Average Error: 59.9 → 1.1
Time: 9.9m
Precision: 64
Internal Precision: 128
\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8}\right)\]
↓
\[\left(\sqrt{2 \cdot \pi} \cdot {\left(0.5 + \left(z - -6\right)\right)}^{z}\right) \cdot \left(\left(\frac{{\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}}{\frac{z \cdot e^{6.5}}{676.5203681218851}} + {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \left(\frac{z \cdot 2473.108663945773}{e^{6.5}} - \frac{1656.8104518737207}{e^{6.5}}\right)\right) + \left({\left(\frac{1}{{6.5}^{7.0}}\right)}^{0.5} \cdot \frac{z \cdot 1099.3455981980633}{e^{6.5}} + \frac{26.020014158534043 \cdot z}{e^{6.5}} \cdot {\left(\frac{1}{{6.5}^{3.0}}\right)}^{0.5}\right)\right)\]

Error

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  4. Applied associate-+l-1.2

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

  5. Applied pow-sub1.2

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

  6. Applied associate-*l/1.2

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

  7. Applied associate-/l/0.9

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

  9. Applied div-inv0.9

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

  10. Applied associate-*l*0.9

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

  11. Taylor expanded around 0 1.2

    \[\leadsto \left({\left(\left(z - -6\right) + 0.5\right)}^{z} \cdot \sqrt{\pi \cdot 2}\right) \cdot \color{blue}{\left(\left(1099.3455981980633 \cdot \left({\left(\frac{1}{{6.5}^{7.0}}\right)}^{0.5} \cdot \frac{z}{e^{6.5}}\right) + \left(26.020014158534043 \cdot \left({\left(\frac{1}{{6.5}^{3.0}}\right)}^{0.5} \cdot \frac{z}{e^{6.5}}\right) + \left(676.5203681218851 \cdot \left({\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \frac{1}{e^{6.5} \cdot z}\right) + 2473.108663945773 \cdot \left({\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \frac{z}{e^{6.5}}\right)\right)\right)\right) - 1656.8104518737207 \cdot \left({\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \frac{1}{e^{6.5}}\right)\right)}\]

  12. Simplified1.1

    \[\leadsto \left({\left(\left(z - -6\right) + 0.5\right)}^{z} \cdot \sqrt{\pi \cdot 2}\right) \cdot \color{blue}{\left(\left(\frac{z \cdot 26.020014158534043}{e^{6.5}} \cdot {\left(\frac{1}{{6.5}^{3.0}}\right)}^{0.5} + \frac{1099.3455981980633 \cdot z}{e^{6.5}} \cdot {\left(\frac{1}{{6.5}^{7.0}}\right)}^{0.5}\right) + \left(\frac{{\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}}{\frac{e^{6.5} \cdot z}{676.5203681218851}} + {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \left(\frac{2473.108663945773 \cdot z}{e^{6.5}} - \frac{1 \cdot 1656.8104518737207}{e^{6.5}}\right)\right)\right)}\]

  13. Final simplification1.1

    \[\leadsto \left(\sqrt{2 \cdot \pi} \cdot {\left(0.5 + \left(z - -6\right)\right)}^{z}\right) \cdot \left(\left(\frac{{\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}}{\frac{z \cdot e^{6.5}}{676.5203681218851}} + {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \left(\frac{z \cdot 2473.108663945773}{e^{6.5}} - \frac{1656.8104518737207}{e^{6.5}}\right)\right) + \left({\left(\frac{1}{{6.5}^{7.0}}\right)}^{0.5} \cdot \frac{z \cdot 1099.3455981980633}{e^{6.5}} + \frac{26.020014158534043 \cdot z}{e^{6.5}} \cdot {\left(\frac{1}{{6.5}^{3.0}}\right)}^{0.5}\right)\right)\]

Reproduce

herbie shell --seed 2019026 
(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)))))