Jmat.Real.gamma, branch z greater than 0.5

Average Error: 61.6 → 0.8

Time: 27.1s

Precision: 64

• could not determine a ground truth for program body

  1. z = 4.595623646030552e+209

Math

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

↓

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

C

double f(double z) {
        double r186483 = atan2(1.0, 0.0);
        double r186484 = 2.0;
        double r186485 = r186483 * r186484;
        double r186486 = sqrt(r186485);
        double r186487 = z;
        double r186488 = 1.0;
        double r186489 = r186487 - r186488;
        double r186490 = 7.0;
        double r186491 = r186489 + r186490;
        double r186492 = 0.5;
        double r186493 = r186491 + r186492;
        double r186494 = r186489 + r186492;
        double r186495 = pow(r186493, r186494);
        double r186496 = r186486 * r186495;
        double r186497 = -r186493;
        double r186498 = exp(r186497);
        double r186499 = r186496 * r186498;
        double r186500 = 0.9999999999998099;
        double r186501 = 676.5203681218851;
        double r186502 = r186489 + r186488;
        double r186503 = r186501 / r186502;
        double r186504 = r186500 + r186503;
        double r186505 = -1259.1392167224028;
        double r186506 = r186489 + r186484;
        double r186507 = r186505 / r186506;
        double r186508 = r186504 + r186507;
        double r186509 = 771.3234287776531;
        double r186510 = 3.0;
        double r186511 = r186489 + r186510;
        double r186512 = r186509 / r186511;
        double r186513 = r186508 + r186512;
        double r186514 = -176.6150291621406;
        double r186515 = 4.0;
        double r186516 = r186489 + r186515;
        double r186517 = r186514 / r186516;
        double r186518 = r186513 + r186517;
        double r186519 = 12.507343278686905;
        double r186520 = 5.0;
        double r186521 = r186489 + r186520;
        double r186522 = r186519 / r186521;
        double r186523 = r186518 + r186522;
        double r186524 = -0.13857109526572012;
        double r186525 = 6.0;
        double r186526 = r186489 + r186525;
        double r186527 = r186524 / r186526;
        double r186528 = r186523 + r186527;
        double r186529 = 9.984369578019572e-06;
        double r186530 = r186529 / r186491;
        double r186531 = r186528 + r186530;
        double r186532 = 1.5056327351493116e-07;
        double r186533 = 8.0;
        double r186534 = r186489 + r186533;
        double r186535 = r186532 / r186534;
        double r186536 = r186531 + r186535;
        double r186537 = r186499 * r186536;
        return r186537;
}

↓

double f(double z) {
        double r186538 = 1.0;
        double r186539 = z;
        double r186540 = 1.0;
        double r186541 = r186539 - r186540;
        double r186542 = exp(r186541);
        double r186543 = r186538 / r186542;
        double r186544 = atan2(1.0, 0.0);
        double r186545 = 2.0;
        double r186546 = r186544 * r186545;
        double r186547 = sqrt(r186546);
        double r186548 = 7.0;
        double r186549 = exp(r186548);
        double r186550 = r186547 / r186549;
        double r186551 = r186543 * r186550;
        double r186552 = r186541 + r186548;
        double r186553 = 0.5;
        double r186554 = r186552 + r186553;
        double r186555 = r186541 + r186553;
        double r186556 = pow(r186554, r186555);
        double r186557 = 0.9999999999998099;
        double r186558 = 676.5203681218851;
        double r186559 = r186541 - r186540;
        double r186560 = r186559 * r186539;
        double r186561 = r186560 / r186559;
        double r186562 = r186558 / r186561;
        double r186563 = r186557 + r186562;
        double r186564 = -1259.1392167224028;
        double r186565 = r186541 + r186545;
        double r186566 = r186564 / r186565;
        double r186567 = r186563 + r186566;
        double r186568 = 771.3234287776531;
        double r186569 = 3.0;
        double r186570 = r186541 + r186569;
        double r186571 = r186568 / r186570;
        double r186572 = r186567 + r186571;
        double r186573 = -176.6150291621406;
        double r186574 = 4.0;
        double r186575 = r186541 + r186574;
        double r186576 = r186573 / r186575;
        double r186577 = r186572 + r186576;
        double r186578 = 12.507343278686905;
        double r186579 = 5.0;
        double r186580 = r186541 + r186579;
        double r186581 = r186578 / r186580;
        double r186582 = r186577 + r186581;
        double r186583 = -0.13857109526572012;
        double r186584 = 6.0;
        double r186585 = r186541 + r186584;
        double r186586 = r186583 / r186585;
        double r186587 = r186582 + r186586;
        double r186588 = 9.984369578019572e-06;
        double r186589 = r186588 / r186552;
        double r186590 = r186587 + r186589;
        double r186591 = 1.5056327351493116e-07;
        double r186592 = 8.0;
        double r186593 = r186541 + r186592;
        double r186594 = r186591 / r186593;
        double r186595 = r186590 + r186594;
        double r186596 = r186556 * r186595;
        double r186597 = exp(r186553);
        double r186598 = r186596 / r186597;
        double r186599 = r186551 * r186598;
        return r186599;
}

Error

Bits error versus z

Derivation

1. Initial program 61.6

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

2. Simplified 61.6

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

4. Applied flip-+ 61.6

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

5. Simplified 0.8

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

7. Applied associate-*l* 0.8

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

9. Applied exp-sum 0.8

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

10. Applied times-frac 0.8

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

12. Applied exp-sum 0.8

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

13. Applied *-un-lft-identity 0.8

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

14. Applied times-frac 0.8

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

15. Final simplification 0.8

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

Reproduce

herbie shell --seed 2020056 +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)))))