Jmat.Real.gamma, branch z less than 0.5

Average Error: 1.8 → 1.8

Time: 1.8m

Precision: 64

• could not determine a ground truth for program body

  1. z = -2.9343974362389973e+108

Math

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

↓

\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7} + \left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{5 + \left(\sqrt[3]{1 \cdot 1} \cdot \mathsf{fma}\left(\sqrt[3]{\mathsf{fma}\left({z}^{2}, 0.83333333333333337, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left({z}^{2}, 0.83333333333333337, 1\right)}, \sqrt[3]{\mathsf{fma}\left({z}^{2}, 0.83333333333333337, 1\right)}, -z \cdot 1\right) + \left(\left(1 \cdot \left(\left(-z\right) + z\right)\right) \cdot \sqrt[3]{1 \cdot 1} - \mathsf{fma}\left({z}^{2}, \sqrt[3]{\frac{1}{{1}^{4}}} \cdot 0.83333333333333326, 1\right)\right)\right)}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right)\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right) \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right)\right)\]

C

double f(double z) {
        double r100519 = atan2(1.0, 0.0);
        double r100520 = z;
        double r100521 = r100519 * r100520;
        double r100522 = sin(r100521);
        double r100523 = r100519 / r100522;
        double r100524 = 2.0;
        double r100525 = r100519 * r100524;
        double r100526 = sqrt(r100525);
        double r100527 = 1.0;
        double r100528 = r100527 - r100520;
        double r100529 = r100528 - r100527;
        double r100530 = 7.0;
        double r100531 = r100529 + r100530;
        double r100532 = 0.5;
        double r100533 = r100531 + r100532;
        double r100534 = r100529 + r100532;
        double r100535 = pow(r100533, r100534);
        double r100536 = r100526 * r100535;
        double r100537 = -r100533;
        double r100538 = exp(r100537);
        double r100539 = r100536 * r100538;
        double r100540 = 0.9999999999998099;
        double r100541 = 676.5203681218851;
        double r100542 = r100529 + r100527;
        double r100543 = r100541 / r100542;
        double r100544 = r100540 + r100543;
        double r100545 = -1259.1392167224028;
        double r100546 = r100529 + r100524;
        double r100547 = r100545 / r100546;
        double r100548 = r100544 + r100547;
        double r100549 = 771.3234287776531;
        double r100550 = 3.0;
        double r100551 = r100529 + r100550;
        double r100552 = r100549 / r100551;
        double r100553 = r100548 + r100552;
        double r100554 = -176.6150291621406;
        double r100555 = 4.0;
        double r100556 = r100529 + r100555;
        double r100557 = r100554 / r100556;
        double r100558 = r100553 + r100557;
        double r100559 = 12.507343278686905;
        double r100560 = 5.0;
        double r100561 = r100529 + r100560;
        double r100562 = r100559 / r100561;
        double r100563 = r100558 + r100562;
        double r100564 = -0.13857109526572012;
        double r100565 = 6.0;
        double r100566 = r100529 + r100565;
        double r100567 = r100564 / r100566;
        double r100568 = r100563 + r100567;
        double r100569 = 9.984369578019572e-06;
        double r100570 = r100569 / r100531;
        double r100571 = r100568 + r100570;
        double r100572 = 1.5056327351493116e-07;
        double r100573 = 8.0;
        double r100574 = r100529 + r100573;
        double r100575 = r100572 / r100574;
        double r100576 = r100571 + r100575;
        double r100577 = r100539 * r100576;
        double r100578 = r100523 * r100577;
        return r100578;
}

↓

double f(double z) {
        double r100579 = atan2(1.0, 0.0);
        double r100580 = z;
        double r100581 = r100579 * r100580;
        double r100582 = sin(r100581);
        double r100583 = r100579 / r100582;
        double r100584 = 9.984369578019572e-06;
        double r100585 = 1.0;
        double r100586 = r100585 - r100580;
        double r100587 = r100586 - r100585;
        double r100588 = 7.0;
        double r100589 = r100587 + r100588;
        double r100590 = r100584 / r100589;
        double r100591 = 0.9999999999998099;
        double r100592 = 676.5203681218851;
        double r100593 = r100587 + r100585;
        double r100594 = r100592 / r100593;
        double r100595 = r100591 + r100594;
        double r100596 = -1259.1392167224028;
        double r100597 = 2.0;
        double r100598 = r100587 + r100597;
        double r100599 = r100596 / r100598;
        double r100600 = r100595 + r100599;
        double r100601 = 771.3234287776531;
        double r100602 = 3.0;
        double r100603 = r100587 + r100602;
        double r100604 = r100601 / r100603;
        double r100605 = r100600 + r100604;
        double r100606 = -176.6150291621406;
        double r100607 = 4.0;
        double r100608 = r100587 + r100607;
        double r100609 = r100606 / r100608;
        double r100610 = r100605 + r100609;
        double r100611 = 12.507343278686905;
        double r100612 = 5.0;
        double r100613 = r100585 * r100585;
        double r100614 = cbrt(r100613);
        double r100615 = 2.0;
        double r100616 = pow(r100580, r100615);
        double r100617 = 0.8333333333333334;
        double r100618 = 1.0;
        double r100619 = fma(r100616, r100617, r100618);
        double r100620 = cbrt(r100619);
        double r100621 = r100620 * r100620;
        double r100622 = r100580 * r100585;
        double r100623 = -r100622;
        double r100624 = fma(r100621, r100620, r100623);
        double r100625 = r100614 * r100624;
        double r100626 = -r100580;
        double r100627 = r100626 + r100580;
        double r100628 = r100585 * r100627;
        double r100629 = r100628 * r100614;
        double r100630 = 4.0;
        double r100631 = pow(r100585, r100630);
        double r100632 = r100618 / r100631;
        double r100633 = cbrt(r100632);
        double r100634 = 0.8333333333333333;
        double r100635 = r100633 * r100634;
        double r100636 = fma(r100616, r100635, r100585);
        double r100637 = r100629 - r100636;
        double r100638 = r100625 + r100637;
        double r100639 = r100612 + r100638;
        double r100640 = r100611 / r100639;
        double r100641 = r100610 + r100640;
        double r100642 = -0.13857109526572012;
        double r100643 = 6.0;
        double r100644 = r100587 + r100643;
        double r100645 = r100642 / r100644;
        double r100646 = r100641 + r100645;
        double r100647 = r100590 + r100646;
        double r100648 = 1.5056327351493116e-07;
        double r100649 = 8.0;
        double r100650 = r100587 + r100649;
        double r100651 = r100648 / r100650;
        double r100652 = r100647 + r100651;
        double r100653 = r100579 * r100597;
        double r100654 = sqrt(r100653);
        double r100655 = 0.5;
        double r100656 = r100589 + r100655;
        double r100657 = r100587 + r100655;
        double r100658 = pow(r100656, r100657);
        double r100659 = r100654 * r100658;
        double r100660 = -r100656;
        double r100661 = exp(r100660);
        double r100662 = r100659 * r100661;
        double r100663 = r100652 * r100662;
        double r100664 = r100583 * r100663;
        return r100664;
}

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

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

4. Applied add-cube-cbrt 1.8

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

5. Applied prod-diff 1.8

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

6. Simplified 1.8

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

7. Simplified 1.8

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

9. Applied add-cube-cbrt 1.8

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

10. Applied unpow-prod-down 1.8

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

11. Applied fma-neg 1.8

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

12. Taylor expanded around 0 1.8

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\color{blue}{\left(\left({\left({1}^{2}\right)}^{\frac{1}{3}} + 0.83333333333333337 \cdot \left({z}^{2} \cdot {\left({1}^{2}\right)}^{\frac{1}{3}}\right)\right) - \left(1 \cdot \left(z \cdot {\left({1}^{2}\right)}^{\frac{1}{3}}\right) + \left(0.83333333333333326 \cdot \left({z}^{2} \cdot {\left(\frac{1}{{1}^{4}}\right)}^{\frac{1}{3}}\right) + 1\right)\right)\right)} + 0\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

13. Simplified 1.8

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\color{blue}{\left(\sqrt[3]{1 \cdot 1} \cdot \left(\mathsf{fma}\left({z}^{2}, 0.83333333333333337, 1\right) - 1 \cdot z\right) - \mathsf{fma}\left({z}^{2}, \sqrt[3]{\frac{1}{{1}^{4}}} \cdot 0.83333333333333326, 1\right)\right)} + 0\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
14. Using strategy rm

15. Applied add-cube-cbrt 1.8

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(\sqrt[3]{1 \cdot 1} \cdot \left(\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left({z}^{2}, 0.83333333333333337, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left({z}^{2}, 0.83333333333333337, 1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left({z}^{2}, 0.83333333333333337, 1\right)}} - 1 \cdot z\right) - \mathsf{fma}\left({z}^{2}, \sqrt[3]{\frac{1}{{1}^{4}}} \cdot 0.83333333333333326, 1\right)\right) + 0\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

16. Applied prod-diff 1.8

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(\sqrt[3]{1 \cdot 1} \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{\mathsf{fma}\left({z}^{2}, 0.83333333333333337, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left({z}^{2}, 0.83333333333333337, 1\right)}, \sqrt[3]{\mathsf{fma}\left({z}^{2}, 0.83333333333333337, 1\right)}, -z \cdot 1\right) + \mathsf{fma}\left(-z, 1, z \cdot 1\right)\right)} - \mathsf{fma}\left({z}^{2}, \sqrt[3]{\frac{1}{{1}^{4}}} \cdot 0.83333333333333326, 1\right)\right) + 0\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

17. Applied distribute-lft-in 1.8

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(\color{blue}{\left(\sqrt[3]{1 \cdot 1} \cdot \mathsf{fma}\left(\sqrt[3]{\mathsf{fma}\left({z}^{2}, 0.83333333333333337, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left({z}^{2}, 0.83333333333333337, 1\right)}, \sqrt[3]{\mathsf{fma}\left({z}^{2}, 0.83333333333333337, 1\right)}, -z \cdot 1\right) + \sqrt[3]{1 \cdot 1} \cdot \mathsf{fma}\left(-z, 1, z \cdot 1\right)\right)} - \mathsf{fma}\left({z}^{2}, \sqrt[3]{\frac{1}{{1}^{4}}} \cdot 0.83333333333333326, 1\right)\right) + 0\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

18. Applied associate--l+ 1.8

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\color{blue}{\left(\sqrt[3]{1 \cdot 1} \cdot \mathsf{fma}\left(\sqrt[3]{\mathsf{fma}\left({z}^{2}, 0.83333333333333337, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left({z}^{2}, 0.83333333333333337, 1\right)}, \sqrt[3]{\mathsf{fma}\left({z}^{2}, 0.83333333333333337, 1\right)}, -z \cdot 1\right) + \left(\sqrt[3]{1 \cdot 1} \cdot \mathsf{fma}\left(-z, 1, z \cdot 1\right) - \mathsf{fma}\left({z}^{2}, \sqrt[3]{\frac{1}{{1}^{4}}} \cdot 0.83333333333333326, 1\right)\right)\right)} + 0\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

19. Simplified 1.8

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(\sqrt[3]{1 \cdot 1} \cdot \mathsf{fma}\left(\sqrt[3]{\mathsf{fma}\left({z}^{2}, 0.83333333333333337, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left({z}^{2}, 0.83333333333333337, 1\right)}, \sqrt[3]{\mathsf{fma}\left({z}^{2}, 0.83333333333333337, 1\right)}, -z \cdot 1\right) + \color{blue}{\left(\left(1 \cdot \left(\left(-z\right) + z\right)\right) \cdot \sqrt[3]{1 \cdot 1} - \mathsf{fma}\left({z}^{2}, \sqrt[3]{\frac{1}{{1}^{4}}} \cdot 0.83333333333333326, 1\right)\right)}\right) + 0\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

20. Final simplification 1.8

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7} + \left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{5 + \left(\sqrt[3]{1 \cdot 1} \cdot \mathsf{fma}\left(\sqrt[3]{\mathsf{fma}\left({z}^{2}, 0.83333333333333337, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left({z}^{2}, 0.83333333333333337, 1\right)}, \sqrt[3]{\mathsf{fma}\left({z}^{2}, 0.83333333333333337, 1\right)}, -z \cdot 1\right) + \left(\left(1 \cdot \left(\left(-z\right) + z\right)\right) \cdot \sqrt[3]{1 \cdot 1} - \mathsf{fma}\left({z}^{2}, \sqrt[3]{\frac{1}{{1}^{4}}} \cdot 0.83333333333333326, 1\right)\right)\right)}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right)\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right) \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right)\right)\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (z)
  :name "Jmat.Real.gamma, branch z less than 0.5"
  (* (/ PI (sin (* PI z))) (* (* (* (sqrt (* PI 2.0)) (pow (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5) (+ (- (- 1.0 z) 1.0) 0.5))) (exp (- (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- (- 1.0 z) 1.0) 1.0))) (/ -1259.1392167224028 (+ (- (- 1.0 z) 1.0) 2.0))) (/ 771.3234287776531 (+ (- (- 1.0 z) 1.0) 3.0))) (/ -176.6150291621406 (+ (- (- 1.0 z) 1.0) 4.0))) (/ 12.507343278686905 (+ (- (- 1.0 z) 1.0) 5.0))) (/ -0.13857109526572012 (+ (- (- 1.0 z) 1.0) 6.0))) (/ 9.984369578019572e-06 (+ (- (- 1.0 z) 1.0) 7.0))) (/ 1.5056327351493116e-07 (+ (- (- 1.0 z) 1.0) 8.0))))))