Average Error: 1.8 → 2.4
Time: 1.7m
Precision: 64
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531346025876700878143310547}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.50734327868690520801919774385169148445}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.1385710952657201178173096423051902092993}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\frac{\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.139216722402807135949842631816864014}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531346025876700878143310547}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.50734327868690520801919774385169148445}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.1385710952657201178173096423051902092993}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)}{e^{\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5}}\]
\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{\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \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)\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)}{e^{\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5}}
double f(double z) {
        double r86873 = atan2(1.0, 0.0);
        double r86874 = z;
        double r86875 = r86873 * r86874;
        double r86876 = sin(r86875);
        double r86877 = r86873 / r86876;
        double r86878 = 2.0;
        double r86879 = r86873 * r86878;
        double r86880 = sqrt(r86879);
        double r86881 = 1.0;
        double r86882 = r86881 - r86874;
        double r86883 = r86882 - r86881;
        double r86884 = 7.0;
        double r86885 = r86883 + r86884;
        double r86886 = 0.5;
        double r86887 = r86885 + r86886;
        double r86888 = r86883 + r86886;
        double r86889 = pow(r86887, r86888);
        double r86890 = r86880 * r86889;
        double r86891 = -r86887;
        double r86892 = exp(r86891);
        double r86893 = r86890 * r86892;
        double r86894 = 0.9999999999998099;
        double r86895 = 676.5203681218851;
        double r86896 = r86883 + r86881;
        double r86897 = r86895 / r86896;
        double r86898 = r86894 + r86897;
        double r86899 = -1259.1392167224028;
        double r86900 = r86883 + r86878;
        double r86901 = r86899 / r86900;
        double r86902 = r86898 + r86901;
        double r86903 = 771.3234287776531;
        double r86904 = 3.0;
        double r86905 = r86883 + r86904;
        double r86906 = r86903 / r86905;
        double r86907 = r86902 + r86906;
        double r86908 = -176.6150291621406;
        double r86909 = 4.0;
        double r86910 = r86883 + r86909;
        double r86911 = r86908 / r86910;
        double r86912 = r86907 + r86911;
        double r86913 = 12.507343278686905;
        double r86914 = 5.0;
        double r86915 = r86883 + r86914;
        double r86916 = r86913 / r86915;
        double r86917 = r86912 + r86916;
        double r86918 = -0.13857109526572012;
        double r86919 = 6.0;
        double r86920 = r86883 + r86919;
        double r86921 = r86918 / r86920;
        double r86922 = r86917 + r86921;
        double r86923 = 9.984369578019572e-06;
        double r86924 = r86923 / r86885;
        double r86925 = r86922 + r86924;
        double r86926 = 1.5056327351493116e-07;
        double r86927 = 8.0;
        double r86928 = r86883 + r86927;
        double r86929 = r86926 / r86928;
        double r86930 = r86925 + r86929;
        double r86931 = r86893 * r86930;
        double r86932 = r86877 * r86931;
        return r86932;
}
↓
double f(double z) {
        double r86933 = atan2(1.0, 0.0);
        double r86934 = z;
        double r86935 = r86933 * r86934;
        double r86936 = sin(r86935);
        double r86937 = r86933 / r86936;
        double r86938 = 2.0;
        double r86939 = r86933 * r86938;
        double r86940 = sqrt(r86939);
        double r86941 = 1.0;
        double r86942 = r86941 - r86934;
        double r86943 = r86942 - r86941;
        double r86944 = 7.0;
        double r86945 = r86943 + r86944;
        double r86946 = 0.5;
        double r86947 = r86945 + r86946;
        double r86948 = r86943 + r86946;
        double r86949 = pow(r86947, r86948);
        double r86950 = r86940 * r86949;
        double r86951 = r86937 * r86950;
        double r86952 = 0.9999999999998099;
        double r86953 = 676.5203681218851;
        double r86954 = r86943 + r86941;
        double r86955 = r86953 / r86954;
        double r86956 = r86952 + r86955;
        double r86957 = -1259.1392167224028;
        double r86958 = r86943 + r86938;
        double r86959 = r86957 / r86958;
        double r86960 = r86956 + r86959;
        double r86961 = 771.3234287776531;
        double r86962 = 3.0;
        double r86963 = r86943 + r86962;
        double r86964 = r86961 / r86963;
        double r86965 = r86960 + r86964;
        double r86966 = -176.6150291621406;
        double r86967 = 4.0;
        double r86968 = r86943 + r86967;
        double r86969 = r86966 / r86968;
        double r86970 = r86965 + r86969;
        double r86971 = 12.507343278686905;
        double r86972 = 5.0;
        double r86973 = r86943 + r86972;
        double r86974 = r86971 / r86973;
        double r86975 = r86970 + r86974;
        double r86976 = -0.13857109526572012;
        double r86977 = 6.0;
        double r86978 = r86943 + r86977;
        double r86979 = r86976 / r86978;
        double r86980 = r86975 + r86979;
        double r86981 = 9.984369578019572e-06;
        double r86982 = r86981 / r86945;
        double r86983 = r86980 + r86982;
        double r86984 = 1.5056327351493116e-07;
        double r86985 = 8.0;
        double r86986 = r86943 + r86985;
        double r86987 = r86984 / r86986;
        double r86988 = r86983 + r86987;
        double r86989 = r86951 * r86988;
        double r86990 = exp(r86947);
        double r86991 = r86989 / r86990;
        return r86991;
}

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 simplification2.4

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

Reproduce

herbie shell --seed 2019298 
(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)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.99999999999980993 (/ 676.520368121885099 (+ (- (- 1 z) 1) 1))) (/ -1259.13921672240281 (+ (- (- 1 z) 1) 2))) (/ 771.32342877765313 (+ (- (- 1 z) 1) 3))) (/ -176.615029162140587 (+ (- (- 1 z) 1) 4))) (/ 12.5073432786869052 (+ (- (- 1 z) 1) 5))) (/ -0.138571095265720118 (+ (- (- 1 z) 1) 6))) (/ 9.98436957801957158e-6 (+ (- (- 1 z) 1) 7))) (/ 1.50563273514931162e-7 (+ (- (- 1 z) 1) 8))))))