) (((double) (2.0 - z)) * 771.3234287776531)))))))) + ((double) (((double) (0.9999999999996199 + ((double) ((676.5203681218851 / ((double) (1.0 - z))) * ((double) ((676.5203681218851 / ((double) (1.0 - z))) - 0.9999999999998099)))))) * ((double) (((double) (2.0 - z)) * ((double) (((double) (3.0 - z)) * -176.6150291621406)))))))) / ((double) (((double) (0.9999999999996199 + ((double) ((676.5203681218851 / ((double) (1.0 - z))) * ((double) ((676.5203681218851 / ((double) (1.0 - z))) - 0.9999999999998099)))))) * ((double) (((double) (2.0 - z)) * ((double) (((double) (4.0 - z)) * ((double) (3.0 - z))))))))) + (12.507343278686905 / ((double) (5.0 - z))))) + (-0.13857109526572012 / ((double) (6.0 - z))))) + (9.984369578019572e-06 / ((double) (7.0 - z))))) + (1.5056327351493116e-07 / ((double) (8.0 - z)))))))));
}

Error

Bits error versus z

Derivation

1. Initial program 1.8

   \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

2. Simplified 1.8

   \[\leadsto \color{blue}{\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{2 - z}\right) + \frac{771.3234287776531}{3 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \frac{12.507343278686905}{5 - z}\right) + \frac{-0.13857109526572012}{6 - z}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{8 - z}\right)\right)}\]
3. Using strategy rm

4. Applied flip3-+_binary64 1.8

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\color{blue}{\frac{{0.9999999999998099}^{3} + {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}}{0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)}} + \frac{-1259.1392167224028}{2 - z}\right) + \frac{771.3234287776531}{3 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \frac{12.507343278686905}{5 - z}\right) + \frac{-0.13857109526572012}{6 - z}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{8 - z}\right)\right)\]

5. Applied frac-add_binary64 1.2

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\frac{\left({0.9999999999998099}^{3} + {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}\right) \cdot \left(2 - z\right) + \left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)\right) \cdot -1259.1392167224028}{\left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)\right) \cdot \left(2 - z\right)}} + \frac{771.3234287776531}{3 - z}\right) + \frac{-176.6150291621406}{4 - z}\right) + \frac{12.507343278686905}{5 - z}\right) + \frac{-0.13857109526572012}{6 - z}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{8 - z}\right)\right)\]

6. Applied frac-add_binary64 1.2

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\frac{\left(\left({0.9999999999998099}^{3} + {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}\right) \cdot \left(2 - z\right) + \left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)\right) \cdot -1259.1392167224028\right) \cdot \left(3 - z\right) + \left(\left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)\right) \cdot \left(2 - z\right)\right) \cdot 771.3234287776531}{\left(\left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)}} + \frac{-176.6150291621406}{4 - z}\right) + \frac{12.507343278686905}{5 - z}\right) + \frac{-0.13857109526572012}{6 - z}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{8 - z}\right)\right)\]

7. Applied frac-add_binary64 0.4

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\left(\color{blue}{\frac{\left(\left(\left({0.9999999999998099}^{3} + {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}\right) \cdot \left(2 - z\right) + \left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)\right) \cdot -1259.1392167224028\right) \cdot \left(3 - z\right) + \left(\left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)\right) \cdot \left(2 - z\right)\right) \cdot 771.3234287776531\right) \cdot \left(4 - z\right) + \left(\left(\left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot -176.6150291621406}{\left(\left(\left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot \left(4 - z\right)}} + \frac{12.507343278686905}{5 - z}\right) + \frac{-0.13857109526572012}{6 - z}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{8 - z}\right)\right)\]

8. Simplified 0.4

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\left(\frac{\color{blue}{\left(4 - z\right) \cdot \left(\left(\left(2 - z\right) \cdot \left(0.9999999999994298 + {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}\right) + -1259.1392167224028 \cdot \left(0.9999999999996199 + \frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} - 0.9999999999998099\right)\right)\right) \cdot \left(3 - z\right) + \left(0.9999999999996199 + \frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} - 0.9999999999998099\right)\right) \cdot \left(\left(2 - z\right) \cdot 771.3234287776531\right)\right) + \left(0.9999999999996199 + \frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} - 0.9999999999998099\right)\right) \cdot \left(\left(2 - z\right) \cdot \left(\left(3 - z\right) \cdot -176.6150291621406\right)\right)}}{\left(\left(\left(0.9999999999998099 \cdot 0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} - 0.9999999999998099 \cdot \frac{676.5203681218851}{1 - z}\right)\right) \cdot \left(2 - z\right)\right) \cdot \left(3 - z\right)\right) \cdot \left(4 - z\right)} + \frac{12.507343278686905}{5 - z}\right) + \frac{-0.13857109526572012}{6 - z}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{8 - z}\right)\right)\]

9. Simplified 0.4

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\left(\frac{\left(4 - z\right) \cdot \left(\left(\left(2 - z\right) \cdot \left(0.9999999999994298 + {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}\right) + -1259.1392167224028 \cdot \left(0.9999999999996199 + \frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} - 0.9999999999998099\right)\right)\right) \cdot \left(3 - z\right) + \left(0.9999999999996199 + \frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} - 0.9999999999998099\right)\right) \cdot \left(\left(2 - z\right) \cdot 771.3234287776531\right)\right) + \left(0.9999999999996199 + \frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} - 0.9999999999998099\right)\right) \cdot \left(\left(2 - z\right) \cdot \left(\left(3 - z\right) \cdot -176.6150291621406\right)\right)}{\color{blue}{\left(0.9999999999996199 + \frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} - 0.9999999999998099\right)\right) \cdot \left(\left(2 - z\right) \cdot \left(\left(3 - z\right) \cdot \left(4 - z\right)\right)\right)}} + \frac{12.507343278686905}{5 - z}\right) + \frac{-0.13857109526572012}{6 - z}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{8 - z}\right)\right)\]
10. Using strategy rm

11. Applied exp-sum_binary64 0.4

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \color{blue}{\left(e^{z} \cdot e^{-7.5}\right)}\right) \cdot \left(\left(\left(\left(\frac{\left(4 - z\right) \cdot \left(\left(\left(2 - z\right) \cdot \left(0.9999999999994298 + {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}\right) + -1259.1392167224028 \cdot \left(0.9999999999996199 + \frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} - 0.9999999999998099\right)\right)\right) \cdot \left(3 - z\right) + \left(0.9999999999996199 + \frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} - 0.9999999999998099\right)\right) \cdot \left(\left(2 - z\right) \cdot 771.3234287776531\right)\right) + \left(0.9999999999996199 + \frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} - 0.9999999999998099\right)\right) \cdot \left(\left(2 - z\right) \cdot \left(\left(3 - z\right) \cdot -176.6150291621406\right)\right)}{\left(0.9999999999996199 + \frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} - 0.9999999999998099\right)\right) \cdot \left(\left(2 - z\right) \cdot \left(\left(3 - z\right) \cdot \left(4 - z\right)\right)\right)} + \frac{12.507343278686905}{5 - z}\right) + \frac{-0.13857109526572012}{6 - z}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{8 - z}\right)\right)\]

12. Applied associate-*r*_binary64 0.4

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\color{blue}{\left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z}\right) \cdot e^{-7.5}\right)} \cdot \left(\left(\left(\left(\frac{\left(4 - z\right) \cdot \left(\left(\left(2 - z\right) \cdot \left(0.9999999999994298 + {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}\right) + -1259.1392167224028 \cdot \left(0.9999999999996199 + \frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} - 0.9999999999998099\right)\right)\right) \cdot \left(3 - z\right) + \left(0.9999999999996199 + \frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} - 0.9999999999998099\right)\right) \cdot \left(\left(2 - z\right) \cdot 771.3234287776531\right)\right) + \left(0.9999999999996199 + \frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} - 0.9999999999998099\right)\right) \cdot \left(\left(2 - z\right) \cdot \left(\left(3 - z\right) \cdot -176.6150291621406\right)\right)}{\left(0.9999999999996199 + \frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} - 0.9999999999998099\right)\right) \cdot \left(\left(2 - z\right) \cdot \left(\left(3 - z\right) \cdot \left(4 - z\right)\right)\right)} + \frac{12.507343278686905}{5 - z}\right) + \frac{-0.13857109526572012}{6 - z}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{8 - z}\right)\right)\]

13. Simplified 0.4

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\left(\sqrt{2 \cdot \pi} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z}\right)} \cdot e^{-7.5}\right) \cdot \left(\left(\left(\left(\frac{\left(4 - z\right) \cdot \left(\left(\left(2 - z\right) \cdot \left(0.9999999999994298 + {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}\right) + -1259.1392167224028 \cdot \left(0.9999999999996199 + \frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} - 0.9999999999998099\right)\right)\right) \cdot \left(3 - z\right) + \left(0.9999999999996199 + \frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} - 0.9999999999998099\right)\right) \cdot \left(\left(2 - z\right) \cdot 771.3234287776531\right)\right) + \left(0.9999999999996199 + \frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} - 0.9999999999998099\right)\right) \cdot \left(\left(2 - z\right) \cdot \left(\left(3 - z\right) \cdot -176.6150291621406\right)\right)}{\left(0.9999999999996199 + \frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} - 0.9999999999998099\right)\right) \cdot \left(\left(2 - z\right) \cdot \left(\left(3 - z\right) \cdot \left(4 - z\right)\right)\right)} + \frac{12.507343278686905}{5 - z}\right) + \frac{-0.13857109526572012}{6 - z}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{8 - z}\right)\right)\]

14. Final simplification 0.4

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot e^{z}\right) \cdot e^{-7.5}\right) \cdot \left(\left(\left(\left(\frac{\left(4 - z\right) \cdot \left(\left(\left(2 - z\right) \cdot \left(0.9999999999994298 + {\left(\frac{676.5203681218851}{1 - z}\right)}^{3}\right) + -1259.1392167224028 \cdot \left(0.9999999999996199 + \frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} - 0.9999999999998099\right)\right)\right) \cdot \left(3 - z\right) + \left(0.9999999999996199 + \frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} - 0.9999999999998099\right)\right) \cdot \left(\left(2 - z\right) \cdot 771.3234287776531\right)\right) + \left(0.9999999999996199 + \frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} - 0.9999999999998099\right)\right) \cdot \left(\left(2 - z\right) \cdot \left(\left(3 - z\right) \cdot -176.6150291621406\right)\right)}{\left(0.9999999999996199 + \frac{676.5203681218851}{1 - z} \cdot \left(\frac{676.5203681218851}{1 - z} - 0.9999999999998099\right)\right) \cdot \left(\left(2 - z\right) \cdot \left(\left(4 - z\right) \cdot \left(3 - z\right)\right)\right)} + \frac{12.507343278686905}{5 - z}\right) + \frac{-0.13857109526572012}{6 - z}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{8 - z}\right)\right)\]

Reproduce

herbie shell --seed 2020219 
(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-06 (+ (- (- 1.0 z) 1.0) 7.0))) (/ 1.5056327351493116e-07 (+ (- (- 1.0 z) 1.0) 8.0))))))