Jmat.Real.gamma, branch z greater than 0.5

Average Error: 61.6 → 1.5

Time: 1.2min

Precision: binary64

• could not determine a ground truth for program body

  1. z = 8.389744943889928e+157

Math

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

↓

\[\left(\left(\left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right) \cdot \left(2581.19179968122216 \cdot \frac{\sqrt{2} \cdot z}{e^{6.5}} + 676.520368121885099 \cdot \frac{\sqrt{2}}{z \cdot e^{6.5}}\right) + 169.130092030471275 \cdot \left(\frac{\sqrt{2} \cdot z}{e^{6.5}} \cdot \left({\left(\frac{1}{{6.5}^{5}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right)\right) + 676.520368121885099 \cdot \left(\sqrt{\pi} \cdot \left(\frac{\log 6.5 \cdot \sqrt{2}}{e^{6.5}} \cdot {\left(\frac{1}{{6.5}^{1}}\right)}^{0.5}\right)\right)\right) + \left(338.260184060942549 \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \left(\frac{{\left(\log 6.5\right)}^{2} \cdot \left(z \cdot \sqrt{2}\right)}{e^{6.5}} \cdot \sqrt{\pi}\right)\right) - \left(\sqrt{\pi} \cdot 1656.8104518737205\right) \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \left(\frac{\log 6.5 \cdot \left(z \cdot \sqrt{2}\right)}{e^{6.5}} + \frac{\sqrt{2}}{e^{6.5}}\right)\right)\right)\]

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. Simplified 61.6

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

3. Taylor expanded around 0 0.9

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

4. Taylor expanded around 0 1.5

   \[\leadsto \color{blue}{\left(338.260184060942549 \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \left(\frac{{\left(\log 6.5\right)}^{2} \cdot \left(z \cdot \sqrt{2}\right)}{e^{6.5}} \cdot \sqrt{\pi}\right)\right) + \left(2581.19179968122216 \cdot \left(\frac{\sqrt{2} \cdot z}{e^{6.5}} \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right) + \left(676.520368121885099 \cdot \left(\frac{\sqrt{2}}{z \cdot e^{6.5}} \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right) + \left(676.520368121885099 \cdot \left(\sqrt{\pi} \cdot \left(\frac{\log 6.5 \cdot \sqrt{2}}{e^{6.5}} \cdot {\left(\frac{1}{{6.5}^{1}}\right)}^{0.5}\right)\right) + 169.130092030471275 \cdot \left(\frac{\sqrt{2} \cdot z}{e^{6.5}} \cdot \left({\left(\frac{1}{{6.5}^{5}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right)\right)\right)\right)\right) - \left(1656.8104518737205 \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \left(\frac{\log 6.5 \cdot \left(z \cdot \sqrt{2}\right)}{e^{6.5}} \cdot \sqrt{\pi}\right)\right) + 1656.8104518737205 \cdot \left(\sqrt{\pi} \cdot \left(\frac{\sqrt{2}}{e^{6.5}} \cdot {\left(\frac{1}{{6.5}^{1}}\right)}^{0.5}\right)\right)\right)}\]

5. Simplified 1.5

   \[\leadsto \color{blue}{338.260184060942549 \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \left(\frac{{\left(\log 6.5\right)}^{2} \cdot \left(z \cdot \sqrt{2}\right)}{e^{6.5}} \cdot \sqrt{\pi}\right)\right) + \left(\left(2581.19179968122216 \cdot \left(\frac{\sqrt{2} \cdot z}{e^{6.5}} \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right) + \left(676.520368121885099 \cdot \left(\frac{\sqrt{2}}{z \cdot e^{6.5}} \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right) + \left(676.520368121885099 \cdot \left(\sqrt{\pi} \cdot \left(\frac{\log 6.5 \cdot \sqrt{2}}{e^{6.5}} \cdot {\left(\frac{1}{{6.5}^{1}}\right)}^{0.5}\right)\right) + 169.130092030471275 \cdot \left(\frac{\sqrt{2} \cdot z}{e^{6.5}} \cdot \left({\left(\frac{1}{{6.5}^{5}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right)\right)\right)\right) - 1656.8104518737205 \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \left(\frac{\log 6.5 \cdot \left(z \cdot \sqrt{2}\right)}{e^{6.5}} \cdot \sqrt{\pi}\right) + \sqrt{\pi} \cdot \left(\frac{\sqrt{2}}{e^{6.5}} \cdot {\left(\frac{1}{{6.5}^{1}}\right)}^{0.5}\right)\right)\right)}\]

6. Simplified 1.5

   \[\leadsto \color{blue}{\left(\left(\left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right) \cdot \left(2581.19179968122216 \cdot \frac{\sqrt{2} \cdot z}{e^{6.5}} + 676.520368121885099 \cdot \frac{\sqrt{2}}{z \cdot e^{6.5}}\right) + 169.130092030471275 \cdot \left(\frac{\sqrt{2} \cdot z}{e^{6.5}} \cdot \left({\left(\frac{1}{{6.5}^{5}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right)\right) + 676.520368121885099 \cdot \left(\sqrt{\pi} \cdot \left(\frac{\log 6.5 \cdot \sqrt{2}}{e^{6.5}} \cdot {\left(\frac{1}{{6.5}^{1}}\right)}^{0.5}\right)\right)\right) + \left(338.260184060942549 \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \left(\frac{{\left(\log 6.5\right)}^{2} \cdot \left(z \cdot \sqrt{2}\right)}{e^{6.5}} \cdot \sqrt{\pi}\right)\right) - \left(\sqrt{\pi} \cdot 1656.8104518737205\right) \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \left(\frac{\log 6.5 \cdot \left(z \cdot \sqrt{2}\right)}{e^{6.5}} + \frac{\sqrt{2}}{e^{6.5}}\right)\right)\right)}\]

7. Final simplification 1.5

   \[\leadsto \left(\left(\left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right) \cdot \left(2581.19179968122216 \cdot \frac{\sqrt{2} \cdot z}{e^{6.5}} + 676.520368121885099 \cdot \frac{\sqrt{2}}{z \cdot e^{6.5}}\right) + 169.130092030471275 \cdot \left(\frac{\sqrt{2} \cdot z}{e^{6.5}} \cdot \left({\left(\frac{1}{{6.5}^{5}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right)\right) + 676.520368121885099 \cdot \left(\sqrt{\pi} \cdot \left(\frac{\log 6.5 \cdot \sqrt{2}}{e^{6.5}} \cdot {\left(\frac{1}{{6.5}^{1}}\right)}^{0.5}\right)\right)\right) + \left(338.260184060942549 \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \left(\frac{{\left(\log 6.5\right)}^{2} \cdot \left(z \cdot \sqrt{2}\right)}{e^{6.5}} \cdot \sqrt{\pi}\right)\right) - \left(\sqrt{\pi} \cdot 1656.8104518737205\right) \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \left(\frac{\log 6.5 \cdot \left(z \cdot \sqrt{2}\right)}{e^{6.5}} + \frac{\sqrt{2}}{e^{6.5}}\right)\right)\right)\]

Reproduce

herbie shell --seed 2020163 
(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)))))