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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. Simplified0.6

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

4. Applied sqrt-prod1.0

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

6. Applied flip-+0.6

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

8. Applied frac-times0.6

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

9. Applied associate-*r/0.6

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

10. Applied frac-sub1.3

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

11. Applied associate-/l/0.6

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

12. Simplified0.6

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

14. Applied flip3--0.6

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

15. Applied flip3--0.6

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

16. Applied frac-times0.6

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

17. Applied flip--0.6

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

18. Applied frac-times0.6

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

19. Applied flip3--0.6

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

20. Applied frac-times0.6

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

21. Applied associate-/r/0.6

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

22. Simplified0.6

    \[\leadsto \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \frac{1}{e^{\left(1 - z\right) - \left(-6 - 0.5\right)}}\right) \cdot \left(\left(\sqrt{2} \cdot \sqrt{\pi}\right) \cdot {\left(\left(1 - z\right) - \left(-6 - 0.5\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \frac{\left(\left(4 - z\right) \cdot \left(4 - z\right)\right) \cdot \frac{771.3234287776531 \cdot 771.3234287776531}{3 - z} - \left(-176.6150291621406 \cdot -176.6150291621406\right) \cdot \left(3 - z\right)}{\left({\left(\frac{771.3234287776531}{1 - \left(z + -2\right)}\right)}^{3} - {\left(\frac{-176.6150291621406}{5 - \left(z + 1\right)}\right)}^{3}\right) \cdot \left(\left(1 \cdot 1 - \left(z + -2\right) \cdot \left(z + -2\right)\right) \cdot \left(\left({5}^{3} - {\left(z + 1\right)}^{3}\right) \cdot \left({5}^{3} - {\left(z + 1\right)}^{3}\right)\right)\right)} \cdot \color{blue}{\left(\left(\left(\left(z + -1\right) \cdot \left(25 + \left(1 + z\right) \cdot \left(z + 6\right)\right)\right) \cdot \left(25 + \left(1 + z\right) \cdot \left(z + 6\right)\right)\right) \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right) \cdot \frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z} \cdot \frac{771.3234287776531}{3 - z}\right)\right)}\right) + \left(\left(\frac{-0.13857109526572012}{7 - \left(z + 1\right)} + \frac{12.507343278686905}{6 - \left(z + 1\right)}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{1 - \left(z + -7\right)} + \frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) + 6}\right)\right)\right)\]

23. Final simplification0.6

    \[\leadsto \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{1 - \left(-7 + z\right)} + \frac{9.984369578019572 \cdot 10^{-06}}{6 + \left(1 - z\right)}\right) + \left(\frac{-0.13857109526572012}{7 - \left(z + 1\right)} + \frac{12.507343278686905}{6 - \left(z + 1\right)}\right)\right) + \left(\frac{\frac{771.3234287776531 \cdot 771.3234287776531}{3 - z} \cdot \left(\left(4 - z\right) \cdot \left(4 - z\right)\right) - \left(-176.6150291621406 \cdot -176.6150291621406\right) \cdot \left(3 - z\right)}{\left(\left(1 - \left(z + -2\right) \cdot \left(z + -2\right)\right) \cdot \left(\left(125 - {\left(z + 1\right)}^{3}\right) \cdot \left(125 - {\left(z + 1\right)}^{3}\right)\right)\right) \cdot \left({\left(\frac{771.3234287776531}{1 - \left(z + -2\right)}\right)}^{3} - {\left(\frac{-176.6150291621406}{5 - \left(z + 1\right)}\right)}^{3}\right)} \cdot \left(\left(\left(\left(-1 + z\right) \cdot \left(25 + \left(z + 1\right) \cdot \left(6 + z\right)\right)\right) \cdot \left(25 + \left(z + 1\right) \cdot \left(6 + z\right)\right)\right) \cdot \left(\left(\frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z}\right) \cdot \frac{-176.6150291621406}{4 - z} + \frac{771.3234287776531}{3 - z} \cdot \frac{771.3234287776531}{3 - z}\right)\right) + \left(\frac{-1259.1392167224028}{1 + \left(1 - z\right)} + \left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right)\right)\right)\right) \cdot \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \frac{1}{e^{\left(1 - z\right) - \left(-6 - 0.5\right)}}\right) \cdot \left({\left(\left(1 - z\right) - \left(-6 - 0.5\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)} \cdot \left(\sqrt{2} \cdot \sqrt{\pi}\right)\right)\right)\]

Reproduce

herbie shell --seed 2019002 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z less than 0.5"
  (* (/ PI (sin (* PI z))) (* (* (* (sqrt (* PI 2)) (pow (+ (+ (- (- 1 z) 1) 7) 0.5) (+ (- (- 1 z) 1) 0.5))) (exp (- (+ (+ (- (- 1 z) 1) 7) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- (- 1 z) 1) 1))) (/ -1259.1392167224028 (+ (- (- 1 z) 1) 2))) (/ 771.3234287776531 (+ (- (- 1 z) 1) 3))) (/ -176.6150291621406 (+ (- (- 1 z) 1) 4))) (/ 12.507343278686905 (+ (- (- 1 z) 1) 5))) (/ -0.13857109526572012 (+ (- (- 1 z) 1) 6))) (/ 9.984369578019572e-06 (+ (- (- 1 z) 1) 7))) (/ 1.5056327351493116e-07 (+ (- (- 1 z) 1) 8))))))

Details

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