Jmat.Real.gamma, branch z less than 0.5

Average Error: 1.8 → 1.2

Time: 1.8m

Precision: 64

• could not determine a ground truth for program body

  1. z = -3.4804422368984588e+159

Math

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

↓

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

C

double f(double z) {
        double r8832182 = atan2(1.0, 0.0);
        double r8832183 = z;
        double r8832184 = r8832182 * r8832183;
        double r8832185 = sin(r8832184);
        double r8832186 = r8832182 / r8832185;
        double r8832187 = 2.0;
        double r8832188 = r8832182 * r8832187;
        double r8832189 = sqrt(r8832188);
        double r8832190 = 1.0;
        double r8832191 = r8832190 - r8832183;
        double r8832192 = r8832191 - r8832190;
        double r8832193 = 7.0;
        double r8832194 = r8832192 + r8832193;
        double r8832195 = 0.5;
        double r8832196 = r8832194 + r8832195;
        double r8832197 = r8832192 + r8832195;
        double r8832198 = pow(r8832196, r8832197);
        double r8832199 = r8832189 * r8832198;
        double r8832200 = -r8832196;
        double r8832201 = exp(r8832200);
        double r8832202 = r8832199 * r8832201;
        double r8832203 = 0.9999999999998099;
        double r8832204 = 676.5203681218851;
        double r8832205 = r8832192 + r8832190;
        double r8832206 = r8832204 / r8832205;
        double r8832207 = r8832203 + r8832206;
        double r8832208 = -1259.1392167224028;
        double r8832209 = r8832192 + r8832187;
        double r8832210 = r8832208 / r8832209;
        double r8832211 = r8832207 + r8832210;
        double r8832212 = 771.3234287776531;
        double r8832213 = 3.0;
        double r8832214 = r8832192 + r8832213;
        double r8832215 = r8832212 / r8832214;
        double r8832216 = r8832211 + r8832215;
        double r8832217 = -176.6150291621406;
        double r8832218 = 4.0;
        double r8832219 = r8832192 + r8832218;
        double r8832220 = r8832217 / r8832219;
        double r8832221 = r8832216 + r8832220;
        double r8832222 = 12.507343278686905;
        double r8832223 = 5.0;
        double r8832224 = r8832192 + r8832223;
        double r8832225 = r8832222 / r8832224;
        double r8832226 = r8832221 + r8832225;
        double r8832227 = -0.13857109526572012;
        double r8832228 = 6.0;
        double r8832229 = r8832192 + r8832228;
        double r8832230 = r8832227 / r8832229;
        double r8832231 = r8832226 + r8832230;
        double r8832232 = 9.984369578019572e-06;
        double r8832233 = r8832232 / r8832194;
        double r8832234 = r8832231 + r8832233;
        double r8832235 = 1.5056327351493116e-07;
        double r8832236 = 8.0;
        double r8832237 = r8832192 + r8832236;
        double r8832238 = r8832235 / r8832237;
        double r8832239 = r8832234 + r8832238;
        double r8832240 = r8832202 * r8832239;
        double r8832241 = r8832186 * r8832240;
        return r8832241;
}

↓

double f(double z) {
        double r8832242 = -0.13857109526572012;
        double r8832243 = 1.0;
        double r8832244 = z;
        double r8832245 = r8832243 - r8832244;
        double r8832246 = r8832245 - r8832243;
        double r8832247 = 6.0;
        double r8832248 = r8832246 + r8832247;
        double r8832249 = r8832242 / r8832248;
        double r8832250 = 9.984369578019572e-06;
        double r8832251 = 7.0;
        double r8832252 = r8832246 + r8832251;
        double r8832253 = r8832250 / r8832252;
        double r8832254 = r8832249 + r8832253;
        double r8832255 = 1.5056327351493116e-07;
        double r8832256 = 8.0;
        double r8832257 = r8832246 + r8832256;
        double r8832258 = r8832255 / r8832257;
        double r8832259 = r8832254 + r8832258;
        double r8832260 = 0.9999999999998099;
        double r8832261 = 676.5203681218851;
        double r8832262 = r8832261 / r8832245;
        double r8832263 = -1259.1392167224028;
        double r8832264 = 2.0;
        double r8832265 = r8832246 + r8832264;
        double r8832266 = r8832263 / r8832265;
        double r8832267 = 771.3234287776531;
        double r8832268 = 3.0;
        double r8832269 = r8832246 + r8832268;
        double r8832270 = r8832267 / r8832269;
        double r8832271 = r8832266 + r8832270;
        double r8832272 = -176.6150291621406;
        double r8832273 = 4.0;
        double r8832274 = r8832246 + r8832273;
        double r8832275 = r8832272 / r8832274;
        double r8832276 = 12.507343278686905;
        double r8832277 = 5.0;
        double r8832278 = r8832246 + r8832277;
        double r8832279 = r8832276 / r8832278;
        double r8832280 = r8832275 + r8832279;
        double r8832281 = r8832271 + r8832280;
        double r8832282 = r8832262 + r8832281;
        double r8832283 = r8832260 + r8832282;
        double r8832284 = r8832259 + r8832283;
        double r8832285 = atan2(1.0, 0.0);
        double r8832286 = r8832285 * r8832264;
        double r8832287 = sqrt(r8832286);
        double r8832288 = 0.5;
        double r8832289 = r8832252 + r8832288;
        double r8832290 = -r8832244;
        double r8832291 = exp(r8832290);
        double r8832292 = sqrt(r8832291);
        double r8832293 = log(r8832292);
        double r8832294 = exp(r8832244);
        double r8832295 = sqrt(r8832294);
        double r8832296 = log(r8832295);
        double r8832297 = -r8832296;
        double r8832298 = r8832293 + r8832297;
        double r8832299 = r8832298 + r8832288;
        double r8832300 = pow(r8832289, r8832299);
        double r8832301 = r8832287 * r8832300;
        double r8832302 = r8832285 * r8832244;
        double r8832303 = sin(r8832302);
        double r8832304 = r8832285 / r8832303;
        double r8832305 = r8832301 * r8832304;
        double r8832306 = exp(r8832289);
        double r8832307 = r8832305 / r8832306;
        double r8832308 = r8832284 * r8832307;
        return r8832308;
}

Error

Bits error versus z

Derivation

1. Initial program 1.8

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

2. Simplified 1.2

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

4. Applied add-log-exp 1.2

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

5. Applied add-log-exp 1.2

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

6. Applied add-log-exp 1.2

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

7. Applied diff-log 1.2

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

8. Applied diff-log 1.2

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

9. Simplified 1.2

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

11. Applied add-sqr-sqrt 1.2

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

12. Applied log-prod 1.2

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

14. Applied exp-neg 1.2

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

15. Applied sqrt-div 1.2

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

16. Applied log-div 1.2

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

17. Simplified 1.2

    \[\leadsto \left(\left(\left(\frac{-0.1385710952657201178173096423051902092993}{\left(\left(1 - z\right) - 1\right) + 6} + \frac{9.984369578019571583242346146658263705831 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.505632735149311617592788074479481785772 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right) + \left(0.9999999999998099298181841732002794742584 + \left(\frac{676.5203681218850988443591631948947906494}{1 - z} + \left(\left(\frac{-1259.139216722402807135949842631816864014}{\left(\left(1 - z\right) - 1\right) + 2} + \frac{771.3234287776531346025876700878143310547}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \left(\frac{-176.6150291621405870046146446838974952698}{\left(\left(1 - z\right) - 1\right) + 4} + \frac{12.50734327868690520801919774385169148445}{\left(\left(1 - z\right) - 1\right) + 5}\right)\right)\right)\right)\right) \cdot \frac{\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(\color{blue}{0} - \log \left(\sqrt{e^{z}}\right)\right) + \log \left(\sqrt{e^{-z}}\right)\right) + 0.5\right)}\right)}{e^{\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5}}\]

18. Final simplification 1.2

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

Reproduce

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