Average Error: 60.1 → 1.1
Time: 1.4m
Precision: 64
\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8}\right)\]
↓
\[\left(\left(e^{\left(-\log \left(z + 6.5\right)\right) \cdot \left(0.5 - z\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \left(\left(2351.6663247613023 \cdot \frac{z}{e^{6.5}} + \frac{676.5203681218851}{e^{6.5} \cdot z}\right) - \frac{1604.7704235566525}{e^{6.5}}\right)\]
\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)
↓
\left(\left(e^{\left(-\log \left(z + 6.5\right)\right) \cdot \left(0.5 - z\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \left(\left(2351.6663247613023 \cdot \frac{z}{e^{6.5}} + \frac{676.5203681218851}{e^{6.5} \cdot z}\right) - \frac{1604.7704235566525}{e^{6.5}}\right)
double f(double z) {
        double r2845802 = atan2(1.0, 0.0);
        double r2845803 = 2.0;
        double r2845804 = r2845802 * r2845803;
        double r2845805 = sqrt(r2845804);
        double r2845806 = z;
        double r2845807 = 1.0;
        double r2845808 = r2845806 - r2845807;
        double r2845809 = 7.0;
        double r2845810 = r2845808 + r2845809;
        double r2845811 = 0.5;
        double r2845812 = r2845810 + r2845811;
        double r2845813 = r2845808 + r2845811;
        double r2845814 = pow(r2845812, r2845813);
        double r2845815 = r2845805 * r2845814;
        double r2845816 = -r2845812;
        double r2845817 = exp(r2845816);
        double r2845818 = r2845815 * r2845817;
        double r2845819 = 0.9999999999998099;
        double r2845820 = 676.5203681218851;
        double r2845821 = r2845808 + r2845807;
        double r2845822 = r2845820 / r2845821;
        double r2845823 = r2845819 + r2845822;
        double r2845824 = -1259.1392167224028;
        double r2845825 = r2845808 + r2845803;
        double r2845826 = r2845824 / r2845825;
        double r2845827 = r2845823 + r2845826;
        double r2845828 = 771.3234287776531;
        double r2845829 = 3.0;
        double r2845830 = r2845808 + r2845829;
        double r2845831 = r2845828 / r2845830;
        double r2845832 = r2845827 + r2845831;
        double r2845833 = -176.6150291621406;
        double r2845834 = 4.0;
        double r2845835 = r2845808 + r2845834;
        double r2845836 = r2845833 / r2845835;
        double r2845837 = r2845832 + r2845836;
        double r2845838 = 12.507343278686905;
        double r2845839 = 5.0;
        double r2845840 = r2845808 + r2845839;
        double r2845841 = r2845838 / r2845840;
        double r2845842 = r2845837 + r2845841;
        double r2845843 = -0.13857109526572012;
        double r2845844 = 6.0;
        double r2845845 = r2845808 + r2845844;
        double r2845846 = r2845843 / r2845845;
        double r2845847 = r2845842 + r2845846;
        double r2845848 = 9.984369578019572e-06;
        double r2845849 = r2845848 / r2845810;
        double r2845850 = r2845847 + r2845849;
        double r2845851 = 1.5056327351493116e-07;
        double r2845852 = 8.0;
        double r2845853 = r2845808 + r2845852;
        double r2845854 = r2845851 / r2845853;
        double r2845855 = r2845850 + r2845854;
        double r2845856 = r2845818 * r2845855;
        return r2845856;
}
↓
double f(double z) {
        double r2845857 = z;
        double r2845858 = 6.5;
        double r2845859 = r2845857 + r2845858;
        double r2845860 = log(r2845859);
        double r2845861 = -r2845860;
        double r2845862 = 0.5;
        double r2845863 = r2845862 - r2845857;
        double r2845864 = r2845861 * r2845863;
        double r2845865 = exp(r2845864);
        double r2845866 = 2.0;
        double r2845867 = sqrt(r2845866);
        double r2845868 = r2845865 * r2845867;
        double r2845869 = atan2(1.0, 0.0);
        double r2845870 = sqrt(r2845869);
        double r2845871 = r2845868 * r2845870;
        double r2845872 = 2351.6663247613023;
        double r2845873 = exp(r2845858);
        double r2845874 = r2845857 / r2845873;
        double r2845875 = r2845872 * r2845874;
        double r2845876 = 676.5203681218851;
        double r2845877 = r2845873 * r2845857;
        double r2845878 = r2845876 / r2845877;
        double r2845879 = r2845875 + r2845878;
        double r2845880 = 1604.7704235566525;
        double r2845881 = r2845880 / r2845873;
        double r2845882 = r2845879 - r2845881;
        double r2845883 = r2845871 * r2845882;
        return r2845883;
}

Error

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 60.1

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

  2. Simplified1.0

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

  3. Taylor expanded around 0 1.1

    \[\leadsto \left({\left(\left(z - -6\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \color{blue}{\left(\left(676.5203681218851 \cdot \frac{1}{e^{6.5} \cdot z} + 2351.6663247613023 \cdot \frac{z}{e^{6.5}}\right) - 1604.7704235566525 \cdot \frac{1}{e^{6.5}}\right)}\]

  4. Simplified1.1

    \[\leadsto \left({\left(\left(z - -6\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \color{blue}{\left(\left(\frac{676.5203681218851}{z \cdot e^{6.5}} + 2351.6663247613023 \cdot \frac{z}{e^{6.5}}\right) - \frac{1604.7704235566525}{e^{6.5}}\right)}\]

  5. Taylor expanded around -inf 1.1

    \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot e^{-1 \cdot \left(\log \left(z + 6.5\right) \cdot \left(0.5 - z\right)\right)}\right) \cdot \sqrt{\pi}\right)} \cdot \left(\left(\frac{676.5203681218851}{z \cdot e^{6.5}} + 2351.6663247613023 \cdot \frac{z}{e^{6.5}}\right) - \frac{1604.7704235566525}{e^{6.5}}\right)\]

  6. Simplified1.1

    \[\leadsto \color{blue}{\left(\left(e^{\left(0.5 - z\right) \cdot \left(-\log \left(z + 6.5\right)\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right)} \cdot \left(\left(\frac{676.5203681218851}{z \cdot e^{6.5}} + 2351.6663247613023 \cdot \frac{z}{e^{6.5}}\right) - \frac{1604.7704235566525}{e^{6.5}}\right)\]

  7. Final simplification1.1

    \[\leadsto \left(\left(e^{\left(-\log \left(z + 6.5\right)\right) \cdot \left(0.5 - z\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot \left(\left(2351.6663247613023 \cdot \frac{z}{e^{6.5}} + \frac{676.5203681218851}{e^{6.5} \cdot z}\right) - \frac{1604.7704235566525}{e^{6.5}}\right)\]

Reproduce

herbie shell --seed 2019156 
(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)))))