Jmat.Real.gamma, branch z greater than 0.5

Average Error: 61.5 → 0.9

Time: 25.4s

Precision: 64

• could not determine a ground truth for program body

  1. z = 5.1572421902127275e+265

Math

\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(z - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(z - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(z - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\]

↓

\[\frac{\left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{z + 0}\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(z - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(z - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{z}\right)}{{\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(1 - 0.5\right)} \cdot e^{\left(\left(z - 1\right) + 7\right) + 0.5}}\]

C

double f(double z) {
        double r156262 = atan2(1.0, 0.0);
        double r156263 = 2.0;
        double r156264 = r156262 * r156263;
        double r156265 = sqrt(r156264);
        double r156266 = z;
        double r156267 = 1.0;
        double r156268 = r156266 - r156267;
        double r156269 = 7.0;
        double r156270 = r156268 + r156269;
        double r156271 = 0.5;
        double r156272 = r156270 + r156271;
        double r156273 = r156268 + r156271;
        double r156274 = pow(r156272, r156273);
        double r156275 = r156265 * r156274;
        double r156276 = -r156272;
        double r156277 = exp(r156276);
        double r156278 = r156275 * r156277;
        double r156279 = 0.9999999999998099;
        double r156280 = 676.5203681218851;
        double r156281 = r156268 + r156267;
        double r156282 = r156280 / r156281;
        double r156283 = r156279 + r156282;
        double r156284 = -1259.1392167224028;
        double r156285 = r156268 + r156263;
        double r156286 = r156284 / r156285;
        double r156287 = r156283 + r156286;
        double r156288 = 771.3234287776531;
        double r156289 = 3.0;
        double r156290 = r156268 + r156289;
        double r156291 = r156288 / r156290;
        double r156292 = r156287 + r156291;
        double r156293 = -176.6150291621406;
        double r156294 = 4.0;
        double r156295 = r156268 + r156294;
        double r156296 = r156293 / r156295;
        double r156297 = r156292 + r156296;
        double r156298 = 12.507343278686905;
        double r156299 = 5.0;
        double r156300 = r156268 + r156299;
        double r156301 = r156298 / r156300;
        double r156302 = r156297 + r156301;
        double r156303 = -0.13857109526572012;
        double r156304 = 6.0;
        double r156305 = r156268 + r156304;
        double r156306 = r156303 / r156305;
        double r156307 = r156302 + r156306;
        double r156308 = 9.984369578019572e-06;
        double r156309 = r156308 / r156270;
        double r156310 = r156307 + r156309;
        double r156311 = 1.5056327351493116e-07;
        double r156312 = 8.0;
        double r156313 = r156268 + r156312;
        double r156314 = r156311 / r156313;
        double r156315 = r156310 + r156314;
        double r156316 = r156278 * r156315;
        return r156316;
}

↓

double f(double z) {
        double r156317 = 0.9999999999998099;
        double r156318 = 676.5203681218851;
        double r156319 = z;
        double r156320 = 0.0;
        double r156321 = r156319 + r156320;
        double r156322 = r156318 / r156321;
        double r156323 = r156317 + r156322;
        double r156324 = -1259.1392167224028;
        double r156325 = 1.0;
        double r156326 = r156319 - r156325;
        double r156327 = 2.0;
        double r156328 = r156326 + r156327;
        double r156329 = r156324 / r156328;
        double r156330 = r156323 + r156329;
        double r156331 = 771.3234287776531;
        double r156332 = 3.0;
        double r156333 = r156326 + r156332;
        double r156334 = r156331 / r156333;
        double r156335 = r156330 + r156334;
        double r156336 = -176.6150291621406;
        double r156337 = 4.0;
        double r156338 = r156326 + r156337;
        double r156339 = r156336 / r156338;
        double r156340 = r156335 + r156339;
        double r156341 = 12.507343278686905;
        double r156342 = 5.0;
        double r156343 = r156326 + r156342;
        double r156344 = r156341 / r156343;
        double r156345 = r156340 + r156344;
        double r156346 = -0.13857109526572012;
        double r156347 = 6.0;
        double r156348 = r156326 + r156347;
        double r156349 = r156346 / r156348;
        double r156350 = r156345 + r156349;
        double r156351 = 9.984369578019572e-06;
        double r156352 = 7.0;
        double r156353 = r156326 + r156352;
        double r156354 = r156351 / r156353;
        double r156355 = r156350 + r156354;
        double r156356 = 1.5056327351493116e-07;
        double r156357 = 8.0;
        double r156358 = r156326 + r156357;
        double r156359 = r156356 / r156358;
        double r156360 = r156355 + r156359;
        double r156361 = atan2(1.0, 0.0);
        double r156362 = r156361 * r156327;
        double r156363 = sqrt(r156362);
        double r156364 = 0.5;
        double r156365 = r156353 + r156364;
        double r156366 = pow(r156365, r156319);
        double r156367 = r156363 * r156366;
        double r156368 = r156360 * r156367;
        double r156369 = r156325 - r156364;
        double r156370 = pow(r156365, r156369);
        double r156371 = exp(r156365);
        double r156372 = r156370 * r156371;
        double r156373 = r156368 / r156372;
        return r156373;
}

Error

Bits error versus z

Derivation

1. Initial program 61.5

   \[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(z - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(z - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(z - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\]
2. Using strategy rm

3. Applied sub-neg 61.5

   \[\leadsto \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\color{blue}{\left(z + \left(-1\right)\right)} + 1}\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(z - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(z - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\]

4. Applied associate-+l+ 0.9

   \[\leadsto \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\color{blue}{z + \left(\left(-1\right) + 1\right)}}\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(z - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(z - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\]

5. Simplified 0.9

   \[\leadsto \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{z + \color{blue}{0}}\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(z - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(z - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\]
6. Using strategy rm

7. Applied exp-neg 1.1

   \[\leadsto \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot \color{blue}{\frac{1}{e^{\left(\left(z - 1\right) + 7\right) + 0.5}}}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{z + 0}\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(z - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(z - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\]

8. Applied associate-+l- 1.1

   \[\leadsto \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\color{blue}{\left(z - \left(1 - 0.5\right)\right)}}\right) \cdot \frac{1}{e^{\left(\left(z - 1\right) + 7\right) + 0.5}}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{z + 0}\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(z - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(z - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\]

9. Applied pow-sub 1.1

   \[\leadsto \left(\left(\sqrt{\pi \cdot 2} \cdot \color{blue}{\frac{{\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{z}}{{\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(1 - 0.5\right)}}}\right) \cdot \frac{1}{e^{\left(\left(z - 1\right) + 7\right) + 0.5}}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{z + 0}\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(z - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(z - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\]

10. Applied associate-*r/ 1.1

    \[\leadsto \left(\color{blue}{\frac{\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{z}}{{\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(1 - 0.5\right)}}} \cdot \frac{1}{e^{\left(\left(z - 1\right) + 7\right) + 0.5}}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{z + 0}\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(z - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(z - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\]

11. Applied frac-times 0.9

    \[\leadsto \color{blue}{\frac{\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{z}\right) \cdot 1}{{\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(1 - 0.5\right)} \cdot e^{\left(\left(z - 1\right) + 7\right) + 0.5}}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{z + 0}\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(z - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(z - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\]

12. Applied associate-*l/ 0.9

    \[\leadsto \color{blue}{\frac{\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{z}\right) \cdot 1\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{z + 0}\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(z - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(z - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)}{{\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(1 - 0.5\right)} \cdot e^{\left(\left(z - 1\right) + 7\right) + 0.5}}}\]

13. Simplified 0.9

    \[\leadsto \frac{\color{blue}{\left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{z + 0}\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(z - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(z - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{z}\right)}}{{\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(1 - 0.5\right)} \cdot e^{\left(\left(z - 1\right) + 7\right) + 0.5}}\]

14. Final simplification 0.9

    \[\leadsto \frac{\left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{z + 0}\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(z - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(z - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \cdot \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{z}\right)}{{\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(1 - 0.5\right)} \cdot e^{\left(\left(z - 1\right) + 7\right) + 0.5}}\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (z)
  :name "Jmat.Real.gamma, branch z greater than 0.5"
  :precision binary64
  (* (* (* (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)))))