Average Error: 1.8 → 1.8
Time: 48.7s
Precision: 64
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
\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(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \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(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)
double f(double z) {
        double r124820 = atan2(1.0, 0.0);
        double r124821 = z;
        double r124822 = r124820 * r124821;
        double r124823 = sin(r124822);
        double r124824 = r124820 / r124823;
        double r124825 = 2.0;
        double r124826 = r124820 * r124825;
        double r124827 = sqrt(r124826);
        double r124828 = 1.0;
        double r124829 = r124828 - r124821;
        double r124830 = r124829 - r124828;
        double r124831 = 7.0;
        double r124832 = r124830 + r124831;
        double r124833 = 0.5;
        double r124834 = r124832 + r124833;
        double r124835 = r124830 + r124833;
        double r124836 = pow(r124834, r124835);
        double r124837 = r124827 * r124836;
        double r124838 = -r124834;
        double r124839 = exp(r124838);
        double r124840 = r124837 * r124839;
        double r124841 = 0.9999999999998099;
        double r124842 = 676.5203681218851;
        double r124843 = r124830 + r124828;
        double r124844 = r124842 / r124843;
        double r124845 = r124841 + r124844;
        double r124846 = -1259.1392167224028;
        double r124847 = r124830 + r124825;
        double r124848 = r124846 / r124847;
        double r124849 = r124845 + r124848;
        double r124850 = 771.3234287776531;
        double r124851 = 3.0;
        double r124852 = r124830 + r124851;
        double r124853 = r124850 / r124852;
        double r124854 = r124849 + r124853;
        double r124855 = -176.6150291621406;
        double r124856 = 4.0;
        double r124857 = r124830 + r124856;
        double r124858 = r124855 / r124857;
        double r124859 = r124854 + r124858;
        double r124860 = 12.507343278686905;
        double r124861 = 5.0;
        double r124862 = r124830 + r124861;
        double r124863 = r124860 / r124862;
        double r124864 = r124859 + r124863;
        double r124865 = -0.13857109526572012;
        double r124866 = 6.0;
        double r124867 = r124830 + r124866;
        double r124868 = r124865 / r124867;
        double r124869 = r124864 + r124868;
        double r124870 = 9.984369578019572e-06;
        double r124871 = r124870 / r124832;
        double r124872 = r124869 + r124871;
        double r124873 = 1.5056327351493116e-07;
        double r124874 = 8.0;
        double r124875 = r124830 + r124874;
        double r124876 = r124873 / r124875;
        double r124877 = r124872 + r124876;
        double r124878 = r124840 * r124877;
        double r124879 = r124824 * r124878;
        return r124879;
}
↓
double f(double z) {
        double r124880 = atan2(1.0, 0.0);
        double r124881 = z;
        double r124882 = r124880 * r124881;
        double r124883 = sin(r124882);
        double r124884 = r124880 / r124883;
        double r124885 = 2.0;
        double r124886 = r124880 * r124885;
        double r124887 = sqrt(r124886);
        double r124888 = 1.0;
        double r124889 = r124888 - r124881;
        double r124890 = r124889 - r124888;
        double r124891 = 7.0;
        double r124892 = r124890 + r124891;
        double r124893 = 0.5;
        double r124894 = r124892 + r124893;
        double r124895 = r124890 + r124893;
        double r124896 = pow(r124894, r124895);
        double r124897 = r124887 * r124896;
        double r124898 = -r124894;
        double r124899 = exp(r124898);
        double r124900 = r124897 * r124899;
        double r124901 = 0.9999999999998099;
        double r124902 = 676.5203681218851;
        double r124903 = r124890 + r124888;
        double r124904 = r124902 / r124903;
        double r124905 = r124901 + r124904;
        double r124906 = -1259.1392167224028;
        double r124907 = r124890 + r124885;
        double r124908 = r124906 / r124907;
        double r124909 = r124905 + r124908;
        double r124910 = 771.3234287776531;
        double r124911 = 3.0;
        double r124912 = r124890 + r124911;
        double r124913 = r124910 / r124912;
        double r124914 = r124909 + r124913;
        double r124915 = -176.6150291621406;
        double r124916 = 4.0;
        double r124917 = r124890 + r124916;
        double r124918 = r124915 / r124917;
        double r124919 = r124914 + r124918;
        double r124920 = 12.507343278686905;
        double r124921 = 5.0;
        double r124922 = r124890 + r124921;
        double r124923 = r124920 / r124922;
        double r124924 = r124919 + r124923;
        double r124925 = -0.13857109526572012;
        double r124926 = 6.0;
        double r124927 = r124890 + r124926;
        double r124928 = r124925 / r124927;
        double r124929 = r124924 + r124928;
        double r124930 = 9.984369578019572e-06;
        double r124931 = r124930 / r124892;
        double r124932 = r124929 + r124931;
        double r124933 = 1.5056327351493116e-07;
        double r124934 = 8.0;
        double r124935 = r124890 + r124934;
        double r124936 = r124933 / r124935;
        double r124937 = r124932 + r124936;
        double r124938 = r124900 * r124937;
        double r124939 = r124884 * r124938;
        return r124939;
}

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

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

  2. Final simplification1.8

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

Reproduce

herbie shell --seed 2020059 
(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))))))