Jmat.Real.gamma, branch z greater than 0.5

Average Error: 60.1 → 0.5

Time: 4.7m

Precision: 64

• could not determine a ground truth for program body

  1. z = 3.517832587491486e+241

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

C

double f(double z) {
        double r6522042 = atan2(1.0, 0.0);
        double r6522043 = 2.0;
        double r6522044 = r6522042 * r6522043;
        double r6522045 = sqrt(r6522044);
        double r6522046 = z;
        double r6522047 = 1.0;
        double r6522048 = r6522046 - r6522047;
        double r6522049 = 7.0;
        double r6522050 = r6522048 + r6522049;
        double r6522051 = 0.5;
        double r6522052 = r6522050 + r6522051;
        double r6522053 = r6522048 + r6522051;
        double r6522054 = pow(r6522052, r6522053);
        double r6522055 = r6522045 * r6522054;
        double r6522056 = -r6522052;
        double r6522057 = exp(r6522056);
        double r6522058 = r6522055 * r6522057;
        double r6522059 = 0.9999999999998099;
        double r6522060 = 676.5203681218851;
        double r6522061 = r6522048 + r6522047;
        double r6522062 = r6522060 / r6522061;
        double r6522063 = r6522059 + r6522062;
        double r6522064 = -1259.1392167224028;
        double r6522065 = r6522048 + r6522043;
        double r6522066 = r6522064 / r6522065;
        double r6522067 = r6522063 + r6522066;
        double r6522068 = 771.3234287776531;
        double r6522069 = 3.0;
        double r6522070 = r6522048 + r6522069;
        double r6522071 = r6522068 / r6522070;
        double r6522072 = r6522067 + r6522071;
        double r6522073 = -176.6150291621406;
        double r6522074 = 4.0;
        double r6522075 = r6522048 + r6522074;
        double r6522076 = r6522073 / r6522075;
        double r6522077 = r6522072 + r6522076;
        double r6522078 = 12.507343278686905;
        double r6522079 = 5.0;
        double r6522080 = r6522048 + r6522079;
        double r6522081 = r6522078 / r6522080;
        double r6522082 = r6522077 + r6522081;
        double r6522083 = -0.13857109526572012;
        double r6522084 = 6.0;
        double r6522085 = r6522048 + r6522084;
        double r6522086 = r6522083 / r6522085;
        double r6522087 = r6522082 + r6522086;
        double r6522088 = 9.984369578019572e-06;
        double r6522089 = r6522088 / r6522050;
        double r6522090 = r6522087 + r6522089;
        double r6522091 = 1.5056327351493116e-07;
        double r6522092 = 8.0;
        double r6522093 = r6522048 + r6522092;
        double r6522094 = r6522091 / r6522093;
        double r6522095 = r6522090 + r6522094;
        double r6522096 = r6522058 * r6522095;
        return r6522096;
}

↓

double f(double z) {
        double r6522097 = z;
        double r6522098 = 6.0;
        double r6522099 = r6522097 + r6522098;
        double r6522100 = 0.5;
        double r6522101 = r6522099 + r6522100;
        double r6522102 = 1.0;
        double r6522103 = r6522097 - r6522102;
        double r6522104 = pow(r6522101, r6522103);
        double r6522105 = exp(r6522101);
        double r6522106 = cbrt(r6522105);
        double r6522107 = r6522106 * r6522106;
        double r6522108 = r6522104 / r6522107;
        double r6522109 = pow(r6522101, r6522100);
        double r6522110 = r6522109 / r6522106;
        double r6522111 = r6522108 * r6522110;
        double r6522112 = -5.0;
        double r6522113 = r6522097 - r6522112;
        double r6522114 = 7.0;
        double r6522115 = r6522114 + r6522097;
        double r6522116 = r6522115 * r6522099;
        double r6522117 = 676.5203681218851;
        double r6522118 = 2.0;
        double r6522119 = r6522097 + r6522118;
        double r6522120 = 0.9999999999998099;
        double r6522121 = -176.6150291621406;
        double r6522122 = 3.0;
        double r6522123 = r6522097 + r6522122;
        double r6522124 = r6522121 / r6522123;
        double r6522125 = r6522120 + r6522124;
        double r6522126 = -1259.1392167224028;
        double r6522127 = r6522097 + r6522102;
        double r6522128 = r6522126 / r6522127;
        double r6522129 = r6522125 - r6522128;
        double r6522130 = r6522119 * r6522129;
        double r6522131 = r6522117 * r6522130;
        double r6522132 = 771.3234287776531;
        double r6522133 = r6522129 * r6522132;
        double r6522134 = r6522125 * r6522125;
        double r6522135 = r6522128 * r6522128;
        double r6522136 = r6522134 - r6522135;
        double r6522137 = r6522119 * r6522136;
        double r6522138 = r6522133 + r6522137;
        double r6522139 = r6522138 * r6522097;
        double r6522140 = r6522131 + r6522139;
        double r6522141 = 4.0;
        double r6522142 = r6522097 + r6522141;
        double r6522143 = r6522140 * r6522142;
        double r6522144 = r6522130 * r6522097;
        double r6522145 = 12.507343278686905;
        double r6522146 = r6522144 * r6522145;
        double r6522147 = r6522143 + r6522146;
        double r6522148 = r6522116 * r6522147;
        double r6522149 = r6522144 * r6522142;
        double r6522150 = 1.5056327351493116e-07;
        double r6522151 = r6522099 * r6522150;
        double r6522152 = 9.984369578019572e-06;
        double r6522153 = r6522115 * r6522152;
        double r6522154 = r6522151 + r6522153;
        double r6522155 = r6522149 * r6522154;
        double r6522156 = r6522148 + r6522155;
        double r6522157 = r6522113 * r6522156;
        double r6522158 = -0.13857109526572012;
        double r6522159 = r6522149 * r6522116;
        double r6522160 = r6522158 * r6522159;
        double r6522161 = r6522157 + r6522160;
        double r6522162 = atan2(1.0, 0.0);
        double r6522163 = r6522162 * r6522118;
        double r6522164 = sqrt(r6522163);
        double r6522165 = r6522161 * r6522164;
        double r6522166 = r6522111 * r6522165;
        double r6522167 = r6522159 * r6522113;
        double r6522168 = r6522166 / r6522167;
        return r6522168;
}

Error

Bits error versus z

Derivation

1. Initial program 60.1

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

2. Simplified 1.0

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

3. Simplified 0.8

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

5. Applied flip-+ 0.8

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

6. Applied frac-add 0.8

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

7. Applied frac-add 0.9

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

8. Applied frac-add 0.9

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

9. Applied frac-add 0.9

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

10. Applied frac-add 0.9

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

11. Applied frac-add 1.0

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

12. Applied associate-*r/ 1.0

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

13. Applied associate-*r/ 0.5

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

15. Applied add-cube-cbrt 1.2

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

16. Applied associate--r- 1.2

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

17. Applied unpow-prod-up 0.5

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

18. Applied times-frac 0.5

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

19. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019149 
(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)))))