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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 flip3-+0.6

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

6. Applied flip--0.6

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

7. Applied associate-/r/0.6

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

8. Applied cube-prod0.6

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

9. Simplified0.6

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

11. Applied flip-+0.6

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

12. Applied frac-add1.0

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

13. Simplified0.6

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

14. Final simplification0.6

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

Reproduce

herbie shell --seed 2019005 
(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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