Average Error: 1.8 → 1.8
Time: 50.3s
Precision: 64
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531346025876700878143310547}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.50734327868690520801919774385169148445}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.1385710952657201178173096423051902092993}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531346025876700878143310547}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.50734327868690520801919774385169148445}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.1385710952657201178173096423051902092993}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531346025876700878143310547}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.50734327868690520801919774385169148445}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.1385710952657201178173096423051902092993}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)
↓
\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531346025876700878143310547}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.50734327868690520801919774385169148445}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.1385710952657201178173096423051902092993}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)
double f(double z) {
        double r174803 = atan2(1.0, 0.0);
        double r174804 = z;
        double r174805 = r174803 * r174804;
        double r174806 = sin(r174805);
        double r174807 = r174803 / r174806;
        double r174808 = 2.0;
        double r174809 = r174803 * r174808;
        double r174810 = sqrt(r174809);
        double r174811 = 1.0;
        double r174812 = r174811 - r174804;
        double r174813 = r174812 - r174811;
        double r174814 = 7.0;
        double r174815 = r174813 + r174814;
        double r174816 = 0.5;
        double r174817 = r174815 + r174816;
        double r174818 = r174813 + r174816;
        double r174819 = pow(r174817, r174818);
        double r174820 = r174810 * r174819;
        double r174821 = -r174817;
        double r174822 = exp(r174821);
        double r174823 = r174820 * r174822;
        double r174824 = 0.9999999999998099;
        double r174825 = 676.5203681218851;
        double r174826 = r174813 + r174811;
        double r174827 = r174825 / r174826;
        double r174828 = r174824 + r174827;
        double r174829 = -1259.1392167224028;
        double r174830 = r174813 + r174808;
        double r174831 = r174829 / r174830;
        double r174832 = r174828 + r174831;
        double r174833 = 771.3234287776531;
        double r174834 = 3.0;
        double r174835 = r174813 + r174834;
        double r174836 = r174833 / r174835;
        double r174837 = r174832 + r174836;
        double r174838 = -176.6150291621406;
        double r174839 = 4.0;
        double r174840 = r174813 + r174839;
        double r174841 = r174838 / r174840;
        double r174842 = r174837 + r174841;
        double r174843 = 12.507343278686905;
        double r174844 = 5.0;
        double r174845 = r174813 + r174844;
        double r174846 = r174843 / r174845;
        double r174847 = r174842 + r174846;
        double r174848 = -0.13857109526572012;
        double r174849 = 6.0;
        double r174850 = r174813 + r174849;
        double r174851 = r174848 / r174850;
        double r174852 = r174847 + r174851;
        double r174853 = 9.984369578019572e-06;
        double r174854 = r174853 / r174815;
        double r174855 = r174852 + r174854;
        double r174856 = 1.5056327351493116e-07;
        double r174857 = 8.0;
        double r174858 = r174813 + r174857;
        double r174859 = r174856 / r174858;
        double r174860 = r174855 + r174859;
        double r174861 = r174823 * r174860;
        double r174862 = r174807 * r174861;
        return r174862;
}
↓
double f(double z) {
        double r174863 = atan2(1.0, 0.0);
        double r174864 = z;
        double r174865 = r174863 * r174864;
        double r174866 = sin(r174865);
        double r174867 = r174863 / r174866;
        double r174868 = 2.0;
        double r174869 = r174863 * r174868;
        double r174870 = sqrt(r174869);
        double r174871 = 1.0;
        double r174872 = r174871 - r174864;
        double r174873 = r174872 - r174871;
        double r174874 = 7.0;
        double r174875 = r174873 + r174874;
        double r174876 = 0.5;
        double r174877 = r174875 + r174876;
        double r174878 = r174873 + r174876;
        double r174879 = pow(r174877, r174878);
        double r174880 = r174870 * r174879;
        double r174881 = -r174877;
        double r174882 = exp(r174881);
        double r174883 = r174880 * r174882;
        double r174884 = 0.9999999999998099;
        double r174885 = 676.5203681218851;
        double r174886 = r174873 + r174871;
        double r174887 = r174885 / r174886;
        double r174888 = r174884 + r174887;
        double r174889 = -1259.1392167224028;
        double r174890 = r174873 + r174868;
        double r174891 = r174889 / r174890;
        double r174892 = r174888 + r174891;
        double r174893 = 771.3234287776531;
        double r174894 = 3.0;
        double r174895 = r174873 + r174894;
        double r174896 = r174893 / r174895;
        double r174897 = r174892 + r174896;
        double r174898 = -176.6150291621406;
        double r174899 = 4.0;
        double r174900 = r174873 + r174899;
        double r174901 = r174898 / r174900;
        double r174902 = r174897 + r174901;
        double r174903 = 12.507343278686905;
        double r174904 = 5.0;
        double r174905 = r174873 + r174904;
        double r174906 = r174903 / r174905;
        double r174907 = r174902 + r174906;
        double r174908 = -0.13857109526572012;
        double r174909 = 6.0;
        double r174910 = r174873 + r174909;
        double r174911 = r174908 / r174910;
        double r174912 = r174907 + r174911;
        double r174913 = 9.984369578019572e-06;
        double r174914 = r174913 / r174875;
        double r174915 = r174912 + r174914;
        double r174916 = 1.5056327351493116e-07;
        double r174917 = 8.0;
        double r174918 = r174873 + r174917;
        double r174919 = r174916 / r174918;
        double r174920 = r174915 + r174919;
        double r174921 = r174883 * r174920;
        double r174922 = r174867 * r174921;
        return r174922;
}

Error

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.8

    \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531346025876700878143310547}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.50734327868690520801919774385169148445}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.1385710952657201178173096423051902092993}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

  2. Final simplification1.8

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531346025876700878143310547}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.50734327868690520801919774385169148445}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.1385710952657201178173096423051902092993}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

Reproduce

herbie shell --seed 2019346 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z less than 0.5"
  :precision binary64
  (* (/ PI (sin (* PI z))) (* (* (* (sqrt (* PI 2)) (pow (+ (+ (- (- 1 z) 1) 7) 0.5) (+ (- (- 1 z) 1) 0.5))) (exp (- (+ (+ (- (- 1 z) 1) 7) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- (- 1 z) 1) 1))) (/ -1259.1392167224028 (+ (- (- 1 z) 1) 2))) (/ 771.3234287776531 (+ (- (- 1 z) 1) 3))) (/ -176.6150291621406 (+ (- (- 1 z) 1) 4))) (/ 12.507343278686905 (+ (- (- 1 z) 1) 5))) (/ -0.13857109526572012 (+ (- (- 1 z) 1) 6))) (/ 9.984369578019572e-06 (+ (- (- 1 z) 1) 7))) (/ 1.5056327351493116e-07 (+ (- (- 1 z) 1) 8))))))