\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8}\right)\]
Test:
Jmat.Real.gamma, branch z greater than 0.5
Bits:
128 bits
Bits error versus z
Time: 4.0 m
Input Error: 61.2
Output Error: 1.6
Log: ⚲
Profile: 🕒
\((\left(\frac{\sqrt{2}}{e^{-1 + 7.5} \cdot z} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right) * \left(\sqrt{\pi} \cdot 676.5203681218851\right) + \left((\left(338.26018406094255 \cdot \sqrt{\pi}\right) * \left(\frac{\left(\left(\sqrt{2} \cdot z\right) \cdot {\left(\log 6.5\right)}^2\right) \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}}{e^{-1 + 7.5}}\right) + \left(\frac{\left(\left(\sqrt{\pi} \cdot 2585.1948787825354\right) \cdot \left(\sqrt{2} \cdot z\right)\right) \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}}{e^{-1 + 7.5}}\right))_*\right))_* - \left(\left(\sqrt{\pi} \cdot 1656.8104518737205\right) \cdot (\left({\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right) * \left(\frac{\sqrt{2} \cdot \left(z \cdot \log 6.5\right)}{e^{-1 + 7.5}}\right) + \left(\frac{\sqrt{2}}{e^{-1 + 7.5}} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right))_* - \left(\frac{\sqrt{2} \cdot \log 6.5}{e^{-1 + 7.5}} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right) \cdot \left(\sqrt{\pi} \cdot 676.5203681218851\right)\right)\)
  1. Started with
    \[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8}\right)\]
    61.2
  2. Applied simplify to get
    \[\color{red}{\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8}\right)} \leadsto \color{blue}{\left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{\left(z + 8\right) - 1} + \frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(z - 1\right)}\right) + \left(\frac{-0.13857109526572012}{\left(z - 1\right) + 6} + \frac{12.507343278686905}{z - \left(1 - 5\right)}\right)\right) + \left(\left(\frac{-1259.1392167224028}{z - \left(1 - 2\right)} + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \left(\left(\frac{676.5203681218851}{z - 0} + 0.9999999999998099\right) + \frac{-176.6150291621406}{\left(z + 4\right) - 1}\right)\right)\right) \cdot \frac{{\left(\left(z - 1\right) + \left(0.5 + 7\right)\right)}^{\left(0.5 + \left(z - 1\right)\right)} \cdot \sqrt{2 \cdot \pi}}{e^{\left(z - 1\right) + \left(0.5 + 7\right)}}}\]
    31.8
  3. Using strategy rm
    31.8
  4. Applied exp-sum to get
    \[\left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{\left(z + 8\right) - 1} + \frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(z - 1\right)}\right) + \left(\frac{-0.13857109526572012}{\left(z - 1\right) + 6} + \frac{12.507343278686905}{z - \left(1 - 5\right)}\right)\right) + \left(\left(\frac{-1259.1392167224028}{z - \left(1 - 2\right)} + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \left(\left(\frac{676.5203681218851}{z - 0} + 0.9999999999998099\right) + \frac{-176.6150291621406}{\left(z + 4\right) - 1}\right)\right)\right) \cdot \frac{{\left(\left(z - 1\right) + \left(0.5 + 7\right)\right)}^{\left(0.5 + \left(z - 1\right)\right)} \cdot \sqrt{2 \cdot \pi}}{\color{red}{e^{\left(z - 1\right) + \left(0.5 + 7\right)}}} \leadsto \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{\left(z + 8\right) - 1} + \frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(z - 1\right)}\right) + \left(\frac{-0.13857109526572012}{\left(z - 1\right) + 6} + \frac{12.507343278686905}{z - \left(1 - 5\right)}\right)\right) + \left(\left(\frac{-1259.1392167224028}{z - \left(1 - 2\right)} + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \left(\left(\frac{676.5203681218851}{z - 0} + 0.9999999999998099\right) + \frac{-176.6150291621406}{\left(z + 4\right) - 1}\right)\right)\right) \cdot \frac{{\left(\left(z - 1\right) + \left(0.5 + 7\right)\right)}^{\left(0.5 + \left(z - 1\right)\right)} \cdot \sqrt{2 \cdot \pi}}{\color{blue}{e^{z - 1} \cdot e^{0.5 + 7}}}\]
    31.8
  5. Applied times-frac to get
    \[\left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{\left(z + 8\right) - 1} + \frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(z - 1\right)}\right) + \left(\frac{-0.13857109526572012}{\left(z - 1\right) + 6} + \frac{12.507343278686905}{z - \left(1 - 5\right)}\right)\right) + \left(\left(\frac{-1259.1392167224028}{z - \left(1 - 2\right)} + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \left(\left(\frac{676.5203681218851}{z - 0} + 0.9999999999998099\right) + \frac{-176.6150291621406}{\left(z + 4\right) - 1}\right)\right)\right) \cdot \color{red}{\frac{{\left(\left(z - 1\right) + \left(0.5 + 7\right)\right)}^{\left(0.5 + \left(z - 1\right)\right)} \cdot \sqrt{2 \cdot \pi}}{e^{z - 1} \cdot e^{0.5 + 7}}} \leadsto \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{\left(z + 8\right) - 1} + \frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(z - 1\right)}\right) + \left(\frac{-0.13857109526572012}{\left(z - 1\right) + 6} + \frac{12.507343278686905}{z - \left(1 - 5\right)}\right)\right) + \left(\left(\frac{-1259.1392167224028}{z - \left(1 - 2\right)} + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \left(\left(\frac{676.5203681218851}{z - 0} + 0.9999999999998099\right) + \frac{-176.6150291621406}{\left(z + 4\right) - 1}\right)\right)\right) \cdot \color{blue}{\left(\frac{{\left(\left(z - 1\right) + \left(0.5 + 7\right)\right)}^{\left(0.5 + \left(z - 1\right)\right)}}{e^{z - 1}} \cdot \frac{\sqrt{2 \cdot \pi}}{e^{0.5 + 7}}\right)}\]
    31.8
  6. Applied associate-*r* to get
    \[\color{red}{\left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{\left(z + 8\right) - 1} + \frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(z - 1\right)}\right) + \left(\frac{-0.13857109526572012}{\left(z - 1\right) + 6} + \frac{12.507343278686905}{z - \left(1 - 5\right)}\right)\right) + \left(\left(\frac{-1259.1392167224028}{z - \left(1 - 2\right)} + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \left(\left(\frac{676.5203681218851}{z - 0} + 0.9999999999998099\right) + \frac{-176.6150291621406}{\left(z + 4\right) - 1}\right)\right)\right) \cdot \left(\frac{{\left(\left(z - 1\right) + \left(0.5 + 7\right)\right)}^{\left(0.5 + \left(z - 1\right)\right)}}{e^{z - 1}} \cdot \frac{\sqrt{2 \cdot \pi}}{e^{0.5 + 7}}\right)} \leadsto \color{blue}{\left(\left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{\left(z + 8\right) - 1} + \frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(z - 1\right)}\right) + \left(\frac{-0.13857109526572012}{\left(z - 1\right) + 6} + \frac{12.507343278686905}{z - \left(1 - 5\right)}\right)\right) + \left(\left(\frac{-1259.1392167224028}{z - \left(1 - 2\right)} + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \left(\left(\frac{676.5203681218851}{z - 0} + 0.9999999999998099\right) + \frac{-176.6150291621406}{\left(z + 4\right) - 1}\right)\right)\right) \cdot \frac{{\left(\left(z - 1\right) + \left(0.5 + 7\right)\right)}^{\left(0.5 + \left(z - 1\right)\right)}}{e^{z - 1}}\right) \cdot \frac{\sqrt{2 \cdot \pi}}{e^{0.5 + 7}}}\]
    31.7
  7. Applied taylor to get
    \[\left(\left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{\left(z + 8\right) - 1} + \frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(z - 1\right)}\right) + \left(\frac{-0.13857109526572012}{\left(z - 1\right) + 6} + \frac{12.507343278686905}{z - \left(1 - 5\right)}\right)\right) + \left(\left(\frac{-1259.1392167224028}{z - \left(1 - 2\right)} + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \left(\left(\frac{676.5203681218851}{z - 0} + 0.9999999999998099\right) + \frac{-176.6150291621406}{\left(z + 4\right) - 1}\right)\right)\right) \cdot \frac{{\left(\left(z - 1\right) + \left(0.5 + 7\right)\right)}^{\left(0.5 + \left(z - 1\right)\right)}}{e^{z - 1}}\right) \cdot \frac{\sqrt{2 \cdot \pi}}{e^{0.5 + 7}} \leadsto \left(676.5203681218851 \cdot \left(\sqrt{\pi} \cdot \left(\frac{\sqrt{2} \cdot \log 6.5}{e^{7.5} \cdot e^{-1}} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + \left(2585.1948787825354 \cdot \left(\sqrt{\pi} \cdot \left(\frac{\sqrt{2} \cdot z}{e^{7.5} \cdot e^{-1}} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + \left(338.26018406094255 \cdot \left(\sqrt{\pi} \cdot \left(\frac{z \cdot \left(\sqrt{2} \cdot {\left(\log 6.5\right)}^2\right)}{e^{-1} \cdot e^{7.5}} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + 676.5203681218851 \cdot \left(\sqrt{\pi} \cdot \left(\frac{\sqrt{2}}{e^{7.5} \cdot \left(e^{-1} \cdot z\right)} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right)\right)\right)\right) - \left(1656.8104518737205 \cdot \left(\sqrt{\pi} \cdot \left(\frac{\sqrt{2}}{e^{7.5} \cdot e^{-1}} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + 1656.8104518737205 \cdot \left(\sqrt{\pi} \cdot \left(\frac{z \cdot \left(\sqrt{2} \cdot \log 6.5\right)}{e^{-1} \cdot e^{7.5}} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right)\right)\]
    1.3
  8. Taylor expanded around 0 to get
    \[\color{red}{\left(676.5203681218851 \cdot \left(\sqrt{\pi} \cdot \left(\frac{\sqrt{2} \cdot \log 6.5}{e^{7.5} \cdot e^{-1}} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + \left(2585.1948787825354 \cdot \left(\sqrt{\pi} \cdot \left(\frac{\sqrt{2} \cdot z}{e^{7.5} \cdot e^{-1}} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + \left(338.26018406094255 \cdot \left(\sqrt{\pi} \cdot \left(\frac{z \cdot \left(\sqrt{2} \cdot {\left(\log 6.5\right)}^2\right)}{e^{-1} \cdot e^{7.5}} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + 676.5203681218851 \cdot \left(\sqrt{\pi} \cdot \left(\frac{\sqrt{2}}{e^{7.5} \cdot \left(e^{-1} \cdot z\right)} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right)\right)\right)\right) - \left(1656.8104518737205 \cdot \left(\sqrt{\pi} \cdot \left(\frac{\sqrt{2}}{e^{7.5} \cdot e^{-1}} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + 1656.8104518737205 \cdot \left(\sqrt{\pi} \cdot \left(\frac{z \cdot \left(\sqrt{2} \cdot \log 6.5\right)}{e^{-1} \cdot e^{7.5}} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right)\right)} \leadsto \color{blue}{\left(676.5203681218851 \cdot \left(\sqrt{\pi} \cdot \left(\frac{\sqrt{2} \cdot \log 6.5}{e^{7.5} \cdot e^{-1}} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + \left(2585.1948787825354 \cdot \left(\sqrt{\pi} \cdot \left(\frac{\sqrt{2} \cdot z}{e^{7.5} \cdot e^{-1}} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + \left(338.26018406094255 \cdot \left(\sqrt{\pi} \cdot \left(\frac{z \cdot \left(\sqrt{2} \cdot {\left(\log 6.5\right)}^2\right)}{e^{-1} \cdot e^{7.5}} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + 676.5203681218851 \cdot \left(\sqrt{\pi} \cdot \left(\frac{\sqrt{2}}{e^{7.5} \cdot \left(e^{-1} \cdot z\right)} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right)\right)\right)\right) - \left(1656.8104518737205 \cdot \left(\sqrt{\pi} \cdot \left(\frac{\sqrt{2}}{e^{7.5} \cdot e^{-1}} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + 1656.8104518737205 \cdot \left(\sqrt{\pi} \cdot \left(\frac{z \cdot \left(\sqrt{2} \cdot \log 6.5\right)}{e^{-1} \cdot e^{7.5}} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right)\right)}\]
    1.3
  9. Applied simplify to get
    \[\color{red}{\left(676.5203681218851 \cdot \left(\sqrt{\pi} \cdot \left(\frac{\sqrt{2} \cdot \log 6.5}{e^{7.5} \cdot e^{-1}} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + \left(2585.1948787825354 \cdot \left(\sqrt{\pi} \cdot \left(\frac{\sqrt{2} \cdot z}{e^{7.5} \cdot e^{-1}} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + \left(338.26018406094255 \cdot \left(\sqrt{\pi} \cdot \left(\frac{z \cdot \left(\sqrt{2} \cdot {\left(\log 6.5\right)}^2\right)}{e^{-1} \cdot e^{7.5}} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + 676.5203681218851 \cdot \left(\sqrt{\pi} \cdot \left(\frac{\sqrt{2}}{e^{7.5} \cdot \left(e^{-1} \cdot z\right)} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right)\right)\right)\right) - \left(1656.8104518737205 \cdot \left(\sqrt{\pi} \cdot \left(\frac{\sqrt{2}}{e^{7.5} \cdot e^{-1}} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + 1656.8104518737205 \cdot \left(\sqrt{\pi} \cdot \left(\frac{z \cdot \left(\sqrt{2} \cdot \log 6.5\right)}{e^{-1} \cdot e^{7.5}} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right)\right)} \leadsto \color{blue}{(\left(\frac{\sqrt{2}}{e^{-1 + 7.5} \cdot z} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right) * \left(\sqrt{\pi} \cdot 676.5203681218851\right) + \left((\left(338.26018406094255 \cdot \sqrt{\pi}\right) * \left(\frac{\left(\left(\sqrt{2} \cdot z\right) \cdot {\left(\log 6.5\right)}^2\right) \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}}{e^{-1 + 7.5}}\right) + \left(\frac{\left(\left(\sqrt{\pi} \cdot 2585.1948787825354\right) \cdot \left(\sqrt{2} \cdot z\right)\right) \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}}{e^{-1 + 7.5}}\right))_*\right))_* - \left(\left(\sqrt{\pi} \cdot 1656.8104518737205\right) \cdot (\left({\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right) * \left(\frac{\sqrt{2} \cdot \left(z \cdot \log 6.5\right)}{e^{-1 + 7.5}}\right) + \left(\frac{\sqrt{2}}{e^{-1 + 7.5}} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right))_* - \left(\frac{\sqrt{2} \cdot \log 6.5}{e^{-1 + 7.5}} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right) \cdot \left(\sqrt{\pi} \cdot 676.5203681218851\right)\right)}\]
    1.6
  10. Removed slow pow expressions

Original test:


(lambda ((z default))
  #: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)))))