Average Error: 61.7 → 1.0
Time: 27.8s
Precision: 64
\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(z - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(z - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(z - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\]
↓
\[\left(\left(\left({\left(z + 6.5\right)}^{\left(z - 0.5\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{z + 0}\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(z - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(z - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\]
\left(\left(\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(\left({\left(z + 6.5\right)}^{\left(z - 0.5\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\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 + 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)
double f(double z) {
        double r198514 = atan2(1.0, 0.0);
        double r198515 = 2.0;
        double r198516 = r198514 * r198515;
        double r198517 = sqrt(r198516);
        double r198518 = z;
        double r198519 = 1.0;
        double r198520 = r198518 - r198519;
        double r198521 = 7.0;
        double r198522 = r198520 + r198521;
        double r198523 = 0.5;
        double r198524 = r198522 + r198523;
        double r198525 = r198520 + r198523;
        double r198526 = pow(r198524, r198525);
        double r198527 = r198517 * r198526;
        double r198528 = -r198524;
        double r198529 = exp(r198528);
        double r198530 = r198527 * r198529;
        double r198531 = 0.9999999999998099;
        double r198532 = 676.5203681218851;
        double r198533 = r198520 + r198519;
        double r198534 = r198532 / r198533;
        double r198535 = r198531 + r198534;
        double r198536 = -1259.1392167224028;
        double r198537 = r198520 + r198515;
        double r198538 = r198536 / r198537;
        double r198539 = r198535 + r198538;
        double r198540 = 771.3234287776531;
        double r198541 = 3.0;
        double r198542 = r198520 + r198541;
        double r198543 = r198540 / r198542;
        double r198544 = r198539 + r198543;
        double r198545 = -176.6150291621406;
        double r198546 = 4.0;
        double r198547 = r198520 + r198546;
        double r198548 = r198545 / r198547;
        double r198549 = r198544 + r198548;
        double r198550 = 12.507343278686905;
        double r198551 = 5.0;
        double r198552 = r198520 + r198551;
        double r198553 = r198550 / r198552;
        double r198554 = r198549 + r198553;
        double r198555 = -0.13857109526572012;
        double r198556 = 6.0;
        double r198557 = r198520 + r198556;
        double r198558 = r198555 / r198557;
        double r198559 = r198554 + r198558;
        double r198560 = 9.984369578019572e-06;
        double r198561 = r198560 / r198522;
        double r198562 = r198559 + r198561;
        double r198563 = 1.5056327351493116e-07;
        double r198564 = 8.0;
        double r198565 = r198520 + r198564;
        double r198566 = r198563 / r198565;
        double r198567 = r198562 + r198566;
        double r198568 = r198530 * r198567;
        return r198568;
}
↓
double f(double z) {
        double r198569 = z;
        double r198570 = 6.5;
        double r198571 = r198569 + r198570;
        double r198572 = 0.5;
        double r198573 = r198569 - r198572;
        double r198574 = pow(r198571, r198573);
        double r198575 = 2.0;
        double r198576 = sqrt(r198575);
        double r198577 = r198574 * r198576;
        double r198578 = atan2(1.0, 0.0);
        double r198579 = sqrt(r198578);
        double r198580 = r198577 * r198579;
        double r198581 = 1.0;
        double r198582 = r198569 - r198581;
        double r198583 = 7.0;
        double r198584 = r198582 + r198583;
        double r198585 = r198584 + r198572;
        double r198586 = -r198585;
        double r198587 = exp(r198586);
        double r198588 = r198580 * r198587;
        double r198589 = 0.9999999999998099;
        double r198590 = 676.5203681218851;
        double r198591 = 0.0;
        double r198592 = r198569 + r198591;
        double r198593 = r198590 / r198592;
        double r198594 = r198589 + r198593;
        double r198595 = -1259.1392167224028;
        double r198596 = r198582 + r198575;
        double r198597 = r198595 / r198596;
        double r198598 = r198594 + r198597;
        double r198599 = 771.3234287776531;
        double r198600 = 3.0;
        double r198601 = r198582 + r198600;
        double r198602 = r198599 / r198601;
        double r198603 = r198598 + r198602;
        double r198604 = -176.6150291621406;
        double r198605 = 4.0;
        double r198606 = r198582 + r198605;
        double r198607 = r198604 / r198606;
        double r198608 = r198603 + r198607;
        double r198609 = 12.507343278686905;
        double r198610 = 5.0;
        double r198611 = r198582 + r198610;
        double r198612 = r198609 / r198611;
        double r198613 = r198608 + r198612;
        double r198614 = -0.13857109526572012;
        double r198615 = 6.0;
        double r198616 = r198582 + r198615;
        double r198617 = r198614 / r198616;
        double r198618 = r198613 + r198617;
        double r198619 = 9.984369578019572e-06;
        double r198620 = r198619 / r198584;
        double r198621 = r198618 + r198620;
        double r198622 = 1.5056327351493116e-07;
        double r198623 = 8.0;
        double r198624 = r198582 + r198623;
        double r198625 = r198622 / r198624;
        double r198626 = r198621 + r198625;
        double r198627 = r198588 * r198626;
        return r198627;
}

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. Using strategy rm

  3. Applied sub-neg61.7

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

  4. Applied associate-+l+1.0

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

  5. Simplified1.0

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

  6. Taylor expanded around inf 1.0

    \[\leadsto \left(\color{blue}{\left(\left({\left(z + 6.5\right)}^{\left(z - 0.5\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right)} \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{z + 0}\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(z - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(z - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\]

  7. Final simplification1.0

    \[\leadsto \left(\left(\left({\left(z + 6.5\right)}^{\left(z - 0.5\right)} \cdot \sqrt{2}\right) \cdot \sqrt{\pi}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{z + 0}\right) + \frac{-1259.13921672240281}{\left(z - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(z - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(z - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(z - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(z - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right)\]

Reproduce

herbie shell --seed 2020065 
(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)))))