Error

Derivation

1. Initial program 59.9

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

2. Simplified1.2

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

3. Taylor expanded around 0 1.5

   \[\leadsto \color{blue}{\left(676.5203681218851 \cdot \left(\frac{\sqrt{2} \cdot \log 6.5}{e^{6.5}} \cdot \left({\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right) + \left(338.26018406094255 \cdot \left(\frac{\sqrt{2} \cdot \left(z \cdot {\left(\log 6.5\right)}^{2}\right)}{e^{6.5}} \cdot \left({\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right) + \left(169.13009203047127 \cdot \left(\frac{\sqrt{2} \cdot z}{e^{6.5}} \cdot \left({\left(\frac{1}{{6.5}^{5.0}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right) + \left(2581.191799681222 \cdot \left(\frac{\sqrt{2} \cdot z}{e^{6.5}} \cdot \left({\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right) + 676.5203681218851 \cdot \left(\frac{\sqrt{2}}{e^{6.5} \cdot z} \cdot \left({\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right)\right)\right)\right)\right) - \left(1656.8104518737205 \cdot \left(\frac{\sqrt{2} \cdot \left(z \cdot \log 6.5\right)}{e^{6.5}} \cdot \left({\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right) + 1656.8104518737205 \cdot \left(\frac{\sqrt{2}}{e^{6.5}} \cdot \left({\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right)\right)}\]

4. Simplified1.7

   \[\leadsto \color{blue}{\left((\left(\frac{169.13009203047127 \cdot \sqrt{2}}{\frac{e^{6.5}}{z}}\right) \cdot \left({\left(\frac{1}{{6.5}^{5.0}}\right)}^{0.5} \cdot \sqrt{\pi}\right) + \left(\left({\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \sqrt{\pi}\right) \cdot \left(\frac{676.5203681218851 \cdot \sqrt{2}}{e^{6.5} \cdot z} + \frac{2581.191799681222 \cdot \sqrt{2}}{\frac{e^{6.5}}{z}}\right)\right))_* + \left({\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \sqrt{\pi}\right) \cdot \left(\frac{676.5203681218851 \cdot \sqrt{2}}{\frac{e^{6.5}}{\log 6.5}} + \frac{338.26018406094255 \cdot \sqrt{2}}{\frac{\frac{e^{6.5}}{z}}{\log 6.5 \cdot \log 6.5}}\right)\right) - \left(\left({\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \sqrt{\pi}\right) \cdot \left(\frac{\sqrt{2}}{e^{6.5}} \cdot \left(\log 6.5 \cdot z\right) + \frac{\sqrt{2}}{e^{6.5}}\right)\right) \cdot 1656.8104518737205}\]

5. Final simplification1.7

   \[\leadsto \left((\left(\frac{169.13009203047127 \cdot \sqrt{2}}{\frac{e^{6.5}}{z}}\right) \cdot \left({\left(\frac{1}{{6.5}^{5.0}}\right)}^{0.5} \cdot \sqrt{\pi}\right) + \left(\left(\sqrt{\pi} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right) \cdot \left(\frac{\sqrt{2} \cdot 2581.191799681222}{\frac{e^{6.5}}{z}} + \frac{676.5203681218851 \cdot \sqrt{2}}{e^{6.5} \cdot z}\right)\right))_* + \left(\sqrt{\pi} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right) \cdot \left(\frac{676.5203681218851 \cdot \sqrt{2}}{\frac{e^{6.5}}{\log 6.5}} + \frac{\sqrt{2} \cdot 338.26018406094255}{\frac{\frac{e^{6.5}}{z}}{\log 6.5 \cdot \log 6.5}}\right)\right) - 1656.8104518737205 \cdot \left(\left(\sqrt{\pi} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right) \cdot \left(\left(\log 6.5 \cdot z\right) \cdot \frac{\sqrt{2}}{e^{6.5}} + \frac{\sqrt{2}}{e^{6.5}}\right)\right)\]

Reproduce

herbie shell --seed 2019026 +o rules:numerics
(FPCore (z)
  :name "Jmat.Real.gamma, branch z greater than 0.5"
  (* (* (* (sqrt (* PI 2)) (pow (+ (+ (- z 1) 7) 0.5) (+ (- z 1) 0.5))) (exp (- (+ (+ (- z 1) 7) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- z 1) 1))) (/ -1259.1392167224028 (+ (- z 1) 2))) (/ 771.3234287776531 (+ (- z 1) 3))) (/ -176.6150291621406 (+ (- z 1) 4))) (/ 12.507343278686905 (+ (- z 1) 5))) (/ -0.13857109526572012 (+ (- z 1) 6))) (/ 9.984369578019572e-06 (+ (- z 1) 7))) (/ 1.5056327351493116e-07 (+ (- z 1) 8)))))