Average Error: 1.8 → 1.8
Time: 52.8s
Precision: 64
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
\frac{\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{\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)
double f(double z) {
        double r114723 = atan2(1.0, 0.0);
        double r114724 = z;
        double r114725 = r114723 * r114724;
        double r114726 = sin(r114725);
        double r114727 = r114723 / r114726;
        double r114728 = 2.0;
        double r114729 = r114723 * r114728;
        double r114730 = sqrt(r114729);
        double r114731 = 1.0;
        double r114732 = r114731 - r114724;
        double r114733 = r114732 - r114731;
        double r114734 = 7.0;
        double r114735 = r114733 + r114734;
        double r114736 = 0.5;
        double r114737 = r114735 + r114736;
        double r114738 = r114733 + r114736;
        double r114739 = pow(r114737, r114738);
        double r114740 = r114730 * r114739;
        double r114741 = -r114737;
        double r114742 = exp(r114741);
        double r114743 = r114740 * r114742;
        double r114744 = 0.9999999999998099;
        double r114745 = 676.5203681218851;
        double r114746 = r114733 + r114731;
        double r114747 = r114745 / r114746;
        double r114748 = r114744 + r114747;
        double r114749 = -1259.1392167224028;
        double r114750 = r114733 + r114728;
        double r114751 = r114749 / r114750;
        double r114752 = r114748 + r114751;
        double r114753 = 771.3234287776531;
        double r114754 = 3.0;
        double r114755 = r114733 + r114754;
        double r114756 = r114753 / r114755;
        double r114757 = r114752 + r114756;
        double r114758 = -176.6150291621406;
        double r114759 = 4.0;
        double r114760 = r114733 + r114759;
        double r114761 = r114758 / r114760;
        double r114762 = r114757 + r114761;
        double r114763 = 12.507343278686905;
        double r114764 = 5.0;
        double r114765 = r114733 + r114764;
        double r114766 = r114763 / r114765;
        double r114767 = r114762 + r114766;
        double r114768 = -0.13857109526572012;
        double r114769 = 6.0;
        double r114770 = r114733 + r114769;
        double r114771 = r114768 / r114770;
        double r114772 = r114767 + r114771;
        double r114773 = 9.984369578019572e-06;
        double r114774 = r114773 / r114735;
        double r114775 = r114772 + r114774;
        double r114776 = 1.5056327351493116e-07;
        double r114777 = 8.0;
        double r114778 = r114733 + r114777;
        double r114779 = r114776 / r114778;
        double r114780 = r114775 + r114779;
        double r114781 = r114743 * r114780;
        double r114782 = r114727 * r114781;
        return r114782;
}
↓
double f(double z) {
        double r114783 = atan2(1.0, 0.0);
        double r114784 = z;
        double r114785 = r114783 * r114784;
        double r114786 = sin(r114785);
        double r114787 = r114783 / r114786;
        double r114788 = 2.0;
        double r114789 = r114783 * r114788;
        double r114790 = sqrt(r114789);
        double r114791 = 1.0;
        double r114792 = r114791 - r114784;
        double r114793 = r114792 - r114791;
        double r114794 = 7.0;
        double r114795 = r114793 + r114794;
        double r114796 = 0.5;
        double r114797 = r114795 + r114796;
        double r114798 = r114793 + r114796;
        double r114799 = pow(r114797, r114798);
        double r114800 = r114790 * r114799;
        double r114801 = -r114797;
        double r114802 = exp(r114801);
        double r114803 = r114800 * r114802;
        double r114804 = 0.9999999999998099;
        double r114805 = 676.5203681218851;
        double r114806 = r114793 + r114791;
        double r114807 = r114805 / r114806;
        double r114808 = r114804 + r114807;
        double r114809 = -1259.1392167224028;
        double r114810 = r114793 + r114788;
        double r114811 = r114809 / r114810;
        double r114812 = r114808 + r114811;
        double r114813 = 771.3234287776531;
        double r114814 = 3.0;
        double r114815 = r114793 + r114814;
        double r114816 = r114813 / r114815;
        double r114817 = r114812 + r114816;
        double r114818 = -176.6150291621406;
        double r114819 = 4.0;
        double r114820 = r114793 + r114819;
        double r114821 = r114818 / r114820;
        double r114822 = r114817 + r114821;
        double r114823 = 12.507343278686905;
        double r114824 = 5.0;
        double r114825 = r114793 + r114824;
        double r114826 = r114823 / r114825;
        double r114827 = r114822 + r114826;
        double r114828 = -0.13857109526572012;
        double r114829 = 6.0;
        double r114830 = r114793 + r114829;
        double r114831 = r114828 / r114830;
        double r114832 = r114827 + r114831;
        double r114833 = 9.984369578019572e-06;
        double r114834 = r114833 / r114795;
        double r114835 = r114832 + r114834;
        double r114836 = 1.5056327351493116e-07;
        double r114837 = 8.0;
        double r114838 = r114793 + r114837;
        double r114839 = r114836 / r114838;
        double r114840 = r114835 + r114839;
        double r114841 = r114803 * r114840;
        double r114842 = r114787 * r114841;
        return r114842;
}

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. Final simplification1.8

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

Reproduce

herbie shell --seed 2020046 
(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))))))