Average Error: 59.9 → 0.8
Time: 5.5m
Precision: 64
\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8}\right)\]
↓
\[\frac{\frac{\left(\frac{-0.13857109526572012}{\left(z - 1\right) + 6} + \left(\frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(z - 1\right)} + \frac{12.507343278686905}{5 + \left(z - 1\right)}\right)\right) + \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \left(\frac{-1259.1392167224028}{\left(z - 1\right) + 2} + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8}\right)}{e^{7 + \left(z - 1\right)}}}{e^{0.5}} \cdot \left(\left(\sqrt{2} \cdot {\left(6.5 + z\right)}^{\left(z - 0.5\right)}\right) \cdot \sqrt{\pi}\right)\]
\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8}\right)
↓
\frac{\frac{\left(\frac{-0.13857109526572012}{\left(z - 1\right) + 6} + \left(\frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(z - 1\right)} + \frac{12.507343278686905}{5 + \left(z - 1\right)}\right)\right) + \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \left(\frac{-1259.1392167224028}{\left(z - 1\right) + 2} + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8}\right)}{e^{7 + \left(z - 1\right)}}}{e^{0.5}} \cdot \left(\left(\sqrt{2} \cdot {\left(6.5 + z\right)}^{\left(z - 0.5\right)}\right) \cdot \sqrt{\pi}\right)
double f(double z) {
        double r8425911 = atan2(1.0, 0.0);
        double r8425912 = 2.0;
        double r8425913 = r8425911 * r8425912;
        double r8425914 = sqrt(r8425913);
        double r8425915 = z;
        double r8425916 = 1.0;
        double r8425917 = r8425915 - r8425916;
        double r8425918 = 7.0;
        double r8425919 = r8425917 + r8425918;
        double r8425920 = 0.5;
        double r8425921 = r8425919 + r8425920;
        double r8425922 = r8425917 + r8425920;
        double r8425923 = pow(r8425921, r8425922);
        double r8425924 = r8425914 * r8425923;
        double r8425925 = -r8425921;
        double r8425926 = exp(r8425925);
        double r8425927 = r8425924 * r8425926;
        double r8425928 = 0.9999999999998099;
        double r8425929 = 676.5203681218851;
        double r8425930 = r8425917 + r8425916;
        double r8425931 = r8425929 / r8425930;
        double r8425932 = r8425928 + r8425931;
        double r8425933 = -1259.1392167224028;
        double r8425934 = r8425917 + r8425912;
        double r8425935 = r8425933 / r8425934;
        double r8425936 = r8425932 + r8425935;
        double r8425937 = 771.3234287776531;
        double r8425938 = 3.0;
        double r8425939 = r8425917 + r8425938;
        double r8425940 = r8425937 / r8425939;
        double r8425941 = r8425936 + r8425940;
        double r8425942 = -176.6150291621406;
        double r8425943 = 4.0;
        double r8425944 = r8425917 + r8425943;
        double r8425945 = r8425942 / r8425944;
        double r8425946 = r8425941 + r8425945;
        double r8425947 = 12.507343278686905;
        double r8425948 = 5.0;
        double r8425949 = r8425917 + r8425948;
        double r8425950 = r8425947 / r8425949;
        double r8425951 = r8425946 + r8425950;
        double r8425952 = -0.13857109526572012;
        double r8425953 = 6.0;
        double r8425954 = r8425917 + r8425953;
        double r8425955 = r8425952 / r8425954;
        double r8425956 = r8425951 + r8425955;
        double r8425957 = 9.984369578019572e-06;
        double r8425958 = r8425957 / r8425919;
        double r8425959 = r8425956 + r8425958;
        double r8425960 = 1.5056327351493116e-07;
        double r8425961 = 8.0;
        double r8425962 = r8425917 + r8425961;
        double r8425963 = r8425960 / r8425962;
        double r8425964 = r8425959 + r8425963;
        double r8425965 = r8425927 * r8425964;
        return r8425965;
}
↓
double f(double z) {
        double r8425966 = -0.13857109526572012;
        double r8425967 = z;
        double r8425968 = 1.0;
        double r8425969 = r8425967 - r8425968;
        double r8425970 = 6.0;
        double r8425971 = r8425969 + r8425970;
        double r8425972 = r8425966 / r8425971;
        double r8425973 = 9.984369578019572e-06;
        double r8425974 = 7.0;
        double r8425975 = r8425974 + r8425969;
        double r8425976 = r8425973 / r8425975;
        double r8425977 = 12.507343278686905;
        double r8425978 = 5.0;
        double r8425979 = r8425978 + r8425969;
        double r8425980 = r8425977 / r8425979;
        double r8425981 = r8425976 + r8425980;
        double r8425982 = r8425972 + r8425981;
        double r8425983 = 0.9999999999998099;
        double r8425984 = 676.5203681218851;
        double r8425985 = r8425984 / r8425967;
        double r8425986 = r8425983 + r8425985;
        double r8425987 = -176.6150291621406;
        double r8425988 = 4.0;
        double r8425989 = r8425969 + r8425988;
        double r8425990 = r8425987 / r8425989;
        double r8425991 = r8425986 + r8425990;
        double r8425992 = -1259.1392167224028;
        double r8425993 = 2.0;
        double r8425994 = r8425969 + r8425993;
        double r8425995 = r8425992 / r8425994;
        double r8425996 = 771.3234287776531;
        double r8425997 = 3.0;
        double r8425998 = r8425969 + r8425997;
        double r8425999 = r8425996 / r8425998;
        double r8426000 = r8425995 + r8425999;
        double r8426001 = r8425991 + r8426000;
        double r8426002 = 1.5056327351493116e-07;
        double r8426003 = 8.0;
        double r8426004 = r8425969 + r8426003;
        double r8426005 = r8426002 / r8426004;
        double r8426006 = r8426001 + r8426005;
        double r8426007 = r8425982 + r8426006;
        double r8426008 = exp(r8425975);
        double r8426009 = r8426007 / r8426008;
        double r8426010 = 0.5;
        double r8426011 = exp(r8426010);
        double r8426012 = r8426009 / r8426011;
        double r8426013 = sqrt(r8425993);
        double r8426014 = 6.5;
        double r8426015 = r8426014 + r8425967;
        double r8426016 = r8425967 - r8426010;
        double r8426017 = pow(r8426015, r8426016);
        double r8426018 = r8426013 * r8426017;
        double r8426019 = atan2(1.0, 0.0);
        double r8426020 = sqrt(r8426019);
        double r8426021 = r8426018 * r8426020;
        double r8426022 = r8426012 * r8426021;
        return r8426022;
}

Error

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 59.9

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

  2. Simplified0.8

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

  4. Applied exp-sum0.8

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

  5. Applied associate-/r*0.8

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

  6. Simplified0.8

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

  7. Taylor expanded around inf 0.8

    \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot {\left(z + 6.5\right)}^{\left(z - 0.5\right)}\right) \cdot \sqrt{\pi}\right)} \cdot \frac{\frac{\left(\frac{-0.13857109526572012}{6 + \left(z - 1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(z - 1\right)} + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right)\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8} + \left(\left(\frac{771.3234287776531}{3 + \left(z - 1\right)} + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \left(\frac{-176.6150291621406}{4 + \left(z - 1\right)} + \left(\frac{676.5203681218851}{z} + 0.9999999999998099\right)\right)\right)\right)}{e^{7 + \left(z - 1\right)}}}{e^{0.5}}\]

  8. Final simplification0.8

    \[\leadsto \frac{\frac{\left(\frac{-0.13857109526572012}{\left(z - 1\right) + 6} + \left(\frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(z - 1\right)} + \frac{12.507343278686905}{5 + \left(z - 1\right)}\right)\right) + \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \left(\frac{-1259.1392167224028}{\left(z - 1\right) + 2} + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8}\right)}{e^{7 + \left(z - 1\right)}}}{e^{0.5}} \cdot \left(\left(\sqrt{2} \cdot {\left(6.5 + z\right)}^{\left(z - 0.5\right)}\right) \cdot \sqrt{\pi}\right)\]

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (z)
  :name "Jmat.Real.gamma, branch z greater than 0.5"
  (* (* (* (sqrt (* PI 2)) (pow (+ (+ (- z 1) 7) 0.5) (+ (- z 1) 0.5))) (exp (- (+ (+ (- z 1) 7) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- z 1) 1))) (/ -1259.1392167224028 (+ (- z 1) 2))) (/ 771.3234287776531 (+ (- z 1) 3))) (/ -176.6150291621406 (+ (- z 1) 4))) (/ 12.507343278686905 (+ (- z 1) 5))) (/ -0.13857109526572012 (+ (- z 1) 6))) (/ 9.984369578019572e-06 (+ (- z 1) 7))) (/ 1.5056327351493116e-07 (+ (- z 1) 8)))))