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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. Initial simplification0.7

   \[\leadsto \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \left({\left(\left(1 - z\right) - \left(-6 - 0.5\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)} \cdot \frac{1}{e^{\left(1 - z\right) - \left(-6 - 0.5\right)}}\right)\right) \cdot \left(\left(\left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{5 - \left(z + 1\right)}\right) + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right)\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{12.507343278686905}{6 - \left(z + 1\right)}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{9 - \left(z + 1\right)} + \frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) - -6}\right)\right)\right)\]
3. Using strategy rm

4. Applied add-log-exp0.8

   \[\leadsto \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \left({\left(\left(1 - z\right) - \left(-6 - 0.5\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)} \cdot \frac{1}{e^{\left(1 - z\right) - \left(-6 - 0.5\right)}}\right)\right) \cdot \left(\left(\left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{5 - \left(z + 1\right)}\right) + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \color{blue}{\log \left(e^{\frac{-1259.1392167224028}{\left(1 - z\right) + 1}}\right)}\right)\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{12.507343278686905}{6 - \left(z + 1\right)}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{9 - \left(z + 1\right)} + \frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) - -6}\right)\right)\right)\]

5. Applied add-log-exp0.9

   \[\leadsto \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \left({\left(\left(1 - z\right) - \left(-6 - 0.5\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)} \cdot \frac{1}{e^{\left(1 - z\right) - \left(-6 - 0.5\right)}}\right)\right) \cdot \left(\left(\left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{5 - \left(z + 1\right)}\right) + \left(\color{blue}{\log \left(e^{0.9999999999998099 + \frac{676.5203681218851}{1 - z}}\right)} + \log \left(e^{\frac{-1259.1392167224028}{\left(1 - z\right) + 1}}\right)\right)\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{12.507343278686905}{6 - \left(z + 1\right)}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{9 - \left(z + 1\right)} + \frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) - -6}\right)\right)\right)\]

6. Applied sum-log0.9

   \[\leadsto \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \left({\left(\left(1 - z\right) - \left(-6 - 0.5\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)} \cdot \frac{1}{e^{\left(1 - z\right) - \left(-6 - 0.5\right)}}\right)\right) \cdot \left(\left(\left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{5 - \left(z + 1\right)}\right) + \color{blue}{\log \left(e^{0.9999999999998099 + \frac{676.5203681218851}{1 - z}} \cdot e^{\frac{-1259.1392167224028}{\left(1 - z\right) + 1}}\right)}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{12.507343278686905}{6 - \left(z + 1\right)}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{9 - \left(z + 1\right)} + \frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) - -6}\right)\right)\right)\]

7. Simplified0.7

   \[\leadsto \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot \left({\left(\left(1 - z\right) - \left(-6 - 0.5\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)} \cdot \frac{1}{e^{\left(1 - z\right) - \left(-6 - 0.5\right)}}\right)\right) \cdot \left(\left(\left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{5 - \left(z + 1\right)}\right) + \log \color{blue}{\left(e^{\left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}\right) + \frac{676.5203681218851}{1 - z}}\right)}\right) + \left(\left(\frac{-0.13857109526572012}{\left(1 - z\right) + 5} + \frac{12.507343278686905}{6 - \left(z + 1\right)}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{9 - \left(z + 1\right)} + \frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) - -6}\right)\right)\right)\]

8. Simplified0.7

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

10. Applied sub-neg0.7

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

11. Applied exp-sum0.7

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

12. Applied add-cube-cbrt0.7

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

13. Applied add-sqr-sqrt0.8

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

14. Applied prod-diff0.8

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

15. Applied unpow-prod-up0.8

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

16. Applied times-frac0.7

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

17. Simplified1.2

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

18. Simplified0.7

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

20. Applied expm1-log1p-u0.7

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

21. Final simplification0.7

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

Runtime

Time bar (total: 9.2m)Debug logProfile

herbie shell --seed 2018263 +o rules:numerics
(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))))))