Jmat.Real.gamma, branch z greater than 0.5

Average Error: 59.9 → 0.6

Time: 2.7m

Precision: 64

• could not determine a ground truth for program body

  1. z = 1.9756105794475683e+22

Math

\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8}\right)\]

↓

\[\frac{\frac{\sqrt{\pi \cdot 2} \cdot {\left(\left(z - -6\right) + 0.5\right)}^{\left(z - \left(1 - 0.5\right)\right)}}{e^{\left(z - -6\right) + 0.5}} \cdot \left(\left(\left(\left(6 + z\right) \cdot 1.5056327351493116 \cdot 10^{-07} + 9.984369578019572 \cdot 10^{-06} \cdot \left(z + 7\right)\right) \cdot \left(z + 5\right) + -0.13857109526572012 \cdot \left(\left(z + 7\right) \cdot \left(6 + z\right)\right)\right) \cdot \left(\left(z + 3\right) \cdot \left(\left(\left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{12.507343278686905}{4 + z} \cdot 0.9999999999998099\right) + \frac{12.507343278686905}{4 + z} \cdot \frac{12.507343278686905}{4 + z}\right) \cdot \left(\left(z \cdot \left(z + 1\right)\right) \cdot \left(z + 2\right)\right)\right)\right) + \left(-176.6150291621406 \cdot \left(\left(\left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{12.507343278686905}{4 + z} \cdot 0.9999999999998099\right) + \frac{12.507343278686905}{4 + z} \cdot \frac{12.507343278686905}{4 + z}\right) \cdot \left(\left(z \cdot \left(z + 1\right)\right) \cdot \left(z + 2\right)\right)\right) + \left(z + 3\right) \cdot \left(\left(\left(z \cdot \left(z + 1\right)\right) \cdot \left(z + 2\right)\right) \cdot \left({0.9999999999998099}^{3} + {\left(\frac{12.507343278686905}{4 + z}\right)}^{3}\right) + \left(\left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{12.507343278686905}{4 + z} \cdot 0.9999999999998099\right) + \frac{12.507343278686905}{4 + z} \cdot \frac{12.507343278686905}{4 + z}\right) \cdot \left(771.3234287776531 \cdot \left(z \cdot \left(z + 1\right)\right) + \left(-1259.1392167224028 \cdot z + 676.5203681218851 \cdot \left(z + 1\right)\right) \cdot \left(z + 2\right)\right)\right)\right) \cdot \left(\left(\left(z + 7\right) \cdot \left(6 + z\right)\right) \cdot \left(z + 5\right)\right)\right)}{\left(\left(z + 3\right) \cdot \left(\left(\left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{12.507343278686905}{4 + z} \cdot 0.9999999999998099\right) + \frac{12.507343278686905}{4 + z} \cdot \frac{12.507343278686905}{4 + z}\right) \cdot \left(\left(z \cdot \left(z + 1\right)\right) \cdot \left(z + 2\right)\right)\right)\right) \cdot \left(\left(\left(z + 7\right) \cdot \left(6 + z\right)\right) \cdot \left(z + 5\right)\right)}\]

C

double f(double z) {
        double r10248612 = atan2(1.0, 0.0);
        double r10248613 = 2.0;
        double r10248614 = r10248612 * r10248613;
        double r10248615 = sqrt(r10248614);
        double r10248616 = z;
        double r10248617 = 1.0;
        double r10248618 = r10248616 - r10248617;
        double r10248619 = 7.0;
        double r10248620 = r10248618 + r10248619;
        double r10248621 = 0.5;
        double r10248622 = r10248620 + r10248621;
        double r10248623 = r10248618 + r10248621;
        double r10248624 = pow(r10248622, r10248623);
        double r10248625 = r10248615 * r10248624;
        double r10248626 = -r10248622;
        double r10248627 = exp(r10248626);
        double r10248628 = r10248625 * r10248627;
        double r10248629 = 0.9999999999998099;
        double r10248630 = 676.5203681218851;
        double r10248631 = r10248618 + r10248617;
        double r10248632 = r10248630 / r10248631;
        double r10248633 = r10248629 + r10248632;
        double r10248634 = -1259.1392167224028;
        double r10248635 = r10248618 + r10248613;
        double r10248636 = r10248634 / r10248635;
        double r10248637 = r10248633 + r10248636;
        double r10248638 = 771.3234287776531;
        double r10248639 = 3.0;
        double r10248640 = r10248618 + r10248639;
        double r10248641 = r10248638 / r10248640;
        double r10248642 = r10248637 + r10248641;
        double r10248643 = -176.6150291621406;
        double r10248644 = 4.0;
        double r10248645 = r10248618 + r10248644;
        double r10248646 = r10248643 / r10248645;
        double r10248647 = r10248642 + r10248646;
        double r10248648 = 12.507343278686905;
        double r10248649 = 5.0;
        double r10248650 = r10248618 + r10248649;
        double r10248651 = r10248648 / r10248650;
        double r10248652 = r10248647 + r10248651;
        double r10248653 = -0.13857109526572012;
        double r10248654 = 6.0;
        double r10248655 = r10248618 + r10248654;
        double r10248656 = r10248653 / r10248655;
        double r10248657 = r10248652 + r10248656;
        double r10248658 = 9.984369578019572e-06;
        double r10248659 = r10248658 / r10248620;
        double r10248660 = r10248657 + r10248659;
        double r10248661 = 1.5056327351493116e-07;
        double r10248662 = 8.0;
        double r10248663 = r10248618 + r10248662;
        double r10248664 = r10248661 / r10248663;
        double r10248665 = r10248660 + r10248664;
        double r10248666 = r10248628 * r10248665;
        return r10248666;
}

↓

double f(double z) {
        double r10248667 = atan2(1.0, 0.0);
        double r10248668 = 2.0;
        double r10248669 = r10248667 * r10248668;
        double r10248670 = sqrt(r10248669);
        double r10248671 = z;
        double r10248672 = -6.0;
        double r10248673 = r10248671 - r10248672;
        double r10248674 = 0.5;
        double r10248675 = r10248673 + r10248674;
        double r10248676 = 1.0;
        double r10248677 = r10248676 - r10248674;
        double r10248678 = r10248671 - r10248677;
        double r10248679 = pow(r10248675, r10248678);
        double r10248680 = r10248670 * r10248679;
        double r10248681 = exp(r10248675);
        double r10248682 = r10248680 / r10248681;
        double r10248683 = 6.0;
        double r10248684 = r10248683 + r10248671;
        double r10248685 = 1.5056327351493116e-07;
        double r10248686 = r10248684 * r10248685;
        double r10248687 = 9.984369578019572e-06;
        double r10248688 = 7.0;
        double r10248689 = r10248671 + r10248688;
        double r10248690 = r10248687 * r10248689;
        double r10248691 = r10248686 + r10248690;
        double r10248692 = 5.0;
        double r10248693 = r10248671 + r10248692;
        double r10248694 = r10248691 * r10248693;
        double r10248695 = -0.13857109526572012;
        double r10248696 = r10248689 * r10248684;
        double r10248697 = r10248695 * r10248696;
        double r10248698 = r10248694 + r10248697;
        double r10248699 = 3.0;
        double r10248700 = r10248671 + r10248699;
        double r10248701 = 0.9999999999998099;
        double r10248702 = r10248701 * r10248701;
        double r10248703 = 12.507343278686905;
        double r10248704 = 4.0;
        double r10248705 = r10248704 + r10248671;
        double r10248706 = r10248703 / r10248705;
        double r10248707 = r10248706 * r10248701;
        double r10248708 = r10248702 - r10248707;
        double r10248709 = r10248706 * r10248706;
        double r10248710 = r10248708 + r10248709;
        double r10248711 = r10248671 + r10248676;
        double r10248712 = r10248671 * r10248711;
        double r10248713 = r10248671 + r10248668;
        double r10248714 = r10248712 * r10248713;
        double r10248715 = r10248710 * r10248714;
        double r10248716 = r10248700 * r10248715;
        double r10248717 = r10248698 * r10248716;
        double r10248718 = -176.6150291621406;
        double r10248719 = r10248718 * r10248715;
        double r10248720 = pow(r10248701, r10248699);
        double r10248721 = pow(r10248706, r10248699);
        double r10248722 = r10248720 + r10248721;
        double r10248723 = r10248714 * r10248722;
        double r10248724 = 771.3234287776531;
        double r10248725 = r10248724 * r10248712;
        double r10248726 = -1259.1392167224028;
        double r10248727 = r10248726 * r10248671;
        double r10248728 = 676.5203681218851;
        double r10248729 = r10248728 * r10248711;
        double r10248730 = r10248727 + r10248729;
        double r10248731 = r10248730 * r10248713;
        double r10248732 = r10248725 + r10248731;
        double r10248733 = r10248710 * r10248732;
        double r10248734 = r10248723 + r10248733;
        double r10248735 = r10248700 * r10248734;
        double r10248736 = r10248719 + r10248735;
        double r10248737 = r10248696 * r10248693;
        double r10248738 = r10248736 * r10248737;
        double r10248739 = r10248717 + r10248738;
        double r10248740 = r10248682 * r10248739;
        double r10248741 = r10248716 * r10248737;
        double r10248742 = r10248740 / r10248741;
        return r10248742;
}

Error

Bits error versus z

Derivation

1. Initial program 59.9

   \[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8}\right)\]

2. Simplified 1.0

   \[\leadsto \color{blue}{\frac{{\left(0.5 + \left(z - -6\right)\right)}^{\left(z - \left(1 - 0.5\right)\right)} \cdot \sqrt{\pi \cdot 2}}{e^{0.5 + \left(z - -6\right)}} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-06}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-07}}{z + 7}\right) + \frac{-0.13857109526572012}{z + 5}\right) + \left(\left(\left(\frac{12.507343278686905}{z + 4} + 0.9999999999998099\right) + \left(\left(\frac{-1259.1392167224028}{1 + z} + \frac{676.5203681218851}{z}\right) + \frac{771.3234287776531}{z + 2}\right)\right) + \frac{-176.6150291621406}{3 + z}\right)\right)}\]
3. Using strategy rm

4. Applied frac-add 1.0

   \[\leadsto \frac{{\left(0.5 + \left(z - -6\right)\right)}^{\left(z - \left(1 - 0.5\right)\right)} \cdot \sqrt{\pi \cdot 2}}{e^{0.5 + \left(z - -6\right)}} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-06}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-07}}{z + 7}\right) + \frac{-0.13857109526572012}{z + 5}\right) + \left(\left(\left(\frac{12.507343278686905}{z + 4} + 0.9999999999998099\right) + \left(\color{blue}{\frac{-1259.1392167224028 \cdot z + \left(1 + z\right) \cdot 676.5203681218851}{\left(1 + z\right) \cdot z}} + \frac{771.3234287776531}{z + 2}\right)\right) + \frac{-176.6150291621406}{3 + z}\right)\right)\]

5. Applied frac-add 1.0

   \[\leadsto \frac{{\left(0.5 + \left(z - -6\right)\right)}^{\left(z - \left(1 - 0.5\right)\right)} \cdot \sqrt{\pi \cdot 2}}{e^{0.5 + \left(z - -6\right)}} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-06}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-07}}{z + 7}\right) + \frac{-0.13857109526572012}{z + 5}\right) + \left(\left(\left(\frac{12.507343278686905}{z + 4} + 0.9999999999998099\right) + \color{blue}{\frac{\left(-1259.1392167224028 \cdot z + \left(1 + z\right) \cdot 676.5203681218851\right) \cdot \left(z + 2\right) + \left(\left(1 + z\right) \cdot z\right) \cdot 771.3234287776531}{\left(\left(1 + z\right) \cdot z\right) \cdot \left(z + 2\right)}}\right) + \frac{-176.6150291621406}{3 + z}\right)\right)\]

6. Applied flip3-+ 1.0

   \[\leadsto \frac{{\left(0.5 + \left(z - -6\right)\right)}^{\left(z - \left(1 - 0.5\right)\right)} \cdot \sqrt{\pi \cdot 2}}{e^{0.5 + \left(z - -6\right)}} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-06}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-07}}{z + 7}\right) + \frac{-0.13857109526572012}{z + 5}\right) + \left(\left(\color{blue}{\frac{{\left(\frac{12.507343278686905}{z + 4}\right)}^{3} + {0.9999999999998099}^{3}}{\frac{12.507343278686905}{z + 4} \cdot \frac{12.507343278686905}{z + 4} + \left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{12.507343278686905}{z + 4} \cdot 0.9999999999998099\right)}} + \frac{\left(-1259.1392167224028 \cdot z + \left(1 + z\right) \cdot 676.5203681218851\right) \cdot \left(z + 2\right) + \left(\left(1 + z\right) \cdot z\right) \cdot 771.3234287776531}{\left(\left(1 + z\right) \cdot z\right) \cdot \left(z + 2\right)}\right) + \frac{-176.6150291621406}{3 + z}\right)\right)\]

7. Applied frac-add 1.3

   \[\leadsto \frac{{\left(0.5 + \left(z - -6\right)\right)}^{\left(z - \left(1 - 0.5\right)\right)} \cdot \sqrt{\pi \cdot 2}}{e^{0.5 + \left(z - -6\right)}} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-06}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-07}}{z + 7}\right) + \frac{-0.13857109526572012}{z + 5}\right) + \left(\color{blue}{\frac{\left({\left(\frac{12.507343278686905}{z + 4}\right)}^{3} + {0.9999999999998099}^{3}\right) \cdot \left(\left(\left(1 + z\right) \cdot z\right) \cdot \left(z + 2\right)\right) + \left(\frac{12.507343278686905}{z + 4} \cdot \frac{12.507343278686905}{z + 4} + \left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{12.507343278686905}{z + 4} \cdot 0.9999999999998099\right)\right) \cdot \left(\left(-1259.1392167224028 \cdot z + \left(1 + z\right) \cdot 676.5203681218851\right) \cdot \left(z + 2\right) + \left(\left(1 + z\right) \cdot z\right) \cdot 771.3234287776531\right)}{\left(\frac{12.507343278686905}{z + 4} \cdot \frac{12.507343278686905}{z + 4} + \left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{12.507343278686905}{z + 4} \cdot 0.9999999999998099\right)\right) \cdot \left(\left(\left(1 + z\right) \cdot z\right) \cdot \left(z + 2\right)\right)}} + \frac{-176.6150291621406}{3 + z}\right)\right)\]

8. Applied frac-add 1.3

   \[\leadsto \frac{{\left(0.5 + \left(z - -6\right)\right)}^{\left(z - \left(1 - 0.5\right)\right)} \cdot \sqrt{\pi \cdot 2}}{e^{0.5 + \left(z - -6\right)}} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-06}}{z + 6} + \frac{1.5056327351493116 \cdot 10^{-07}}{z + 7}\right) + \frac{-0.13857109526572012}{z + 5}\right) + \color{blue}{\frac{\left(\left({\left(\frac{12.507343278686905}{z + 4}\right)}^{3} + {0.9999999999998099}^{3}\right) \cdot \left(\left(\left(1 + z\right) \cdot z\right) \cdot \left(z + 2\right)\right) + \left(\frac{12.507343278686905}{z + 4} \cdot \frac{12.507343278686905}{z + 4} + \left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{12.507343278686905}{z + 4} \cdot 0.9999999999998099\right)\right) \cdot \left(\left(-1259.1392167224028 \cdot z + \left(1 + z\right) \cdot 676.5203681218851\right) \cdot \left(z + 2\right) + \left(\left(1 + z\right) \cdot z\right) \cdot 771.3234287776531\right)\right) \cdot \left(3 + z\right) + \left(\left(\frac{12.507343278686905}{z + 4} \cdot \frac{12.507343278686905}{z + 4} + \left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{12.507343278686905}{z + 4} \cdot 0.9999999999998099\right)\right) \cdot \left(\left(\left(1 + z\right) \cdot z\right) \cdot \left(z + 2\right)\right)\right) \cdot -176.6150291621406}{\left(\left(\frac{12.507343278686905}{z + 4} \cdot \frac{12.507343278686905}{z + 4} + \left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{12.507343278686905}{z + 4} \cdot 0.9999999999998099\right)\right) \cdot \left(\left(\left(1 + z\right) \cdot z\right) \cdot \left(z + 2\right)\right)\right) \cdot \left(3 + z\right)}}\right)\]

9. Applied frac-add 1.3

   \[\leadsto \frac{{\left(0.5 + \left(z - -6\right)\right)}^{\left(z - \left(1 - 0.5\right)\right)} \cdot \sqrt{\pi \cdot 2}}{e^{0.5 + \left(z - -6\right)}} \cdot \left(\left(\color{blue}{\frac{9.984369578019572 \cdot 10^{-06} \cdot \left(z + 7\right) + \left(z + 6\right) \cdot 1.5056327351493116 \cdot 10^{-07}}{\left(z + 6\right) \cdot \left(z + 7\right)}} + \frac{-0.13857109526572012}{z + 5}\right) + \frac{\left(\left({\left(\frac{12.507343278686905}{z + 4}\right)}^{3} + {0.9999999999998099}^{3}\right) \cdot \left(\left(\left(1 + z\right) \cdot z\right) \cdot \left(z + 2\right)\right) + \left(\frac{12.507343278686905}{z + 4} \cdot \frac{12.507343278686905}{z + 4} + \left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{12.507343278686905}{z + 4} \cdot 0.9999999999998099\right)\right) \cdot \left(\left(-1259.1392167224028 \cdot z + \left(1 + z\right) \cdot 676.5203681218851\right) \cdot \left(z + 2\right) + \left(\left(1 + z\right) \cdot z\right) \cdot 771.3234287776531\right)\right) \cdot \left(3 + z\right) + \left(\left(\frac{12.507343278686905}{z + 4} \cdot \frac{12.507343278686905}{z + 4} + \left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{12.507343278686905}{z + 4} \cdot 0.9999999999998099\right)\right) \cdot \left(\left(\left(1 + z\right) \cdot z\right) \cdot \left(z + 2\right)\right)\right) \cdot -176.6150291621406}{\left(\left(\frac{12.507343278686905}{z + 4} \cdot \frac{12.507343278686905}{z + 4} + \left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{12.507343278686905}{z + 4} \cdot 0.9999999999998099\right)\right) \cdot \left(\left(\left(1 + z\right) \cdot z\right) \cdot \left(z + 2\right)\right)\right) \cdot \left(3 + z\right)}\right)\]

10. Applied frac-add 1.3

    \[\leadsto \frac{{\left(0.5 + \left(z - -6\right)\right)}^{\left(z - \left(1 - 0.5\right)\right)} \cdot \sqrt{\pi \cdot 2}}{e^{0.5 + \left(z - -6\right)}} \cdot \left(\color{blue}{\frac{\left(9.984369578019572 \cdot 10^{-06} \cdot \left(z + 7\right) + \left(z + 6\right) \cdot 1.5056327351493116 \cdot 10^{-07}\right) \cdot \left(z + 5\right) + \left(\left(z + 6\right) \cdot \left(z + 7\right)\right) \cdot -0.13857109526572012}{\left(\left(z + 6\right) \cdot \left(z + 7\right)\right) \cdot \left(z + 5\right)}} + \frac{\left(\left({\left(\frac{12.507343278686905}{z + 4}\right)}^{3} + {0.9999999999998099}^{3}\right) \cdot \left(\left(\left(1 + z\right) \cdot z\right) \cdot \left(z + 2\right)\right) + \left(\frac{12.507343278686905}{z + 4} \cdot \frac{12.507343278686905}{z + 4} + \left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{12.507343278686905}{z + 4} \cdot 0.9999999999998099\right)\right) \cdot \left(\left(-1259.1392167224028 \cdot z + \left(1 + z\right) \cdot 676.5203681218851\right) \cdot \left(z + 2\right) + \left(\left(1 + z\right) \cdot z\right) \cdot 771.3234287776531\right)\right) \cdot \left(3 + z\right) + \left(\left(\frac{12.507343278686905}{z + 4} \cdot \frac{12.507343278686905}{z + 4} + \left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{12.507343278686905}{z + 4} \cdot 0.9999999999998099\right)\right) \cdot \left(\left(\left(1 + z\right) \cdot z\right) \cdot \left(z + 2\right)\right)\right) \cdot -176.6150291621406}{\left(\left(\frac{12.507343278686905}{z + 4} \cdot \frac{12.507343278686905}{z + 4} + \left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{12.507343278686905}{z + 4} \cdot 0.9999999999998099\right)\right) \cdot \left(\left(\left(1 + z\right) \cdot z\right) \cdot \left(z + 2\right)\right)\right) \cdot \left(3 + z\right)}\right)\]

11. Applied frac-add 1.1

    \[\leadsto \frac{{\left(0.5 + \left(z - -6\right)\right)}^{\left(z - \left(1 - 0.5\right)\right)} \cdot \sqrt{\pi \cdot 2}}{e^{0.5 + \left(z - -6\right)}} \cdot \color{blue}{\frac{\left(\left(9.984369578019572 \cdot 10^{-06} \cdot \left(z + 7\right) + \left(z + 6\right) \cdot 1.5056327351493116 \cdot 10^{-07}\right) \cdot \left(z + 5\right) + \left(\left(z + 6\right) \cdot \left(z + 7\right)\right) \cdot -0.13857109526572012\right) \cdot \left(\left(\left(\frac{12.507343278686905}{z + 4} \cdot \frac{12.507343278686905}{z + 4} + \left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{12.507343278686905}{z + 4} \cdot 0.9999999999998099\right)\right) \cdot \left(\left(\left(1 + z\right) \cdot z\right) \cdot \left(z + 2\right)\right)\right) \cdot \left(3 + z\right)\right) + \left(\left(\left(z + 6\right) \cdot \left(z + 7\right)\right) \cdot \left(z + 5\right)\right) \cdot \left(\left(\left({\left(\frac{12.507343278686905}{z + 4}\right)}^{3} + {0.9999999999998099}^{3}\right) \cdot \left(\left(\left(1 + z\right) \cdot z\right) \cdot \left(z + 2\right)\right) + \left(\frac{12.507343278686905}{z + 4} \cdot \frac{12.507343278686905}{z + 4} + \left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{12.507343278686905}{z + 4} \cdot 0.9999999999998099\right)\right) \cdot \left(\left(-1259.1392167224028 \cdot z + \left(1 + z\right) \cdot 676.5203681218851\right) \cdot \left(z + 2\right) + \left(\left(1 + z\right) \cdot z\right) \cdot 771.3234287776531\right)\right) \cdot \left(3 + z\right) + \left(\left(\frac{12.507343278686905}{z + 4} \cdot \frac{12.507343278686905}{z + 4} + \left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{12.507343278686905}{z + 4} \cdot 0.9999999999998099\right)\right) \cdot \left(\left(\left(1 + z\right) \cdot z\right) \cdot \left(z + 2\right)\right)\right) \cdot -176.6150291621406\right)}{\left(\left(\left(z + 6\right) \cdot \left(z + 7\right)\right) \cdot \left(z + 5\right)\right) \cdot \left(\left(\left(\frac{12.507343278686905}{z + 4} \cdot \frac{12.507343278686905}{z + 4} + \left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{12.507343278686905}{z + 4} \cdot 0.9999999999998099\right)\right) \cdot \left(\left(\left(1 + z\right) \cdot z\right) \cdot \left(z + 2\right)\right)\right) \cdot \left(3 + z\right)\right)}}\]

12. Applied associate-*r/ 0.6

    \[\leadsto \color{blue}{\frac{\frac{{\left(0.5 + \left(z - -6\right)\right)}^{\left(z - \left(1 - 0.5\right)\right)} \cdot \sqrt{\pi \cdot 2}}{e^{0.5 + \left(z - -6\right)}} \cdot \left(\left(\left(9.984369578019572 \cdot 10^{-06} \cdot \left(z + 7\right) + \left(z + 6\right) \cdot 1.5056327351493116 \cdot 10^{-07}\right) \cdot \left(z + 5\right) + \left(\left(z + 6\right) \cdot \left(z + 7\right)\right) \cdot -0.13857109526572012\right) \cdot \left(\left(\left(\frac{12.507343278686905}{z + 4} \cdot \frac{12.507343278686905}{z + 4} + \left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{12.507343278686905}{z + 4} \cdot 0.9999999999998099\right)\right) \cdot \left(\left(\left(1 + z\right) \cdot z\right) \cdot \left(z + 2\right)\right)\right) \cdot \left(3 + z\right)\right) + \left(\left(\left(z + 6\right) \cdot \left(z + 7\right)\right) \cdot \left(z + 5\right)\right) \cdot \left(\left(\left({\left(\frac{12.507343278686905}{z + 4}\right)}^{3} + {0.9999999999998099}^{3}\right) \cdot \left(\left(\left(1 + z\right) \cdot z\right) \cdot \left(z + 2\right)\right) + \left(\frac{12.507343278686905}{z + 4} \cdot \frac{12.507343278686905}{z + 4} + \left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{12.507343278686905}{z + 4} \cdot 0.9999999999998099\right)\right) \cdot \left(\left(-1259.1392167224028 \cdot z + \left(1 + z\right) \cdot 676.5203681218851\right) \cdot \left(z + 2\right) + \left(\left(1 + z\right) \cdot z\right) \cdot 771.3234287776531\right)\right) \cdot \left(3 + z\right) + \left(\left(\frac{12.507343278686905}{z + 4} \cdot \frac{12.507343278686905}{z + 4} + \left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{12.507343278686905}{z + 4} \cdot 0.9999999999998099\right)\right) \cdot \left(\left(\left(1 + z\right) \cdot z\right) \cdot \left(z + 2\right)\right)\right) \cdot -176.6150291621406\right)\right)}{\left(\left(\left(z + 6\right) \cdot \left(z + 7\right)\right) \cdot \left(z + 5\right)\right) \cdot \left(\left(\left(\frac{12.507343278686905}{z + 4} \cdot \frac{12.507343278686905}{z + 4} + \left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{12.507343278686905}{z + 4} \cdot 0.9999999999998099\right)\right) \cdot \left(\left(\left(1 + z\right) \cdot z\right) \cdot \left(z + 2\right)\right)\right) \cdot \left(3 + z\right)\right)}}\]

13. Final simplification 0.6

    \[\leadsto \frac{\frac{\sqrt{\pi \cdot 2} \cdot {\left(\left(z - -6\right) + 0.5\right)}^{\left(z - \left(1 - 0.5\right)\right)}}{e^{\left(z - -6\right) + 0.5}} \cdot \left(\left(\left(\left(6 + z\right) \cdot 1.5056327351493116 \cdot 10^{-07} + 9.984369578019572 \cdot 10^{-06} \cdot \left(z + 7\right)\right) \cdot \left(z + 5\right) + -0.13857109526572012 \cdot \left(\left(z + 7\right) \cdot \left(6 + z\right)\right)\right) \cdot \left(\left(z + 3\right) \cdot \left(\left(\left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{12.507343278686905}{4 + z} \cdot 0.9999999999998099\right) + \frac{12.507343278686905}{4 + z} \cdot \frac{12.507343278686905}{4 + z}\right) \cdot \left(\left(z \cdot \left(z + 1\right)\right) \cdot \left(z + 2\right)\right)\right)\right) + \left(-176.6150291621406 \cdot \left(\left(\left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{12.507343278686905}{4 + z} \cdot 0.9999999999998099\right) + \frac{12.507343278686905}{4 + z} \cdot \frac{12.507343278686905}{4 + z}\right) \cdot \left(\left(z \cdot \left(z + 1\right)\right) \cdot \left(z + 2\right)\right)\right) + \left(z + 3\right) \cdot \left(\left(\left(z \cdot \left(z + 1\right)\right) \cdot \left(z + 2\right)\right) \cdot \left({0.9999999999998099}^{3} + {\left(\frac{12.507343278686905}{4 + z}\right)}^{3}\right) + \left(\left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{12.507343278686905}{4 + z} \cdot 0.9999999999998099\right) + \frac{12.507343278686905}{4 + z} \cdot \frac{12.507343278686905}{4 + z}\right) \cdot \left(771.3234287776531 \cdot \left(z \cdot \left(z + 1\right)\right) + \left(-1259.1392167224028 \cdot z + 676.5203681218851 \cdot \left(z + 1\right)\right) \cdot \left(z + 2\right)\right)\right)\right) \cdot \left(\left(\left(z + 7\right) \cdot \left(6 + z\right)\right) \cdot \left(z + 5\right)\right)\right)}{\left(\left(z + 3\right) \cdot \left(\left(\left(0.9999999999998099 \cdot 0.9999999999998099 - \frac{12.507343278686905}{4 + z} \cdot 0.9999999999998099\right) + \frac{12.507343278686905}{4 + z} \cdot \frac{12.507343278686905}{4 + z}\right) \cdot \left(\left(z \cdot \left(z + 1\right)\right) \cdot \left(z + 2\right)\right)\right)\right) \cdot \left(\left(\left(z + 7\right) \cdot \left(6 + z\right)\right) \cdot \left(z + 5\right)\right)}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z greater than 0.5"
  (* (* (* (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)))))