Jmat.Real.gamma, branch z greater than 0.5

Average Error: 60.0 → 1.3

Time: 3.6m

Precision: 64

• could not determine a ground truth for program body

  1. z = 5.0362447883134526e+148

Math

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

↓

\[\frac{\sqrt{\pi \cdot 2} \cdot {\left(0.5 + \left(z - -6\right)\right)}^{\left(0.5 + \left(z - 1\right)\right)}}{\sqrt{e^{0.5 + \left(z - -6\right)}}} \cdot \left(\mathsf{fma}\left(676.5203681218851, \sqrt{e^{-6.5}} \cdot \frac{1}{z}, \left(z \cdot \sqrt{e^{-6.5}}\right) \cdot 1633.8461589982119\right) - 1266.51023949571 \cdot \sqrt{e^{-6.5}}\right)\]

C

double f(double z) {
        double r10754805 = atan2(1.0, 0.0);
        double r10754806 = 2.0;
        double r10754807 = r10754805 * r10754806;
        double r10754808 = sqrt(r10754807);
        double r10754809 = z;
        double r10754810 = 1.0;
        double r10754811 = r10754809 - r10754810;
        double r10754812 = 7.0;
        double r10754813 = r10754811 + r10754812;
        double r10754814 = 0.5;
        double r10754815 = r10754813 + r10754814;
        double r10754816 = r10754811 + r10754814;
        double r10754817 = pow(r10754815, r10754816);
        double r10754818 = r10754808 * r10754817;
        double r10754819 = -r10754815;
        double r10754820 = exp(r10754819);
        double r10754821 = r10754818 * r10754820;
        double r10754822 = 0.9999999999998099;
        double r10754823 = 676.5203681218851;
        double r10754824 = r10754811 + r10754810;
        double r10754825 = r10754823 / r10754824;
        double r10754826 = r10754822 + r10754825;
        double r10754827 = -1259.1392167224028;
        double r10754828 = r10754811 + r10754806;
        double r10754829 = r10754827 / r10754828;
        double r10754830 = r10754826 + r10754829;
        double r10754831 = 771.3234287776531;
        double r10754832 = 3.0;
        double r10754833 = r10754811 + r10754832;
        double r10754834 = r10754831 / r10754833;
        double r10754835 = r10754830 + r10754834;
        double r10754836 = -176.6150291621406;
        double r10754837 = 4.0;
        double r10754838 = r10754811 + r10754837;
        double r10754839 = r10754836 / r10754838;
        double r10754840 = r10754835 + r10754839;
        double r10754841 = 12.507343278686905;
        double r10754842 = 5.0;
        double r10754843 = r10754811 + r10754842;
        double r10754844 = r10754841 / r10754843;
        double r10754845 = r10754840 + r10754844;
        double r10754846 = -0.13857109526572012;
        double r10754847 = 6.0;
        double r10754848 = r10754811 + r10754847;
        double r10754849 = r10754846 / r10754848;
        double r10754850 = r10754845 + r10754849;
        double r10754851 = 9.984369578019572e-06;
        double r10754852 = r10754851 / r10754813;
        double r10754853 = r10754850 + r10754852;
        double r10754854 = 1.5056327351493116e-07;
        double r10754855 = 8.0;
        double r10754856 = r10754811 + r10754855;
        double r10754857 = r10754854 / r10754856;
        double r10754858 = r10754853 + r10754857;
        double r10754859 = r10754821 * r10754858;
        return r10754859;
}

↓

double f(double z) {
        double r10754860 = atan2(1.0, 0.0);
        double r10754861 = 2.0;
        double r10754862 = r10754860 * r10754861;
        double r10754863 = sqrt(r10754862);
        double r10754864 = 0.5;
        double r10754865 = z;
        double r10754866 = -6.0;
        double r10754867 = r10754865 - r10754866;
        double r10754868 = r10754864 + r10754867;
        double r10754869 = 1.0;
        double r10754870 = r10754865 - r10754869;
        double r10754871 = r10754864 + r10754870;
        double r10754872 = pow(r10754868, r10754871);
        double r10754873 = r10754863 * r10754872;
        double r10754874 = exp(r10754868);
        double r10754875 = sqrt(r10754874);
        double r10754876 = r10754873 / r10754875;
        double r10754877 = 676.5203681218851;
        double r10754878 = 6.5;
        double r10754879 = -r10754878;
        double r10754880 = exp(r10754879);
        double r10754881 = sqrt(r10754880);
        double r10754882 = r10754869 / r10754865;
        double r10754883 = r10754881 * r10754882;
        double r10754884 = r10754865 * r10754881;
        double r10754885 = 1633.8461589982119;
        double r10754886 = r10754884 * r10754885;
        double r10754887 = fma(r10754877, r10754883, r10754886);
        double r10754888 = 1266.51023949571;
        double r10754889 = r10754888 * r10754881;
        double r10754890 = r10754887 - r10754889;
        double r10754891 = r10754876 * r10754890;
        return r10754891;
}

Error

Bits error versus z

Derivation

1. Initial program 60.0

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

2. Simplified 1.1

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

4. Applied add-sqr-sqrt 1.1

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

5. Applied times-frac 0.9

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

7. Applied frac-times 1.1

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

8. Applied associate-*l/ 0.9

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

10. Applied times-frac 0.9

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

11. Taylor expanded around 0 1.3

    \[\leadsto \frac{{\left(\left(z - -6\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot \sqrt{\pi \cdot 2}}{\sqrt{e^{\left(z - -6\right) + 0.5}}} \cdot \color{blue}{\left(\left(676.5203681218851 \cdot \left(\sqrt{\frac{1}{e^{6.5}}} \cdot \frac{1}{z}\right) + 1633.8461589982119 \cdot \left(\sqrt{\frac{1}{e^{6.5}}} \cdot z\right)\right) - 1266.51023949571 \cdot \sqrt{\frac{1}{e^{6.5}}}\right)}\]

12. Simplified 1.3

    \[\leadsto \frac{{\left(\left(z - -6\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot \sqrt{\pi \cdot 2}}{\sqrt{e^{\left(z - -6\right) + 0.5}}} \cdot \color{blue}{\left(\mathsf{fma}\left(676.5203681218851, \frac{1}{z} \cdot \sqrt{e^{-6.5}}, \left(z \cdot \sqrt{e^{-6.5}}\right) \cdot 1633.8461589982119\right) - 1266.51023949571 \cdot \sqrt{e^{-6.5}}\right)}\]

13. Final simplification 1.3

    \[\leadsto \frac{\sqrt{\pi \cdot 2} \cdot {\left(0.5 + \left(z - -6\right)\right)}^{\left(0.5 + \left(z - 1\right)\right)}}{\sqrt{e^{0.5 + \left(z - -6\right)}}} \cdot \left(\mathsf{fma}\left(676.5203681218851, \sqrt{e^{-6.5}} \cdot \frac{1}{z}, \left(z \cdot \sqrt{e^{-6.5}}\right) \cdot 1633.8461589982119\right) - 1266.51023949571 \cdot \sqrt{e^{-6.5}}\right)\]

Reproduce

herbie shell --seed 2019152 +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)))))