Jmat.Real.gamma, branch z greater than 0.5

Average Error: 61.6 → 0.9

Time: 6.0m

Precision: 64

• could not determine a ground truth for program body

  1. z = 1.8832230334659124e+24

Math

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

↓

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

C

double f(double z) {
        double r274151 = atan2(1.0, 0.0);
        double r274152 = 2.0;
        double r274153 = r274151 * r274152;
        double r274154 = sqrt(r274153);
        double r274155 = z;
        double r274156 = 1.0;
        double r274157 = r274155 - r274156;
        double r274158 = 7.0;
        double r274159 = r274157 + r274158;
        double r274160 = 0.5;
        double r274161 = r274159 + r274160;
        double r274162 = r274157 + r274160;
        double r274163 = pow(r274161, r274162);
        double r274164 = r274154 * r274163;
        double r274165 = -r274161;
        double r274166 = exp(r274165);
        double r274167 = r274164 * r274166;
        double r274168 = 0.9999999999998099;
        double r274169 = 676.5203681218851;
        double r274170 = r274157 + r274156;
        double r274171 = r274169 / r274170;
        double r274172 = r274168 + r274171;
        double r274173 = -1259.1392167224028;
        double r274174 = r274157 + r274152;
        double r274175 = r274173 / r274174;
        double r274176 = r274172 + r274175;
        double r274177 = 771.3234287776531;
        double r274178 = 3.0;
        double r274179 = r274157 + r274178;
        double r274180 = r274177 / r274179;
        double r274181 = r274176 + r274180;
        double r274182 = -176.6150291621406;
        double r274183 = 4.0;
        double r274184 = r274157 + r274183;
        double r274185 = r274182 / r274184;
        double r274186 = r274181 + r274185;
        double r274187 = 12.507343278686905;
        double r274188 = 5.0;
        double r274189 = r274157 + r274188;
        double r274190 = r274187 / r274189;
        double r274191 = r274186 + r274190;
        double r274192 = -0.13857109526572012;
        double r274193 = 6.0;
        double r274194 = r274157 + r274193;
        double r274195 = r274192 / r274194;
        double r274196 = r274191 + r274195;
        double r274197 = 9.984369578019572e-06;
        double r274198 = r274197 / r274159;
        double r274199 = r274196 + r274198;
        double r274200 = 1.5056327351493116e-07;
        double r274201 = 8.0;
        double r274202 = r274157 + r274201;
        double r274203 = r274200 / r274202;
        double r274204 = r274199 + r274203;
        double r274205 = r274167 * r274204;
        return r274205;
}

↓

double f(double z) {
        double r274206 = atan2(1.0, 0.0);
        double r274207 = 2.0;
        double r274208 = r274206 * r274207;
        double r274209 = sqrt(r274208);
        double r274210 = 0.5;
        double r274211 = z;
        double r274212 = 7.0;
        double r274213 = r274211 + r274212;
        double r274214 = r274210 + r274213;
        double r274215 = 1.0;
        double r274216 = r274214 - r274215;
        double r274217 = sqrt(r274216);
        double r274218 = r274211 - r274215;
        double r274219 = r274210 + r274218;
        double r274220 = pow(r274217, r274219);
        double r274221 = exp(r274211);
        double r274222 = r274220 / r274221;
        double r274223 = r274215 - r274210;
        double r274224 = r274223 - r274212;
        double r274225 = -r274224;
        double r274226 = exp(r274225);
        double r274227 = r274220 / r274226;
        double r274228 = r274222 * r274227;
        double r274229 = r274209 * r274228;
        double r274230 = -0.13857109526572012;
        double r274231 = 6.0;
        double r274232 = r274218 + r274231;
        double r274233 = r274230 / r274232;
        double r274234 = 1.5056327351493116e-07;
        double r274235 = 8.0;
        double r274236 = r274235 + r274218;
        double r274237 = r274234 / r274236;
        double r274238 = 9.984369578019572e-06;
        double r274239 = r274218 + r274212;
        double r274240 = r274238 / r274239;
        double r274241 = r274237 + r274240;
        double r274242 = 12.507343278686905;
        double r274243 = 5.0;
        double r274244 = r274218 + r274243;
        double r274245 = r274242 / r274244;
        double r274246 = r274241 + r274245;
        double r274247 = r274233 + r274246;
        double r274248 = 0.9999999999998099;
        double r274249 = 676.5203681218851;
        double r274250 = r274249 / r274211;
        double r274251 = r274248 + r274250;
        double r274252 = -176.6150291621406;
        double r274253 = 4.0;
        double r274254 = r274253 + r274218;
        double r274255 = r274252 / r274254;
        double r274256 = r274251 + r274255;
        double r274257 = 771.3234287776531;
        double r274258 = 3.0;
        double r274259 = r274218 + r274258;
        double r274260 = r274257 / r274259;
        double r274261 = r274256 + r274260;
        double r274262 = r274247 + r274261;
        double r274263 = -1259.1392167224028;
        double r274264 = r274218 + r274207;
        double r274265 = r274263 / r274264;
        double r274266 = r274262 + r274265;
        double r274267 = r274229 * r274266;
        return r274267;
}

Error

Bits error versus z

Derivation

1. Initial program 61.6

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

2. Simplified 1.0

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

4. Applied sub-neg 1.0

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

5. Applied exp-sum 0.9

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

6. Applied add-sqr-sqrt 0.9

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

7. Applied unpow-prod-down 0.9

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

8. Applied times-frac 0.9

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

9. Simplified 0.9

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

10. Simplified 0.9

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

11. Final simplification 0.9

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

Reproduce

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