2nthrt (problem 3.4.6)

Average Error: 29.1 → 22.3

Time: 9.8s

Precision: 64

• could not determine a ground truth for program body

  1. x = 1.3063160135701974e+105
  2. n = 1.297677126524779e-204

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -235.54342409600352:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\log \left(e^{\frac{0.5}{{x}^{2} \cdot n}}\right) - \frac{\log x \cdot 1}{e^{\log \left(x \cdot {n}^{2}\right)}}\right)\\ \mathbf{elif}\;n \le 0.2317853494525819:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{e^{\left(\sqrt[3]{\log \left(x \cdot {n}^{2}\right)} \cdot \sqrt[3]{\log \left(x \cdot {n}^{2}\right)}\right) \cdot \sqrt[3]{\log \left(x \cdot {n}^{2}\right)}}}\right)\\ \end{array}\]

C

double f(double x, double n) {
        double r39784 = x;
        double r39785 = 1.0;
        double r39786 = r39784 + r39785;
        double r39787 = n;
        double r39788 = r39785 / r39787;
        double r39789 = pow(r39786, r39788);
        double r39790 = pow(r39784, r39788);
        double r39791 = r39789 - r39790;
        return r39791;
}

↓

double f(double x, double n) {
        double r39792 = n;
        double r39793 = -235.54342409600352;
        bool r39794 = r39792 <= r39793;
        double r39795 = 1.0;
        double r39796 = r39795 / r39792;
        double r39797 = x;
        double r39798 = r39796 / r39797;
        double r39799 = 0.5;
        double r39800 = 2.0;
        double r39801 = pow(r39797, r39800);
        double r39802 = r39801 * r39792;
        double r39803 = r39799 / r39802;
        double r39804 = exp(r39803);
        double r39805 = log(r39804);
        double r39806 = log(r39797);
        double r39807 = r39806 * r39795;
        double r39808 = pow(r39792, r39800);
        double r39809 = r39797 * r39808;
        double r39810 = log(r39809);
        double r39811 = exp(r39810);
        double r39812 = r39807 / r39811;
        double r39813 = r39805 - r39812;
        double r39814 = r39798 - r39813;
        double r39815 = 0.2317853494525819;
        bool r39816 = r39792 <= r39815;
        double r39817 = r39797 + r39795;
        double r39818 = pow(r39817, r39796);
        double r39819 = r39796 / r39800;
        double r39820 = pow(r39797, r39819);
        double r39821 = r39820 * r39820;
        double r39822 = r39818 - r39821;
        double r39823 = r39799 / r39792;
        double r39824 = r39823 / r39801;
        double r39825 = cbrt(r39810);
        double r39826 = r39825 * r39825;
        double r39827 = r39826 * r39825;
        double r39828 = exp(r39827);
        double r39829 = r39807 / r39828;
        double r39830 = r39824 - r39829;
        double r39831 = r39798 - r39830;
        double r39832 = r39816 ? r39822 : r39831;
        double r39833 = r39794 ? r39814 : r39832;
        return r39833;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if n < -235.54342409600352

   1. Initial program 44.7

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

   2. Taylor expanded around inf 33.2

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

   3. Simplified 32.7

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

   5. Applied add-exp-log 64.0

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

   6. Applied pow-exp 64.0

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

   7. Applied add-exp-log 64.0

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

   8. Applied prod-exp 64.0

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

   9. Simplified 32.7

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

   11. Applied add-log-exp 32.8

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

   12. Simplified 32.8

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

   if -235.54342409600352 < n < 0.2317853494525819

   1. Initial program 7.6

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

   3. Applied sqr-pow 7.6

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

   if 0.2317853494525819 < n

   1. Initial program 44.2

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

   2. Taylor expanded around inf 33.4

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

   3. Simplified 32.8

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

   5. Applied add-exp-log 32.8

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

   6. Applied pow-exp 32.8

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

   7. Applied add-exp-log 32.8

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

   8. Applied prod-exp 32.8

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

   9. Simplified 32.8

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

   11. Applied add-cube-cbrt 32.8

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

4. Final simplification 22.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -235.54342409600352:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\log \left(e^{\frac{0.5}{{x}^{2} \cdot n}}\right) - \frac{\log x \cdot 1}{e^{\log \left(x \cdot {n}^{2}\right)}}\right)\\ \mathbf{elif}\;n \le 0.2317853494525819:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{e^{\left(\sqrt[3]{\log \left(x \cdot {n}^{2}\right)} \cdot \sqrt[3]{\log \left(x \cdot {n}^{2}\right)}\right) \cdot \sqrt[3]{\log \left(x \cdot {n}^{2}\right)}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))