Jmat.Real.gamma, branch z less than 0.5

Average Error: 1.8 → 0.5

Time: 2.9m

Precision: 64

• could not determine a ground truth for program body

  1. z = -1.7033689761779905e+269

Math

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

↓

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

C

double f(double z) {
        double r7692750 = atan2(1.0, 0.0);
        double r7692751 = z;
        double r7692752 = r7692750 * r7692751;
        double r7692753 = sin(r7692752);
        double r7692754 = r7692750 / r7692753;
        double r7692755 = 2.0;
        double r7692756 = r7692750 * r7692755;
        double r7692757 = sqrt(r7692756);
        double r7692758 = 1.0;
        double r7692759 = r7692758 - r7692751;
        double r7692760 = r7692759 - r7692758;
        double r7692761 = 7.0;
        double r7692762 = r7692760 + r7692761;
        double r7692763 = 0.5;
        double r7692764 = r7692762 + r7692763;
        double r7692765 = r7692760 + r7692763;
        double r7692766 = pow(r7692764, r7692765);
        double r7692767 = r7692757 * r7692766;
        double r7692768 = -r7692764;
        double r7692769 = exp(r7692768);
        double r7692770 = r7692767 * r7692769;
        double r7692771 = 0.9999999999998099;
        double r7692772 = 676.5203681218851;
        double r7692773 = r7692760 + r7692758;
        double r7692774 = r7692772 / r7692773;
        double r7692775 = r7692771 + r7692774;
        double r7692776 = -1259.1392167224028;
        double r7692777 = r7692760 + r7692755;
        double r7692778 = r7692776 / r7692777;
        double r7692779 = r7692775 + r7692778;
        double r7692780 = 771.3234287776531;
        double r7692781 = 3.0;
        double r7692782 = r7692760 + r7692781;
        double r7692783 = r7692780 / r7692782;
        double r7692784 = r7692779 + r7692783;
        double r7692785 = -176.6150291621406;
        double r7692786 = 4.0;
        double r7692787 = r7692760 + r7692786;
        double r7692788 = r7692785 / r7692787;
        double r7692789 = r7692784 + r7692788;
        double r7692790 = 12.507343278686905;
        double r7692791 = 5.0;
        double r7692792 = r7692760 + r7692791;
        double r7692793 = r7692790 / r7692792;
        double r7692794 = r7692789 + r7692793;
        double r7692795 = -0.13857109526572012;
        double r7692796 = 6.0;
        double r7692797 = r7692760 + r7692796;
        double r7692798 = r7692795 / r7692797;
        double r7692799 = r7692794 + r7692798;
        double r7692800 = 9.984369578019572e-06;
        double r7692801 = r7692800 / r7692762;
        double r7692802 = r7692799 + r7692801;
        double r7692803 = 1.5056327351493116e-07;
        double r7692804 = 8.0;
        double r7692805 = r7692760 + r7692804;
        double r7692806 = r7692803 / r7692805;
        double r7692807 = r7692802 + r7692806;
        double r7692808 = r7692770 * r7692807;
        double r7692809 = r7692754 * r7692808;
        return r7692809;
}

↓

double f(double z) {
        double r7692810 = atan2(1.0, 0.0);
        double r7692811 = z;
        double r7692812 = r7692811 * r7692810;
        double r7692813 = sin(r7692812);
        double r7692814 = r7692810 / r7692813;
        double r7692815 = 12.507343278686905;
        double r7692816 = 5.0;
        double r7692817 = r7692816 - r7692811;
        double r7692818 = r7692815 / r7692817;
        double r7692819 = -176.6150291621406;
        double r7692820 = 2.0;
        double r7692821 = 1.0;
        double r7692822 = r7692821 - r7692811;
        double r7692823 = r7692820 + r7692822;
        double r7692824 = r7692823 + r7692821;
        double r7692825 = r7692819 / r7692824;
        double r7692826 = 771.3234287776531;
        double r7692827 = r7692826 / r7692823;
        double r7692828 = r7692825 + r7692827;
        double r7692829 = -0.13857109526572012;
        double r7692830 = 6.0;
        double r7692831 = r7692830 - r7692811;
        double r7692832 = r7692829 / r7692831;
        double r7692833 = r7692828 + r7692832;
        double r7692834 = 0.9999999999998099;
        double r7692835 = 676.5203681218851;
        double r7692836 = r7692835 / r7692822;
        double r7692837 = -1259.1392167224028;
        double r7692838 = r7692820 - r7692811;
        double r7692839 = r7692837 / r7692838;
        double r7692840 = r7692836 + r7692839;
        double r7692841 = r7692834 + r7692840;
        double r7692842 = r7692833 + r7692841;
        double r7692843 = 3.0;
        double r7692844 = pow(r7692842, r7692843);
        double r7692845 = 1.5056327351493116e-07;
        double r7692846 = 8.0;
        double r7692847 = r7692846 - r7692811;
        double r7692848 = r7692845 / r7692847;
        double r7692849 = 9.984369578019572e-06;
        double r7692850 = 7.0;
        double r7692851 = r7692850 - r7692811;
        double r7692852 = r7692849 / r7692851;
        double r7692853 = r7692848 + r7692852;
        double r7692854 = pow(r7692853, r7692843);
        double r7692855 = r7692844 + r7692854;
        double r7692856 = r7692842 * r7692842;
        double r7692857 = r7692842 * r7692853;
        double r7692858 = r7692856 - r7692857;
        double r7692859 = r7692853 * r7692853;
        double r7692860 = r7692858 + r7692859;
        double r7692861 = r7692855 / r7692860;
        double r7692862 = r7692818 + r7692861;
        double r7692863 = 0.5;
        double r7692864 = r7692863 + r7692851;
        double r7692865 = -r7692811;
        double r7692866 = r7692863 + r7692865;
        double r7692867 = pow(r7692864, r7692866);
        double r7692868 = exp(r7692864);
        double r7692869 = r7692867 / r7692868;
        double r7692870 = r7692862 * r7692869;
        double r7692871 = r7692814 * r7692870;
        double r7692872 = r7692820 * r7692810;
        double r7692873 = sqrt(r7692872);
        double r7692874 = r7692871 * r7692873;
        return r7692874;
}

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. Simplified 0.5

   \[\leadsto \color{blue}{\sqrt{2 \cdot \pi} \cdot \left(\left(\frac{{\left(\left(7 - z\right) + 0.5\right)}^{\left(0.5 + \left(0 - z\right)\right)}}{e^{\left(7 - z\right) + 0.5}} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) + \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{1 + \left(\left(1 - z\right) + 2\right)}\right)\right) + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right) + \frac{12.507343278686905}{5 - z}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)}\]
3. Using strategy rm

4. Applied flip3-+ 0.5

   \[\leadsto \sqrt{2 \cdot \pi} \cdot \left(\left(\frac{{\left(\left(7 - z\right) + 0.5\right)}^{\left(0.5 + \left(0 - z\right)\right)}}{e^{\left(7 - z\right) + 0.5}} \cdot \left(\color{blue}{\frac{{\left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right)}^{3} + {\left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{1 + \left(\left(1 - z\right) + 2\right)}\right)\right) + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)}^{3}}{\left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) + \left(\left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{1 + \left(\left(1 - z\right) + 2\right)}\right)\right) + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{1 + \left(\left(1 - z\right) + 2\right)}\right)\right) + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right) - \left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right) \cdot \left(\left(\frac{-0.13857109526572012}{6 - z} + \left(\frac{771.3234287776531}{\left(1 - z\right) + 2} + \frac{-176.6150291621406}{1 + \left(\left(1 - z\right) + 2\right)}\right)\right) + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right)\right)\right)}} + \frac{12.507343278686905}{5 - z}\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\]

5. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019158 +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))))))