Average Error: 1.8 → 0.4
Time: 9.0m
Precision: 64
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\frac{\left(\left(\pi \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(0.5 + \left(7 - z\right)\right)}^{0.5}\right) \cdot \left(\frac{-0.138571095265720118}{6 - z} + \left(\frac{771.32342877765313}{3 - z} + \left(\left(\frac{12.5073432786869052}{5 - z} + \left(0.99999999999980993 + \left(\left(\frac{676.520368121885099}{1 - z} + \frac{-1259.13921672240281}{2 - z}\right) + \frac{-176.615029162140587}{4 - z}\right)\right)\right) + \left(\frac{1.50563273514931162 \cdot 10^{-7}}{8 - z} + \frac{9.98436957801957158 \cdot 10^{-6}}{7 - z}\right)\right)\right)\right)}{e^{0.5 + \left(7 - z\right)} \cdot \left(\sin \left(\pi \cdot z\right) \cdot {\left(0.5 + \left(7 - z\right)\right)}^{z}\right)}\]
\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{\left(\left(\pi \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(0.5 + \left(7 - z\right)\right)}^{0.5}\right) \cdot \left(\frac{-0.138571095265720118}{6 - z} + \left(\frac{771.32342877765313}{3 - z} + \left(\left(\frac{12.5073432786869052}{5 - z} + \left(0.99999999999980993 + \left(\left(\frac{676.520368121885099}{1 - z} + \frac{-1259.13921672240281}{2 - z}\right) + \frac{-176.615029162140587}{4 - z}\right)\right)\right) + \left(\frac{1.50563273514931162 \cdot 10^{-7}}{8 - z} + \frac{9.98436957801957158 \cdot 10^{-6}}{7 - z}\right)\right)\right)\right)}{e^{0.5 + \left(7 - z\right)} \cdot \left(\sin \left(\pi \cdot z\right) \cdot {\left(0.5 + \left(7 - z\right)\right)}^{z}\right)}
double f(double z) {
        double r162774 = atan2(1.0, 0.0);
        double r162775 = z;
        double r162776 = r162774 * r162775;
        double r162777 = sin(r162776);
        double r162778 = r162774 / r162777;
        double r162779 = 2.0;
        double r162780 = r162774 * r162779;
        double r162781 = sqrt(r162780);
        double r162782 = 1.0;
        double r162783 = r162782 - r162775;
        double r162784 = r162783 - r162782;
        double r162785 = 7.0;
        double r162786 = r162784 + r162785;
        double r162787 = 0.5;
        double r162788 = r162786 + r162787;
        double r162789 = r162784 + r162787;
        double r162790 = pow(r162788, r162789);
        double r162791 = r162781 * r162790;
        double r162792 = -r162788;
        double r162793 = exp(r162792);
        double r162794 = r162791 * r162793;
        double r162795 = 0.9999999999998099;
        double r162796 = 676.5203681218851;
        double r162797 = r162784 + r162782;
        double r162798 = r162796 / r162797;
        double r162799 = r162795 + r162798;
        double r162800 = -1259.1392167224028;
        double r162801 = r162784 + r162779;
        double r162802 = r162800 / r162801;
        double r162803 = r162799 + r162802;
        double r162804 = 771.3234287776531;
        double r162805 = 3.0;
        double r162806 = r162784 + r162805;
        double r162807 = r162804 / r162806;
        double r162808 = r162803 + r162807;
        double r162809 = -176.6150291621406;
        double r162810 = 4.0;
        double r162811 = r162784 + r162810;
        double r162812 = r162809 / r162811;
        double r162813 = r162808 + r162812;
        double r162814 = 12.507343278686905;
        double r162815 = 5.0;
        double r162816 = r162784 + r162815;
        double r162817 = r162814 / r162816;
        double r162818 = r162813 + r162817;
        double r162819 = -0.13857109526572012;
        double r162820 = 6.0;
        double r162821 = r162784 + r162820;
        double r162822 = r162819 / r162821;
        double r162823 = r162818 + r162822;
        double r162824 = 9.984369578019572e-06;
        double r162825 = r162824 / r162786;
        double r162826 = r162823 + r162825;
        double r162827 = 1.5056327351493116e-07;
        double r162828 = 8.0;
        double r162829 = r162784 + r162828;
        double r162830 = r162827 / r162829;
        double r162831 = r162826 + r162830;
        double r162832 = r162794 * r162831;
        double r162833 = r162778 * r162832;
        return r162833;
}
↓
double f(double z) {
        double r162834 = atan2(1.0, 0.0);
        double r162835 = 2.0;
        double r162836 = r162834 * r162835;
        double r162837 = sqrt(r162836);
        double r162838 = r162834 * r162837;
        double r162839 = 0.5;
        double r162840 = 7.0;
        double r162841 = z;
        double r162842 = r162840 - r162841;
        double r162843 = r162839 + r162842;
        double r162844 = pow(r162843, r162839);
        double r162845 = r162838 * r162844;
        double r162846 = -0.13857109526572012;
        double r162847 = 6.0;
        double r162848 = r162847 - r162841;
        double r162849 = r162846 / r162848;
        double r162850 = 771.3234287776531;
        double r162851 = 3.0;
        double r162852 = r162851 - r162841;
        double r162853 = r162850 / r162852;
        double r162854 = 12.507343278686905;
        double r162855 = 5.0;
        double r162856 = r162855 - r162841;
        double r162857 = r162854 / r162856;
        double r162858 = 0.9999999999998099;
        double r162859 = 676.5203681218851;
        double r162860 = 1.0;
        double r162861 = r162860 - r162841;
        double r162862 = r162859 / r162861;
        double r162863 = -1259.1392167224028;
        double r162864 = r162835 - r162841;
        double r162865 = r162863 / r162864;
        double r162866 = r162862 + r162865;
        double r162867 = -176.6150291621406;
        double r162868 = 4.0;
        double r162869 = r162868 - r162841;
        double r162870 = r162867 / r162869;
        double r162871 = r162866 + r162870;
        double r162872 = r162858 + r162871;
        double r162873 = r162857 + r162872;
        double r162874 = 1.5056327351493116e-07;
        double r162875 = 8.0;
        double r162876 = r162875 - r162841;
        double r162877 = r162874 / r162876;
        double r162878 = 9.984369578019572e-06;
        double r162879 = r162878 / r162842;
        double r162880 = r162877 + r162879;
        double r162881 = r162873 + r162880;
        double r162882 = r162853 + r162881;
        double r162883 = r162849 + r162882;
        double r162884 = r162845 * r162883;
        double r162885 = exp(r162843);
        double r162886 = r162834 * r162841;
        double r162887 = sin(r162886);
        double r162888 = pow(r162843, r162841);
        double r162889 = r162887 * r162888;
        double r162890 = r162885 * r162889;
        double r162891 = r162884 / r162890;
        return r162891;
}

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. Simplified1.1

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

  4. Applied pow-sub1.1

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

  5. Applied associate-*l/1.1

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

  6. Applied frac-times1.2

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

  7. Applied associate-*l/1.0

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

  8. Applied associate-/l/0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(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))))))