Average Error: 1.8 → 0.9
Time: 2.8m
Precision: 64
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\sqrt{2 \cdot \pi} \cdot \left(\left(\left(\left(\left(\frac{9.984369578019572 \cdot 10^{-06}}{7 - z} + \left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \left(\frac{-1259.1392167224028}{2 - z} + 0.9999999999998099\right)\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \cdot e^{\left(z + 0.5 \cdot \log \left(7.5 - z\right)\right) - \left(7.5 + z \cdot \log \left(7.5 - z\right)\right)}\right)\]
\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(0.9999999999998099 + \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)
↓
\sqrt{2 \cdot \pi} \cdot \left(\left(\left(\left(\left(\frac{9.984369578019572 \cdot 10^{-06}}{7 - z} + \left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \left(\frac{-1259.1392167224028}{2 - z} + 0.9999999999998099\right)\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \cdot e^{\left(z + 0.5 \cdot \log \left(7.5 - z\right)\right) - \left(7.5 + z \cdot \log \left(7.5 - z\right)\right)}\right)
double f(double z) {
        double r11768721 = atan2(1.0, 0.0);
        double r11768722 = z;
        double r11768723 = r11768721 * r11768722;
        double r11768724 = sin(r11768723);
        double r11768725 = r11768721 / r11768724;
        double r11768726 = 2.0;
        double r11768727 = r11768721 * r11768726;
        double r11768728 = sqrt(r11768727);
        double r11768729 = 1.0;
        double r11768730 = r11768729 - r11768722;
        double r11768731 = r11768730 - r11768729;
        double r11768732 = 7.0;
        double r11768733 = r11768731 + r11768732;
        double r11768734 = 0.5;
        double r11768735 = r11768733 + r11768734;
        double r11768736 = r11768731 + r11768734;
        double r11768737 = pow(r11768735, r11768736);
        double r11768738 = r11768728 * r11768737;
        double r11768739 = -r11768735;
        double r11768740 = exp(r11768739);
        double r11768741 = r11768738 * r11768740;
        double r11768742 = 0.9999999999998099;
        double r11768743 = 676.5203681218851;
        double r11768744 = r11768731 + r11768729;
        double r11768745 = r11768743 / r11768744;
        double r11768746 = r11768742 + r11768745;
        double r11768747 = -1259.1392167224028;
        double r11768748 = r11768731 + r11768726;
        double r11768749 = r11768747 / r11768748;
        double r11768750 = r11768746 + r11768749;
        double r11768751 = 771.3234287776531;
        double r11768752 = 3.0;
        double r11768753 = r11768731 + r11768752;
        double r11768754 = r11768751 / r11768753;
        double r11768755 = r11768750 + r11768754;
        double r11768756 = -176.6150291621406;
        double r11768757 = 4.0;
        double r11768758 = r11768731 + r11768757;
        double r11768759 = r11768756 / r11768758;
        double r11768760 = r11768755 + r11768759;
        double r11768761 = 12.507343278686905;
        double r11768762 = 5.0;
        double r11768763 = r11768731 + r11768762;
        double r11768764 = r11768761 / r11768763;
        double r11768765 = r11768760 + r11768764;
        double r11768766 = -0.13857109526572012;
        double r11768767 = 6.0;
        double r11768768 = r11768731 + r11768767;
        double r11768769 = r11768766 / r11768768;
        double r11768770 = r11768765 + r11768769;
        double r11768771 = 9.984369578019572e-06;
        double r11768772 = r11768771 / r11768733;
        double r11768773 = r11768770 + r11768772;
        double r11768774 = 1.5056327351493116e-07;
        double r11768775 = 8.0;
        double r11768776 = r11768731 + r11768775;
        double r11768777 = r11768774 / r11768776;
        double r11768778 = r11768773 + r11768777;
        double r11768779 = r11768741 * r11768778;
        double r11768780 = r11768725 * r11768779;
        return r11768780;
}
↓
double f(double z) {
        double r11768781 = 2.0;
        double r11768782 = atan2(1.0, 0.0);
        double r11768783 = r11768781 * r11768782;
        double r11768784 = sqrt(r11768783);
        double r11768785 = 9.984369578019572e-06;
        double r11768786 = 7.0;
        double r11768787 = z;
        double r11768788 = r11768786 - r11768787;
        double r11768789 = r11768785 / r11768788;
        double r11768790 = 12.507343278686905;
        double r11768791 = 5.0;
        double r11768792 = r11768791 - r11768787;
        double r11768793 = r11768790 / r11768792;
        double r11768794 = -0.13857109526572012;
        double r11768795 = 6.0;
        double r11768796 = r11768795 - r11768787;
        double r11768797 = r11768794 / r11768796;
        double r11768798 = r11768793 + r11768797;
        double r11768799 = r11768789 + r11768798;
        double r11768800 = -176.6150291621406;
        double r11768801 = 4.0;
        double r11768802 = r11768801 - r11768787;
        double r11768803 = r11768800 / r11768802;
        double r11768804 = r11768799 + r11768803;
        double r11768805 = 1.5056327351493116e-07;
        double r11768806 = 8.0;
        double r11768807 = r11768806 - r11768787;
        double r11768808 = r11768805 / r11768807;
        double r11768809 = 676.5203681218851;
        double r11768810 = 1.0;
        double r11768811 = r11768810 - r11768787;
        double r11768812 = r11768809 / r11768811;
        double r11768813 = 771.3234287776531;
        double r11768814 = 3.0;
        double r11768815 = r11768814 - r11768787;
        double r11768816 = r11768813 / r11768815;
        double r11768817 = -1259.1392167224028;
        double r11768818 = r11768781 - r11768787;
        double r11768819 = r11768817 / r11768818;
        double r11768820 = 0.9999999999998099;
        double r11768821 = r11768819 + r11768820;
        double r11768822 = r11768816 + r11768821;
        double r11768823 = r11768812 + r11768822;
        double r11768824 = r11768808 + r11768823;
        double r11768825 = r11768804 + r11768824;
        double r11768826 = r11768787 * r11768782;
        double r11768827 = sin(r11768826);
        double r11768828 = r11768782 / r11768827;
        double r11768829 = r11768825 * r11768828;
        double r11768830 = 0.5;
        double r11768831 = 7.5;
        double r11768832 = r11768831 - r11768787;
        double r11768833 = log(r11768832);
        double r11768834 = r11768830 * r11768833;
        double r11768835 = r11768787 + r11768834;
        double r11768836 = r11768787 * r11768833;
        double r11768837 = r11768831 + r11768836;
        double r11768838 = r11768835 - r11768837;
        double r11768839 = exp(r11768838);
        double r11768840 = r11768829 * r11768839;
        double r11768841 = r11768784 * r11768840;
        return r11768841;
}

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

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

  2. Simplified1.3

    \[\leadsto \color{blue}{\sqrt{2 \cdot \pi} \cdot \left(\frac{{\left(\left(7 - z\right) + 0.5\right)}^{\left(\left(0 - z\right) + 0.5\right)}}{e^{\left(7 - z\right) + 0.5}} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(\left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}\right) + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-176.6150291621406}{4 - z} + \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{12.507343278686905}{5 - z}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right)}\]

  3. Taylor expanded around inf 1.3

    \[\leadsto \sqrt{2 \cdot \pi} \cdot \left(\color{blue}{\frac{e^{\log \left(7.5 - z\right) \cdot \left(0.5 - z\right)}}{e^{7.5 - z}}} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(\left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}\right) + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-176.6150291621406}{4 - z} + \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{12.507343278686905}{5 - z}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right)\]

  4. Simplified0.9

    \[\leadsto \sqrt{2 \cdot \pi} \cdot \left(\color{blue}{e^{\log \left(7.5 - z\right) \cdot \left(0.5 - z\right) - \left(7.5 - z\right)}} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(\left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}\right) + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-176.6150291621406}{4 - z} + \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{12.507343278686905}{5 - z}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right)\]

  5. Taylor expanded around inf 0.9

    \[\leadsto \sqrt{2 \cdot \pi} \cdot \left(\color{blue}{e^{\left(0.5 \cdot \log \left(7.5 - z\right) + z\right) - \left(\log \left(7.5 - z\right) \cdot z + 7.5\right)}} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(\left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}\right) + \frac{771.3234287776531}{3 - z}\right)\right)\right) + \left(\frac{-176.6150291621406}{4 - z} + \left(\left(\frac{-0.13857109526572012}{6 - z} + \frac{12.507343278686905}{5 - z}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right)\]

  6. Final simplification0.9

    \[\leadsto \sqrt{2 \cdot \pi} \cdot \left(\left(\left(\left(\left(\frac{9.984369578019572 \cdot 10^{-06}}{7 - z} + \left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right) + \frac{-176.6150291621406}{4 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \left(\frac{676.5203681218851}{1 - z} + \left(\frac{771.3234287776531}{3 - z} + \left(\frac{-1259.1392167224028}{2 - z} + 0.9999999999998099\right)\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(z \cdot \pi\right)}\right) \cdot e^{\left(z + 0.5 \cdot \log \left(7.5 - z\right)\right) - \left(7.5 + z \cdot \log \left(7.5 - z\right)\right)}\right)\]

Reproduce

herbie shell --seed 2019158 
(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))))))