2nthrt (problem 3.4.6)

Average Error: 29.6 → 19.7

Time: 31.1s

Precision: 64

• could not determine a ground truth for program body

  1. x = 6.446454094549423e+237
  2. n = 2.396068957455706e-282

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 -1.2161219111661097 \cdot 10^{-07}:\\ \;\;\;\;\mathsf{fma}\left(-\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right) + \sqrt[3]{\left(\mathsf{fma}\left(1, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}, -\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right) \cdot \mathsf{fma}\left(1, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}, -\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right)\right) \cdot \mathsf{fma}\left(1, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}, -\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 4.362000941685765 \cdot 10^{-12}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} + \frac{\frac{\log x}{n \cdot n}}{x}\right) - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \le 1.1080811788905206 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(-\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right) + \sqrt[3]{\left(\mathsf{fma}\left(1, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}, -\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right) \cdot \mathsf{fma}\left(1, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}, -\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right)\right) \cdot \mathsf{fma}\left(1, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}, -\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(\frac{\mathsf{log1p}\left(x\right)}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

C

double f(double x, double n) {
        double r2126859 = x;
        double r2126860 = 1.0;
        double r2126861 = r2126859 + r2126860;
        double r2126862 = n;
        double r2126863 = r2126860 / r2126862;
        double r2126864 = pow(r2126861, r2126863);
        double r2126865 = pow(r2126859, r2126863);
        double r2126866 = r2126864 - r2126865;
        return r2126866;
}

↓

double f(double x, double n) {
        double r2126867 = 1.0;
        double r2126868 = n;
        double r2126869 = r2126867 / r2126868;
        double r2126870 = -1.2161219111661097e-07;
        bool r2126871 = r2126869 <= r2126870;
        double r2126872 = x;
        double r2126873 = pow(r2126872, r2126869);
        double r2126874 = cbrt(r2126873);
        double r2126875 = -r2126874;
        double r2126876 = r2126874 * r2126874;
        double r2126877 = r2126874 * r2126876;
        double r2126878 = fma(r2126875, r2126876, r2126877);
        double r2126879 = r2126872 + r2126867;
        double r2126880 = pow(r2126879, r2126869);
        double r2126881 = -r2126877;
        double r2126882 = fma(r2126867, r2126880, r2126881);
        double r2126883 = r2126882 * r2126882;
        double r2126884 = r2126883 * r2126882;
        double r2126885 = cbrt(r2126884);
        double r2126886 = r2126878 + r2126885;
        double r2126887 = 4.362000941685765e-12;
        bool r2126888 = r2126869 <= r2126887;
        double r2126889 = r2126869 / r2126872;
        double r2126890 = log(r2126872);
        double r2126891 = r2126868 * r2126868;
        double r2126892 = r2126890 / r2126891;
        double r2126893 = r2126892 / r2126872;
        double r2126894 = r2126889 + r2126893;
        double r2126895 = 0.5;
        double r2126896 = r2126895 / r2126868;
        double r2126897 = r2126872 * r2126872;
        double r2126898 = r2126896 / r2126897;
        double r2126899 = r2126894 - r2126898;
        double r2126900 = 1.1080811788905206e+146;
        bool r2126901 = r2126869 <= r2126900;
        double r2126902 = log1p(r2126872);
        double r2126903 = r2126902 / r2126868;
        double r2126904 = /* ERROR: no posit support in C */;
        double r2126905 = /* ERROR: no posit support in C */;
        double r2126906 = exp(r2126905);
        double r2126907 = r2126906 - r2126873;
        double r2126908 = r2126901 ? r2126886 : r2126907;
        double r2126909 = r2126888 ? r2126899 : r2126908;
        double r2126910 = r2126871 ? r2126886 : r2126909;
        return r2126910;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1 n) < -1.2161219111661097e-07 or 4.362000941685765e-12 < (/ 1 n) < 1.1080811788905206e+146

   1. Initial program 3.2

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

   3. Applied add-cube-cbrt 3.3

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

   4. Applied *-un-lft-identity 3.3

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

   5. Applied prod-diff 3.3

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

   7. Applied add-cbrt-cube 3.4

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

   if -1.2161219111661097e-07 < (/ 1 n) < 4.362000941685765e-12

   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 32.0

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

   3. Simplified 31.4

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

   if 1.1080811788905206e+146 < (/ 1 n)

   1. Initial program 41.0

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

   3. Applied add-exp-log 41.0

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

   4. Simplified 0.0

      \[\leadsto e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
   5. Using strategy rm

   6. Applied insert-posit16 7.9

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

4. Final simplification 19.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.2161219111661097 \cdot 10^{-07}:\\ \;\;\;\;\mathsf{fma}\left(-\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right) + \sqrt[3]{\left(\mathsf{fma}\left(1, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}, -\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right) \cdot \mathsf{fma}\left(1, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}, -\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right)\right) \cdot \mathsf{fma}\left(1, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}, -\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 4.362000941685765 \cdot 10^{-12}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} + \frac{\frac{\log x}{n \cdot n}}{x}\right) - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \le 1.1080811788905206 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(-\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right) + \sqrt[3]{\left(\mathsf{fma}\left(1, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}, -\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right) \cdot \mathsf{fma}\left(1, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}, -\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right)\right) \cdot \mathsf{fma}\left(1, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}, -\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(\frac{\mathsf{log1p}\left(x\right)}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))