Average Error: 1.8 → 1.8
Time: 53.3s
Precision: 64
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531346025876700878143310547}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.50734327868690520801919774385169148445}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.1385710952657201178173096423051902092993}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531346025876700878143310547}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.50734327868690520801919774385169148445}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.1385710952657201178173096423051902092993}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
\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)
↓
\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)
double f(double z) {
        double r145695 = atan2(1.0, 0.0);
        double r145696 = z;
        double r145697 = r145695 * r145696;
        double r145698 = sin(r145697);
        double r145699 = r145695 / r145698;
        double r145700 = 2.0;
        double r145701 = r145695 * r145700;
        double r145702 = sqrt(r145701);
        double r145703 = 1.0;
        double r145704 = r145703 - r145696;
        double r145705 = r145704 - r145703;
        double r145706 = 7.0;
        double r145707 = r145705 + r145706;
        double r145708 = 0.5;
        double r145709 = r145707 + r145708;
        double r145710 = r145705 + r145708;
        double r145711 = pow(r145709, r145710);
        double r145712 = r145702 * r145711;
        double r145713 = -r145709;
        double r145714 = exp(r145713);
        double r145715 = r145712 * r145714;
        double r145716 = 0.9999999999998099;
        double r145717 = 676.5203681218851;
        double r145718 = r145705 + r145703;
        double r145719 = r145717 / r145718;
        double r145720 = r145716 + r145719;
        double r145721 = -1259.1392167224028;
        double r145722 = r145705 + r145700;
        double r145723 = r145721 / r145722;
        double r145724 = r145720 + r145723;
        double r145725 = 771.3234287776531;
        double r145726 = 3.0;
        double r145727 = r145705 + r145726;
        double r145728 = r145725 / r145727;
        double r145729 = r145724 + r145728;
        double r145730 = -176.6150291621406;
        double r145731 = 4.0;
        double r145732 = r145705 + r145731;
        double r145733 = r145730 / r145732;
        double r145734 = r145729 + r145733;
        double r145735 = 12.507343278686905;
        double r145736 = 5.0;
        double r145737 = r145705 + r145736;
        double r145738 = r145735 / r145737;
        double r145739 = r145734 + r145738;
        double r145740 = -0.13857109526572012;
        double r145741 = 6.0;
        double r145742 = r145705 + r145741;
        double r145743 = r145740 / r145742;
        double r145744 = r145739 + r145743;
        double r145745 = 9.984369578019572e-06;
        double r145746 = r145745 / r145707;
        double r145747 = r145744 + r145746;
        double r145748 = 1.5056327351493116e-07;
        double r145749 = 8.0;
        double r145750 = r145705 + r145749;
        double r145751 = r145748 / r145750;
        double r145752 = r145747 + r145751;
        double r145753 = r145715 * r145752;
        double r145754 = r145699 * r145753;
        return r145754;
}
↓
double f(double z) {
        double r145755 = atan2(1.0, 0.0);
        double r145756 = z;
        double r145757 = r145755 * r145756;
        double r145758 = sin(r145757);
        double r145759 = r145755 / r145758;
        double r145760 = 2.0;
        double r145761 = r145755 * r145760;
        double r145762 = sqrt(r145761);
        double r145763 = 1.0;
        double r145764 = r145763 - r145756;
        double r145765 = r145764 - r145763;
        double r145766 = 7.0;
        double r145767 = r145765 + r145766;
        double r145768 = 0.5;
        double r145769 = r145767 + r145768;
        double r145770 = r145765 + r145768;
        double r145771 = pow(r145769, r145770);
        double r145772 = r145762 * r145771;
        double r145773 = -r145769;
        double r145774 = exp(r145773);
        double r145775 = r145772 * r145774;
        double r145776 = 0.9999999999998099;
        double r145777 = 676.5203681218851;
        double r145778 = r145765 + r145763;
        double r145779 = r145777 / r145778;
        double r145780 = r145776 + r145779;
        double r145781 = -1259.1392167224028;
        double r145782 = r145765 + r145760;
        double r145783 = r145781 / r145782;
        double r145784 = r145780 + r145783;
        double r145785 = 771.3234287776531;
        double r145786 = 3.0;
        double r145787 = r145765 + r145786;
        double r145788 = r145785 / r145787;
        double r145789 = r145784 + r145788;
        double r145790 = -176.6150291621406;
        double r145791 = 4.0;
        double r145792 = r145765 + r145791;
        double r145793 = r145790 / r145792;
        double r145794 = r145789 + r145793;
        double r145795 = 12.507343278686905;
        double r145796 = 5.0;
        double r145797 = r145765 + r145796;
        double r145798 = r145795 / r145797;
        double r145799 = r145794 + r145798;
        double r145800 = -0.13857109526572012;
        double r145801 = 6.0;
        double r145802 = r145765 + r145801;
        double r145803 = r145800 / r145802;
        double r145804 = r145799 + r145803;
        double r145805 = 9.984369578019572e-06;
        double r145806 = r145805 / r145767;
        double r145807 = r145804 + r145806;
        double r145808 = 1.5056327351493116e-07;
        double r145809 = 8.0;
        double r145810 = r145765 + r145809;
        double r145811 = r145808 / r145810;
        double r145812 = r145807 + r145811;
        double r145813 = r145775 * r145812;
        double r145814 = r145759 * r145813;
        return r145814;
}

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. Final simplification1.8

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

Reproduce

herbie shell --seed 2019344 +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)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 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))))))