Jmat.Real.gamma, branch z greater than 0.5

Average Error: 61.6 → 0.9

Time: 6.8m

Precision: 64

• could not determine a ground truth for program body

  1. z = 1.8832230334659124e+24

Math

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

↓

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

C

double f(double z) {
        double r18922929 = atan2(1.0, 0.0);
        double r18922930 = 2.0;
        double r18922931 = r18922929 * r18922930;
        double r18922932 = sqrt(r18922931);
        double r18922933 = z;
        double r18922934 = 1.0;
        double r18922935 = r18922933 - r18922934;
        double r18922936 = 7.0;
        double r18922937 = r18922935 + r18922936;
        double r18922938 = 0.5;
        double r18922939 = r18922937 + r18922938;
        double r18922940 = r18922935 + r18922938;
        double r18922941 = pow(r18922939, r18922940);
        double r18922942 = r18922932 * r18922941;
        double r18922943 = -r18922939;
        double r18922944 = exp(r18922943);
        double r18922945 = r18922942 * r18922944;
        double r18922946 = 0.9999999999998099;
        double r18922947 = 676.5203681218851;
        double r18922948 = r18922935 + r18922934;
        double r18922949 = r18922947 / r18922948;
        double r18922950 = r18922946 + r18922949;
        double r18922951 = -1259.1392167224028;
        double r18922952 = r18922935 + r18922930;
        double r18922953 = r18922951 / r18922952;
        double r18922954 = r18922950 + r18922953;
        double r18922955 = 771.3234287776531;
        double r18922956 = 3.0;
        double r18922957 = r18922935 + r18922956;
        double r18922958 = r18922955 / r18922957;
        double r18922959 = r18922954 + r18922958;
        double r18922960 = -176.6150291621406;
        double r18922961 = 4.0;
        double r18922962 = r18922935 + r18922961;
        double r18922963 = r18922960 / r18922962;
        double r18922964 = r18922959 + r18922963;
        double r18922965 = 12.507343278686905;
        double r18922966 = 5.0;
        double r18922967 = r18922935 + r18922966;
        double r18922968 = r18922965 / r18922967;
        double r18922969 = r18922964 + r18922968;
        double r18922970 = -0.13857109526572012;
        double r18922971 = 6.0;
        double r18922972 = r18922935 + r18922971;
        double r18922973 = r18922970 / r18922972;
        double r18922974 = r18922969 + r18922973;
        double r18922975 = 9.984369578019572e-06;
        double r18922976 = r18922975 / r18922937;
        double r18922977 = r18922974 + r18922976;
        double r18922978 = 1.5056327351493116e-07;
        double r18922979 = 8.0;
        double r18922980 = r18922935 + r18922979;
        double r18922981 = r18922978 / r18922980;
        double r18922982 = r18922977 + r18922981;
        double r18922983 = r18922945 * r18922982;
        return r18922983;
}

↓

double f(double z) {
        double r18922984 = -1259.1392167224028;
        double r18922985 = cbrt(r18922984);
        double r18922986 = r18922985 * r18922985;
        double r18922987 = z;
        double r18922988 = 1.0;
        double r18922989 = r18922987 - r18922988;
        double r18922990 = 2.0;
        double r18922991 = r18922989 + r18922990;
        double r18922992 = cbrt(r18922991);
        double r18922993 = r18922992 * r18922992;
        double r18922994 = r18922986 / r18922993;
        double r18922995 = r18922985 / r18922992;
        double r18922996 = 0.9999999999998099;
        double r18922997 = 676.5203681218851;
        double r18922998 = r18922997 / r18922987;
        double r18922999 = r18922996 + r18922998;
        double r18923000 = -176.6150291621406;
        double r18923001 = 4.0;
        double r18923002 = r18922989 + r18923001;
        double r18923003 = r18923000 / r18923002;
        double r18923004 = r18922999 + r18923003;
        double r18923005 = 771.3234287776531;
        double r18923006 = 3.0;
        double r18923007 = r18922989 + r18923006;
        double r18923008 = r18923005 / r18923007;
        double r18923009 = r18923004 + r18923008;
        double r18923010 = 12.507343278686905;
        double r18923011 = 5.0;
        double r18923012 = r18922989 + r18923011;
        double r18923013 = r18923010 / r18923012;
        double r18923014 = -0.13857109526572012;
        double r18923015 = 6.0;
        double r18923016 = r18922989 + r18923015;
        double r18923017 = r18923014 / r18923016;
        double r18923018 = r18923013 + r18923017;
        double r18923019 = 9.984369578019572e-06;
        double r18923020 = 7.0;
        double r18923021 = r18922989 + r18923020;
        double r18923022 = r18923019 / r18923021;
        double r18923023 = 1.5056327351493116e-07;
        double r18923024 = 8.0;
        double r18923025 = r18922989 + r18923024;
        double r18923026 = r18923023 / r18923025;
        double r18923027 = r18923022 + r18923026;
        double r18923028 = r18923018 + r18923027;
        double r18923029 = r18923009 + r18923028;
        double r18923030 = fma(r18922994, r18922995, r18923029);
        double r18923031 = r18922988 - r18923020;
        double r18923032 = 0.5;
        double r18923033 = r18923031 - r18923032;
        double r18923034 = exp(r18923033);
        double r18923035 = r18922989 + r18923032;
        double r18923036 = r18923021 + r18923032;
        double r18923037 = log(r18923036);
        double r18923038 = atan2(1.0, 0.0);
        double r18923039 = r18923038 * r18922990;
        double r18923040 = sqrt(r18923039);
        double r18923041 = log(r18923040);
        double r18923042 = fma(r18923035, r18923037, r18923041);
        double r18923043 = r18922987 - r18923042;
        double r18923044 = exp(r18923043);
        double r18923045 = r18923034 / r18923044;
        double r18923046 = r18923030 * r18923045;
        return r18923046;
}

Error

Bits error versus z

Derivation

1. Initial program 61.6

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

2. Simplified 1.0

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

4. Applied add-exp-log 1.0

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

5. Applied add-exp-log 1.0

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

6. Applied prod-exp 1.0

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

7. Simplified 1.0

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

9. Applied associate-+l- 1.0

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

10. Applied associate-+l- 1.0

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

11. Applied exp-diff 1.3

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

12. Applied associate-/r/ 1.0

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

13. Simplified 1.0

    \[\leadsto \color{blue}{\frac{\frac{-1259.139216722402807135949842631816864014}{\left(z - 1\right) + 2} + \left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{z}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(z - 1\right) + 4}\right) + \frac{771.3234287776531346025876700878143310547}{\left(z - 1\right) + 3}\right) + \left(\left(\frac{12.50734327868690520801919774385169148445}{\left(z - 1\right) + 5} + \frac{-0.1385710952657201178173096423051902092993}{\left(z - 1\right) + 6}\right) + \left(\frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(z - 1\right) + 7} + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\right)\right)}{\frac{e^{z}}{e^{\mathsf{fma}\left(\left(z - 1\right) + 0.5, \log \left(\left(\left(z - 1\right) + 7\right) + 0.5\right), \log \left(\sqrt{\pi \cdot 2}\right)\right)}}}} \cdot e^{\left(1 - 7\right) - 0.5}\]
14. Using strategy rm

15. Applied div-inv 1.0

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

16. Applied associate-*l* 0.9

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

17. Simplified 0.9

    \[\leadsto \left(\frac{-1259.139216722402807135949842631816864014}{\left(z - 1\right) + 2} + \left(\left(\left(\left(0.9999999999998099298181841732002794742584 + \frac{676.5203681218850988443591631948947906494}{z}\right) + \frac{-176.6150291621405870046146446838974952698}{\left(z - 1\right) + 4}\right) + \frac{771.3234287776531346025876700878143310547}{\left(z - 1\right) + 3}\right) + \left(\left(\frac{12.50734327868690520801919774385169148445}{\left(z - 1\right) + 5} + \frac{-0.1385710952657201178173096423051902092993}{\left(z - 1\right) + 6}\right) + \left(\frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(z - 1\right) + 7} + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\right)\right)\right) \cdot \color{blue}{\frac{e^{\left(1 - 7\right) - 0.5}}{e^{z - \mathsf{fma}\left(\left(z - 1\right) + 0.5, \log \left(\left(\left(z - 1\right) + 7\right) + 0.5\right), \log \left(\sqrt{\pi \cdot 2}\right)\right)}}}\]
18. Using strategy rm

19. Applied add-cube-cbrt 0.9

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

20. Applied add-cube-cbrt 0.9

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

21. Applied times-frac 0.9

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

22. Applied fma-def 0.9

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

23. Final simplification 0.9

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (z)
  :name "Jmat.Real.gamma, branch z greater than 0.5"
  (* (* (* (sqrt (* PI 2.0)) (pow (+ (+ (- z 1.0) 7.0) 0.5) (+ (- z 1.0) 0.5))) (exp (- (+ (+ (- z 1.0) 7.0) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- z 1.0) 1.0))) (/ -1259.1392167224028 (+ (- z 1.0) 2.0))) (/ 771.3234287776531 (+ (- z 1.0) 3.0))) (/ -176.6150291621406 (+ (- z 1.0) 4.0))) (/ 12.507343278686905 (+ (- z 1.0) 5.0))) (/ -0.13857109526572012 (+ (- z 1.0) 6.0))) (/ 9.984369578019572e-06 (+ (- z 1.0) 7.0))) (/ 1.5056327351493116e-07 (+ (- z 1.0) 8.0)))))