\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8}\right)\]
Test:
Jmat.Real.gamma, branch z greater than 0.5
Bits:
128 bits
Bits error versus z
Time: 1.4 m
Input Error: 28.4
Output Error: 0.2
Log: ⚲
Profile: 🕒
\(\left(2585.1948787825354 \cdot \left(\frac{z \cdot \sqrt{2}}{e^{6.5}} \cdot \left(\sqrt{\pi} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + \left(338.26018406094255 \cdot \left(\frac{z \cdot \left(\sqrt{2} \cdot {\left(\log 6.5\right)}^2\right)}{e^{6.5}} \cdot \left(\sqrt{\pi} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + \left(676.5203681218851 \cdot \left(\frac{\sqrt{2}}{e^{6.5} \cdot z} \cdot \left(\sqrt{\pi} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + 676.5203681218851 \cdot \left(\frac{\sqrt{2} \cdot \log 6.5}{e^{6.5}} \cdot \left(\sqrt{\pi} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right)\right)\right)\right) - \left(1656.8104518737205 \cdot \left(\frac{\sqrt{2}}{e^{6.5}} \cdot \left(\sqrt{\pi} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + 1656.8104518737205 \cdot \left(\frac{z \cdot \left(\sqrt{2} \cdot \log 6.5\right)}{e^{6.5}} \cdot \left(\sqrt{\pi} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right)\right)\)
  1. Started with
    \[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8}\right)\]
    28.4
  2. Applied simplify to get
    \[\color{red}{\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8}\right)} \leadsto \color{blue}{\left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8} + \frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(z - 1\right)}\right) + \left(\frac{-0.13857109526572012}{\left(z - 1\right) + 6} + \frac{12.507343278686905}{\left(5 + z\right) - 1}\right)\right) + \left(\left(\left(\frac{676.5203681218851}{z - 0} + 0.9999999999998099\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \left(\frac{-1259.1392167224028}{z - \left(1 - 2\right)} + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right)\right)\right) \cdot \frac{{\left(\left(7 + z\right) - \left(1 - 0.5\right)\right)}^{\left(0.5 + \left(z - 1\right)\right)} \cdot \sqrt{2 \cdot \pi}}{e^{\left(7 + z\right) - \left(1 - 0.5\right)}}}\]
    16.4
  3. Applied taylor to get
    \[\left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8} + \frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(z - 1\right)}\right) + \left(\frac{-0.13857109526572012}{\left(z - 1\right) + 6} + \frac{12.507343278686905}{\left(5 + z\right) - 1}\right)\right) + \left(\left(\left(\frac{676.5203681218851}{z - 0} + 0.9999999999998099\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \left(\frac{-1259.1392167224028}{z - \left(1 - 2\right)} + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right)\right)\right) \cdot \frac{{\left(\left(7 + z\right) - \left(1 - 0.5\right)\right)}^{\left(0.5 + \left(z - 1\right)\right)} \cdot \sqrt{2 \cdot \pi}}{e^{\left(7 + z\right) - \left(1 - 0.5\right)}} \leadsto \left(2585.1948787825354 \cdot \left(\frac{z \cdot \sqrt{2}}{e^{6.5}} \cdot \left(\sqrt{\pi} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + \left(338.26018406094255 \cdot \left(\frac{z \cdot \left(\sqrt{2} \cdot {\left(\log 6.5\right)}^2\right)}{e^{6.5}} \cdot \left(\sqrt{\pi} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + \left(676.5203681218851 \cdot \left(\frac{\sqrt{2}}{e^{6.5} \cdot z} \cdot \left(\sqrt{\pi} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + 676.5203681218851 \cdot \left(\frac{\sqrt{2} \cdot \log 6.5}{e^{6.5}} \cdot \left(\sqrt{\pi} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right)\right)\right)\right) - \left(1656.8104518737205 \cdot \left(\frac{\sqrt{2}}{e^{6.5}} \cdot \left(\sqrt{\pi} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + 1656.8104518737205 \cdot \left(\frac{z \cdot \left(\sqrt{2} \cdot \log 6.5\right)}{e^{6.5}} \cdot \left(\sqrt{\pi} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right)\right)\]
    0.2
  4. Taylor expanded around 0 to get
    \[\color{red}{\left(2585.1948787825354 \cdot \left(\frac{z \cdot \sqrt{2}}{e^{6.5}} \cdot \left(\sqrt{\pi} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + \left(338.26018406094255 \cdot \left(\frac{z \cdot \left(\sqrt{2} \cdot {\left(\log 6.5\right)}^2\right)}{e^{6.5}} \cdot \left(\sqrt{\pi} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + \left(676.5203681218851 \cdot \left(\frac{\sqrt{2}}{e^{6.5} \cdot z} \cdot \left(\sqrt{\pi} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + 676.5203681218851 \cdot \left(\frac{\sqrt{2} \cdot \log 6.5}{e^{6.5}} \cdot \left(\sqrt{\pi} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right)\right)\right)\right) - \left(1656.8104518737205 \cdot \left(\frac{\sqrt{2}}{e^{6.5}} \cdot \left(\sqrt{\pi} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + 1656.8104518737205 \cdot \left(\frac{z \cdot \left(\sqrt{2} \cdot \log 6.5\right)}{e^{6.5}} \cdot \left(\sqrt{\pi} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right)\right)} \leadsto \color{blue}{\left(2585.1948787825354 \cdot \left(\frac{z \cdot \sqrt{2}}{e^{6.5}} \cdot \left(\sqrt{\pi} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + \left(338.26018406094255 \cdot \left(\frac{z \cdot \left(\sqrt{2} \cdot {\left(\log 6.5\right)}^2\right)}{e^{6.5}} \cdot \left(\sqrt{\pi} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + \left(676.5203681218851 \cdot \left(\frac{\sqrt{2}}{e^{6.5} \cdot z} \cdot \left(\sqrt{\pi} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + 676.5203681218851 \cdot \left(\frac{\sqrt{2} \cdot \log 6.5}{e^{6.5}} \cdot \left(\sqrt{\pi} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right)\right)\right)\right) - \left(1656.8104518737205 \cdot \left(\frac{\sqrt{2}}{e^{6.5}} \cdot \left(\sqrt{\pi} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + 1656.8104518737205 \cdot \left(\frac{z \cdot \left(\sqrt{2} \cdot \log 6.5\right)}{e^{6.5}} \cdot \left(\sqrt{\pi} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right)\right)}\]
    0.2
  5. 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)))))