Average Error: 1.8 → 0.6
Time: 2.5m
Precision: 64
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\left(\left(\sqrt{2 \cdot \pi} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(\sqrt[3]{\frac{{\left(\left(7 - z\right) + 0.5\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}}{e^{\left(7 - z\right) + 0.5}}} \cdot \left(\sqrt[3]{\frac{{\left(\left(7 - z\right) + 0.5\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}}{e^{\left(7 - z\right) + 0.5}}} \cdot \sqrt[3]{\frac{{\left(\left(7 - z\right) + 0.5\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}}{e^{\left(7 - z\right) + 0.5}}}\right)\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{2 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right)\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right)\right) + \left(\frac{12.507343278686905}{-1 + \left(6 - z\right)} + \frac{-176.6150291621406}{4 - z}\right)\right)\]
\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)
↓
\left(\left(\sqrt{2 \cdot \pi} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right) \cdot \left(\sqrt[3]{\frac{{\left(\left(7 - z\right) + 0.5\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}}{e^{\left(7 - z\right) + 0.5}}} \cdot \left(\sqrt[3]{\frac{{\left(\left(7 - z\right) + 0.5\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}}{e^{\left(7 - z\right) + 0.5}}} \cdot \sqrt[3]{\frac{{\left(\left(7 - z\right) + 0.5\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}}{e^{\left(7 - z\right) + 0.5}}}\right)\right)\right) \cdot \left(\left(\left(\left(\frac{-1259.1392167224028}{2 - z} + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \frac{771.3234287776531}{\left(1 - z\right) + 2}\right)\right)\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right)\right) + \left(\frac{12.507343278686905}{-1 + \left(6 - z\right)} + \frac{-176.6150291621406}{4 - z}\right)\right)
double f(double z) {
        double r4390868 = atan2(1.0, 0.0);
        double r4390869 = z;
        double r4390870 = r4390868 * r4390869;
        double r4390871 = sin(r4390870);
        double r4390872 = r4390868 / r4390871;
        double r4390873 = 2.0;
        double r4390874 = r4390868 * r4390873;
        double r4390875 = sqrt(r4390874);
        double r4390876 = 1.0;
        double r4390877 = r4390876 - r4390869;
        double r4390878 = r4390877 - r4390876;
        double r4390879 = 7.0;
        double r4390880 = r4390878 + r4390879;
        double r4390881 = 0.5;
        double r4390882 = r4390880 + r4390881;
        double r4390883 = r4390878 + r4390881;
        double r4390884 = pow(r4390882, r4390883);
        double r4390885 = r4390875 * r4390884;
        double r4390886 = -r4390882;
        double r4390887 = exp(r4390886);
        double r4390888 = r4390885 * r4390887;
        double r4390889 = 0.9999999999998099;
        double r4390890 = 676.5203681218851;
        double r4390891 = r4390878 + r4390876;
        double r4390892 = r4390890 / r4390891;
        double r4390893 = r4390889 + r4390892;
        double r4390894 = -1259.1392167224028;
        double r4390895 = r4390878 + r4390873;
        double r4390896 = r4390894 / r4390895;
        double r4390897 = r4390893 + r4390896;
        double r4390898 = 771.3234287776531;
        double r4390899 = 3.0;
        double r4390900 = r4390878 + r4390899;
        double r4390901 = r4390898 / r4390900;
        double r4390902 = r4390897 + r4390901;
        double r4390903 = -176.6150291621406;
        double r4390904 = 4.0;
        double r4390905 = r4390878 + r4390904;
        double r4390906 = r4390903 / r4390905;
        double r4390907 = r4390902 + r4390906;
        double r4390908 = 12.507343278686905;
        double r4390909 = 5.0;
        double r4390910 = r4390878 + r4390909;
        double r4390911 = r4390908 / r4390910;
        double r4390912 = r4390907 + r4390911;
        double r4390913 = -0.13857109526572012;
        double r4390914 = 6.0;
        double r4390915 = r4390878 + r4390914;
        double r4390916 = r4390913 / r4390915;
        double r4390917 = r4390912 + r4390916;
        double r4390918 = 9.984369578019572e-06;
        double r4390919 = r4390918 / r4390880;
        double r4390920 = r4390917 + r4390919;
        double r4390921 = 1.5056327351493116e-07;
        double r4390922 = 8.0;
        double r4390923 = r4390878 + r4390922;
        double r4390924 = r4390921 / r4390923;
        double r4390925 = r4390920 + r4390924;
        double r4390926 = r4390888 * r4390925;
        double r4390927 = r4390872 * r4390926;
        return r4390927;
}
↓
double f(double z) {
        double r4390928 = 2.0;
        double r4390929 = atan2(1.0, 0.0);
        double r4390930 = r4390928 * r4390929;
        double r4390931 = sqrt(r4390930);
        double r4390932 = z;
        double r4390933 = r4390929 * r4390932;
        double r4390934 = sin(r4390933);
        double r4390935 = r4390929 / r4390934;
        double r4390936 = r4390931 * r4390935;
        double r4390937 = 7.0;
        double r4390938 = r4390937 - r4390932;
        double r4390939 = 0.5;
        double r4390940 = r4390938 + r4390939;
        double r4390941 = 1.0;
        double r4390942 = r4390941 - r4390932;
        double r4390943 = r4390941 - r4390939;
        double r4390944 = r4390942 - r4390943;
        double r4390945 = pow(r4390940, r4390944);
        double r4390946 = exp(r4390940);
        double r4390947 = r4390945 / r4390946;
        double r4390948 = cbrt(r4390947);
        double r4390949 = r4390948 * r4390948;
        double r4390950 = r4390948 * r4390949;
        double r4390951 = r4390936 * r4390950;
        double r4390952 = -1259.1392167224028;
        double r4390953 = r4390928 - r4390932;
        double r4390954 = r4390952 / r4390953;
        double r4390955 = -0.13857109526572012;
        double r4390956 = 6.0;
        double r4390957 = r4390956 - r4390932;
        double r4390958 = r4390955 / r4390957;
        double r4390959 = r4390954 + r4390958;
        double r4390960 = 676.5203681218851;
        double r4390961 = r4390960 / r4390942;
        double r4390962 = 0.9999999999998099;
        double r4390963 = 771.3234287776531;
        double r4390964 = r4390942 + r4390928;
        double r4390965 = r4390963 / r4390964;
        double r4390966 = r4390962 + r4390965;
        double r4390967 = r4390961 + r4390966;
        double r4390968 = r4390959 + r4390967;
        double r4390969 = 1.5056327351493116e-07;
        double r4390970 = 8.0;
        double r4390971 = r4390970 - r4390932;
        double r4390972 = r4390969 / r4390971;
        double r4390973 = 9.984369578019572e-06;
        double r4390974 = r4390973 / r4390938;
        double r4390975 = r4390972 + r4390974;
        double r4390976 = r4390968 + r4390975;
        double r4390977 = 12.507343278686905;
        double r4390978 = -1.0;
        double r4390979 = r4390978 + r4390957;
        double r4390980 = r4390977 / r4390979;
        double r4390981 = -176.6150291621406;
        double r4390982 = 4.0;
        double r4390983 = r4390982 - r4390932;
        double r4390984 = r4390981 / r4390983;
        double r4390985 = r4390980 + r4390984;
        double r4390986 = r4390976 + r4390985;
        double r4390987 = r4390951 * r4390986;
        return r4390987;
}

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. Simplified2.0

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

  4. Applied add-cube-cbrt0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019151 
(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))))))