Average Error: 61.7 → 1.0
Time: 29.1s
Precision: 64
\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(z - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(z - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(z - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\]
↓
\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{z}\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(z - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(z - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\]
\left(\left(\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(\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}\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)
double f(double z) {
        double r193395 = atan2(1.0, 0.0);
        double r193396 = 2.0;
        double r193397 = r193395 * r193396;
        double r193398 = sqrt(r193397);
        double r193399 = z;
        double r193400 = 1.0;
        double r193401 = r193399 - r193400;
        double r193402 = 7.0;
        double r193403 = r193401 + r193402;
        double r193404 = 0.5;
        double r193405 = r193403 + r193404;
        double r193406 = r193401 + r193404;
        double r193407 = pow(r193405, r193406);
        double r193408 = r193398 * r193407;
        double r193409 = -r193405;
        double r193410 = exp(r193409);
        double r193411 = r193408 * r193410;
        double r193412 = 0.9999999999998099;
        double r193413 = 676.5203681218851;
        double r193414 = r193401 + r193400;
        double r193415 = r193413 / r193414;
        double r193416 = r193412 + r193415;
        double r193417 = -1259.1392167224028;
        double r193418 = r193401 + r193396;
        double r193419 = r193417 / r193418;
        double r193420 = r193416 + r193419;
        double r193421 = 771.3234287776531;
        double r193422 = 3.0;
        double r193423 = r193401 + r193422;
        double r193424 = r193421 / r193423;
        double r193425 = r193420 + r193424;
        double r193426 = -176.6150291621406;
        double r193427 = 4.0;
        double r193428 = r193401 + r193427;
        double r193429 = r193426 / r193428;
        double r193430 = r193425 + r193429;
        double r193431 = 12.507343278686905;
        double r193432 = 5.0;
        double r193433 = r193401 + r193432;
        double r193434 = r193431 / r193433;
        double r193435 = r193430 + r193434;
        double r193436 = -0.13857109526572012;
        double r193437 = 6.0;
        double r193438 = r193401 + r193437;
        double r193439 = r193436 / r193438;
        double r193440 = r193435 + r193439;
        double r193441 = 9.984369578019572e-06;
        double r193442 = r193441 / r193403;
        double r193443 = r193440 + r193442;
        double r193444 = 1.5056327351493116e-07;
        double r193445 = 8.0;
        double r193446 = r193401 + r193445;
        double r193447 = r193444 / r193446;
        double r193448 = r193443 + r193447;
        double r193449 = r193411 * r193448;
        return r193449;
}
↓
double f(double z) {
        double r193450 = atan2(1.0, 0.0);
        double r193451 = 2.0;
        double r193452 = r193450 * r193451;
        double r193453 = sqrt(r193452);
        double r193454 = z;
        double r193455 = 1.0;
        double r193456 = r193454 - r193455;
        double r193457 = 7.0;
        double r193458 = r193456 + r193457;
        double r193459 = 0.5;
        double r193460 = r193458 + r193459;
        double r193461 = r193456 + r193459;
        double r193462 = pow(r193460, r193461);
        double r193463 = r193453 * r193462;
        double r193464 = -r193460;
        double r193465 = exp(r193464);
        double r193466 = r193463 * r193465;
        double r193467 = 0.9999999999998099;
        double r193468 = 676.5203681218851;
        double r193469 = r193468 / r193454;
        double r193470 = r193467 + r193469;
        double r193471 = -1259.1392167224028;
        double r193472 = r193456 + r193451;
        double r193473 = r193471 / r193472;
        double r193474 = r193470 + r193473;
        double r193475 = 771.3234287776531;
        double r193476 = 3.0;
        double r193477 = r193456 + r193476;
        double r193478 = r193475 / r193477;
        double r193479 = r193474 + r193478;
        double r193480 = -176.6150291621406;
        double r193481 = 4.0;
        double r193482 = r193456 + r193481;
        double r193483 = r193480 / r193482;
        double r193484 = r193479 + r193483;
        double r193485 = 12.507343278686905;
        double r193486 = 5.0;
        double r193487 = r193456 + r193486;
        double r193488 = r193485 / r193487;
        double r193489 = r193484 + r193488;
        double r193490 = -0.13857109526572012;
        double r193491 = 6.0;
        double r193492 = r193456 + r193491;
        double r193493 = r193490 / r193492;
        double r193494 = r193489 + r193493;
        double r193495 = 9.984369578019572e-06;
        double r193496 = r193495 / r193458;
        double r193497 = r193494 + r193496;
        double r193498 = 1.5056327351493116e-07;
        double r193499 = 8.0;
        double r193500 = r193456 + r193499;
        double r193501 = r193498 / r193500;
        double r193502 = r193497 + r193501;
        double r193503 = r193466 * r193502;
        return r193503;
}

Error

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 61.7

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

  2. Taylor expanded around 0 1.0

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

  3. Final simplification1.0

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

Reproduce

herbie shell --seed 2020065 +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)))))