Jmat.Real.gamma, branch z less than 0.5

Average Error: 1.8 → 1.9

Time: 2.0m

Precision: 64

• could not determine a ground truth for program body

  1. z = -2.9250842867993404e+30

Math

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

↓

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

C

double f(double z) {
        double r2297747 = atan2(1.0, 0.0);
        double r2297748 = z;
        double r2297749 = r2297747 * r2297748;
        double r2297750 = sin(r2297749);
        double r2297751 = r2297747 / r2297750;
        double r2297752 = 2.0;
        double r2297753 = r2297747 * r2297752;
        double r2297754 = sqrt(r2297753);
        double r2297755 = 1.0;
        double r2297756 = r2297755 - r2297748;
        double r2297757 = r2297756 - r2297755;
        double r2297758 = 7.0;
        double r2297759 = r2297757 + r2297758;
        double r2297760 = 0.5;
        double r2297761 = r2297759 + r2297760;
        double r2297762 = r2297757 + r2297760;
        double r2297763 = pow(r2297761, r2297762);
        double r2297764 = r2297754 * r2297763;
        double r2297765 = -r2297761;
        double r2297766 = exp(r2297765);
        double r2297767 = r2297764 * r2297766;
        double r2297768 = 0.9999999999998099;
        double r2297769 = 676.5203681218851;
        double r2297770 = r2297757 + r2297755;
        double r2297771 = r2297769 / r2297770;
        double r2297772 = r2297768 + r2297771;
        double r2297773 = -1259.1392167224028;
        double r2297774 = r2297757 + r2297752;
        double r2297775 = r2297773 / r2297774;
        double r2297776 = r2297772 + r2297775;
        double r2297777 = 771.3234287776531;
        double r2297778 = 3.0;
        double r2297779 = r2297757 + r2297778;
        double r2297780 = r2297777 / r2297779;
        double r2297781 = r2297776 + r2297780;
        double r2297782 = -176.6150291621406;
        double r2297783 = 4.0;
        double r2297784 = r2297757 + r2297783;
        double r2297785 = r2297782 / r2297784;
        double r2297786 = r2297781 + r2297785;
        double r2297787 = 12.507343278686905;
        double r2297788 = 5.0;
        double r2297789 = r2297757 + r2297788;
        double r2297790 = r2297787 / r2297789;
        double r2297791 = r2297786 + r2297790;
        double r2297792 = -0.13857109526572012;
        double r2297793 = 6.0;
        double r2297794 = r2297757 + r2297793;
        double r2297795 = r2297792 / r2297794;
        double r2297796 = r2297791 + r2297795;
        double r2297797 = 9.984369578019572e-06;
        double r2297798 = r2297797 / r2297759;
        double r2297799 = r2297796 + r2297798;
        double r2297800 = 1.5056327351493116e-07;
        double r2297801 = 8.0;
        double r2297802 = r2297757 + r2297801;
        double r2297803 = r2297800 / r2297802;
        double r2297804 = r2297799 + r2297803;
        double r2297805 = r2297767 * r2297804;
        double r2297806 = r2297751 * r2297805;
        return r2297806;
}

↓

double f(double z) {
        double r2297807 = 1.5056327351493116e-07;
        double r2297808 = 8.0;
        double r2297809 = 1.0;
        double r2297810 = z;
        double r2297811 = r2297809 - r2297810;
        double r2297812 = r2297811 - r2297809;
        double r2297813 = r2297808 + r2297812;
        double r2297814 = r2297807 / r2297813;
        double r2297815 = -0.13857109526572012;
        double r2297816 = 6.0;
        double r2297817 = r2297812 + r2297816;
        double r2297818 = r2297815 / r2297817;
        double r2297819 = -176.6150291621406;
        double r2297820 = 4.0;
        double r2297821 = r2297812 + r2297820;
        double r2297822 = r2297819 / r2297821;
        double r2297823 = 771.3234287776531;
        double r2297824 = 3.0;
        double r2297825 = r2297824 + r2297812;
        double r2297826 = r2297823 / r2297825;
        double r2297827 = 0.9999999999998099;
        double r2297828 = 676.5203681218851;
        double r2297829 = r2297812 + r2297809;
        double r2297830 = r2297828 / r2297829;
        double r2297831 = r2297827 + r2297830;
        double r2297832 = -1259.1392167224028;
        double r2297833 = 2.0;
        double r2297834 = r2297812 + r2297833;
        double r2297835 = r2297832 / r2297834;
        double r2297836 = r2297831 + r2297835;
        double r2297837 = r2297826 + r2297836;
        double r2297838 = r2297822 + r2297837;
        double r2297839 = 12.507343278686905;
        double r2297840 = sqrt(r2297811);
        double r2297841 = sqrt(r2297840);
        double r2297842 = r2297841 - r2297809;
        double r2297843 = r2297841 + r2297809;
        double r2297844 = r2297842 * r2297843;
        double r2297845 = r2297809 + r2297840;
        double r2297846 = r2297844 * r2297845;
        double r2297847 = 5.0;
        double r2297848 = r2297846 + r2297847;
        double r2297849 = r2297839 / r2297848;
        double r2297850 = r2297838 + r2297849;
        double r2297851 = r2297818 + r2297850;
        double r2297852 = 9.984369578019572e-06;
        double r2297853 = 7.0;
        double r2297854 = r2297853 + r2297812;
        double r2297855 = r2297852 / r2297854;
        double r2297856 = r2297851 + r2297855;
        double r2297857 = r2297814 + r2297856;
        double r2297858 = atan2(1.0, 0.0);
        double r2297859 = r2297833 * r2297858;
        double r2297860 = sqrt(r2297859);
        double r2297861 = 0.5;
        double r2297862 = r2297854 + r2297861;
        double r2297863 = r2297861 + r2297812;
        double r2297864 = pow(r2297862, r2297863);
        double r2297865 = r2297860 * r2297864;
        double r2297866 = -r2297862;
        double r2297867 = exp(r2297866);
        double r2297868 = r2297865 * r2297867;
        double r2297869 = r2297857 * r2297868;
        double r2297870 = r2297858 * r2297810;
        double r2297871 = sin(r2297870);
        double r2297872 = r2297858 / r2297871;
        double r2297873 = r2297869 * r2297872;
        return r2297873;
}

Error

Bits error versus z

Derivation

1. Initial program 1.8

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

3. Applied *-un-lft-identity 1.8

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

4. Applied add-sqr-sqrt 1.9

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

5. Applied difference-of-squares 1.9

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

7. Applied *-un-lft-identity 1.9

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

8. Applied add-sqr-sqrt 1.9

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

9. Applied sqrt-prod 1.9

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

10. Applied difference-of-squares 1.9

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

11. Final simplification 1.9

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

Reproduce

herbie shell --seed 2019155 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z less than 0.5"
  (* (/ 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))))))