2nthrt (problem 3.4.6)

Average Error: 29.5 → 23.0

Time: 32.3s

Precision: 64

• could not determine a ground truth for program body

  1. x = 2.2545226427089226e+284
  2. n = 1.8034259511601163e-130

Math

\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.05925712577091127308825946329307043924928:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right) \cdot \sqrt[3]{e^{1 \cdot \frac{\log 1}{n}} - {x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \le 2.255271015860772358417468981097305856376 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}} \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right) \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right)}\right) \cdot \sqrt[3]{e^{1 \cdot \frac{\log 1}{n}} - {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

C

double f(double x, double n) {
        double r66862 = x;
        double r66863 = 1.0;
        double r66864 = r66862 + r66863;
        double r66865 = n;
        double r66866 = r66863 / r66865;
        double r66867 = pow(r66864, r66866);
        double r66868 = pow(r66862, r66866);
        double r66869 = r66867 - r66868;
        return r66869;
}

↓

double f(double x, double n) {
        double r66870 = 1.0;
        double r66871 = n;
        double r66872 = r66870 / r66871;
        double r66873 = -0.05925712577091127;
        bool r66874 = r66872 <= r66873;
        double r66875 = x;
        double r66876 = r66875 + r66870;
        double r66877 = pow(r66876, r66872);
        double r66878 = pow(r66875, r66872);
        double r66879 = r66877 - r66878;
        double r66880 = cbrt(r66879);
        double r66881 = sqrt(r66877);
        double r66882 = 2.0;
        double r66883 = r66872 / r66882;
        double r66884 = pow(r66875, r66883);
        double r66885 = r66881 + r66884;
        double r66886 = r66881 - r66884;
        double r66887 = r66885 * r66886;
        double r66888 = cbrt(r66887);
        double r66889 = r66880 * r66888;
        double r66890 = log(r66870);
        double r66891 = r66890 / r66871;
        double r66892 = r66870 * r66891;
        double r66893 = exp(r66892);
        double r66894 = r66893 - r66878;
        double r66895 = cbrt(r66894);
        double r66896 = r66889 * r66895;
        double r66897 = 2.2552710158607724e-27;
        bool r66898 = r66872 <= r66897;
        double r66899 = r66870 / r66875;
        double r66900 = 1.0;
        double r66901 = r66900 / r66871;
        double r66902 = log(r66875);
        double r66903 = -r66902;
        double r66904 = pow(r66871, r66882);
        double r66905 = r66903 / r66904;
        double r66906 = r66901 - r66905;
        double r66907 = r66899 * r66906;
        double r66908 = 0.5;
        double r66909 = pow(r66875, r66882);
        double r66910 = r66909 * r66871;
        double r66911 = r66908 / r66910;
        double r66912 = r66907 - r66911;
        double r66913 = cbrt(r66886);
        double r66914 = r66913 * r66913;
        double r66915 = r66914 * r66913;
        double r66916 = r66885 * r66915;
        double r66917 = cbrt(r66916);
        double r66918 = r66880 * r66917;
        double r66919 = r66918 * r66895;
        double r66920 = r66898 ? r66912 : r66919;
        double r66921 = r66874 ? r66896 : r66920;
        return r66921;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -0.05925712577091127

   1. Initial program 0.1

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 0.1

      \[\leadsto \color{blue}{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\]
   4. Using strategy rm

   5. Applied sqr-pow 0.1

      \[\leadsto \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{{x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\]

   6. Applied add-sqr-sqrt 0.1

      \[\leadsto \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\color{blue}{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\]

   7. Applied difference-of-squares 0.1

      \[\leadsto \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\color{blue}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\]

   8. Taylor expanded around 0 0.1

      \[\leadsto \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right) \cdot \sqrt[3]{\color{blue}{e^{1 \cdot \frac{\log 1}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\]

   if -0.05925712577091127 < (/ 1.0 n) < 2.2552710158607724e-27

   1. Initial program 44.9

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]

   2. Taylor expanded around inf 33.9

      \[\leadsto \color{blue}{1 \cdot \frac{1}{x \cdot n} - \left(0.5 \cdot \frac{1}{{x}^{2} \cdot n} + 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)}\]

   3. Simplified 33.3

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}}\]

   if 2.2552710158607724e-27 < (/ 1.0 n)

   1. Initial program 28.1

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 28.2

      \[\leadsto \color{blue}{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\]
   4. Using strategy rm

   5. Applied sqr-pow 28.1

      \[\leadsto \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{{x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\]

   6. Applied add-sqr-sqrt 28.1

      \[\leadsto \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\color{blue}{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\]

   7. Applied difference-of-squares 28.1

      \[\leadsto \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\color{blue}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\]

   8. Taylor expanded around 0 28.2

      \[\leadsto \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right) \cdot \sqrt[3]{\color{blue}{e^{1 \cdot \frac{\log 1}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 28.2

       \[\leadsto \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}} \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right) \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right)}}\right) \cdot \sqrt[3]{e^{1 \cdot \frac{\log 1}{n}} - {x}^{\left(\frac{1}{n}\right)}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 23.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.05925712577091127308825946329307043924928:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right) \cdot \sqrt[3]{e^{1 \cdot \frac{\log 1}{n}} - {x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \le 2.255271015860772358417468981097305856376 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}} \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right) \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right)}\right) \cdot \sqrt[3]{e^{1 \cdot \frac{\log 1}{n}} - {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))