Average Error: 60.0 → 0.8
Time: 3.8m
Precision: 64
\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8}\right)\]
↓
\[\left(\left(\left(\frac{-0.13857109526572012}{\left(z - -6\right) + -1} + \frac{12.507343278686905}{z + 4}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{z + 7} + \frac{9.984369578019572 \cdot 10^{-06}}{z - -6}\right)\right) + \left(\left(\frac{-1259.1392167224028}{z - -1} + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{771.3234287776531}{z + 2}\right)\right)\right) + \frac{-176.6150291621406}{z + 3}\right)\right) \cdot \left(\frac{e^{-\log \left(z + 6.5\right) \cdot \left(0.5 - z\right)} \cdot \sqrt{2}}{e^{6.5} \cdot e^{z}} \cdot \sqrt{\pi}\right)\]
\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)
↓
\left(\left(\left(\frac{-0.13857109526572012}{\left(z - -6\right) + -1} + \frac{12.507343278686905}{z + 4}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{z + 7} + \frac{9.984369578019572 \cdot 10^{-06}}{z - -6}\right)\right) + \left(\left(\frac{-1259.1392167224028}{z - -1} + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{771.3234287776531}{z + 2}\right)\right)\right) + \frac{-176.6150291621406}{z + 3}\right)\right) \cdot \left(\frac{e^{-\log \left(z + 6.5\right) \cdot \left(0.5 - z\right)} \cdot \sqrt{2}}{e^{6.5} \cdot e^{z}} \cdot \sqrt{\pi}\right)
double f(double z) {
        double r11237698 = atan2(1.0, 0.0);
        double r11237699 = 2.0;
        double r11237700 = r11237698 * r11237699;
        double r11237701 = sqrt(r11237700);
        double r11237702 = z;
        double r11237703 = 1.0;
        double r11237704 = r11237702 - r11237703;
        double r11237705 = 7.0;
        double r11237706 = r11237704 + r11237705;
        double r11237707 = 0.5;
        double r11237708 = r11237706 + r11237707;
        double r11237709 = r11237704 + r11237707;
        double r11237710 = pow(r11237708, r11237709);
        double r11237711 = r11237701 * r11237710;
        double r11237712 = -r11237708;
        double r11237713 = exp(r11237712);
        double r11237714 = r11237711 * r11237713;
        double r11237715 = 0.9999999999998099;
        double r11237716 = 676.5203681218851;
        double r11237717 = r11237704 + r11237703;
        double r11237718 = r11237716 / r11237717;
        double r11237719 = r11237715 + r11237718;
        double r11237720 = -1259.1392167224028;
        double r11237721 = r11237704 + r11237699;
        double r11237722 = r11237720 / r11237721;
        double r11237723 = r11237719 + r11237722;
        double r11237724 = 771.3234287776531;
        double r11237725 = 3.0;
        double r11237726 = r11237704 + r11237725;
        double r11237727 = r11237724 / r11237726;
        double r11237728 = r11237723 + r11237727;
        double r11237729 = -176.6150291621406;
        double r11237730 = 4.0;
        double r11237731 = r11237704 + r11237730;
        double r11237732 = r11237729 / r11237731;
        double r11237733 = r11237728 + r11237732;
        double r11237734 = 12.507343278686905;
        double r11237735 = 5.0;
        double r11237736 = r11237704 + r11237735;
        double r11237737 = r11237734 / r11237736;
        double r11237738 = r11237733 + r11237737;
        double r11237739 = -0.13857109526572012;
        double r11237740 = 6.0;
        double r11237741 = r11237704 + r11237740;
        double r11237742 = r11237739 / r11237741;
        double r11237743 = r11237738 + r11237742;
        double r11237744 = 9.984369578019572e-06;
        double r11237745 = r11237744 / r11237706;
        double r11237746 = r11237743 + r11237745;
        double r11237747 = 1.5056327351493116e-07;
        double r11237748 = 8.0;
        double r11237749 = r11237704 + r11237748;
        double r11237750 = r11237747 / r11237749;
        double r11237751 = r11237746 + r11237750;
        double r11237752 = r11237714 * r11237751;
        return r11237752;
}
↓
double f(double z) {
        double r11237753 = -0.13857109526572012;
        double r11237754 = z;
        double r11237755 = -6.0;
        double r11237756 = r11237754 - r11237755;
        double r11237757 = -1.0;
        double r11237758 = r11237756 + r11237757;
        double r11237759 = r11237753 / r11237758;
        double r11237760 = 12.507343278686905;
        double r11237761 = 4.0;
        double r11237762 = r11237754 + r11237761;
        double r11237763 = r11237760 / r11237762;
        double r11237764 = r11237759 + r11237763;
        double r11237765 = 1.5056327351493116e-07;
        double r11237766 = 7.0;
        double r11237767 = r11237754 + r11237766;
        double r11237768 = r11237765 / r11237767;
        double r11237769 = 9.984369578019572e-06;
        double r11237770 = r11237769 / r11237756;
        double r11237771 = r11237768 + r11237770;
        double r11237772 = r11237764 + r11237771;
        double r11237773 = -1259.1392167224028;
        double r11237774 = r11237754 - r11237757;
        double r11237775 = r11237773 / r11237774;
        double r11237776 = 0.9999999999998099;
        double r11237777 = 676.5203681218851;
        double r11237778 = r11237777 / r11237754;
        double r11237779 = 771.3234287776531;
        double r11237780 = 2.0;
        double r11237781 = r11237754 + r11237780;
        double r11237782 = r11237779 / r11237781;
        double r11237783 = r11237778 + r11237782;
        double r11237784 = r11237776 + r11237783;
        double r11237785 = r11237775 + r11237784;
        double r11237786 = -176.6150291621406;
        double r11237787 = 3.0;
        double r11237788 = r11237754 + r11237787;
        double r11237789 = r11237786 / r11237788;
        double r11237790 = r11237785 + r11237789;
        double r11237791 = r11237772 + r11237790;
        double r11237792 = 6.5;
        double r11237793 = r11237754 + r11237792;
        double r11237794 = log(r11237793);
        double r11237795 = 0.5;
        double r11237796 = r11237795 - r11237754;
        double r11237797 = r11237794 * r11237796;
        double r11237798 = -r11237797;
        double r11237799 = exp(r11237798);
        double r11237800 = sqrt(r11237780);
        double r11237801 = r11237799 * r11237800;
        double r11237802 = exp(r11237792);
        double r11237803 = exp(r11237754);
        double r11237804 = r11237802 * r11237803;
        double r11237805 = r11237801 / r11237804;
        double r11237806 = atan2(1.0, 0.0);
        double r11237807 = sqrt(r11237806);
        double r11237808 = r11237805 * r11237807;
        double r11237809 = r11237791 * r11237808;
        return r11237809;
}

Error

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 60.0

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

  2. Simplified1.0

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

  3. Taylor expanded around -inf 0.8

    \[\leadsto \color{blue}{\left(\frac{\sqrt{2} \cdot e^{-1 \cdot \left(\log \left(z + 6.5\right) \cdot \left(0.5 - z\right)\right)}}{e^{z + 6.5}} \cdot \sqrt{\pi}\right)} \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-06}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-07}}{7 + z}\right) + \left(\frac{-0.13857109526572012}{-1 + \left(z - -6\right)} + \frac{12.507343278686905}{z + 4}\right)\right) + \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{771.3234287776531}{2 + z}\right)\right) + \frac{-1259.1392167224028}{z - -1}\right) + \frac{-176.6150291621406}{z + 3}\right)\right)\]
  4. Using strategy rm

  5. Applied exp-sum0.8

    \[\leadsto \left(\frac{\sqrt{2} \cdot e^{-1 \cdot \left(\log \left(z + 6.5\right) \cdot \left(0.5 - z\right)\right)}}{\color{blue}{e^{z} \cdot e^{6.5}}} \cdot \sqrt{\pi}\right) \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-06}}{z - -6} + \frac{1.5056327351493116 \cdot 10^{-07}}{7 + z}\right) + \left(\frac{-0.13857109526572012}{-1 + \left(z - -6\right)} + \frac{12.507343278686905}{z + 4}\right)\right) + \left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{771.3234287776531}{2 + z}\right)\right) + \frac{-1259.1392167224028}{z - -1}\right) + \frac{-176.6150291621406}{z + 3}\right)\right)\]

  6. Final simplification0.8

    \[\leadsto \left(\left(\left(\frac{-0.13857109526572012}{\left(z - -6\right) + -1} + \frac{12.507343278686905}{z + 4}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{z + 7} + \frac{9.984369578019572 \cdot 10^{-06}}{z - -6}\right)\right) + \left(\left(\frac{-1259.1392167224028}{z - -1} + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{771.3234287776531}{z + 2}\right)\right)\right) + \frac{-176.6150291621406}{z + 3}\right)\right) \cdot \left(\frac{e^{-\log \left(z + 6.5\right) \cdot \left(0.5 - z\right)} \cdot \sqrt{2}}{e^{6.5} \cdot e^{z}} \cdot \sqrt{\pi}\right)\]

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(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)))))