Average Error: 61.6 → 1.0
Time: 43.6s
Precision: binary64
\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(z - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(z - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(z - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\]
↓
\[\frac{\sqrt{\pi \cdot 2} \cdot \left(\frac{0.99999999999980993 + \left(\frac{676.520368121885099}{z} + \left(\frac{-1259.13921672240281}{2 + \left(z - 1\right)} + \left(\frac{771.32342877765313}{z + \left(3 - 1\right)} + \left(\frac{-176.615029162140587}{z + \left(4 - 1\right)} + \left(\frac{12.5073432786869052}{z + \left(5 - 1\right)} + \left(\frac{-0.138571095265720118}{z + \left(6 - 1\right)} + \left(\frac{9.98436957801957158 \cdot 10^{-6}}{z + \left(7 - 1\right)} + \frac{1.50563273514931162 \cdot 10^{-7}}{z + \left(8 - 1\right)}\right)\right)\right)\right)\right)\right)\right)}{e^{z + 0.5}} \cdot {\left(7 + \left(z + \left(0.5 - 1\right)\right)\right)}^{\left(z + \left(0.5 - 1\right)\right)}\right)}{e^{7 - 1}}\]

Error

Bits error versus z

Derivation

  1. Initial program 61.6

    \[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(z - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(z - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(z - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\]

  2. Simplified1.0

    \[\leadsto \color{blue}{\sqrt{\pi \cdot 2} \cdot \left({\left(z + \left(7 - \left(1 - 0.5\right)\right)\right)}^{\left(z + \left(0.5 - 1\right)\right)} \cdot \frac{0.99999999999980993 + \left(\left(\frac{676.520368121885099}{z} + \left(\frac{-1259.13921672240281}{2 + \left(z - 1\right)} + \left(\frac{771.32342877765313}{z + \left(3 - 1\right)} + \frac{-176.615029162140587}{z + \left(4 - 1\right)}\right)\right)\right) + \left(\frac{12.5073432786869052}{z + \left(5 - 1\right)} + \left(\frac{-0.138571095265720118}{z + \left(6 - 1\right)} + \left(\frac{9.98436957801957158 \cdot 10^{-6}}{z + \left(7 - 1\right)} + \frac{1.50563273514931162 \cdot 10^{-7}}{z + \left(8 - 1\right)}\right)\right)\right)\right)}{e^{z + \left(7 - \left(1 - 0.5\right)\right)}}\right)}\]
  3. Using strategy rm

  4. Applied exp-sum0.9

    \[\leadsto \sqrt{\pi \cdot 2} \cdot \left({\left(z + \left(7 - \left(1 - 0.5\right)\right)\right)}^{\left(z + \left(0.5 - 1\right)\right)} \cdot \frac{0.99999999999980993 + \left(\left(\frac{676.520368121885099}{z} + \left(\frac{-1259.13921672240281}{2 + \left(z - 1\right)} + \left(\frac{771.32342877765313}{z + \left(3 - 1\right)} + \frac{-176.615029162140587}{z + \left(4 - 1\right)}\right)\right)\right) + \left(\frac{12.5073432786869052}{z + \left(5 - 1\right)} + \left(\frac{-0.138571095265720118}{z + \left(6 - 1\right)} + \left(\frac{9.98436957801957158 \cdot 10^{-6}}{z + \left(7 - 1\right)} + \frac{1.50563273514931162 \cdot 10^{-7}}{z + \left(8 - 1\right)}\right)\right)\right)\right)}{\color{blue}{e^{z} \cdot e^{7 - \left(1 - 0.5\right)}}}\right)\]

  5. Applied associate-/r*0.9

    \[\leadsto \sqrt{\pi \cdot 2} \cdot \left({\left(z + \left(7 - \left(1 - 0.5\right)\right)\right)}^{\left(z + \left(0.5 - 1\right)\right)} \cdot \color{blue}{\frac{\frac{0.99999999999980993 + \left(\left(\frac{676.520368121885099}{z} + \left(\frac{-1259.13921672240281}{2 + \left(z - 1\right)} + \left(\frac{771.32342877765313}{z + \left(3 - 1\right)} + \frac{-176.615029162140587}{z + \left(4 - 1\right)}\right)\right)\right) + \left(\frac{12.5073432786869052}{z + \left(5 - 1\right)} + \left(\frac{-0.138571095265720118}{z + \left(6 - 1\right)} + \left(\frac{9.98436957801957158 \cdot 10^{-6}}{z + \left(7 - 1\right)} + \frac{1.50563273514931162 \cdot 10^{-7}}{z + \left(8 - 1\right)}\right)\right)\right)\right)}{e^{z}}}{e^{7 - \left(1 - 0.5\right)}}}\right)\]

  6. Simplified0.9

    \[\leadsto \sqrt{\pi \cdot 2} \cdot \left({\left(z + \left(7 - \left(1 - 0.5\right)\right)\right)}^{\left(z + \left(0.5 - 1\right)\right)} \cdot \frac{\color{blue}{\frac{0.99999999999980993 + \left(\frac{676.520368121885099}{z} + \left(\frac{-1259.13921672240281}{2 + \left(z - 1\right)} + \left(\left(\frac{771.32342877765313}{z + \left(3 - 1\right)} + \frac{-176.615029162140587}{z + \left(4 - 1\right)}\right) + \left(\frac{12.5073432786869052}{z + \left(5 - 1\right)} + \left(\frac{-0.138571095265720118}{z + \left(6 - 1\right)} + \left(\frac{9.98436957801957158 \cdot 10^{-6}}{z + \left(7 - 1\right)} + \frac{1.50563273514931162 \cdot 10^{-7}}{z + \left(8 - 1\right)}\right)\right)\right)\right)\right)\right)}{e^{z}}}}{e^{7 - \left(1 - 0.5\right)}}\right)\]
  7. Using strategy rm

  8. Applied associate-*r/0.9

    \[\leadsto \sqrt{\pi \cdot 2} \cdot \color{blue}{\frac{{\left(z + \left(7 - \left(1 - 0.5\right)\right)\right)}^{\left(z + \left(0.5 - 1\right)\right)} \cdot \frac{0.99999999999980993 + \left(\frac{676.520368121885099}{z} + \left(\frac{-1259.13921672240281}{2 + \left(z - 1\right)} + \left(\left(\frac{771.32342877765313}{z + \left(3 - 1\right)} + \frac{-176.615029162140587}{z + \left(4 - 1\right)}\right) + \left(\frac{12.5073432786869052}{z + \left(5 - 1\right)} + \left(\frac{-0.138571095265720118}{z + \left(6 - 1\right)} + \left(\frac{9.98436957801957158 \cdot 10^{-6}}{z + \left(7 - 1\right)} + \frac{1.50563273514931162 \cdot 10^{-7}}{z + \left(8 - 1\right)}\right)\right)\right)\right)\right)\right)}{e^{z}}}{e^{7 - \left(1 - 0.5\right)}}}\]

  9. Applied associate-*r/0.9

    \[\leadsto \color{blue}{\frac{\sqrt{\pi \cdot 2} \cdot \left({\left(z + \left(7 - \left(1 - 0.5\right)\right)\right)}^{\left(z + \left(0.5 - 1\right)\right)} \cdot \frac{0.99999999999980993 + \left(\frac{676.520368121885099}{z} + \left(\frac{-1259.13921672240281}{2 + \left(z - 1\right)} + \left(\left(\frac{771.32342877765313}{z + \left(3 - 1\right)} + \frac{-176.615029162140587}{z + \left(4 - 1\right)}\right) + \left(\frac{12.5073432786869052}{z + \left(5 - 1\right)} + \left(\frac{-0.138571095265720118}{z + \left(6 - 1\right)} + \left(\frac{9.98436957801957158 \cdot 10^{-6}}{z + \left(7 - 1\right)} + \frac{1.50563273514931162 \cdot 10^{-7}}{z + \left(8 - 1\right)}\right)\right)\right)\right)\right)\right)}{e^{z}}\right)}{e^{7 - \left(1 - 0.5\right)}}}\]

  10. Simplified0.9

    \[\leadsto \frac{\color{blue}{\sqrt{\pi \cdot 2} \cdot \left(\frac{0.99999999999980993 + \left(\frac{676.520368121885099}{z} + \left(\frac{-1259.13921672240281}{2 + \left(z - 1\right)} + \left(\frac{771.32342877765313}{z + \left(3 - 1\right)} + \left(\frac{-176.615029162140587}{z + \left(4 - 1\right)} + \left(\frac{12.5073432786869052}{z + \left(5 - 1\right)} + \left(\frac{-0.138571095265720118}{z + \left(6 - 1\right)} + \left(\frac{9.98436957801957158 \cdot 10^{-6}}{z + \left(7 - 1\right)} + \frac{1.50563273514931162 \cdot 10^{-7}}{z + \left(8 - 1\right)}\right)\right)\right)\right)\right)\right)\right)}{e^{z}} \cdot {\left(z + \left(7 + \left(0.5 - 1\right)\right)\right)}^{\left(z + \left(0.5 - 1\right)\right)}\right)}}{e^{7 - \left(1 - 0.5\right)}}\]
  11. Using strategy rm

  12. Applied associate--r-0.9

    \[\leadsto \frac{\sqrt{\pi \cdot 2} \cdot \left(\frac{0.99999999999980993 + \left(\frac{676.520368121885099}{z} + \left(\frac{-1259.13921672240281}{2 + \left(z - 1\right)} + \left(\frac{771.32342877765313}{z + \left(3 - 1\right)} + \left(\frac{-176.615029162140587}{z + \left(4 - 1\right)} + \left(\frac{12.5073432786869052}{z + \left(5 - 1\right)} + \left(\frac{-0.138571095265720118}{z + \left(6 - 1\right)} + \left(\frac{9.98436957801957158 \cdot 10^{-6}}{z + \left(7 - 1\right)} + \frac{1.50563273514931162 \cdot 10^{-7}}{z + \left(8 - 1\right)}\right)\right)\right)\right)\right)\right)\right)}{e^{z}} \cdot {\left(z + \left(7 + \left(0.5 - 1\right)\right)\right)}^{\left(z + \left(0.5 - 1\right)\right)}\right)}{e^{\color{blue}{\left(7 - 1\right) + 0.5}}}\]

  13. Applied exp-sum0.9

    \[\leadsto \frac{\sqrt{\pi \cdot 2} \cdot \left(\frac{0.99999999999980993 + \left(\frac{676.520368121885099}{z} + \left(\frac{-1259.13921672240281}{2 + \left(z - 1\right)} + \left(\frac{771.32342877765313}{z + \left(3 - 1\right)} + \left(\frac{-176.615029162140587}{z + \left(4 - 1\right)} + \left(\frac{12.5073432786869052}{z + \left(5 - 1\right)} + \left(\frac{-0.138571095265720118}{z + \left(6 - 1\right)} + \left(\frac{9.98436957801957158 \cdot 10^{-6}}{z + \left(7 - 1\right)} + \frac{1.50563273514931162 \cdot 10^{-7}}{z + \left(8 - 1\right)}\right)\right)\right)\right)\right)\right)\right)}{e^{z}} \cdot {\left(z + \left(7 + \left(0.5 - 1\right)\right)\right)}^{\left(z + \left(0.5 - 1\right)\right)}\right)}{\color{blue}{e^{7 - 1} \cdot e^{0.5}}}\]

  14. Applied times-frac0.9

    \[\leadsto \color{blue}{\frac{\sqrt{\pi \cdot 2}}{e^{7 - 1}} \cdot \frac{\frac{0.99999999999980993 + \left(\frac{676.520368121885099}{z} + \left(\frac{-1259.13921672240281}{2 + \left(z - 1\right)} + \left(\frac{771.32342877765313}{z + \left(3 - 1\right)} + \left(\frac{-176.615029162140587}{z + \left(4 - 1\right)} + \left(\frac{12.5073432786869052}{z + \left(5 - 1\right)} + \left(\frac{-0.138571095265720118}{z + \left(6 - 1\right)} + \left(\frac{9.98436957801957158 \cdot 10^{-6}}{z + \left(7 - 1\right)} + \frac{1.50563273514931162 \cdot 10^{-7}}{z + \left(8 - 1\right)}\right)\right)\right)\right)\right)\right)\right)}{e^{z}} \cdot {\left(z + \left(7 + \left(0.5 - 1\right)\right)\right)}^{\left(z + \left(0.5 - 1\right)\right)}}{e^{0.5}}}\]

  15. Simplified0.9

    \[\leadsto \frac{\sqrt{\pi \cdot 2}}{e^{7 - 1}} \cdot \color{blue}{\frac{\left(0.99999999999980993 + \left(\frac{676.520368121885099}{z} + \left(\frac{-1259.13921672240281}{2 + \left(z - 1\right)} + \left(\frac{771.32342877765313}{z + \left(3 - 1\right)} + \left(\frac{-176.615029162140587}{z + \left(4 - 1\right)} + \left(\frac{12.5073432786869052}{z + \left(5 - 1\right)} + \left(\frac{-0.138571095265720118}{z + \left(6 - 1\right)} + \left(\frac{9.98436957801957158 \cdot 10^{-6}}{z + \left(7 - 1\right)} + \frac{1.50563273514931162 \cdot 10^{-7}}{z + \left(8 - 1\right)}\right)\right)\right)\right)\right)\right)\right)\right) \cdot {\left(z + \left(7 + \left(0.5 - 1\right)\right)\right)}^{\left(z + \left(0.5 - 1\right)\right)}}{e^{z + 0.5}}}\]
  16. Using strategy rm

  17. Applied associate-*l/1.0

    \[\leadsto \color{blue}{\frac{\sqrt{\pi \cdot 2} \cdot \frac{\left(0.99999999999980993 + \left(\frac{676.520368121885099}{z} + \left(\frac{-1259.13921672240281}{2 + \left(z - 1\right)} + \left(\frac{771.32342877765313}{z + \left(3 - 1\right)} + \left(\frac{-176.615029162140587}{z + \left(4 - 1\right)} + \left(\frac{12.5073432786869052}{z + \left(5 - 1\right)} + \left(\frac{-0.138571095265720118}{z + \left(6 - 1\right)} + \left(\frac{9.98436957801957158 \cdot 10^{-6}}{z + \left(7 - 1\right)} + \frac{1.50563273514931162 \cdot 10^{-7}}{z + \left(8 - 1\right)}\right)\right)\right)\right)\right)\right)\right)\right) \cdot {\left(z + \left(7 + \left(0.5 - 1\right)\right)\right)}^{\left(z + \left(0.5 - 1\right)\right)}}{e^{z + 0.5}}}{e^{7 - 1}}}\]

  18. Simplified1.0

    \[\leadsto \frac{\color{blue}{\sqrt{\pi \cdot 2} \cdot \left(\frac{0.99999999999980993 + \left(\frac{676.520368121885099}{z} + \left(\frac{-1259.13921672240281}{2 + \left(z - 1\right)} + \left(\frac{771.32342877765313}{z + \left(3 - 1\right)} + \left(\frac{-176.615029162140587}{z + \left(4 - 1\right)} + \left(\frac{12.5073432786869052}{z + \left(5 - 1\right)} + \left(\frac{-0.138571095265720118}{z + \left(6 - 1\right)} + \left(\frac{9.98436957801957158 \cdot 10^{-6}}{\left(7 - 1\right) + z} + \frac{1.50563273514931162 \cdot 10^{-7}}{z + \left(8 - 1\right)}\right)\right)\right)\right)\right)\right)\right)}{e^{z + 0.5}} \cdot {\left(7 + \left(z + \left(0.5 - 1\right)\right)\right)}^{\left(z + \left(0.5 - 1\right)\right)}\right)}}{e^{7 - 1}}\]

  19. Final simplification1.0

    \[\leadsto \frac{\sqrt{\pi \cdot 2} \cdot \left(\frac{0.99999999999980993 + \left(\frac{676.520368121885099}{z} + \left(\frac{-1259.13921672240281}{2 + \left(z - 1\right)} + \left(\frac{771.32342877765313}{z + \left(3 - 1\right)} + \left(\frac{-176.615029162140587}{z + \left(4 - 1\right)} + \left(\frac{12.5073432786869052}{z + \left(5 - 1\right)} + \left(\frac{-0.138571095265720118}{z + \left(6 - 1\right)} + \left(\frac{9.98436957801957158 \cdot 10^{-6}}{z + \left(7 - 1\right)} + \frac{1.50563273514931162 \cdot 10^{-7}}{z + \left(8 - 1\right)}\right)\right)\right)\right)\right)\right)\right)}{e^{z + 0.5}} \cdot {\left(7 + \left(z + \left(0.5 - 1\right)\right)\right)}^{\left(z + \left(0.5 - 1\right)\right)}\right)}{e^{7 - 1}}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z greater than 0.5"
  :precision binary64
  (* (* (* (sqrt (* PI 2.0)) (pow (+ (+ (- z 1.0) 7.0) 0.5) (+ (- z 1.0) 0.5))) (exp (neg (+ (+ (- z 1.0) 7.0) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- z 1.0) 1.0))) (/ -1259.1392167224028 (+ (- z 1.0) 2.0))) (/ 771.3234287776531 (+ (- z 1.0) 3.0))) (/ -176.6150291621406 (+ (- z 1.0) 4.0))) (/ 12.507343278686905 (+ (- z 1.0) 5.0))) (/ -0.13857109526572012 (+ (- z 1.0) 6.0))) (/ 9.984369578019572e-06 (+ (- z 1.0) 7.0))) (/ 1.5056327351493116e-07 (+ (- z 1.0) 8.0)))))