Average Error: 61.6 → 1.4
Time: 20.7s
Precision: 64
\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(z - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(z - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(z - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\]
↓
\[169.130092030471275 \cdot \left(\frac{\sqrt{2} \cdot z}{e^{6.5}} \cdot \left({\left(\frac{1}{{6.5}^{5}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right) + {\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \left(\left(\left(\frac{{\left(\log 6.5\right)}^{2} \cdot \left(z \cdot \sqrt{2}\right)}{e^{6.5}} \cdot \sqrt{\pi}\right) \cdot 338.260184060942549 - \left(\frac{\log 6.5 \cdot \left(z \cdot \sqrt{2}\right)}{e^{6.5}} + \frac{\sqrt{2}}{e^{6.5}}\right) \cdot \left(\sqrt{\pi} \cdot 1656.8104518737205\right)\right) + \left(\left(676.520368121885099 \cdot \sqrt{\pi}\right) \cdot \frac{\log 6.5 \cdot \sqrt{2}}{e^{6.5}} + \sqrt{\pi} \cdot \left(2581.19179968122216 \cdot \frac{\sqrt{2} \cdot z}{e^{6.5}} + 676.520368121885099 \cdot \frac{\sqrt{2}}{z \cdot e^{6.5}}\right)\right)\right)\]
\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)
↓
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) + {\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \left(\left(\left(\frac{{\left(\log 6.5\right)}^{2} \cdot \left(z \cdot \sqrt{2}\right)}{e^{6.5}} \cdot \sqrt{\pi}\right) \cdot 338.260184060942549 - \left(\frac{\log 6.5 \cdot \left(z \cdot \sqrt{2}\right)}{e^{6.5}} + \frac{\sqrt{2}}{e^{6.5}}\right) \cdot \left(\sqrt{\pi} \cdot 1656.8104518737205\right)\right) + \left(\left(676.520368121885099 \cdot \sqrt{\pi}\right) \cdot \frac{\log 6.5 \cdot \sqrt{2}}{e^{6.5}} + \sqrt{\pi} \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)\right)\right)
double code(double z) {
	return (((sqrt((((double) M_PI) * 2.0)) * pow((((z - 1.0) + 7.0) + 0.5), ((z - 1.0) + 0.5))) * exp(-(((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))));
}
↓
double code(double z) {
	return ((169.13009203047127 * (((sqrt(2.0) * z) / exp(6.5)) * (pow((1.0 / pow(6.5, 5.0)), 0.5) * sqrt(((double) M_PI))))) + (pow((1.0 / pow(6.5, 1.0)), 0.5) * ((((((pow(log(6.5), 2.0) * (z * sqrt(2.0))) / exp(6.5)) * sqrt(((double) M_PI))) * 338.26018406094255) - ((((log(6.5) * (z * sqrt(2.0))) / exp(6.5)) + (sqrt(2.0) / exp(6.5))) * (sqrt(((double) M_PI)) * 1656.8104518737205))) + (((676.5203681218851 * sqrt(((double) M_PI))) * ((log(6.5) * sqrt(2.0)) / exp(6.5))) + (sqrt(((double) M_PI)) * ((2581.191799681222 * ((sqrt(2.0) * z) / exp(6.5))) + (676.5203681218851 * (sqrt(2.0) / (z * exp(6.5))))))))));
}

Error

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Simplified61.6

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

  3. Taylor expanded around 0 1.5

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

  4. Simplified1.5

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

  5. Simplified1.5

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

  6. Simplified1.4

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

  7. Simplified1.4

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

  8. Final simplification1.4

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

Reproduce

herbie shell --seed 2020092 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z greater than 0.5"
  :precision binary64
  (* (* (* (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)))))