Jmat.Real.gamma, branch z less than 0.5

Average Error: 1.8 → 1.8

Time: 53.2s

Precision: 64

• could not determine a ground truth for program body

  1. z = -5.705042820112819e+286

Math

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

↓

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

C

double f(double z) {
        double r106402 = atan2(1.0, 0.0);
        double r106403 = z;
        double r106404 = r106402 * r106403;
        double r106405 = sin(r106404);
        double r106406 = r106402 / r106405;
        double r106407 = 2.0;
        double r106408 = r106402 * r106407;
        double r106409 = sqrt(r106408);
        double r106410 = 1.0;
        double r106411 = r106410 - r106403;
        double r106412 = r106411 - r106410;
        double r106413 = 7.0;
        double r106414 = r106412 + r106413;
        double r106415 = 0.5;
        double r106416 = r106414 + r106415;
        double r106417 = r106412 + r106415;
        double r106418 = pow(r106416, r106417);
        double r106419 = r106409 * r106418;
        double r106420 = -r106416;
        double r106421 = exp(r106420);
        double r106422 = r106419 * r106421;
        double r106423 = 0.9999999999998099;
        double r106424 = 676.5203681218851;
        double r106425 = r106412 + r106410;
        double r106426 = r106424 / r106425;
        double r106427 = r106423 + r106426;
        double r106428 = -1259.1392167224028;
        double r106429 = r106412 + r106407;
        double r106430 = r106428 / r106429;
        double r106431 = r106427 + r106430;
        double r106432 = 771.3234287776531;
        double r106433 = 3.0;
        double r106434 = r106412 + r106433;
        double r106435 = r106432 / r106434;
        double r106436 = r106431 + r106435;
        double r106437 = -176.6150291621406;
        double r106438 = 4.0;
        double r106439 = r106412 + r106438;
        double r106440 = r106437 / r106439;
        double r106441 = r106436 + r106440;
        double r106442 = 12.507343278686905;
        double r106443 = 5.0;
        double r106444 = r106412 + r106443;
        double r106445 = r106442 / r106444;
        double r106446 = r106441 + r106445;
        double r106447 = -0.13857109526572012;
        double r106448 = 6.0;
        double r106449 = r106412 + r106448;
        double r106450 = r106447 / r106449;
        double r106451 = r106446 + r106450;
        double r106452 = 9.984369578019572e-06;
        double r106453 = r106452 / r106414;
        double r106454 = r106451 + r106453;
        double r106455 = 1.5056327351493116e-07;
        double r106456 = 8.0;
        double r106457 = r106412 + r106456;
        double r106458 = r106455 / r106457;
        double r106459 = r106454 + r106458;
        double r106460 = r106422 * r106459;
        double r106461 = r106406 * r106460;
        return r106461;
}

↓

double f(double z) {
        double r106462 = atan2(1.0, 0.0);
        double r106463 = z;
        double r106464 = r106462 * r106463;
        double r106465 = sin(r106464);
        double r106466 = r106462 / r106465;
        double r106467 = 2.0;
        double r106468 = r106462 * r106467;
        double r106469 = sqrt(r106468);
        double r106470 = 1.0;
        double r106471 = r106470 - r106463;
        double r106472 = r106471 - r106470;
        double r106473 = 7.0;
        double r106474 = r106472 + r106473;
        double r106475 = 0.5;
        double r106476 = r106474 + r106475;
        double r106477 = r106472 + r106475;
        double r106478 = pow(r106476, r106477);
        double r106479 = r106469 * r106478;
        double r106480 = -r106476;
        double r106481 = exp(r106480);
        double r106482 = r106479 * r106481;
        double r106483 = 0.9999999999998099;
        double r106484 = 676.5203681218851;
        double r106485 = r106472 + r106470;
        double r106486 = r106484 / r106485;
        double r106487 = r106483 + r106486;
        double r106488 = -1259.1392167224028;
        double r106489 = r106472 + r106467;
        double r106490 = r106488 / r106489;
        double r106491 = r106487 + r106490;
        double r106492 = 771.3234287776531;
        double r106493 = 3.0;
        double r106494 = r106472 + r106493;
        double r106495 = r106492 / r106494;
        double r106496 = r106491 + r106495;
        double r106497 = -176.6150291621406;
        double r106498 = 4.0;
        double r106499 = r106472 + r106498;
        double r106500 = r106497 / r106499;
        double r106501 = r106496 + r106500;
        double r106502 = 12.507343278686905;
        double r106503 = 5.0;
        double r106504 = r106472 + r106503;
        double r106505 = r106502 / r106504;
        double r106506 = r106501 + r106505;
        double r106507 = -0.13857109526572012;
        double r106508 = cbrt(r106472);
        double r106509 = r106508 * r106508;
        double r106510 = r106509 * r106508;
        double r106511 = cbrt(r106510);
        double r106512 = r106508 * r106511;
        double r106513 = -1.0;
        double r106514 = 3.0;
        double r106515 = pow(r106463, r106514);
        double r106516 = r106513 * r106515;
        double r106517 = cbrt(r106516);
        double r106518 = cbrt(r106517);
        double r106519 = r106508 * r106518;
        double r106520 = r106519 * r106508;
        double r106521 = cbrt(r106520);
        double r106522 = r106512 * r106521;
        double r106523 = 6.0;
        double r106524 = r106522 + r106523;
        double r106525 = r106507 / r106524;
        double r106526 = r106506 + r106525;
        double r106527 = 9.984369578019572e-06;
        double r106528 = r106527 / r106474;
        double r106529 = r106526 + r106528;
        double r106530 = 1.5056327351493116e-07;
        double r106531 = 8.0;
        double r106532 = r106472 + r106531;
        double r106533 = r106530 / r106532;
        double r106534 = r106529 + r106533;
        double r106535 = r106482 * r106534;
        double r106536 = r106466 * r106535;
        return r106536;
}

Error

Bits error versus z

Derivation

1. Initial program 1.8

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

3. Applied add-cube-cbrt 1.8

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

5. Applied add-cube-cbrt 1.8

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

7. Applied add-cube-cbrt 1.8

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

9. Applied add-cbrt-cube 1.8

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

10. Simplified 1.8

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

11. Final simplification 1.8

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

Reproduce

herbie shell --seed 2020056 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z less than 0.5"
  :precision binary64
  (* (/ PI (sin (* PI z))) (* (* (* (sqrt (* PI 2)) (pow (+ (+ (- (- 1 z) 1) 7) 0.5) (+ (- (- 1 z) 1) 0.5))) (exp (- (+ (+ (- (- 1 z) 1) 7) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- (- 1 z) 1) 1))) (/ -1259.1392167224028 (+ (- (- 1 z) 1) 2))) (/ 771.3234287776531 (+ (- (- 1 z) 1) 3))) (/ -176.6150291621406 (+ (- (- 1 z) 1) 4))) (/ 12.507343278686905 (+ (- (- 1 z) 1) 5))) (/ -0.13857109526572012 (+ (- (- 1 z) 1) 6))) (/ 9.984369578019572e-06 (+ (- (- 1 z) 1) 7))) (/ 1.5056327351493116e-07 (+ (- (- 1 z) 1) 8))))))