2nthrt (problem 3.4.6)

Average Error: 29.4 → 19.0

Time: 36.9s

Precision: 64

• could not determine a ground truth for program body

  1. x = 8.351211544149733e+102
  2. n = 4.308249766459817e-192

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 -3.147233419817972 \cdot 10^{-08}:\\ \;\;\;\;\sqrt[3]{\left(\left(\sqrt[3]{\mathsf{fma}\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)} + \mathsf{fma}\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\right) \cdot \left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \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)\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 2.220885236034127 \cdot 10^{-08}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} + \frac{\log x}{\left(n \cdot n\right) \cdot x}\right) - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

C

double f(double x, double n) {
        double r3622841 = x;
        double r3622842 = 1.0;
        double r3622843 = r3622841 + r3622842;
        double r3622844 = n;
        double r3622845 = r3622842 / r3622844;
        double r3622846 = pow(r3622843, r3622845);
        double r3622847 = pow(r3622841, r3622845);
        double r3622848 = r3622846 - r3622847;
        return r3622848;
}

↓

double f(double x, double n) {
        double r3622849 = 1.0;
        double r3622850 = n;
        double r3622851 = r3622849 / r3622850;
        double r3622852 = -3.147233419817972e-08;
        bool r3622853 = r3622851 <= r3622852;
        double r3622854 = x;
        double r3622855 = r3622854 + r3622849;
        double r3622856 = pow(r3622855, r3622851);
        double r3622857 = sqrt(r3622856);
        double r3622858 = cbrt(r3622854);
        double r3622859 = pow(r3622858, r3622851);
        double r3622860 = -r3622859;
        double r3622861 = r3622858 * r3622858;
        double r3622862 = pow(r3622861, r3622851);
        double r3622863 = r3622860 * r3622862;
        double r3622864 = fma(r3622857, r3622857, r3622863);
        double r3622865 = cbrt(r3622864);
        double r3622866 = r3622865 * r3622865;
        double r3622867 = r3622866 * r3622865;
        double r3622868 = r3622859 * r3622862;
        double r3622869 = fma(r3622860, r3622862, r3622868);
        double r3622870 = r3622867 + r3622869;
        double r3622871 = pow(r3622854, r3622851);
        double r3622872 = r3622856 - r3622871;
        double r3622873 = cbrt(r3622872);
        double r3622874 = r3622873 * r3622873;
        double r3622875 = r3622873 * r3622874;
        double r3622876 = r3622872 * r3622875;
        double r3622877 = r3622870 * r3622876;
        double r3622878 = cbrt(r3622877);
        double r3622879 = 2.220885236034127e-08;
        bool r3622880 = r3622851 <= r3622879;
        double r3622881 = r3622854 * r3622850;
        double r3622882 = r3622849 / r3622881;
        double r3622883 = log(r3622854);
        double r3622884 = r3622850 * r3622850;
        double r3622885 = r3622884 * r3622854;
        double r3622886 = r3622883 / r3622885;
        double r3622887 = r3622882 + r3622886;
        double r3622888 = 0.5;
        double r3622889 = r3622888 / r3622850;
        double r3622890 = r3622854 * r3622854;
        double r3622891 = r3622889 / r3622890;
        double r3622892 = r3622887 - r3622891;
        double r3622893 = log1p(r3622854);
        double r3622894 = r3622893 / r3622850;
        double r3622895 = exp(r3622894);
        double r3622896 = r3622895 - r3622871;
        double r3622897 = r3622880 ? r3622892 : r3622896;
        double r3622898 = r3622853 ? r3622878 : r3622897;
        return r3622898;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1 n) < -3.147233419817972e-08

   1. Initial program 0.8

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

   3. Applied add-cbrt-cube 0.9

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

   5. Applied add-cube-cbrt 0.9

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

   6. Applied unpow-prod-down 0.9

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

   7. Applied add-sqr-sqrt 0.9

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

   8. Applied prod-diff 0.9

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

   10. Applied add-cube-cbrt 0.9

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

   12. Applied add-cube-cbrt 0.9

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

   if -3.147233419817972e-08 < (/ 1 n) < 2.220885236034127e-08

   1. Initial program 45.2

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

   2. Taylor expanded around inf 32.6

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

   3. Simplified 32.5

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

   if 2.220885236034127e-08 < (/ 1 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-exp-log 25.7

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

   4. Applied pow-exp 25.7

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

   5. Simplified 1.9

      \[\leadsto e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 19.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -3.147233419817972 \cdot 10^{-08}:\\ \;\;\;\;\sqrt[3]{\left(\left(\sqrt[3]{\mathsf{fma}\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)} + \mathsf{fma}\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\right) \cdot \left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \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)\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 2.220885236034127 \cdot 10^{-08}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} + \frac{\log x}{\left(n \cdot n\right) \cdot x}\right) - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))