Jmat.Real.gamma, branch z greater than 0.5

Average Error: 59.8 → 0.6

Time: 2.0m

Precision: 64

• could not determine a ground truth for program body

  1. z = 8.43616807864734e+75

Math

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

↓

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

C

double f(double z) {
        double r5302073 = atan2(1.0, 0.0);
        double r5302074 = 2.0;
        double r5302075 = r5302073 * r5302074;
        double r5302076 = sqrt(r5302075);
        double r5302077 = z;
        double r5302078 = 1.0;
        double r5302079 = r5302077 - r5302078;
        double r5302080 = 7.0;
        double r5302081 = r5302079 + r5302080;
        double r5302082 = 0.5;
        double r5302083 = r5302081 + r5302082;
        double r5302084 = r5302079 + r5302082;
        double r5302085 = pow(r5302083, r5302084);
        double r5302086 = r5302076 * r5302085;
        double r5302087 = -r5302083;
        double r5302088 = exp(r5302087);
        double r5302089 = r5302086 * r5302088;
        double r5302090 = 0.9999999999998099;
        double r5302091 = 676.5203681218851;
        double r5302092 = r5302079 + r5302078;
        double r5302093 = r5302091 / r5302092;
        double r5302094 = r5302090 + r5302093;
        double r5302095 = -1259.1392167224028;
        double r5302096 = r5302079 + r5302074;
        double r5302097 = r5302095 / r5302096;
        double r5302098 = r5302094 + r5302097;
        double r5302099 = 771.3234287776531;
        double r5302100 = 3.0;
        double r5302101 = r5302079 + r5302100;
        double r5302102 = r5302099 / r5302101;
        double r5302103 = r5302098 + r5302102;
        double r5302104 = -176.6150291621406;
        double r5302105 = 4.0;
        double r5302106 = r5302079 + r5302105;
        double r5302107 = r5302104 / r5302106;
        double r5302108 = r5302103 + r5302107;
        double r5302109 = 12.507343278686905;
        double r5302110 = 5.0;
        double r5302111 = r5302079 + r5302110;
        double r5302112 = r5302109 / r5302111;
        double r5302113 = r5302108 + r5302112;
        double r5302114 = -0.13857109526572012;
        double r5302115 = 6.0;
        double r5302116 = r5302079 + r5302115;
        double r5302117 = r5302114 / r5302116;
        double r5302118 = r5302113 + r5302117;
        double r5302119 = 9.984369578019572e-06;
        double r5302120 = r5302119 / r5302081;
        double r5302121 = r5302118 + r5302120;
        double r5302122 = 1.5056327351493116e-07;
        double r5302123 = 8.0;
        double r5302124 = r5302079 + r5302123;
        double r5302125 = r5302122 / r5302124;
        double r5302126 = r5302121 + r5302125;
        double r5302127 = r5302089 * r5302126;
        return r5302127;
}

↓

double f(double z) {
        double r5302128 = 0.5;
        double r5302129 = z;
        double r5302130 = 1.0;
        double r5302131 = r5302129 - r5302130;
        double r5302132 = 7.0;
        double r5302133 = r5302131 + r5302132;
        double r5302134 = r5302128 + r5302133;
        double r5302135 = r5302131 + r5302128;
        double r5302136 = pow(r5302134, r5302135);
        double r5302137 = atan2(1.0, 0.0);
        double r5302138 = 2.0;
        double r5302139 = r5302137 * r5302138;
        double r5302140 = sqrt(r5302139);
        double r5302141 = r5302136 * r5302140;
        double r5302142 = exp(r5302134);
        double r5302143 = r5302141 / r5302142;
        double r5302144 = 9.984369578019572e-06;
        double r5302145 = r5302144 / r5302133;
        double r5302146 = 1.5056327351493116e-07;
        double r5302147 = r5302129 + r5302132;
        double r5302148 = r5302146 / r5302147;
        double r5302149 = r5302145 + r5302148;
        double r5302150 = r5302143 * r5302149;
        double r5302151 = 0.9999999999998099;
        double r5302152 = r5302151 * r5302151;
        double r5302153 = -1259.1392167224028;
        double r5302154 = -1.0;
        double r5302155 = r5302129 - r5302154;
        double r5302156 = r5302153 / r5302155;
        double r5302157 = r5302156 * r5302156;
        double r5302158 = r5302156 * r5302151;
        double r5302159 = r5302157 - r5302158;
        double r5302160 = r5302152 + r5302159;
        double r5302161 = r5302129 + r5302138;
        double r5302162 = r5302161 * r5302129;
        double r5302163 = r5302160 * r5302162;
        double r5302164 = -5.0;
        double r5302165 = r5302129 - r5302164;
        double r5302166 = r5302163 * r5302165;
        double r5302167 = 4.0;
        double r5302168 = r5302131 + r5302167;
        double r5302169 = r5302166 * r5302168;
        double r5302170 = 12.507343278686905;
        double r5302171 = r5302169 * r5302170;
        double r5302172 = r5302167 + r5302129;
        double r5302173 = -0.13857109526572012;
        double r5302174 = r5302163 * r5302173;
        double r5302175 = 3.0;
        double r5302176 = pow(r5302151, r5302175);
        double r5302177 = pow(r5302156, r5302175);
        double r5302178 = r5302176 + r5302177;
        double r5302179 = r5302162 * r5302178;
        double r5302180 = 771.3234287776531;
        double r5302181 = r5302180 * r5302129;
        double r5302182 = 676.5203681218851;
        double r5302183 = r5302182 * r5302161;
        double r5302184 = r5302181 + r5302183;
        double r5302185 = r5302160 * r5302184;
        double r5302186 = r5302179 + r5302185;
        double r5302187 = r5302165 * r5302186;
        double r5302188 = r5302174 + r5302187;
        double r5302189 = r5302188 * r5302168;
        double r5302190 = -176.6150291621406;
        double r5302191 = r5302190 * r5302166;
        double r5302192 = r5302189 + r5302191;
        double r5302193 = r5302172 * r5302192;
        double r5302194 = r5302171 + r5302193;
        double r5302195 = pow(r5302134, r5302131);
        double r5302196 = pow(r5302134, r5302128);
        double r5302197 = r5302140 * r5302196;
        double r5302198 = r5302195 * r5302197;
        double r5302199 = r5302194 * r5302198;
        double r5302200 = r5302169 * r5302172;
        double r5302201 = r5302142 * r5302200;
        double r5302202 = r5302199 / r5302201;
        double r5302203 = r5302150 + r5302202;
        return r5302203;
}

Error

Bits error versus z

Derivation

1. Initial program 59.8

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

2. Simplified 1.1

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

4. Applied unpow-prod-up 1.5

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

5. Applied associate-*l* 0.8

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

7. Applied flip3-+ 0.8

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

8. Applied frac-add 0.8

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

9. Applied frac-add 1.0

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

10. Applied frac-add 0.9

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

11. Applied frac-add 1.0

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

12. Applied frac-add 1.0

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

13. Applied frac-times 0.6

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

14. Final simplification 0.6

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

Reproduce

herbie shell --seed 2019143 
(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)))))