Average Error: 1.8 → 1.8
Time: 2.3m
Precision: 64
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\left(\left(\left(\sqrt{2 \cdot \pi} \cdot {\left(\left(7 + \left(\left(1 - z\right) - 1\right)\right) + 0.5\right)}^{\left(0.5 + \left(\left(1 - z\right) - 1\right)\right)}\right) \cdot e^{-\left(\left(7 + \left(\left(1 - z\right) - 1\right)\right) + 0.5\right)}\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(\left(1 - z\right) - 1\right)} + \left(\left(\left(\frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4} + \left(\frac{771.3234287776531}{3 + \left(\left(1 - z\right) - 1\right)} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{6 + \left(\left(1 - z\right) - 1\right)}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(\left(1 - z\right) - 1\right)\right) + 8}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\]
\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(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)
↓
\left(\left(\left(\sqrt{2 \cdot \pi} \cdot {\left(\left(7 + \left(\left(1 - z\right) - 1\right)\right) + 0.5\right)}^{\left(0.5 + \left(\left(1 - z\right) - 1\right)\right)}\right) \cdot e^{-\left(\left(7 + \left(\left(1 - z\right) - 1\right)\right) + 0.5\right)}\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(\left(1 - z\right) - 1\right)} + \left(\left(\left(\frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4} + \left(\frac{771.3234287776531}{3 + \left(\left(1 - z\right) - 1\right)} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{6 + \left(\left(1 - z\right) - 1\right)}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(\left(1 - z\right) - 1\right)\right) + 8}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}
double f(double z) {
        double r2409705 = atan2(1.0, 0.0);
        double r2409706 = z;
        double r2409707 = r2409705 * r2409706;
        double r2409708 = sin(r2409707);
        double r2409709 = r2409705 / r2409708;
        double r2409710 = 2.0;
        double r2409711 = r2409705 * r2409710;
        double r2409712 = sqrt(r2409711);
        double r2409713 = 1.0;
        double r2409714 = r2409713 - r2409706;
        double r2409715 = r2409714 - r2409713;
        double r2409716 = 7.0;
        double r2409717 = r2409715 + r2409716;
        double r2409718 = 0.5;
        double r2409719 = r2409717 + r2409718;
        double r2409720 = r2409715 + r2409718;
        double r2409721 = pow(r2409719, r2409720);
        double r2409722 = r2409712 * r2409721;
        double r2409723 = -r2409719;
        double r2409724 = exp(r2409723);
        double r2409725 = r2409722 * r2409724;
        double r2409726 = 0.9999999999998099;
        double r2409727 = 676.5203681218851;
        double r2409728 = r2409715 + r2409713;
        double r2409729 = r2409727 / r2409728;
        double r2409730 = r2409726 + r2409729;
        double r2409731 = -1259.1392167224028;
        double r2409732 = r2409715 + r2409710;
        double r2409733 = r2409731 / r2409732;
        double r2409734 = r2409730 + r2409733;
        double r2409735 = 771.3234287776531;
        double r2409736 = 3.0;
        double r2409737 = r2409715 + r2409736;
        double r2409738 = r2409735 / r2409737;
        double r2409739 = r2409734 + r2409738;
        double r2409740 = -176.6150291621406;
        double r2409741 = 4.0;
        double r2409742 = r2409715 + r2409741;
        double r2409743 = r2409740 / r2409742;
        double r2409744 = r2409739 + r2409743;
        double r2409745 = 12.507343278686905;
        double r2409746 = 5.0;
        double r2409747 = r2409715 + r2409746;
        double r2409748 = r2409745 / r2409747;
        double r2409749 = r2409744 + r2409748;
        double r2409750 = -0.13857109526572012;
        double r2409751 = 6.0;
        double r2409752 = r2409715 + r2409751;
        double r2409753 = r2409750 / r2409752;
        double r2409754 = r2409749 + r2409753;
        double r2409755 = 9.984369578019572e-06;
        double r2409756 = r2409755 / r2409717;
        double r2409757 = r2409754 + r2409756;
        double r2409758 = 1.5056327351493116e-07;
        double r2409759 = 8.0;
        double r2409760 = r2409715 + r2409759;
        double r2409761 = r2409758 / r2409760;
        double r2409762 = r2409757 + r2409761;
        double r2409763 = r2409725 * r2409762;
        double r2409764 = r2409709 * r2409763;
        return r2409764;
}
↓
double f(double z) {
        double r2409765 = 2.0;
        double r2409766 = atan2(1.0, 0.0);
        double r2409767 = r2409765 * r2409766;
        double r2409768 = sqrt(r2409767);
        double r2409769 = 7.0;
        double r2409770 = 1.0;
        double r2409771 = z;
        double r2409772 = r2409770 - r2409771;
        double r2409773 = r2409772 - r2409770;
        double r2409774 = r2409769 + r2409773;
        double r2409775 = 0.5;
        double r2409776 = r2409774 + r2409775;
        double r2409777 = r2409775 + r2409773;
        double r2409778 = pow(r2409776, r2409777);
        double r2409779 = r2409768 * r2409778;
        double r2409780 = -r2409776;
        double r2409781 = exp(r2409780);
        double r2409782 = r2409779 * r2409781;
        double r2409783 = 9.984369578019572e-06;
        double r2409784 = r2409783 / r2409774;
        double r2409785 = -176.6150291621406;
        double r2409786 = 4.0;
        double r2409787 = r2409773 + r2409786;
        double r2409788 = r2409785 / r2409787;
        double r2409789 = 771.3234287776531;
        double r2409790 = 3.0;
        double r2409791 = r2409790 + r2409773;
        double r2409792 = r2409789 / r2409791;
        double r2409793 = 0.9999999999998099;
        double r2409794 = 676.5203681218851;
        double r2409795 = r2409773 + r2409770;
        double r2409796 = r2409794 / r2409795;
        double r2409797 = r2409793 + r2409796;
        double r2409798 = -1259.1392167224028;
        double r2409799 = r2409773 + r2409765;
        double r2409800 = r2409798 / r2409799;
        double r2409801 = r2409797 + r2409800;
        double r2409802 = r2409792 + r2409801;
        double r2409803 = r2409788 + r2409802;
        double r2409804 = 12.507343278686905;
        double r2409805 = 5.0;
        double r2409806 = r2409773 + r2409805;
        double r2409807 = r2409804 / r2409806;
        double r2409808 = r2409803 + r2409807;
        double r2409809 = -0.13857109526572012;
        double r2409810 = 6.0;
        double r2409811 = r2409810 + r2409773;
        double r2409812 = r2409809 / r2409811;
        double r2409813 = r2409808 + r2409812;
        double r2409814 = r2409784 + r2409813;
        double r2409815 = 1.5056327351493116e-07;
        double r2409816 = /* ERROR: no posit support in C */;
        double r2409817 = /* ERROR: no posit support in C */;
        double r2409818 = 8.0;
        double r2409819 = r2409817 + r2409818;
        double r2409820 = r2409815 / r2409819;
        double r2409821 = r2409814 + r2409820;
        double r2409822 = r2409782 * r2409821;
        double r2409823 = r2409766 * r2409771;
        double r2409824 = sin(r2409823);
        double r2409825 = r2409766 / r2409824;
        double r2409826 = r2409822 * r2409825;
        return r2409826;
}

Error

Bits error versus z

Derivation

  1. Initial program 1.8

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

  3. Applied insert-posit161.8

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

  4. Final simplification1.8

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (z)
  :name "Jmat.Real.gamma, branch z less than 0.5"
  (* (/ 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))))))