Average Error: 1.8 → 1.8
Time: 2.0m
Precision: 64
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\left(\left(\left(\sqrt{2 \cdot \pi} \cdot {\left(\left(7 + \left(\left(1 - z\right) - 1\right)\right) + 0.5\right)}^{\left(0.5 + \left(\left(1 - z\right) - 1\right)\right)}\right) \cdot e^{-\left(\left(7 + \left(\left(1 - z\right) - 1\right)\right) + 0.5\right)}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 + \left(\left(1 - z\right) - 1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(\left(1 - z\right) - 1\right)} + \left(\left(\left(\frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4} + \left(\frac{771.3234287776531}{3 + \left(\left(1 - z\right) - 1\right)} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{6 + \left(\left(1 - z\right) - 1\right)}\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\]
\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(\frac{1.5056327351493116 \cdot 10^{-07}}{8 + \left(\left(1 - z\right) - 1\right)} + \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)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}
double f(double z) {
        double r1678740 = atan2(1.0, 0.0);
        double r1678741 = z;
        double r1678742 = r1678740 * r1678741;
        double r1678743 = sin(r1678742);
        double r1678744 = r1678740 / r1678743;
        double r1678745 = 2.0;
        double r1678746 = r1678740 * r1678745;
        double r1678747 = sqrt(r1678746);
        double r1678748 = 1.0;
        double r1678749 = r1678748 - r1678741;
        double r1678750 = r1678749 - r1678748;
        double r1678751 = 7.0;
        double r1678752 = r1678750 + r1678751;
        double r1678753 = 0.5;
        double r1678754 = r1678752 + r1678753;
        double r1678755 = r1678750 + r1678753;
        double r1678756 = pow(r1678754, r1678755);
        double r1678757 = r1678747 * r1678756;
        double r1678758 = -r1678754;
        double r1678759 = exp(r1678758);
        double r1678760 = r1678757 * r1678759;
        double r1678761 = 0.9999999999998099;
        double r1678762 = 676.5203681218851;
        double r1678763 = r1678750 + r1678748;
        double r1678764 = r1678762 / r1678763;
        double r1678765 = r1678761 + r1678764;
        double r1678766 = -1259.1392167224028;
        double r1678767 = r1678750 + r1678745;
        double r1678768 = r1678766 / r1678767;
        double r1678769 = r1678765 + r1678768;
        double r1678770 = 771.3234287776531;
        double r1678771 = 3.0;
        double r1678772 = r1678750 + r1678771;
        double r1678773 = r1678770 / r1678772;
        double r1678774 = r1678769 + r1678773;
        double r1678775 = -176.6150291621406;
        double r1678776 = 4.0;
        double r1678777 = r1678750 + r1678776;
        double r1678778 = r1678775 / r1678777;
        double r1678779 = r1678774 + r1678778;
        double r1678780 = 12.507343278686905;
        double r1678781 = 5.0;
        double r1678782 = r1678750 + r1678781;
        double r1678783 = r1678780 / r1678782;
        double r1678784 = r1678779 + r1678783;
        double r1678785 = -0.13857109526572012;
        double r1678786 = 6.0;
        double r1678787 = r1678750 + r1678786;
        double r1678788 = r1678785 / r1678787;
        double r1678789 = r1678784 + r1678788;
        double r1678790 = 9.984369578019572e-06;
        double r1678791 = r1678790 / r1678752;
        double r1678792 = r1678789 + r1678791;
        double r1678793 = 1.5056327351493116e-07;
        double r1678794 = 8.0;
        double r1678795 = r1678750 + r1678794;
        double r1678796 = r1678793 / r1678795;
        double r1678797 = r1678792 + r1678796;
        double r1678798 = r1678760 * r1678797;
        double r1678799 = r1678744 * r1678798;
        return r1678799;
}
↓
double f(double z) {
        double r1678800 = 2.0;
        double r1678801 = atan2(1.0, 0.0);
        double r1678802 = r1678800 * r1678801;
        double r1678803 = sqrt(r1678802);
        double r1678804 = 7.0;
        double r1678805 = 1.0;
        double r1678806 = z;
        double r1678807 = r1678805 - r1678806;
        double r1678808 = r1678807 - r1678805;
        double r1678809 = r1678804 + r1678808;
        double r1678810 = 0.5;
        double r1678811 = r1678809 + r1678810;
        double r1678812 = r1678810 + r1678808;
        double r1678813 = pow(r1678811, r1678812);
        double r1678814 = r1678803 * r1678813;
        double r1678815 = -r1678811;
        double r1678816 = exp(r1678815);
        double r1678817 = r1678814 * r1678816;
        double r1678818 = 1.5056327351493116e-07;
        double r1678819 = 8.0;
        double r1678820 = r1678819 + r1678808;
        double r1678821 = r1678818 / r1678820;
        double r1678822 = 9.984369578019572e-06;
        double r1678823 = r1678822 / r1678809;
        double r1678824 = -176.6150291621406;
        double r1678825 = 4.0;
        double r1678826 = r1678808 + r1678825;
        double r1678827 = r1678824 / r1678826;
        double r1678828 = 771.3234287776531;
        double r1678829 = 3.0;
        double r1678830 = r1678829 + r1678808;
        double r1678831 = r1678828 / r1678830;
        double r1678832 = 0.9999999999998099;
        double r1678833 = 676.5203681218851;
        double r1678834 = r1678808 + r1678805;
        double r1678835 = r1678833 / r1678834;
        double r1678836 = r1678832 + r1678835;
        double r1678837 = -1259.1392167224028;
        double r1678838 = r1678808 + r1678800;
        double r1678839 = r1678837 / r1678838;
        double r1678840 = r1678836 + r1678839;
        double r1678841 = r1678831 + r1678840;
        double r1678842 = r1678827 + r1678841;
        double r1678843 = 12.507343278686905;
        double r1678844 = 5.0;
        double r1678845 = r1678808 + r1678844;
        double r1678846 = r1678843 / r1678845;
        double r1678847 = r1678842 + r1678846;
        double r1678848 = -0.13857109526572012;
        double r1678849 = 6.0;
        double r1678850 = r1678849 + r1678808;
        double r1678851 = r1678848 / r1678850;
        double r1678852 = r1678847 + r1678851;
        double r1678853 = r1678823 + r1678852;
        double r1678854 = r1678821 + r1678853;
        double r1678855 = r1678817 * r1678854;
        double r1678856 = r1678801 * r1678806;
        double r1678857 = sin(r1678856);
        double r1678858 = r1678801 / r1678857;
        double r1678859 = r1678855 * r1678858;
        return r1678859;
}

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. Final simplification1.8

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

Reproduce

herbie shell --seed 2019153 +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))))))