Average Error: 1.8 → 0.7
Time: 4.4m
Precision: 64
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531346025876700878143310547}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.50734327868690520801919774385169148445}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.1385710952657201178173096423051902092993}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[e^{\log \left(\left(\left(\left(\frac{771.3234287776531346025876700878143310547}{3 - z} + \left(\frac{-176.6150291621405870046146446838974952698}{4 - z} + \left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{1 - z}\right)\right)\right) + \frac{12.50734327868690520801919774385169148445}{5 - z}\right) + \left(\frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{8 - z} + \left(\frac{-0.1385710952657201178173096423051902092993}{6 - z} + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(-z\right) + 7}\right)\right)\right) + \frac{-1259.139216722402807135949842631816864014}{\left(-z\right) + 2}\right) - \left(0.5 + \left(\left(-z\right) + 7\right)\right)} \cdot \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(0.5 + \left(\left(-z\right) + 7\right)\right)}^{\left(\left(-z\right) + 0.5\right)}\right)\]
\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(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531346025876700878143310547}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.50734327868690520801919774385169148445}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.1385710952657201178173096423051902092993}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)
↓
e^{\log \left(\left(\left(\left(\frac{771.3234287776531346025876700878143310547}{3 - z} + \left(\frac{-176.6150291621405870046146446838974952698}{4 - z} + \left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{1 - z}\right)\right)\right) + \frac{12.50734327868690520801919774385169148445}{5 - z}\right) + \left(\frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{8 - z} + \left(\frac{-0.1385710952657201178173096423051902092993}{6 - z} + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(-z\right) + 7}\right)\right)\right) + \frac{-1259.139216722402807135949842631816864014}{\left(-z\right) + 2}\right) - \left(0.5 + \left(\left(-z\right) + 7\right)\right)} \cdot \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(0.5 + \left(\left(-z\right) + 7\right)\right)}^{\left(\left(-z\right) + 0.5\right)}\right)
double f(double z) {
        double r152740 = atan2(1.0, 0.0);
        double r152741 = z;
        double r152742 = r152740 * r152741;
        double r152743 = sin(r152742);
        double r152744 = r152740 / r152743;
        double r152745 = 2.0;
        double r152746 = r152740 * r152745;
        double r152747 = sqrt(r152746);
        double r152748 = 1.0;
        double r152749 = r152748 - r152741;
        double r152750 = r152749 - r152748;
        double r152751 = 7.0;
        double r152752 = r152750 + r152751;
        double r152753 = 0.5;
        double r152754 = r152752 + r152753;
        double r152755 = r152750 + r152753;
        double r152756 = pow(r152754, r152755);
        double r152757 = r152747 * r152756;
        double r152758 = -r152754;
        double r152759 = exp(r152758);
        double r152760 = r152757 * r152759;
        double r152761 = 0.9999999999998099;
        double r152762 = 676.5203681218851;
        double r152763 = r152750 + r152748;
        double r152764 = r152762 / r152763;
        double r152765 = r152761 + r152764;
        double r152766 = -1259.1392167224028;
        double r152767 = r152750 + r152745;
        double r152768 = r152766 / r152767;
        double r152769 = r152765 + r152768;
        double r152770 = 771.3234287776531;
        double r152771 = 3.0;
        double r152772 = r152750 + r152771;
        double r152773 = r152770 / r152772;
        double r152774 = r152769 + r152773;
        double r152775 = -176.6150291621406;
        double r152776 = 4.0;
        double r152777 = r152750 + r152776;
        double r152778 = r152775 / r152777;
        double r152779 = r152774 + r152778;
        double r152780 = 12.507343278686905;
        double r152781 = 5.0;
        double r152782 = r152750 + r152781;
        double r152783 = r152780 / r152782;
        double r152784 = r152779 + r152783;
        double r152785 = -0.13857109526572012;
        double r152786 = 6.0;
        double r152787 = r152750 + r152786;
        double r152788 = r152785 / r152787;
        double r152789 = r152784 + r152788;
        double r152790 = 9.984369578019572e-06;
        double r152791 = r152790 / r152752;
        double r152792 = r152789 + r152791;
        double r152793 = 1.5056327351493116e-07;
        double r152794 = 8.0;
        double r152795 = r152750 + r152794;
        double r152796 = r152793 / r152795;
        double r152797 = r152792 + r152796;
        double r152798 = r152760 * r152797;
        double r152799 = r152744 * r152798;
        return r152799;
}
↓
double f(double z) {
        double r152800 = 771.3234287776531;
        double r152801 = 3.0;
        double r152802 = z;
        double r152803 = r152801 - r152802;
        double r152804 = r152800 / r152803;
        double r152805 = -176.6150291621406;
        double r152806 = 4.0;
        double r152807 = r152806 - r152802;
        double r152808 = r152805 / r152807;
        double r152809 = 0.9999999999998099;
        double r152810 = 676.5203681218851;
        double r152811 = 1.0;
        double r152812 = r152811 - r152802;
        double r152813 = r152810 / r152812;
        double r152814 = r152809 + r152813;
        double r152815 = r152808 + r152814;
        double r152816 = r152804 + r152815;
        double r152817 = 12.507343278686905;
        double r152818 = 5.0;
        double r152819 = r152818 - r152802;
        double r152820 = r152817 / r152819;
        double r152821 = r152816 + r152820;
        double r152822 = 1.5056327351493116e-07;
        double r152823 = 8.0;
        double r152824 = r152823 - r152802;
        double r152825 = r152822 / r152824;
        double r152826 = -0.13857109526572012;
        double r152827 = 6.0;
        double r152828 = r152827 - r152802;
        double r152829 = r152826 / r152828;
        double r152830 = 9.984369578019572e-06;
        double r152831 = -r152802;
        double r152832 = 7.0;
        double r152833 = r152831 + r152832;
        double r152834 = r152830 / r152833;
        double r152835 = r152829 + r152834;
        double r152836 = r152825 + r152835;
        double r152837 = r152821 + r152836;
        double r152838 = -1259.1392167224028;
        double r152839 = 2.0;
        double r152840 = r152831 + r152839;
        double r152841 = r152838 / r152840;
        double r152842 = r152837 + r152841;
        double r152843 = log(r152842);
        double r152844 = 0.5;
        double r152845 = r152844 + r152833;
        double r152846 = r152843 - r152845;
        double r152847 = exp(r152846);
        double r152848 = atan2(1.0, 0.0);
        double r152849 = r152848 * r152802;
        double r152850 = sin(r152849);
        double r152851 = r152848 / r152850;
        double r152852 = r152848 * r152839;
        double r152853 = sqrt(r152852);
        double r152854 = r152851 * r152853;
        double r152855 = r152831 + r152844;
        double r152856 = pow(r152845, r152855);
        double r152857 = r152854 * r152856;
        double r152858 = r152847 * r152857;
        return r152858;
}

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

    \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531346025876700878143310547}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.50734327868690520801919774385169148445}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.1385710952657201178173096423051902092993}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

  2. Simplified2.2

    \[\leadsto \color{blue}{\frac{\frac{-1259.139216722402807135949842631816864014}{\left(-z\right) + 2} + \left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{1 - z}\right) + \frac{-176.6150291621405870046146446838974952698}{4 + \left(-z\right)}\right) + \frac{771.3234287776531346025876700878143310547}{3 + \left(-z\right)}\right) + \left(\frac{12.50734327868690520801919774385169148445}{5 + \left(-z\right)} + \left(\frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{8 + \left(-z\right)} + \left(\frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(-z\right) + 7} + \frac{-0.1385710952657201178173096423051902092993}{6 + \left(-z\right)}\right)\right)\right)\right)}{e^{0.5 + \left(\left(-z\right) + 7\right)}} \cdot \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(0.5 + \left(\left(-z\right) + 7\right)\right)}^{\left(\left(-z\right) + 0.5\right)}\right)}\]
  3. Using strategy rm

  4. Applied add-exp-log2.8

    \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{-1259.139216722402807135949842631816864014}{\left(-z\right) + 2} + \left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{1 - z}\right) + \frac{-176.6150291621405870046146446838974952698}{4 + \left(-z\right)}\right) + \frac{771.3234287776531346025876700878143310547}{3 + \left(-z\right)}\right) + \left(\frac{12.50734327868690520801919774385169148445}{5 + \left(-z\right)} + \left(\frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{8 + \left(-z\right)} + \left(\frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(-z\right) + 7} + \frac{-0.1385710952657201178173096423051902092993}{6 + \left(-z\right)}\right)\right)\right)\right)\right)}}}{e^{0.5 + \left(\left(-z\right) + 7\right)}} \cdot \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(0.5 + \left(\left(-z\right) + 7\right)\right)}^{\left(\left(-z\right) + 0.5\right)}\right)\]

  5. Applied div-exp2.8

    \[\leadsto \color{blue}{e^{\log \left(\frac{-1259.139216722402807135949842631816864014}{\left(-z\right) + 2} + \left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{1 - z}\right) + \frac{-176.6150291621405870046146446838974952698}{4 + \left(-z\right)}\right) + \frac{771.3234287776531346025876700878143310547}{3 + \left(-z\right)}\right) + \left(\frac{12.50734327868690520801919774385169148445}{5 + \left(-z\right)} + \left(\frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{8 + \left(-z\right)} + \left(\frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(-z\right) + 7} + \frac{-0.1385710952657201178173096423051902092993}{6 + \left(-z\right)}\right)\right)\right)\right)\right) - \left(0.5 + \left(\left(-z\right) + 7\right)\right)}} \cdot \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(0.5 + \left(\left(-z\right) + 7\right)\right)}^{\left(\left(-z\right) + 0.5\right)}\right)\]

  6. Simplified0.7

    \[\leadsto e^{\color{blue}{\log \left(\left(\left(\left(\frac{771.3234287776531346025876700878143310547}{3 - z} + \left(\frac{-176.6150291621405870046146446838974952698}{4 - z} + \left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{1 - z}\right)\right)\right) + \frac{12.50734327868690520801919774385169148445}{5 - z}\right) + \left(\frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{8 - z} + \left(\frac{-0.1385710952657201178173096423051902092993}{6 - z} + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(-z\right) + 7}\right)\right)\right) + \frac{-1259.139216722402807135949842631816864014}{\left(-z\right) + 2}\right) - \left(0.5 + \left(\left(-z\right) + 7\right)\right)}} \cdot \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(0.5 + \left(\left(-z\right) + 7\right)\right)}^{\left(\left(-z\right) + 0.5\right)}\right)\]

  7. Final simplification0.7

    \[\leadsto e^{\log \left(\left(\left(\left(\frac{771.3234287776531346025876700878143310547}{3 - z} + \left(\frac{-176.6150291621405870046146446838974952698}{4 - z} + \left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{1 - z}\right)\right)\right) + \frac{12.50734327868690520801919774385169148445}{5 - z}\right) + \left(\frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{8 - z} + \left(\frac{-0.1385710952657201178173096423051902092993}{6 - z} + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(-z\right) + 7}\right)\right)\right) + \frac{-1259.139216722402807135949842631816864014}{\left(-z\right) + 2}\right) - \left(0.5 + \left(\left(-z\right) + 7\right)\right)} \cdot \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot {\left(0.5 + \left(\left(-z\right) + 7\right)\right)}^{\left(\left(-z\right) + 0.5\right)}\right)\]

Reproduce

herbie shell --seed 2019306 +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)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.99999999999980993 (/ 676.520368121885099 (+ (- (- 1 z) 1) 1))) (/ -1259.13921672240281 (+ (- (- 1 z) 1) 2))) (/ 771.32342877765313 (+ (- (- 1 z) 1) 3))) (/ -176.615029162140587 (+ (- (- 1 z) 1) 4))) (/ 12.5073432786869052 (+ (- (- 1 z) 1) 5))) (/ -0.138571095265720118 (+ (- (- 1 z) 1) 6))) (/ 9.98436957801957158e-6 (+ (- (- 1 z) 1) 7))) (/ 1.50563273514931162e-7 (+ (- (- 1 z) 1) 8))))))