2nthrt (problem 3.4.6)

Average Error: 29.4 → 19.9

Time: 30.9s

Precision: 64

• could not determine a ground truth for program body

  1. x = 3.701835359357904e+165
  2. n = 2.9165008331150025e-106

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -5.516643619668029 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + \left(\left(\frac{\log x}{n \cdot n} \cdot \frac{\frac{1}{3}}{x} - \frac{\frac{1}{2}}{n \cdot \left(x \cdot x\right)}\right) + \left(\frac{\frac{1}{x}}{n} + \frac{\frac{\log x \cdot \frac{2}{3}}{x}}{n \cdot n}\right)\right)\\ \mathbf{elif}\;n \le -1.9461952197688902 \cdot 10^{-306}:\\ \;\;\;\;\mathsf{fma}\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + \log \left(e^{\mathsf{fma}\left({\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}, \left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)}\right)\\ \mathbf{elif}\;n \le 590284494.5165114:\\ \;\;\;\;e^{\mathsf{log1p}\left(x\right) \cdot \frac{1}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) - \frac{-\log x}{\left(x \cdot n\right) \cdot n}\\ \end{array}\]

C

double f(double x, double n) {
        double r950817 = x;
        double r950818 = 1.0;
        double r950819 = r950817 + r950818;
        double r950820 = n;
        double r950821 = r950818 / r950820;
        double r950822 = pow(r950819, r950821);
        double r950823 = pow(r950817, r950821);
        double r950824 = r950822 - r950823;
        return r950824;
}

↓

double f(double x, double n) {
        double r950825 = n;
        double r950826 = -5.516643619668029e+38;
        bool r950827 = r950825 <= r950826;
        double r950828 = x;
        double r950829 = cbrt(r950828);
        double r950830 = 1.0;
        double r950831 = r950830 / r950825;
        double r950832 = pow(r950829, r950831);
        double r950833 = -r950832;
        double r950834 = r950829 * r950829;
        double r950835 = pow(r950834, r950831);
        double r950836 = r950835 * r950832;
        double r950837 = fma(r950833, r950835, r950836);
        double r950838 = log(r950828);
        double r950839 = r950825 * r950825;
        double r950840 = r950838 / r950839;
        double r950841 = 0.3333333333333333;
        double r950842 = r950841 / r950828;
        double r950843 = r950840 * r950842;
        double r950844 = 0.5;
        double r950845 = r950828 * r950828;
        double r950846 = r950825 * r950845;
        double r950847 = r950844 / r950846;
        double r950848 = r950843 - r950847;
        double r950849 = r950830 / r950828;
        double r950850 = r950849 / r950825;
        double r950851 = 0.6666666666666666;
        double r950852 = r950838 * r950851;
        double r950853 = r950852 / r950828;
        double r950854 = r950853 / r950839;
        double r950855 = r950850 + r950854;
        double r950856 = r950848 + r950855;
        double r950857 = r950837 + r950856;
        double r950858 = -1.9461952197688902e-306;
        bool r950859 = r950825 <= r950858;
        double r950860 = r950830 + r950828;
        double r950861 = cbrt(r950860);
        double r950862 = r950861 * r950861;
        double r950863 = pow(r950862, r950831);
        double r950864 = pow(r950861, r950831);
        double r950865 = r950833 * r950835;
        double r950866 = fma(r950863, r950864, r950865);
        double r950867 = exp(r950866);
        double r950868 = log(r950867);
        double r950869 = r950837 + r950868;
        double r950870 = 590284494.5165114;
        bool r950871 = r950825 <= r950870;
        double r950872 = log1p(r950828);
        double r950873 = r950872 * r950831;
        double r950874 = exp(r950873);
        double r950875 = pow(r950828, r950831);
        double r950876 = r950874 - r950875;
        double r950877 = r950828 * r950825;
        double r950878 = r950830 / r950877;
        double r950879 = r950844 / r950825;
        double r950880 = r950879 / r950845;
        double r950881 = r950878 - r950880;
        double r950882 = -r950838;
        double r950883 = r950877 * r950825;
        double r950884 = r950882 / r950883;
        double r950885 = r950881 - r950884;
        double r950886 = r950871 ? r950876 : r950885;
        double r950887 = r950859 ? r950869 : r950886;
        double r950888 = r950827 ? r950857 : r950887;
        return r950888;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 4 regimes

2. if n < -5.516643619668029e+38

   1. Initial program 43.8

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

   3. Applied add-cube-cbrt 43.8

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

   4. Applied unpow-prod-down 43.8

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

   5. Applied add-cube-cbrt 43.8

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

   6. Applied unpow-prod-down 43.8

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

   7. Applied prod-diff 43.8

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

   8. Taylor expanded around inf 32.9

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

   9. Simplified 32.4

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

   if -5.516643619668029e+38 < n < -1.9461952197688902e-306

   1. Initial program 6.0

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

   3. Applied add-cube-cbrt 6.0

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

   4. Applied unpow-prod-down 6.0

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

   5. Applied add-cube-cbrt 6.0

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

   6. Applied unpow-prod-down 6.0

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

   7. Applied prod-diff 6.0

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

   9. Applied add-log-exp 6.1

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

   if -1.9461952197688902e-306 < n < 590284494.5165114

   1. Initial program 25.1

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

   3. Applied add-exp-log 25.1

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

   4. Simplified 2.9

      \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]

   if 590284494.5165114 < n

   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 32.7

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

   3. Simplified 32.7

      \[\leadsto \color{blue}{\left(\frac{1}{n \cdot x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) - \left(-\frac{\log x}{n \cdot \left(n \cdot x\right)}\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 19.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -5.516643619668029 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + \left(\left(\frac{\log x}{n \cdot n} \cdot \frac{\frac{1}{3}}{x} - \frac{\frac{1}{2}}{n \cdot \left(x \cdot x\right)}\right) + \left(\frac{\frac{1}{x}}{n} + \frac{\frac{\log x \cdot \frac{2}{3}}{x}}{n \cdot n}\right)\right)\\ \mathbf{elif}\;n \le -1.9461952197688902 \cdot 10^{-306}:\\ \;\;\;\;\mathsf{fma}\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + \log \left(e^{\mathsf{fma}\left({\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{1 + x}\right)}^{\left(\frac{1}{n}\right)}, \left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)}\right)\\ \mathbf{elif}\;n \le 590284494.5165114:\\ \;\;\;\;e^{\mathsf{log1p}\left(x\right) \cdot \frac{1}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) - \frac{-\log x}{\left(x \cdot n\right) \cdot n}\\ \end{array}\]

Reproduce

herbie shell --seed 2019155 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))