2nthrt (problem 3.4.6)

Average Error: 29.4 → 22.4

Time: 10.8s

Precision: 64

• could not determine a ground truth for program body

  1. x = 1.3007964282682276e+285
  2. n = 1.5934329611832436e-130

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -934833342441455353856:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{0.5}{n}}{{x}^{2}}\right) - \left(-\frac{1 \cdot \log x}{x \cdot {n}^{2}}\right)\\ \mathbf{elif}\;n \le 59.535473331148146769464801764115691185:\\ \;\;\;\;{\left({\left(x + 1\right)}^{\left(\frac{\sqrt{1}}{\sqrt[3]{n} \cdot \sqrt[3]{n}}\right)}\right)}^{\left(\frac{\sqrt{1}}{\sqrt[3]{n}}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\sqrt{1}}{\sqrt{n}}}{\sqrt{x}} + \frac{\sqrt{\frac{0.5}{n}}}{x}\right) \cdot \left(\frac{\frac{\sqrt{1}}{\sqrt{n}}}{\sqrt{x}} - \frac{\sqrt{\frac{0.5}{n}}}{x}\right) - \left(-\frac{1 \cdot \log x}{x \cdot {n}^{2}}\right)\\ \end{array}\]

C

double f(double x, double n) {
        double r70868 = x;
        double r70869 = 1.0;
        double r70870 = r70868 + r70869;
        double r70871 = n;
        double r70872 = r70869 / r70871;
        double r70873 = pow(r70870, r70872);
        double r70874 = pow(r70868, r70872);
        double r70875 = r70873 - r70874;
        return r70875;
}

↓

double f(double x, double n) {
        double r70876 = n;
        double r70877 = -9.348333424414554e+20;
        bool r70878 = r70876 <= r70877;
        double r70879 = 1.0;
        double r70880 = r70879 / r70876;
        double r70881 = x;
        double r70882 = r70880 / r70881;
        double r70883 = 0.5;
        double r70884 = r70883 / r70876;
        double r70885 = 2.0;
        double r70886 = pow(r70881, r70885);
        double r70887 = r70884 / r70886;
        double r70888 = r70882 - r70887;
        double r70889 = log(r70881);
        double r70890 = r70879 * r70889;
        double r70891 = pow(r70876, r70885);
        double r70892 = r70881 * r70891;
        double r70893 = r70890 / r70892;
        double r70894 = -r70893;
        double r70895 = r70888 - r70894;
        double r70896 = 59.53547333114815;
        bool r70897 = r70876 <= r70896;
        double r70898 = r70881 + r70879;
        double r70899 = sqrt(r70879);
        double r70900 = cbrt(r70876);
        double r70901 = r70900 * r70900;
        double r70902 = r70899 / r70901;
        double r70903 = pow(r70898, r70902);
        double r70904 = r70899 / r70900;
        double r70905 = pow(r70903, r70904);
        double r70906 = pow(r70881, r70880);
        double r70907 = r70905 - r70906;
        double r70908 = sqrt(r70876);
        double r70909 = r70899 / r70908;
        double r70910 = sqrt(r70881);
        double r70911 = r70909 / r70910;
        double r70912 = sqrt(r70884);
        double r70913 = r70912 / r70881;
        double r70914 = r70911 + r70913;
        double r70915 = r70911 - r70913;
        double r70916 = r70914 * r70915;
        double r70917 = r70916 - r70894;
        double r70918 = r70897 ? r70907 : r70917;
        double r70919 = r70878 ? r70895 : r70918;
        return r70919;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if n < -9.348333424414554e+20

   1. Initial program 44.8

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

   2. Taylor expanded around inf 32.3

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

   3. Simplified 31.8

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

   5. Applied sub-neg 31.8

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

   6. Applied associate--r+ 31.8

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

   if -9.348333424414554e+20 < n < 59.53547333114815

   1. Initial program 9.4

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

   3. Applied add-cube-cbrt 9.4

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

   4. Applied add-sqr-sqrt 9.4

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

   5. Applied times-frac 9.4

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

   6. Applied pow-unpow 9.4

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

   if 59.53547333114815 < n

   1. Initial program 44.0

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

   2. Taylor expanded around inf 33.1

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

   3. Simplified 32.5

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

   5. Applied sub-neg 32.5

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

   6. Applied associate--r+ 32.5

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

   8. Applied add-sqr-sqrt 32.5

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

   9. Applied unpow-prod-down 32.5

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

   10. Applied add-sqr-sqrt 32.5

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

   11. Applied times-frac 32.4

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

   12. Applied add-sqr-sqrt 32.5

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

   13. Applied add-sqr-sqrt 32.5

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

   14. Applied add-sqr-sqrt 32.5

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

   15. Applied times-frac 32.5

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

   16. Applied times-frac 32.5

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

   17. Applied difference-of-squares 32.5

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

   18. Simplified 32.5

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

   19. Simplified 32.5

       \[\leadsto \left(\frac{\frac{\sqrt{1}}{\sqrt{n}}}{\sqrt{x}} + \frac{\sqrt{\frac{0.5}{n}}}{x}\right) \cdot \color{blue}{\left(\frac{\frac{\sqrt{1}}{\sqrt{n}}}{\sqrt{x}} - \frac{\sqrt{\frac{0.5}{n}}}{x}\right)} - \left(-\frac{1 \cdot \log x}{x \cdot {n}^{2}}\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 22.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -934833342441455353856:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{0.5}{n}}{{x}^{2}}\right) - \left(-\frac{1 \cdot \log x}{x \cdot {n}^{2}}\right)\\ \mathbf{elif}\;n \le 59.535473331148146769464801764115691185:\\ \;\;\;\;{\left({\left(x + 1\right)}^{\left(\frac{\sqrt{1}}{\sqrt[3]{n} \cdot \sqrt[3]{n}}\right)}\right)}^{\left(\frac{\sqrt{1}}{\sqrt[3]{n}}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\sqrt{1}}{\sqrt{n}}}{\sqrt{x}} + \frac{\sqrt{\frac{0.5}{n}}}{x}\right) \cdot \left(\frac{\frac{\sqrt{1}}{\sqrt{n}}}{\sqrt{x}} - \frac{\sqrt{\frac{0.5}{n}}}{x}\right) - \left(-\frac{1 \cdot \log x}{x \cdot {n}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))