2nthrt (problem 3.4.6)

Average Error: 29.3 → 22.1

Time: 31.5s

Precision: 64

• could not determine a ground truth for program body

  1. x = 3.440447948190826e+244
  2. n = 2.0347801024726467e-255

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -7.177444174248558 \cdot 10^{-20}:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 3.904316443592393 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{\frac{\log x}{n}}{x \cdot n} - \frac{\frac{1}{2}}{x \cdot \left(x \cdot n\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(e^{\log \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\\ \end{array}\]

C

double f(double x, double n) {
        double r1810741 = x;
        double r1810742 = 1.0;
        double r1810743 = r1810741 + r1810742;
        double r1810744 = n;
        double r1810745 = r1810742 / r1810744;
        double r1810746 = pow(r1810743, r1810745);
        double r1810747 = pow(r1810741, r1810745);
        double r1810748 = r1810746 - r1810747;
        return r1810748;
}

↓

double f(double x, double n) {
        double r1810749 = 1.0;
        double r1810750 = n;
        double r1810751 = r1810749 / r1810750;
        double r1810752 = -7.177444174248558e-20;
        bool r1810753 = r1810751 <= r1810752;
        double r1810754 = x;
        double r1810755 = r1810754 + r1810749;
        double r1810756 = pow(r1810755, r1810751);
        double r1810757 = pow(r1810754, r1810751);
        double r1810758 = r1810756 - r1810757;
        double r1810759 = exp(r1810758);
        double r1810760 = log(r1810759);
        double r1810761 = 3.904316443592393e-11;
        bool r1810762 = r1810751 <= r1810761;
        double r1810763 = r1810751 / r1810754;
        double r1810764 = log(r1810754);
        double r1810765 = r1810764 / r1810750;
        double r1810766 = r1810754 * r1810750;
        double r1810767 = r1810765 / r1810766;
        double r1810768 = 0.5;
        double r1810769 = r1810754 * r1810766;
        double r1810770 = r1810768 / r1810769;
        double r1810771 = r1810767 - r1810770;
        double r1810772 = r1810763 + r1810771;
        double r1810773 = sqrt(r1810756);
        double r1810774 = sqrt(r1810757);
        double r1810775 = r1810773 - r1810774;
        double r1810776 = log(r1810775);
        double r1810777 = exp(r1810776);
        double r1810778 = r1810774 + r1810773;
        double r1810779 = r1810777 * r1810778;
        double r1810780 = r1810758 * r1810779;
        double r1810781 = r1810780 * r1810758;
        double r1810782 = cbrt(r1810781);
        double r1810783 = r1810762 ? r1810772 : r1810782;
        double r1810784 = r1810753 ? r1810760 : r1810783;
        return r1810784;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1 n) < -7.177444174248558e-20

   1. Initial program 2.5

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

   3. Applied add-sqr-sqrt 2.5

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

   5. Applied add-log-exp 2.7

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

   6. Simplified 2.7

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

   if -7.177444174248558e-20 < (/ 1 n) < 3.904316443592393e-11

   1. Initial program 44.4

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

   3. Applied add-sqr-sqrt 44.4

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

   4. Taylor expanded around inf 32.0

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

   5. Simplified 31.4

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

   if 3.904316443592393e-11 < (/ 1 n)

   1. Initial program 25.7

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

   3. Applied add-sqr-sqrt 25.7

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

   5. Applied add-cbrt-cube 25.8

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

   6. Simplified 25.8

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

   8. Applied add-sqr-sqrt 25.8

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

   9. Applied add-sqr-sqrt 25.8

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

   10. Applied difference-of-squares 25.8

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

   12. Applied add-exp-log 25.8

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

4. Final simplification 22.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -7.177444174248558 \cdot 10^{-20}:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 3.904316443592393 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{\frac{\log x}{n}}{x \cdot n} - \frac{\frac{1}{2}}{x \cdot \left(x \cdot n\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(e^{\log \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))