Average Error: 59.9 → 0.7
Time: 3.0m
Precision: 64
\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8}\right)\]
↓
\[\left(\left(\left(\frac{676.5203681218851}{z} + \left(\left(\frac{-1259.1392167224028}{1 + z} + \frac{771.3234287776531}{z + 2}\right) + \frac{-176.6150291621406}{z - -3}\right)\right) + 0.9999999999998099\right) + \left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{7 + z} + \frac{9.984369578019572 \cdot 10^{-06}}{z + 6}\right) + \left(\frac{-0.13857109526572012}{\left(z + 6\right) - 1} + \frac{12.507343278686905}{4 + z}\right)\right)\right) \cdot \left(\frac{\sqrt{\pi \cdot 2} \cdot {\left(\left(0.5 + z\right) - -6\right)}^{\left(0.5 + \left(z - 1\right)\right)}}{e^{0.5 + z}} \cdot e^{-6}\right)\]
\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(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8}\right)
↓
\left(\left(\left(\frac{676.5203681218851}{z} + \left(\left(\frac{-1259.1392167224028}{1 + z} + \frac{771.3234287776531}{z + 2}\right) + \frac{-176.6150291621406}{z - -3}\right)\right) + 0.9999999999998099\right) + \left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{7 + z} + \frac{9.984369578019572 \cdot 10^{-06}}{z + 6}\right) + \left(\frac{-0.13857109526572012}{\left(z + 6\right) - 1} + \frac{12.507343278686905}{4 + z}\right)\right)\right) \cdot \left(\frac{\sqrt{\pi \cdot 2} \cdot {\left(\left(0.5 + z\right) - -6\right)}^{\left(0.5 + \left(z - 1\right)\right)}}{e^{0.5 + z}} \cdot e^{-6}\right)
double f(double z) {
        double r7023785 = atan2(1.0, 0.0);
        double r7023786 = 2.0;
        double r7023787 = r7023785 * r7023786;
        double r7023788 = sqrt(r7023787);
        double r7023789 = z;
        double r7023790 = 1.0;
        double r7023791 = r7023789 - r7023790;
        double r7023792 = 7.0;
        double r7023793 = r7023791 + r7023792;
        double r7023794 = 0.5;
        double r7023795 = r7023793 + r7023794;
        double r7023796 = r7023791 + r7023794;
        double r7023797 = pow(r7023795, r7023796);
        double r7023798 = r7023788 * r7023797;
        double r7023799 = -r7023795;
        double r7023800 = exp(r7023799);
        double r7023801 = r7023798 * r7023800;
        double r7023802 = 0.9999999999998099;
        double r7023803 = 676.5203681218851;
        double r7023804 = r7023791 + r7023790;
        double r7023805 = r7023803 / r7023804;
        double r7023806 = r7023802 + r7023805;
        double r7023807 = -1259.1392167224028;
        double r7023808 = r7023791 + r7023786;
        double r7023809 = r7023807 / r7023808;
        double r7023810 = r7023806 + r7023809;
        double r7023811 = 771.3234287776531;
        double r7023812 = 3.0;
        double r7023813 = r7023791 + r7023812;
        double r7023814 = r7023811 / r7023813;
        double r7023815 = r7023810 + r7023814;
        double r7023816 = -176.6150291621406;
        double r7023817 = 4.0;
        double r7023818 = r7023791 + r7023817;
        double r7023819 = r7023816 / r7023818;
        double r7023820 = r7023815 + r7023819;
        double r7023821 = 12.507343278686905;
        double r7023822 = 5.0;
        double r7023823 = r7023791 + r7023822;
        double r7023824 = r7023821 / r7023823;
        double r7023825 = r7023820 + r7023824;
        double r7023826 = -0.13857109526572012;
        double r7023827 = 6.0;
        double r7023828 = r7023791 + r7023827;
        double r7023829 = r7023826 / r7023828;
        double r7023830 = r7023825 + r7023829;
        double r7023831 = 9.984369578019572e-06;
        double r7023832 = r7023831 / r7023793;
        double r7023833 = r7023830 + r7023832;
        double r7023834 = 1.5056327351493116e-07;
        double r7023835 = 8.0;
        double r7023836 = r7023791 + r7023835;
        double r7023837 = r7023834 / r7023836;
        double r7023838 = r7023833 + r7023837;
        double r7023839 = r7023801 * r7023838;
        return r7023839;
}
↓
double f(double z) {
        double r7023840 = 676.5203681218851;
        double r7023841 = z;
        double r7023842 = r7023840 / r7023841;
        double r7023843 = -1259.1392167224028;
        double r7023844 = 1.0;
        double r7023845 = r7023844 + r7023841;
        double r7023846 = r7023843 / r7023845;
        double r7023847 = 771.3234287776531;
        double r7023848 = 2.0;
        double r7023849 = r7023841 + r7023848;
        double r7023850 = r7023847 / r7023849;
        double r7023851 = r7023846 + r7023850;
        double r7023852 = -176.6150291621406;
        double r7023853 = -3.0;
        double r7023854 = r7023841 - r7023853;
        double r7023855 = r7023852 / r7023854;
        double r7023856 = r7023851 + r7023855;
        double r7023857 = r7023842 + r7023856;
        double r7023858 = 0.9999999999998099;
        double r7023859 = r7023857 + r7023858;
        double r7023860 = 1.5056327351493116e-07;
        double r7023861 = 7.0;
        double r7023862 = r7023861 + r7023841;
        double r7023863 = r7023860 / r7023862;
        double r7023864 = 9.984369578019572e-06;
        double r7023865 = 6.0;
        double r7023866 = r7023841 + r7023865;
        double r7023867 = r7023864 / r7023866;
        double r7023868 = r7023863 + r7023867;
        double r7023869 = -0.13857109526572012;
        double r7023870 = r7023866 - r7023844;
        double r7023871 = r7023869 / r7023870;
        double r7023872 = 12.507343278686905;
        double r7023873 = 4.0;
        double r7023874 = r7023873 + r7023841;
        double r7023875 = r7023872 / r7023874;
        double r7023876 = r7023871 + r7023875;
        double r7023877 = r7023868 + r7023876;
        double r7023878 = r7023859 + r7023877;
        double r7023879 = atan2(1.0, 0.0);
        double r7023880 = r7023879 * r7023848;
        double r7023881 = sqrt(r7023880);
        double r7023882 = 0.5;
        double r7023883 = r7023882 + r7023841;
        double r7023884 = -6.0;
        double r7023885 = r7023883 - r7023884;
        double r7023886 = r7023841 - r7023844;
        double r7023887 = r7023882 + r7023886;
        double r7023888 = pow(r7023885, r7023887);
        double r7023889 = r7023881 * r7023888;
        double r7023890 = exp(r7023883);
        double r7023891 = r7023889 / r7023890;
        double r7023892 = exp(r7023884);
        double r7023893 = r7023891 * r7023892;
        double r7023894 = r7023878 * r7023893;
        return r7023894;
}

Error

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 59.9

    \[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8}\right)\]

  2. Simplified1.0

    \[\leadsto \color{blue}{\frac{{\left(\left(z + 0.5\right) - -6\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot \sqrt{\pi \cdot 2}}{e^{\left(z + 0.5\right) - -6}} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(6 + z\right) - 1} + \frac{12.507343278686905}{z + 4}\right) + \left(\frac{9.984369578019572 \cdot 10^{-06}}{6 + z} + \frac{1.5056327351493116 \cdot 10^{-07}}{7 + z}\right)\right) + \left(\left(\frac{676.5203681218851}{z} + \left(\frac{-176.6150291621406}{z - -3} + \left(\frac{771.3234287776531}{2 + z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + 0.9999999999998099\right)\right)}\]
  3. Using strategy rm

  4. Applied exp-diff1.0

    \[\leadsto \frac{{\left(\left(z + 0.5\right) - -6\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot \sqrt{\pi \cdot 2}}{\color{blue}{\frac{e^{z + 0.5}}{e^{-6}}}} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(6 + z\right) - 1} + \frac{12.507343278686905}{z + 4}\right) + \left(\frac{9.984369578019572 \cdot 10^{-06}}{6 + z} + \frac{1.5056327351493116 \cdot 10^{-07}}{7 + z}\right)\right) + \left(\left(\frac{676.5203681218851}{z} + \left(\frac{-176.6150291621406}{z - -3} + \left(\frac{771.3234287776531}{2 + z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + 0.9999999999998099\right)\right)\]

  5. Applied associate-/r/0.7

    \[\leadsto \color{blue}{\left(\frac{{\left(\left(z + 0.5\right) - -6\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot \sqrt{\pi \cdot 2}}{e^{z + 0.5}} \cdot e^{-6}\right)} \cdot \left(\left(\left(\frac{-0.13857109526572012}{\left(6 + z\right) - 1} + \frac{12.507343278686905}{z + 4}\right) + \left(\frac{9.984369578019572 \cdot 10^{-06}}{6 + z} + \frac{1.5056327351493116 \cdot 10^{-07}}{7 + z}\right)\right) + \left(\left(\frac{676.5203681218851}{z} + \left(\frac{-176.6150291621406}{z - -3} + \left(\frac{771.3234287776531}{2 + z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right) + 0.9999999999998099\right)\right)\]

  6. Final simplification0.7

    \[\leadsto \left(\left(\left(\frac{676.5203681218851}{z} + \left(\left(\frac{-1259.1392167224028}{1 + z} + \frac{771.3234287776531}{z + 2}\right) + \frac{-176.6150291621406}{z - -3}\right)\right) + 0.9999999999998099\right) + \left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{7 + z} + \frac{9.984369578019572 \cdot 10^{-06}}{z + 6}\right) + \left(\frac{-0.13857109526572012}{\left(z + 6\right) - 1} + \frac{12.507343278686905}{4 + z}\right)\right)\right) \cdot \left(\frac{\sqrt{\pi \cdot 2} \cdot {\left(\left(0.5 + z\right) - -6\right)}^{\left(0.5 + \left(z - 1\right)\right)}}{e^{0.5 + z}} \cdot e^{-6}\right)\]

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (z)
  :name "Jmat.Real.gamma, branch z greater than 0.5"
  (* (* (* (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)))))