Jmat.Real.gamma, branch z greater than 0.5

Average Error: 61.6 → 1.0

Time: 4.7min

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

• could not determine a ground truth for program body

  1. z = 1.770577532995958e+251

Math

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

↓

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

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 1.0

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

4. Applied associate-*r/ 1.1

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

6. Applied associate-/l* 1.0

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

8. Applied associate-/r/ 1.0

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

9. Final simplification 1.0

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

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)))))