Jmat.Real.gamma, branch z less than 0.5

Average Error: 1.8 → 1.8

Time: 53.4s

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 r123492 = atan2(1.0, 0.0);
        double r123493 = z;
        double r123494 = r123492 * r123493;
        double r123495 = sin(r123494);
        double r123496 = r123492 / r123495;
        double r123497 = 2.0;
        double r123498 = r123492 * r123497;
        double r123499 = sqrt(r123498);
        double r123500 = 1.0;
        double r123501 = r123500 - r123493;
        double r123502 = r123501 - r123500;
        double r123503 = 7.0;
        double r123504 = r123502 + r123503;
        double r123505 = 0.5;
        double r123506 = r123504 + r123505;
        double r123507 = r123502 + r123505;
        double r123508 = pow(r123506, r123507);
        double r123509 = r123499 * r123508;
        double r123510 = -r123506;
        double r123511 = exp(r123510);
        double r123512 = r123509 * r123511;
        double r123513 = 0.9999999999998099;
        double r123514 = 676.5203681218851;
        double r123515 = r123502 + r123500;
        double r123516 = r123514 / r123515;
        double r123517 = r123513 + r123516;
        double r123518 = -1259.1392167224028;
        double r123519 = r123502 + r123497;
        double r123520 = r123518 / r123519;
        double r123521 = r123517 + r123520;
        double r123522 = 771.3234287776531;
        double r123523 = 3.0;
        double r123524 = r123502 + r123523;
        double r123525 = r123522 / r123524;
        double r123526 = r123521 + r123525;
        double r123527 = -176.6150291621406;
        double r123528 = 4.0;
        double r123529 = r123502 + r123528;
        double r123530 = r123527 / r123529;
        double r123531 = r123526 + r123530;
        double r123532 = 12.507343278686905;
        double r123533 = 5.0;
        double r123534 = r123502 + r123533;
        double r123535 = r123532 / r123534;
        double r123536 = r123531 + r123535;
        double r123537 = -0.13857109526572012;
        double r123538 = 6.0;
        double r123539 = r123502 + r123538;
        double r123540 = r123537 / r123539;
        double r123541 = r123536 + r123540;
        double r123542 = 9.984369578019572e-06;
        double r123543 = r123542 / r123504;
        double r123544 = r123541 + r123543;
        double r123545 = 1.5056327351493116e-07;
        double r123546 = 8.0;
        double r123547 = r123502 + r123546;
        double r123548 = r123545 / r123547;
        double r123549 = r123544 + r123548;
        double r123550 = r123512 * r123549;
        double r123551 = r123496 * r123550;
        return r123551;
}

↓

double f(double z) {
        double r123552 = atan2(1.0, 0.0);
        double r123553 = z;
        double r123554 = r123552 * r123553;
        double r123555 = sin(r123554);
        double r123556 = r123552 / r123555;
        double r123557 = 2.0;
        double r123558 = r123552 * r123557;
        double r123559 = sqrt(r123558);
        double r123560 = 1.0;
        double r123561 = r123560 - r123553;
        double r123562 = r123561 - r123560;
        double r123563 = 7.0;
        double r123564 = r123562 + r123563;
        double r123565 = 0.5;
        double r123566 = r123564 + r123565;
        double r123567 = r123562 + r123565;
        double r123568 = pow(r123566, r123567);
        double r123569 = r123559 * r123568;
        double r123570 = -r123566;
        double r123571 = exp(r123570);
        double r123572 = r123569 * r123571;
        double r123573 = 0.9999999999998099;
        double r123574 = 676.5203681218851;
        double r123575 = r123562 + r123560;
        double r123576 = r123574 / r123575;
        double r123577 = r123573 + r123576;
        double r123578 = -1259.1392167224028;
        double r123579 = r123562 + r123557;
        double r123580 = r123578 / r123579;
        double r123581 = r123577 + r123580;
        double r123582 = 771.3234287776531;
        double r123583 = 3.0;
        double r123584 = r123562 + r123583;
        double r123585 = r123582 / r123584;
        double r123586 = r123581 + r123585;
        double r123587 = -176.6150291621406;
        double r123588 = 4.0;
        double r123589 = r123562 + r123588;
        double r123590 = r123587 / r123589;
        double r123591 = r123586 + r123590;
        double r123592 = 12.507343278686905;
        double r123593 = 5.0;
        double r123594 = r123562 + r123593;
        double r123595 = r123592 / r123594;
        double r123596 = r123591 + r123595;
        double r123597 = -0.13857109526572012;
        double r123598 = cbrt(r123562);
        double r123599 = r123598 * r123598;
        double r123600 = r123599 * r123598;
        double r123601 = cbrt(r123600);
        double r123602 = r123598 * r123601;
        double r123603 = -1.0;
        double r123604 = 3.0;
        double r123605 = pow(r123553, r123604);
        double r123606 = r123603 * r123605;
        double r123607 = cbrt(r123606);
        double r123608 = cbrt(r123607);
        double r123609 = r123598 * r123608;
        double r123610 = r123609 * r123598;
        double r123611 = cbrt(r123610);
        double r123612 = r123602 * r123611;
        double r123613 = 6.0;
        double r123614 = r123612 + r123613;
        double r123615 = r123597 / r123614;
        double r123616 = r123596 + r123615;
        double r123617 = 9.984369578019572e-06;
        double r123618 = r123617 / r123564;
        double r123619 = r123616 + r123618;
        double r123620 = 1.5056327351493116e-07;
        double r123621 = 8.0;
        double r123622 = r123562 + r123621;
        double r123623 = r123620 / r123622;
        double r123624 = r123619 + r123623;
        double r123625 = r123572 * r123624;
        double r123626 = r123556 * r123625;
        return r123626;
}

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))))))