Jmat.Real.gamma, branch z greater than 0.5

Average Error: 60.0 → 1.1

Time: 3.3m

Precision: 64

• could not determine a ground truth for program body

  1. z = 4.74163504483872e+48

Math

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

↓

\[\mathsf{fma}\left(676.5203681218851, {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \left(\frac{\sqrt{2} \cdot \log 6.5}{e^{0.5 + 6}} \cdot \sqrt{\pi}\right), \mathsf{fma}\left(169.13009203047127, {\left(\frac{1}{{6.5}^{5.0}}\right)}^{0.5} \cdot \left(\frac{\sqrt{2} \cdot z}{e^{0.5 + 6}} \cdot \sqrt{\pi}\right), \mathsf{fma}\left(676.5203681218851, {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \left(\frac{\sqrt{2}}{\left(z \cdot e^{6}\right) \cdot e^{0.5}} \cdot \sqrt{\pi}\right), \mathsf{fma}\left(2581.191799681222, \sqrt{\pi} \cdot \left(\frac{\sqrt{2} \cdot z}{e^{0.5 + 6}} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right), 338.26018406094255 \cdot \left(\frac{\sqrt{\pi} \cdot \left(\left(\log 6.5 \cdot \log 6.5\right) \cdot \left(\sqrt{2} \cdot z\right)\right)}{e^{0.5 + 6}} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right)\right)\right)\right) - 1656.8104518737205 \cdot \left(\left(\frac{\log 6.5 \cdot \left(\sqrt{2} \cdot z\right)}{e^{0.5 + 6}} \cdot \sqrt{\pi}\right) \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} + \left(\sqrt{\pi} \cdot \frac{\sqrt{2}}{e^{0.5 + 6}}\right) \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\]

C

double f(double z) {
        double r5397721 = atan2(1.0, 0.0);
        double r5397722 = 2.0;
        double r5397723 = r5397721 * r5397722;
        double r5397724 = sqrt(r5397723);
        double r5397725 = z;
        double r5397726 = 1.0;
        double r5397727 = r5397725 - r5397726;
        double r5397728 = 7.0;
        double r5397729 = r5397727 + r5397728;
        double r5397730 = 0.5;
        double r5397731 = r5397729 + r5397730;
        double r5397732 = r5397727 + r5397730;
        double r5397733 = pow(r5397731, r5397732);
        double r5397734 = r5397724 * r5397733;
        double r5397735 = -r5397731;
        double r5397736 = exp(r5397735);
        double r5397737 = r5397734 * r5397736;
        double r5397738 = 0.9999999999998099;
        double r5397739 = 676.5203681218851;
        double r5397740 = r5397727 + r5397726;
        double r5397741 = r5397739 / r5397740;
        double r5397742 = r5397738 + r5397741;
        double r5397743 = -1259.1392167224028;
        double r5397744 = r5397727 + r5397722;
        double r5397745 = r5397743 / r5397744;
        double r5397746 = r5397742 + r5397745;
        double r5397747 = 771.3234287776531;
        double r5397748 = 3.0;
        double r5397749 = r5397727 + r5397748;
        double r5397750 = r5397747 / r5397749;
        double r5397751 = r5397746 + r5397750;
        double r5397752 = -176.6150291621406;
        double r5397753 = 4.0;
        double r5397754 = r5397727 + r5397753;
        double r5397755 = r5397752 / r5397754;
        double r5397756 = r5397751 + r5397755;
        double r5397757 = 12.507343278686905;
        double r5397758 = 5.0;
        double r5397759 = r5397727 + r5397758;
        double r5397760 = r5397757 / r5397759;
        double r5397761 = r5397756 + r5397760;
        double r5397762 = -0.13857109526572012;
        double r5397763 = 6.0;
        double r5397764 = r5397727 + r5397763;
        double r5397765 = r5397762 / r5397764;
        double r5397766 = r5397761 + r5397765;
        double r5397767 = 9.984369578019572e-06;
        double r5397768 = r5397767 / r5397729;
        double r5397769 = r5397766 + r5397768;
        double r5397770 = 1.5056327351493116e-07;
        double r5397771 = 8.0;
        double r5397772 = r5397727 + r5397771;
        double r5397773 = r5397770 / r5397772;
        double r5397774 = r5397769 + r5397773;
        double r5397775 = r5397737 * r5397774;
        return r5397775;
}

↓

double f(double z) {
        double r5397776 = 676.5203681218851;
        double r5397777 = 1.0;
        double r5397778 = 6.5;
        double r5397779 = 1.0;
        double r5397780 = pow(r5397778, r5397779);
        double r5397781 = r5397777 / r5397780;
        double r5397782 = 0.5;
        double r5397783 = pow(r5397781, r5397782);
        double r5397784 = 2.0;
        double r5397785 = sqrt(r5397784);
        double r5397786 = log(r5397778);
        double r5397787 = r5397785 * r5397786;
        double r5397788 = 6.0;
        double r5397789 = r5397782 + r5397788;
        double r5397790 = exp(r5397789);
        double r5397791 = r5397787 / r5397790;
        double r5397792 = atan2(1.0, 0.0);
        double r5397793 = sqrt(r5397792);
        double r5397794 = r5397791 * r5397793;
        double r5397795 = r5397783 * r5397794;
        double r5397796 = 169.13009203047127;
        double r5397797 = 5.0;
        double r5397798 = pow(r5397778, r5397797);
        double r5397799 = r5397777 / r5397798;
        double r5397800 = pow(r5397799, r5397782);
        double r5397801 = z;
        double r5397802 = r5397785 * r5397801;
        double r5397803 = r5397802 / r5397790;
        double r5397804 = r5397803 * r5397793;
        double r5397805 = r5397800 * r5397804;
        double r5397806 = exp(r5397788);
        double r5397807 = r5397801 * r5397806;
        double r5397808 = exp(r5397782);
        double r5397809 = r5397807 * r5397808;
        double r5397810 = r5397785 / r5397809;
        double r5397811 = r5397810 * r5397793;
        double r5397812 = r5397783 * r5397811;
        double r5397813 = 2581.191799681222;
        double r5397814 = r5397803 * r5397783;
        double r5397815 = r5397793 * r5397814;
        double r5397816 = 338.26018406094255;
        double r5397817 = r5397786 * r5397786;
        double r5397818 = r5397817 * r5397802;
        double r5397819 = r5397793 * r5397818;
        double r5397820 = r5397819 / r5397790;
        double r5397821 = r5397820 * r5397783;
        double r5397822 = r5397816 * r5397821;
        double r5397823 = fma(r5397813, r5397815, r5397822);
        double r5397824 = fma(r5397776, r5397812, r5397823);
        double r5397825 = fma(r5397796, r5397805, r5397824);
        double r5397826 = fma(r5397776, r5397795, r5397825);
        double r5397827 = 1656.8104518737205;
        double r5397828 = r5397786 * r5397802;
        double r5397829 = r5397828 / r5397790;
        double r5397830 = r5397829 * r5397793;
        double r5397831 = r5397830 * r5397783;
        double r5397832 = r5397785 / r5397790;
        double r5397833 = r5397793 * r5397832;
        double r5397834 = r5397833 * r5397783;
        double r5397835 = r5397831 + r5397834;
        double r5397836 = r5397827 * r5397835;
        double r5397837 = r5397826 - r5397836;
        return r5397837;
}

Error

Bits error versus z

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. Simplified 1.0

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

4. Applied sub-neg 1.0

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

5. Applied exp-sum 1.0

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

6. Applied associate-/r* 0.8

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

7. Taylor expanded around 0 1.1

   \[\leadsto \color{blue}{\left(676.5203681218851 \cdot \left({\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \left(\frac{\sqrt{2} \cdot \log 6.5}{e^{6} \cdot e^{0.5}} \cdot \sqrt{\pi}\right)\right) + \left(169.13009203047127 \cdot \left(\sqrt{\pi} \cdot \left(\frac{\sqrt{2} \cdot z}{e^{0.5} \cdot e^{6}} \cdot {\left(\frac{1}{{6.5}^{5.0}}\right)}^{0.5}\right)\right) + \left(676.5203681218851 \cdot \left({\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \left(\frac{\sqrt{2}}{e^{6} \cdot \left(z \cdot e^{0.5}\right)} \cdot \sqrt{\pi}\right)\right) + \left(2581.191799681222 \cdot \left(\sqrt{\pi} \cdot \left(\frac{\sqrt{2} \cdot z}{e^{0.5} \cdot e^{6}} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right) + 338.26018406094255 \cdot \left({\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \left(\sqrt{\pi} \cdot \frac{\sqrt{2} \cdot \left(z \cdot {\left(\log 6.5\right)}^{2}\right)}{e^{6} \cdot e^{0.5}}\right)\right)\right)\right)\right)\right) - \left(1656.8104518737205 \cdot \left({\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \left(\frac{\sqrt{2}}{e^{6} \cdot e^{0.5}} \cdot \sqrt{\pi}\right)\right) + 1656.8104518737205 \cdot \left({\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \left(\sqrt{\pi} \cdot \frac{\sqrt{2} \cdot \left(z \cdot \log 6.5\right)}{e^{6} \cdot e^{0.5}}\right)\right)\right)}\]

8. Simplified 1.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(676.5203681218851, {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \left(\sqrt{\pi} \cdot \frac{\log 6.5 \cdot \sqrt{2}}{e^{0.5 + 6}}\right), \mathsf{fma}\left(169.13009203047127, \left(\sqrt{\pi} \cdot \frac{z \cdot \sqrt{2}}{e^{0.5 + 6}}\right) \cdot {\left(\frac{1}{{6.5}^{5.0}}\right)}^{0.5}, \mathsf{fma}\left(676.5203681218851, {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \left(\sqrt{\pi} \cdot \frac{\sqrt{2}}{\left(e^{6} \cdot z\right) \cdot e^{0.5}}\right), \mathsf{fma}\left(2581.191799681222, \left({\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \frac{z \cdot \sqrt{2}}{e^{0.5 + 6}}\right) \cdot \sqrt{\pi}, \left({\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \frac{\sqrt{\pi} \cdot \left(\left(z \cdot \sqrt{2}\right) \cdot \left(\log 6.5 \cdot \log 6.5\right)\right)}{e^{0.5 + 6}}\right) \cdot 338.26018406094255\right)\right)\right)\right) - 1656.8104518737205 \cdot \left({\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \left(\sqrt{\pi} \cdot \frac{\sqrt{2}}{e^{0.5 + 6}}\right) + {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \left(\sqrt{\pi} \cdot \frac{\left(z \cdot \sqrt{2}\right) \cdot \log 6.5}{e^{0.5 + 6}}\right)\right)}\]

9. Final simplification 1.1

   \[\leadsto \mathsf{fma}\left(676.5203681218851, {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \left(\frac{\sqrt{2} \cdot \log 6.5}{e^{0.5 + 6}} \cdot \sqrt{\pi}\right), \mathsf{fma}\left(169.13009203047127, {\left(\frac{1}{{6.5}^{5.0}}\right)}^{0.5} \cdot \left(\frac{\sqrt{2} \cdot z}{e^{0.5 + 6}} \cdot \sqrt{\pi}\right), \mathsf{fma}\left(676.5203681218851, {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} \cdot \left(\frac{\sqrt{2}}{\left(z \cdot e^{6}\right) \cdot e^{0.5}} \cdot \sqrt{\pi}\right), \mathsf{fma}\left(2581.191799681222, \sqrt{\pi} \cdot \left(\frac{\sqrt{2} \cdot z}{e^{0.5 + 6}} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right), 338.26018406094255 \cdot \left(\frac{\sqrt{\pi} \cdot \left(\left(\log 6.5 \cdot \log 6.5\right) \cdot \left(\sqrt{2} \cdot z\right)\right)}{e^{0.5 + 6}} \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\right)\right)\right)\right) - 1656.8104518737205 \cdot \left(\left(\frac{\log 6.5 \cdot \left(\sqrt{2} \cdot z\right)}{e^{0.5 + 6}} \cdot \sqrt{\pi}\right) \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5} + \left(\sqrt{\pi} \cdot \frac{\sqrt{2}}{e^{0.5 + 6}}\right) \cdot {\left(\frac{1}{{6.5}^{1.0}}\right)}^{0.5}\right)\]

Reproduce

herbie shell --seed 2019135 +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)))))