Average Error: 61.8 → 1.0
Time: 47.9s
Precision: binary64
\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(z - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(z - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(z - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\]
↓
\[\frac{\sqrt{\pi \cdot 2} \cdot \left(\frac{-0.138571095265720118}{z + \left(6 - 1\right)} + \left(\frac{9.98436957801957158 \cdot 10^{-6}}{z + \left(7 - 1\right)} + \left(\frac{1.50563273514931162 \cdot 10^{-7}}{z + \left(8 - 1\right)} + \left(\left(\frac{-1259.13921672240281}{2 + \left(z - 1\right)} + \left(\frac{771.32342877765313}{z + \left(3 - 1\right)} + \left(\frac{-176.615029162140587}{z + \left(4 - 1\right)} + \left(\frac{12.5073432786869052}{z + \left(5 - 1\right)} + \frac{676.520368121885099}{z}\right)\right)\right)\right) + 0.99999999999980993\right)\right)\right)\right)}{\frac{e^{z + \left(\left(7 - 1\right) + 0.5\right)}}{{\left(7 + \left(\left(z - 1\right) + 0.5\right)\right)}^{\left(\left(z - 1\right) + 0.5\right)}}}\]

Error

Bits error versus z

Derivation

  1. Initial program 61.8

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

  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 associate-*r/1.0

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

  5. Applied associate-*r/1.0

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

  6. Simplified1.0

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

  8. Applied exp-sum0.9

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

  9. Applied times-frac0.9

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

  10. Simplified0.9

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

  12. Applied frac-times0.9

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

  13. Simplified0.9

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

  14. Simplified1.0

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

  15. Final simplification1.0

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

Reproduce

herbie shell --seed 2020179 
(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)))))