Average Error: 61.6 → 1.0
Time: 26.6s
Precision: 64
\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(z - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(z - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(z - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\]
↓
\[\left(\left(169.130092030471275 \cdot \left(\left(\sqrt{2} \cdot \left(z \cdot e^{-6.5}\right)\right) \cdot \left({\left(\frac{1}{{6.5}^{5}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right) + \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right) \cdot \left(676.520368121885099 \cdot \frac{\sqrt{2} \cdot e^{-6.5}}{z} + 2581.19179968122216 \cdot \left(z \cdot \left(\sqrt{2} \cdot e^{-6.5}\right)\right)\right)\right) + 676.520368121885099 \cdot \left(\left(\log 6.5 \cdot \left(\sqrt{2} \cdot e^{-6.5}\right)\right) \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right)\right) - \left(\left(1656.8104518737205 \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right) \cdot \left(\log 6.5 \cdot \left(z \cdot \left(\sqrt{2} \cdot e^{-6.5}\right)\right) + \sqrt{2} \cdot e^{-6.5}\right) - \left(338.260184060942549 \cdot \left({\left(\log 6.5\right)}^{2} \cdot \left(z \cdot \left(\sqrt{2} \cdot e^{-6.5}\right)\right)\right)\right) \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right)\]
\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)
↓
\left(\left(169.130092030471275 \cdot \left(\left(\sqrt{2} \cdot \left(z \cdot e^{-6.5}\right)\right) \cdot \left({\left(\frac{1}{{6.5}^{5}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right) + \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right) \cdot \left(676.520368121885099 \cdot \frac{\sqrt{2} \cdot e^{-6.5}}{z} + 2581.19179968122216 \cdot \left(z \cdot \left(\sqrt{2} \cdot e^{-6.5}\right)\right)\right)\right) + 676.520368121885099 \cdot \left(\left(\log 6.5 \cdot \left(\sqrt{2} \cdot e^{-6.5}\right)\right) \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right)\right) - \left(\left(1656.8104518737205 \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right) \cdot \left(\log 6.5 \cdot \left(z \cdot \left(\sqrt{2} \cdot e^{-6.5}\right)\right) + \sqrt{2} \cdot e^{-6.5}\right) - \left(338.260184060942549 \cdot \left({\left(\log 6.5\right)}^{2} \cdot \left(z \cdot \left(\sqrt{2} \cdot e^{-6.5}\right)\right)\right)\right) \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\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) * sqrt(((double) M_PI))) * ((676.5203681218851 * ((sqrt(2.0) * exp(-6.5)) / z)) + (2581.191799681222 * (z * (sqrt(2.0) * exp(-6.5))))))) + (676.5203681218851 * ((log(6.5) * (sqrt(2.0) * exp(-6.5))) * (pow((1.0 / pow(6.5, 1.0)), 0.5) * sqrt(((double) M_PI)))))) - (((1656.8104518737205 * (pow((1.0 / pow(6.5, 1.0)), 0.5) * sqrt(((double) M_PI)))) * ((log(6.5) * (z * (sqrt(2.0) * exp(-6.5)))) + (sqrt(2.0) * exp(-6.5)))) - ((338.26018406094255 * (pow(log(6.5), 2.0) * (z * (sqrt(2.0) * exp(-6.5))))) * (pow((1.0 / pow(6.5, 1.0)), 0.5) * sqrt(((double) M_PI))))));
}

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. Using strategy rm

  3. Applied *-un-lft-identity61.6

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

  4. Applied *-un-lft-identity61.6

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

  5. Applied distribute-lft-out61.6

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

  6. Simplified0.9

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

  7. Taylor expanded around 0 1.0

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

  8. Simplified1.0

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

  9. Simplified1.0

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

  10. Final simplification1.0

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

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(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)))))