2nthrt (problem 3.4.6)

Average Error: 29.8 → 22.8

Time: 1.5m

Precision: 64

• could not determine a ground truth for program body

  1. x = 2.1948653002835354e+303
  2. n = 8.152068102250278e-21

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -62.577189808211486:\\ \;\;\;\;\frac{1}{\sqrt{x}} \cdot \frac{\frac{1}{n}}{\sqrt{x}} + \frac{\frac{1}{n}}{x} \cdot \left(\frac{\log x}{n} - \frac{\frac{1}{2}}{x}\right)\\ \mathbf{elif}\;n \le 7.520631394347698 \cdot 10^{+26}:\\ \;\;\;\;{e}^{\left(\log \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\log x}{n} - \frac{\frac{1}{2}}{x}\right) \cdot \log \left(e^{\frac{\frac{1}{n}}{x}}\right) + \frac{\frac{1}{n}}{x}\\ \end{array}\]

C

double f(double x, double n) {
        double r8291027 = x;
        double r8291028 = 1.0;
        double r8291029 = r8291027 + r8291028;
        double r8291030 = n;
        double r8291031 = r8291028 / r8291030;
        double r8291032 = pow(r8291029, r8291031);
        double r8291033 = pow(r8291027, r8291031);
        double r8291034 = r8291032 - r8291033;
        return r8291034;
}

↓

double f(double x, double n) {
        double r8291035 = n;
        double r8291036 = -62.577189808211486;
        bool r8291037 = r8291035 <= r8291036;
        double r8291038 = 1.0;
        double r8291039 = x;
        double r8291040 = sqrt(r8291039);
        double r8291041 = r8291038 / r8291040;
        double r8291042 = r8291038 / r8291035;
        double r8291043 = r8291042 / r8291040;
        double r8291044 = r8291041 * r8291043;
        double r8291045 = r8291042 / r8291039;
        double r8291046 = log(r8291039);
        double r8291047 = r8291046 / r8291035;
        double r8291048 = 0.5;
        double r8291049 = r8291048 / r8291039;
        double r8291050 = r8291047 - r8291049;
        double r8291051 = r8291045 * r8291050;
        double r8291052 = r8291044 + r8291051;
        double r8291053 = 7.520631394347698e+26;
        bool r8291054 = r8291035 <= r8291053;
        double r8291055 = exp(1.0);
        double r8291056 = r8291038 + r8291039;
        double r8291057 = pow(r8291056, r8291042);
        double r8291058 = pow(r8291039, r8291042);
        double r8291059 = r8291057 - r8291058;
        double r8291060 = log(r8291059);
        double r8291061 = pow(r8291055, r8291060);
        double r8291062 = exp(r8291045);
        double r8291063 = log(r8291062);
        double r8291064 = r8291050 * r8291063;
        double r8291065 = r8291064 + r8291045;
        double r8291066 = r8291054 ? r8291061 : r8291065;
        double r8291067 = r8291037 ? r8291052 : r8291066;
        return r8291067;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if n < -62.577189808211486

   1. Initial program 45.6

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

   3. Applied add-exp-log 46.6

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

   5. Applied *-un-lft-identity 46.6

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

   6. Applied exp-prod 46.6

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

   7. Simplified 46.6

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

   8. Taylor expanded around -inf 63.2

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

   9. Simplified 33.2

      \[\leadsto \color{blue}{\left(\frac{\frac{1}{n}}{x} + 0\right) + \frac{\frac{1}{n}}{x} \cdot \left(\frac{\log x}{n} - \frac{\frac{1}{2}}{x}\right)}\]
   10. Using strategy rm

   11. Applied add-sqr-sqrt 33.2

       \[\leadsto \left(\frac{\frac{1}{n}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} + 0\right) + \frac{\frac{1}{n}}{x} \cdot \left(\frac{\log x}{n} - \frac{\frac{1}{2}}{x}\right)\]

   12. Applied *-un-lft-identity 33.2

       \[\leadsto \left(\frac{\frac{1}{\color{blue}{1 \cdot n}}}{\sqrt{x} \cdot \sqrt{x}} + 0\right) + \frac{\frac{1}{n}}{x} \cdot \left(\frac{\log x}{n} - \frac{\frac{1}{2}}{x}\right)\]

   13. Applied *-un-lft-identity 33.2

       \[\leadsto \left(\frac{\frac{\color{blue}{1 \cdot 1}}{1 \cdot n}}{\sqrt{x} \cdot \sqrt{x}} + 0\right) + \frac{\frac{1}{n}}{x} \cdot \left(\frac{\log x}{n} - \frac{\frac{1}{2}}{x}\right)\]

   14. Applied times-frac 33.2

       \[\leadsto \left(\frac{\color{blue}{\frac{1}{1} \cdot \frac{1}{n}}}{\sqrt{x} \cdot \sqrt{x}} + 0\right) + \frac{\frac{1}{n}}{x} \cdot \left(\frac{\log x}{n} - \frac{\frac{1}{2}}{x}\right)\]

   15. Applied times-frac 33.3

       \[\leadsto \left(\color{blue}{\frac{\frac{1}{1}}{\sqrt{x}} \cdot \frac{\frac{1}{n}}{\sqrt{x}}} + 0\right) + \frac{\frac{1}{n}}{x} \cdot \left(\frac{\log x}{n} - \frac{\frac{1}{2}}{x}\right)\]

   16. Simplified 33.3

       \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x}}} \cdot \frac{\frac{1}{n}}{\sqrt{x}} + 0\right) + \frac{\frac{1}{n}}{x} \cdot \left(\frac{\log x}{n} - \frac{\frac{1}{2}}{x}\right)\]

   if -62.577189808211486 < n < 7.520631394347698e+26

   1. Initial program 10.3

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

   3. Applied add-exp-log 10.4

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

   5. Applied *-un-lft-identity 10.4

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

   6. Applied exp-prod 10.4

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

   7. Simplified 10.4

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

   if 7.520631394347698e+26 < n

   1. Initial program 44.3

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

   3. Applied add-exp-log 44.3

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

   5. Applied *-un-lft-identity 44.3

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

   6. Applied exp-prod 44.3

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

   7. Simplified 44.3

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

   8. Taylor expanded around -inf 62.7

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

   9. Simplified 31.7

      \[\leadsto \color{blue}{\left(\frac{\frac{1}{n}}{x} + 0\right) + \frac{\frac{1}{n}}{x} \cdot \left(\frac{\log x}{n} - \frac{\frac{1}{2}}{x}\right)}\]
   10. Using strategy rm

   11. Applied add-log-exp 31.4

       \[\leadsto \left(\frac{\frac{1}{n}}{x} + 0\right) + \color{blue}{\log \left(e^{\frac{\frac{1}{n}}{x}}\right)} \cdot \left(\frac{\log x}{n} - \frac{\frac{1}{2}}{x}\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 22.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -62.577189808211486:\\ \;\;\;\;\frac{1}{\sqrt{x}} \cdot \frac{\frac{1}{n}}{\sqrt{x}} + \frac{\frac{1}{n}}{x} \cdot \left(\frac{\log x}{n} - \frac{\frac{1}{2}}{x}\right)\\ \mathbf{elif}\;n \le 7.520631394347698 \cdot 10^{+26}:\\ \;\;\;\;{e}^{\left(\log \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\log x}{n} - \frac{\frac{1}{2}}{x}\right) \cdot \log \left(e^{\frac{\frac{1}{n}}{x}}\right) + \frac{\frac{1}{n}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))