2nthrt (problem 3.4.6)

Average Error: 33.0 → 24.3

Time: 17.4s

Precision: 64

• could not determine a ground truth for program body

  1. x = 4.60915386116327e+88
  2. n = 1.8004234423016248e-156

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -4044640181538.08936:\\ \;\;\;\;{\left(e^{\sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)} \cdot \sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)}}\right)}^{\left(\sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 1.46973031667613272 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{1 \cdot \log x}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \frac{{\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3} - {\left({\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left({\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\\ \end{array}\]

C

double f(double x, double n) {
        double r86154 = x;
        double r86155 = 1.0;
        double r86156 = r86154 + r86155;
        double r86157 = n;
        double r86158 = r86155 / r86157;
        double r86159 = pow(r86156, r86158);
        double r86160 = pow(r86154, r86158);
        double r86161 = r86159 - r86160;
        return r86161;
}

↓

double f(double x, double n) {
        double r86162 = 1.0;
        double r86163 = n;
        double r86164 = r86162 / r86163;
        double r86165 = -4044640181538.0894;
        bool r86166 = r86164 <= r86165;
        double r86167 = x;
        double r86168 = r86167 + r86162;
        double r86169 = pow(r86168, r86164);
        double r86170 = sqrt(r86167);
        double r86171 = 2.0;
        double r86172 = r86171 * r86164;
        double r86173 = pow(r86170, r86172);
        double r86174 = r86169 - r86173;
        double r86175 = log(r86174);
        double r86176 = cbrt(r86175);
        double r86177 = r86176 * r86176;
        double r86178 = exp(r86177);
        double r86179 = pow(r86178, r86176);
        double r86180 = 1.4697303166761327e-07;
        bool r86181 = r86164 <= r86180;
        double r86182 = r86164 / r86167;
        double r86183 = 0.5;
        double r86184 = r86183 / r86163;
        double r86185 = pow(r86167, r86171);
        double r86186 = r86184 / r86185;
        double r86187 = log(r86167);
        double r86188 = r86162 * r86187;
        double r86189 = pow(r86163, r86171);
        double r86190 = r86167 * r86189;
        double r86191 = r86188 / r86190;
        double r86192 = r86186 - r86191;
        double r86193 = r86182 - r86192;
        double r86194 = r86164 / r86171;
        double r86195 = pow(r86168, r86194);
        double r86196 = pow(r86170, r86164);
        double r86197 = r86195 + r86196;
        double r86198 = 3.0;
        double r86199 = pow(r86195, r86198);
        double r86200 = pow(r86196, r86198);
        double r86201 = r86199 - r86200;
        double r86202 = r86196 + r86195;
        double r86203 = r86196 * r86202;
        double r86204 = r86169 + r86203;
        double r86205 = r86201 / r86204;
        double r86206 = r86197 * r86205;
        double r86207 = r86181 ? r86193 : r86206;
        double r86208 = r86166 ? r86179 : r86207;
        return r86208;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -4044640181538.0894

   1. Initial program 0

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

   3. Applied add-sqr-sqrt 0

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

   4. Applied unpow-prod-down 0

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

   5. Applied sqr-pow 0

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

   6. Applied difference-of-squares 0

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

   8. Applied add-exp-log 0

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

   9. Applied add-exp-log 0

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

   10. Applied prod-exp 0

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

   11. Simplified 0

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

   13. Applied add-cube-cbrt 0

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

   14. Applied exp-prod 0

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

   if -4044640181538.0894 < (/ 1.0 n) < 1.4697303166761327e-07

   1. Initial program 44.4

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

   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 32.5

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

   if 1.4697303166761327e-07 < (/ 1.0 n)

   1. Initial program 6.5

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

   3. Applied add-sqr-sqrt 6.5

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

   4. Applied unpow-prod-down 6.6

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

   5. Applied sqr-pow 6.5

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

   6. Applied difference-of-squares 6.5

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

   8. Applied flip3-- 6.5

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

   9. Simplified 6.5

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

4. Final simplification 24.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -4044640181538.08936:\\ \;\;\;\;{\left(e^{\sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)} \cdot \sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)}}\right)}^{\left(\sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 1.46973031667613272 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{1 \cdot \log x}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \frac{{\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3} - {\left({\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left({\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))