2nthrt (problem 3.4.6)

Average Error: 29.6 → 22.1

Time: 18.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

• could not determine a ground truth for program body

  1. x = 7.113723716624242e+188
  2. n = 3.1812150306990062e-208

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.00289957062036440716551455309968332585413:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\left({\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 5.120424455825518505286756233089616913316 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\left({\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot \left({\left(\sqrt{\sqrt[3]{x + 1}}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(\sqrt{\sqrt[3]{x + 1}}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\\ \end{array}\]

C

double f(double x, double n) {
        double r49853 = x;
        double r49854 = 1.0;
        double r49855 = r49853 + r49854;
        double r49856 = n;
        double r49857 = r49854 / r49856;
        double r49858 = pow(r49855, r49857);
        double r49859 = pow(r49853, r49857);
        double r49860 = r49858 - r49859;
        return r49860;
}

↓

double f(double x, double n) {
        double r49861 = 1.0;
        double r49862 = n;
        double r49863 = r49861 / r49862;
        double r49864 = -0.002899570620364407;
        bool r49865 = r49863 <= r49864;
        double r49866 = x;
        double r49867 = r49866 + r49861;
        double r49868 = pow(r49867, r49863);
        double r49869 = pow(r49866, r49863);
        double r49870 = r49868 - r49869;
        double r49871 = cbrt(r49870);
        double r49872 = r49871 * r49871;
        double r49873 = cbrt(r49867);
        double r49874 = r49873 * r49873;
        double r49875 = 2.0;
        double r49876 = r49863 / r49875;
        double r49877 = pow(r49874, r49876);
        double r49878 = pow(r49873, r49876);
        double r49879 = r49877 * r49878;
        double r49880 = pow(r49866, r49876);
        double r49881 = r49879 + r49880;
        double r49882 = pow(r49867, r49876);
        double r49883 = r49882 - r49880;
        double r49884 = r49881 * r49883;
        double r49885 = cbrt(r49884);
        double r49886 = r49872 * r49885;
        double r49887 = 5.1204244558255185e-17;
        bool r49888 = r49863 <= r49887;
        double r49889 = r49861 / r49866;
        double r49890 = 1.0;
        double r49891 = r49890 / r49862;
        double r49892 = log(r49866);
        double r49893 = -r49892;
        double r49894 = pow(r49862, r49875);
        double r49895 = r49893 / r49894;
        double r49896 = r49891 - r49895;
        double r49897 = r49889 * r49896;
        double r49898 = 0.5;
        double r49899 = pow(r49866, r49875);
        double r49900 = r49899 * r49862;
        double r49901 = r49898 / r49900;
        double r49902 = r49897 - r49901;
        double r49903 = sqrt(r49873);
        double r49904 = pow(r49903, r49876);
        double r49905 = r49904 * r49904;
        double r49906 = r49877 * r49905;
        double r49907 = r49906 + r49880;
        double r49908 = r49907 * r49883;
        double r49909 = cbrt(r49908);
        double r49910 = r49872 * r49909;
        double r49911 = r49888 ? r49902 : r49910;
        double r49912 = r49865 ? r49886 : r49911;
        return r49912;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -0.002899570620364407

   1. Initial program 0.3

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

   3. Applied add-cube-cbrt 0.3

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

   5. Applied sqr-pow 0.3

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

   6. Applied sqr-pow 0.3

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

   7. Applied difference-of-squares 0.3

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

   9. Applied add-cube-cbrt 0.3

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

   10. Applied unpow-prod-down 0.3

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

   if -0.002899570620364407 < (/ 1.0 n) < 5.1204244558255185e-17

   1. Initial program 45.1

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

   2. Taylor expanded around inf 32.5

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

   3. Simplified 31.9

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

   if 5.1204244558255185e-17 < (/ 1.0 n)

   1. Initial program 25.7

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

   3. Applied add-cube-cbrt 25.7

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

   5. Applied sqr-pow 25.7

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

   6. Applied sqr-pow 25.7

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

   7. Applied difference-of-squares 25.7

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

   9. Applied add-cube-cbrt 25.8

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

   10. Applied unpow-prod-down 25.8

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

   12. Applied add-sqr-sqrt 25.8

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

   13. Applied unpow-prod-down 25.8

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

4. Final simplification 22.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.00289957062036440716551455309968332585413:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\left({\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 5.120424455825518505286756233089616913316 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\left({\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot \left({\left(\sqrt{\sqrt[3]{x + 1}}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(\sqrt{\sqrt[3]{x + 1}}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))