Jmat.Real.gamma, branch z greater than 0.5

Average Error: 61.7 → 1.3

Time: 22.4s

Precision: 64

• could not determine a ground truth for program body

  1. z = 1.0551194375545526e+290

Math

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

↓

\[\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot \left(\left(676.520368121885099 \cdot \frac{1}{z \cdot e^{6.5}} + 2351.6663247613023 \cdot \frac{z}{e^{6.5}}\right) - 1604.7704235566525 \cdot \frac{1}{e^{6.5}}\right)\]

C

double f(double z) {
        double r177735 = atan2(1.0, 0.0);
        double r177736 = 2.0;
        double r177737 = r177735 * r177736;
        double r177738 = sqrt(r177737);
        double r177739 = z;
        double r177740 = 1.0;
        double r177741 = r177739 - r177740;
        double r177742 = 7.0;
        double r177743 = r177741 + r177742;
        double r177744 = 0.5;
        double r177745 = r177743 + r177744;
        double r177746 = r177741 + r177744;
        double r177747 = pow(r177745, r177746);
        double r177748 = r177738 * r177747;
        double r177749 = -r177745;
        double r177750 = exp(r177749);
        double r177751 = r177748 * r177750;
        double r177752 = 0.9999999999998099;
        double r177753 = 676.5203681218851;
        double r177754 = r177741 + r177740;
        double r177755 = r177753 / r177754;
        double r177756 = r177752 + r177755;
        double r177757 = -1259.1392167224028;
        double r177758 = r177741 + r177736;
        double r177759 = r177757 / r177758;
        double r177760 = r177756 + r177759;
        double r177761 = 771.3234287776531;
        double r177762 = 3.0;
        double r177763 = r177741 + r177762;
        double r177764 = r177761 / r177763;
        double r177765 = r177760 + r177764;
        double r177766 = -176.6150291621406;
        double r177767 = 4.0;
        double r177768 = r177741 + r177767;
        double r177769 = r177766 / r177768;
        double r177770 = r177765 + r177769;
        double r177771 = 12.507343278686905;
        double r177772 = 5.0;
        double r177773 = r177741 + r177772;
        double r177774 = r177771 / r177773;
        double r177775 = r177770 + r177774;
        double r177776 = -0.13857109526572012;
        double r177777 = 6.0;
        double r177778 = r177741 + r177777;
        double r177779 = r177776 / r177778;
        double r177780 = r177775 + r177779;
        double r177781 = 9.984369578019572e-06;
        double r177782 = r177781 / r177743;
        double r177783 = r177780 + r177782;
        double r177784 = 1.5056327351493116e-07;
        double r177785 = 8.0;
        double r177786 = r177741 + r177785;
        double r177787 = r177784 / r177786;
        double r177788 = r177783 + r177787;
        double r177789 = r177751 * r177788;
        return r177789;
}

↓

double f(double z) {
        double r177790 = atan2(1.0, 0.0);
        double r177791 = 2.0;
        double r177792 = r177790 * r177791;
        double r177793 = sqrt(r177792);
        double r177794 = z;
        double r177795 = 1.0;
        double r177796 = r177794 - r177795;
        double r177797 = 7.0;
        double r177798 = r177796 + r177797;
        double r177799 = 0.5;
        double r177800 = r177798 + r177799;
        double r177801 = r177796 + r177799;
        double r177802 = pow(r177800, r177801);
        double r177803 = r177793 * r177802;
        double r177804 = 676.5203681218851;
        double r177805 = 1.0;
        double r177806 = 6.5;
        double r177807 = exp(r177806);
        double r177808 = r177794 * r177807;
        double r177809 = r177805 / r177808;
        double r177810 = r177804 * r177809;
        double r177811 = 2351.6663247613023;
        double r177812 = r177794 / r177807;
        double r177813 = r177811 * r177812;
        double r177814 = r177810 + r177813;
        double r177815 = 1604.7704235566525;
        double r177816 = r177805 / r177807;
        double r177817 = r177815 * r177816;
        double r177818 = r177814 - r177817;
        double r177819 = r177803 * r177818;
        return r177819;
}

Error

Bits error versus z

Derivation

1. Initial program 61.7

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

2. Simplified 61.7

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

4. Applied associate-+l- 1.0

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

5. Simplified 1.0

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

7. Applied *-un-lft-identity 1.0

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

8. Applied times-frac 1.0

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

9. Simplified 1.0

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

10. Taylor expanded around 0 1.3

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot \color{blue}{\left(\left(676.520368121885099 \cdot \frac{1}{z \cdot e^{6.5}} + 2351.6663247613023 \cdot \frac{z}{e^{6.5}}\right) - 1604.7704235566525 \cdot \frac{1}{e^{6.5}}\right)}\]

11. Final simplification 1.3

    \[\leadsto \left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot \left(\left(676.520368121885099 \cdot \frac{1}{z \cdot e^{6.5}} + 2351.6663247613023 \cdot \frac{z}{e^{6.5}}\right) - 1604.7704235566525 \cdot \frac{1}{e^{6.5}}\right)\]

Reproduce

herbie shell --seed 2020060 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z greater than 0.5"
  :precision binary64
  (* (* (* (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)))))