Average Error: 61.6 → 1.1
Time: 21.7s
Precision: 64
\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(z - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(z - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(z - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\]
↓
\[\left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right) \cdot \left(\left(676.520368121885099 \cdot \frac{\sqrt{2} \cdot e^{-6.5}}{z} + 2581.19179968122216 \cdot \left(\sqrt{2} \cdot \left(z \cdot e^{-6.5}\right)\right)\right) + 676.520368121885099 \cdot \left(\log 6.5 \cdot \left(\sqrt{2} \cdot e^{-6.5}\right)\right)\right) + \left(169.130092030471275 \cdot \left(\left(\sqrt{2} \cdot \left(z \cdot e^{-6.5}\right)\right) \cdot \left({\left(\frac{1}{{6.5}^{5}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right) - \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right) \cdot \left(\left(\left(\log 6.5 \cdot z + 1\right) \cdot \left(\sqrt{2} \cdot e^{-6.5}\right)\right) \cdot 1656.8104518737205 - 338.260184060942549 \cdot \left({\left(\log 6.5\right)}^{2} \cdot \left(z \cdot \left(\sqrt{2} \cdot e^{-6.5}\right)\right)\right)\right)\right)\]
\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)
↓
\left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right) \cdot \left(\left(676.520368121885099 \cdot \frac{\sqrt{2} \cdot e^{-6.5}}{z} + 2581.19179968122216 \cdot \left(\sqrt{2} \cdot \left(z \cdot e^{-6.5}\right)\right)\right) + 676.520368121885099 \cdot \left(\log 6.5 \cdot \left(\sqrt{2} \cdot e^{-6.5}\right)\right)\right) + \left(169.130092030471275 \cdot \left(\left(\sqrt{2} \cdot \left(z \cdot e^{-6.5}\right)\right) \cdot \left({\left(\frac{1}{{6.5}^{5}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right) - \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right) \cdot \left(\left(\left(\log 6.5 \cdot z + 1\right) \cdot \left(\sqrt{2} \cdot e^{-6.5}\right)\right) \cdot 1656.8104518737205 - 338.260184060942549 \cdot \left({\left(\log 6.5\right)}^{2} \cdot \left(z \cdot \left(\sqrt{2} \cdot e^{-6.5}\right)\right)\right)\right)\right)
double f(double z) {
        double r217333 = atan2(1.0, 0.0);
        double r217334 = 2.0;
        double r217335 = r217333 * r217334;
        double r217336 = sqrt(r217335);
        double r217337 = z;
        double r217338 = 1.0;
        double r217339 = r217337 - r217338;
        double r217340 = 7.0;
        double r217341 = r217339 + r217340;
        double r217342 = 0.5;
        double r217343 = r217341 + r217342;
        double r217344 = r217339 + r217342;
        double r217345 = pow(r217343, r217344);
        double r217346 = r217336 * r217345;
        double r217347 = -r217343;
        double r217348 = exp(r217347);
        double r217349 = r217346 * r217348;
        double r217350 = 0.9999999999998099;
        double r217351 = 676.5203681218851;
        double r217352 = r217339 + r217338;
        double r217353 = r217351 / r217352;
        double r217354 = r217350 + r217353;
        double r217355 = -1259.1392167224028;
        double r217356 = r217339 + r217334;
        double r217357 = r217355 / r217356;
        double r217358 = r217354 + r217357;
        double r217359 = 771.3234287776531;
        double r217360 = 3.0;
        double r217361 = r217339 + r217360;
        double r217362 = r217359 / r217361;
        double r217363 = r217358 + r217362;
        double r217364 = -176.6150291621406;
        double r217365 = 4.0;
        double r217366 = r217339 + r217365;
        double r217367 = r217364 / r217366;
        double r217368 = r217363 + r217367;
        double r217369 = 12.507343278686905;
        double r217370 = 5.0;
        double r217371 = r217339 + r217370;
        double r217372 = r217369 / r217371;
        double r217373 = r217368 + r217372;
        double r217374 = -0.13857109526572012;
        double r217375 = 6.0;
        double r217376 = r217339 + r217375;
        double r217377 = r217374 / r217376;
        double r217378 = r217373 + r217377;
        double r217379 = 9.984369578019572e-06;
        double r217380 = r217379 / r217341;
        double r217381 = r217378 + r217380;
        double r217382 = 1.5056327351493116e-07;
        double r217383 = 8.0;
        double r217384 = r217339 + r217383;
        double r217385 = r217382 / r217384;
        double r217386 = r217381 + r217385;
        double r217387 = r217349 * r217386;
        return r217387;
}
↓
double f(double z) {
        double r217388 = 1.0;
        double r217389 = 6.5;
        double r217390 = 1.0;
        double r217391 = pow(r217389, r217390);
        double r217392 = r217388 / r217391;
        double r217393 = 0.5;
        double r217394 = pow(r217392, r217393);
        double r217395 = atan2(1.0, 0.0);
        double r217396 = sqrt(r217395);
        double r217397 = r217394 * r217396;
        double r217398 = 676.5203681218851;
        double r217399 = 2.0;
        double r217400 = sqrt(r217399);
        double r217401 = -r217389;
        double r217402 = exp(r217401);
        double r217403 = r217400 * r217402;
        double r217404 = z;
        double r217405 = r217403 / r217404;
        double r217406 = r217398 * r217405;
        double r217407 = 2581.191799681222;
        double r217408 = r217404 * r217402;
        double r217409 = r217400 * r217408;
        double r217410 = r217407 * r217409;
        double r217411 = r217406 + r217410;
        double r217412 = log(r217389);
        double r217413 = r217412 * r217403;
        double r217414 = r217398 * r217413;
        double r217415 = r217411 + r217414;
        double r217416 = r217397 * r217415;
        double r217417 = 169.13009203047127;
        double r217418 = 5.0;
        double r217419 = pow(r217389, r217418);
        double r217420 = r217388 / r217419;
        double r217421 = pow(r217420, r217393);
        double r217422 = r217421 * r217396;
        double r217423 = r217409 * r217422;
        double r217424 = r217417 * r217423;
        double r217425 = r217412 * r217404;
        double r217426 = r217425 + r217388;
        double r217427 = r217426 * r217403;
        double r217428 = 1656.8104518737205;
        double r217429 = r217427 * r217428;
        double r217430 = 338.26018406094255;
        double r217431 = 2.0;
        double r217432 = pow(r217412, r217431);
        double r217433 = r217404 * r217403;
        double r217434 = r217432 * r217433;
        double r217435 = r217430 * r217434;
        double r217436 = r217429 - r217435;
        double r217437 = r217397 * r217436;
        double r217438 = r217424 - r217437;
        double r217439 = r217416 + r217438;
        return r217439;
}

Error

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 61.6

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

  3. Applied flip-+61.6

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

  4. Simplified0.9

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

  5. Taylor expanded around 0 1.1

    \[\leadsto \color{blue}{\left(338.260184060942549 \cdot \left(\left({\left(\log 6.5\right)}^{2} \cdot \left(z \cdot \left(\sqrt{2} \cdot e^{-6.5}\right)\right)\right) \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right) + \left(169.130092030471275 \cdot \left(\left(\sqrt{2} \cdot \left(z \cdot e^{-6.5}\right)\right) \cdot \left({\left(\frac{1}{{6.5}^{5}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right) + \left(676.520368121885099 \cdot \left(\frac{\sqrt{2} \cdot e^{-6.5}}{z} \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right) + \left(2581.19179968122216 \cdot \left(\left(\sqrt{2} \cdot \left(z \cdot e^{-6.5}\right)\right) \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right) + 676.520368121885099 \cdot \left(\left(\log 6.5 \cdot \left(\sqrt{2} \cdot e^{-6.5}\right)\right) \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right)\right)\right)\right)\right) - \left(1656.8104518737205 \cdot \left(\left(\log 6.5 \cdot \left(z \cdot \left(\sqrt{2} \cdot e^{-6.5}\right)\right)\right) \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right) + 1656.8104518737205 \cdot \left(\left(\sqrt{2} \cdot e^{-6.5}\right) \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right)\right)}\]

  6. Simplified1.1

    \[\leadsto \color{blue}{338.260184060942549 \cdot \left(\left({\left(\log 6.5\right)}^{2} \cdot \left(z \cdot \left(\sqrt{2} \cdot e^{-6.5}\right)\right)\right) \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right) + \left(\left(169.130092030471275 \cdot \left(\left(\sqrt{2} \cdot \left(z \cdot e^{-6.5}\right)\right) \cdot \left({\left(\frac{1}{{6.5}^{5}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right) + \left(676.520368121885099 \cdot \left(\frac{\sqrt{2} \cdot e^{-6.5}}{z} \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right) + \left(2581.19179968122216 \cdot \left(\left(\sqrt{2} \cdot \left(z \cdot e^{-6.5}\right)\right) \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right) + 676.520368121885099 \cdot \left(\left(\log 6.5 \cdot \left(\sqrt{2} \cdot e^{-6.5}\right)\right) \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right)\right)\right)\right) - 1656.8104518737205 \cdot \left(\left(\log 6.5 \cdot \left(z \cdot \left(\sqrt{2} \cdot e^{-6.5}\right)\right)\right) \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right) + \left(\sqrt{2} \cdot e^{-6.5}\right) \cdot \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right)\right)}\]

  7. Simplified1.1

    \[\leadsto \color{blue}{\left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right) \cdot \left(\left(676.520368121885099 \cdot \frac{\sqrt{2} \cdot e^{-6.5}}{z} + 2581.19179968122216 \cdot \left(\sqrt{2} \cdot \left(z \cdot e^{-6.5}\right)\right)\right) + 676.520368121885099 \cdot \left(\log 6.5 \cdot \left(\sqrt{2} \cdot e^{-6.5}\right)\right)\right) + \left(169.130092030471275 \cdot \left(\left(\sqrt{2} \cdot \left(z \cdot e^{-6.5}\right)\right) \cdot \left({\left(\frac{1}{{6.5}^{5}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right) - \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right) \cdot \left(\left(\left(\log 6.5 \cdot z + 1\right) \cdot \left(\sqrt{2} \cdot e^{-6.5}\right)\right) \cdot 1656.8104518737205 - 338.260184060942549 \cdot \left({\left(\log 6.5\right)}^{2} \cdot \left(z \cdot \left(\sqrt{2} \cdot e^{-6.5}\right)\right)\right)\right)\right)}\]

  8. Final simplification1.1

    \[\leadsto \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right) \cdot \left(\left(676.520368121885099 \cdot \frac{\sqrt{2} \cdot e^{-6.5}}{z} + 2581.19179968122216 \cdot \left(\sqrt{2} \cdot \left(z \cdot e^{-6.5}\right)\right)\right) + 676.520368121885099 \cdot \left(\log 6.5 \cdot \left(\sqrt{2} \cdot e^{-6.5}\right)\right)\right) + \left(169.130092030471275 \cdot \left(\left(\sqrt{2} \cdot \left(z \cdot e^{-6.5}\right)\right) \cdot \left({\left(\frac{1}{{6.5}^{5}}\right)}^{0.5} \cdot \sqrt{\pi}\right)\right) - \left({\left(\frac{1}{{6.5}^{1}}\right)}^{0.5} \cdot \sqrt{\pi}\right) \cdot \left(\left(\left(\log 6.5 \cdot z + 1\right) \cdot \left(\sqrt{2} \cdot e^{-6.5}\right)\right) \cdot 1656.8104518737205 - 338.260184060942549 \cdot \left({\left(\log 6.5\right)}^{2} \cdot \left(z \cdot \left(\sqrt{2} \cdot e^{-6.5}\right)\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2020064 
(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)))))