Jmat.Real.gamma, branch z greater than 0.5

Average Error: 61.6 → 1.0

Time: 6.1m

Precision: 64

• could not determine a ground truth for program body

  1. z = 1.8832230334659124e+24

Math

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

↓

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

C

double f(double z) {
        double r17502166 = atan2(1.0, 0.0);
        double r17502167 = 2.0;
        double r17502168 = r17502166 * r17502167;
        double r17502169 = sqrt(r17502168);
        double r17502170 = z;
        double r17502171 = 1.0;
        double r17502172 = r17502170 - r17502171;
        double r17502173 = 7.0;
        double r17502174 = r17502172 + r17502173;
        double r17502175 = 0.5;
        double r17502176 = r17502174 + r17502175;
        double r17502177 = r17502172 + r17502175;
        double r17502178 = pow(r17502176, r17502177);
        double r17502179 = r17502169 * r17502178;
        double r17502180 = -r17502176;
        double r17502181 = exp(r17502180);
        double r17502182 = r17502179 * r17502181;
        double r17502183 = 0.9999999999998099;
        double r17502184 = 676.5203681218851;
        double r17502185 = r17502172 + r17502171;
        double r17502186 = r17502184 / r17502185;
        double r17502187 = r17502183 + r17502186;
        double r17502188 = -1259.1392167224028;
        double r17502189 = r17502172 + r17502167;
        double r17502190 = r17502188 / r17502189;
        double r17502191 = r17502187 + r17502190;
        double r17502192 = 771.3234287776531;
        double r17502193 = 3.0;
        double r17502194 = r17502172 + r17502193;
        double r17502195 = r17502192 / r17502194;
        double r17502196 = r17502191 + r17502195;
        double r17502197 = -176.6150291621406;
        double r17502198 = 4.0;
        double r17502199 = r17502172 + r17502198;
        double r17502200 = r17502197 / r17502199;
        double r17502201 = r17502196 + r17502200;
        double r17502202 = 12.507343278686905;
        double r17502203 = 5.0;
        double r17502204 = r17502172 + r17502203;
        double r17502205 = r17502202 / r17502204;
        double r17502206 = r17502201 + r17502205;
        double r17502207 = -0.13857109526572012;
        double r17502208 = 6.0;
        double r17502209 = r17502172 + r17502208;
        double r17502210 = r17502207 / r17502209;
        double r17502211 = r17502206 + r17502210;
        double r17502212 = 9.984369578019572e-06;
        double r17502213 = r17502212 / r17502174;
        double r17502214 = r17502211 + r17502213;
        double r17502215 = 1.5056327351493116e-07;
        double r17502216 = 8.0;
        double r17502217 = r17502172 + r17502216;
        double r17502218 = r17502215 / r17502217;
        double r17502219 = r17502214 + r17502218;
        double r17502220 = r17502182 * r17502219;
        return r17502220;
}

↓

double f(double z) {
        double r17502221 = atan2(1.0, 0.0);
        double r17502222 = 2.0;
        double r17502223 = r17502221 * r17502222;
        double r17502224 = sqrt(r17502223);
        double r17502225 = 9.984369578019572e-06;
        double r17502226 = z;
        double r17502227 = 1.0;
        double r17502228 = r17502226 - r17502227;
        double r17502229 = 7.0;
        double r17502230 = r17502228 + r17502229;
        double r17502231 = r17502225 / r17502230;
        double r17502232 = 0.9999999999998099;
        double r17502233 = r17502231 + r17502232;
        double r17502234 = -1259.1392167224028;
        double r17502235 = r17502228 + r17502222;
        double r17502236 = r17502234 / r17502235;
        double r17502237 = 676.5203681218851;
        double r17502238 = r17502237 / r17502226;
        double r17502239 = r17502236 + r17502238;
        double r17502240 = 771.3234287776531;
        double r17502241 = 3.0;
        double r17502242 = r17502228 + r17502241;
        double r17502243 = r17502240 / r17502242;
        double r17502244 = -176.6150291621406;
        double r17502245 = 4.0;
        double r17502246 = r17502228 + r17502245;
        double r17502247 = r17502244 / r17502246;
        double r17502248 = r17502243 + r17502247;
        double r17502249 = r17502239 + r17502248;
        double r17502250 = r17502233 + r17502249;
        double r17502251 = 12.507343278686905;
        double r17502252 = 5.0;
        double r17502253 = r17502228 + r17502252;
        double r17502254 = r17502251 / r17502253;
        double r17502255 = -0.13857109526572012;
        double r17502256 = 6.0;
        double r17502257 = r17502228 + r17502256;
        double r17502258 = r17502255 / r17502257;
        double r17502259 = r17502254 + r17502258;
        double r17502260 = 1.5056327351493116e-07;
        double r17502261 = 8.0;
        double r17502262 = r17502228 + r17502261;
        double r17502263 = r17502260 / r17502262;
        double r17502264 = r17502259 + r17502263;
        double r17502265 = r17502250 + r17502264;
        double r17502266 = r17502224 * r17502265;
        double r17502267 = 0.5;
        double r17502268 = r17502230 + r17502267;
        double r17502269 = sqrt(r17502268);
        double r17502270 = r17502228 + r17502267;
        double r17502271 = pow(r17502269, r17502270);
        double r17502272 = exp(r17502268);
        double r17502273 = r17502271 / r17502272;
        double r17502274 = r17502271 * r17502273;
        double r17502275 = r17502266 * r17502274;
        return r17502275;
}

Error

Bits error versus z

Derivation

1. Initial program 61.6

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

2. Simplified 1.1

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

4. Applied *-un-lft-identity 1.1

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

5. Applied times-frac 1.0

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

6. Simplified 1.0

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

8. Applied *-un-lft-identity 1.0

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

9. Applied add-sqr-sqrt 1.0

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

10. Applied unpow-prod-down 1.0

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

11. Applied times-frac 1.0

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

12. Simplified 1.0

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

13. Final simplification 1.0

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

Reproduce

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